Maharashtra State Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9
(I) Select the appropriate option from the given alternatives.
Question 1
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If y = , then is (A) (B) (C) (D)
Solution & Step-by-Step Answer:
(C) Hint:
Question 2
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If y = ,then = (A) (B) (C) (D)
Solution & Step-by-Step Answer:
(D) Hint:
Question 3
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If y = , then = (A) (B) (C) (D)
Solution & Step-by-Step Answer:
(C) Hint:
Question 4
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If y = , then = (A) (B) (C) (D)
Solution & Step-by-Step Answer:
(B) Hint:
Question 5
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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)
Solution & Step-by-Step 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
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If f(x) = 2x + 6, for 0 ≤ x ≤ 2 = ax2 + bx, for 2 < x ≤ 4 is differentiable at x = 2, then the values of a and b are (A) a = , b = 3 (B) a = , b = 8 (C) a = , b = 8 (D) a = , b = 8
Solution & Step-by-Step Answer:
(D) a = , b = 8 Hint: f(x) = 2x + 6, 0 ≤ x ≤ 2 = ax2 + 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. ∴ ∴ 4a + 2b = 2(2) + 6 ∴ 4a + 2b = 10 ∴ 2a + b = 5 …..(ii) Solving (i) and (ii), we get a = , b = 8
Question 7
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If f(x) = x2 + sin x + 1, for x ≤ 0 = x2 – 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
Solution & Step-by-Step Answer:
(A) f is continuous at x = 0, but not differentiable at x = 0 Hint:
Question 8
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If f(x) = , then f'(1) = (A) 48 (B) 49 (C) 50 (D) 51
Solution & Step-by-Step Answer:
(C) 50 Hint:
(II).
Question 1
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Determine whether the following function is differentiable at x = 3 where, f(x) = x2 + 2, for x ≥ 3 = 6x – 7, for x < 3.
Solution & Step-by-Step Answer:
f(x) = x2 + 2, x ≥ 3 = 6x – 7, x < 3 Differentiability at x = 3 Here, Lf'(3) = Rf'(3) ∴ f is differentiable at x = 3.
Question 2
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Find the values of p and q that make function f(x) differentiable everywhere on R. f(x) = 3 – x, for x < 1 = px2 + qx, for x ≥ 1.
Solution & Step-by-Step Answer:
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
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Determine the values of p and q that make the function f(x) differentiable on R where f(x) = px3, for x < 2 = x2 + q, for x ≥ 2
Solution & Step-by-Step Answer:
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 = Substituting p = in (i), we get 8( – q = 4 ∴ q = – 4 =
Question 4
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Determine all real values of p and q that ensure the function f(x) = px + q, for x ≤ 1 = tan(), for 1 < x < 2 is differentiable at x = 1.
Solution & Step-by-Step Answer:
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
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Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R.
Solution & Step-by-Step Answer:
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
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Test whether the function f(x) = 2x – 3, for x ≥ 2 = x – 1, for x < 2 is differentiable at x = 2.
Solution & Step-by-Step Answer:
Question 7
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Test whether the function f(x) = x2 + 1, for x ≥ 2 = 2x + 1, for x < 2 is differentiable at x = 2.
Solution & Step-by-Step Answer:
Question 8
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Test whether the function f(x) = 5x – 3x2, for x ≥ 1 = 3 – x, for x < 1 is differentiable at x = 1.
Solution & Step-by-Step Answer:
Here, Lf'(1) = Rf'(1) ∴ f(x) is differentiable at x = 1.
Question 9
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If f(2) = 4, f'(2) = 1, then find
Solution & Step-by-Step Answer:
Question 10
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If y = , find when x = 1.
Solution & Step-by-Step Answer:
