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ASSIGNMENT–I (AOD)
Q.1 Find the equations of the tangents drawn to the curve y
2
3
- 4y + 8 = 0 from the point (1, 2).
Q.2 Find the point of intersection of the tangents drawn to the curve x
2 y = 1 – y at the points where it is
intersected by the curve xy = 1 – y.
Q.3 Find all the lines that pass through the point (1, 1) and are tangent to the curve represented parametrically
as x = 2t – t
2 and y = t + t
2 .
Q.4 In the curve x
a y
b = K
a+b , prove that the portion of the tangent intercepted between the coordinate axes
is divided at its point of contact into segments which are in a constant ratio. (All the constants being
positive).
Q.5 A straight line is drawn through the origin and parallel to the tangent to a curve
a
x a y
2 2
= l n
y
a a y
2 2
at an arbitary point M. Show that the locus of the point P of
intersection of the straight line through the origin & the straight line parallel to the x-axis & passing
through the point M is x
2
2 = a
2 .
Q.6 Prove that the segment of the tangent to the curve y =
a
l n
2 2
2 2
a a x
a a x
2 2
a x contained between
the y-axis & the point of tangency has a constant length.
Q.7 A function is defined parametrically by the equations
f(t) = x =
2
t t
t
if t
if t
sin
and g(t) = y =
2
t
t if t
o if t
sin
Find the equation of the tangent and normal at the point for t = 0 if exist.
Q.8 Find all the tangents to the curve y = cos (x + y), 2 x 2 , that are parallel to the line x + 2y = 0.
Q.9 (a) Find the value of n so that the subnormal at any point on the curve xy
n = a
n + 1 may be constant.
(b) Show that in the curve y = a. l n (x² a²), sum of the length of tangent & subtangent varies as the
product of the coordinates of the point of contact.
Q.10 Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos
2 t ; y = a cos
3 t contained
between the co-ordinate axes is equal to 2a.
Q.11 Show that the normals to the curve x = a (cos t + t sin t) ; y = a (sin t t cos t) are tangent lines to the
circle x
2
2 = a
2 .
Q.12 The tangent to the graph of the function y = f(x) at the point with abscissa x = a forms with the x-axis an
angle of /6 and at the point with abscissa x = b an angle of /4, then find the value of the integral
b
a
f (x)f (x)dx^
Q.13 If the tangent at the point (x 1
, y 1
) to the curve x
3
3 = a
3 (a 0) meets the curve again in (x 2
, y 2
) then
show that
1
2
1
2
y
y
x
x
Q.14 Determine a differentiable function y = f (x) which satisfies f ' (x) = [f(x)]
2 and f (0) = –
. Find also the
equation of the tangent at the point where the curve crosses the y-axis.
Q.15 If p 1
& p 2
be the lengths of the perpendiculars from the origin on the tangent & normal respectively at
any point (x, y) on a curve, then show that
p xcos y sin
p xsin y cos
2
1
where tan =
dx
dy
. If in the
above case, the curve be x
2/
2/ = a
2/ then show that : 4 p 1
2
2 = a
2 .
Q.16 The curve y = ax
3
2
- cx + 5 , touches the x - axis at P ( 2 , 0) & cuts the y-axis at a point Q where
its gradient is 3. Find a , b , c.
Q.17 The tangent at a variable point P of the curve y = x
2 x
3 meets it again at Q. Show that the locus of the
middle point of PQ is y = 1 9x + 28x
2 28x
3 .
Q.18 Show that the distance from the origin of t he normal at any point of the curve
x = a e
2 cos
sin & y = a e
2 sin
cos is twice the distance of the tangent at the point
from the origin.
Q. 19 Show that the condition that the curves x
2/
2/ = c
2/ & (x
2 /a
2 ) + (y
2 /b
2 ) = 1 may touch if c = a + b.
Q.20 The graph of a certain function f contains the point (0, 2) and has the property that for each number ' p '
the line tangent to y = f (x) at (^) p, f (p)intersect the x-axis at p + 2. Find f (x).
Q.21 A curve is given by the equations x = at
2 & y = at
3
. A variable pair of perpendicular lines through the
origin 'O' meet the curve at P & Q. Show that the locus of the point of intersection of the tangents at P &
Q is 4y
2 = 3ax a
2 .
Q.22(a) Show that the curves
1
2
2
1
2
2
b K
y
a K
x
2
2
2
2
2
2
b K
y
a K
x
= 1 intersect orthogonally..
(b) Find the condition that the curves
b
y
a
x
2 2
b
y
a
x
2 2
= 1 may cut orthogonally..
Q.23 Show that the curve x
3 3xy
2 = a and 3x
2 y y
3 = b cut each other orthogonally, where a and b are
constants.
Q.24 For the curve x
2/
2/ = a
2/ , show that
2
z + 3p
2 = a
2 where z = x + iy & p is the length of the
perpendicular from (0 , 0) to the tangent at (x , y) on the curve.
Q.25 A and B are points of the parabola y = x
2
. The tangents at A and B meet at C. The median of the triangle
ABC from C has length 'm' units. Find the area of the triangle in terms of 'm'.
ASSIGNMENT–I
Q.1 2 3 x y = 2 (^) 3 ^1 or 2 3 x + y = 2 (^) 3 ^1 Q.2 (0, 1)
Q.3 x = 1 when t = 1, m ; 5x – 4y = 1 if t 1, m = 1/3]
Q.7 T : x – 2y = 0 ; N : 2x + y = 0 Q.8 x + 2 y = /2 & x + 2 y = 3 /
Q.9 (a) n = 2 Q.12 –1 Q.14 –
x 2
; x – 4y = 2 Q.16 a = 1/2 ; b = 3/4 ; c = 3
Q.20 2e
–x/ Q.22 (b) a b = a
b Q.
m m
Q.12 Water is flowing out at the rate of 6 m
3 /min from a reservoir shaped like a hemispherical bowl of radius
R = 13 m. The volume of water in the hemispherical bowl is given by V =
·y ( 3 R y)
2
when the
water is y meter deep. Find
(a) At what rate is the water level changing when the water is 8 m deep.
(b) At what rate is the radius of the water surface changing when the water is 8 m deep.
Q.13 If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining elements are
changed slightly, show that
cosB
db
cos A
da
Q.14 At time t > 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At
t = 0, the radius of the sphere is 1 unit and at t = 15 the radius is 2 units.
(a) Find the radius of the sphere as a function of time t.
(b) At what time t will the volume of the sphere be 27 times its volume at t = 0.
Q.15 Use differentials to a approximate the values of ; (a) 25. 2 and (b)
3
ASSIGNMENT–II
Q.1 1/9 m/min Q.2 (i) 6 km/h (ii) 2 km/hr Q.3 (4 , 11) & ( 4, 31/3)
Q.4 3/8 cm/min Q.5 1 + 36 cu. cm/sec Q.6 1/48 cm/s Q.7 0.05 cm/sec
Q.
2
4
cm/s Q.9 200
r
3 / (r + 5)² km² / h Q.
66
7
Q.
1
4
cm/sec.
Q.12 (a) –
m/min., (b) –
m/min. Q.14 (a) r = (1 + t)
1/ , (b) t = 80 Q.15 (a) 5.02, (b)
80
27
ASSIGNMENT–I
Q. 1 Find the intervals of monotonocity for the following functions & represent your solution set on thenumber line.
(a) f(x) = 2.
x 4 x
2
e
(b) f(x) = e
x /x (c) f(x) = x
2 e
x (d) f (x) = 2x
2
Also plot the graphs in each case.
Q.2 Let f (x) = 1 – x – x
3
. Find all real values of x satisfying the inequality, 1 – f (x) – f
3 (x) > f (1 – 5x)
Q.3 Find the intervals of monotonocity of the function
(a) f (x) = sin x – cos x in x [0 , 2 ] (b) g (x) = 2 sinx + cos 2x in (0 x 2 ).
Q.4 Show that, x
3 3x
2 9 x + 20 is positive for all values of x > 4.
Q.5 Let f (x) = x
3 x
2
3 x , 1 x 2
max{f(t): 0 t x} , 0 x 1
Discuss the conti. & differentiability of g(x) in the interval (0,2).
Q.6 Find the set of all values of the parameter 'a' for which the function,
f(x) = sin 2x – 8(a + 1)sin x + (4a
2
- 8a – 14)x increases for all x R and has no critical points
for all x R.
Q.7 Find the greatest & the least values of the following functions in the given interval if they exist.
(a) f (x) = sin
1
x 1
x
2
l n x in
(b) y = x
x in (0, ) (c) y = x
5
4
3
Q.8 Find the values of 'a' for which the function f(x) = sinx a sin2x
sin3x + 2ax increases throughout the
number line.
Q.9 Prove that f (x) =
x e
2
2
9 cos ( 2 l n t) 25 cos( 2 l nt) 17 dt is always an increasing function of x,^ ^ xR
Q.10 If f(x) =
a 1
2
x
3
2
- 2x + 1 is monotonic increasing for every x R then find the range of
values of ‘a’.
Q.11 Find the set of values of 'a' for which the function,
f(x) = 1
21 4
1
2
a a
a
x
3
- 5x + 7 is increasing at every point of its domain.
Q.12 Find the intervals in which the function f (x) = 3 cos
4 x + 10 cos
3 x + 6 cos
2 x – 3, 0 x ; is
monotonically increasing or decreasing.
Q.13 Investigate for maxima and minima for the function f (x) =
x
3 2 2
1
2 t 1 t 2 3 t 1 t 2 dt
Q.14 Find the value of x > 1 for which the function F (x) =
2 x
x
dt
t 1
n
t
l is increasing and decreasing.
ASSIGNMENT–II
Q.1 Verify Rolles throrem for f(x) = (x a)
m (x b)
n on [a, b] ; m, n being positive integer.
Q.2 Let f : [a, b] R be continuous on [a, b] and differentiable on (a, b). If f (a) < f (b), then show that
f ' (c) > 0 for some c (a, b).
Q.3 Let f (x) = 4x
3 3x
2 2x + 1, use Rolle's theorem to prove that there exist c, 0< c <1 such that f(c) = 0.
Q.4 Using LMVT prove that : (a) tan x > x in
, (b) sin x < x for x > 0
Q.5 Prove that if f is differentiable on [a, b] and if f (a) = f (b) = 0 then for any real there is an x (a, b)
such that f (x) + f ' (x) = 0.
Q.6 For what value of a, m and b does the function f (x) =
mx b 1 x 2
x 3 x a 0 x 1
3 x 0
2
satisfy the hypothesis of the mean value theorem for the interval [0, 2].
Q.7 Suppose that on the interval [–2, 4] the function f is differentiable, f (–2) = 1 and | f ' (x) | 5. Find the
bounding functions of f on [–2, 4], using LMVT.
Q.8 Let f, g be differentiable on R and suppose that f (0) = g (0) and f ' (x) g ' (x) for all x 0. Show that
f (x) g (x) for all x 0.
Q.9 Let f be continuous on [a, b] and differentiable on (a, b). If f (a) = a and f (b) = b, show that there exist
distinct c 1
, c 2
in (a, b) such that f ' (c 1
) + f '(c 2
Q.10 Let f (x) and g (x) be differentiable functions such that f ' (x) g (x) f (x) g ' (x) for any real x. Show that
between any two real solutions of f (x) = 0, there is at least one real solution of g (x) = 0.
Q.11 Let f defined on [0, 1] be a twice differentiable function such that, | f " (x) | 1 for all x [0, 1]
If f (0) = f (1), then show that, | f ' (x) | < 1 for all x [0, 1]
Q.12 f (x) and g (x) are differentiable functions for 0 x 2 such that f (0) = 5, g (0) = 0, f (2) = 8, g (2) = 1.
Show that there exists a number c satisfying 0 < c < 2 and f ' (c) = 3 g' (c).
Q.13 If f, , are continuous in [a, b] and derivable in ]a, b[ then show that there is a value of c lying between
a & b such that,
(a) (b) (c )
(a) (b) (c )
f(a) f(b) f(c )
Q.14 For what real values of a and b are all the extrema of the function
f(x) =
2
5a
x
3
2
- 9x + b are positive and the maximum is at the point x 0
Q.15 Let a > 0 and f be continuous in [–a, a]. Suppose that f ' (x) exists and f ' (x) 1 for all x (–a, a). If
f (a) = a and f (– a) = – a, show that f (0) = 0.
