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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Choose the correct or the most suitable answer from the given four alternative
Question 1.
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 1
Solution:
(b)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 2

Question 2.
If y = f(x2 + 2) and f'(3) = 5, then \(\frac{d y}{d x}\) at x = 1 is …………….
(a) 5
(b) 25
(c) 15
(d) 10
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 3
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 4

Question 3.
If y = \(\frac{1}{4}\)u4, u = \(\frac{2}{3}\)x3 + 5, then \(\frac{d y}{d x}\) is ……………
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 5
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 6

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Question 4.
If f(x) = x2 – 3x, then the points at which f(x) = f'(x) are …………………..
(a) both positive integers
(b) both negative integers
(c) both irrational
(d) one rational and another irrational
Solution:
(c)
f(x) = x2 – 3x
f'(x) = 2x – 3
Given f(x) = f'(x)
⇒ x2 – 3x = 2x – 3
⇒ x2 – 5x + 3 = 0
x = \(\frac{5 \pm \sqrt{25-12}}{2}=\frac{5 \pm \sqrt{13}}{2}\)
⇒ The roots are irrational

Question 5.
If y = \(\frac{1}{a-z}\), then \(\frac{d z}{d y}\) is ……………….
(a) (a – z)2
(b) -(z – a)2
(c) (z + a)2
(d) -(z + a)2
Solution:
(a)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 7

Question 6.
If y = cos (sin x2), then \(\frac{d y}{d x}\) at x = \(\sqrt{\frac{\pi}{2}}\) is …………..
(a) -2
(b) 2
(c) -2\(\sqrt{\frac{\pi}{2}}\)
(d) 0
Solution:
(d)
y = cos (sin x2)
\(\frac{d y}{d x}\) = – sin (sin x2) [cos (x2)] (2x)
∴ \(\frac{d y}{d x}\) at x = \(\sqrt{\frac{\pi}{2}}\) = -sin (1) [0] = 0

Question 7.
If y = mx + c and f(0) = f'(0) = 1, then f(2) is ………………
(a) 1
(b) 2
(c) 3
(d) -3
Solution:
(c)
y = mx+c
\(\frac{d y}{d x}\) = m
y = x + c (i.e.) f(x) = x + c
y(a tx = 0) = f(0) 0 + c = 1 ⇒ c = 1
y = x + 1 ⇒ f(x) = x + 1
f(2) = 2 + 1 = 3

Question 8.
If f(x) = x tan-1x, then f'(1) is ……………
(a) 1 + \(\sqrt{\frac{\pi}{4}}\)
(b) \(\frac{1}{2}+\frac{\pi}{4}\)
(c) \(\frac{1}{2}-\frac{\pi}{4}\)
(d) 2
Solution:
(b)
f(x) = x tan-1 x
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 8

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Question 9.
\(\frac{d}{d x}\)(ex+5logx) is ……………..
(a) ex.x4 (x + 5)
(b) ex.x (x + 5)
(c) ex + \(\frac{5}{x}\)
(d) ex – \(\frac{5}{x}\)
Solution:
(a)
y = ex+5logx = ex.e5logx = ex.elogx5
= x5 ex
∴ \(\frac{d y}{d x}\) = x5 (ex) + ex (5x4)
= ex. x4 (x + 5)

Question 10.
If the derivative of (ax – 5) e3x at x = 0 is -13, then the value of a is …………….
(a) 8
(b) -2
(c) 5
(d) 2
Solution:
(d)
y = (ax – 5)e3x
\(\frac{d y}{d x}\) = y’ = (ax – 5) (3e3x) + e3x (a)
= e3x[3ax – 15 + a]
Given \(\frac{d y}{d x}\) = -13 at x = 0
⇒ [-15 + a] = -13
⇒ a = -13 + 15
a = 2

Question 11.
x = \(\frac{1-t^{2}}{1+t^{2}}\), y = \(\frac{2 t}{1+t^{2}}\) then \(\frac{d y}{d x}\) is …………..
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 9
Solution:
(c)
Given x = \(\frac{1-t^{2}}{1+t^{2}}\) and y = \(\frac{2 t}{1+t^{2}}\)
when we put t = tan θ
Then x = cos 2θ and y = sin 2θ
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 10

Question 12.
If x = a sin θ and y = b cos θ, then \(\frac{d^{2} y}{d x^{2}}\) is …………..
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 11
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 12

Question 13.
The differential coefficient of log10x with respect to logx 10 is …………….
(a) 1
(b) -(log10x)2
(c) (logx 10)2
(d) \(\frac{x^{2}}{100}\)
Solution:
(b)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 13
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 14

Question 14.
If f(x) = x + 2, then f'(f(x)) at x = 4 is ……………..
(a) 8
(b) 1
(c) 4
(d) 5
Solution:
(b)
f(x) = x + 2
f'(x) = 1
f'(x) (at x = 4) = 1

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Question 15.
If y = \(\frac{(1-x)^{2}}{x^{2}}\), then \(\frac{d y}{d x}\) is ………………
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 15
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 16

Question 16.
If pv = 81, then \(\frac{d p}{d v}\) at v = 9 is ………….
(a) 1
(b) -1
(c) 2
(d) -2
Solution:
(b)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 17

Question 17.
If f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 18 then the right hand derivative of f(x) at x = 2 is ……………….
(a) 0
(b) 2
(c) 3
(d) 4
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 19

Question 18.
It is given that f'(a) exists, then Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 20 is ……………..
(a) f(a) – af'(a)
(b) f ‘(a)
(c) -f ‘(a)
(d) f(a) + af ‘(a)
Solution:
(a)

Question 19.
If f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 21 then f ‘(2) is ………………
(a) 0
(b) 1
(c) 2
(d) does not exist
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 22
∴ f ‘(2) does not exist

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Question 20.
If g(x) = (x2 + 2x + 3) f(x) and f(0) = 5 and Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 23 then g ‘(θ) is ……………
(a) 20
(b) 22
(c) 18
(d) 12
Solution:
(b) 22
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 24

Question 21.
If f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 25, then at x = 3, f ‘(x) is ………………
(a) 1
(b) -1
(c) 0
(d) does not exist
Solution:
(d)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 26
as LHS ≠ RHS limit does not exist

Question 22.
The derivative of f(x) = x|x| at x = -3 is …………..
(a) 6
(b) -6
(c) does not exist
(d) 0
Solution:
(a)
f(x) = x|x|
f(x) = x(-x) ⇒ f(x) = – x2
f ‘(x) = -(2x)
f ‘(-3) = -(2) (-3) = 6

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Question 23.
If f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 27, then which one of the following is true?
(a) f(x) is not differentiable at x = a
(b) f(x) is discontinuous at x = a
(c) f(x) is continuous for all x in R
(d) f(x) is differentiable for all x ≥ a
Solution:
(a)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 28
f(x) is not differentiable at x = a

Question 24.
If f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 29 is differentiable at x = 1, then ………………
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 30
Solution:
(c)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5 31

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.5

Question 25.
Then number of points in R in which the function f(x) = |x – 1| + |x – 3| + sin x is not differentiable, is ……………..
(a) 3
(b) 2
(c) 1
(d) 4
Solution:
(b) 2
f(x) = |x – 1| + |x – 3| + sin x is not differentiable at x = 1, and x = 3

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