Balbharti Maharashtra State Board 11th Maths Book Solutions Pdf Chapter 9 Differentiation Miscellaneous Exercise 9 Questions and Answers.

## Maharashtra State Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

(I) Select the appropriate option from the given alternatives.

Question 1.

If y = \(\frac{x-4}{\sqrt{x+2}}\), then \(\frac{d y}{d x}\) is

(A) \(\frac{1}{x+4}\)

(B) \(\frac{\sqrt{x}}{\left(\sqrt{x+2)^{2}}\right.}\)

(C) \(\frac{1}{2 \sqrt{x}}\)

(D) \(\frac{x}{(\sqrt{x}+2)^{2}}\)

Answer:

(C) \(\frac{1}{2 \sqrt{x}}\)

Hint:

Question 2.

If y = \(\frac{a x+b}{c x+d}\),then \(\frac{d y}{d x}\) =

(A) \(\frac{a b-c d}{(c x+d)^{2}}\)

(B) \(\frac{a x-c}{(c x+d)^{2}}\)

(C) \(\frac{a c-b d}{(c x+d)^{2}}\)

(D) \(\frac{a d-b c}{(c x+d)^{2}}\)

Answer:

(D) \(\frac{a d-b c}{(c x+d)^{2}}\)

Hint:

Question 3.

If y = \(\frac{3 x+5}{4 x+5}\), then \(\frac{d y}{d x}\) =

(A) \(-\frac{15}{(3 x+5)^{2}}\)

(B) \(-\frac{15}{(4 x+5)^{2}}\)

(C) \(-\frac{5}{(4 x+5)^{2}}\)

(D) \(-\frac{13}{(4 x+5)^{2}}\)

Answer:

(C) \(-\frac{5}{(4 x+5)^{2}}\)

Hint:

Question 4.

If y = \(\frac{5 \sin x-2}{4 \sin x+3}\), then \(\frac{d y}{d x}\) =

(A) \(\frac{7 \cos x}{(4 \sin x+3)^{2}}\)

(B) \(\frac{23 \cos x}{(4 \sin x+3)^{2}}\)

(C) \(-\frac{7 \cos x}{(4 \sin x+3)^{2}}\)

(D) \(-\frac{15 \cos x}{(4 \sin x+3)^{2}}\)

Answer:

(B) \(\frac{23 \cos x}{(4 \sin x+3)^{2}}\)

Hint:

Question 5.

Suppose f(x) is the derivative of g(x) and g(x) is the derivative of h(x).

If h(x) = a sin x + b cos x + c, then f(x) + h(x) =

(A) 0

(B) c

(C) -c

(D) -2(a sin x + b cos x)

Answer:

(B) c

Hint:

h(x) = a sin x + b cos x + c

Differentiating w.r.t. x, we get

h'(x) = a cos x – b sin x = g(x) …..[given]

Differentiating w.r.t. x, we get

g'(x) = -a sin x – b cos x = f(x) …..[given]

∴ f(x) + h(x) = -a sin x – b cos x + a sin x + b cos x + c

∴ f(x) + h(x) = c

Question 6.

If f(x) = 2x + 6, for 0 ≤ x ≤ 2

= ax^{2} + bx, for 2 < x ≤ 4

is differentiable at x = 2, then the values of a and b are

(A) a = \(-\frac{3}{2}\), b = 3

(B) a = \(\frac{3}{2}\), b = 8

(C) a = \(\frac{1}{2}\), b = 8

(D) a = \(-\frac{3}{2}\), b = 8

Answer:

(D) a = \(-\frac{3}{2}\), b = 8

Hint:

f(x) = 2x + 6, 0 ≤ x ≤ 2

= ax^{2} + bx, 2 < x ≤ 4

Lf'(2) = 2, Rf'(2) = 4a + b

Since f is differentiable at x = 2,

Lf'(2) = Rf'(2)

∴ 2 = 4a + b …..(i)

f is continuous at x = 2.

∴ \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)=\lim _{x \rightarrow 2^{-}} f(x)\)

∴ 4a + 2b = 2(2) + 6

∴ 4a + 2b = 10

∴ 2a + b = 5 …..(ii)

Solving (i) and (ii), we get

a = \(-\frac{3}{2}\), b = 8

Question 7.

