Practice questions( calculus ) xii

ASSIGNMENT
Differential Calculus I
Q. 1. Discuss the continuity of
Q. 2. If
Q. 3. Differentiate the following function w.r.t. x….
Q. 4. If then showthat
Q. 5.
Q. 6. Differentiate withrespect to

Q. 7. If showthat
Q. 8. Find the value of a and b such that the function defined by
Differential Calculus II
Q. 2. Find the relationshipbetweena and b so that the function f
defined by
Q. 3. If
then show that
Q. 4.
Q. 5.If cos y = x Cos (a + y), with
prove that

Q. 6. Differentiate w.r.to x.
Q. 7.
Q. 8. If for some c > 0, prove that
is a constant independent of a and b.
Differential Calculus III
Q. 2. For what value of is the function defined by
continuous at x = 0? What about continuity at x = 1?
Q. 3.

Q. 4.
Q. 5.
Q. 6.
Q. 7.
Q. 8.
Continuity & Differentiation
Q. 1. Find the values of a and b suchthat the function defined by f(x) =
( 5, if x ≤ 2 ax + b if 2<x<10 21, if x 10 ) is a continuous function.
Q. 2. Find of sin2y + cos (xy) = p
Q. 3. Differentiate w.r.t. x ( x cosx)x + (x sinx)1/x
Q. 4. If x = , = show that
Q. 5. If y = (tan-1x)2, showthat (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2.

Q. 6. Differentiate sin-1 w.r.t. x
Q. 7. If x for -1<x<1, showthat
Q. 8. Find if y = a t + 1/t , x = ( t + 1/t)a
Q. 9. Discuss the continuity of the function given by :-
Q. 10. If the function f(x) is givenby f(x) =
is continuous at x = 1, find the values of a and b.
Q. 11. If y = [x + ]n, then prove that
Q. 12. Prove :
Q. 13. Find when y = sec-1
Q. 1. Find the value of the following :
i.
ii.

iii.
Q. 2. Prove That
i.
ii.
iii.
Q. 3. Solve
i.
ii.
Q. 4. Simplify :
i.
ii.
Q. 5.
Q. 6.

Q. 7.
Q. 8.
Q. 14. If ex + ey = ex+y, prove that
Q. 15. Given that cos
prove that
Q. 16. If x=a(q+ sinq),y= a(1+ cosq),prove that
Q. 17.
Q. 18. Find the value of‘k’ if

is continuous at x =
Q. 19. If
Q. 20. If Cos y = x Cos2 ( a + y ) , with Cos a ≠ 1, prove that
Question 3 The function f is defined as {
𝒙² + 𝒂𝒙 + 𝒃 , 𝟎 ≤ 𝒙 < 2
𝟑𝒙 + 𝟐 , 𝟐 ≤ 𝒙 ≤ 𝟒
𝟐𝒂𝒙 + 𝟓𝒃 , 𝟒 < 𝑥 ≤ 8
If f(x) is continuous on [0,8], find the values of a and b. Answer
[a=3,b=-2]
Rate of Change of Quantities.
Q.1. A point source of light along a straight roadis at a height of ‘a’
metres.A boy ‘b’ metres in height is walking along the road. Howfast
is his shadow increasing if he is walking away from the light at the
rate of c metres per minute?
Solution:
Fig.
Let lamp-post be AB and CD be the boy whose distance from lamp-post
at any time t be x m, let CE = y m be its shadow. Then

dx/dt = c m/m.
As, ∆ BAE ~ ∆ DCE, AB/CD = AE/CE
=> a/b = (x + y)/y
=> ay = b(x + y)
=> (a – b) y = bx
=> (a – b)dy/dt = b dx/dt = bc
Therefore,dy/dt = bc/(a – b). [Ans.]
Q.2. The two equal sides ofan isosceles triangle withfixedbase b cm
are decreasing at the rate of 3 cm/sec.Howfast is the areadecreasing
when the two equal sides are equal to the base?
Solution:
Fig.
Q.3. The volume of a cube is increasing at the rate of 7 cubic
centimeters per second. Howfast is the surface area of the cube
increasing when the length of an edge is 12 centimeters?
6.2. Increasing and Decreasing Function.
Q.1. Find the intervals in which the function f(x) = x3 – 12 x2 + 36 x +
17 is
i. increasing,
ii. decreasing.
i. x ε ] – ∞,2[ U ] 6, ∞ [ . [Ans.]
(ii) x ε ]2, 6[ [Ans.]
Q.2. Find the intervals in which the function f(x) = 2x3 – 9x2 + 12x + 15
is (i) increasing and (ii) decreasing.
Solution:

Therefore,disjoint intervals on real number line are (– ∞,1), (1, 2),
(2, ∞)
Intervals Test
Value
Nature of f’(x)
f’(x) = 6(x –
2)(x – 1)
f(x)
( – ∞, 1) x = 0 ( + ) (– ) (– ) =
( + ) > 0
↑
(1, 2) x = 1.5 ( + )( – )( – ) =
( – ) < 0
↓
(2, ∞) x = 3 ( + )( + )( + ) =
( + ) > 0
↑
Therefore,f(x) is increasing in ( – ∞, 1), (2, ∞) and decreasing in (1,
2). [Ans.]
Tangents and Normals.
Q.1. If x = a sin 2t (1+ cos 2t) and y = b cos 2t (1 – cos 2t),show
that [dy/dx]at t=π/4= b/a.
Q.2. If x = a(cos θ + log tan θ/2) and y = a sin θ, find the value of dy/dx
at θ = π/4.
Q.3. Find the slope ofthe tangent to the curve y = 3x4 – 4x at x = 1.
Q.4. For the curve y = 3x2 + 4x, find the slope of the tangent to the
curve at the point whose x-coordinate is – 2.
Q.5. Find the equationof the tangent and the normal to the curve y =
x3 at the point P(1,1).
Q.6. Find the equationof the tangent to the curve:x = θ + sinθ, y = 1 +
cos θ at θ = π/4.
Q.7. Find the equationof the tangent to the curve x = sin 3t, y = cos 2t,
at t = π/4.
2√2 x – 3y – 2 = 0. [Ans.]

Q.8. At what points will the tangent to the curve y = 2x3 – 15x2 + 36x –
21 be parallel to x-axis? Also,find the equations of tangents to the
curve at those points.
6.4. Approximation.
Q.1. If f(x) = 3x2 + 15x + 5, then find the approximate value of f(3.02),
using differentials.
77.66. [Ans.]
Q. 1. An open box, witha square base, is to be made out ofa given
quantity of metal sheet ofarea C2. Show that the maximum volume of
the box is C3/6√3.
Q.2. A window is in the form of a rectangle surmountedby a semi-
circle.If the total perimeter ofthe window is 30 m, find the
dimensions of the window so that maximum light is admitted.
Solution:
AB = 30/(π + 4) m and BC = 30/(π + 4) m. [Ans.]
Q.3. Find the point on the curve y2 = 4x which is nearest to the point
(2, –8).
the nearest point is (4, – 4) [Ans.]
Q.4. Find the largest possible areaofthe right-angledtriangle whose
hypotenuse is 5 cm.
Solution:

.
= 25/4 sq. units. [Ans.]
Q.5. Prove that the radius of the right circular cylinder of the greatest
curved surface that can be inscribedin a given cone is half of the
radius of the cone.
Solution:
Q.6. A right-angledtriangle withconstant areaS is given. Prove that
the hypotenuse ofthe triangle is least when the triangle is isosceles.
Solution:
Q.7. Three sides ofa trapezium are equal, each being 10 cm. Find the
area ofthe trapezium when it is maximum.

Solution:
the maximum area ofthe trapezium is 75√3. [Ans.]
Q.8. Showthat the semi-vertical angle of the right circular cone of
given total surface area and maximum volume is sin –11/3.
Solution:
Q.9. Showthat a rectangle of maximum perimeter whichcan be
inscribedin a circle ofradius r is a square of side √2r.
Solution:
Fig.
Let ABCD be the rectangle inscribedin a circle of radius r and centre
O. BD is the diameter = 2r. Let LOBA = θ, 0 < θ < π/2.
Now, AB = 2r cos θ and AD = 2r sin θ.
Perimeter ofthe rectangle,p = 2(AB + CD) = 2(2r cos θ + 2r sin θ)
= 4r (cos θ + sin θ)

Therefore,dp/dθ = 4r (– sin θ + cos θ)
and d2p/dθ2= 4r (– cos θ – sin θ) = – 4r(cos θ + sinθ).
Now, dp/dθ = 0 => 4r (– sin θ + cos θ) = 0
Or, tan θ = 1 => θ = π/4. [As, 0 < θ < π/2]
Also [d2p/dθ2]θ = π/4= – 4r ( sin π/4 + cos π/4)
= – 4r (1/√2+ 1/√2) = – 4r.2/√2= – 4√2 r < 0.
Therefore,p is maximum when θ = π/4.
That is when BC = 2r sin π/4= 2r. 1/√2 = √2 r and AB = 2r cos π/4=
2r. 1/√2= √2 r.
AB and BC are adjacent sides, hence ABCD is a square.
Hence, perimeter ofABCD is maximum when it is a square.[Proved.]
Q.10. Showthat the rectangle of maximum areathat can be inscribed
in a circle is a square.
Q.12. Showthat the height of a cylinder of maximum volume that can
be inscribedin a sphere of radius R is 2R/√3.
Or,
Prove that the height of a right circular cylinder of maximum volume
that can be inscribedin a sphere of radius R is 2R/√3.Also find the
maximum volume.
Solution:
Q.13. Find the altitude of a right circular cone of maximum curved
surface which can be inscribed in a sphere of radius r.
Solution:

Q.14. A wire oflength 20 m is available to fence off a flower bed in the
form of a sector ofa circle.What must be the radius of the circle,if we
wish to have a flower bed withthe greatest possible area?
Solution:
Q.15. Showthat the height of a cylinder of maximum volume that can
be inscribedin a cone of height h is 1/3h.
Solution:
Q.16. Showthat the volume of the greatest cylinder that can be
inscribedin a cone of height h and semi-vertical α is 4/27 πh3tan2α.
Solution:

Q.17. Showthat the semi-vertical angle of a cone ofmaximum volume
and ofgiven slant height is tan –1(√2).
Solution:
Q.18. Find the volume of the largest cone that can be inscribedin a
sphere of radius R.
Solution:
Fig.
Let
Q.19. Prove that the area of right-angledtriangle of a given
hypotenuse is maximum when the triangle is isosceles.
Solution:

Fig.
Q.20. A closedcircular cylinder has a volume of 2156 c.c. What will be
the radius of its base so that its total surface area is minimum. Find
the height of the cylinder when its total surface area is minimum.
Or
Showthat the height of the closedright circular cylinder,of given
volume and minimum total surface area, is equal to its diameter.
Q.21. Three numbers are given whose sum is 180 and the ratio
between first two of them is 1:2. if the product of the number is
greatest,find the numbers.
[ numbers are 40 , 80 , 60 . [Ans.]
Q.22. ABC is a right-angledtriangle of given areaS. Find the sides of
the triangle for whichthe area of the circumscribedcircle is least.
Solution:
Q.24. A box is to be constructedfrom a square metal sheet of side 60
cm by cutting out identical squares from the four corners and turning
up the sides. Find the lengthof the side of the square to be cut out so
that the box has maximum volume.
Solution:

Q.25. Find the shortest distance of the point C (0.c) from the parabola
y = x2, c > 1/2.
Solution:
Let P(x,y) be any point on the given parabolay = x2, then
| CP | = √{(x – 0)2 + (y – c)2} = √{y + (y – c)2} [ writing y for x2 as, y = x2]
=√{y2 – (2c – 1)y + c2}.
Or, | CP |2 = y2 – (2c – 1)y + c2
Now, | CP | is the shortest ifand only if | CP | 2 is the shortest.
Writing, | CP |2 as f(y),we get
f(y) = y2 – (2c – 1)y + c2 ------------ (i)
f’(y) = 2y – (2c – 1) and f”(y) = 2.
Now, f’(y) = 0 => 2y – (2c – 1) = 0
Or, y = (2c – 1)/2.
Hence, f”{(2c – 1)/2}= 2 > 0.
Therefore,f(y) is minimum when y = (2c – 1)/2
i.e. | CP | is minimum when y = (2c – 1)/2
and the minimum value of| CP | = √[{(2c – 1)/2} + {(2c – 1)/2– c}2]
= √[(2c – 1)/2 + 1/4]
= √[(4c – 1)/2]. [Ans.]
Q.26. An enemy vehicle is moving along the curve y = x2 + 2. Find the
shortest distance between the vehicle and our artillery locatedat (3,
2). Also find the co-ordinates ofthe vehicle when the distance is
shortest.
Solution:

[when x = 1, y = 12 + 2 = 3. Thus the co-ordinates of the vehicle when
the distance is the shortest are (1, 3). [Ans.]
Q.27. Given the sum of the perimeters ofa square and a circle, that the
sum of their areas is least when the side of the square is equal to the
diameter of the circle.
ASSIGNMENT(continuity & differentiability) (XII)
**Question 1 Determine a and b so that the function f given by
f(x) =
𝟏−𝒔𝒊𝒏²𝒙
𝟑𝒄𝒐𝒔²𝒙
, x<п/2
=a, x=п/2
=
𝒃(𝟏−𝒔𝒊𝒏𝒙)
(п−𝟐𝒙)²
, x>п/2
Is continuous at x=п/2.
Answer [a = 1/3 , b = 8/3]
**Question 2 Find k such that following functions are continuous
at indicated point
(i) f(x) ={
𝟏−𝒄𝒐𝒔𝟒𝒙
𝟖𝒙²
, 𝒙 ≠ 𝟎
𝒌 , 𝒙 = 𝟎
at x=0
(ii) f(x) = (2x+2 - 16)/(4x – 16) , x≠2
= k, x = 0 at x=2.
Answer [ (i) k=1,(ii) k=1/2]
**Question 3 The function f is defined as
{
𝒙² + 𝒂𝒙 + 𝒃 , 𝟎 ≤ 𝒙 < 2
𝟑𝒙 + 𝟐 , 𝟐 ≤ 𝒙 ≤ 𝟒
𝟐𝒂𝒙 + 𝟓𝒃 , 𝟒 < 𝑥 ≤ 8

If f(x) is continuous on [0,8], find the values of a and b.
Answer [a=3,b=-2]
** Question 4 If f(x) = {
√ 𝟏+𝒑𝒙− √ 𝟏−𝒑𝒙
𝒙
, −𝟏 ≤ 𝒙 < 0
𝟐𝒙+𝟏
𝒙−𝟏
, 𝟎 ≤ 𝒙 ≤ 𝟏
is continuous
in the [-1,1], find p.
Answer [p=-1]
**Question 5 Find the value of a and b such that the f(x) defined as
f(x) = {
𝒙 + 𝒂√ 𝟐𝒔𝒊𝒏𝒙 , 𝟎 ≤ 𝒙 < п/4
𝟐𝒙𝒄𝒐𝒕𝒙 + 𝒃 , п/𝟒 ≤ 𝒙 ≤ п/𝟐
𝒂𝒄𝒐𝒔𝟐𝒙 − 𝒃𝒔𝒊𝒏𝒙 ,
п
𝟐
< 𝑥 ≤ п
is continuous for
all values of x in [0,п].
ANSWER [a=п/6 , b=-п/12]
** Question 6 Prove that 𝐥𝐢𝐦
𝒙→𝝅/𝟒
𝒕𝒂𝒏 𝟑 𝒙−𝒕𝒂𝒏𝒙
𝐜𝐨𝐬(𝒙+
√ 𝝅
𝟒
)
= -4
[ Hint: Nr. Can be written as tanx(tanx-1)(tanx+1) =- [tanx(cosx-
sinx)(tanx+1)]/cosx
Cosx-sinx = √𝟐 cos(𝒙 +
√ 𝝅
𝟒
) ]
**Question 7 Prove that (i) 𝐥𝐢𝐦
𝒙→
𝟏
√𝟐
𝒙−𝐜𝐨𝐬(𝒔𝒊𝒏−𝟏 𝒙)
𝟏−𝐭𝐚𝐧(𝒔𝒊𝒏−𝟏 𝒙)
= −
𝟏
√𝟐
[ Hint: put
x= sinѲ]
(ii) 𝐥𝐢𝐦
𝒙→∞
𝒙(𝒕𝒂𝒏−𝟏 𝒙+𝟏
𝒙+𝟐
−
𝝅
𝟒
) = -3/2. [Hint:
𝝅
𝟒
= 𝒕𝒂𝒏−𝟏
𝟏 & use
formula of 𝒕𝒂𝒏−𝟏
𝒙 − 𝒕𝒂𝒏−𝟏
𝒚 ]
Question 8 f(x) =
𝒂𝒙 𝟐+𝒃
𝒙 𝟐+𝟏
, 𝐥𝐢𝐦
𝒙→𝟎
𝒇(𝒙) =1 & 𝐥𝐢𝐦
𝒙→∞
𝒇(𝒙) =1, then p.t. f(-
2)=f(2)=1. [ Hint: 𝐥𝐢𝐦
𝒙→∞
𝟏
𝒙 𝟐 =0]

Question 9 𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒙−𝟏
√ 𝟏−𝒄𝒐𝒔𝒙
[Dr. = √2|sinx/2| &𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒙−𝟏
𝒙
=1
|sinx/2| =+ve & -ve as x→0+ & x→0- , ⇨ limit does not exist]
Question 10 Show that the function
f(x)=
{
𝒔𝒊𝒏𝟑𝒙
𝒕𝒂𝒏𝟑𝒙
, 𝒙 < 0
𝟑
𝟐
, 𝒙 = 𝟎
𝐥𝐨𝐠(𝟏+𝟑𝒙)
𝒆 𝟐𝒙−𝟏
, 𝒙 > 0
is continuous at x=0.
[Hint: use 𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒙−𝟏
𝒙
=1 , 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐨𝐠(𝟏+𝒙)
𝒙
=1]
Question11 Show that f(x) = |x-3|,x∊R is cts. But not diff. at x=3.
[Hint:showL.H.lt=R.H.lt by |x-3| = x-3, if x ≥3 and –x+3, if x<3, L.hd=-
1≠1(R.h.d)
QUESTION 12 Discuss the continuity of the fn. f(x) = |x+1|+|x+2|,
at x = -1 & -2 [Hint:f(x) = {
−𝟐𝒙 − 𝟑, 𝒘𝒉𝒆𝒏 𝒙 < −2
𝟏, 𝒘𝒉𝒆𝒏 − 𝟐 ≤ 𝒙 < −1
𝟐𝒙 + 𝟑, 𝒘𝒉𝒆𝒏 𝒙 ≥ −𝟏
yes cts. At x=-1,-2
Question 13 Find the values of p and q so that f(x)
={
𝒙² + 𝟑𝒙 + 𝒑, 𝒊𝒇 𝒙 ≤ 𝟏
𝟐𝒙 + 𝟐, 𝒊𝒇 𝒙 > 1
is diff. at x = 1. [ answer is p=3 , q=5]
Question 14 For what choice of a, b, c if any , does the function
F(x) = {
𝒂𝒙² + 𝒃𝒙 + 𝒄, 𝟎 ≤ 𝒙 ≤ 𝟏
𝒃𝒙 − 𝒄, 𝟏 < 𝑥 ≤ 2
𝒄, 𝒙 > 2
becomes diff at x=1,2 & show that
a=b=c=0.
Question15For what values a,b f(x)={
𝒆 𝟐𝒙
− 𝟏, 𝒘𝒉𝒆𝒏 𝒙 ≤ 𝟎
𝒂𝒙 +
𝒃𝒙²
𝟐
, 𝒘𝒉𝒆𝒏 𝒙 > 0
is diff.at
x=0
[Hint:L.H.d= 2 𝒃𝒚 𝒖𝒔𝒊𝒏𝒈 𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒙−𝟏
𝒙
=1& R.H.d=a, since f‘(x)=0exists, a=2,b∊R]
Q. 16 Discuss the diff. Of f(x) = | x-1| + |x-2|

[ Hint: we have f(x)= {
−𝟐𝒙 + 𝟑, 𝒙 < 1
𝟏 , 𝟏 ≤ 𝒙 < 2
𝟐𝒙 − 𝟑, 𝒙 ≥ 𝟐
to be examined not diff. At x=1,2]
ASSIGMENT OF DIFFERENTITION
Question 1 Show that y = aex
and y = be –x
cut at right
angles ab=1 [ by equating ,we get ex
= √
𝒃
𝒂
⇨ x= ½ log ( b/a) ,
find slopes(dy/dx)at pt. of intersection is (½ log ( b/a , √ 𝒂𝒃).
Question 2 (i) If y√ 𝟏 − 𝒙² + x√𝟏 − 𝒚² = 1, prove that
dy/dx= (-1)√
𝟏−𝒚²
𝟏−𝒙²
[Hint: put y=sinѲ & x= sin𝝋 , use formula of sin(𝜽 + 𝝋)]
(ii) If cos-1
(
𝐱²−𝐲²
𝐱²+𝐲²
) = tan-1
a , find dy/dx.
[let cos(tan-1
a )= k(constant), then assume c= 1-k/1+k , dy/dx= y/x]
(iii) If 𝒚 𝒙
= 𝒆 𝒚−𝒙
, prove that dy/dx =
( 𝟏+𝒍𝒐𝒈𝒚)²
𝒍𝒐𝒈𝒚
(iv) If xm
.yn
= (x+y)m+n
, then find dy/dx. [ y/x]
Question 3 Differentiate w.r.t. x :
**(i) Using logarithmic differentiation, differentiate:
Solution:

(ii) 𝒙 𝒕𝒂𝒏𝒙
+ √
𝐱²+𝟏
𝐱
(iii) (iogx)x
+ xlogx
Question 4 (i) If 𝒚 𝒙
= 𝒆 𝒚−𝒙
, prove that dy/dx =
( 𝟏+𝒍𝒐𝒈𝒚)²
𝒍𝒐𝒈𝒚
(ii) If f(1)= 4,f’(1)=2,find d/dx{logf(ex
)} at the point x =0.[1/2]
(iii)If y = √ 𝒙 + √ 𝒙 + √ 𝒙 + ⋯ … . ∞ ,show that (2y – 1)dy/dx =1.
(iv) If x = (t+1/t)
a
, y= a
(t+1/t)
where a>0,a≠1,t≠0, find dy/dx.
[Hint: take dy/dt & dx/dt , then find dy/dx = ylogy/ax. ]
Question5
(i)differentiate: Sec-1
(1/(2x2
– 1)),w.r.t.sin-1
(3x –4x3
).
[Hint: let u=1st
fn. & v= 2nd
fn. , find du/dv = 1]
(ii)differentiate: tan-1
(
√ 𝟏+𝒙²−𝟏
𝒙
),w.r.t. sin-1
(
𝟐𝒙
𝟏+𝒙²
) if -
1<x<1;x≠0
[ du/dv= ¼, put x=tanѲ⇨ u=Ѳ/2, v=2Ѳ , u&v as assumed above]
(iii) If y = e
(msin-1x)
, show that (1-x2)y2 – xy1 – m2y= 0.
Question 6 Water is driping out from a conical funnel, at the
uniform rate of 2cm3
/sec. through a tiny hole at the vertex at the
bottom. When the slant height of the water is 4cm.,find the rate of
decrease of the slant height of the water given that the vertical
angle of the funnel is 1200
.
[Hint: Let l is slant height ,V = 1/3.𝝅 .l(√𝟑/2)2
.l/2= 𝝅l3
/8(vertical
angle will be 600
(half cone), take dv/dt=-2cm3
/sec. ⇨l=-1/3𝝅
cm/s.]
**Question 7(i) Let f be differentiable for all x. If f(1)=-2 and if f `(x)
≥2 ∀ x∊[1, 6], then prove f(6) ≥8.[ use L.M.V.Thm.,f`(c)≥2,c∊[1, 6]]
(ii) If the function f(x)= x3
– 6x2
+ax+b defined on [1, 3] satisfies
the rolle’s theorem for c = (2√𝟑 +i)/ √𝟑 , then p.t. a = 11 & b∊R.
[Hint: Take f(1)=f(3) , use rolle’s thm. f`(c)=0⇨ a=11]
Question 8 (i) Show that f(x)= x/sinx is increasing in (0, п/2)
[HINT: f’(x)>0 , tanx >x]
(ii) Find the intervals of increase and decrease for f(x) = x3
+ 2x2
– 1.
[Answer is increasing in (-∞, -4/3)U(0, ∞) & decreasing in (-4/3, 0)]
(iii) Find the interval of increase & decrease for f(x) =log(1+x)-

(x/1+x)
OR
Prove that x/1+x < log(1+x) < x for x > 0.
[ Hint: f(x)strictly ↑ in [0, ∞) , x>0 ⇨f(x)>f(0), let g(x)=x-log(1+x)
g(x)>0 ↑ in [0,∞) & f(x) ↓ in (-∞, 0].]
(iv) For which value of a , f(x)=a(x+sinx)+a is increasing.
[Hint: f’(x) a(1+cosx) ≥0 ⇨ a>0 ∵ -1≤cosx≤1]
**Question 9 Problem:Using differentials, approximate the expression
Solution: We let
Hence, x = 0.05 and y = /4.
Differentiating, we obtain
Substituting, we get
Question10 For the curve y = 4x3
− 2x5
, find all the points at whichthe
tangents passes throughthe origin.
[Hint: eqn. Of tangent at (x0,y0) , put x,y=0,(x0,y0)lies on given
curve]

Question 11 Find the stationary points of the function f(x) = 3x4 –
8x3
+6x2
and distinguish b/w them. Also find the local max. And
local mini. Values, if they exist.
[ f’(x)=0⇨ x=0,1 f has local mini. At x=0∵f’’>0 & f’’(1)=0, f has
point of inflexion at x=1,f(1)=1]
Question 12 Show that the semi – vertical angle of right circular
cone of given total surface area and max. Volume is sin-1
1/3.
[Hint: take S=Пr(l+r) ⇨ l= S/пr – r , take derivative of V² OR can use
trigonometric functions for l & h]
Question 13 A window has the shape of a rectangle surmounted by
an equilateral ∆. If the perimeter of the window is 12 m., find the
dimensions of the rectangle so that it may produce the largest area
of the window.
[Hint: let x=length, y=breadth, then y=6 – 3y/2, A= xy+√ 𝟑x2
/4, take
derivative of A & it is max. ,x=4(6+√ 𝟑)/11 ,y=6(5−√ 𝟑)/11]
Application ofDerivatives

Q. 1. The volume of a cube is increasing aa constant rate. Prove that
the increase in surface area varies inversely as the lengthof the edge
of the cube.
Q. 2. Use differentials to find the approximate value of
Q. 3. It is given that for the function f(x) = x3 – 6x2 + ax + b on [1, 3],
Rolle’s theorem holds withc = 2+ . Find the values of a and b if
f(1)= f(3) = 0
Q. 4. Find a point on the curve y = (x – 3)2, where the tangent is
parallel to the line joining (4, 1) and (3, 0).
Q. 5. Find the intervals in which the function f(x) = x4 – 8x3 + 22x2 –
24x + 21 is decreasing or increasing.
Q. 6. Find the local maximum or local minimum of the
function.
Q. 7. Find the point on the curve y2 = 4x which is nearest to the point
(2, 1).
Q. 8. A figure consists ofa semi-circle witha rectangle on its diameter.
Given the perimeter of the figure,find its dimensions in order that the
area may be maximum.
Q. 9. A balloonwhich always remainspherical has a variable
diameter . Find the rate ofchange of its volume with respect to
x.
Q. 10. Find the intervals in which f(x) = (x+1)3 (x – 3)3 is strictly
increasing or decreasing.
Q. 11. Prove that the curves x = y2 and xy = k cut at right angles if 8k2 =
1
Q. 12. Using differentials, find the approximate value of(26.57)1/3
Q. 13. Showthat of all the rectangles inscribedin a givenfixed circle,
the square has the maximum area.

