Class 12 Mathematics Topic Wise Line by Line Chapter 5 Applications of Derivatives

APPLICATIONS OF DERIVATIVES
Chapter 04
1. DERIVATIVE AS RATE OF CHANGE
In various fields of applied mathematics one has the quest
to know the rate at which one variable is changing, with
respect to other. The rate of change naturally refers to time.
But we can have rate of change with respect to other
variables also.
An economist may want to study how the investment
changes with respect to variations in interest rates.
Aphysician may want to know, how small changes in dosage
can affect the body’s response to a drug.
A physicist may want to know the rate of change of distance
with respect to time.
All questions of the above type can be interpreted and
represented using derivatives.
Definition :
The average rate ofchange ofa function f (x) withrespect to
x over an interval [a, a + h] is defined as
a + h - a
h
f f
.
Definition :
The instantaneous rate of change of f with respect to x is
defined as
h 0
a h a
´ x lim
h
®
+ -
=
f f
f , provided the limit exists.
NOTES:
To use the word ‘instantaneous’, x may not be representing
time. We usually use the word ‘rate of change’ to mean
‘instantaneous rate of change’.
2. EQUATIONS OF TANGENT & NORMAL
(I) The value of the derivative at P (x1
, y1
) gives the
slope of the tangent to the curve at P. Symbolically
f´(x1
) =
x , y
1 1
dy
dx = Slope of tangent at
P (x1
, y1
) = m (say).
(II) Equation of tangent at (x1
, y1
) is ;
1 1
x , y
1 1
dy
y y x x
dx
æ ö
- = ´ -
ç ÷
è ø
(III) Equation of normal at (x1
, y1
) is ;
1 1
x , y
1 1
1
y y x x
dy
dx
æ ö
ç ÷
-
- = ´ -
ç ÷
ç ÷
è ø
NOTES:
1. The point P (x1
, y1
) will satisfy the equation of the curve &
the equation of tangent & normal line.
2. Ifthetangent at any point P on the curve is parallelto X-axis
then dy/dx = 0 at the point P.
3. If the tangent at any point on the curve is parallel to
Y-axis, thendy/dx= ¥ or dx/dy= 0.
4. If the tangent at any point on the curve is equally inclined
to both the axes then dy/dx = +1.
5. If the tangent at any point makes equal intercept on the
coordinate axes then dy/dx = +1.
6. Tangent to a curve at the point P (x1
, y1
) can be drawn
even though dy/dx at P does not exist. e.g. x = 0 is a
tangent to y = x2/3
at (0, 0).
7. If a curve passing through the origin be given by a rational
integral algebraic equation, the equation of the tangent
(or tangents) at the origin is obtained by equating to zero
the terms of the lowest degree in the equation. e.g. If the
equation of a curve be x2
– y2
+ x3
+ 3x2
y – y3
= 0, the
tangents at the origin are given by x2
– y2
= 0 i.e. x + y = 0
and x – y = 0.
187

APPLICATIONS OF DERIVATIVES 188
(IV) (a) Length of the tangent (PT) =
2
1 1
1
y 1 ´ x
´ x
+ é ù
ë û
f
f
(b) Length of Subtangent (MT) =
1
1
y
´ x
f
(c) Length ofNormal (PN) =
2
1 1
y 1 ´ x
+ é ù
ë û
f
(d) Length of Subnormal (MN) = y1
f ´ (x1
)
(V) Differential :
The differential of a function is equal to its derivative
multiplied by the differential of the independent variable.
Thus if, y = tan x then dy = sec2
x dx.
In general dy = f ´ (x) dx.
NOTES:
d (c) = 0 where ‘c’ is a constant.
d (u + v – w) = du + dv – dw
d (uv) = udv + vdu
* The relation dy = f´(x) dx can be written as
´ x ;
=
dy
f
dx
thus the quotient of the differentials
of‘y’and ‘x’isequalto thederivativeof‘y’w.r.t. ‘x’.
3. TANGENT FROM AN EXTERNAL POINT
Given a point P (a, b) which does not lie on the curve
y = f (x), then the equation of possible tangents to the curve
y = f (x), passing through (a, b) can be found by solving for
the point of contact Q.
And equation of tangent is
h b
y b x a
h a
-
- = -
-
f
4. ANGLE BETWEEN THE CURVES
Angle between two intersecting curves is defined as the
acute angle between their tangents or the normals at the
point of intersection of two curves.
1 2
1 2
m m
tan
1 m m
-
q =
+
where m1
& m2
are the slopes of tangents at the intersection
point (x1
, y1
).
NOTES:
(i) The angle is defined between two curves if the
curves are intersecting. This can be ensured by
finding their point of intersection or bygraphically.
(ii) If the curves intersect at more than one point then
angle between curves is found out with respect to
the point of intersection.
(iii) Two curves are said to be orthogonal if angle
between them at each point of intersection is right
angle i.e. m1
m2
= –1.
5. SHORTEST DISTANCE BETWEEN TWO CURVES
Shortest distance between two non-intersecting
differentiable curves is always along their common normal.
(Wherever defined)
6. ERRORS AND APPROXIMATIONS
(a) Errors
Let y = f (x)
From definition of derivative, x 0
y dy
lim
x dx
d ®
d
=
d
y dy
x dx
d
=
d
approximately
or
dy
y . x approximately
dx
æ ö
d = d
ç ÷
è ø
Definition :
(i) dx is known as absolute error in x.
(ii)
x
x
d
is known as relative error in x.

(iii)
x
100
x
d
´ is known as percentage error in x.
NOTES:
dx and dy are known as differentials.
(b) Approximations
From definition of derivative,
Derivative of f (x) at (x = a) = f ´(a)
or f ´(a) = x 0
(a x) (a)
lim
x
d ®
+ d -
d
f f
or
(a x) (a)
'(a)
x
+ d -
®
d
f f
f (approximately)
f (a +dx)= f(a)+ dx f´(a) (approximately)
7. DEFINITIONS
1. Afunctionf(x)iscalledanIncreasingFunctionatapointx=a
if in a sufficiently small neighbourhood around x = a we
have
f (a + h) > f (a)
f (a – h) < f (a)
Similarly Decreasing Function if
f (a + h) < f (a)
f (a – h) > f (a)
Above statements hold true irrespective of whether f is non
derivable or even discontinuous at x = a
2. A differentiable function is called increasing in an interval
(a, b) if it is increasing at every point within the interval (but
not necessarily at the end points). A function decreasing in
an interval (a, b) is similarly defined.
3. A function which in a given interval is increasing or
decreasing is called "Monotonic" in that interval.
4. Tests for increasing and decreasing of a function at a point :
If the derivative f ´(x) is positive at a point x = a, then the
function f (x) at this point is increasing. If it is negative, then
the function is decreasing.
NOTES:
Even if f ´(a) is not defined, f can still be increasing or
decreasing. (Look at the cases below).
NOTES:
If f ´ (a) = 0, then for x = a the function may be still increasing
or it may be decreasing as shown. It has to be identified by a
separate rule.
e.g. f (x) = x3
is increasing at every point.
Note that, dy/dx = 3x2
.
NOTES:
1. If a function is invertible it has to be either increasing or
decreasing.
2. If a function is continuous, the intervals in which it rises
and falls may be separated by points at which its
derivative fails to exist.
3. If f is increasing in [a, b] and is continuous then
f (b) is the greatest and f (a) is the least value of
f in [a,b]. Similarly if f is decreasing in [a, b] then f (a)is the
greatest value and f (b) is the least value.
5. (a) ROLLE'STheorem:
Let f (x) be a function of x subject to the following
conditions :
(i) f (x) is a continuousfunction of x in theclosed interval
of a < x < b.
(ii) f ´ (x) exists for every point in the open interval
a < x < b.
(iii) f (a) = f (b).
Then there exists at least one point x = c such that
a < c < b where f ´ (c) = 0.
(b) LMVTTheorem:
Let f (x) be a function of x subject to the following
conditions :

(i) f (x) is a continuousfunction of x in theclosed interval
of a < x < b.
(ii) f ´ (x) exists for every point in the open interval
a < x < b.
Then there exists at least one point x = c such that
a < c < b where f ´ (c) =
(b) (a)
b a
-
-
f f
Geometrically, the slope of the secant line joining the curve
at x = a & x = b is equal to the slope of the tangent line drawn
to the curve at x = c.
Notethefollowing:Rollestheoremisaspecial caseof LMVT
since
(b) (a)
(a) (b) ´(c) 0
b a
-
= Þ = =
-
f f
f f f
NOTES:
PhysicalInterpretationofLMVT :
Now [ f (b) – f (a)] is the change in the function f as x changes
from a to b so that
(b) (a)
b a
-
-
f f
is the average rate of change
of the function over the interval [a, b]. Also f ´ (c) is the actual
rate of change of the function for x = c. Thus, the theorem
states that the average rate of change of a function over an
interval is also the actual rate of change of the function at
some point ofthe interval. In particular, for instance,theaverage
velocity of a particle over an interval of time is equal to the
velocity at some instant belonging to the interval.
This interpretation of the theorem justifies the name "Mean
Value" for the theorem.
(c) Application of rolles theorem for isolating the real roots of
an equation f (x) = 0
Suppose a & b are two real numbers such that ;
(i) f (x) & its first derivative f ´ (x) are continuous for
a < x < b.
(ii) f (a) & f (b) have opposite signs.
(iii) f ´ (x) isdifferent fromzero forallvalues ofxbetween
a & b.
Then there is one & only one real root of the equation
f (x) = 0 between a & b.
8. HOW MAXIMA & MINIMA ARE CLASSIFIED
1. Maxima & Minima
A function f (x) is said to have a local maximumat x = a if f (a)
is greater than every other value assumed by f (x) in the
immediate neighbourhood of x = a. Symbolically
a a h
x=a gives maxima
a a h
> + ù
Þ
ú
> - ú
û
f f
f f
for a sufficiently small positive h.
Similarly,afunctionf(x)issaid to havealocal minimumvalue
atx =b iff(b) isleastthaneveryother valueassumed byf(x)in
the immediateneighbourhood atx =b.Symbolicallyif
b b h
x=b
b b h
< + ù
Þ
ú
< - ú
û
f f
f f
gives minima for a sufficiently
small positive h.
NOTES:
(i) The local maximum& local minimum values of a function
are also known as local/relative maxima or local/relative
minima as these are the greatest & least values of the
function relative to some neighbourhood of the point in
question.
(ii) The term‘extremum’is used both for maxima or a minima.
(iii) A local maximum(local minimum) value ofa function may
not be the greatest (least) value in a finite interval.
(iv) A function can have several local maximum & local
minimum values & a local minimum value may even be
greater than a local maximum value.
(v) Maxima & minima of a continuous function occur
alternately & between two consecutive maxima there is a
minima & vice versa.

2. A necessary condition for maxima & minima
If f (x)is a maxima or minima at x = c & if f ´ (c) exists then
f ´ (c) = 0.
NOTES:
(i) The set of values ofx for which f ´ (x) = 0 are often
called as stationary points. The rate of change of
function is zero at a stationary point.
(ii) In case f ´ (c) does not exist f (c) may be a maxima
or a minima & in this case left hand and right hand
derivatives are of opposite signs.
(iii) The greatest (global maxima) and the least (global
minima) values of a function f inan interval[a,b]are
f (a) or f (b) orare given by the values of x which are
critical points.
(iv) Critical points are those where :
(i)
dy
0,
dx
= if it exists; (ii) or it fails to exist
3. Sufficient condition for extreme values
First Derivative Test
´ c h 0
x c
´ c h 0
- > ù
Þ =
ú
+ < ú
û
f
f
is a point of local maxima,
where h is a sufficiently small positive quantity
Similarly
´ c h 0
x c
´ c h 0
- < ù
Þ =
ú
+ > ú
û
f
f
is apointof local minima,
where h is a sufficiently small positive quantity
Note: f ´(c) in both the cases may or may not exist. If it
exists, then f ´ (c) = 0.
NOTES:
If f´ (x) does not change sign i.e. has the same sign in a
certain complete neighbourhood of c, then f (x) is either
strictly increasing or decreasing throughout this
neighbourhood implying that f (c) is not an extreme value
of f .
4. Use of second order derivative in
ascertaining the maxima or minima
(a) f (c) is a minima of the function f, if
f ´ (c) = 0 & f ´´ (c)> 0.
(b) f (c) is a maxima of the function f, if
f´(c)=0&f´´(c)<0.
NOTES:
If f ´
´(c) = 0 then the test fails. Revert back to the first
order derivative check for ascertaining the maxima or
minima.
5. Summary-working rule
First : When possible, draw a figure to illustrate them
problem & label those parts that are important in the
problem. Constants & variables should be clearly
distinguished.
Second : Write an equation for the quantity that is to be
maximised or minimised. If this quantity is denoted by ‘y’, it
must be expressed in terms of a single independent variable
x. This may require some algebraic manipulations.
Third : If y = f (x) is a quantity to be maximum or minimum,
find those values of x for which dy/dx = f ´ (x) = 0.
Fourth: Testeachvaluesofxforwhichf´ (x)=0 to determine
whether it provides a maxima or minima or neither. The usual
tests are :
(a) If d2
y/dx2
is positive when dy/dx = 0
Þ y is minima.
If d2
y/dx2
is negative when dy/dx = 0
Þ yis maxima.
If d2
y/dx2
= 0 when dy/dx = 0, the test fails.
(b)
dy
If is
dx
0
0 0
0
positive for x x
zero for x x a maxima occurs at x x .
negative for x x
< ù
ú
= Þ =
ú
ú
> û
But if dy/dx changes sign from negative to zero to positive
as x advances through x0
, there is a minima. If dy/dx does
not change sign, neither a maxima nor a minima. Such points
arecalledINFLECTIONPOINTS.
Fifth : If the function y = f (x) is defined for only a limited
range of values a £ x £ b then examine x = a & x = b for
possible extreme values.
Sixth : If the derivative fails to exist at some point, examine
this point as possible maxima or minima.
(In general, check at all Critical Points).
NOTES:
= If the sum of two positive numbers x and y is
constant than their product is maximum if they are
equal, i.e. x + y = c, x > 0, y > 0, then
2 2
1
xy x y x y
4
é ù
= + - -
ë û

= If the product of two positive numbers is constant
then their sum is least if they are equal.
i.e. (x + y)2
= (x – y)2
+ 4xy
9. USEFUL FORMULAE OF MENSURATION TO
REMEMBER
= Volume of a cuboid = lbh.
= Surface area of a cuboid = 2 (lb + bh + hl).
= Volume of a prism = area of the base × height.
= Lateralsurfaceofaprism=perimeterofthebase×height.
= Total surface of a prism = lateral surface + 2 area of
the base
(Notethatlateralsurfaces ofaprismare all rectangles).
= Volume of a pyramid =
1
3
area of the base × height.
= Curved surface of a pyramid =
1
2
(perimeter of the
base) × slant height.
(Note that slant surfaces of a pyramid are triangles).
= Volume of a cone = 2
1
r h.
3
p
= Curved surface of a cylinder = 2prh.
= Total surface of a cylinder = 2prh + 2pr2
.
= Volume of a sphere = 3
4
r .
3
p
= Surface area of a sphere = 4pr2
.
= Area of a circular sector = 2
1
r ,
2
q where q is in
radians.
10. SIGNIFICANCE OF THE SIGN OF 2ND ORDER
DERIVATIVE AND POINTS OF INFLECTION
The sign of the 2nd
order derivative determinesthe concavity
of the curve. Such point such as C & E on the graph where
the concavity of the curve changes are called the points of
inflection. From the graph we find that if :
(i)
2
2
d y
0 concave upwards
dx
> Þ
(ii)
2
2
d y
0 concave downwards.
dx
< Þ
At the point of inflection we find that
2 2
2 2
d y d y
0 and
dx dx
=
changes sign.
Inflection points can also occur if
2
2
d y
dx
fails to exist (but
changes its sign). For example, consider the graph of the
function defined as,
3/5
2
x for x ,1
x
2 x for x 1,
é Î -¥
= ê
- Î ¥
ê
ë
f
NOTES:
The graph below exhibits two critical points one is a point
of local maximum (x = c) & the other a point of inflection
(x = 0). This implies that not every Critical Point is a point
ofextrema.

SOLVED EXAMPLES
SOLVED EXAMPLES
Example – 1
If the function f (x) = 2x3
– 9ax2
+ 12a2
x + 1, where a > 0,
attains its maximum and minimum at p and q respectively
such that p2
= q, then a equals
(a) 1 (b) 2
(c)
2
1
(d) 3
Ans. (b)
Sol. For maximumand minima ' x 0
f =
2 2
6x -18ax+12a =0
Þ
a,2a
x
Þ =
Also, f’’(x)=12x -18x
''( ) 0 max ' '
f a at a
< Þ
"(2 ) 0 min '2 '
f a at a
> Þ
So, p = a and q = 2 a
Given p2
= q
2 2
a =2a a -2a = 0
Þ Þ
a(a-2)=0 a = 0, a = 2
Þ Þ
Example – 2
The real number x when added to its inverse gives the
minimum value of the sum at x equal to
(a) 1 (b) – 1
(c) – 2 (d) 2
Ans. (a)
Sol.
1
(x) = x+
2
f
2 3
1 2
'(x) 1- and "(x) =
x x
f f
=
Now '(x) = 0
f
1
" 1 0
x
f
Þ = ±
>
Q
x = 1
Þ is point of minima.
Example – 3
A function y = f (x) has a second order derivative
f ” = 6(x–1). If its graph passes through the point (2, 1)
and at that point the tangent to the graph is y = 3x – 5,
then the function is
(a) (x– 1)2
(b) (x– 1)3
(c) (x+1)3
(d)(x+ 1)2
Ans. (b)
Sol. Given "( )=6 ( x - 1)
f x
2
6(x-1)
'(x) = +c
2
f
Þ
tangent 2 3 5
3 3
' 2 3
0
at x is y x
c
f
c
= = +
Þ = + é
ê Þ =
Þ = ë
Q
2
so '(x) = 3 (x-1)
f
3
1
f (x)=(x-1) +c
Þ as curve passes through (2,1)
3
1
1= ( 2 - 1 ) c
Þ +
3
1 0 ( ) ( 1)
c hence f x x
Þ = = -
Example – 4
Find the maximum surface area of a cylinder that can be
inscribed in a given sphere of radius R.
Sol.
Let r be the radius and h be the height of cylinder. Consider

the right triangle shown in the figure.
2r = 2R cos q and h = 2 R sin q
Surface area of the cylinder = 2 p rh + 2 p r2
Þ S (q) = 4 p R2
sin q cos q + 2 p R2
cos2
q
Þ S (q) = 2 p R2
sin 2q + 2 p R2
cos2
q
Þ S’ (q) = 4 p R2
cos 2q – 2 p R2
sin 2q
S´ (q) = 0 Þ 2 cos 2q – sin 2q = 0
Þ tan 2q = 2 Þ q = q0
= 1/2 tan–1
2
S´
´(q0
) = – 8 p R2
sin 2q – 4 p R2
cos 2q
S´
´(q) = – 8 p R2
÷
÷
ø
ö
ç
ç
è
æ
5
2
– 4 pR2 0
5
1
<
÷
÷
ø
ö
ç
ç
è
æ
Hence surface area is maximum for q = q0
= 1/2 tan–1
2
Smax
= 2 p R2
sin 2 q0
+ 2 p R2
cos2
q0
Þ ÷
÷
ø
ö
ç
ç
è
æ +
p
+
÷
÷
ø
ö
ç
ç
è
æ
p
=
2
5
/
1
1
R
2
5
2
R
2
S 2
2
max
Þ )
5
1
(
R
S 2
max +
p
=
Example – 5
Find the semi-vertical angle of the cone of maximum curved
surface areathat canbe inscribed in a givensphere ofradius R.
Sol.
Let h be the height of cone and r be the radius of the cone.
Consider the right DOMC where O is the centre of sphere
and AM is perpendicular to the base BC of cone.
OM = h – R, OC = R, MC = r
R2
= (h – R)2
+ r2
...(i)
and r2
+ h2
= l2
...(ii)
where l is the slant height of cone.
Curve surface area = C = p r l
Using (i) and (ii), express C in terms of h only.
hR
2
h
hR
2
C
h
r
r
C 2
2
2
-
p
=
Þ
+
p
=
We willmaximise C2
.
Let C2
= f (h) = 2 p2
h R (2hR – h2
)
2 2 3
2 2
R h R h
p
= -
Þ f ’(h) = 2 p2
R (4hR – 3h2
)
f’ (h) = 0 Þ 4hR – 3h2
= 0 4 3 0
h R h
Þ - =
Þ h = 4R/3.
f ´
´(h) = 2 p2
R (4R – 6h)
f ´
´ ÷
ø
ö
ç
è
æ
3
R
4
=2 pR2
(4R – 8R) < 0
Hence curved surface area is maximum for
3
R
4
h =
Using (i), we get :
R
3
2
2
r
9
R
8
h
hR
2
r
2
2
2
=
Þ
=
-
=
Semi–vertical angle = q = tan–1
r/h = tan–1
1/ 2 .
Example – 6
If f and g are differentiable functions in [0, 1] satisfying
f(0)=2=g(1),g(0)=0and f(1)= 6,thenforsomecÎ]0,1[:
(a) f’(c) = 2g’(c) (b) 2f’(c) = g’(c)
(c) 2f’(c) = 3g’(c) (d) f’(c) = g’(c)
Ans. (a)
Sol. ByLMVT
(1) (0)
'( )
1 0
f f
f c
-
=
-
6 2
4
1
-
= =
(1) (0)
'( )
1 0
g g
g c
-
=
-
2 0
2
1
-
= Þ
' 2 '( )
f c g c
Þ =
Example –7
If 2a + 3b + 6c = 0 (a,b, c, ÎR), thenthe quadratic equation
ax2
+ bx + c = 0 has
(a) at least one root in (0, 1) (b) at least one root in [2, 3]
(c) at least one root in [4, 5] (d) none of the above
Ans. (a)

Sol.
3 2
ax bx
Let usconsiderf x = + +cx
3 2
a b
0 =0and 1 = + +c
3 2
2a+3b+6c
= =0(given).
6
f f

As 0 = 1 = 0
f f and x
f is continuous and
differentiable also in 0,1 .
By Rolle’s theorem x =0
f
2
ax +bx+c=0
Þ has at least one root in the interval (0, 1).
Example – 8
Find the approximate value of (0.007)1/3.
Sol. Let f(x) =(x)1/3
Now, 2/3
x
f (x + x) f(x) = f (x). x
3x
d
¢
d - d =
we maywrite, 0.007 =0.008 – 0.001
Taking x = 0.008 and d x = – 0.001, we have
f (0.007)– f(0.008)= 2/3
0.001
3 0.008
-
or f (0.007)– (0.008)1/3
= 2
0.001
3 0.2
- or
f (0.007) = 0.2 –
0.001
3 0.04
=0.2
1 23
120 120
- =
Hence (0.007)1/3
=
23
120
.
Example – 9
Discuss concavity and convexity and find points of
inflexion ofy = x2
e–x
.
Sol. Let f(x) = x2
e–x
.
Differentiate w.r.t.x to get :
f ´ (x) = e–x
(2x) + (–e–x
)x2
=xe–x
[2 – x]
Differentiate again w.r.t. x to get :
f ´
´(x) = (2 – 2x) e–x
+ (2x – x2
)(–e–x
)
= e–x
(2 – 2x – 2x + x2
)
= e–x
(x2
– 4x+ 2)
= ))
2
2
(
x
(
))
2
2
(
x
(
e x
+
-
-
-
-
See the figure and observe how the sign of f ´
´ (x) changes.
Sign of f ´
´(x) is changing at .
2
2
x ±
=
Therefore points ofinflextion of f (x) are .
2
2
x ±
=
]
,
2
2
[
]
2
2
,
[
x
0
)
x
(
´ ¥
+
È
-
-¥
Î
"
³
´
f
Therefore f (x) is “Concave upward”
)
,
2
2
[
]
2
2
,
(
x ¥
+
È
-
-¥
Î
"
Similarly we can observe
]
2
2
,
2
2
[
x
0
)
x
(
´ +
-
Î
"
£
´
f
Therefore f (x) is “Convex downwards”
]
2
2
,
2
2
[
x +
-
Î
"
Example –10
Prove that the minimum intercept made by axes on the
tangent to the ellipse 1
b
y
a
x
2
2
2
2
=
+ is a + b. Also find the
ratio in which the point of contact divides this intercept.
Sol.
Intercept made by the axes on the tangent is the length of
the portion of the tangent intercepted between the axes.
Consider a point P on the ellipse whose coordinates are
x = a cost, y = b sint (where t is the parameter)
sint
cost
dx
a
dt
dy
b
dt
= -
=
The equation of the tangent is :

Þ equation is y – y0
=
1
3
0
0
0
y
x x
x
æ ö
- -
ç ÷
è ø
Þ 1/3 1/3 1/3 1/3
0 0 0 0 0 0
x y y x xy x y
- = - +
Þ 1/3 1/3 1/3 1/3
0 0 0 0 0 0
x y yx x y y x
+ = +
Þ
1/3 1/3
2/3 2/3
0 0
0 0
1/3 1/3 1/3 1/3
0 0 0 0
x y y x
x y
x y x y
+ = +
Þ equation of tangent is : 2/3
1/3 1/3
0 0
x y
a
x y
+ =
Length intercepted between the axes :
length = 2 2
(x intercept) (y intercept)
+
x intercept 1/3 2/3
0
x a
=
y intercept = 1/3 2/3
0
y a
=
2 2
1/3 2/3 1/3 2/3
0 0
x a y a
= +
2/3 4/3 2/3 4/3
0 0
x a y a
= +
2/3 2/3 2/3
0 0
a x y
= +
2/3 2/3
a a
=
= a i.e. constant.
Method2:
Express the equation in parametric form
x = a sin3
t, y = a cos3
t
2 2
3 sin cost, 3 cos tsin
dx dy
a t a t
dt dt
= = -
Equation of tangent is :
(y – a cos3
t) =
2
2
3 a cos t sin t
3 a sin t cos t
-
( x – a sin3
t)
Þ y sin t – a sin t cos3
t = – x cos t + a sin3
t cos t
Þ x cos t + y sin t = a sin t cos t
Þ
x y
a
sin t cos t
+ =
in terms of (x0
, y0
) equation is :
1/3 1/3
0 0
x y
a
x / a y / a
+ =
Length of tangent intercepted between axes
t
cos
a
x
t
sin
a
t
cos
b
t
sin
b
y -
-
=
-
Þ 1
t
sin
b
y
t
cos
a
x
=
+
Þ
t
sin
b
OB
,
t
cos
a
OA =
=
Length of intercept = l = AB =
t
sin
b
t
cos
a
2
2
2
2
+
We will minimise l 2
.
Let l 2
= f (t) = a2
sec2
t + cosec2
t
Þ f´(t) = 2a2
sec2
t tan t – 2b2
cosec2
t cot t
f´(t)= 0 Þ a2
sin4
t = b2
cos4
t
Þ t = tan–1
b/a
f ´
´(t) = 2a2
(sec4
t + 2 tan2
t sec2
t)
+ 2b2
(cosec4
t + 2 cosec2
t cot2
t), which is positive.
Hence f (t) is minimum for tan t =
b
a
.
Þ )
b
/
a
1
(
b
)
a
/
b
1
(
a 2
2
min +
+
+
=
l
Þ lmin
= a + b
2
2 2 2
a
PA a cost b sin t
cos t
æ ö
= - +
ç ÷
è ø
t
sin
b
t
cos
t
sin
a 2
2
2
4
2
+
=
= (a2
tan2
t + b2
) sin2
t
2
2
b
b
a
b
)
b
ab
( =
+
+
= Þ PA= b
a
b
PB
PA
Hence = Þ P divides AB in the ratio b : a
Example – 11
Find the equation of tangent to the curve
x2/3
+ y2/3
= a2/3
at (x0
, y0
). Hence prove that the length of the
portion oftangent intercepted between the axes is constant.
Sol. Method1:
x2/3
+ y2/3
= a2/3
Differentiating wrt x,
1 1
3 3
2 2 dy
x y 0
3 3 dx
- -
+ =
Þ
1
3
0
x ,y 0
0 0
x
dy
dx y
-
æ ö
= -ç ÷
è ø
1
3
0
, 0
0 0
x y
y
dy
dx x
æ ö
ù
Þ = -ç ÷
ú
û è ø

NOTES:
2 2
int int
x y
= +
2 2 2 2
a sin t a cos t a
= + = which is constant
1. The parametric form is very useful in these type of
problems.
2. Equation of tangent can also be obtained by
substituting b = a and m = 2/3 in the result
m 1 m 1
0 0
x y
x y
1.
a a b b
- -
æ ö æ ö
+ =
ç ÷ ç ÷
è ø è ø
Example – 12
For the curve xy = c2
, prove that
(i) the intercept between the axes on the tangent at
any point is bisected at the point of contact.
(ii) the tangent at any point makes with the co-ordinate
axes a triangle of constant area.
Sol. Lettheequationofthecurveinparametricformbex=ct,y=c/t
2
dx
c
dt
dy c
dt t
=
-
=
Let the point of contact be (ct, c/t)
Equation of tangent is :
y – c/t =
2
c/ t
c
-
(x – ct)
Þ t2
y – ct = –x + ct
Þ x + t2
y = 2 ct .......(i)
(i) Let the tangent cut the x and y axes atAand B respectively.
Writing the equations as :
x y
1
2ct 2c/ t
+ =
Þ xintercept
= 2ct, yintercept
= 2 c/t
Þ
2c
A (2ct, 0) and B 0,
t
æ ö
º º ç ÷
è ø
mid point of
2ct 0 0 2c/ t
AB ,
2 2
+ +
æ ö
º ç ÷
è ø
(ct, c/ t)
º
Hence, the point of contact bisects AB.
(ii) If O is the origin,
Area of triangle D OAB = 1/2 (OA) (OB)
2c
1
2ct
2 t
=
= 2 c2
i.e. constant for all tangents because it is independent of t.
Example – 13
Find critical points of f (x)= x2/3
(2x – 1).
Sol. f (x) =2x5/3
–x2/3
Differentiate w.r.t. x to get,
.
x
)
1
x
5
(
3
2
x
3
2
x
3
10
)
x
´( 3
/
1
3
/
1
3
/
2 -
=
-
= -
f
For critical points,
f ´ (x) = 0 or f ´ (x) is not defined.
Put f ´(x) = 0 to get .
5
1
x =
f ´(x) is not defined when denominator = 0.
Þ x1/3
=0 Þ x=0
Now we can say that x = 0 and
5
1
x = are critical points as
f (x) exists at both x = 0 and .
5
1
x =
Þ Critical points of f (x) are x = 0, .
5
1
x =
Example – 14
The ends A and B of a rod of length 5 are sliding along
the curve y = 2x2
. Let xA
and xB
be the x-coordinate of the
ends. At the moment when Ais at (0, 0) and B is at (1, 2),
find the value of the derivative B
A
dx
dx
.
Sol. We have y = 2x2
(AB)2
=(xB
– xA
)2
+(2x2
B
– 2x2
A
)2
=5 5
As AB =
or (xB
– xA
)2
+4 (x2
B
– x2
A
)2
=5

Y
A
X
( )
2
, 2
B B
B x x
( )
2
,2
A A
x x
(0, 0) (1,2)
Differentiating w.r.t. xA
and denoting
B
A
dx
D
dx
=
2 (xB
– xA
) (D – 1) + 8 (x2
B
– x2
A
) (2xB
D – 2xA
) = 0
Put xA
=0, xB
=1
2 (1 – 0) (D – 1) + 8 (1 – 0) (2D –0) = 0
2D – 2 +16D = 0 Þ D = 1/9
1
9
B
A
dx
dx
Þ =
Example – 15
The equation of the tangent to the curve 2
4
y x ,
x
= + that
is parallel to the x-axis, is
(a) y = 0 (b) y = 1
(c) y = 2 (d) y = 3
Ans. (d)
Sol. Tangent is parallel to x-axis
3
dy 8
= 0 1- = 0 x = 2 y = 3
dx x
Þ Þ Þ Þ
Example – 16
For 0 x
2
p
< £ , show that
3
x
x sin x x
6
- < < .
Sol. Let f (x)= sin x – x
f ´(x) = cos x – 1 = – (1 – cos x) = – 2 sin2
x/2 < 0
f (x) is a decreasing function
for x > 0
f (x) < f (0) Þ sin x – x < 0 (Q f (0)= 0)
Þ sinx < x ......(1)
Now let g (x) =
3
x
x sin x
6
- -
2
x
(x) =1 cos x
2
¢ - -
g
To find sign of (x)
¢
g we consider
2
x
(x) =1 cos x
2
f - -
(x) = x sin x 0
¢
f - + < [From(1)]
f (x) is a decreasing function Þ (x) < 0
¢
g
Þ g (x) is a decreasing function Q x > 0
Þ g (x)< g (0)
Þ
3
x
x sin x 0
6
- - < (Q g (0) = 0)
Þ
3
x
x sin x
6
- < ......(2)
Combining (1) and (2) we get
3
x
x sin x x
6
- < < .
Example – 17
Show that x / (1 + x) < log (1 + x) < x
for x > 0.
Sol. Let
x
1
x
)
x
1
(
log
)
x
(
+
-
+
=
f
2
)
x
1
(
x
)
x
1
(
x
1
1
)
x
(
+
-
+
-
+
=
´
f
2
( ) 0 0
(1 )
x
f ´ x for x
x
= > >
+
Þ f (x) is increasing.
Hence x > 0 Þ f (x) > f (0) by the definition of the increasing
function.
Þ
0
1
0
)
0
1
(
log
x
1
x
)
x
1
log(
+
-
+
>
+
-
+
Þ 0
x
1
x
)
x
1
(
log >
+
-
+
Þ
x
1
x
)
x
1
(
log
+
>
+ ...(i)
Now, let g (x) = x – log (1 + x)
0
x
for
0
x
1
x
x
1
1
1
)
x
´( >
>
+
=
+
-
=
g
Þ g (x) is increasing.
Hence x > 0 Þ g (x) > g (0)
Þ x – log (1 + x) > 0 – log (1 + 0)

Þ x – log (1 + x) > 0
Þ x> log (1 + x) ...(ii)
Combining (i) and (ii), we get :
x
)
x
1
(
log
x
1
x
<
+
<
+
Example – 18
Angle between the tangents to the curve y = x2
– 5x + 6 at
the points (2, 0) and (3, 0) is
(a) p/2 (b) p/3
(c) p/6 (d) p/4
Ans. (a)
Sol. Given equation 2
y = x - 5x + 6 ,given point (2,0),(3,0)
dy
= 2x – 5
dx

say 1 x 2
y 0
dy
m 4 5 1
dx =
=
æ ö
= = - = -
ç ÷
è ø
and 2 x 3
y 0
dy
m 6 5 1
dx =
=
æ ö
= = - =
ç ÷
è ø
since 1 2
m m 1
= -
Þ tangents are at right angle i.e .
2
p
Example – 19
Determine the absolute extrema for the following function
and interval.
g (t) = 2t3
+ 3t2
– 12t + 4 on [0, 2]
Sol. Differentiate w.r.t. t
g ´ (t) = 6t2
+ 6t – 12 = 6 (t + 2) (t – 1)
Note that this problem is almost identical to the first problem.
The only difference is the interval that we were working on.
The first step is to again find the critical points. From the
first example we know these are t = – 2 and t = 1.At this point
it’s important to recall that we only want the critical points
that actually fall in the interval in question. This means that
we only want t = 1 since t = – 2 falls outside the interval so
reject it.
Nowfor absolute maxima
We have,
Max {g (1), g (0), g (2)}
i.e., Max {–3,4, 8}
On comparing all these values we get g (t) has absolute max.
as 8 at t = 2 and similarly absolute minimum of g (t) is – 3 at
t = 1.
Example – 20
Let f be differentiable for all x.
If f (1) = – 2 and f ´(x) ³ 2 for x Î [1, 6], then
(a) f (6)< 8 (b) f (6) ³ 8
(c) f (6)= 5 (d) f (6) <5
Ans. (b)
Sol. Using LMVT,
6 1
' 1,6
6 1
f f
f c for somec
-
= Î
-
6 2
2
5
f - -
Þ ³
6 8
f
Þ ³
Example – 21
Find points of local maximum and local minimum of
f (x)=x2/3
(2x– 1).
Sol. Let f (x) =2x5/3
–x2/3
Differentiate w.r.t. x to get :
3
/
1
3
/
1
3
/
2
x
)
1
x
5
(
3
2
x
3
2
x
3
5
2
)
x
´(
-
=
-
÷
ø
ö
ç
è
æ
= -
f
By taking f’(x) = 0 or f’(x) is not defined.
Critical points of f (x) are
5
1
x = and x = 0.
Using the following figure, we can determine how sign of
f’(x) is changing at x = 0 and .
5
1
x =
fromfigure,

x= 0 is point oflocal maximumassign of f´ (x)changesfrom
positive to negative and
5
1
x = is a point of local minimum
as sign of f’(x) is changing from negative to positive.
Example – 22
Find the local maximum and local minimum values of the
function y = xx
.
Sol. Let f (x) =y = xx
Þ log y = x log x
Þ x
log
x
1
x
dx
dy
y
1
+
=
Þ )
x
log
1
(
x
dx
dy x
+
=
f ´ (x) = 0 Þ xx
(1 + log x) = 0
Þ log x = –1 Þ x = e–1
= 1/e.
Method 1 : (First Derivative Test)
f ´(x)= xx
(1 +log x)
f ´(x) = xx
logx
x< 1/e Þ ex < 1
Þ f ´(x)< 0
x> 1/e Þ ex > 1
Þ f ´(x)> 0
The sign of f ´(x) changes from – ve to + ve around
x=1/e.
In other words, f (x) changes from decreasing to increasing
at x = 1/e.
Hence x = 1/e is a point of local minimum.
Local minimum value = (1/e)1/e
= e–1/e
.
Method II : (Second Derivative Test)
÷
ø
ö
ç
è
æ
+
+
=
x
1
x
x
dx
d
)
x
log
1
(
)
x
(
´ x
x
´
f
= xx
(1 + log x)2
+ xx –1
f ´
´(1/e) = 0 + (e)(e – 1)/e
> 0.
Hence x = 1/e is a point of local minimum.
Local minimum value is (1/e)1/e
= e–1/e
.
Example – 23
The function g (x)
x
2
2
x
+
= has a local minimum at
(a) x= 2 (b) x = – 2
(c) x= 0 (d)x= 1
Ans. (a)
Sol.
x 2
Let g(x) = +
2 x
2
1 2
g' (x) = -
2 x

for maximaand minima g' (x) = 0 x = 2
Þ ±
Again 3
4
g " (x) = 0 for x = 2
x
>
0for x = - 2 x 2
< = is point of minima
Example – 24
Suppose the cubic x3
– px + q has three distinct real roots
where p > 0 and q > 0. Then which one of the following
holds?
(a) The cubic has maxima at both
p
3
and –
p
3
(b) The cubic has minima at
p
3
and maxima at –
p
3
(c) The cubic has minima at –
p
3
and maxima at
p
3
(d) The cubic has minima at both
p
3
and –
p
3
Ans. (b)
Sol. Let 3
(x) = x
f px q
- +

Now 2
'(x) = 0, i.e. 3x - p = 0
f
p p
x = - ,
3 3
Þ
Also,
p p
"(x) = 6x " - 0 " 0
3 3
f f and f
æ ö æ ö
Þ < >
ç ÷ ç ÷
ç ÷ ç ÷
è ø è ø
Thus maxima at
p
-
3
and minima at
p
3
Example – 25
Given P (x) = x4
+ ax3
+ bx2
+ cx + d such that x = 0 is the
only real root of P’(x) = 0. If P(–1) < P(1), then in the
interval [–1, 1]
(a) P (–1) is the minimum and P(1) is the maximum of P
(b) P (–1) is not minimumbut P(1) is the maximum of P
(c) P(–1) isthe minimumand P(1) is notthemaximumofP
(d) neitherP(–1)istheminimumnorP(1)isthemaximumofP
Ans. (b)
Sol. 4 3 2
P(x) = x + ax + bx + cx + d
3 2
P'(x) = 4x + 3ax + 2bx + c
P'(0) 0 c 0
= Þ =
Now, 2
P'(x) = x (4x +3ax+2b)
As P'(x) = 0 has no real roots except
x = 0 , we have
Discriminant of 2
4x + 3ax + 2b is less than zero.
i.e., (3a)2
– (4) (4) (2b) < 0
then 2
4x + 3ax + 2b 0 x R
> " Î
2 2
( If a > 0,b - 4ac < 0 then ax + bx+ c 0 x )
> " ÎR
So P'(x) 0if x 1,0
< Î - i.e.,decreasing
and P'(x) 0if x 0,1
> Î i.e., increasing
Max.of P(x) = P(1)
But minimum of P(x) doesn’t occur at x 1
= - ,i.e., P (-1)
is not the minimum.
Example – 26
For
5
x 0, ,
2
p
æ ö
Îç ÷
è ø
define
x
0
f (x) t
= ò sin t dt. Then, f has
(a) local minimum at pand 2p
(b) local minimumat pand local maximumat 2p
(c) local maximumat pand local minimum at 2p
(d)local maximumat pand 2p
Ans. (c)
Sol. '(x) x sin x, '(x) 0
f f
= =
x 0or sin x = 0
Þ =
5
x 2 , x 0,
2
p
p p
æ ö
æ ö
Þ = Î
ç ÷
ç ÷
è ø
è ø
Q
1 1
"(x) x cos x+ sin x = (2x cos x + sin x)
2 2
f
x x
=
"( ) < 0 and "(2 ) 0
f f
p p >
Þ Local maxima at x p
= and local minima at x 2p
=
Example – 27
Let f : R ® R be defined by
k 2x, if x 1
f(x)
2x 3, if x 1
- £ -
ì
= í
+ > -
î
If f has a local minimum at x = –1, then a possible value of
k is
(a) 1 (b)0
(c) 1
2
- (d) –1
Ans. (d)
Sol. 1
lim
x® +
(x) = 1
f
As ( 1) 2
f k
- = +
As f has a local minimum at 1
x = -
( 1 ) ( 1) ( 1 ) 1 k+2
f f f
+ -
- ³ - £ - Þ ³
k+2 1. k -1
Þ £ £
Thus 1
k = - is a possible value.

