# CBSE Class 12th Math 5 – Continuity and Differentiability MCQs

#### If A = e2n - en and B = en, then the value of (A - B2)2 is ____.

Correct! Wrong!

Given that A = e2n - en and B = en, Now A - B2 = (e2n - en) - (en)2 = e2n - en - e2n = - en And (A - B2)2 = (-en)2 = e2n Hence the value of value of (A - B2)2 is e2n.

#### If y = 7x3 - 5x2+ 1 then y'' = ____.

Correct! Wrong!

Given y = 7x3 - 5x2 +1 Differentiate both sides with respect to x, we get y' = 21 x2 - 10x Again, differentiate both sides with respect to x, we get y'' = 42x – 10 Hence, y'' = 42x – 10

#### If area of a circle of radius R is A, then the second derivative of A with respect to R is ____.

Correct! Wrong!

Area of a circle of radius R is A = π R2 Differentiate both sides with respect to R, we get A' = 2πR Again, differentiate both sides with respect to R, we get A'' = 2π Hence, the second derivative of A with respect to R is 2π.

#### If y = sin x, then y'' = ____ y.

Correct! Wrong!

Given function is y = sin x Differentiate both sides with respect to x, we get y'= d(sinx)/dx ⇒ y' = cos x Again, differentiate both sides with respect to x, we get y' ' = d(cosx)/dx ⇒y'' = -sin x ⇒ y'' = -y

#### If a function f (x) = x3 - 7 for x ∈ [–4, 2] is verified by Rolle’s theorem, the possible value/values of c in (-4, 2) is/are ____.

Correct! Wrong!

Rolle’s Theorem states: If a function f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ’(c) = 0. The given function is f (x) = x3 - 7. and the derivative is f ‘(x) = 3 x2. derivative at c is f ‘(c) = 3 c2 f ‘(c) = 3 c2 = 0 (By Rolle’s Theorem) Hence, c = 0.

#### If a function f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ’(c) = 0. This is called ____ theorem.

Correct! Wrong!

Rolle's Theorem states: If a function f: [a, b] →R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ’(c) = 0.

#### A function f is verified by Mean Value Theorem at c in(a, b), if f (x) = 3 x2 – 4x in the interval [a, b], where a = 1 and b = 4. Then, the value of f'(c) is ____.

Correct! Wrong!

Mean Value Theorem states: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that f '(c) = f(b) – f(a) /b-a. Here, f(b) = f(4) = (3 × 42) - (4 × 4) = 32 f(a) = f(1) = (3 ×12) - (4 × 1) = -1 Now, f '(c) = f(b) – f(a)/b-a =32—(1) /(4 - 1) = 11 Hence, the value of f ‘(c) is 11.

#### If y = log x, then x2 D2y + x Dy = ____.

Correct! Wrong!

Given function is y = log x. Differentiate both sides with respect to x, we get Dy = 1/x --------- (1) Differentiate equation (1) both sides with respect to x, we get D2y=-1/x2 ----------- (2) Now, x2D2y + xDy = x2(-1/x2) + x(1/x) ( from equations (1) and (2)) = -1 + 1 = 0 ⇒ x2 D2y + x Dy = 0 Hence, x2 D2y + x Dy = 0.

#### If a function f (x) = sin (2x) for x ∈ [0, π/2] is verified by Rolle’s theorem, then all possible values of c in (0, π/2) is/are ____.

Correct! Wrong!

Rolle’s Theorem states: If a function f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ’(c) = 0. The given function is f (x) = sin (2x) And the derivative is f ‘(x) = 2cos (2x) f ‘(x) = 2cos (2x) = 0 (By Rolle’s Theorem) ⇒ cos (2x) = 0 ⇒ cos (2x) = cos π/2 ⇒2x=2nπ ± π/2 ⇒nπ ± π/4 So, the possible value of x in (0, π/2) is π/4 Hence, all possible values of c in (0, π/2) is/are π/4.

#### If a function f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) and verified by Rolle’s Theorem, there exists some c in (a, b), such that f ‘(c) = 0,. Then, the value of f(a)/f(b) is ____.

Correct! Wrong!

Rolle's Theorem states: If a function f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ’(c) = 0. Therefore f (a) = f (b). ⇒f(a)/f(b) = 1 Hence, the value of f(a)/f(b) is 1

#### If a function f (x) = x3 - 3x for x ∈ [– 4, 2] is verified by Rolle’s theorem, then the number of all possible values of c in (-4, 2) is/are ____.

Correct! Wrong!

Rolle’s Theorem states: If a function f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ’(c) = 0. The given function is f (x) = x3 - 3x and the derivative is f ‘(x) = 3 x2- 3 derivative at c is f ‘(c) = 3 c2 – 3 f ‘(c) = 3 c2 - 3 = 0 (By Rolle’s Theorem) ⇒ c2 - 1 = 0 ⇒ c = -1and 1 ⇒ (c + 1)(c - 1) = 0 So, we get two values of c (-4, 2). Hence, the number of all possible values of c in (-4, 2) is/are 2.

#### If e(log x)= 1/3, then the value of x is ____.

Correct! Wrong!

Given e(log x)= 1/3 ⇒log(1/3) = logx ⇒x =1/3 Hence, the value of x is 1/3.

#### If n = log 2 + log sin A + log cos A, then the value of ℮n is ____.

Correct! Wrong!

Given n = log 2 + log sin A + log cos A ⇒ n = log (2 sin A cos A) ⇒ n = log sin(2A) ⇒ ℮n = sin 2A Hence, the value of ℮n is sin 2A.

#### If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that f'(c) = f(b) – f(a) /k. The value of k is ____.

Correct! Wrong!

Mean Value Theorem states: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that f '(c) =f(b) – f(a) /b-a. Given that f '(c) = f '(c) =f(b) – f(a) /b-a. ⇒k = b - a the value of k is b - a.

#### If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that f'(c) = f(b) – f(a)/b-a and it is called__________theorem.

Correct! Wrong!

Mean Value Theorem states: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that f'(c) = f(b) – f(a)/b-a

#### If y = 3log x – 5x, then D2y = ____.

Correct! Wrong!

Given function is y = 3log x – 5x. Differentiate both sides with respect to x. We get Dy = (3/x) - 5. Again, differentiate both sides with respect to x. We get D2y= -3/ x2. Hence, D2y = -3/ x2.

#### If f(x) = (3 - 2x)2, then f''(x) = ____.

Correct! Wrong!

Given function is f(x) = (3 - 2x)2 Differentiate both sides with respect to x, we get ⇒f'x = d(3-2x)2/dx = d(3-2x)2/d(3-2x) × d(3-2x/dx) = 2(3 - 2x) × (-2) = -4(3 - 2x) ⇒ f'(x) = 8x – 12 -----(i) Differentiate equation (i) with respect to x ⇒ f''(x) = 8 Hence, f''(x) = 8.

#### If en + e-n = x, then the value of e2n + e-2nis ____.

Correct! Wrong!

Given that en + e-n = x By squaring on both sides, we get (en + e-n)2 = x2 ⇒ (en)2 + (e-n)2 + 2 (en)(e-n) = x2 ⇒ e2n + e-2n + 2 = x2 ⇒ e2n + e-2n = x2 – 2 Hence, the value of e2n + e-2n is x2 - 2.

#### If a function f (x) = cos (2x) for x ∈ [0, π/2] is verified by Mean Value theorem, then all possible values of c in (0, π/2) is/are ____.

Correct! Wrong!

Mean Value Theorem states: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that f'(c)=f(b) – f(a)/b-a. The given function is f (x) = cos (2x) a = 0, b = π/2 f‘(x) = -2sin (2x) f‘(c) = -2sin (2c) Here, f(b) = f(π/2) = cos(2 × π/2) = -1 f(a) = f(0) = cos (2 × 0) = 1 Now, f '(c) = f(b) – f(a)/b-a -2sin(2c) =(-1-1)/(π/2-0) sin2c = 2/π c=1/2arcsin2/π Hence, all possible values of c in (0, π/2) is/are 1/2arcsin2/π.

#### If volume of a cylindrical container of radius R and height H is V, then the second derivative of V with respect to R is ____ H.

Correct! Wrong!

Volume of a cylindrical container of radius R and height H is V = πR2H Differentiate both sides with respect to R, we get V' = 2πRH Again, differentiate both sides with respect to R, we get V'' = 2πH Hence, the second derivative of V with respect to R is 2πH.

#### If Volume of a spherical ball of radius R is V, then the second derivative of A with respect to R is ____ R.

Correct! Wrong!

Volume of a Spherical Ball of radius R is V = 4/3πR3. Differentiate both sides with respect to R, we get V' = 3 × 4/3π R2 = 4π R2 Again, differentiate both sides with respect to R, we get V'' = 8πR Hence, the second derivative of V with respect to R is 8πR.

#### If en + e-n = 2, then the value of n is ____.

Correct! Wrong!

Given that en + e-n = 2 By multiplying both sides by ℮n, we get (en)2 + (en)(e-n) = 2en ⇒ (en)2 + 1 = 2en ⇒ (en)2 - 2en+ 1 = 0 ⇒ (en - 1)2= 0 ⇒ en = 1 ⇒ en = e0 ⇒ n = 0 Hence, the value of n is 0.

#### A polynomial function is differentiable at ____.

Correct! Wrong!

A polynomial function is differentiable at every point of real number R, i.e. the polynomial function is differentiable at R. Hence, a polynomial function is differentiable at (-∞, ∞).

#### A function is said to be differentiable in an interval [a, b] if it is differentiable at ____.

Correct! Wrong!

From the definition, a function is said to be differentiable in the interval [a, b], if it is differentiable at every point of [a, b]. Hence, a function is said to be differentiable in an interval [a, b], if it is differentiable at every point of [a, b].

#### If A = 2 log sin θ and B = 2 log cos θ, then the value of ℮A + ℮B is ____.

Correct! Wrong!

Given A = 2 log sin θ = log sin2 θ ⇒ eA = sin2 θ --------- (1) Given B = 2 log cos θ = log cos2 θ ⇒ eB = cos2θ -------- (2) By adding equations (1) and (2), we get ℮A + ℮B = sin2 θ + cos2 θ ⇒ ℮A + ℮B = 1 Hence, the value of ℮A + ℮B is 1.

CBSE Class 12th Math 5 - Continuity and Differentiability MCQs
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