If f(x) = x^{2} + sin x + 1, for x ≤ 0

= x^{2} – 2x + 1, for x ≤ 0, then

(A) f is continuous at x = 0, but not differentiable at x = 0

(B) f is neither continuous nor differentiable at x = 0

(C) f is not continuous at x = 0, but differentiable at x = 0

(D) f is both continuous and differentiable at x = 0

Answer:

(A) f is continuous at x = 0, but not differentiable at x = 0

Hint:

Question 8.

If f(x) = \(\frac{x^{50}}{50}+\frac{x^{49}}{49}+\frac{x^{48}}{48}+\ldots .+\frac{x^{2}}{2}+x+1\), then f'(1) =

(A) 48

(B) 49

(C) 50

(D) 51

Answer:

(C) 50

Hint:

(II).

Question 1.

Determine whether the following function is differentiable at x = 3 where,

f(x) = x^{2} + 2, for x ≥ 3

= 6x – 7, for x < 3.

Solution:

f(x) = x^{2} + 2, x ≥ 3

= 6x – 7, x < 3

Differentiability at x = 3

Here, Lf'(3) = Rf'(3)

∴ f is differentiable at x = 3.

Question 2.

Find the values of p and q that make function f(x) differentiable everywhere on R.

f(x) = 3 – x, for x < 1

= px^{2} + qx, for x ≥ 1.

Solution:

f(x) is differentiable everywhere on R.

∴ f(x) is differentiable at x = 1.

∴ f(x) is continuous at x = 1.

f(x) is differentiable at x = 1.

∴ Lf'(1) = Rf'(1)

∴ -1 = 2p + q …..(ii)

Subtracting (i) from (ii), we get

p = -3

Substituting p = -3 in (i), we get

p + q = 2

∴ -3 + q = 2

∴ q = 5

Question 3.

Determine the values of p and q that make the function f(x) differentiable on R where

f(x) = px^{3}, for x < 2

= x^{2} + q, for x ≥ 2

Solution:

f(x) is differentiable on R.

∴ f(x) is differentiable at x = 2.

∴ f(x) is continuous at x = 2.

Continuity at x = 2:

f(x) is continuous at x = 2.

f(x) is differentiable at x = 2.

∴ Lf'(2) = Rf'(2)

∴ 12p = 4

∴ p = \(\frac{1}{3}\)

Substituting p = \(\frac{1}{3}\) in (i), we get

8(\(\frac{1}{3}\) – q = 4

∴ q = \(\frac{8}{3}\) – 4 = \(\frac{-4}{3}\)

Question 4.

Determine all real values of p and q that ensure the function

f(x) = px + q, for x ≤ 1

= tan(\(\frac{\pi x}{4}\)), for 1 < x < 2

is differentiable at x = 1.

Solution:

f(x) is differentiable at x = 1.

∴ f(x) is continuous at x = 1.

Continuity at x= 1:

f(x) is continuous at x = 1.

Question 5.

Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R.

Solution:

Here, Lf'(1) ≠ Rf'(1)

∴ f is not differentiable at x = 1.

∴ f is not differentiable at x = -1 and x = 1.

∴ f is not differentiable ∀ x ∈ R.

Question 6.

Test whether the function

f(x) = 2x – 3, for x ≥ 2

= x – 1, for x < 2

is differentiable at x = 2.

Solution:

Question 7.

Test whether the function

f(x) = x^{2} + 1, for x ≥ 2

= 2x + 1, for x < 2

is differentiable at x = 2.

Solution:

Question 8.

Test whether the function

f(x) = 5x – 3x^{2}, for x ≥ 1

= 3 – x, for x < 1

is differentiable at x = 1.

Solution:

Here, Lf'(1) = Rf'(1)

∴ f(x) is differentiable at x = 1.

Question 9.

If f(2) = 4, f'(2) = 1, then find \(\lim _{x \rightarrow 2}\left[\frac{x f(2)-2 f(x)}{x-2}\right]\)

Solution:

Question 10.

If y = \(\frac{\mathbf{e}^{x}}{\sqrt{x}}\), find \(\frac{d y}{d x}\) when x = 1.

Solution: