Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 9 Applications of Integration Ex 9.10 Textbook Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 9 Applications of Integration Ex 9.10

Choose the most suitable answer from the given four alternatives:

Question 1.
The value of $$\int_{0}^{2/3}$$ $$\frac { dx }{ \sqrt{4-9x^2} }$$
(a) $$\frac { π }{ 6 }$$
(b) $$\frac { π }{ 2 }$$
(c) $$\frac { π }{ 4 }$$
(d) π
Solution:
(a) $$\frac { π }{ 6 }$$
Hint:

Question 2.
The value of $$\int_{-1}^{2}$$ |x| dx
(a) $$\frac { 1 }{ 2 }$$
(b) $$\frac { 3 }{ 2 }$$
(c) $$\frac { 5 }{ 2 }$$
(d) $$\frac { 7 }{ 2 }$$
Solution:
(c) $$\frac { 5 }{ 2 }$$
Hint:
|x| = x if x > 0
= -x if x < 0

Question 3.
For any value of n ∈ Z $$\int_{0}^{π}$$ ecos²x cos3 [(2n + 1) x] dx
(a) $$\frac { π }{ 2 }$$
(b) π
(c) 0
(d) 2
Solution:
(c) 0
Hint:

Question 4.
The value of $$\int_{-π/2}^{π/2}$$ sin² x cos x dx
(a) $$\frac { 3 }{ 2 }$$
(b) $$\frac { 1 }{ 2 }$$
(c) 0
(d) $$\frac { 2 }{ 3 }$$
Solution:
(d) $$\frac { 2 }{ 3 }$$
Hint:
f(x) = sin²x cos x
f(-x) = sin²(-x) cos(-x) = sin²x cos x
f(x) = f(-x)
f is an even function

Question 5.
The value of $$\int_{-4}^{4}$$[tan-1($$\frac { x^2 }{ x^4+1 }$$) + tan-1($$\frac { x^4+1 }{ x^2 }$$)] dx
(a) π
(b) 2π
(c) 3π
(d) 4π
Solution:
(d) 4π
Hint:

Question 6.
The value of $$\int_{-π/4}^{π/4}$$ ($$\frac { 2x^7-3x^5+7x^3-x+1 }{ cos^2x }$$) dx is
(a) 4
(b) 3
(c) 2
(d) 0
Solution:
(c) 2
Hint:

Odd function – 3 Odd function + 7 Odd function – Odd function + even function
= 0 + 2 $$\int_{0}^{π/4}$$ sec² x dx
= 2 [tan x]$$_{0}^{π/4}$$
= 2(1 – 0)
= 2

Question 7.
If f(x) = $$\int_{0}^{x}$$ cos t dt, then $$\frac { df }{ dx }$$ =
(a) cos x – x sin x
(b) sin x + x cos x
(c) x cos x
(d) x sin x
Solution:
(c) x cos x
Hint:
f(x) = $$\int_{0}^{x}$$ cos t dt
u = t, dv = cos t dt
u’ = 1, v = sin t
v1 = -cos t
f(x) = [t sin t + cos t]$$_{0}^{x}$$
f(x) = x sin x + cos x – 1
$$\frac { df }{ dx }$$ = x cos + sin x – sin x
$$\frac { df }{ dx }$$ = x cos x

Question 8.
The area between y² = 4x and its latus rectum is
(a) $$\frac { 2 }{ 3 }$$
(b) $$\frac { 4 }{ 3 }$$
(c) $$\frac { 8 }{ 3 }$$
(d) $$\frac { 5 }{ 3 }$$
Solution:
(c) $$\frac { 8 }{ 3 }$$
Hint:
y² = 4x
y = 2√x

Question 9.
The value of $$\int_{0}^{1}$$ x (1 – x)99 dx is
(a) $$\frac { 1 }{ 11000 }$$
(b) $$\frac { 1 }{ 10100}$$
(c) $$\frac { 1 }{ 10010 }$$
(d) $$\frac { 1 }{ 10001 }$$
Solution:
(b) $$\frac { 1 }{ 10100}$$
Hint:

Question 10.
The value of $$\int_{0}^{π}$$ $$\frac { dx }{ 1+5^{cosx} }$$ is
(a) $$\frac { π }{ 2 }$$
(b) π
(c) $$\frac { 3π }{ 2 }$$
(d) 2π
Solution:
(a) $$\frac { π }{ 2 }$$
Hint:

Adding (1) and (2)

Question 11.
If $$\frac { Γ(n+2) }{ Γn }$$ = 90 then n is
(a) 10
(b) 5
(c) 8
(d) 9
Solution:
(d) 9
Hint:

n² + n = 90
n² + n – 90 = 0
(n + 10) (n – 9) = 0
n = 9

Question 12.
The value of $$\int_{0}^{π/6}$$ cos³ 3x dx is
(a) $$\frac { 2 }{ 3 }$$
(b) $$\frac { 2 }{ 9}$$
(c) $$\frac { 1 }{ 9 }$$
(d) $$\frac { 1 }{ 3 }$$
Solution:
(b) $$\frac { 1 }{ 10100}$$
Hint:
$$\int_{0}^{π/6}$$ cos³ 3xdx = $$\frac { 1 }{ 3 }$$ × $$\frac { 2 }{ 3 }$$ × 1
= $$\frac { 2 }{ 9 }$$

Question 13.
The value of $$\int_{0}^{π}$$ sin4 x dx
(a) $$\frac { 3π }{ 10 }$$
(b) $$\frac { 3π }{ 8}$$
(c) $$\frac { 3π }{ 4 }$$
(d) $$\frac { 3π }{ 2 }$$
Solution:
(b) $$\frac { 3π }{ 8}$$
Hint:
(x) = sin4x dx
f(π – x) = sin4 (π – x) = sin4x

Question 14.
The value of $$\int_{0}^{∞}$$ e-3x x² dx
(a) $$\frac { 7 }{ 27 }$$
(b) $$\frac { 5 }{ 27 }$$
(c) $$\frac { 4 }{ 27 }$$
(d) $$\frac { 2 }{ 27 }$$
Solution:
(d) $$\frac { 2 }{ 27 }$$
Hint:

Question 15.
If $$\int_{0}^{a}$$ $$\frac { 1 }{ 4+x^2 }$$ dx = $$\frac { π }{ 8 }$$ then a is
(a) 4
(b) 1
(c) 3
(d) 2
Solution:
(d) 2
Hint:

Question 16.
The volume of solid of revolution of the region bounded by y² = x(a – x) about the x-axis is
(a) πa³
(b) $$\frac { πa^3 }{ 4 }$$
(c) $$\frac { πa^3 }{ 5 }$$
(d) $$\frac { πa^3 }{ 6 }$$
Solution:
(d) $$\frac { πa^3 }{ 6 }$$
Hint:
y² = x(a – x)
To find limit put y = 0
x (a – x) = 0
x = 0, x = a

Question 17.
If f(x) = $$\int_{1}^{x}$$ $$\frac { e^{sinx} }{ u }$$ du, x > 1 and $$\int_{1}^{3}$$ $$\frac { e^{sinx^2} }{ x }$$ dx = $$\frac { πa^3 }{ 6 }$$ [f(a) – f(1)] then one of the possible value of a is
(a) 3
(b) 6
(c) 9
(d) 5
Solution:
(c) 9
Hint:

Question 18.
The value of $$\int_{0}^{1}$$ (sin-1 x)² dx is
(a) $$\frac { π^2 }{ 4 }$$ – 1
(b) $$\frac { π^2 }{ 4 }$$ + 2
(c) $$\frac { π^2 }{ 4 }$$ + 1
(d) $$\frac { π^2 }{ 4 }$$ – 2
Solution:
(d) $$\frac { π^2 }{ 4 }$$ – 2
Hint:

Question 19.
The value of $$\int_{0}^{a}$$ ($$\sqrt{a^2-x^2}$$)³ dx is
(a) $$\frac { πa^3 }{ 16 }$$
(b) $$\frac { 3πa^4 }{ 16 }$$
(c) $$\frac { 3πa^2 }{ 8 }$$
(d) $$\frac { 3πa^4 }{ 8 }$$
Solution:
(b) $$\frac { 3πa^4 }{ 16 }$$
Hint:

Question 20.
If $$\int_{0}^{x}$$ f(t) dt = x + $$\int_{x}^{1}$$ f(t) dt, then the value of f(1) is
(a) $$\frac { 1 }{ 2 }$$
(b) 2
(c) 1
(d) $$\frac { 3 }{ 4 }$$
Solution:
(a) $$\frac { 1 }{ 2 }$$
Hint:

= 1 + 0 – x f(x)
f(x) + x f(x) = 1
f(x)[1 + x] = 1
when x = 1,
f(1)(2) = 1
f(1) = $$\frac { 1 }{ 2 }$$