Q.16 Let a, b, c be three real number such that a < b < c, f (x) is continuous in [a, c] and differentiable
in (a, c). Also f ' (x) is strictly increasing in (a, c). Prove that
(c – b) f (a) + (b – a) f (c) > (c – a) f (b)
Q.17 Use the mean value theorem to prove,
x
x 1
< l n x < x – 1, x > 1
Q.18 Use mean value theorem to evaluate,
Lim x 1 x
x
Q.19 Using L.M.V.T. or otherwise prove that difference of square root of two consecutive natural numbers
greater than N
2 is less than
2 N
Q.20 Prove the inequality e
x
(1 + x) using LMVT for all x R 0
and use it to determine which of the two
numbers e
and
e is greater.
ASSIGNMENT–II
Q.1 c =
mb na
m n
which lies between a & b
Q.6 a = 3, b = 4 and m = 1
Q.7 y = – 5x – 9 and y = 5x + 11
Q. 14 If a =
(^) , then b
and If a =
then b
Q.18 0
Q.13 A window of fixed perimeter (including the base of the arch) is in the form of a rectangle surmounted by
a semicircle. The semicircular portion is fitted with coloured glass while the rectangular part is fitted with
clean glass. The clear glass transmits three times as much light per square meter as the coloured glass
does. What is the ratio of the sides of the rectangle so that the window transmits the maximum light?
Q.14 A closed rectangular box with a square base is to be made to contain 1000 cubic feet. The cost of the
material per square foot for the bottom is 15 paise, for the top 25 paise and for the sides 20 paise. The
labour charges for making the box are Rs. 3/-. Find the dimensions of the box when the cost is minimum.
Q.15 Find the area of the largest rectangle with lower base on the x-axis & upper vertices on the
curve y = 12 x
2 .
Q.16 A trapezium ABCD is inscribed into a semicircle of radius l so that the base AD of the trapezium is a
diameter and the vertices B & C lie on the circumference. Find the base angle of the trapezium ABCD
which has the greatest perimeter.
Q.17 If y =
ax b
x x
( 1 ) ( 4 )
has a turning value at (2, 1) find a & b and show that the turning value is a
maximum.
Q.18 Prove that among all triangles with a given perimeter, the equilateral triangle has the maximum area.
Q.19 A sheet of poster has its area 18 m². The margin at the top & bottom are 75 cms and at the sides
50 cms. What are the dimensions of the poster if the area of the printed space is maximum?
Q.20 A perpendicular is drawn from the centre to a tangent to an ellipse
x
a
2
2
y
b
2
2
= 1. Find the greatest value
of the intercept between the point of contact and the foot of the perpendicular.
Q.21 Consider the function, F (x) =
x
1
2
( t t)dt , x R.
(a) Find the x and y intercept of F if they exist.
(b) Derivatives F ' (x) and F '' (x).
(c) The intervals on which F is an increasing and the invervals on which F is decreasing.
(d) Relative maximum and minimum points.
(e) Any inflection point.
Q.22 A beam of rectangular cross section must be sawn from a round log of diameter d. What should the
width x and height y of the cross section be for the beam to offer the greatest resistance (a) to compression;
(b) to bending. Assume that the compressive strength of a beam is proportional to the area of the cross
section and the bending strength is proportional to the product of the width of section by the square of its
height.
Q.23 What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangle
of base 36 ft. & altitude 12 ft? Assume that one side of the rectangle lies on the base of the triangle.
Q.24 The flower bed is to be in the shape of a circular sector of radius r & central angle . If the area is fixed
& perimeter is minimum, find r and .
Q.25 The circle x
2
2 = 1 cuts the x-axis at P & Q. Another circle with centre at Q and varable radius
intersects the first circle at R above the x-axis & the line segment PQ at S. Find the maximum area of
the triangle QSR.
ASSIGNMENT–II
Q.1 The mass of a cell culture at time t is given by, M (t) = t
1 4 e
(a) Find LimM(t)
t
and LimM(t)
t
(b) Show that
dt
dM
= M(^3 M)
(c) Find the maximum rate of growth of M and also the vlaue of t at which occurs.
Q.2 Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given
constant length l of the median drawn to its lateral side.
Q.3 From a fixed point A on the circumference of a circle of radius 'a', let the perpendicular AY fall on the
tangent at a point P on the circle, prove that the greatest area which the APY can have
is 3 3
a
2
sq. units.
Q.4 Given two points A ( 2 , 0) & B (0 , 4) and a line y = x. Find the co-ordinates of a point M on this line
so that the perimeter of the AMB is least.