Q. 14. Find the equation ofthe tangent and normal to the hyperbola
at the point (x0,y0)
Q. 15. Find the intervals ofthe function is
strictly increasing or strictly decreasing.
Q. 16. An open topped box is to be constructedby removing equal
squares from each corner of a 3 metre by 8 metre rectangular sheet of
aluminium and folding up the sides.Find the volume of the largest
such box.
Q. 17. Prove that the volume ofthe largest cone that can be inscribed
in a sphere of radius R is of the volume of the sphere.
Q. 18. Showthat the right circular cylinder ofgiven surface area and
maximum volume is such that its height is equal to the diameter of the
base.
Q. 19. The sum ofthe perimeter of a circle,and square is k, where k is
some constant.Prove that the sum of their areas is least when the side
of square is double the radius of the circle.
Q. 20. A window is in the form of a rectangle surmoundedby a
semicircular opening.The total perimeter of the windowis 10 m. Find
the dimensions of the window to admit maximum light throughthe
whole opening.
Q. 21. Sand is pouring from a pipe at the rate of 12 cm3/s.The falling
sand forms a cone on the groundin such a way that the height of the
cone is always one-sixth of the radius of the base. Howfast is the
height of the sand cone increasing when the height is 4 cm?
Q. 22. For the curve y = 4 x3– 2 x5 , find all the points at which the
tangent passes throughthe origin.
Q. 23. An Apache helicopter ofenemy is flying along the curve given
by y = x2+ 7. A soldier,placedat (3, 7), wants to shoot down the
helicopter when it is nearest to him. Find the nearest distance.

Q. 24. A rectangular sheet of tin 45 cm by 24 cm is to be made into a
box without top, by cutting off square from each corner and folding up
the flaps. What should be the side of the square to be cut off so that the
volume of the box is maximum ?
Q. 25. A wire of length28 m is to be cut into two pieces. One of the
pieces is to be made into a square and the other into a circle.What
should be the lengthof the two pieces so that the combinedarea of the
square and the circle is minimum?
Q. 26. A tank withrectangular base and rectangular sides, open at the
top is to be constructedso that its depth is 2 m and volume is 8 m3. If
building of tank costs Rs 70 per sqmetres for the base and Rs 45 per
square metre for sides.What is the cost of least expensive tank
Q. 1. If y = x4 - 10 and if x changes from 2 to 1.99, what is the
approximate change in y.
Q. 2. A circular plate expands under heating so that its radius
increases by 2%.Find the approximate increase in the area of the
plate if the radius of the plate before heating is 10 cm.
Q. 3. Find the approximate value of f(3.02) when f(x) = 3x2+5x+3.
Using differentials find the approximate value of
Q. 4.
Q. 5.
Q. 6. tan46o. given 1o = 0.01745 radians.
Answers
1. 5.68
2. 4p
3. 45.46
4. 5.02
5. 0.1925
6. 1.03490

Q. 1. Find intervals in which the function given
by is
(a) strictly increasing (b) strictly decreasing.
Q. 2. Showthat the function f given by f (x) = tan–1(sinx + cos x), x > 0 is
always an strictly increasing functionin
Find the intervals in which the following functions are increasing or
decreasing
Q. 1. f(x) = -x2 - 2x +15.
Q. 2. f(x) = 2x3 +9x2 +12x +20
Q. 3. (x+1)3(x-3)3.
Q. 4. x4 - .
Q. 5. f(x) sin3x.
Q. 6. f(x) = sinx + cosx.
Q. 7. f(x) sin4x + cos4 x on [0, p/2]
Q. 8. f(x) = log(1+x) - .
Answers
1.
2.
3.
4.
5.
6.
7.
8.

Q. 1. Find the maximum slope ofthe curve y = -x3 + 3x2+ 2x - 27. and
what point is it
Q. 2. A right circular cone of maximum volume is inscribedin a sphere
of radius r. find its altitude. Also show that the maximum volume of
the cone is 8/27 times the volume of the sphere.
Q. 3. A point on the hypotenuse of a triangle is at a distance a and b
from the sides of the triangle.Show that the maximum lengthof the
hypotenuse is
Q. 4. From a piece of tin 20cm. in square, a simple box without top is
made by cutting a square from each corner and folding up the
remaining rectangular tips to form the sides of the box. What is the
dimensionof the squares is cut in order that the volume of the box is
maximum.
Q. 5. If lengthof three sides of a trapezium other than base are equal
to 10cm, then find the area of the trapezium when it is maximum.
Q. 6. Find the shortest distance ofthe point (o,c) from the parabolay =
x2, where 0 £ x £ 5.
Q. 7. A window consists ofa rectangle surmountedby a semicircle.If
the perimeter ofthe window is p centimetres,showthat the window
will allow the maximum possible light when the radius of the semi
circles cm.
Q. 8. Showthat the semi vertical angle of the cone of given surface
area and maximum volume is .
Q. 9. A wire oflength a is cut into two parts which are bent
respectively inthe form ofa square and a circle.Showthat the least
value of the areas so formedis .
Q. 10. Showthat the volume of the greatest cylinder which can be
inscribedin a cone of height h and semi vertical angle a is

Q. 11. An open tank with square base and vertical sides is to be
constructedfrom metal sheet so as to hold a given quantity ofwater.
Showthat the cost of material will be least when the depth of the tank
is half the width.
Q. 12. Find the areaof the greatest isosceles triangle that canbe
inscribedin a givenellipse
having its vertex coincident with one end of the major axis.
Q. 13. Find the maximum and minimum points for the following:
i.
ii.
iii.
Q. 14. The sectionof a window consists ofa rectangle surmountedby
an equilateral triangle. Ifthe perimeters be givenas 16m. find the
dimensions of the window in order that the maximum amount of light
may be admitted.
Q. 15. A square tank of capacity 250 cu.m has to be dug out. The cost of
land is Rs.50.per sq.m. The cost ofdigging increases withthe depth
and for the whole tank is 400(depth)2 rupees. Find the dimensions of
the tank for the least total cost.
Q. 16. Find the dimensions ofthe rectangle of greatest areathat can be
inscribedin a semi circle ofradius r.
Q. 17. A running trackof 440 ft is to be laid out enclosing a football
field, the shape of which is rectangle with semicircle at each end. Ifthe
area of the rectangular portionis to be maximum find the length of
the sides.
Q. 18. Find the maximum and minimum values ofy = |4-x2|, -3 £ x £ 3.
Also determine the greatest and least values.

Q. 1. Whether
.
If so find the point of contact.
Q. 2. Find points at which the tangent to the curve y = x3 – 3x2– 9x + 7 is
parallel to the x-axis.
Q. 3. Showthat the tangents to the curve y = 7x3 + 11 at the points
where x = 2 andx = – 2 are parallel.
Q. 4. Find the points on the curve y = x3 at which the slope of the
tangent is equal to the y-coordinate ofthe point.
Q. 5. Find the equationof the normals to the curve y = x3 + 2x + 6 which
are parallel to the line x + 14y + 4 = 0.
Q. 6. Find the equationof the tangent to the curve which is
parallel to the line 4x - 2y + 5 = 0
Q. 7. Find the equationof the tangent to
.
Q. 8. Find the equationof the tangent and normal to
Q. 9. Find the tangent and normal to
Q. 10. Showthat the normal at q to x = acosq+aqsinqand y = asinq-
aqcosqis at constant distance from the origin.
Q. 11. Find the equation ofthe normal to x3 +y3 = 8xy where it meet
y2 = 4x other than the origin.

Q. 12. Showthat touches at the point where the curve
crosses y-axis.
Q. 13. Find the angle of intersectionof the curves xy = a2
and
Q. 14. Find the equation(s) ofnormal(s) to the curve 3x2 - y2 = 8 which
is (are) parallel to the line x+3y = 4.
Ans: x+3y-8 = 0 and x+3y+8 = 0}
Q. 15. For the curve y = 4x3 -2x5, find all the points at which the
tangent passes throughthe origin.
Ans: (0,0),(1,2),(-1,-2)
Q. 16. Prove that the sum of intercepts ofthe tangent to the
curve with the co-ordinate axes is constant.
ASSIGNMENT OF INTEGRATION
Question1 Evaluate:(i)** Integrate .[Use the power
substitution
Put ]
 ** (iii) Integrate . [ Use the power substitution
 Put ]
 (iii) ∫ 𝒔𝒆𝒄𝒙 𝒕𝒂𝒏 𝟑𝝅/𝟒
𝟎
𝒙 𝒅𝒙 [answer is (2 - √2)/3 ]
(iv) ∫ dx [multiply&divide by sin(a-b)] (v)∫ √
𝟏−√ 𝒙
𝟏+√ 𝒙
dx
[multiply & divide by √ 𝟏 − √ 𝒙 ] (Vi)∫
𝒙
𝒙 𝟑−𝟏
dx [by partial fraction]

(v)∫
(𝒙−𝟒)𝒆 𝒙
(𝒙−𝟐) 𝟑 dx [ use ∫ex
(f(x)+f’(x))dx] (vi)∫
𝐝𝐱
𝟑+𝟐𝐬𝐢𝐧𝐱+𝐜𝐨𝐬𝐱
𝛑/𝟐
𝐨
dx [put sinx=
𝟐𝒕𝒂𝒏𝒙/𝟐
𝟏+𝒕𝒂𝒏²𝒙/𝟐
, cosx=
𝟏−𝒕𝒂𝒏²𝒙/𝟐
𝟏+𝒕𝒂𝒏²𝒙/𝟐
, thenputt=tanx/2. Answeris 𝐭𝐚𝐧−𝟏
𝟐 –п/𝟐]
(vii) ∫ |𝒙𝒄𝒐𝒔𝝅𝒙|
𝟑/𝟐
𝟎
dx [∫ |𝒙𝒄𝒐𝒔𝝅𝒙|
𝟏/𝟐
𝟎
+ ∫ |𝒙𝒄𝒐𝒔𝝅𝒙|
𝟑/𝟐
𝟏/𝟐
= ∫+ve dx+∫ -ve dx ,
answer is 5/2п- 1/п2
] (viii) [ write sin2
x =1-cos2
x answer is
п/6](ix) ∫ √ 𝒕𝒂𝒏𝒙
𝝅/𝟐
𝟎
+ √ 𝒄𝒐𝒕𝒙 dx [ answer is √2𝝅] (x) ∫ 𝐬𝐢𝐧−𝟏
√
𝒙
𝒂+𝒙
𝒂
𝟎
dx [ put
x=atan2
Ѳ , answeris a/2(п-2)] (xi) ∫
𝒙
𝟏+𝒔𝒊𝒏²𝒙
𝝅
𝟎
dx[ useproperty∫ 𝒇( 𝒙)
𝒂
𝟎
dx
= ∫ 𝒇( 𝒂 − 𝒙)
𝒂
𝟎
dx , ∫ 𝒇( 𝒙)
𝟐𝒂
𝟎
dx=𝟐 ∫ 𝒇( 𝒙)
𝒂
𝟎
dx∵f(2a-x) =f(x), thenputt=tanx,
answer isп²/2√2] (xii) ∫ 𝒇( 𝒙)
𝟎
−𝟓
dx, wheref(x)=|x|+|x+2|+|x+5|.
[∫ (−𝒙 + 𝟑)
−𝟐
−𝟓
dx+ ∫ (𝒙 + 𝟕)
𝟎
−𝟐
dx, answer is31.5] (xiii) Evaluate ∫
𝒆 𝒙( 𝟏−𝒙)²
( 𝒙²+𝟏)²
dx
[use ∫ 𝒆 𝒙
(f(x)+f’(x))dx
Question2 Usingintegration, findtheareaof theregions: (i) { (x,y): |x-1|
≤y ≤√ 𝟓 − 𝒙² }
(ii) {(x,y):0≤y≤x2+3;0≤y≤2x+3; 0≤x≤3}
[(i) A= ∫ √ 𝟓 − 𝒙²
𝟐
−𝟏
dx-∫ (−𝒙 + 𝟏)
𝟏
−𝟏
dx- ∫ (𝒙 − 𝟏)
𝟐
𝟏
dx= 5/2[ 𝐬𝐢𝐧−𝟏
(
𝟐
√𝟓
)
+𝐬𝐢𝐧−𝟏
(
𝟏
√𝟓
)] – ½ ] [(ii) 𝑨 = ∫ (𝒙² + 𝟑
𝟐
𝟎
) dx+∫ (𝟐𝒙 + 𝟑)
𝟑
𝟐
dx, answeris 50/3]
(iii) Findtheareaboundedby thecurvex 2 =4y & thelinex = 4y– 2.
[A = ∫
𝒙+𝟐
𝟒
𝟐
−𝟏
dx - ∫
𝒙²
𝟒
𝟐
−𝟏
dx= 9/8sq. Unit.]
**(iv)Sketchthegraph of f(x)= {
| 𝐱 − 𝟐| + 𝟐, 𝐱 ≤ 𝟐
𝐱² − 𝟐, 𝐱 > 2
,evaluate∫ 𝒇(𝒙)
𝟒
𝟎
dx
[hint: ∫ 𝒇(𝒙)
𝟒
𝟎
dx= ∫ (𝟒 − 𝒙)
𝟐
𝟎
dx+ ∫ (𝒙² − 𝟐)
𝟒
𝟐
dx=62/3.]
**Question3 evaluate ∫
√𝟏+𝒙
√𝒙
dx [ mult. & divideby √ 𝟏 + 𝒙 , put1+x
=A.(d/dx)(x2+x)+B,findA=B=1/2, integrate]

Definite integral as the limit of a sum , use formula : ∫ 𝒇( 𝒙)
𝒃
𝒂
dx
𝐥𝐢𝐦
𝒉→𝟎
∑ 𝒇(𝒂 + 𝒓𝒉)𝒏
𝒓=𝟏 , where nh=b-a & n→∞ Question 4 Evaluate
( 𝒊) ∫ (𝒙 + 𝒆 𝒙𝟒
𝟎
) dx (ii) ∫ (𝒙² − 𝟐𝒙 + 𝟐)
𝟑
𝟎
dx
[ use 𝐥𝐢𝐦
𝒉→𝟎
𝒆 𝒉−𝟏
𝒉
= 1 forpart(i) , useformulasof specialsequences, answer is
6]
Some special case :
(1) Evaluate: ∫
𝒅𝒙
(𝒙−𝟑)√ 𝒙+𝟏
[ put x+1=t²] (2) ∫
𝒅𝒙
(𝒙²−𝟒)√ 𝒙+𝟏
[ put x+1 = t² ]
(3) Evaluate: ∫
𝒅𝒙
(𝒙+𝟏)√ 𝒙²−𝟏
(4) Evaluate: ∫
𝒅𝒙
𝒙²√ 𝒙²+𝟏
[ put x=1/t for both]
(5) Evaluate: ∫
(𝒙²+𝟏)𝒅𝒙
𝒙 𝟒+𝟏
[ divide Nr. & Dr. By x2
, thenwrite x²+1/x²=(x-1/x)²
+2 according to Nr. , let x-1/x=t]
(6) Evaluate ∫ 𝒙√ 𝟏 + 𝒙 − 𝒙² dx [ let x=A(d/dx) ( 1+x-x²) +B]
(7) Integrating by parts evaluate ∫
𝒙²
( 𝒙𝒔𝒊𝒏𝒙+𝒄𝒐𝒔𝒙)²
= ∫( 𝒙𝒔𝒆𝒄𝒙).
𝒙𝒄𝒐𝒔𝒙
( 𝒙𝒔𝒊𝒏𝒙+𝒄𝒐𝒔𝒙)²
(8) Evaluate ∫
𝟏
𝟏+𝒄𝒐𝒕𝒙
dx =∫
𝒔𝒊𝒏𝒙
𝒔𝒊𝒏𝒙+𝒄𝒐𝒔𝒙
dx [ put
sinx=Ad/dx(sinx+cosx)+B(sinx+cosx)+C
If Nr. Is constant termthenuse formulas of sinx,cosx as Ques. No. 1 (vi) part]
The important discussions in Differential Equations are as follows:

ASSESSMENTOF DIFFERENTIAL EQUATIONS FOR
CLASS—XII Level--1
Q.1 Find the order and degree of the following differential
equations. State also whether they are linear or non-linear.

(i) X2
(
𝒅²𝒚
𝒅𝒙²
)3
+ y (
𝒅𝒚
𝒅𝒙
)4
y4
=0. (ii)
𝒅²𝒚
𝒅𝒙²
= √𝟏 + (
𝒅𝒚
𝒅𝒙
𝟑
)² .
Q.2 Form the differential equation corresponding to y2
= a (b –
x)(b+ x) by eliminating parameters a and b.
Q.3 Solve the differential equation (1+e2x
) dy + (1+y2
) ex
dx = 0,
when x= 0, y =1.
Q.4 Solve the differential equation:
𝒅𝒚
𝒅𝒙
=
𝟏−𝒄𝒐𝒔𝒙
𝟏+𝒄𝒐𝒔𝒙
.
Q.5 Verify that y = A cosx – Bsinx is a solution of the differential
equation
𝒅²𝒚
𝒅𝒙²
+ y = 0.
Answersof Level—1
1. (i) order =2 , degree=3 , non linear( because degree is more
than 1 ) , (ii) order 2 , degree 3 , non-linear .
2. y2
= a(b – x)(b+ x) = a (b2
– x2
), 2y
𝒅𝒚
𝒅𝒙
=-2ax ⇨ y
𝒅𝒚
𝒅𝒙
= -ax, again
differentiate Y
𝒅²𝒚
𝒅𝒙²
+ (
𝒅𝒚
𝒅𝒙
)2 = -a , by using the value of a from above
step , we will get , x{ Y
𝒅²𝒚
𝒅𝒙²
+ (
𝒅𝒚
𝒅𝒙
)2
} = y
𝒅𝒚
𝒅𝒙
.
3.
𝒅𝒚
𝟏+𝒚²
= -
𝒆 𝒙 𝒅𝒙
𝟏+𝒆 𝟐𝒙 , Integrating both sides, we get
𝐭𝐚𝐧−𝟏
𝒚 = - ∫
𝒆 𝒙 𝒅𝒙
𝟏+𝒆 𝟐𝒙 , put ex
= t⇨ 𝐭𝐚𝐧−𝟏
𝒚 = - 𝐭𝐚𝐧−𝟏
𝒕 +c
Using x=0, y=1, we have y = 1/ex.
4. y = 2 tan(x/2) – x +c , put tan(x/2) =
𝟏−𝒄𝒐𝒔𝒙
𝟏+𝒄𝒐𝒔𝒙
5.
𝒅𝒚
𝒅𝒙
= - A sinx – B cosx ,
𝒅²𝒚
𝒅𝒙²
= - A cosx + B sinx = -y.
Level---2
Q.1 Solve: y dx + x log (
𝒚
𝒙
) dy – 2x dy = 0 .
Q. 2 Which of following transformations reduce the differential
equation
𝒅𝒛
𝒅𝒙
+
𝒛
𝒙
𝐥𝐨𝐠 𝒛 =
𝒛
𝒙²
( 𝐥𝐨𝐠 𝒛)² into the form
𝒅𝒖
𝒅𝒙
+ P(x) u = Q(x) ?
(i) u = 𝐥𝐨𝐠 𝒙 (ii) u = ex
(iii) u = (𝐥𝐨𝐠 𝒛)-1
(iv) u = (𝐥𝐨𝐠 𝒛)2

Q.3 Solve:
𝒅𝒚
𝒅𝒙
+xy = xy3
Q.4 Solve:
𝒅𝒚
𝒅𝒙
= 𝐜𝐨𝐬(𝒙 + 𝒚) + 𝐬𝐢𝐧(𝒙 + 𝒚)
Q.5 Solve:
𝒅𝒚
𝒅𝒙
+ x 𝐬𝐢𝐧 𝟐𝒚 = x3
cos2
y
Answersof Level ---2 1. put x = vy , answer = 1+ log (
𝒚
𝒙
) = ky .
2. (iii) differentiate w.r.t. x
𝒅𝒖
𝒅𝒙
= -
𝟏
(𝐥𝐨𝐠 𝒛)²
.
𝟏
𝒛
𝒅𝒛
𝒅𝒙
, put the value of
𝒅𝒛
𝒅𝒙
in the given differential equation. 3. put
𝟏
𝒚²
= t, answer is
𝟏
𝒚²
= 1
+ c𝒆 𝒙²
. 4. put x+y = v, answer is 𝐥𝐨𝐠(𝟏 +
𝐭𝐚𝐧( 𝒙+𝒚)
𝟐
) = x + c.
5. put tan y = v, I.F. = 𝒆 𝒙²
, also use 𝐬𝐢𝐧 𝟐𝒚 = 2siny cosy.
Order of a Differential Equation.
Q.1. Write the order and degree of the following differential equation :
(dy/dx)4 + 3yd2y/dx2 = 0.
Solution:
We have, (dy/dx)4 + 3yd2y/dx2 = 0
Order of the differential equation = order of highest derivative = 2.
[Ans.]
Degree of the differential equation= degree of highest derivative = 1.
[Ans.]
Q.2. Write the order and degree of the differential equation :
(d2y/dx2)2 + (dy/dx)3 + 2y = 0
Solution:
We have, (d2y/dx2)2 + (dy/dx)3+2y = 0
Order = 2, Degree = 2. [Ans.]
Q.1. (i)Verify that y = A cos x – B sin x is a solutionofthe differential
equationd2y/dx2 + y = 0.

(ii) Form the differential equation of the family of curves y = a sin (x +
b), where a and b are arbitrary constants.
Q.2. Form the differential equationof the family ofcircles having
centre on x-axis and passing throughthe origin.
Q.3. Verify that y = 3 cos (log x) + 4 sin (log x) is a solutionof the
differential equation,
x2 d2y/dx2 + x dy/dx + y = 0.
[Hence, y = 3 cos (log x) + 4 sin (log x) is a solutionofx2 d2y/dx2 + x
dy/dx + y = 0.
[Proved.]
9.4. Differential Equations with Variables Separable.
Q.1. Solve the following differential equation :
dy/dx = log (x + 1).
[ (x + 1) log (x + 1) – x + c. [Ans.]
dy/dx = ex+yx+y + x2.ey.
[– ye–y = ex + x3/3+ c => ex + ey + x3/3+ c = 0 [Ans.]
x(1 + y2)dx – y(1 + x2)dy = 0, given that y = 0 when x = 1.
[ x2 – 2y2 – 1 = 0 [Ans.]
Q.4. Solve the following differential equation : (1 + e2x)dy + (1 + y2)ex
dx = 0.
tan-1y + tan -1ex = c. [Ans.]
9.5. Homogeneous Differential Equations.
Q.1. Solve the following differential equation : (y2 – x2) dy = 3xy dx.
[ – 1/4log(y/x) – 3/8log |4 – y2/x2| = log x + c. [Ans.]

Q.2. Solve the following differential equation : 2xy dx + (x2 + 2y2) dy =
0. [3x2y + 2y3 = c. [Ans.]
Q.3. Solve the following differential equation : x dy/dx – y + x tan(y/x)
= 0. [ x sin v = c => x sin(y/x) = c. [Ans.]
Q.4. Solve the following differential equation : (x2 – y2)dx + 2 xydy = 0,
given that y = 1 when x = 1.
Q.6. Solve the following differential equation : x2 dy/dx = y2 +
2xy. Given that y = 1 when x = 1. [ y = x2/(2– x). [Ans.]
Linear Differential Equations.
Q.1. Solve the following differential equation : sin x dy/dx + cos x.y =
cos x.sin2x
[y = (1/3) sin2 x + c cosec x. [Ans.]
Q.2. Solve the following differential equation : dy/dx – y/x = 2x2.
[Thus y = x3 + cx. [Ans.]
Q.3. Solve the following differential equation : dy/dx + (sec x).y = tan
x.
[y = sec x + tan x – x + c. [Ans.]
Q.4. Solve the following differential equation : dy/dx + 2tan x .y = sinx.
[ y = cos x + c cos2x. Ans. ]
Q.6. Solve the differential equation : (1 – x2)dy/dx + xy = ax.
[ y = a + c√(1 – x2). [Ans.]
Q.7. Solve the differential equation : dy/dx + 2y tan x = sin x, giventhat
y = 0 if x = π/3.
[ y = cos x – 2 cos2 x. [Ans.]
Q.8. Solve the following differential equation : cos2 x dy/dx + y = tan x.
[ y = tan x – 1 + ce-tan x . [Ans.]

Q.9. Solve the following differential equation : (x2 + 1) dy/dx + 2xy =
√(x2 + 4).
[ x/2√(x2 +4) + 2log|x + √(x2 + 4)| + c. [Ans.
Area of The Region Bounded by a Curve and a Line.
Q.1. Find the areaof the regionbounded by the parabolax2 = 4y and
the line x = 4y – 2.
Solution:
Q.2. Find the areaof the regionbounded by y2 = 4x, x = 1, x = 4 and x-
axis in the first quadrant.
Solution:
Area of triangle.
Q.1. Using integration,find the areaof the triangle ABC, the
coordinates ofwhose vertices are A(2, 0), B(4, 5) and C(6,3).
Solution:

= 7 sq. units. [Ans.]
Q.2. Using integrationfind the area of the triangular regionwhose
vertices are (1, 0), (2, 2) and (3, 1).
Solution: [Ans. = 3/2]
Q. 1. Evaluate as a limit of a sum.
Q. 2. Evaluate .
Q. 3. Evaluate .
Q. 4. Evaluate the integral .
Q. 5. Evaluate .
Q. 6. Evaluate
Q. 7. Evaluate
Q. 8. Evaluate
Q. 9. Prove that .

Q. 10. Evaluate .
Q. 1. Find the area lying above the x-axis and included between the
circle x2+y2=8x and the parabola y2 =4x.
Q. 2. Find the area lying above the x-axis and included between the
circle x2 +y2=16a2 and the parabola y2 =6ax.
Q. 3. Find the area of the smaller region bounded by the
ellipse and the line
Q. 4. Find the area of the region included between x2 =4y , y = 2 , y = 4
and the y-axis in the first quadrant.
Q. 5. Find the area between the parabolas 4ay = x2 and y2 = 4ax.
Q. 6. Find the area bounded by the curve y2 = 4ax and the line y = 2a
and y-axis.
Q. 7. Find the area bounded by the parabola y2 = 8x and its latus
rectum
Q. 8. Find the area of the circle x2 + y2 = 16, which is exterior to the
parabola y2 = 6x.
Q. 9. Sketch the region common to the circle x2 + y2 = 8 and the
parabola x2 = 4y. Also find the area of the common region using
integration.
Q. 10. Draw the rough sketch of the region
and find the area enclosed by the region using method of integration.
Q. 11. Using integration, find the area of the triangle ABC whose
vertices are A(2,3), B(2,8) and C(6,5).
vertices are A(2,5), B(4,7) and C(6,2).
vertices are A(-1,1), B(0.5) and C(3,2).

Q. 14. Compute the area bounded by the lines x+2y = 2, y-x=1 and 2x+y
= 7.
Q. 15. Compute the area bounded by the lines y = 4x+5, y = 5-x, and 4y
= x+5.
Q. 16. Compute the area bounded by the lines 2x+y = 4, 3x-2y = 6, and
x-3y+5=0.
Q. 17. Using integration, find the area of the region bounded by x-
7y+19=0, and y =çxú.
Q. 18. Using integration, find the area of the region bounded by the
line
i. 2y= -x+8, x-axis and the lines x = 2 and x = 4.
ii. y -1 = x, x-axis and the lines x = -2 and x = 3
iii. y = , line y = x and the positive x- axis.
Q. 19. Find the areaof the regionenclosedbetween the two circles x2 +
y2 = 1 and (x-4)2 + y2 =16.
Q. 20. Find the area of the region in the first quadrant enclosed by the
x-axis, the line y = 4x and the circle x2 + y2 = 32
Q. 21. Find the area of the smaller part of the circle x2 + y2 =a2 cut off
by the line .
Q. 22. Find the area of the region
Q. 23. Sketch the graph of the curve y = and evaluate
Q. 24. Sketch the graph of the curve and find the area bounded by y =
, x=-2, x=3, y=0.
Q. 25. Find the area bounded by the line y = sin2x and y = cos2x
between x = 0 and x=p/4
Q. 26. Prove that the curves y2 = 4x and x2 = 4y divide the area of the
square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

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