# MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers

Question 1.
If f (x) = 2x and g (x) = $$\frac{x^2}{2}$$ + 1, then’which of the following can be a discontinuous function
(a) f(x) + g(x)
(b) f(x) – g(x)
(c) f(x).g(x)
(d) $$\frac{g(x)}{f(x)}$$

Answer

Answer: (d) $$\frac{g(x)}{f(x)}$$

Question 2.
The function f(x) = $$\frac{4-x^2}{4x-x^3}$$ is
(a) discontinuous at only one point at x = 0
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) None of these

Answer

Answer: (a) discontinuous at only one point at x = 0

Question 3.
The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is
(a) R
(b) R = {$$\frac{1}{2}$$}
(c) (0, ∞)
(d) None of these

Answer

Answer: (b) R = {$$\frac{1}{2}$$}

Question 4.
The function f(x) = cot x is discontinuous on the set
(a) {x = nπ, n ∈ Z}
(b) {x = 2nπ, n ∈ Z}
(c) {x = (2n + 1) $$\frac{π}{2}$$ n ∈ Z}
(d) {x – $$\frac{nπ}{2}$$ n ∈ Z}

Answer

Answer: (a) {x = nπ, n ∈ Z}

Question 5.
The function f(x) = e|x| is
(a) continuous everywhere but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) None of these

Answer

Answer: (a) continuous everywhere but not differentiable at x = 0

Question 6.
If f(x) = x² sin$$\frac{1}{x}$$, where x ≠ 0, then the value of the function f(x) at x = 0, so that the function is continuous at x = 0 is
(a) 0
(b) -1
(c) 1
(d) None of these

Answer

Answer: (a) 0

Question 7.
If f(x) =is continuous at x = $$\frac{π}{2}$$, then
(a) m = 1, n = 0
(b) m = $$\frac{nπ}{2}$$ + 1
(c) n = $$\frac{mπ}{2}$$
(d) m = n = $$\frac{π}{2}$$

Answer

Answer: (c) n = $$\frac{mπ}{2}$$

Question 8.
If y = log($$\frac{1-x^2}{1+x^2}$$), then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{4x^3}{1-x^4}$$
(b) $$\frac{-4x}{1-x^4}$$
(c) $$\frac{1}{4-x^4}$$
(d) $$\frac{-4x^3}{1-x^4}$$

Answer

Answer: (b) $$\frac{-4x}{1-x^4}$$

Question 9.
Let f(x) = |sin x| Then
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
(c) f is everywhere continuous but no differentiable at x = (2n + 1) $$\frac{π}{2}$$ n ∈ Z
(d) None of these

Answer

Answer: (b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z

Question 10.
If y = $$\sqrt{sin x+y}$$ then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{cosx}{2y-1}$$
(b) $$\frac{cosx}{1-2y}$$
(c) $$\frac{sinx}{1-xy}$$
(d) $$\frac{sinx}{2y-1}$$

Answer

Answer: (a) $$\frac{cosx}{2y-1}$$

Question 11.
The derivative of cos-1 (2x² – 1) w.r.t cos-1 x is
(a) 2
(b) $$\frac{-1}{2\sqrt{1-x^2}}$$
(c) $$\frac{2}{x}$$
(d) 1 – x²

Answer

Answer: (a) 2

Question 12.
If x = t², y = t³, then $$\frac{d^2y}{dx^2}$$
(a) $$\frac{3}{2}$$
(b) $$\frac{3}{4t}$$
(c) $$\frac{3}{2t}$$
(d) $$\frac{3}{4t}$$

Answer

Answer: (b) $$\frac{3}{4t}$$

Question 13.
The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval [o, √3] is
(a) 1
(b) -1
(c) $$\frac{3}{2}$$
(d) $$\frac{1}{3}$$

Answer

Answer: (a) 1

Question 14.
For the function f(x) = x + $$\frac{1}{x}$$, x ∈ [1, 3] the value of c for mean value theorem is
(a) 1
(b) √3
(c) 2
(d) None of these

Answer

Answer: (b) √3

Question 15.
Let f be defined on [-5, 5] as
f(x) = {$$_{-x, if x is irrational}^{x, if x is rational}$$ Then f(x) is
(a) continuous at every x except x = 0
(b) discontinuous at everyx except x = 0
(c) continuous everywhere
(d) discontinuous everywhere

Answer

Answer: (b) discontinuous at everyx except x = 0

Question 16.
Let function f (x) =
(a) continuous at x = 1
(b) differentiable at x = 1
(c) continuous at x = -3
(d) All of these

Answer

Answer: (d) All of these

Question 17.
If f(x) = $$\frac{\sqrt{4+x}-2}{x}$$ x ≠ 0 be continuous at x = 0, then f(o) =
(a) $$\frac{1}{2}$$
(b) $$\frac{1}{4}$$
(c) 2
(d) $$\frac{3}{2}$$

Answer

Answer: (b) $$\frac{1}{4}$$

Question 18.
let f(2) = 4 then f”(2) = 4 then $$_{x→2}^{lim}$$ $$\frac{xf(2)-2f(x)}{x-2}$$ is given by
(a) 2
(b) -2
(c) -4
(d) 3

Answer

Answer: (c) -4

Question 19.
It is given that f'(a) exists, then $$_{x→2}^{lim}$$ [/latex] $$\frac{xf(a)-af(x)}{(x-a)}$$ is equal to
(a) f(a) – af'(a)
(b) f'(a)
(c) -f’(a)
(d) f (a) + af'(a)

Answer

Answer: (a) f(a) – af'(a)

Question 20.
If f(x) = $$\sqrt{25-x^2}$$, then $$_{x→2}^{lim}$$$$\frac{f(x)-f(1)}{x-1}$$ is equal to
(a) $$\frac{1}{24}$$
(b) $$\frac{1}{5}$$
(c) –$$\sqrt{24}$$
(d) $$\frac{1}{\sqrt{24}}$$

Answer

Answer: (d) $$\frac{1}{\sqrt{24}}$$

Question 21.
If y = ax² + b, then $$\frac{dy}{dx}$$ at x = 2 is equal to ax
(a) 4a
(b) 3a
(c) 2a
(d) None of these

Answer

Answer: (a) 4a

Question 22.
If x sin (a + y) = sin y, then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{sin^2(a+y)}{sin a}$$
(b) $$\frac{sin a}{sin^2(a+y)}$$
(c) $$\frac{sin(a+y)}{sin a}$$
(d) $$\frac{sin a}{sin(a+y)}$$

Answer

Answer: (a) $$\frac{sin^2(a+y)}{sin a}$$

Question 23.
If x $$\sqrt{1+y}+y\sqrt{1+x}$$ = 0, then $$\frac{dy}{dx}$$ =
(a) $$\frac{x+1}{x}$$
(b) $$\frac{1}{1+x}$$
(c) $$\frac{-1}{(1+x)^2}$$
(d) $$\frac{x}{1+x}$$

Answer

Answer: (c) $$\frac{-1}{(1+x)^2}$$

Question 24.
If y = x tan y, then $$\frac{dy}{dx}$$ =
(a) $$\frac{tan x}{x-x^2-y^2}$$
(b) $$\frac{y}{x-x^2-y^2}$$
(c) $$\frac{tan y}{y-x}$$
(d) $$\frac{tan x}{x-y^2}$$

Answer

Answer: (b) $$\frac{y}{x-x^2-y^2}$$

Question 25.
If y = (1 + x) (1 + x²) (1 + x4) …….. (1 + x2n), then the value of $$\frac{dy}{dx}$$ at x = 0 is
(a) 0
(b) -1
(c) 1
(d) None of these

Answer

Answer: (c) 1

Question 26.
If f(x) = $$\frac{5x}{(1-x)^{2/3}}$$ + cos² (2x + 1), then f'(0) =
(a) 5 + 2 sin 2
(b) 5 + 2 cos 2
(c) 5 – 2 sin 2
(d) 5 – 2 cos 2

Answer

Answer: (c) 5 – 2 sin 2

Question 27.
If sec($$\frac{x^2-2x}{x^2+1}$$) – y then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{y*2}{x^2}$$
(b) $$\frac{2y\sqrt{y^2-1}(x^2+x-1)}{(x^2+1)^2}$$
(c) $$\frac{(x^2+x-1)}{y\sqrt{y^2-1}}$$
(d) $$\frac{x^2-y^2}{x^2+y^2}$$

Answer

Answer: (b) $$\frac{2y\sqrt{y^2-1}(x^2+x-1)}{(x^2+1)^2}$$

Question 28.
If f(x) = $$\sqrt{1+cos^2(x^2)}$$, then the value of f’ ($$\frac{√π}{2}$$) is
(a) $$\frac{√π}{6}$$
(b) –$$\frac{√π}{6}$$
(c) $$\frac{1}{√6}$$
(d) $$\frac{π}{√6}$$

Answer

Answer: (b) –$$\frac{√π}{6}$$

Question 29.
Differential coefficient of $$\sqrt{sec√x}$$ is
(a) $$\frac{1}{4√x}$$ = sec √x sin √x
(b) $$\frac{1}{4√x}$$ = (sec√x)3/2 sin√x
(c) $$\frac{1}{2}$$ √x sec√x sin √x.
(d) $$\frac{1}{2}$$√x (sec√x)3/2 sin√x

Answer

Answer: (b) $$\frac{1}{4√x}$$ = (sec√x)3/2 sin√x

Question 30.
Let f(x)={$$_{1-cos x, for x ≤ 0}^{sin x, for x > 0}$$ and g (x) = ex. Then the value of (g o f)’ (0) is
(a) 1
(b) -1
(c) 0
(d) None of these

Answer

Answer: (c) 0

Question 31.
If xmyn = (x + y)m+n, then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{x+y}{xy}$$
(b) xy
(c) $$\frac{x}{y}$$
(d) $$\frac{y}{x}$$

Answer

Answer: (d) $$\frac{y}{x}$$

Question 32.
If $$\sqrt{(x+y)}$$ + $$\sqrt{(y-x)}$$ = a, then $$\frac{dy}{dx}$$

Answer

Answer: (a) $$\frac{\sqrt{(x+y)}-\sqrt{(y-x)}}{\sqrt{y-x}+\sqrt{x+y}}$$

Question 33.
If ax² + 2hxy + by² = 1, then $$\frac{dy}{dx}$$equals
(a) $$\frac{hx+by}{ax+by}$$
(b) $$\frac{ax+by}{hx+by}$$
(c) $$\frac{ax+hy}{hx+hy}$$
(d) $$\frac{-(ax+hy)}{hx+by}$$

Answer

Answer: (d) $$\frac{-(ax+hy)}{hx+by}$$

Question 34.
If sec ($$\frac{x-y}{x+y}$$) = a then $$\frac{dy}{dx}$$ is
(a) –$$\frac{y}{x}$$
(b) $$\frac{x}{y}$$
(c) –$$\frac{x}{y}$$
(d) $$\frac{y}{x}$$

Answer

Answer: (d) $$\frac{y}{x}$$

Question 35.
If y = tan-1($$\frac{sinx+cosx}{cox-sinx}$$) then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{1}{2}$$
(b) $$\frac{π}{4}$$
(c) 0
(d) 1

Answer

Answer: (d) 1

Question 36.
If y = tan-1($$\frac{√x-x}{1+x^{3/2}}$$), then y'(1) is equal to
(a) 0
(b) ($$\frac{√x-x}{1+x^{3/2}}$$)
(c) -1
(d) –$$\frac{1}{4}$$

Answer

Answer: (d) –$$\frac{1}{4}$$

Question 37.
The differential coefficient of tan-1($$\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$$) is
(a) $$\sqrt{1-x^2}$$
(b) $$\frac{1}{\sqrt{1-x^2}}$$
(c) $$\frac{1}{2\sqrt{1-x^2}}$$
(d) x

Answer

Answer: (c) $$\frac{1}{2\sqrt{1-x^2}}$$

Question 38.
$$\frac{d}{dx}$$[tan-1($$\frac{a-x}{1+ax}$$)] is equal to

Answer

Answer: (a) –$$\frac{1}{1+x^2}$$

Question 39.
$$\frac{d}{dx}$$(x$$\sqrt{a^2-x^2}+a^2 sin^{-1}(\frac{x}{a})$$) is equal to
(a) $$\sqrt{a^2-x^2}$$
(b) 2$$\sqrt{a^2-x^2}$$
(c) $$\frac{1}{\sqrt{a^2-x^2}}$$
(d) None of these

Answer

Answer: (b) 2$$\sqrt{a^2-x^2}$$

Question 40.
If f(x) = tan-1($$\sqrt{\frac{1+sinx}{1-sinx}}$$), 0 ≤ x ≤ $$\frac{π}{2}$$, then f'($$\frac{π}{6}$$) is
(a) –$$\frac{1}{4}$$
(b) –$$\frac{1}{2}$$
(c) $$\frac{1}{4}$$
(d) $$\frac{1}{2}$$

Answer

Answer: (d) $$\frac{1}{2}$$

Question 41.
If y = sin-1($$\frac{√x-1}{√x+1}$$) + sec-1($$\frac{√x+1}{√x-1}$$), x > 0, then $$\frac{dy}{dx}$$ is equal to
(a) 1
(b) 0
(c) $$\frac{π}{2}$$
(d) None of these

Answer

Answer: (b) 0

Question 42.
If x = exp {tan-1($$\frac{y-x^2}{x^2}$$)}, then $$\frac{dy}{dx}$$ equals
(a) 2x [1 + tan (log x)] + x sec² (log x)
(b) x [1 + tan (log x)] + sec² (log x)
(c) 2x [1 + tan (logx)] + x² sec² (log x)
(d) 2x [1 + tan (log x)] + sec² (log x)

Answer

Answer: (a) 2x [1 + tan (log x)] + x sec² (log x)

Question 43.
If y = e3x+n, then the value of $$\frac{dy}{dx}$$|x=0 is
(a) 1
(b) 0
(c) -1
(d) 3e7

Answer

Answer: (d) 3e7

Question 44.
Let f (x) = ex, g (x) = sin-1 x and h (x) = f |g(x)|, then $$\frac{h'(x)}{h(x)}$$ is equal to
(a) esin-1x
(b) $$\frac{1}{\sqrt{1-x^2}}$$
(c) sin-1x
(d) $$\frac{1}{(1-x^2)}$$

Answer

Answer: (b) $$\frac{1}{\sqrt{1-x^2}}$$

Question 45.
If y = aex+ be-x + c Where a, b, c are parameters, they y’ is equal to
(a) aex – be-x
(b) aex + be-x
(c) -(aex + be-x)
(d) aex – bex

Answer

Answer: (a) aex – be-x

Question 46.
If sin y + e-xcos y = e, then $$\frac{dy}{dx}$$ at (1, π) is equal to
(a) sin y
(b) -x cos y
(c) e
(d) sin y – x cos y

Answer

Answer: (c) e

Question 47.
Derivative of the function f (x) = log5 (Iog,x), x > 7 is
(a) $$\frac{1}{x(log5)(log7)(log7-x)}$$
(b) $$\frac{1}{x(log5)(log7)}$$
(c) $$\frac{1}{x(logx)}$$
(d) None of these

Answer

Answer: (a) $$\frac{1}{x(log5)(log7)(log7-x)}$$

Question 48.
If y = log10x + log y, then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{y}{y-1}$$
(b) $$\frac{y}{x}$$
(c) $$\frac{log_{10}e}{x}$$($$\frac{y}{y-1}$$)
(d) None of these

Answer

Answer: (c) $$\frac{log_{10}e}{x}$$($$\frac{y}{y-1}$$)

Question 49.
If y = log [ex($$\frac{x-1}{x-2}$$)$$^{1/2}$$], then $$\frac{dy}{dx}$$ is equal to
(a) 7
(b) $$\frac{3}{x-2}$$
(c) $$\frac{3}{(x-1)}$$
(d) None of these

Answer

Answer: (d) None of these

Question 50.
If y = e$$\frac{1}{2}$$ log(1+tan²x), then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{1}{2}$$ sec² x
(b) sec² x
(c) sec x tan x
(d) e$$\frac{1}{2}$$ log(1+tan²x)

Answer

Answer: (c) sec x tan x

Question 51.
If y = 2x32x-1 then $$\frac{dy}{dx}$$ is equal to dx
(a) (log 2) (log 3)
(b) (log lg)
(c) (log 18²) y²
(d) y (log 18)

Answer

Answer: (d) y (log 18)

Question 52.
If xx = yy, then $$\frac{dy}{dx}$$ is equal to
(a) –$$\frac{y}{x}$$
(b) –$$\frac{x}{y}$$
(c) 1 + log ($$\frac{x}{y}$$ )
(d) $$\frac{1+logx}{1+logy}$$

Answer

Answer: (d) $$\frac{1+logx}{1+logy}$$

Question 53.
If y = (tan x)sin x, then $$\frac{dy}{dx}$$ is equal to
(a) sec x + cos x
(b) sec x+ log tan x
(c) (tan x)sin x
(d) None of these

Answer

Answer: (d) None of these

Question 54.
If xy = ex-y then $$\frac{dy}{dx}$$ is
(a) $$\frac{1+x}{1+log x}$$
(b) $$\frac{1-log x}{1+log y}$$
(c) not defined
(d) $$\frac{-y}{(1+log x)^2}$$

Answer

Answer: (d) $$\frac{-y}{(1+log x)^2}$$

Question 55.
The derivative of y = (1 – x) (2 – x)…. (n – x) at x = 1 is equal to
(a) 0
(b) (-1) (n – 1)!
(c) n ! – 1
(d) (-1)n-1 (n – 1)!

Answer

Answer: (b) (-1) (n – 1)!

Question 56.
If f(x) = cos x, cos 2 x, cos 4 x, cos 8 x, cos 16 x, then the value of'($$\frac{π}{4}$$) is
(a) 1
(b) √2
(c) $$\frac{1}{√2}$$
(d) 0

Answer

Answer: (b) (-1) (n – 1)!

Question 57.
xy. yx = 16, then the value of $$\frac{dy}{dx}$$ at (2, 2) is
(a) -1
(b) 0
(c) -1
(d) None of these

Answer

Answer: (a) -1

Question 58.
If y = ex+ex+ex+….to∞ find $$\frac{dy}{dx}$$ =
(a) $$\frac{y^2}{1-y}$$
(b) $$\frac{y^2}{y-1}$$
(c) $$\frac{y}{y-1}$$
(d) $$\frac{-y}{y-1}$$

Answer

Answer: (c) $$\frac{y}{y-1}$$

Question 59.
If x = $$\frac{1-t^2}{1+t^2}$$ and y = $$\frac{2t}{1+t^2}$$ then $$\frac{dy}{dx}$$ is equal to dx
(a) –$$\frac{y}{x}$$
(b) $$\frac{y}{x}$$
(c) –$$\frac{x}{y}$$
(d) $$\frac{x}{y}$$

Answer

Answer: (c) –$$\frac{x}{y}$$

Question 60.
If x = a cos4 θ, y = a sin4 θ. then $$\frac{dy}{dx}$$ at θ = $$\frac{3π}{4}$$ is
(a) -1
(b) 1
(c) -a²
(d) a²

Answer

Answer: (a) -1

Question 61.
If x = sin-1 (3t – 4t³) and y = cos-1 ($$\sqrt{1-t^2}$$) then $$\frac{dy}{dx}$$ is equal to
(a) $$\frac{1}{2}$$
(b) $$\frac{2}{5}$$
(c) $$\frac{3}{2}$$
(d) $$\frac{1}{3}$$

Answer

Answer: (d) $$\frac{1}{3}$$

Question 62.
Let y = t10 + 1 and x = t8 + 1, then $$\frac{d^2y}{dx^2}$$, is equal to
(a) $$\frac{d^2y}{dx^2}$$
(b) 20t8
(c) $$\frac{5}{16t^6}$$
(d) None of these

Answer

Answer: (d) $$\frac{1}{3}$$

Question 63.
The derivative of ex3 with respect to log x is
(a) ee3
(b) 3x22ex3
(c) 3x3ex3
(d) 3x2ex3+ 3x2

Answer

Answer: (c) 3x3ex3

Question 64.
If x = et sin t, y = etcos t, t is a parameter, then $$\frac{dy}{dx}$$ at (1, 1) is equal to
(a) –$$\frac{1}{2}$$
(b) –$$\frac{1}{4}$$
(c) 0
(d) $$\frac{1}{2}$$

Answer

Answer: (c) 0

Question 65.
The derivative of sin-1 ($$\frac{2x}{1+x^2}$$) with respect to cos-1 ($$\frac{1-x^2}{1+x^2}$$) is
(a) -1
(b) 1
(c) 2
(d) 4

Answer

Answer: (b) 1