Q.5 A given quantity of metal is to be casted into a half cylinder i.e. with a rectangular base and semicircular
ends. Show that in order that total surface area may be minimum , the ratio of the height of the cylinder
to the diameter of the semi circular ends is /(
Q.6 Depending on the values of p R, find the value of 'a' for which the equation x
3
2
distinct real roots.
Q.7 Show that for each a > 0 the function e
ax
. x
a² has a maximum value say F (a), and that F (x) has a
minimum value, e
e/ .
Q.8 For a > 0, find the minimum value of the integral
1 a
0
3 5 2 ax
(a 4 x a x )e dx.
Q.9 Let f(x) =
3 2
2
2
x x 10x 5 x 1
2x log b 2 x 1
. Find the values of b for which f(x) has greatest value at x =1.
Q.10 Consider the function f (x) =
0 for x 0
x nx whenx 0
l
(a) Find whether f is continuous at x = 0 or not.
(b) Find the minima and maxima if they exist.
(c) Does f ' (0)? Find Lim '(x)
x 0
f
(d) Find the inflection points of the graph of y = f (x)..
Q.11 Consider the function y = f (x) = l n (1 + sin x) with – 2 x 2 . Find
(a) the zeroes of f (x)
(b) inflection points if any on the graph
(c) local maxima and minima of f (x)
(d) asymptotes of the graph
(e) sketch the graph of f (x) and compute the value of the definite integral
2
2
f (x) dx.
ASSIGNMENT–I
Q.1 f (x) = x
3
2 x + 2 Q.2 max. at x = 1 ; f(1) = 0 , min. at x = 7/5 ; f(7/5) = 108/
Q.3 (a) Max at x = 2 , Max value = 2 , Min. at x = 0 , Min value = 0
(b) Max at x = /6 & also at x = 5 /6 and
Max value = 3/2 , Min at x = /2 , Min value = 3
Q.4 f (x) =
2
3
x
6
12
5
x
5
4 Q.5 P max
= a (^1)
cos ec
Q.6 75 3 sq. units
Q.7 r =
2
4
A
s =
2
4
A
Q.9 3x + 4y – 9 = 0 ; 3x – 4y + 9 = 0
Q.10 4 2 m Q.11 1/ cu m
Q.12 110 ' , 70 ' Q.13 6/(6 + ) Q.14 side 10', height 10'
Q.15 32 sq. units Q.16 = 60
0 Q.17 a = 1, b = 0
Q.19 width 2 3 m, length 3 3 m Q.20 a b
Q.21 (a) (–1, 0), (0, 5/6) ; (b) F ' (x) = (x
2
- x), F '' (x) = 2x – 1, (c) increasing (– , 0) (1, ),
decreasing (0, 1) ; (d) (0, 5/6) ; (1, 2/3) ; (e) x = 1/
Q.22 (a) x = y =
d
2
, (b) x =
d
3
, y =
2
3
d Q.23 6' × 18'
Q.24 r = (^) A , = 2 radians Q.
ASSIGNMENT–II
Q.1 (a) 0, 3, (c)
, t = l n 4 Q.2 cos A = 0.8 Q.4 (0 , 0)
Q.6 p < a <
32
27
3 p
32
27
3 p
- p < a < p if p < 0 Q.8 4 when a = 2
Q.9 b 130,^2 2,^130
Q.10 (a) f is continuous at x = 0 ; (b) –
e
; (c) does not exist, does not exist ; (d) pt. of inflection x = 1
Q.11 (a) x = – 2, – , 0, , 2, (b) no inflection point, (c) maxima at x =
and –
and no minima,
(d) x =
and x = –
, (e) – l n 2
Q.12 4 Q.13 (0 , 2) & max. distance = 4 Q.14 m
Q.
4
Q.17 ( , 3) (3 , 29/7) Q.18 H = x =
1 / 3
4 V
Q.
(c ab)(abc )
Q.20 L/4 Q.
Q.23 (a) increasing in (0, 2) and decreasing in (–, 0) (2, ), local min. value = 0 and local max. value = 2
(b) concave up for (– , 2 – 2 ) (2 + 2 , ) and concave down in (2 – 2 ), (2 + 2 )
(c) f (x) =
2 ·x 2
e ·x
Q.24 2 2 1