Example – 28
Let a, b Î R be such that the function f given by
f (x) = log |x| + bx2
+ ax, x ¹ 0 has extreme values at
x = –1 and x = 2.
Statement I f haslocal maximumat x = –1 and x = 2.
StatementII
1 1
a and b
2 4
.
-
= =
(a) Statement I is false, Statement II is true.
(b) Statement I is true, Statement II is true;
Statement II is a correct explanation for Statement I.
(c) Statement I is true, Statement II is true;
Statement II is not a correct explanation for Statement I.
(d) Statement I is true, Statement II is false.
Ans. (c)
Sol. Given 2
(x)=In +bx +ax
f x
1
'(x) = + 2bx + a
f
x

at x = -1, '(-1) = - 1 - 2b + a = 0
f
a - 2b = 1 ...(i)
Þ
1
at x = 2 , '(-2) = + 4b + a = 0
2
f
1
a + 4b = - ...(ii)
2
Þ
Solving (i) and (ii) we get,
1 1
a = , b = -
2 4
.
2
1 x 1 2-x +x -(x+1)(x-2)
'(x) = - + = =
x 2 2 2x 2x
f
Þ
Þ maximaas x = - 1.2
Hence both statement are true but statement II.
is not correct explanation of statement I.
Example – 29
The normal to the curve x = a (cos q + q sin q),
y =a (sin q –q cos q) at any point q is such that
(a) it makes angle
2
p
+q with x–axis
(b) it passes through the origin
(c) it is a constant distance from the origin
(d) it passes through a , a
2
p
æ ö
-
ç ÷
è ø
Ans. (a,c)
Sol.
dy dy dθ
= . = tanθ =
dx dx dx
Slope of tangent
Slope of normal to the curve
= - cot θ tan
2
p
q
æ ö
æ ö
= +
ç ÷
ç ÷
è ø
è ø
Now, equation of normal to the curve
cosθ
[y-a (sinθ-θcosθ)] - (x - a (cosθ + sin θ))
sinθ
q
=
x cos θ+ ysin θ = a(1)
Þ
Now, distance from (0,0) to x cos θ + y sin θ = a is
distance
(0 + 0 - a)
(d) =
1
distance is constant = |a|.
Example – 30
A point P (x, y) moves along the line whose equation is
x – 2y + 4 = 0 in such a way that y increases at the rate of
3 units/sec.The pointA(0, 6) is joined to Pand the segment
AP is prolonged to meet the x-axis in a point Q. Find how
fast the distance from the origin to Q is changing when P
reaches the point (4, 4).
Sol. The rate of change of y is given and it is desired to find the
rate of change of OQ, which we denote by z. If MP is
perpendicular to the x-axis, MP = y and OM = x.
The triangles OAQ and MPQ are similar, hence
z z x 6x
yz 6z 6x z
6 y 6 y
-
= Þ = - Þ =
-

Substituting the value of x from the equation of the given
line, we have
12 (y 2)
z
6 y
-
=
-
2
dz 48 dy
dt dt
(6 y)
=
-
Setting y = 4 and
dy
3,
dt
= we obtain
dz
36
dt
= that is, z is
increasing at the rate of 36 units/sec.
Example – 31
The maximum distance from origin of a point on the curve
x = a sin t – b sin
at
b
æ ö
ç ÷
è ø
y = a cos t – b cos
at
b
æ ö
ç ÷
è ø
, both a, b > 0, is
(a) a – b (b) a + b
(c) 2 2
a b
+ (d) 2 2
a b
-
Ans. (b)
Sol. LetA(0, 0) and B(x, y)
2 2 2
AB x y
= +
2 2 2 2 2
2
at
a (sin t+cos t)+b sin
b
AB=
at at
+cos -2ab cos t-
b b
æ æ æ ö
ç ç ç ÷
è ø
è
è
Þ
ö
ö
æ ö æ ö
÷
÷
ç ÷ ç ÷
è ø è ø
ø ø
2 2
= a +b -2ab cosa 2 2
2
a b ab
= + +
(Qexpression will take max value when as cos 1
a = - )
= (a + b)
Example – 32
The greatest value of f (x) = (x +1)1/3
– (x – 1)1/3
on [0, 1] is
(a) 1 (b) 2
(c) 3 (d)
3
1
Ans. (b)
Sol. We have
1 1
3 3
x = x+1 - x-1
f
2/3 2/3
1 1 1
x = - .
3
x+1 x-1
f
é ù
¢
ê ú
ê ú
ë û
2/3 2/3
2/3
2
x-1 - x+1
3 x -1
=
for critical points : f’(x) = 0 or not defined.
Clearly, x
f ¢ does not exist at x = ±1
Now,
2/3 2/3
x = 0 x-1 = x+1 x=0
f ¢ Þ Þ
Clearly, so x= 0, + 1 are critical point in [0, 1].
f(0) = 2 and f(1) = 21/3
Hence greatest value = 2
Example – 33
If x is real, the maximum value of
7
x
9
x
3
17
x
9
x
3
2
2
+
+
+
+
is
(a)
4
1
(b)41
(c) 1 (d)
7
17
Ans. (b)
Sol. For the range of the expression
2 2
2 2
3x + 9x + 17 ax + bc + c
y ,
3x + 9x + 7 px + qx + r
= =
[ find the solution of the inequality 2
Ay +By+K 0
³
Where 2
A = q - 4pr = - 3 , B = 4ar + 4pc - 2bq = 126
2
K = b - 4ac = - 123
i.e., solve 2
3y - 126 + y - 123 0
³
2 2
3 126 123 0 y - 42y + 41 0
y y
Þ - + £ Þ £
( y - 1) ( y - 42) 0 1 y 42
Þ £ Þ £ £
Þ Maximum value ofy is 42
Example – 34
If p and q are positive real numbers such that p2
+ q2
=1,
then the maximum value of (p + q) is
(a)
2
1
(b)
2
1
(c) 2 (d)2
Ans. (c)
Sol. st
I solution :
Let p = cos θ , q = sin θ where 0
2
p
q
£ £
p + q = cos θ + sin θ
Þ maximumvalue of (p + q) 2
=

nd
II solution :
By using
2 2
p +q 1
A.M G.M, pq pq
2 2
³ ³ Þ £
2 2 2
(p + q) p + q 2pq (p+q) 2
= + Þ £
Example – 35
Let f be a function defined by
tan x
, x 0
f (x) x
1, x 0
ì
¹
ï
= í
ï =
î
Statement I x = 0 is point ofminima of f.
StatementIIf’ (0)=0
(a) Statement I is false, Statement II is true.
(b) Statement I is true, Statement II is true;
Statement II is correct explanation for Statement I.
(c) Statement I is true, Statement II is true;
Statement II is not a correct explanation for Statement I.
(d) Statement I is true, Statement II is false.
Ans. (c)
Sol.
tan
, 0
1, 0
x
x
f x x
x
ì
¹
ï
= í
ï =
î
In right neighbourhood of ‘0’
tanx
tan x> x >1
x
Þ
In left neighbourhood of ‘0’
tanx
tan x x 1( tanx 0)
x
< Þ > <
Q
at x 0, ( ) 1
f x
= =
x 0
Þ = is point of minima
0 0
tan h
1
0 0
' 0 lim lim
h h
f h f h
f
h h
® ®
-
+ -
= =
2
0
tan
lim 0
h
h h
h
®
-
= =
hence f’(0) = 0
Þ statement I is true and statement II is true.
Example – 36
If
1 cos 1
f sin 1 cos
1 sin 1
q
q = - q - q
- q
and A and B are
respectively the maximum and the minimum values of
f ,
q then (A, B) is equal to:
(a) (3,-1) (b) 4,2 2
-
(c) 2 2,2 2
+ - (d) 2 2, 1
+ -
Ans. (c)
Sol.
1 cos 1
( ) sin 1 cos
1 sin 1
f
q
q q q
q
= - -
-
(1 sin cos ) cos .( sin cos )
q q q q q
Þ + - - - 2
sin 1
q
+ - +
( ) 2 sin 2 cos2
f q q q
Þ = + +
min
( ) 2 2
f q
Þ = -
max
( ) 2 2
f q
Þ = +
Example – 37
Find the interval in which
f (x) = x4
– 8x3
+ 22 x2
– 24x + 5 is increasing.
Sol. Given f (x) =x4
– 8x3
+22 x2
– 24x+ 5
(x)
¢
f =4x3
– 24 x2
+44x – 24
=4 (x3
– 6x2
+ 11 x – 6)
=4 (x – 1) (x– 2) (x – 3)
For increasing function f ´ (x) > 0
or 4 (x – 1) (x – 2)(x – 3)> 0
or (x – 1) (x – 2) (x – 3) > 0
x Î (1, 2) È (3, ¥ )

Example – 38
Find the interval in which f (x) = x – 2 sin x, 0 x 2
   is
increasing
Sol. Given f (x)= x – 2 sin x
 f ´(x) = 1 – 2 cos x
f ´(x) > 0 or 1 – 2 cos x > 0  cos x <
1
2
or – cos x > –
1
2
or cos
2
( x) cos
3

  
or
2 2
2n x 2n , n I
3 3
 
        
or
5
2n x 2n
3 3
 
     
For n = 1,
5
x
3 3
 
  which is true ( 0 x 2 )
  

Hence,
5
x ,
3 3
 
 
 
 
Example – 39
Find the intervals of monotonicity of the function
.
x
|
1
x
|
)
x
( 2


f
Sol. The given function f (x) can be written as :













1
x
;
x
1
x
0
x
,
1
x
;
x
x
1
x
|
1
x
|
)
x
(
2
2
2
f
Consider x < 1
3
2
3
x
2
x
x
1
x
2
)
x
´(





f
For increasing, f ´ (x) > 0  0
x
2
x
3


 x(x– 2)> 0 [as x2
is positive]
 x (– , 0) (2, ).
Combining with x < 1, we get f (x) is increasing in x < 0 and
decreasing in x (0, 1) ...(i)
Consider x 1
2 3 3
1 2 2 x
´(x)
x x x
 
  
f
+ – + –
0 1 2
For increasing f ´ (x) > 0
 (2 – x) > 0 [as x3
is positive]
 (x– 2) < 0.
 x<2.
Combining with x > 1, f (x) is increasing in x (1, 2) and
decreasing in x (2, ) ...(ii)
Combining (i) and (ii), we get :
f (x) is strictlyincreasing on x (– , 0) (1, 2) and strictly
decreasing on x (0, 1) (2, ).
Example – 40
The function f (x) =log (x– 2)2
– x2
+ 4x +1 increaseson the
interval
(a)(1,2) (b) (2, 3)
(c) (5/2, 3) (d) (2, 4)
Ans. (b,c)
Sol. f (x) = 2 log(x – 2) – x2
+ 4x + 1
 4
x
2
2
x
2
)
x
´( 



f
 2
x
)
3
x
(
)
1
x
(
2
2
x
)
2
x
(
1
2
)
x
´(
2















f
 2
2(x 1) (x 3) (x 2)
´(x)
(x 2)
  
 

f
 f ´ (x) > 0 – 2 (x – 1) (x– 3) (x – 2) > 0
 (x– 1) (x – 2) (x– 3) < 0
 x (– , 1) (2, 3).
– + – +
1 2 3

Example – 41
A function is matched below against an interval where it is
supposed to be increasing. Which of the following pairs
is incorrectly matched ?
Interval Function
(a) (–¥, –4) x3
+6x2
+6
(b)
1
,
3
æ ù
-¥
ç ú
è û
3x2
–2x+1
(c) [2,¥) 2x3
–3x2
–12x+6
(d) (–¥, ¥) x3
–3x2
+3x+3
Ans. (b)
Sol. For function to be increasing, f’ (x) > 0
(a) f’(x) = 3x(x + 4) Þ increasing in , 4 0,
-¥ - È ¥
(b) f’(x) = 2(3x – 1) Þ decreasing in 1
,
3
-¥
(c) f’(x)=6(x+1)(x– 2) Þincreasingin , 1 2,
-¥ - È ¥
(d) f’(x) = 3(x – 1)2
Þ increasing in ,
-¥ ¥
so (b) match is incorrect.
Example – 42
The function f (x) = tan–1
(sin x + cos x) is an increasing
function in
(a) ÷
ø
ö
ç
è
æ p
2
,
0 (b) ÷
ø
ö
ç
è
æ p
p
-
2
,
2
(c) ÷
ø
ö
ç
è
æ p
p
2
,
4
(d) ÷
ø
ö
ç
è
æ p
p
-
4
,
2
Ans. (d)
Sol. 2
1
'(x) = . (cos x - sin x )
1+ (sin x + cos x)
f
cos x - sin x
'( )=
2 + sin 2 x
f x
If '(x) >0
f then '(x)
f is increasing function
For
π
- x ,cosx sinx
2 4
p
< < >
Hence y '(x)
f
= is increasing in
π π
- ,
2 4
æ ö
ç ÷
è ø
Example – 43
The function f (x) = cot–1
x + x increases in the interval
(a) (1, ¥) (b) (–1, ¥)
(c) (–¥, ¥) (d)(0, ¥)
Ans. (c)
Sol. 1
cot
f x x x
-
= +
2
2 2
-1 x
'(x)= +1= 0
1+x 1+x
f R
> "´Î
Example – 44
A spherical balloon is filled with 4500p cu m of helium
gas. If a leak in the balloon causes the gas to escape at
the rate of 72pcu m/ min, then the rate (in m/min)at which
the radius ofthe balloon decreases 49 min after the leakage
began is
(a)
9
7
(b)
7
9
(c)
2
9
(d)
9
2
Ans. (c)
Sol.
3
0
dv
=-72πm / min,v 4500π
dt
=
3 2
4 dv 4 dr
v= πr = ×3r ×
3 dt 3 dt
p

After 49min , 0
dv
v= v 49. 4500π - 49 72
dt
p
+ = ´
= 4500π - 3528π=972π
3 3
4
972π = πr r = 243×3 = 36 r = 9
3
Þ Þ Þ
dr 18 2
72 π = 4π × 81× = - = -
dt 81 9

Thus ,radius decreases at a rate of
2
m/min
9
Example – 45
A point on the parabola y2
= 18x at which the ordinate
increases at twice the rate of the abscissa, is
(a) (2,4) (b)(2, –4)
(c)
9 9
,
8 2
æ ö
-
ç ÷
è ø
(d)
9 9
,
8 2
æ ö
ç ÷
è ø
Ans. (d)

Sol.
2 dy dy 9
y 18x 2y 18
dx dx y
= Þ = Þ =
dy 9 9 9
Given =2 2
dx y 2 8
y x
Þ = Þ = Þ =
Example – 46
If the volume of a spherical ball is increasing at the rate of
4p cc/sec, then the rate of increase of its radius
(in cm/sec), when the volume is 288 cc,
p is:
(a)
1
6
(b)
1
9
(c)
1
36
(d)
1
24
Ans. (c)
Sol. 4 /
dV
cc sec
dt
p
=
we know 3
4
3
V r
p
=
2 2
4
3 4
3
dV dr dr
r r
dt dt dt
p p
= =
when 288
V cc
p
=
3
216
r
Þ =
6
r
Þ =
2
4 .
dV dr
r
dt dt
p
=
4 4 36
dr
dt
p p
Þ = ´ ´
1
36
dr
dt
Þ =
Example – 47
The period T of a simple pendulum is
T 2
g
= p
l
Find the maximum error in T due to possible errors upto
1% in l and 2.5% in g.
Sol. Since
1/2
T 2 2
g g
æ ö
= p = pç ÷
è ø
l l
Taking logarithm on both sides, we get
ln T = ln
1
2
2
p+ ln l –
1
2
ln g
Differentiating both sides, we get
dT 1 d 1 dg
0 . .
T 2 2 g
= + -
l
l
or
dT 1 d 1 dg
100 100 100
T 2 2 g
æ ö
æ ö æ ö
´ = ´ - ´
ç ÷
ç ÷ ç ÷
è ø è ø è ø
l
l
100
dT
T
æ ö
´ =
ç ÷
è ø
1
2
( 1 ± 2.5)
d dg
100 1and 100 2.5
g
æ ö
´ = ´ =
ç ÷
è ø
Q
l
l
MaximumerrorinT =1.75%.
Example – 48
If the Rolle’s theorem holds for the function
f(x) = 2x3
+ ax2
+ bx in the interval [-1, 1] for the point
1
c ,
2
= then the value of 2a + b is
(a) 1 (b)-1
(c) 2 (d)-2
Ans. (b)
Sol. 3
(x) = 2x + ax + bx
f
Given Rolle’s theorem is applicable
( 1) (1)
f f
Þ - =
2 2
a b a b
Þ - + - = + +
2
b
Þ = -
2
'( ) 6 2
f x x ax b
= + +
2
6 2 2
x ax
= + -
1
' = 0
2
f
æ ö
ç ÷
è ø
1
a =
2
Þ
2 a + b = - 1
Þ

Example – 49
Find the equation of the tangent to
m m
m m
x y
1
a b
+ = at the
point (x0
, y0
).
Sol.
m m
m m
x y
1
a b
+ = Differentiating wrt x,
Þ
m 1 m 1
m m
mx my dy
0
dx
a b
- -
+ =
Þ
m 1
m
m
dy b x
dx y
a
-
æ ö
= - ç ÷
è ø
Þ at the given point (x0
, y0
), slope of tangent is
m 1
m
0
x ,y 0
0 0
x
dy b
dx a y
-
æ ö
æ ö
= - ç ÷
ç ÷
è ø è ø
Þ the equation of tangent is
m 1
m
0
0 0
0
x
b
y y x x
a y
-
æ ö
æ ö
- = - -
ç ÷
ç ÷
è ø è ø
m m 1 m m m m 1 m m
0 0 0 0
a yy a y b x x b x
- -
- = - +
m m 1 m m 1 m m m m
0 0 0 0
a yy b x x a y b x
- -
+ = +
using the equation of given curve, the right side can be
replaced by am
bm
.

m m 1 m m 1 m m
0 0
a yy b x x a b
- -
+ =
Þ the equation of tangent is
m 1 m 1
0 0
x y
x y
1
a a b b
- -
æ ö æ ö
+ =
ç ÷ ç ÷
è ø è ø
Example – 50
Find the equation of the tangent to x3
= ay2
at the point A
(at2
, at3
). Find also the Point where this tangent meets the
curve again.
Sol. Equation of tangent to : x = at2
, y = at3
is
2
2 , 3
dx dy
at at
dt dt
= =
2
3 2
3
2
at
y at x at
at
- = -
Þ 2y – 2at3
= 3tx – 3at3
i.e. 3tx – 2y – at 3
= 0
Let B
2 3
1 1
at , at be the point where it again meets the curve.
Þ slope of tangent at A = slope of AB
3 3
2
1
2 2
1
a t t
3at
2at a t t
-
=
-
Þ
2 2
1 1
1
t t t t
3t
2 t t
+ +
=
+
Þ 3t2
+ 3 tt1
= 2t2
+ 2t1
2
+ 2 t t1
Þ 2t1
2
– t t1
– t2
= 0
Þ (t1
– t) (2t1
+ t) = 0
Þ t1
= t or t1
= – t/2
The relevant value is t1
= – t/2
Hence the meeting point B is
2 3
t t
a , a
2 2
é ù
- -
æ ö æ ö
= ê ú
ç ÷ ç ÷
è ø è ø
ê ú
ë û
2 3
at at
,
4 8
é ù
-
= ê ú
ë û
Example – 51
The normal to the curve x = a (1 + cos q), y = a sin q at q
always passes through the fixed point
(a) (a, 0) (b) (0, a)
(c) (0,0) (d) (a, a)
Ans. (a)
Sol. sin cos
dx dy
a and a
d d
q q
q q
=- = cot
dy
dx
q
Þ =-
slope of normal at tan
q q
=
the equation of normal at q is
sin tan ( cos )
y a x a a
q q q
- = - -
sin cos sin
x y a
q q q
Þ - =
( )tan
y x a q
Þ = -
which always pasess through (a,0)
Example –52
Two shipsA and B are sailing straight away from a fixed
point O along routes such that AOB
Ð is always 120°.
At a certain instance, OA = 8 km, OB = 6 kmand the ship
A is sailing at the rate of20 km/hr while the ship B sailing
at the rate of 30 km/hr. Then the distance between A and
B is changing at the rate (in km/hr):
(a)
260
37
(b)
260
37
(c)
80
37
(d)
80
37
Ans. (a)

Sol.
Let OA = x and OB = y
20 / ,
dx
km hr
dt
= 30 / .
dy
km hr
dt
=
When OA = 8, OB = 6
Applying cosine formula in AOB
D .
2
2 2
cos 120
2
x y AB
xy
+ -
° =
2
64 36
1
8 2 8 6
AB
+ -
- =
´ ´
Þ -48 = 64 + 36 – (AB)2
2 37
AB
Þ =
Again applying cosine formula in DAOB
When OA = x and OB = y
2
2 2
1
2 2
x y AB
xy
+ -
Þ - =
2 2 2
AB x y xy
Þ = + +
AB = distance between Aand B = Z (let)
z2
= x2
+ y2
+ xy
differentiate w.r.t. “t”
2 . 2 . 2 .
dz dx dy dy dx
z x y x y
dt dt dt dt dt
= + + +
2 2 37 16 20 12 30 240 120
dz
dt
Þ ´ = ´ + ´ + +
4 37 1040
dz
dt
Þ =
260
/
37
dz
km hr
dt
Þ =
Example – 53
If 2a + 3b + 6c = 0, a, b, c Î R then show that the equation
ax2
+ bx + c = 0 has at least one root between 0 and 1.
Sol. Given 2a + 3b + 6c = 0
or
a b
c 0
3 2
+ + = ....(i)
Let
2
(x) = ax bx c
f ¢ + +
On integrating both sides, we get
3 2
ax bx
(x) = cx k
3 2
+ + +
f
Now,
a b
(1) = c k
3 2
+ + +
f [From(i)]
= 0 + k = k
and f (0)= 0 + 0 + 0 + k = k
Since f (x) is a polynomial of three degree, it is continuous
and differentiable and f (0) = f (1), then by Rolle’s theorem
(x) = 0
¢
f i.e., ax2
+ bx + c = 0 has at least one real root
between 0 and 1.
Example – 54
If f (x) = (x – 1) (x– 2) (x – 3) and a =0, b = 4., find ‘c’ using
Lagrange’s mean value theorem.
Sol. We have f (x) = (x – 1) (x – 2) (x – 3) = x3
– 6x2
+ 11 x – 6
f (a) = f (0) = (0 – 1) (0 – 2) (x – 3) = – 6
and f (b) = f (4) = (4 – 1) (4 – 2) (4 – 3) = 6

(b) – (a) 6 ( 6) 12
3
b a 4 0 4
- -
= = =
- -
f f
....(1)
Also
2
(x) = 3x 12x 11
¢ - +
f
gives 2
(c) = 3c 12c 11
¢ - +
f
FromLMVT,
(b) – (a)
(c)
b a
¢
=
-
f f
f ....(2)
Þ 3 = 3c2
– 12c + 11 {From(1) and (2)}
Þ 3c2
– 12c + 8 = 0

12 144 96 2 3
c 2
6 3
± -
= = ±
As both of these values of c lie in the open interval (0, 4).
Hence both of these are required values of c.

Example – 55
Avalue ofc for which conclusion ofMean Value Theorem
holds for the function f (x) = loge
x on the interval [1, 3], is
(a) log3
e (b) loge
3
(c) 2 log3
e (d) 3
log
2
1
e
Ans. (c)
Sol. By LMVT
( ) ( ) (3) (1)
'( )
3-1
f b f a f f
f c
b a
- -
= =
-
e e
e
log 3 log 1 1
'(c) log 3
2 2
f
-
= =
e 3
3
1 1 1
= log 3 c=2log e
c 2 2log e
Þ =

EXERCISE - 1 : BASIC OBJECTIVE QUESTIONS
Derivative asrate Measure
1. Gas is being pumped into a spherical balloon at the rate of
30 ft3
/min. Then, the rate at which the radius increases when
it reaches the value 15 ft, is
(a) min
/
ft
30
1
p
(b) min
/
ft
15
1
p
(c) min
/
ft
20
1
(d) min
/
ft
15
1
2. The position of a point in time ‘t’ is given by x = a + bt–ct2
,
y = at + bt2
. Its acceleration at time ‘t’ is
(a) b – c (b) b + c
(c) 2b – 2c (d) 2
2
c
b
2 +
3. Asphericalironball10cminradiusiscoatedwithalayeroficeof
uniform thickness that melts at a rate of 50 cm3
/min. When the
thicknessoficeis5cm,thentherateatwhichthethicknessofice
decreases, is
(a)
1
cm/ min
18p
(b)
1
cm/ min
36p
(c)
5
cm/ min
6p
(d)
1
cm/ min
54p
4. The rate of change of the surface area of a sphere of radius
r, when the radius is increasing at the rate of 2 cm/s is
proportional to
(a)
r
1
(b) 2
r
1
(c) r (d) r2
5. For what values of x is the rate of increase of x3
– 5x2
+ 5x + 8 is
twice the rate of increase of x ?
(a)
1
3,
3
- - (b)
1
3,
3
-
(c)
1
3,
3
- (d)
1
3,
3
6. If a particle moving along a line follows the law s 1 t,
= +
then the acceleration is proportional to
(a) square of the velocity
(b) cube of the displacement
(c) cube of the velocity
(d) square of the displacement
7. If a particle is moving such that the velocity acquired is
proportional to the square root of the distance covered,
then its acceleration is
(a) a constant (b) µ s2
(c) 2
1
s
µ (d)
1
s
µ
ErrorsandApproximations
8. If y = xn
, then the ratio of relative errors in y and x is
(a) 1 : 1 (b) 2 : 1
(c) 1 : n (d) n : 1
9. If the ratio of base radius and height of a cone is 1 : 2 and
percentage error in radius is l %, then the error in its volume
is
(a) l % (b)2l%
(c)3l% (d) none of these
10. The height of a cylinder is equal to the radius. If an error of
a % is made inthe height, then percentage error inits volume
is
(a) a % (b) 2a%
(c) 3a% (d) none of these
EquationofTangentsandNormals
11. For the curve y = 3 sin q cos q , x e sin ,0 ,
q
= q £ q £ p
the tangent is parallel to x-axis when q is:
(a)
3
4
p
(b)
2
p
(c)
4
p
(d)
6
p

12. The curve y – exy
+ x = 0 has a vertical tangent at
(a) (1,1) (b)(0, 1)
(c) (1,0) (d) no point
13. If the line ax + by + c = 0 is a tangent to the curve xy = 4, then
the possible answer is
(a) a > 0, b > 0 (b) a > 0, b < 0
(c) a < 0, b > 0 (d) none of these
14. The tangent to the curve 5x2
+ y2
= 1 at
1 2
,
3 3
æ ö
-
ç ÷
è ø
passes
through the point
(a) (0,0) (b)(1, –1)
(c) (–1, 1) (d) none of these
15. The equation of the tangent to the curve
2
y 9 2x
= - at the
point where the ordinate and the abscissa are equal, is
(a) 2x y 3 3 0
+ - = (b) 2x y 3 3 0
+ + =
(c) 2x y 3 3 0
- - = (d) none of these
16. The tangent to the curve x2
+ y2
= 25 is parallel to the line 3x
– 4y = 7 at the point
(a) (–3, –4) (b) (3, –4)
(c) (3,4) (d) none of these
17. If the tangent at each point of the curve
y =
2
3
x3
– 2ax2
+ 2x + 5 makes an acute angle with the
positive direction of x-axis, then
(a) a ³ 1 (b) –1 £ a £ 1
(c) a £ – 1 (d) none of these
18. The equation of the tangent to the curve (1 + x2
) y = 2 –x,
where it crosses the x-axis, is
(a) x + 5y = 2 (b) x – 5y = 2
(c) 5x – y = 2 (d) 5x + y – 2 = 0
19. The intercepts on x-axis made by tangents to the curve,
x
0
y t dt x R
| | , ,
= Î
ò which are parallel to the line y = 2x, are
equal to
(a) ± 1 (b) ± 2
(c) ± 3 (d) ± 4
Length oftangent,normal, subtangentand subnormal
20. The length of subtangent to the curve x2
y2
= a4
at the point
(–a, a) is
(a) 3a (b) 2a
(c) a (d) 4a
21. For the parabola y2
= 4ax, the ratio of the sub-tangent to the
abscissa is
(a) 1 : 1 (b) 2 : 1
(c) 1 : 2 (d) 3 : 1
22. The length of subtangent to the curve x2
y2
= a4
at the point
(–a, a) is
(a) 3a (b) 2a
(c) a (d) 4a
23. The product of the lengths of subtangent and subnormal at
any point of a curve is
(a) square of the abscissae (b) square of the ordinate
(c) constant (d) None of these
Angle of intersection between the curves
24. The curves x3
+ p xy2
= –2 and 3x2
y – y3
= 2 are orthogonal
for
(a) p = 3 (b) p = –3
(c) no value of p (d) p = +3
25. The two tangents to the curve ax2
+ 2hxy + by2
= 1, a > 0 at
the points where it crosses x-axis, are
(a) parallel (b) perpendicular
(c) inclined at an angle
4
p
(d) none of these
26. The lines y =
3
x
2
- and
2
y x
5
= - intersect the curve
3x2
+ 4xy + 5y2
– 4 = 0 atthe points P and Q respectively. The
tangents drawn to the curve at P and Q
(a) intersect each other at angle of 45°
(b) are parallel to each other
(c) are perpendicular to each other
(d) none of these
27. The angle between the curves y = sin x and y = cos x is
(a) )
2
2
(
tan 1
-
(b) )
2
3
(
tan 1
-
(c) )
3
3
(
tan 1
-
(d) )
2
5
(
tan 1
-

28. The angle between the tangents to the curve y2
= 2ax at the
points where
a
x ,
2
= is
(a) p/6 (b) p/4
(c) p/3 (d) p/2
29. The angle between the tangents at those points on the curve
y = (x + 1) (x – 3) where it meets x-axis, is
(a)
1 15
tan
8
- æ ö
ç ÷
è ø
(b)
1 8
tan
15
- æ ö
ç ÷
è ø
(c)
4
p
(d) none of these
30. The angle at which the curves y = sin x and y = cosx intersect
in [0, p],is
(a) 1
tan 2 2
- (b) 1
tan 2
-
(c)
1 1
tan
2
- æ ö
ç ÷
è ø
(d) none of these
31. The two curves x3
– 3xy2
+ 2 = 0 and 3x2
y – y3
– 2 = 0
(a) cut at right angles (b) touch each other
(c) cut at an angle
3
p
(d) cut at an angle
4
p
32. The two curves y = 3x
and y = 5x
intersect at an angle
(a)
1 log5 log3
tan
1 log3.log5
- æ ö
-
ç ÷
+
è ø
(b)
1 log3 log5
tan
1 log3.log5
- æ ö
+
ç ÷
-
è ø
(c)
1 log3 log5
tan
1 log3.log5
- æ ö
+
ç ÷
+
è ø
(d) none of these
33. The angle of intersection of the curve y = x2
& 6y = 7 – x3
at
(1, 1)is
(a) p/5 (b) p/4
(c) p/3 (d) p/2
Increasing and Decreasing Functions
34. The function f (x) = 2x2
– log |x |monotonically decreases for
(a) x Î ( –¥, – 1/2] È (0, 1/2]
(b) x Î (– ¥, 1/2]
(c) x Î [– 1/2, 0) È [1/2, ¥)
(d) none of these
35. The interval in which the function x3
increases less rapidly
than 6x2
+ 15x + 5 is :
(a) (–¥, –1) (b) (– 5, 1)
(c) (–1, 5) (d) (5, ¥)
36. The function
2
4
2x 1
y
x
-
= is
(a) a decreasing function for all x Î R – {0}
(b) a increasing function for all x Î R – {0}
(c) increasing for x > 0
(d) none of these
37. The function
sin x
x
x
=
f is decreasing in the interval
(a) , 0
2
p
æ ö
-
ç ÷
è ø
(b) 0,
2
p
æ ö
ç ÷
è ø
(c) (0, p) (d) none of these
38. If
1
x
1
)
x
(
f
+
= – log (1 + x), x > 0, then f is
(a) an increasing function
(b) a decreasing function
(c) both increasing and decreasing function
(d) None of the above
39. Let f (x) = ò
x
1
x
e (x – 1) (x – 2) dx. Then, f decreases in the
interval
(a) (–¥, 2) (b) (–2, –1)
(c) (1,2) (d)(2, ¥)
40. If f(x) = x3
+ 4x2
+ lx + 1 is a strictly decreasing function ofx
in the largest possible interval [–2, –2/3] then
(a) l =4 (b) l =2
(c) l =–1 (d) l has no real value
41. The length of the longest interval, in which the function
3 sin x – 4 sin3
x is increasing, is
(a)
3
p
(b)
2
p
(c)
2
3p
(d)p

42. The function f (x) = x + cos x is
(a) always increasing
(b) always decreasing
(c) increasing for certain range of x
(d) None of the above
43. How many real solutions does the equation
x7
+14x5
+16x3
+30x– 560 = 0 have?
(a) 5 (b) 7
(c) 1 (d) 3
Maximaandminima
– 3x2
– 12x + 4 has
(a) no maxima and minima
(b) one maximaand one minima
(c) two maxima
(d)two minima
45. The greatest value of f (x) = (x +1)1/3
– (x –1)1/3
on [0, 1] is :
(a) 1 (b) 2
(c) 3 (d) 21/3
46. The function f (x) = x2
(x –2)2
(a) decreases on (0, 1) È (2, ¥)
(b) increase on (–¥, 0) È (1, 2)
(c) has a local maximum value 0
(d) has a local maximum value 1
47. Themaximumvalueofthefunctiony=x(x–1)2
,0£x£2is
(a) 0 (b)4/27
(c) –4 (d) none of these
48. The point in the interval [0, 2p], where f (x) = ex
sin x has
maximumslope,is
(a) 0 (b)
2
p
(c) 2p (d)
3
2
p
49. The minimumvalue of xx
is attained (where x is positive real
number) when x is equal to :
(a) e (b) e–1
(c) 1 (d) e2
50. The maximumvalue of x3
– 3x in the intveral [0, 2], is
(a) –2 (b)0
(c) 2 (d) None of these
51. IfA> 0, B > 0 andA+ B
3
,
p
= then the maximum value of
tan A tan B is
(a)
1
3
(b)
1
3
(c) 3 (d) 3
52. The maximum slope of the curve y = –x3
+ 3x2
+ 9x – 27 is
(a) 0 (b) 12
(c)16 (d)32
53. The function
x
3 2 2
1
x 2 t 1 t 2 3 t 1 t 2 dt
f = - - + - -
ò
attains its local maximum value at x =
(a) 1 (b)2
(c) 3 (d)4
54. The maximum area of the rectangle that can be inscribed in
a circle of radius r, is
(a) pr2
(b) r2
(c) pr2
/4 (d) 2r2
55. A triangular park is enclosed on two sides by a fence and on
the third side by a straight river bank. The two sides having
fence are of same length x. The maximum area enclosed by
the park is
(a)
2
x
2
3
(b)
8
x3
(c)
2
x
2
1
(d)px2
56. The greatest and the least value of the function,
f (x) = 2 2
1–2x+x – 1+2x+x , x Î (-¥, ¥) are
(a) 2, –2 (b) 2, –1
(c) 2, 0(d) none
– 15x2
+ 36x + 4 has local maxima at
(a) x= 2 (b) x= 4
(c) x= 0 (d) x= 3

58. The maximum value of xy subject to x + y = 8, is
(a) 8 (b)16
(c)20 (d)24
59. Let f (x) = (1+b2
)x2
+2bx + 1 and m(b) the minimumvalueof
f (x) for a given b. As b varies, the range of m (b) is
(a) [0,1] (b)(0, 1/2]
(c)
1
,1
2
é ù
ê ú
ë û
(d) (0, 1]
60. f (x) = 1 + [cos x] x, in 0 £ x £
2
p
(a) has a minimum value 0
(b) has a maximum value 2
(c) is continuous in 0,
2
p
é ù
ê ú
ë û
(d) is not differentiable at x =
2
p
61. The minimum value of
3
2
x 3 27
2 ,
- +
is
(a) 227
(b) 2
(c) 1 (d) 4
62. Area of the greatest rectangle that can be inscribed in the
ellipse 1
b
y
a
x
2
2
2
2
=
+ is
(a) ab (b) 2 ab
(c) a/b (d) ab
63. The difference between the greatest and least values of the
function, f (x) = cos x +
1
2
cos 2x –
1
3
cos 3x is :
(a) 4/3 (b) 1
(c) 9/4 (d) 1/6
64. A line is drawn through the point(1, 2) to meet the coordinate
axes at P and Q such that it forms a D OPQ, where O is the
origin, if the area of the D OPQ is least, then the slope of the
line PQ is
(a)
1
4
- (b) – 4
(c) – 2 (d)
1
2
-
Numerical ValueType Questions
65. The radius of the base of a cone is increasing at the rate of
3 cm/minute and the altitude is decreasing at the rate of
4 cm/minute. The rate of change of lateral surface when the
radius = 7 cm and altitude = 24 cm, in cm2
/min is:
66. A ladder 10 metres long rests with one end against a vertical
wall, the other end on the floor. The lower end moves away
fromthe wall at the rate of 2 metres/minute. The rate at which
the upper end falls when its base is 6 metres away from the
wall, in M/min is :
67. If the distance ‘s’ metres travelled by a particle in t seconds
is given by s = t3
– 3t2
, then the velocity of the particle when
the acceleration is zero in m/s is
68. An object is moving in the clockwise direction around the
unit circle x2
+ y2
= 1. As it passes through the point
,
2
3
,
2
1
÷
÷
ø
ö
ç
ç
è
æ
its y-coordinate is decreasing at the rate of
3 unit persecond. The rate at whichthe x-coordinate changes
at this point is (in unit per second)
69. If 3
4
V r ,
3
= p at what rate in cubic units is V increasing
when r = 10 and
dr
0.01
dt
= ?
70. Side of an equilateral triangle expands at the rate of2 cm/s. The
rateofincreaseofitsareawheneachsideis10cm,in cm2
/sec is:
71. The radius ofa sphere is changing at the rate of0.1 cm/s. The
rate of change of its surface area when the radius is 200 cm,
in cm2
/sec is:
72. The surface area of a sphere when its volume is increasing
at the same rate as its radius, in sq. unit is :
73. The surface area of a cube is increasing at the rate of 2 cm2
/s.
When its edge is 90 cm, the volume is increasing at the rate
of (in cm3
/sec)
74. The sides of an equilateral triangle are increasing at the rate
of 2 cm/s. The rate at which the area increases, when the
side is 10 cm, in cm2
/s is:

75. The distance moved by the particle in time t is given by
x = t3
– 12t2
+ 6t + 8. At the instant when its acceleration is
zero, the velocity is
76. The circumference of a circle is measured as 28 cm with an
error of 0.01 cm. The percentage error in the area is
77. The triangle formed by the tangent to the curve
f (x) = x2
+ bx – b at the point (1, 1) and the coordinate axes,
lies in the first quadrant. If its area is 2, then the value of b is
78. If the normal to the curve y = f (x) at the point (3, 4)makes an
angle
3
4
p
with the positive x-axis, then f ’ (3) is equal to
79. Find the shortest distance between the line y = x - 2 and the
parabola y = x2
+ 3x + 2.
80. If f (x) is differentiable in the interval [2, 5], where
f (2)
1
5
= and f (5)
1
2
= , then there exists a number
c, 2 < c < 5 for which f ´ (c) is equal to

EXERCISE - 2 : PREVIOUS YEAR JEE MAINS QUESTIONS
1. The normal to the curve, x2
+ 2xy –3y2
= 0, at (1, 1):
(2015)
(a) meets the curve again in the third quadrant.
(b) meets the curve again in the fourth quadrant.
(c) does not meet the curve again.
(d) meets the curve again in the second quadrant.
2. Let f(x) be a polynomial of degree four having extreme
value at x = 1 and x = 2. If 2
x 0
f(x)
lim 1
x
®
é ù
+
ê ú
ë û
= 3, then f(2) is
equal to: (2015)
(a) 0 (b)4
(c) –8 (d) –4
3. If Rolle’s theorem holds for the function f(x) = 2x3
+ bx2
+
cx,x Î [–1, 1], at the point x =
1
2
, then 2b + c equals :
(2015/Online Set–1)
(a) 1 (b)2
(c) –1 (d) –3
4. Let k and K be the minimum and the maximum values of
the function f (x)=
0.6
0.6
(1 x)
1 x
+
+
in [0, 1] respectively, then
the ordered pair (k, K) is equal to: (2015/Online Set–2)
(a) 0.4
(2 ,1)
-
(b) 0.4 0.6
(2 ,2 )
-
(c) 0.6
(2 ,1)
-
(d) 0.6
(1, 2 )
5. A wire of length 2 units is cut into two parts which are
bent respectively to form a square of side =x unit and a
circle of radius = r units. If the sum of the areas of the
square and the circle so formed is minimum, then :
(2016)
(a) (4 – p) x = pr (b) x = 2r
(c) 2x = r (d)2x =(p+ 4)r
6. Consider 1 1 sin x
f x tan ,x 0,
1 sin x 2
- æ ö
+ p
æ ö
= Î
ç ÷ ç ÷
ç ÷
- è ø
è ø
.
A normal to y = f (x) at x
6
p
= also passes through the
point : (2016)
(a)
2
0,
3
p
æ ö
ç ÷
è ø
(b) , 0
6
p
æ ö
ç ÷
è ø
(c) , 0
4
p
æ ö
ç ÷
è ø
(d) (0, 0)
7. If the tangent at a point P, with parameter t, on the curve
x = 4t2
+ 3, y = 8t3
–1, t R,
Î meets the curve again at a
point Q, then the coordinates of Q are :
(a) (t2
+ 3, –t3
– 1) (b) (4t2
+ 3, –8t3
– 1)
(c) (t2
+ 3, t3
– 1) (d) (16t2
+ 3, –64t3
– 1)
8. The minimum distance of a point on the curve
y = x2
– 4 from the origin is : (2016/Online Set–1)
(a)
19
2
(b)
15
2
(c)
15
2
(d)
19
2
9. The normal to the curve y(x – 2) (x – 3) = x + 6 atthe point
where the curve intersects the y-axis passes through the
point: (2017)
(a)
1 1
,
2 2
æ ö
- -
ç ÷
è ø
(b)
1 1
,
2 2
æ ö
ç ÷
è ø
(c)
1 1
,
2 3
æ ö
-
ç ÷
è ø
(d)
1 1
,
2 3
æ ö
ç ÷
è ø
10. Twenty meters ofwire is available for fencing off a flower-
bed in the form of a circular sector. Then the maximum
area (in sq. m) of the flower-bed, is: (2017)
(a)12.5 (b)10
(c)25 (d)30
11. The tangent at the point (2,–2) to the curve, x2
y2
– 2x = 4
(1 – y) does not pass through the point :
(a)
1
4,
3
æ ö
ç ÷
è ø
(b) (8, 5)
(c) (–4, –9) (d) (–2, – 7)

12. The function f defined by f (x) = x3
– 3x2
+ 5x + 7, is
(a) increasing in R.
(b) decreasing in R.
(c) decreasing in (0, ¥ ) and increasing in ( , 0).
-¥
(d) increasing in (0, ¥ ) and decreasing in ( , 0).
-¥
13. If the curves 2 2 2
y 6x,9x by 16
= + = intersect each other
at right angles, then the value of b is (2018)
(a)
9
2
(b)6
(c)
7
2
(d)4
14. Let 2
2
1 1
f x x and g x x ,
x
x
= + = -
x R 1,0,1
Î - - .If
f x
h x
g x
= ,thenthelocalminimum
value of h(x) is : (2018)
(a) 2 2 (b)3
(c) -3 (d) 2 2
-
15. If a right circular cone, having maximum volume, is
inscribed in a sphere of radius 3 cm, then the curved
surface area (in cm2
) of this cone is :
(a) 6 2p (b) 6 3p
(c) 8 2p (d) 8 3p
16. Let f(x) be a polynomial of degree 4 having extreme values
at x = 1 and x = 2. If lim 1 3
f x
x
®
æ ö
ç ÷
è ø
2
x 0
+ = then f (-1) is equal
to: (2018/Online Set–2)
(a)
9
2
(b)
5
2
(c)
3
2
(d)
1
2
17. Let M and m be respectively the absolute maximum and
the absolute minimum values of the function, f (x)= 2x3
-
9x2
+12x+5 in the interval [0, 3]. Then M-m is equal to :
(a) 5 (b)9
(c) 4 (d)1
18. The shortest distance between the line y = x and the curve
y2
= x – 2 is: (2019-04-08/Shift-1)
(a) 2 (b)
7
8
(c)
7
4 2
(d)
11
4 2
19. If S1
and S2
are respectively the sets of local minimum and
local maximum points of the function,
4 3 2
( ) 9 12 36 25,
f x x x x x R
= + - + Î then
(2019-04-08/Shift-1)
(a) S1
= {–2}; S2
= {0, 1} (b) S1
={–2,0}; S2
= {1}
(c) S1
= {–2, 1}; S2
= {0} (d) S1
= {–1}; S2
= {0, 2}
20. Let f : [0, 2] ® R be a twice differentiable function such
that f ’ (x) > 0, for all x Î(0, 2). If f (x) = f(x) + f(2 – x),then
f(x)is: (2019-04-08/Shift-1)
(a) increasing on (0, 1) and decreasing on (1, 2).
(b) decreasing on (0, 2)
(c) decreasing on (0, 1) and increasing on (1, 2).
(d) increasing on (0, 2)
21. The height ofa right circular cylinder of maximum volume
inscribed in a sphere of radius 3 is :
(2019-04-08/Shift-2)
(a) 6 (b)
2
3
3
(c) 2 3 (d) 3
22. If the tangent to the curve, 3
y x ax b
= + - at the point
(1, -5) is perpendicular to the line, 4 0
x y
- + + = , then
which one of the following points lies on the curve?
(2019-04-09/Shift-1)
(a) (-2, 1) (b) (-2, 2)
(c) (2, -1) (d) (2, -2)

23. Let S be the set of all values of x for which the tangent to
the curve 3 2
( ) 2 ( , )
y f x x x xat x y
= = - - is parallel to
the line segment joining the points
(1, (1)) ( 1, ( 1))
f and f
- - then S is equal to:
(2019-04-09/Shift-1)
(a)
1
,1
3
ì ü
í ý
î þ
(b)
1
, 1
3
ì ü
- -
í ý
î þ
(c)
1
, 1
3
ì ü
-
í ý
î þ
(d)
1
,1
3
ì ü
-
í ý
î þ
24. If f (x) is a non-zero polynomial of degree four, having
local extreme points at 1,0,1
x = - then the set
{ : ( ) (0)}
S x R f x f
= Î = contains exactly k real values,
then k is (2019-04-09/Shift-1)
25. A water tank has the shape of an inverted right circular
cone, whose semi-vertical angle is 1 1
tan
2
- æ ö
ç ÷
è ø
.Water is
poured into it at a constantrate of5 cubic meter per minute.
Then the rate (in m/min.), at which the level of water is
rising at the instant when the depth of water in the tank is
10m;is: (2019-04-09/Shift-2)
(a)
1
15p
(b)
1
10p
(c)
2
p
(d)
1
5p
26. Let f (x) = ex
– x and g(x) = x2
– x, x R
" Î . Then the set of
all R
x Î ,where the function h(x) = (fog) (x) is increasing,
is: (2019-04-10/Shift-1)
(a)
1
1
1,
2
,
2
-
é ù
÷
- ú
é ö
È ¥
ê
ë
ë û ø
ê (b)
1
0,
2
1,
È ¥
é ù
ê ú
ë û
(c) 0, ¥ (d)
1
,0 1,
2
é ù
- È ¥
ê ú
ë û
27. If the tangent to the curve 2
, 3
3
x
y x R x
x
= Î ¹ ±
-
,
at a point , 0,0
a b ¹ on it is parallel to the line
2 6 11 0
x y
+ - = then: (2019-04-10/Shift-2)
(a) 6 2 19
a b
+ = (b) 6 2 9
a b
+ =
(c) 2 6 19
a b
+ = (d) 2 6 9
a b
+ =
28. A spherical iron ball of radius 10 cm is coated with a layer
of ice of uniform thickness that melts at a rate of
3
50cm / min When the thickness of the ice is 5 cm, then
the rate at which the thickness (in cm/min) of the ice
decreases, is: (2019-04-10/Shift-2)
(a)
1
18p
(b)
1
36p
(c)
5
6p
(d)
1
9p
29. Let a1
,a2
,a3
..... be an A.P. with a6
= 2 then the common
difference of thisA.P., which maximises the product a1
.a4
.a5
is: (2019-04-10/Shift-2)
(a)
3
2
(b)
8
5
(c)
6
5
(d)
2
3
30. A 2 m ladder leans against a vertical wall. If the top of the
ladder begins to slide down the wall at the rate 25 cm/
sec., then the rate (in cm / sec.) at which the bottom of the
ladder slides away from the wall on the horizontal ground
when the top of the ladder is 1 m above the ground is:
(2019-04-12/Shift-1)
(a) 25 3 (b)
25
3
(c)
25
3
(d)25

31. The maximum volume (in cu.m) of the right circular cone
having slant height 3 m is: (2019-01-09/Shift-1)
(a) 6p (b) 3 3p
(c)
4
3
p (d) 2 3p
32. If q denotes the acute angle between the curves, y = 10 -
x2
and y = 2 + x2
at a point of their intersection, then |tan q|
is equal to: (2019-01-09/Shift-1)
(a)
4
9
(b)
8
15
(c)
7
17
(d)
8
17
33. The shortest distance between the point
3
,0
2
æ ö
ç ÷
è ø
and the
curve ,( 0)
y x x
= > , is: (2019-01-10/Shift-1)
(a)
5
2
(b)
3
2
(c)
3
2
(d)
5
4
34. The tangent to the curve,
2
x
y xe
= passing through the
point (1, e) also passes through the point:
(2019-01-10/Shift-2)
(a) (2, 3e) (b)
4
,2
3
e
æ ö
ç ÷
è ø
(c)
5
,2
3
e
æ ö
ç ÷
è ø
(d) (3, 6e)
35. A helicopter is flying along the curve given by
3/2
7, 0
y x x
- = ³ . A soldier positioned at the point
1
,7
2
æ ö
ç ÷
è ø
wants to shoot down the helicopter when it is
nearest to him. Then this nearest distance is:
(2019-01-10/Shift-2)
(a)
5
6
(b)
1 7
3 3
(c)
1 7
6 3
(d)
1
2
36. The maximum value of the function
3 2
3 18 27 40
f x x x x
= - + - on the set
2
: 30 11
S x R x x
= Î + £ is : (2019-01-11/Shift-1)
37. Let
2 2 2
2
, R,
x d x
f x x
a x b d x
-
= - Î
+ + -
where a, b
and d are non-zero real constants. Then:
(2019-01-11/Shift-2)
(a) f is an increasing function of x
(b) f is a decreasing function of x
(c) f’ is not a continuous function of x
(d) f is neither increasing nor decreasing function of x
38. If the function f given by
3 2
3 2 3 7, 0 7
f x x a x ax f
= - - + + = for some
a R
Î is increasing in (0,1] and decreasing in [1,5) , then
a root of the equation, 2
14
0 1
1
f x
x
x
-
= ¹
-
(2019-01-12/Shift-2)
(a) -7 (b)5
(c) 7 (d)6
39. Let P (h, k) be a point on the curve 2
y x 7x 2,
= + +
nearest to the line, y =3x – 3. Then the equation of the
normal to the curve at P is : (2020-09-02/Shift-1)
(a) x + 3y – 62 = 0 (b) x – 3y – 11 = 0
(c) x – 3y + 22 = 0 (d) x + 3y + 26 = 0
40. If p(x) be a polynomial of degree three that has a local
maximum value 8 at x = 1 and a local minimum value 4 at
x = 2; then p (0) is equal to : (2020-09-02/Shift-1)
(a)12 (b) – 12
(c) –24 (d) 6
41. If the tangent to the curve y = x + sin y at a point (a, b) is
parallel to the line joining
3
0,
2
æ ö
ç ÷
è ø
and
1
,2 ,
2
æ ö
ç ÷
è ø
then :
(2020-09-02/Shift-1)
(a)
2
b a
p
= + (b) | | 1
a b
+ =
(c) | | 1
b a
- = (d) b = a

42. The equation of the normal to the curve
2y 2 1
y (1 x) cos (sin x)

   at x = 0 is :
(2020-09-02/Shift-2)
(a) y + 4x = 2 (b) 2y + x = 4
(c) x + 4y = 8 (d) y = 4x + 2
43. Let  
: 1,
f R
   be defined by f (0) = 1 and
   
1
log 1 , 0
e
f x x x
x
   .Then the functionf :
(2020-09-02/Shift-2)
(a) increases in ( 1, )
 
(b) decreases in (–1, 0) and increases in (0, )

(c) increases in (–1, 0) and decreases in (0, )

(d) decreases in ( 1, ).
 
44. The function, 2/3
( ) (3 7) ,
f x x x x R
   is increasing
for all x lying in : (2020-09-03/Shift-1)
(a)  
14
, 0,
15
 
   
 
 
(b)
14
,
15
 

 
 
(c)  
14
,0 ,
15
 
  
 
 
(d)  
3
,0 ,
7
 
  
 
 
45. Suppose f (x) is a polynomial of degree four, having critical
points at –1, 0, 1. If    
 
T | 0 ,
x R f x f
   then the
sum of squares of all the element of T is :
(2020-09-03/Shift-2)
(a) 6 (b) 2
(c) 8 (d) 4
46. If the surface area of a cube is increasing at a rate of 3.6
cm2
/sec, retaining its shape; then the rate of change of its
volume (in cm3
/sec), when the length of a side of the cube
is 10cm, is : (2020-09-03/Shift-2)
(a) 9 (b)10
(c)18 (d)20
47. The area (in sq. units) of the largest rectangle ABCD
whose vertices A and B lie on the x-axis and vertices C
and D lie on the parabola, y = x2
– 1 below the x-axis, is:
(2020-09-04/Shift-2)
(a)
2
3 3
(b)
4
3
(c)
1
3 3
(d)
4
3 3
48. If x = 1 is a critical point of the function
f (x) = (3x2
+ax – 2 – a)ex
, then: (2020-09-05/Shift-2)
(a) x=1 is a local minima and
2
3
x   isa localmaxima off.
(b) x=1 is alocal maxima and
2
3
x   is a localminima of f.
(c) x=1 and
2
3
x   are local minima of f.
(d) x=1 and
2
3
x   are local maxima of f.
49. Which of the following points lies on the tangent to the
curve 4
2 1 3
y
x e y
   at the point (1,0)?
(2020-09-05/Shift-2)
(a) (2,6) (b) (2,2)
(c) (–2,6) (d) (–2,4)
50. The position of a moving car at time t is given by
  2
, 0,
f t at bt c t
    where a, b and c are real
numbers greater than 1. Then the average speed of the
car over the time interval  
1 2
,
t t is attained at the point:
(2020-09-06/Shift-1)
(a)  
1 2 / 2
t t
 (b)  
1 2
2a t t b
 
(c)  
2 1 / 2
t t
 (d)  
2 1
a t t b
 
51. Let AD and BC be two vertical poles at A and B
respectively on a horizontal ground. If AD = 8m, BC =
11m and AB = 10m; then the distance (in meters) of a
point M on AB from the point A such that MD2
+ MC2
is
minimum is _____. (2020-09-06/Shift-1)

52. The set of all real values of l for which the function
2
1 cos . sin , ,
2 2
f x x x x
p p
l
æ ö
= - + Î -
ç ÷
è ø
,
has exactly one maxima and exactly one minima, is:
(2020-09-06/Shift-2)
(a)
3 3
, 0
2 2
æ ö
- -
ç ÷
è ø
(b)
1 1
, 0
2 2
æ ö
- -
ç ÷
è ø
(c)
3 3
,
2 2
æ ö
-
ç ÷
è ø
(d)
1 1
,
2 2
æ ö
-
ç ÷
è ø
53. If the tangent to the curve, log , 0
e
y f x x x x
= = > at
a point (c,f(c)) is parallel to the line-segment joining the
points (1, 0) and (e, e), then c is equal to :
(2020-09-06/Shift-2)
(a)
1
1 e
e
æ ö
ç ÷
-
è ø (b)
1
e
e
-
(c)
1
1
e -
(d)
1
1
e
e
æ ö
ç ÷
-
è ø
54. For all twice differentiable functions f : R ® R, with
f(0)= f(1) =f’(0) =0, (2020-09-06/Shift-2)
(a) f”(x) = 0, at every point x Î(0,1)
(b) f”(x) ¹ 0, at every point x Î(0,1)
(c) f”(x) = 0, for some x Î (0,1)
(d) f”(0) =0
55. Let the function, f : [-7, 0] ® R be continuous on [-7,0]
and differentiable on (-7,0). If f (-7) =-3 and
f ’(x) £ 2 for all xÎ(-7, 0), then for all such functions
f, f(-1) + f (0) lies in the interval: (7-1-2020/Shift-1)
(a) [-6, 20] (b) (-¥,20]
(c)(-¥,11] (d)[-3,11]
56. Let f (x) be a polynomial of degree 5 such that x = ±1 are
its critical points. If 3
0
lim 2 4
x
f x
x
®
æ ö
+ =
ç ÷
è ø
, then which one
of the following is not true? (7-1-2020/Shift-2)
(a) f (1) – 4f (–1) = 4
(b) x = 1 is a point of maxima and x = –1 is a point of
minimumof f.
(c) f is an odd function.
(d) x = 1 is a point of minima and x = –1 is a point of
maxima of f.
57. The value of c in Lagrange’s mean value theorem for the
function f (x) = x3
- 4x2
+ 8x + 11, where 0,1
x Î is :
(7-1-2020/Shift-2)
(a)
4 7
3
-
(b)
2
3
(c)
7 2
3
-
(d)
4 5
3
-
58. If c is a point at which Rolle’s theorem holds for the
function,
2
log
7
e
x
f x
x
a
æ ö
+
= ç ÷
è ø
in the interval [3, 4],
where R
a Î then ''
f c is equal to:
(8-1-2020/Shift-1)
(a)
1
24
- (b)
1
12
-
(c)
3
7
(d)
1
12
59. Let 1
cos sin , ,
2 2
f x x x x
p p
- æ ö
= - Î -
ç ÷
è ø
,
then which of the following is true? (8-1-2020/Shift-1)
(a) ' 0
2
f
p
= -
(b) '
f is decreasing in ,0
2
p
æ ö
-
ç ÷
è ø
and increasing in 0,
2
p
æ ö
ç ÷
è ø
(c) f is not differentiable at x = 0
(d) '
f is increasing in ,0
2
p
æ ö
-
ç ÷
è ø
and decreasing in 0,
2
p
æ ö
ç ÷
è ø
60. Let the normal at a P on the curve y2
– 3x2
+ y + 10 = 0
intersect the y-axis at
3
0,
2
æ ö
ç ÷
è ø
. If m is the slope of the
tangent at P to the curve, then |m| is equal to _____.
(8-1-2020/Shift-1)

61. The length of the perpendicular from the origin, on the
normal to the curve, 2 2
2 3 0
x xy y
+ - = at the point (2,2)
is: (8-1-2020/Shift-2)
(a) 2 (b) 2 2
(c) 4 2 (d) 2
62. Let f (x) be a polynomial of degree 3 such that f (-1) = 10,
f (1) = - 6, f (x) has a critical point at x = -1 and f ’ (x) has a
critical point at x = 1. Then the local minima at x =______
(8-1-2020/Shift-2)
63. A spherical iron ball of 10 cm radius is coated with a layer
of ice of uniform thickness that melts at the rate of
50 cm3
/min. When the thickness of ice is 5 cm, then the
rate (in cm/min.) at which the thickness of ice decreases,
is: (9-1-2020/Shift-1)
(a)
5
6p
(b)
1
54p
(c)
1
36p
(d)
1
19p
64. Let f be any function continuous on [a.b] and twice
differentiable on (a,b). If for all
, , 0 and 0,
x a b f x f x
¢ ¢¢
Î > < then for any
, ,
f c f a
c a b
f b f c
-
Î
-
is greater than:
(9-1-2020/Shift-1)
(a)
b c
c a
-
-
(b)1
(c)
c a
b c
-
-
(d)
b a
b a
+
-
65. Let a function : 0, 5
f R
® , be continuous, f (1)=3 and
F be defined as: 2
1
x
F x t g t dt
= ò ,
where
1
( )
t
g t f u du
= ò . Then for the function F, the
point x = 1 is (9-1-2020/Shift-2)
(a) a point of inflection (b) a point of local maxima
(c) a point of local minima (d) not a critical point
66. The function
3 2
4x 3x
f x 2sin x 2x 1 cosx
6
-
= - + -
(24-02-2021/Shift-1)
(a) Decreases in
1
,
2
æ ù
-¥
ç ú
è û
(b) Increases in
1
,
2
æ ù
-¥
ç ú
è û
(c) Increases in
1
,
2
é ö
¥÷
ê
ë ø
(d) Decreases in
1
,
2
é ö
¥÷
ê
ë ø
67. The minimum value of a for which the equation
4 1
sin x 1 sin x
+ = a
-
has at least one solution in 0,
2
p
æ ö
ç ÷
è ø
is ________. (24-02-2021/Shift-1)
68. Let f : R R
® be defined as
3 2
3 2
55x, if x 5
f (x) 2x 3x 120x, if 5 x 4
2x 3x 36x 336, if x 4,
- < -
ì
ï
= - - - £ £
í
ï - - - >
î
Let A x R : f is increasing .
= Î Then A is equal to
(24-02-2021/Shift-2)
(a) , 5 4,
-¥ - È ¥ (b) 5,
- ¥
(c) 5, 4 4,
- - È ¥ (d) , 5 4,
-¥ - È - ¥
69. If P Is a point on the parabola y = x
2
+ 4 which is closest to
the straight line y = 4x - 1, then the co-ordinates of P are
(24-02-2021/Shift-2)
(a)(3,13) (b)(2,8)
(c)(-2,8) (d)(1,5)
70. If the curve 2
y ax bx c,x R,
= + + Î passes through the
point (1, 2) and the tangent line to this curve at origin is
y = x, then the possible values of a, b, c are :
(24-02-2021/Shift-2)
(a) a 1,b 1,c 0
= = = (b)
1 1
a ,b ,c 1
2 2
= = =
(c) a 1,b 1,c 1
= - = = (d) a 1,b 0,c 1
= = =

71. If the curves,
2 2
x y
1
a b
+ = and
2 2
x y
1
c d
+ = intersect each
other at an angle of 90°, then which of the following
relations is TRUE? (25-02-2021/Shift-1)
(a)
c d
ab
a b
+
=
+
(b) a c b d
- = +
(c) a b c d
+ = + (d) a b c d
- = -
72. If Rolle's theorem holds for the function
3 2
f x x ax bx 4, x 1,2
= - + - Î with
4
f 0,
3
æ ö
¢ =
ç ÷
è ø
then
ordered pair a,b is equal to: (25-02-2021/Shift-1)
(a) (5,8) (b) (–5, 8)
(c) (5, –8) (d) (–5, –8)
73. Let f x be a polynomial of degree 6 in x, in which the
coefficient of 6
x is unity and it has extrema at x 1
= - and
x 1.
= If 3
x 0
f x
lim 1,
x
®
= then 5 f 2
× is equal to _____.
(25-02-2021/Shift-1)
74. The shortest distance between the line x y 1
- = and the
curve 2
x 2y
= is : (25-02-2021/Shift-2)
(a)
1
2 2
(b)
1
2
(c)
1
2
(d) 0
75. If the curves 4
x y
= and xy k
= cut at right angles, then
6
4k is equal to: (25-02-2021/Shift-2)
76. The maximum slope of the curve
4 3 2
1
y x 5x 18x 19x
2
= - + - occurs at the point:
(26-02-2021/Shift-1)
(a)
21
3,
2
æ ö
ç ÷
è ø
(b) 0,0
(c) 2,9 (d) 2,2
77. The triangle of maximum area that can be inscribed in a
given circle of radius ‘r’ is (26-02-2021/Shift-2)
(a) A right angle triangle having two of its sides of length
2r and r.
(b) An equilateral triangle of height
2r
.
3
(c) An isosceles triangle with base equal to 2r.
(d)An equilateral triangle having each of its side of length
3r.
78. Let a be an integer such that all the real roots of the
polynomial 5 4 3 2
2x 5x 10x 10x 10x 10
+ + + + + lie in the
interval a, a 1 .
+ (26-02-2021/Shift-2)
Then, a is equal to _________.
79. The range of a R
Î for which the function
2
e
x x
f x 4a 3 x log 5 2 a 7 cot sin ,
2 2
æ ö æ ö
= - + + - ç ÷ ç ÷
è ø è ø
x 2n , n N
¹ p Î has critical points is:
(16-03-2021/Shift-1)
(a) , 1
-¥ - (b) 3, 1
-
(c)
4
,2
3
é ù
-
ê ú
ë û
(d) 1,¥
80. Let f be a real valued function, defined on R - {-1, 1} and
given by e
x 1 2
f(x) 3log
x 1 x 1
-
= -
+ -
Then in which of the following intervals, function f(x) is
increasing? (16-03-2021/Shift-2)
(a) , 1,1
-¥ ¥ - -
(b)
1
, 1 , 1
2
æ ö
é ö
-¥ - È ¥ -
ç ÷
÷
ê
ë ø
è ø
(c)
1
1,
2
æ ù
-
ç ú
è û
(d)
1
, 1
2
æ ù
-¥ - -
ç ú
è û

81. Consider the function f : R R
® defined by
1
2 sin | x |, x 0
f (x) .
x
0 , x 0
ì æ ö
æ ö
- ¹
ï ç ÷
ç ÷
= è ø
í è ø
ï =
î
Then f is :
(17-03-2021/Shift-2)
(a) monotonic on ( , 0) (0, )
-¥ È ¥
(b) not monotonic on ( , 0)
-¥ and (0, )
¥
(c) monotonic on ( , 0)
-¥ only
(d) monotonic on (0, )
¥ only
82. Let f :[ 1,1] R
- ® be defined as 2
f (x) ax bx c
= + + for
all x [ 1,1],
Î - where f (x)
¢¢ is
1
.
2
If f (x) ,
£ a
x [ 1,1],
Î - then the least value of a is equal to ................
(17-03-2021/Shift-2)
83. Let a tangent be drawn to the ellipse
2
2
x
y 1
27
+ = at
(3 3cos , sin )
q q where 0, .
2
p
æ ö
qÎç ÷
è ø
Then the value of
q such that the sum of intercepts on axes made by this
tangent is minimum is equal to: (18-03-2021/Shift-2)
(a)
3
p
(b)
6
p
(c)
8
p
(d)
4
p
84. Let ‘a’ be a real number such that the function
2
f x ax 6x 15,x R
= + - Î is increasing in
3
,
4
æ ö
-¥
ç ÷
è ø
and
decreasing in
3
, .
4
æ ö
¥
ç ÷
è ø
Then the function
2
g x ax 6x 15,x R
= - + Î has a: (20-07-2021/Shift-1)
(a) local minimumat
3
x
4
= -
(b) localmaximum at
3
x
4
=
(c) local minimumat . .
(d)local maximumat
3
x
4
= -
85. The sum of all the local minimum values of the twice
differentiable function f : R R
® defined by
3 2 3f " 2
f x x 3x x f " 1
2
= - - + is?
(20-07-2021/Shift-2)
(a) –22 (b)0
(c) –27 (d)5
86. Let f : R R
® be defined as
3 2
x
4
x 2x 3x, x 0
f x 3
3xe , x 0
ì
- + + >
ï
= í
ï £
î
Then f is is increasing function in the interval.
(22-07-2021/Shift-2)
(a)
3
1,
2
æ ö
-
ç ÷
è ø
(b)
1
,2
2
-
æ ö
ç ÷
è ø
(c) 0,2 (d) 3, 1
- -
87. Let
4 3 2
f x 3sin x 10sin x 6sin x 3,x , .
6 2
p p
é ù
= + + - Î -
ê ú
ë û
Then, f is? (25-07-2021/Shift-1)
(a) Increasing in ,0
6
p
æ ö
-
ç ÷
è ø
(b) Decreasing in 0,
2
p
æ ö
ç ÷
è ø
(c) Decreasing in ,0
6
p
æ ö
-
ç ÷
è ø
(d) Increasing in ,
6 2
p p
æ ö
-
ç ÷
è ø

88. The number of real roots of the equation
6x 4x 3x 2x x
e e 2e 12e e 1 0
- - - + + = is ?
(25-07-2021/Shift-1)
(a) 1 (b) 6
(c) 4 (d) 2
89. If a rectangle is inscribed in an equilateral triangle of side
length 2 2 as shown in the figure, then the square of the
largest area of such a rectangle is -
(25-07-2021/Shift-2)
90. Let f : a,b R
® be twice differentiable function such
that
x
a
f x g t dt
= ò for a differentiable function g x .
If f x 0
= has exactly five distinct roots in (a, b), then
g x g' x 0
= has at least: (27-07-2021/Shift-2)
(a) seven roots in (a, b) (b) five roots in (a, b)
(c) three roots in (a, b) (d) twelve roots in (a, b)
91. A wire of length 36 m is cut into two pieces, one of the
pieces is bent to form a square and the other is bent to
form a circle. If the sum of the areas of the two figures is
minimum, and the circumference of the circle is K (meter),
then
4
1 k
æ ö
+
ç ÷
p
è ø
is equal to _______.
(26-08-2021/Shift-1)
92. The local maximum value of the function
2
x
2
f x , x 0
x
æ ö
= >
ç ÷
è ø
is : (26-08-2021/Shift-2)
(a)
2
e
e (b)
e
4
4
e
æ ö
ç ÷
è ø
(c)
1
e
2 e (d) 1
93. A wire of length 20 m is to be cut into two pieces. One of
the pieces is to be made into a square and the other into a
regular hexagon. Then the length of the side (in meters) of
the hexagon, so that the combined area of the square and
the hexagon is minimum, is: (27-08-2021/Shift-1)
(a)
10
3 2 3
+
(b)
5
3 3
+
(c)
10
2 3 3
+
(d)
5
2 3
+
94. The number of distinct real roots of the equation is
4 3 2
3x 4x –12x 4 0
+ + = _______.
(27-08-2021/Shift-1)
95. A box open from top is made from a rectangular sheet of
dimension x from each of the four corners and folding up
the flaps. If the volume of the box is maximum, then x is
equal to : (27-08-2021/Shift-2)
(a)
2 2
a b a b ab
6
+ - + -
(b)
2 2
a b a b ab
6
+ + + -
(c)
2 2
a b a b ab
12
+ - + -
(d)
2 2
a b a b ab
6
+ - + +
96. The number of real roots of the equation
4x 3x x
e 2e e 6 0
+ - - = is? (31-08-2021/Shift-1)
(a) 0 (b)1
(c) 4 (d)2
97. If 'R ' is the least value of 'a ' such that the function
2
f x x ax 1
= + + is increasing on 1,2 and 'S'' is the
greatest value of 'a ' such that the function
2
f x x ax 1
= + + is decreasing on 1,2 , then the value
R S
- is ________ ? (31-08-2021/Shift-1)

98. Let f be any continuous function on 0,2 and twice
differentiable on 0,2 . If f 0 0,f 1 1
= = and
f 2 2,
= then: (31-08-2021/Shift-2)
(a) f x 0
¢¢ > for all x 0,2
Î
(b) f x 0
¢ = for some x 0,2
Î
(c) f x 0
¢¢ = for all x 0,2
Î
(d) f x 0
¢¢ = for some x 0,2
Î
99. An angle of intersection of the curves
2 2
2 2
x y
1
a b
+ = and
2 2
x y ab,a b
+ = < is (31-08-2021/Shift-2)
(a)
1 a b
tan
2 ab
- -
æ ö
ç ÷
è ø
(b)
1 a b
tan
ab
- +
æ ö
ç ÷
è ø
(c)
1 a b
tan
ab
- -
æ ö
ç ÷
è ø
(d)
1
tan 2 ab
-
100. Let f x be a cubic polynomial with
f 1 10,f 1 6,
= - - = and has a local minima at x 1.
= -
Then f 3 is equal to ___ (31-08-2021/Shift-2)

EXERCISE - 3 : ADVANCED OBJECTIVE QUESTIONS
Objective Questions I [Onlyonecorrect option]
1. The two curves y2
= 4x and x2
+ y2
– 6x + 1 = 0 at the point
(1,2)
(a) intersect orthogonally (b) intersect at an angle
3
p
(c) touch each other (d) none of these
2. The function ‘ f ’ is defined by f (x) = xp
(1 –x)q
for all
xÎ R, where p,q are positive integers, has a local maximum
value, for x equal to :
(a)
pq
p+q
(b)1
(c) 0 (d)
p
p+q
3. The triangle formed by the tangent to the parabola y = x2
at
the point with abscissa x1
, the y-axis and the straight line
y = x1
2
has the greatest area where x1
Î [1, 3]. Then x1
equals:
(a) 3 (b) 2
(c) 1 (d) none
4. Let f be a differentiable function with f (2) = 3 and
f ¢(2) = 5, and let g be the function defined by g(x) = x f (x).
y-intercept of the tangent line to the graph of ‘g’ at point
with abscissa 2, is
(a)20 (b)8
(c) – 20 (d) – 18
5. If px2
+ qx + r = 0, p, q, r Î R has no real zero and the line
y + 2 = 0 is tangent to f (x) = px2
+ qx + r then
(a) p + q + r > 0 (b) p – q + r > 0
(c) r < 0 (d) None of these
6. If P (x) = a0
+ a1
x2
+ a2
x4
+ .... + an
x2n
be a polynomial in
x Î R with 0 < a1
< a2
<... < an
, then P(x) has
(a) no point of minima
(b) only one point of minima
(c) only two points of minima
(d) none of these
7. Consider the following statements S and R :
S : Both sin x and cos x are decreasing functions in the
interval ,
2
p
æ ö
p
ç ÷
è ø
R : If a differentiable function decreases in an interval
(a, b), then its derivative also decreases in (a, b).
Which of the following is true ?
(a) Both S and R are wrong.
(b) Both S and R are correct, but R is not the correct
explanation for S.
(c) S is correct and R is the correct explanation for S.
(d) S is correct and R is wrong.
8. If the function f (x) increases in the interval (a, b) then the
function f(x) = [ f (x)]2
.
(a) Increases in (a, b)
(b) decreases in (a, b)
(c) we cannot say that f (x) increases or decreases in (a, b)
(d) none of these
9. If at anypointon a curve the sub-tangent and sub-normal are
equal, then the length of the normal is equal to
(a) 2 ordinate (b) ordinate
(c) 2 ordinate (d) none of these
10. A curve passes through the point (2, 0) and the slope of
the tangent at any point (x, y) is x2
– 2x for all values of x
then 3ylocal max
is equal to
(a) 4 (b) 3
(c) 1 (d)2
11. The radius of a right circular cylinder increases at a
constant rate. Its altitude is a linear function of the radius
and increases three times as fast as radius. When the
radius is 1 cm the altitude is 6 cm. When the radius is 6 cm,
the volume is increasing at the rate of 1 cu cm/sec. When
the radius is 36 cm, the volume is increasing at a rate of n
cu cm/sec. The value of ‘n’ is equal to
(a)12 (b)22
(c)30 (d)33

12. Slope of tangent to the curve
y = 2ex
sin ÷
ø
ö
ç
è
æ
-
p
2
x
4
cos ,
2
x
4
÷
ø
ö
ç
è
æ
-
p
where 0 £ x £ 2p is
minimumatx=
(a) 0 (b) p
(c)2p (d) none of these
13. Let f (x) =
3 2
2
2
x – x 10x –5, x 1
–2x log b – 2 , x 1
é + £
ê
+ >
ê
ë
the set of values of b
for which f (x) have greatest value at x = 1 is given by :
(a) 1 £ b £ 2
(b) b = {1, 2}
(c) b Î (–¥, –1)
(d) – 130,– 2 2, 130
é ù
È
ë û
14. A curve is represented parametrically by the equation
x = t + eat
and y = –t + eat
when t Î R and a > 0. If the curve
touches the axis of x at the point A, then the coordinates
of the point A are
(a) (1,0) (b)(1/e,0)
(c) (e, 0) (d) (2e, 0)
15. If ax +
b
x
³ c for all positive x, where a, b, c > 0, then
(a) ab <
2
c
4
(b) ab ³
2
c
4
(c) ab ³
c
4
(d) none of these
16. If f (x) is a differentiable function and f (x) is twice
differentiable function and aand b are roots of the equation
f (x) = 0 and f¢ (x) = 0 respectively, then which of the
following statement is true ? (a < b).
(a) there exists exactly one root of the equation
f¢ (x). f ¢(x) + f¢¢(x). f (x) = 0 and (a, b)
(b) there exists at least one root of the equation
f¢ (x). f ¢(x) + f¢¢(x). f (x) = 0 and (a, b)
(c) there exists odd number of roots of the equation
f¢ (x). f ¢(x) + f¢¢(x). f (x) = 0 and (a, b)
(d) None of these
17. The sub-normal at any point of the curve
x2
y2
= a2
(x2
– a2
) varies as
(a) (abscissa)–3
(b) (abscissa)3
(c) (ordinate)–3
(d) none of these
18. The sub-tangent at any point of the curve xm
yn
= am + n
varies as
(a) (abscissa)2
(b) (abscissa)3
(c) abscissa (d) ordinate
19. Thelengthoftheperpendicular fromthe originto the normal
of curve x = a (cos q+ qsin q), y = a (sin q – q cos q) at any
point q is
(a) a (b) a/2
(c) a/3 (d) none of these
20. If t, n, t´, n´ are the lengths of tangent, normal, subtangent
& subnormal at a point P (x1
, y1
) on any curve y = f (x) then
(a) t2
+ n2
= t´n´ (b) 2 2
1 1 1
t'n'
t n
+ =
(c) t´n´ = tn (d) nt´ = n´t
21. Find the shortest distance between xy = 9 and x2
+y2
= 1.
(a) 3 2 1
+ (b) 2
(c) 4 (d) 3 2 1
-
22. The largest area of a rectangle which has one side on the
x-axis and the two vertices on the curve y =
2
–x
e is
(a) –1/2
2 e (b) 2 e–1/2
(c) e–1/2
(d) none
23. If (x – a)2n
(x –b)2m +1
, where m and n are positive integers
and a > b, is the derivative of a function f, then
(a) x = a gives neither a maximumnor a minimum
(b) x = a gives a maximum
(c) x = b gives neither a maximumnor a minimum
(d) none of these
24. Let (h, k) be a fixed point, where h > 0, k > 0.Astraight line
passing through this point cuts the positive direction of
the coordinate axes at the points P and Q. The minimum
area of the D OPQ, O being the origin, is
(a) 2 kh (b) kh
(c) 4kh (d) none of these

25. The set of all values of the parameters a for which the
points of local minimum of the function y = 1 + a2
x – x3
satisfy the inequality
2
2
x x 2
0
x 5x 6
+ +
£
+ +
is
(a) an empty set
(b) 3 3 2 3
,
- -
(c) 2 3 3 3
,
(d) 3 3 2 3 2 3 3 3
, ,
- - È
26. The volume of the largest possible right circular cylinder
that can be inscribed in a sphere of radius 3
= is:
(a)
4
3
3
p (b)
8
3
3
p
(c) 4p (d) 2p
27. Tangent of acute angle between the curves y = |x2
–1| and
2
y 7 x
= - at their points of intersection is
(a)
5 3
2
(b)
3 5
2
(c)
5 3
4
(d)
3 5
4
28. A tangent to the curve y = 1 – x2
is drawn so that the
abscissa x0
of the point of tangency belongs to the interval
[0, 1]. The tangent at x0
meets the x-axis and y-axis at
A& B respectively. The minimum area of the triangle OAB,
where O is the origin is
(a)
2 3
9
(b)
4 3
9
(c)
2 2
9
(d) none
29. If the polynomial equation
an
xn
+an–1
xn–1
+ .... + a2
x2
+ a1
x+ a0
= 0,n positiveinteger, has
two different real roots a and b, then between a and b, the
equation
nan
xn–1
+ (n – 1) an–1
xn–2
+ ... + a1
= 0 has
(a) exactly one root (b) atmost one root
(c) atleast one root (d) no root
30. If
sin x sin a sin b
x cos x cos a cos b
tan x tan a tan b
=
f ,where 0 a b
2
p
< < < ,
then the equation f´ (x) = 0 has, in the interval (a, b)
(a) atleast one root (b) atmost one root
(c) no root (d) none of these
31. If
2 2
x x
x ; x
2 2cosx 6x 6sin x
= =
- -
f g where0<x<1,
then :
(a) both ‘ f ’ and ‘g’ are increasing functions
(b) ‘ f ’ is decreasing and ‘g’ is increasing function
(c) ‘ f ’ is increasing and ‘g’ is decreasing function
(d) both ‘ f ’ and ‘g’ are decreasing function
32. For 1 5
x 0, tan
2
-
æ ö
Îç ÷
ç ÷
è ø
, the function
f (x) = cot–1
2 sin x 5 cosx
7
æ ö
+
ç ÷
ç ÷
è ø
(a) increases in 1 5
0, tan
2
-
æ ö
ç ÷
ç ÷
è ø
(b) decreases in 1 5
0, tan
2
-
æ ö
ç ÷
ç ÷
è ø
(c) increases in 1 2
0, tan
5
-
æ ö
ç ÷
ç ÷
è ø
and decreases in
1 1
2 5
tan ,tan
5 2
- -
æ ö
ç ÷
ç ÷
è ø
(d) increases in
1 1
2 5
tan ,tan
5 2
- -
æ ö
ç ÷
ç ÷
è ø
and decreases in
1 2
0, tan
5
-
æ ö
ç ÷
ç ÷
è ø

33. If
|x| |x|
a sgnx a sgn x
x a ; x a
é ù
ê ú
ë û
= =
f g for a > 1 and
x Î R, where{ } & [] denote the fractional part and integral
part functions respectively, then which of the following
statements hold good for the function h (x),
where (lna) h (x) = (ln f (x) + ln g (x)).
(a) ‘h’ is even and increasing
(b) ‘h’ is odd and decreasing
(b) ‘h’ is even and decreasing
(d) ‘h’ is odd and increasing
34. The sum of tangent and sub-tangent at any point of the
curve y = a log (x2
– a2
) varies as
(a) abscissa
(b) product of the coordinates
(c) ordinate
(d) none of these
35. For the curve xm + n
= am –n
y2n
, where a is a positive constant
and m, n are positive integers
(a) (sub-tangent)m
µ (sub-normal)n
(b) (sub-normal)m
µ (sub-tangent)n
(c) the ratio of subtangent and subnormal is constant
(d) none of the above
36. |sin 2x| – |x| – a = 0 does not have solution if a lies in
(a)
3 3
6
,
æ ö
-p
¥
ç ÷
ç ÷
è ø
(b)
3 3
6
,
æ ö
+p
¥
ç ÷
ç ÷
è ø
(c) (1, ¥) (d) None of these
37. Let f (x) =
2
2
x for x 0
x 8 for x 0
,
.
,
ì- <
í
+ ³
î
Then the x-intercept of
the line that is tangent to both portions of the graph of
y = f (x) is
(a) zero (b) –1
(c) –3 (d) –4
38. The least area of a circle circumscribing any right triangle
of area S is :
(a) pS (b)2pS
(c) 2 pS (d)4pS
39. Theminimumvalueofa tan2
x+b cot2
xequalsthemaximum
value of a sin2
q + b cos2
q where a > b > 0, when
(a) a = b (b) a = 2b
(c) a = 3b (d) a = 4b
40. A function fsuch that f ´(a) = f ´
´(a) = ... f 2n
(a) = 0 and f has
a local maximum value b at x = a, if f (x) is
(a) (x – a)2n+2
(b) b –1 –(x +1 –a)2n+1
(c) b – (x – a)2n+2
(d) (x–a)2n+2
– b.
41. A truck is to be driven 300 km on a highway at a constant
speed of x kmph. Speed rules of the highway required that
30 £ x £ 60. The fuel costs Rs. 10 per litre and is consumed
at the rate of
600
x
2
2
+ liters per hour. The wages of the
driver are Rs. 200 per hour. The most economical speed to
drive the truck, in kmph, is
(a)30 (b)60
(c) 3
.
3
30 (d) 3
.
3
20
42. The curve 2
x
1
x
2
y
+
= has
(a) exactly three points of inflection separated by a point
of maximumanda point of minimum
(b) exactly two points of inflection with a point ofmaximum
lying between them
(c) exactly two points ofinflection with apointof minimum
lying between them
(d) exactly three points of inflection separated by two
points ofmaximum
43. Let f (x) =
3 2
x +x +3x+sinx 3+sin1/x x 0
0 x 0
ì ¹
ï
í
=
ï
î
,
,
then
number of points (where f (x) attains its minimum value) is
(a) 1 (b)2
(c) 3 (d) infinite many
44. The number of points with integral coordinates where
tangent exists in the curve y = sin–1
2x
2
1 x
- is
(a) 0 (b)1
(c) 3 (d) None

Objective Questions II [One or more than one correct option]
45. The abscissa of a point on the curve xy = (a + x)2
,
the tangent at which cuts off equal intercepts on the
coordinate axes is
(a) a / 2
- (b) 2 a
(c) 2 a /2 (d) 2 a
-
46. If f is an even function then
(a) f 2
increases on (a, b)
(b) f cannot be monotonic
(c) f 2
need not increases on (a, b)
(d) f has inverse
47. The function y =
2x 1
x 2
-
-
(x ¹ 2) with codomain = R – {2}
(a) is its own inverse
(b) decreases at all values of x in the domain
(c) has a graph entirely above x–axis
(d) is bound for all x.
48. Let g´ (x) > 0 and f ’ (x) < 0, " x Î R, then
(a) g ( f (x+1)) > g ( f (x– 1))
(b) f (g (x–1)) > f (g (x + 1))
(c) g (f (x+1)) < g ( f (x– 1))
(d) g (g (x + 1)) < g (g (x – 1))
49. If f (x)=x3
– x2
+100x+ 1001, then
(a) f (2000)> f (2001)
(b)
1 1
1999 2000
æ ö æ ö
>
ç ÷ ç ÷
è ø è ø
f f
(c) f (x + 1) > f (x – 1)
(d) f (3x – 5) > f (3x)
50. An extremum of the function,
p
-
=
x
2
)
x
(
f cos p(x+3)+ 2
1
p
sinp(x+3)0<x<4 occurs
at :
(a) x= 1 (b) x= 2
(c) x= 3 (d) x= p
51. Thelengthoftheperpendicular fromthe originto the normal
of curve x = a (cosq + q sin q), y = a (sin q – q cos q)
at a point q is ‘a’, if q =
(a) p/4 (b)p/3
(c) p/2 (d)p/6
52. The points on the curve y = x 2
1 x ,
- –1 < x < 1 at which
the tangent line is vertical are
(a) (–1, 0) (b)
1 1
2
2
,
æ ö
- -
ç ÷
è ø
(c) (1,0) (d)
1 1
2 2
,
æ ö
ç ÷
è ø
53. Let the parabolas y = x (c – x) and y = – x2
– ax + b touch
each other at the point (1, 0), then
(a) a + b + c = 0 (b) a + b = 2
(c) b + c = 1 (d) a – c = –2
54. The value of parameter a so that the line
(3 – a) x+ ay+ (a2
– 1) = 0 is normal to thecurvexy=1, may
lie in the interval
(a) (-¥, 0) (b) (1, 3)
(c) (0, 3) (d) (3, ¥)
55. Which of the following pair (s) of curves is/are orthogonal.
(a) y2
= 4ax ; y = e–x/2a
(b) y2
= 4ax ; x2
= 4ay at (0, 0)
(c) xy = a2
; x2
– y2
= b2
(d) y = ax ; x2
+ y2
= c2
56. If f (x) = f (x) + f (2a – x) and f ’’ (x) > 0, a > 0,
0 < x < 2a then
(a) f(x) increases in[a, 2a]
(b) f(x) increases in [0, a]
(c) f(x) decreases in [0, a]
(d) f (x) decreases in [a, 2a]
57. Let f(x) = xm/n
for x Î R where m and n are integers, m even
and n odd and 0 < m < n. Then
(a) f (x) decreases on (–¥, 0]
(b) f (x) increases on [0, ¥)
(c) f (x) increases on (–¥, 0]
(d) f (x) decreases on [0, ¥)

58. For function
n x
f(x) ,
x
=
l
which of the following
statements are true.
(a) f (x) has horizontal tangent at x = e
(b) f (x) cuts the x–axis only at one point
(c) f (x) is many – one function
(d) f (x) has one vertical tangent
59. If f (x) = ,
2
,
0
x
,
x
tan
x
1
x
÷
ø
ö
ç
è
æ p
Î
+
then
(a) f (x) has exactly one point of minimum
(b) f (x) has exactlyone point of maximum
(c) f (x) is increasing in ÷
ø
ö
ç
è
æ p
2
,
0
(d) maximum occurs at x0
where x0
= cosx0
60. Let f (x) = (x – 1)4
(x – 2)n
, n Î N. then f (x)has
(a) local minimumat x = 2 if n is even
(b) local minimumat x = 1 if n is odd
(c) local maximum at x = 1 if n is odd
(d) local minimumat x = 1 if n is even
61. The angle between the tangent at any point P and the line
joining P to the origin, where P is a point on the curve
ln(x2
+ y2
) = c tan–1
y
x
, c is a constant, is
(a) independent of x and y
(b) dependent on c
(c) independent of c but dependent on x
(d) none of these
62. The point on the curve xy2
= 1, which is nearest to the
origin is
(a) (21/3
, 21/6
) (b) (2–1/3
, 21/6
)
(c) (2–1/3
, – 21/6
) (d) (–2–1/3
, 21/6
)
63. Let g(x) =
1
2
f -
- x2
(x – 1) – f (0) (x2
– 1)
1
2
f
+ x2
(x + 1) – f ¢ (0)x (x – 1) (x +1) where
f is a thrice differentiable function. Then the correct
statements are
(a) there exists x Î (–1, 0) such that f ¢ (x) = g¢ (x)
(b) there exists x Î (0, 1) such that f ¢¢ (x) = g¢¢ (x)
(c) there exists x Î (–1, 1) such that f ¢¢¢ (x) = g¢¢¢ (x)
(d)thereexistsxÎ(–1,1)suchthatf¢¢¢(x)=3f(1)–3f(–1)–6f¢(0)
64. If f : [–1, 1] ® R is a continuously differentiable function
such that f (1) > f (–1) and | f ¢(y)| < 1 for all y Î [–1, 1] then
(a) there exists an x Î [–1, 1] such that f ¢(x) > 0
(b) there exists an x Î [–1, 1] such that f ¢(x) < 0
(c) f (1) < f (–1) + 2
(d) f (–1) . f (1) < 0
65. In a triangleABC
(a) sinA sin B sin C
3 3
8
£
(b) sin2
A+ sin2
B + sin2
C
9
4
£
(c) sin A sin B sin C is always positive
(d) sin2
A+ sin2
B = 1 + cos C
66. The diagram shows the graph of the derivative ofa function
f (x) for0 < x <4 with f(0)=0. Whichofthefollowing could
be correct statements for y = f (x) ?
(a) Tangent line to y = f (x) at x = 0 makes an angle of
sec–1
5 with the x-axis.
(b) f is strictly increasing in (0, 3)
(c) x = 1 is both an inflection point as well as point of local
extremum.
(d) Number of critical point on y = f (x) is two.

67. IfAis the area of the triangle formed by positive x-axis and
the normal and the tangent to the circle x2
+ y2
= 4 at
1 3
, then A 3
/ is equal to
68. A cylinderical vessel of volume
1
25
7
cu metres, open at
the top is to be manufactured from a sheet of metal. (The
value of p is taken as 22/7). If r and h are the radius and
height of the vessel so that amount of metal is used in the
least possible then rh is equal to
69. Let a be the angle in radians between
2 2
x y
1
36 4
+ = and the
circle x2
+ y2
= 12 at their points of intersection. If
1 k
tan ,
2 3
-
a = then find the value of k2
.
70. If a is an integer satisfying |a| £ 5 – | [x] |, where x is a real
number for which 2x tan–1
x is greater than or equal to
ln(1 + x2
), thenfind thenumber ofmaximumpossiblevalues
of a. (where [ . ] represents the greatest integer function)
71. The circle x2
+ y2
= 1 cuts the x-axis at P and Q. Another
circle with centre at Q and variable radius intersects the
first circle at R above the x-axis and the line segment PQ at
S. IfAis the maximum area of the triangle QSR then 3 3
A is equal to _____.
72. If f (x) is a twice differentiable function such that
f (a) = 0, f (b) = 2, f (c) = –1, f (d) = 2, f (e) = 0, where
a < b < c < d <e, find the minimum number of zeroes of
g (x) = (f ´ (x))2
+ f ´
´(x) f (x) in the interval [a, e].
73. If the length of the interval of ‘a’ such that the inequality
3 – x2
> |x – a| has atleast one negative solution is k then
find 4k.
74. If k is a positive integer, such that
(i) cos2
x sin
7
x ,
k
> - for all x
(ii) cos2
xsin
7
x
k 1
< -
+
for somex, then k must be equal
to
Assertion & Reason
(A) If ASSERTION is true, REASON is true, REASON is a
correctexplanationforASSERTION.
(B) IfASSERTIONistrue,REASONistrue,REASONisnot
acorrectexplanationforASSERTION.
(C) If ASSERTION is true, REASON is false.
(D) If ASSERTION is false, REASON is true.
75. Assertion : Let f (x) = 5 – 4 (x – 2)2/3
, then at x = 2 the
function f (x) attains neither least value nor greatest value.
Reason : x = 2 is the only critical point of f (x).
(a)A (b) B
(c) C (d) D
76. Assertion : for any triangleABC
sin
3
C
sin
B
sin
A
sin
3
C
B
A +
+
³
÷
ø
ö
ç
è
æ +
+
Reason : y = sin x is concave downward for x Î (0, p].
(a)A (b) B
(c) C (d) D
77. Assertion : The minimum distance of the fixed point
(0, y0
), where ,
2
1
y
0 0 £
£ from the curve y = x2
is y0
.
Reason : Maxima andminima of a function isalways a root
of the equation f ´ (x) = 0.
(a)A (b) B
(c) C (d) D
78. Assertion : The equation 3x2
+ 4ax + b = 0 has at least one
root in (0, 1), if 3 +4a = 0.
Reason: f (x)=3x2
+4ax+b iscontinuousanddifferentiable
in the interval (0, 1).
(a)A (b) B
(c) C (d) D
79. Assertion : Let f : [0, ¥)®[0, ¥) and g : [0, ¥)®[0, ¥) be
non-increasing and non-decreasing functions respectively
and h (x) = g ( f (x)).If f and g are differentiablefor all points
in their respective domains and h (0) = 0 then h (x) is
constant function.
Reason : g (x) Î [0, ¥) Þ h (x) ³ 0 and h´ (x) £ 0.
(a)A (b) B
(c) C (d) D

80. Assertion : The ratio of length of tangent to length of
normal is directlyproportional to the ordinateof the point
of tangency at the curve y2
= 4ax.
Reason : Length of normal & tangent to a curve
y = f (x) is
2
2 y 1 m
y 1 m and ,
m
+
+ where
dy
m
dx
= .
(a)A (b)B
(c) C (d) D
81. Assertion : Among all the rectangles of given perimeter,
the square has the largest area. Also among all the
rectangles of given area, the square has the least perimeter.
Reason : For x > 0, y > 0, if x + y = const, then xy will be
maximum for y = x and if xy = const, then x + y will be
minimumfory= x.
(a)A (b) B
(c) C (d) D
82. Assertion : If g (x) is a differentiable function g(1) ¹ 0,
g (–1) ¹ 0 and Rolles theorem is not applicable to
2
x 1
(x)
g(x)
-
=
f in [–1, 1], then g(x) has atleast one root
in (–1, 1)
Reason : If f (a) = f (b), then Rolles theorem is applicable
for x Î (a, b)
(a)A (b) B
(c) C (d) D
83. Assertion : The tangent at x = 1 to the curve
y = x3
– x2
– x + 2 again meets the curve at x = – 2.
Reason : When a equation of a tangent solved with the
curve, repeated roots are obtained at point of tangency.
(a)A (b) B
(c) C (d) D
84. Assertion : Tangent drawn at the point (0, 1) to the curve
y = x3
– 3x + 1 meets the curve thrice at one point only.
Reason : Tangent drawn at the point (1, –1) to the curve y
= x3
– 3x + 1 meets the curve at 1 point only.
(a)A (b) B
(c) C (d) D
85. Assertion : Shortest distance between
| x | + | y | = 2 & x2
+ y2
= 16 is 4 2
-
Reason : Shortest distance between the two non
intersecting differentiable curves lies along the common
normal.
(a)A (b) B
(c) C (d) D
86. Assertion : If f (x) is increasing function with concavity
upwards, then concavity of f –1
(x) is also upwards.
Reason : If f (x) is decreasing function with concavity
upwards, then concavity of f –1
(x) is also upwards.
(a)A (b) B
(c) C (d) D
87. Assertion : The largest term in the sequence
.
600
)
400
(
is
N
n
,
200
n
n
a
3
/
2
3
2
n Î
+
=
Reason : ,
0
x
,
200
x
x
)
x
( 3
2
>
+
=
f then at x = (400)1/3
,
f(x)ismaximum.
(a)A (b) B
(c) C (d) D
MatchtheFollowing
Each question has two columns. Four options are given
representing matching of elements from Column-I and
Column-II. Only one of these four options corresponds
to acorrectmatching.Foreachquestion,choosetheoption
corresponding to the correct matching.
88. Column–I Column–II
(A) Circular plate is expanded by (P) 4
heat from radius 5 cm to 5.06 cm.
Approximate increase in area is
(B) If an edge of a cube increases by (Q) 0.6p
1% then percentage increase in
volume is
(C) If the rate of decrease of (R) 3
2
x
2x 5
2
- + is twice the rate
of decrease of x, then x is equal to
(rate of decrease is non-zero)
(D) Rate of increase in area of (S) 3 3 / 4
equilateral triangle of side 15cm,
when each side is increasing at
the rate of 0.1 cm/sec; is
The correct matching is :
(a) (A–Q; B–R; C–P; D–S)
(b) (A–R; B–P; C–Q; D–S)
(c) (A–S; B–Q; C–P; D–S)
(d) (A–P; B–Q; C–R; D–S)

89. Column–I Column–II
(A) If portion of the tangent at any (P) 0
point on the curve x = at
3
, y=at
4
between the axes is divided by
the abscissa of the point of
contact in the ratio m : n externally,
then |n + m| is equal to
(m and n are coprime)
(B) The area of triangle formed by (Q) 1/2
normal at the point (1, 0) on the
curve x = e
siny
with axes is
(C) If the angle between curves x
2
y=1 (R) 7
and y = e
2(1–x)
at the point (1, 1) is
q then tan q is equal to
(D) The length of sub-tangent at any (S) 3
point on the curve y = be
x/3
is
equal to
(a) (A–R; B–Q; C–P; D–S)
(b) (A–Q; B–R; C–P; D–S)
(c) (A–P; B–Q; C–R; D–S)
(d) (A–S; B–P; C–Q; D–S)
90. Column - I Column-II
(A) The dimensions of the rectangle (P) 6
of perimeter 36 cm, which sweeps
out the largest volume when
revolved about one of its sides, are
(B) LetA(–1, 2) and B (2, 3) be two (Q) 12
fixed points, A point P lying on
y = x such that perimeter of
triangle PAB is minimum, then
sum of the abscissa and ordinate
of point P, is
(C) If x1
and x2
are abscissae of two (R) 4
points on the curve f (x) = x – x
2
in the interval [0, 1], then maximum
value of expression
(x1
+x2
) – )
x
x
( 2
2
2
1 + is
(D) The number of non-zero integral (S) 1/2
values of ‘a’ for which the function
f (x) = x
4
+ ax
3
+
2
x
3 2
+1 is concave
upward along the entire real line is
(T) 2
(a) (A–R; B–P; C–S; D–Q)
(b) (A–S; B–R; C–P; D–Q)
(c) (A–P,Q; B–R;C–S; D–R)
(d) (A–Q; B–S; C–P; D–R)
91. Column-I Column-II
(A) The equation x log x = 3 – x has (P) (0,1)
at least one root in
(B) If 27a + 9b + 3c + d = 0, then the (Q) (1,3)
equation 4ax
3
+ 3bx
2
+ 2cx + d = 0
has at least one root in
(C) If c 3
= & f (x) =
1
x
x
+ then (R) (0,3)
interval of x in which LMVT
is applicable for f (x), is
(D) If
1
c
2
= & f (x) = 2x – x
2
, then (S) (–1,1)
interval of x in which LMVT is
applicable for f (x), is
(a) (A–P; B–R; C–Q; D–P)
(b) (A–R; B–S; C–Q; D–P)
(c) (A–Q; B–S; C–R; D–P)
(d) (A–R; B–S; C–P; D–P)

92. Column - I Column-II
(A) If x is real, then the greatest and (P) 3
least value of the expression
6
x
3
x
2
2
x
2
+
+
+
is
(B) If a + b = 1; a > 0, b > 0, then the (Q)
3
1
minimumvalue of
÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
+
b
1
1
a
1
1 is
(C) The maximumvalue attained by (R) 5
y = 10 – |x–10|, – 1 £ x £ 3, is
(D) If P (t2
, 2t), t Î [0, 2] is an (S)
13
1
-
arbitrary point on parabola y2
=4x.
Q is foot of perpendicular from
focus S on the tangent at P, then
maximumarea of triangle PQS is
(a) (A–S; B–P; C–P; D–R)
(b) (A–Q; B–S; C–P; D–R)
(c) (A–R; B–Q; C–P; D–S)
(d) (A–S; B–R; C–P; D–Q)
ParagraphType Questions
Using the following, solve Q.93 to Q. 95
Passage
If ,
= ò
v x
u x
y f t dt let us define
dy
dx
in a different manner
as
2 2
' '
dy
v x f v x u x f u x
dx
= - and the
equation of the tangent at ,
a b as
,
a b
dy
y b x a
dx
æ ö
- = -
ç ÷
è ø
93. If
2
x
2
x
y t
= ò dt, then equation of tangent at x = 1 is
(a) y = x + 1 (b) x + y = 1
(c) y = x – 1 (d) y = x
94. If F (x)
x
2
t /2
1
e
= ò (1 – t2
) dt, then
d
dx
F (x) at x = 1 is
(a) 0 (b) 1
(c) 2 (d) –1
95. If
4
x
x 0
3
x
dy
y nt dt, then lim is
dx
+
®
= ò l
(a) 0 (b) 1
(c) 2 (d) –1
Using the following passage, solve Q.96 to Q.98
Passage
Consider a function ÷
ø
ö
ç
è
æ
-
a
-
a
= x
1
)
x
(
f (4 – 3x2
) where
‘a’ is a positive parameter
96. Number of points of extrema of f (x) for a given value of a
is
(a) 0 (b)1
(c) 2 (d)3
97. Absolute difference between local maximum and local
minimum values of f (x) in terms of a is
(a)
3
1
9
4
÷
ø
ö
ç
è
æ
a
+
a (b)
3
1
9
2
÷
ø
ö
ç
è
æ
a
+
a
(c)
3
1
÷
ø
ö
ç
è
æ
a
+
a (d) independent of a
98. Least possible value of the absolute difference between
local maximumand local minimumvalues of f (x) is
(a)
9
32
(b)
9
16
(c)
9
8
(d)
9
1

Passage
Considerthefunctionf(x)=max{x2
,(1– x)2
,2x(1–x)}
where 0 £ x £ 1.
99. The interval in which f (x) is increasing is
(a)
1 2
,
3 3
æ ö
ç ÷
è ø
(b)
1 1
,
3 2
æ ö
ç ÷
è ø
(c)
1 1 1 2
, ,
3 2 2 3
æ ö æ ö
È
ç ÷ ç ÷
è ø è ø
(d)
1 1 2
, ,1
3 2 3
æ ö æ ö
È
ç ÷ ç ÷
è ø è ø
100. The interval in which f (x) is decreasing is
(a)
1 2
,
3 3
æ ö
ç ÷
è ø
(b)
1 1
,
3 2
æ ö
ç ÷
è ø
(c)
1 1 2
0, ,
3 2 3
æ ö æ ö
È
ç ÷ ç ÷
è ø è ø
(d)
1 2
0, ,1
2 3
æ ö æ ö
È
ç ÷ ç ÷
è ø è ø
101. LetRMVT is applicable for f(x) on (a,b)then a + b + c is
(where c is point such that f ´ (c) = 0)
(a)
2
3
(b)
1
3
(c)
1
2
(d)
3
2
Passage
Lety=a x +bx be curve,(2x – y)+ l (2x+ y–4)= 0 be
familyoflines.
102. If curve has slope
2
1
- at (9, 0) then a tangent belonging
to family of lines is
(a) x + 2y – 5 = 0 (b) x – 2y + 3 = 0
(c) 3x – y – 1 = 0 (d) 3x + y – 5 = 0
103. A line of the family cutting positive intercepts on axes
and forming triangle with coordinate axes, then minimum
length of the line segment between axes is
(a) (22/3
– 1)3/2
(b) (22/3
+1)3/2
(c) 73/2
(d)27
104. Two perpendicular chords of curve y2
– 4x – 4y + 4 = 0
belonging to family of lines form diagonals of a
quadrilateral. Minimum area of quadrilateral is
(a)16 (b) 32
(c)64 (d) 50
Passage
If y f x
= is a curve and if there exists two points
1 1
,
A x f x and 2 2
B ,
x f x on it such that
2 1
1
2 2 1
1
'
'
f x f x
f x
f x x x
-
= - =
- then the tangent
at 1
x 1
is normal at 2
x for that curve.
105. Number of such lines on the curve y = sinx is
(a) 1 (b) 0
(c) 2 (d) infinite
106. Number of such lines on the curve y = |ln x| is
(a) 1 (b) 2
(c) 0 (d) infinite
107. Number of such line on the curve y2
= x3
is
(a) 1 (b) 2
(c) 3 (d) 0
Passage
Let f ´ (sin x) < 0 and f ´
´(sin x) > 0 x 0,
2
p
æ ö
" Îç ÷
è ø
Now consider a function g (x) = f (sin x) + f (cos x)
108. g (x) decreases if x belongs to
(a) 0,
4
p
æ ö
ç ÷
è ø
(b) ,
4 2
p p
æ ö
ç ÷
è ø
(c) ,
6 3
p p
æ ö
ç ÷
è ø
(d) none of these
109. g (x) increase if x belongs to
(a) 0,
4
p
æ ö
ç ÷
è ø
(b) ,
4 2
p p
æ ö
ç ÷
è ø
(c) ,
8 3
p p
æ ö
ç ÷
è ø
(d) ,
6 3
p p
æ ö
ç ÷
è ø
110. The set of critical points of g (x) is
(a) ,
8 6
p p
ì ü
í ý
î þ
(b) , ,
8 6 3
p p p
ì ü
í ý
î þ
(c) , ,
8 6 4
p p p
ì ü
í ý
î þ
(d) none of these

EXERCISE - 4 : PREVIOUS YEAR JEE ADVANCED QUESTIONS
Objective Questions I [Onlyonecorrect option]
1. For all x Î(0, 1) (2000)
(a) e
x
< 1 + x (b) loge
(1 + x) < x
(c) sin x > x (d) loge
x > x
2. Let f (x) = x
e x 1 x 2 dx.
- -
ò Then f decreases in the
interval (2000)
(a) (-¥, -2) (b) (-2, -1)
(c) (1,2) (d) (2, ¥)
3. Let f (x) =
x for 0 x 2
1 for x 0
| |, | |
,
< £
ì
í
=
î
Then, at x = 0, f has
(2000)
(a) alocal maximum (b)no local maximum
(c) a local minimum (d) no extremum
4. If the normal to the curve, y = f (x) at the point (3, 4) makes
an angle 3p/4 with the positive x–axis,
then f ´ (3) is equal to (2000)
(a) –1 (b) –3/4
(c) 4/3 (d) 1
5. If f (x) = xex (1–x)
, then f (x) is (2001)
(a) increasing in
1
,1
2
é ù
-
ê ú
ë û
(b) decreasing in R
(c) increasing in R (d) decreasing in
1
,1
2
é ù
-
ê ú
ë û
6. The maximumvalue of(cos a1
) . (cos a2
) ....
. (cos an
),under
the restrictions 0 < a1
, a2
, .... an
<
2
p
and
(cot a1
) . (cot a2
) ....
. (cot an
) = 1 is (2001)
(a) n 2
1
2 / (b) n
1
2
(c)
1
2n
(d) 1
7. The length of a longest interval in which the function
3sin x – 4 sin3
x is increasing, is (2002)
(a)
3
p
(b)
2
p
(c)
3
2
p
(d)p
8. The point(s) on the curve y3
+ 3x2
= 12 y where the tangent
is vertical, is (are) (2002)
(a)
4
, 2
3
æ ö
± -
ç ÷
è ø
(b)
11
, 0
3
æ ö
±
ç ÷
ç ÷
è ø
(c) ( , )
0 0 (d)
4
, 2
3
æ ö
±
ç ÷
è ø
9. The equation of the common tangent to the curves
y2
= 8x and xy = –1 is (2002)
(a) 3y= 9x + 2 (b) y = 2x + 1
(c) 2y =x + 8 (d) y = x + 2
10. Iff(x)=x2
+2bx+2c2
andg(x) =–x2
–2cx+b2
suchthatmin
f (x) > max g (x), then the relation between
b and c, is – (2003)
(a) no real value of b & c (b) 0 c b 2
< <
(c) c b 2
< (d) c b 2
>
11. If f (x) = x3
+ bx2
+ cx + d and 0 < b2
< c, then in (-¥, ¥)
(2004)
(a) f (x) is strictly increasing function
(b) f (x) has a local maxima
(c) f (x) is strictly decreasing function
(d) f (x) is bounded.
12. If f (x) is differentiable and strictly increasing function,
then the value of
2
x 0
x (x)
im
(x) (0)
f f
l
f f
®
-
-
is (2004)
(a) 1 (b) 0
(c) –1 (d) 2

13. Tangents are drawn to the ellipse x2
+ 2y2
= 2, then the
locus ofthe mid point of the intercept made by the tangents
between the coordinate axes is (2004)
(a)
1
2
1
4
1
2 2
x y
+ = (b)
1
4
1
2
1
2 2
x y
+ =
(c)
x y
2 2
2 4
1
+ = (d)
x y
2 2
4 2
1
+ =
14. The angle between the tangents drawn from the point
(1, 4) to the parabola y2
= 4x is (2004)
(a) p/6 (b) p/4
(c) p/3 (d) p/2
15. The second degree polynomial f (x), satisfying f (0) = 0,
f(1)=1, f ´(x)>0forallxÎ(0,1): (2005)
(a) f(x)=f
(b) f (x) = ax+ (1 – a) x2
; a
" Î (0, ¥)
(c) f (x) = ax+ (1 – a) x2
; a Î (0, 2)
(d) No such polynomial
16. The tangent at (1,7) to the curvex2
= y– 6 touches the circle
x2
+y2
+16x+12y+ c= 0 at (2005)
(a) (6,7) (b)(–6, 7)
(c) (6, –7) (d) (–6, –7)
17. The tangent to the curve y = ex
drawn at the point (c, ec
)
intersects the line joining the points (c – 1, ec – 1
) and
(c + 1, ec + 1
) (2007)
(a) on the left of x = c (b) on the right of x = c
(c) at no point (d) at all points
18. Let the function g : (–¥, ¥) ® ÷
ø
ö
ç
è
æ p
p
-
2
,
2
be given by
g (u) = 2 tan–1
(eu
) – .
2
p
Then, g is (2008)
(a) even and is strictly increasing in (0, ¥)
(b) odd and is strictly decreasing in (–¥, ¥)
(c) odd and is strictly increasing in (–¥, ¥)
(d) neither even nor odd, but is strictly increasing in
(–¥, ¥)
19. The total number of local maxima and local minima of the
function
3
2
3
(2 x) , 3 x 1
(x)
x , 1 x 2
ì + - < £ -
ï
= í
ï - < <
î
f is (2008)
(a) 0 (b)1
(c) 2 (d)3
20. Let f, g and h be real-valued functions defined on the
interval [0, 1] by f(x) =
2 2
x x
e e ,
-
+
2 2
x x
g(x) xe e-
= + and
2 2
2 x x
h(x) x e e-
= + . If a, b and c denote respectively, the
absolute maximum of f, g and h on [0, 1], then (2010)
(a) a = b and c ¹ b (b) a = c and a ¹ b
(c) a ¹ b and c ¹ b (d) a = b = c
21. The number of points in (-¥, ¥), for which
x2
– x sin x – cos x = 0, is (2013)
(a) 6 (b)4
(c) 2 (d)0
22. Consider all rectangles lying in the region
( , ) : 0 0 2sin (2 )
2
x y R R x and y x
p
ì ü
Î ´ £ £ £ £
í ý
î þ
and having one side on the x-axis. The area of the rectangle
which has the maximum perimeter among all such
rectangles, is (2020)
(a)
3
2
p
(b) p
(c)
2 3
p
(d)
3
2
p
Objective Questions II [One or more than one correct option]
23. If f (x) is cubic polynomial which has local maximum at
x= –1. If f(2)= 18, f(1)=–1 and f ’ (x)haslocal minimumat
x = 0, then (2006)
(a) the distance between (–1, 2) and (a, f (a)) where x = a is
the point of local minima, is 2 5 .
(b) f (x) is increasing for x [1, 2 5]
Î
(c) f (x) has local minima at x = 1
(d) the value of f (0) = 5

24. If
x
x 1
e , 0 x 1
f(x) 2 e , 1 x 2
x e, 2 x 3
-
ì £ £
ï
= - < £
í
ï - < £
î
and
x
0
g(x) f(t)dt,
= ò x Î [1, 3], then (2006)
(a) g (x) has local maxima at x = 1 + loge
2 and local minima
at x = e
(b) f(x)haslocalmaximaatx=1andlocalminimaatx=2
(c) g (x) has no local minima
(d) f (x) has no local maxima
25. For the function f (x) = x cos
1
,
x
x ³ 1. (2009)
(a) for at least one x in the interval
[1, ¥), f (x + 2) – f (x) < 2
(b) x
lim f (x) 1
®¥
¢ =
(c) forall x in the interval [1, ¥), f(x + 2) – f(x) > 2
(d) f’ (x) is strictly decreasing in the interval [1, ¥)
26. Let f be a real-valued function defined on the interval
(0, ¥), by f (x) =
x
0
n x 1 sin t dt
+ +
ò
l . Then which of the
following statement(s) is (are) true ? (2010)
(a) f ”(x) exists for all x Î (0, ¥)
(b) f ’(x) exists for all x Î (0, ¥) and f ’ is continuous on
(0, ¥), but not differentiable on (0, ¥)
(c) there exists a > 1 such that |f ’ (x)| < | f(x)| for all
x Î (a, ¥)
(d) there exists b > 0 such that |f (x) | + | f ’(x)| £ b from all
xÎ(0, ¥)
27. A rectangular sheet of fixed perimeter with sides having
their lengths in the ratio 8 : 15 is converted into an open
rectangular box by folding after removing squares of equal
area from all four corners. If the total area of removed
squares is 100, the resulting box has maximum volume.
The lengths of the sides of the rectangular sheet are
(2013)
(a)24 (b)32
(c)45 (d)60
28. The function f (x) = 2|x| + |x + 2| – ||x + 2| – 2|x|| has a local
minimum or a local maximumat x is equal to
(2013)
(a) –2 (b)
2
3
-
(c) 2 (d)
2
3
29. Let a Î R and let f : R ® R be given by
f (x) = x5
– 5 x + a.
Then (2014)
(a) f (x) has three real roots if a > 4
(b) f (x) has only one real root if a > 4
(c) f (x) has three real roots if a < – 4
(d) f (x) has three real roots if – 4 < a < 4
30. Let f : R ® (0, ¥) and g : R R,
® be twice differentiable
functions such that f ¢¢ and g¢¢ are continuous functions
on . Suppose f ¢(2) = g(2) = 0, f ¢¢ (2) ¹ 0 and g¢ (2) ¹ 0. If
then (2016)
(a) f has a local minimum at x = 2
(b) f has a local maximum at x = 2
(c) f ¢¢(2) = f (2)
(d) f (x) – f ¢¢(x) = 0 for at least one x Î
31. Let f: R R
® be given by
5 4 3 2
2
3 2
5 10 10 3 1 0
1 0 1
( )
(2 / 3) 4 7 (8 / 3) 1 3
( 2) ( 2) (10 / 3) 3
x x x x x x
x x x
f x
x x x x
x ln x x x
ì + + + + + <
ï
- + £ <
ï
= í
- + - £ <
ï
ï - - - + ³
î
Then which of the following options is/are correct?
(2019)
(a) f ’ is not differentiable at x=1
(b) f is increasing on ( ,0)
-¥
(c) f is onto
(d) f ’ has a local maximumat x=1

32. Let f: R ® R be given by f(x) = (x – 1) (x – 2)(x – 5). Define
f(x) =
0
x
f t
ò dt, x > 0. Thenwhich of the following options
is/are correct? (2019)
(a) f(x) has a local maximum at x = 2
(b) f(x) has a local minimum at x = 1
(c) f(x) has two local maxima and one local minimum in
(0,¥)
(d) f(x) ¹ 0, for all x Î (0, 5)
33. Let 2
sin
( ) , 0.
x
f x x
x
p
= >
Let x1
< x2
< x3
.... < xn
< ..... be all points of local maximumof
f and y1
< y2
< y3
< ...... < yn
< ....... be all the points of local
minimum of f Then which of the following options is/are
correct? (2019)
(a) |xn
– yn
| > 1 for every n
(b) x1
< y1
(c) xn +1
– xn
> 2 for every n
(d)
1
2 ,2
2
n
x n n
æ ö
Î +
ç ÷
è ø
for every n
34. Let f : R R
® be defined by
2
2
x 3x 6
f x
x 2x 4
- -
=
+ +
Then which of the following statements is (are) TRUE?
(2021)
(a) f is decreasing in the interval (-2, -1)
(b) f is increasing in the interval (1, 2)
(c) f is onto
(d) Range of f is 3
,2
2
é ù
-
ê ú
ë û
35. A straight line L with negative slope passes through the
point (8, 2) and cuts the positive coordinate axes at points
P and Q. Find the absolute minimumvalue of OP + OQ, as
L varies, where O is the origin. (2002)
36. Find a point on the curve x2
+ 2y2
= 6 whose distance from
the line x + y = 7, is minimum. (2003)
37. For the circle x2
+ y2
= r2
, find the value of r for which the
area enclosed by the tangents drawn from the point
P (6, 8) to the circle and the chord of contact is maximum.
(2003)
38. If f (x) is twice differentiable function such that
f (a) = 0, f (b) = 2, f (c) = –1, f (d) = 2, f (e) = 0, where
a < b < c < d <e, then the minimum number of zeroes of
g (x) = { f ´ (x)}2
+ f ´
´(x) . f (x) in the interval [a, e] is ?
(2006)
39. Themaximumvalueofthefunctionf(x)=2x3
–15x2
+36x–48
onthesetA={x|x2
+20 <9x}is.........
(2009)
40. The maximum value of the expression
2 2
1
sin 3sin cos 5cos
q+ q q+ q is...... (2010)
41. Let f be a function defined on R (the set ofall real numbers)
such that f ¢ (x) = 2010 (x – 2009) (x – 2010)2
(x – 2011)3
(x – 2012)4
, for all xÎ R.Ifg isafunction defined onR with
values intheinterval (0, ¥) such that f (x) = 1n (g(x)), forall
x Î R, then the number of points in R at which g has a local
maximumis... (2010)
42. The number of distinct real roots of
x4
– 4x3
+ 12x2
+ x – 1 =0 is .... (2011)
43. Let p (x) be a real polynomial of least degree which has a
local maximum at x = 1 and a local minimum at x = 3. If
p (1) = 6 and p (3) = 2, then p¢ (0) is (2012)
44. Let f : R ® R be defined as f (x) = |x| + |x2
– 1|. The total
number of points at which f attains either a local maximum
or alocal minimum is (2012)
45. A vertical line passing through the point (h, 0) intersects
the ellipse
2 2
x y
1
4 3
+ = at the points P and Q. Let the
tangents to the ellipse at P and Q meet at the point R. If
D(h) = area of the DPQR, D1
= 1/2 h 1
max
£ £
D(h) and
D2
= 1/2 h 1
min
£ £
D(h), then 8
5
D1
– 8D2
is equal to (2013)
46. The slope of the tangent to the curve (y–x5
)2
= x(1 + x2
)2
at
the point (1, 3) is (2014)
47. For a polynomial g (x) with real coefficient, let mg
denote
the number of distinct real roots of g (x). Suppose S is the
set of polynomials with real coefficient defined by
2 2 2 3
0 1 2 3 0 1 2 3
{( 1) ( ) : , , , }.
S x a a x a x a x a a a a R
= - + + + Î
For a polynomial f, let f’ and f’’ denote its first and second
order derivatives, respectively.Then the minimum possible
value of ( ),
f f
m m
¢ ¢¢
+ where fÎS, is …….. . (2020)

48. Let the function :(0, )
f R
p ® be defined by
( ) (sin cos ) (sin cos )
f q q q q q
-
2 4
= + +
Suppose the function g has a local minimum atq precisely
when 1
{ ,....., }
r
q lp l p
Î
where 1
0 ...... 1.
r
l l
< < < < Then the value of
1 ..... r
l l
+ + is …….. . (2020)
Assertion & Reason
49. Consider the folloiwng statement S and R :
S : Both sin x & cos x are decreasing functions in the
interval (p/2, p).
R : If a differentiable function decreases in an interval
(a, b), then its derivative also decreases in (a, b).
Which of the following is true ? (2000)
(a) both S and R are wrong
(b) both S and R are correct, but R is not the correct
explanation for S.
(c) S is correct and R is the correct explanation for S
(d) S is correct and R is wrong.
MatchtheFollowing
Each question has two columns. Four options are given
representing matching of elements from Column-I and
Column-II. Only one of these four options corresponds
to acorrectmatching.Foreachquestion,choosetheoption
corresponding to the correct matching.
50. Let the functions defined in Column I have domain
(-p/2, p/2)
Column I Column II
(A) x +sin x (p) increasing
(B) sec x (q) decreasing
(r) neither increasing nor decreasing
(2008)
ParagraphType Questions
Passage
Consider the function f : (-¥, ¥) ® (-¥, ¥) defined by
2
2
x ax 1
x ; 0<a <2
x ax 1
f .
- +
=
+ +
(2008)
51. Which of the following is true ?
(a) (2 + a)2
f ¢¢ (1) + (2 – a)2
f ¢¢ (–1) = 0
(b) (2 – a)2
f ¢¢ (1) – (2 + a)2
f ¢¢ (–1) = 0
(c) f ¢ (1) f ¢ (–1) = (2 – a)2
(d) f ¢ (1) f ¢ (–1) = – (2 + a)2
52. Which of the following is true ?
(a)f(x)isdecreasingon(–1,1)andhasalocalminimumatx=1.
(b)f(x)isincreasingon(–1,1)andhasalocalmaximumatx=1.
(c) f (x) is increasing on (–1, 1) but has neither a local
maximum nora local minimumatx = 1.
(d) f (x) is decreasing on (–1, 1) but has neither a local
maximum nora local minimumatx = 1.
53. Let g (x) =
x
e
2
0
t
dt
1 t
¢
+
ò
f
. Which of the following is true ?
(a) g¢ (x) is positive on (-¥, 0) and negative on (0, ¥)
(b) g¢ (x) is negative on (-¥, 0) and positive on (0, ¥)
(c) g¢ (x) changes sign on both (-¥, 0) and (0, ¥)
(d) g¢ (x) does not change sign (-¥, ¥)
Passage
Consider the polynomial f (x) = 1 + 2x + 3x
2
+ 4x
3
. Let s be
the sum of all distinct real roots of f (x) and let t = |s|
(2010)
54. The real numbers s lies in the interval
(a)
1
,0
4
æ ö
-
ç ÷
è ø
(b)
3
11,
4
æ ö
- -
ç ÷
è ø
(c)
3 1
,
4 2
æ ö
- -
ç ÷
è ø
(d)
1
0,
4
æ ö
ç ÷
è ø
55. The area bounded by the curve y = f(x) and the lines x = 0,
y = 0 and x = t, lies in the interval
(a)
3
,3
4
æ ö
ç ÷
è ø
(b)
21 11
,
64 16
æ ö
ç ÷
è ø
(c)(9,10) (d)
21
0,
64
æ ö
ç ÷
è ø
56. The function f’ (x) is
(a) increasing in
1
t,
4
æ ö
- -
ç ÷
è ø
and decreasing in
1
, t
4
æ ö
-
ç ÷
è ø
(b) decreasing in
1
t,
4
æ ö
- -
ç ÷
è ø
and increasing in
1
,t
4
æ ö
-
ç ÷
è ø
(c) increasing in (–t, t)
(d) decreasing in (–t, t)

Using the following passage, solve Q.57 and Q.58
Passage
Let f (x) = (1–x)2
sin2
x + x2
for all x Î R and let
x
1
2(t 1)
g(x) ln t
t 1
-
æ ö
= -
ç ÷
+
è ø
ò f (t) dt for all x Î (1, ¥)
57. Which of the following is true ? (2012)
(a) g is increasing on (1, ¥)
(b) g is decreasing on (1, ¥)
(c) g is increasing on (1, 2) and decreasing on (2, ¥)
(d) g is decreasing on (1, 2) and increasing on (2, ¥)
58. Consider the statements
P : There exists some x Î R such that
f(x)+ 2x =2(1 +x2
)
Q : There exists some x Î R such that
2f(x)+1=2x(1+x)
Then, (2012)
(a) Both P and Q are true (b) P is true and Q is false
(c) P is false and Q is true (d) Both P and Q are false
Passage
Let f : [0, 1] ® R (the set of all real numbers) be a function.
Suppose the function f is twice differentiable, f (0) = f(1)=0
and satisfies f’’ (x) – 2f’(x) + f(x) ³ ex
, x Î [0, 1].
59. Which of the following is true for 0 < x < 1 ? (2013)
(a) 0 <f(x)< ¥ (b)
1 1
f(x)
2 2
- < <
(c)
1
f(x) 1
4
- < < (d) – ¥ < f(x) < 0
60. If the function e–x
f (x) assumes its minimum in the interval
[0, 1] at
1
x ,
4
= which of the following is true ? (2013)
(a)
1 3
f (x) f(x), x
4 4
¢ < < <
(b)
1
f (x) f(x), 0 x
4
¢ > < <
(c)
1
f (x) f(x), 0 x
4
¢ < < <
(d)
3
f (x) f(x), x 1
4
¢ < < <
Passage
Let f(x) = x + loge
x – x loge
x, x 0,
Î ¥ .
Column 1 contains information about zeros of f(x), f’(x)
and f’’(x).
Column 2 contains information about the limiting behavior
of f(x), f’(x)and f’’(x) at infinity.
Column 3 contains information about increasing/
decreasing nature of f(x) and f’(x).
Column 1 Column 2 Column 3
(I) f(x) = 0 for some (i)
x
lim f(x) 0
®¥
= (P) fisincreasingin(0,1)
2
x (1, e )
Î
(II) f’(x) = 0 for some (ii)
x
lim f (x)
®¥
= -¥ (Q)f isdecreasingin(e,e2
)
x (1, e)
Î
(III)f’(x)=0forsome (iii)
x
lim f '(x)
®¥
= -¥ (R)f’isincreasingin(0,1)
x (0, 1)
Î
(IV)f’’(x)=0 for some (iv)
x
lim f ''(x) 0
®¥
= (S)f’isdecreasingin(e,e2
)
x (1,e)
Î (2017)
61. Which of the following options is the only CORRECT
combination ?
(a) (I)(ii) (R) (b) (IV)(i)(S)
(c) (III)(iv) (P) (d) (II)(iii)(S)
62. Which of the following options is the only CORRECT
combination ?
(a) (I) (i) (P) (b) (II)(ii)(Q)
(c) (III)(iii) (R) (d)(IV)(iv)(S)
63. Which of the following options is the only INCORRECT
combination ?
(a) (II)(iii)(P) (b)(I) (iii) (P)
(c) (III)(i)(R) (d)(II) (iv) (Q)

Passage
Let 1
f : 0, R
¥ ® and 2
f : 0, R
¥ ® be defined by
x 21
j
1
j 1
0
f x t j dt,
=
= -
Õ
ò x >0
and
50 49
2
f x 98 x 1 600 x 1 2450,
= - - - + x > 0,
Where, for any positive integer n and real numbers
n
1 2 n i
i 1
a , a , ......, a , a
=
Õ denotes the product of
1 2 n
a , a , ......, a . Let mi
and ni
, respectively, denote the
number of points of local minima and the number of points
oflocal maximaoffunction fi
, i=1, 2,intheinterval 0, .
¥
(2021)
64. The value of 1 1 1 1
2m 3n m n
+ + is-------.
65. The value of 2 2 2 2
6m 4n 8m n
+ + is ---------.
Text
66. Let – 1 < p < 1. Show that the equation 4x3
– 3x – p = 0 has
a unique root in the interval
1
, 1
2
é ù
ê ú
ë û
and identify it.
(2001)
67. If P (1) = 0 and
dP x
dx
> P (x) for all x > 1, then prove that
P (x)> 0 forall x > 1. (2003)
68. Using the relation 2 (1 – cos x) < x2
, x ¹ 0 or otherwise,
prove that sin (tan x) ³ x, .
4
,
0
x ú
û
ù
ê
ë
é p
Î
" (2003)
69. Prove that sin x + 2x >
3x . x 1
x 0,
2
+ p
é ù
" Îê ú
p ë û
.
(Justify the inequality, if any used). (2004)
70. If|f(x1
)– f(x2
)|<(x1
–x2
)2
,forallx1
,x2
ÎR.Find theequation
of tangent to the curve y = f (x) at the point
(1,2). (2005)

Answer Key
DIRECTION TO USE -
Scan the QR code and check detailed solutions.
DIRECTION TO USE -
CHAPTER -4 APPLICATION OF DERIVATIVE
EXERCISE - 1 :
BASIC OBJECTIVE QUESTIONS
EXERCISE - 2 :
PREVIOUS YEAR JEE MAIN QUESTIONS
1. (b) 2. (a) 3. (c) 4. (a) 5. (b)
6. (a) 7. (a) 8. (c) 9. (b) 10. (c)
11. (d) 12. (a) 13. (a) 14. (a) 15. (d)
16. (a) 17. (b) 18. (c) 19. (c) 20. (c)
21. (c) 22. (d) 23. (d) 24. 3.00 25. (d)
26. (b) 27. (a) 28. (a) 29. (b) 30. (b)
31. (d) 32. (b) 33. (a) 34. (b) 35. (c)
36. 122.00 37. (a) 38. (c) 39. (d) 40. (b)
41. (c) 42. (c) 43. (d) 44. (c) 45. (d)
46. (a) 47. (d) 48. (a) 49. (c) 50. (a)
51. 5 52. (a) 53. (d) 54. (c) 55. (b)
56. (d) 57. (a) 58. (d) 59. (b) 60. 4.00
61. (b) 62. (3.00) 63. (d) 64. (c) 65. (c)
66. (c) 67. (9.00) 68. (c) 69. (b) 70. (a)
71. (d) 72. (a) 73. (144.00) 74. (a)
75. (4.00) 76. (d) 77. (d) 78. (2.00) 79. (c)
80. (b) 81. (b) 82. (5.00) 83. (b) 84. (d)
85. (c) 86. (a) 87. (c) 88. (d) 89. (3.0)
90. (a) 91. (36.00) 92. (a) 93. (a) 94. (4.0)
95. (a) 96. (b) 97. (2.00) 98. (d) 99. (c)
100. (22.00)
1. (a) 2. (d) 3. (a) 4. (c) 5. (d)
6. (c) 7. (a) 8. (d) 9. (c) 10. (c)
11. (c) 12. (c) 13. (a) 14. (d) 15. (a)
16. (b) 17. (b) 18. (a) 19. (a) 20. (c)
21. (b) 22. (c) 23. (b) 24. (b) 25. (a)
26. (c) 27. (a) 28. (d) 29. (b) 30. (a)
31. (a) 32. (a) 33. (d) 34. (a) 35. (c)
36. (d) 37. (b) 38. (b) 39. (c) 40. (a)
41. (a) 42. (a) 43. (c) 44. (b) 45. (b)
46. (d) 47. (d) 48. (c) 49. (b) 50. (c)
51. (b) 52. (b) 53. (a) 54. (d) 55. (c)
56. (a) 57. (a) 58. (b) 59. (d) 60. (b)
61. (c) 62. (b) 63. (c) 64. (c)
65. (169.65) 66. (1.5) 67. (-3) 68. (5.2)
69. (12.57) 70. (17.32) 71. (502.65) 72. (1)
73. (45) 74. (17.32) 75. (-42) 76. (0.07)
77. (-3) 78. (1) 79. (2.12) 80. (0.1)

ANSWER KEY 253
DIRECTION TO USE -
DIRECTION TO USE -
EXERCISE - 3 :
ADVANCED OBJECTIVE QUESTIONS
EXERCISE - 4 :
PREVIOUS YEAR JEE ADVANCED QUESTIONS
1. (c) 2. (d) 3. (a) 4. (c) 5. (c)
6. (b) 7. (d) 8. (c) 9. (a) 10. (a)
11. (d) 12. (b) 13. (d) 14. (d) 15. (b)
16. (b) 17. (a) 18. (c) 19. (a) 20. (b)
21. (d) 22. (a) 23. (a) 24. (a) 25. (d)
26. (c) 27. (c) 28. (b) 29. (c) 30. (a)
31. (c) 32. (d) 33. (d) 34. (b) 35. (a)
36. (a) 37. (b) 38. (a) 39. (d) 40. (c)
41. (b) 42. (a) 43. (a) 44. (c) 45. (a,c)
46. (b,c) 47. (a,b) 48. (b,c) 49. (b,c) 50. (b,d)
51. (a,b,c,d) 52. (a,c) 53. (a,c,d) 54. (a,d)
55. (a,b,c,d) 56. (a,c) 57. (a,b)
58. (a,b,c) 59. (a,c) 60. (a,c,d) 61. (a,b) 62. (b,c)
63. (a,b,c,d) 64. (a,c) 65. (a,b,c)
66. (a,b,d)67. (2) 68. (4) 69. (16) 70. (11)
71. (4) 72. (6) 73. (25) 74. (18) 75. (d)
76. (a) 77. (c) 78. (d) 79. (a) 80. (a)
81. (a) 82. (c) 83. (d) 84. (c) 85. (d)
86. (d) 87. (d) 88. (a) 89. (a) 90. (c)
91. (a) 92. (a) 93. (c) 94. (a) 95. (a)
96. (d) 97. (a) 98. (a) 99. (d) 100. (c)
101. (d) 102. (b) 103. (b) 104. (b) 105. (b)
106. (c) 107. (b) 108. (b) 109. (b) 110. (d)
1. (b) 2. (c) 3. (a) 4. (d) 5. (a)
6. (a) 7. (a) 8. (d) 9. (d) 10. (d)
11. (a) 12. (c) 13. (c) 14. (c) 15. (c)
16. (d) 17. (a) 18. (c) 19. (c) 20. (d)
21. (c) 22. (c) 23. (b,c) 24. (a,b)
25. (b,c,d) 26. (b,c) 27. (a,c) 28. (a,b) 29. (b,d)
30. (a,d) 31. (a,c,d) 32. (a,b,d) 33. (a,c,d) 34. (a,b )
35. (18) 36. (2,1) 37. (5 unit) 38. (6)
39. (7) 40. (2) 41. (1) 42. (2) 43. (2)
44. (5) 45. (9) 46. (8) 47. (5.00)
48. (0.50) 49. (d) 50. (A–p; B–r)
51. (a) 52. (a) 53. (b) 54. (c) 55. (a)
56. (b) 57. (b) 58. (c) 59. (d) 60. (c)
61. (d) 62. (b) 63. (c) 64. (57.00)
65. (6.00) 66.
1
1
cos cos p
3
-
æ ö
ç ÷
è ø
70. y-2 = 0
CHAPTER -4 APPLICATION OF DERIVATIVE

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