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Continuity and Differentiability Class 12 MCQs Questions with Answers
Students are advised to solve the Continuity and Differentiability Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Continuity and Differentiability Class 12 with answers will boost your confidence thereby helping you score well in the exam.
Explore numerous MCQ Questions of Continuity and Differentiability Class 12 with answers provided with detailed solutions by looking below.
Question 1.
The function
f(x) = 
is continuous at x = 0, then the value of ‘k’ is:
(a) 3
(b) 2
(c) 1
(d) 1.5.
Answer
Answer: (b) 2
Question 2.
The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at:
(a) 4
(b)-2
(c) 1
(d) 1.5.
Answer
Answer: (d) 1.5.
Question 3.
The value of ‘k’ which makes the function defined by
continuous at x = 0 is
(a) -8
(b) 1
(c) -1
(d) None of these.
Answer
Answer: (d) None of these.
Question 4.
Differential coefficient of sec (tan-1 x) w.r.t. x is
(a) \(\frac { x }{\sqrt{1+x^2}}\)
(b) \(\frac { x}{1+x^2}\)
(c) x\(\sqrt { 1+x^2}\)
(d) \(\frac { 1 }{\sqrt{1+x^2}}\)
Answer
Answer: (a) \(\frac { x }{\sqrt{1+x^2}}\)
Question 5.
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 6.
If y = \(\sqrt { sin x+ y}\), then \(\frac { dy }{dx}\) is equal to
(a) \(\frac { cos x }{2y-1}\)
(b) \(\frac { cos x}{1-2y}\)
(c) \(\frac {sin x}{1-2y}\)
(d) \(\frac { sin x }{2y-1}\)
Answer
Answer: (a) \(\frac { cos x }{2y-1}\)
Question 7.
If u = sin-1 (\(\frac { 2x }{1+x^2}\)) and u = tan-1 (\(\frac { 2x }{1-x^2}\)) then \(\frac { dy }{dx}\) is
(a) \(\frac { 1 }{2}\)
(b) x
(c) \(\frac {1-x^2}{1+x^2}\)
(d) 1
Answer
Answer: (d) 1
Question 8.
If x = t², y = t³, then \(\frac { d^2y }{dx^2}\) is
(a) \(\frac { 3 }{2}\)
(b) \(\frac { 3 }{4t}\)
(c) \(\frac {3}{2t}\)
(d) \(\frac { 3t }{2}\)
Answer
Answer: (b) \(\frac { 3 }{4t}\)
Question 9.
The value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in the interval [0, √3] is
(a) 1
(b) -1
(c) \(\frac {3}{2}\)
(d) \(\frac {1}{3}\)
Answer
Answer: (a) 1
Question 10.
The value of ‘c’ in Mean Value Theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is
(a) \(\frac {3}{2}\)
(b) \(\frac {2}{3}\)
(c) \(\frac {1}{2}\)
(d) \(\frac {3}{4}\)
Answer
Answer: (a) \(\frac {3}{2}\)
Question 11.
Let f : (- 1, 1) → R be a differentiable function with f(0) = – 1 and f'(0) = 1.
Let g(x) = [f (2f(x) + 2)]². Then g'(0) =
(a) 4
(b) -4
(c) log 2
(d) -log 2.
Answer
Answer: (b) -4
Hint:
Here g (x)= [f (2 f(x) + 2)]²
g'(x) = 2[f(2f(x) + 2)] \(\frac { d }{dx}\) [2f(x) + 2 ]
= 2f(2f(x) + 2) . [2 f'(x)]
∴ g'(0) = 2f(2f(0) + 2) . [2f'(0)]
= 2f(2 (-1) +2). 2f’/(0)
= 2f(0) . 2f'(0) = 4f(0) f'(0)
= 4 (-1) (1) = -4.
Question 12.
\(\frac { d^2x }{dy^2}\) equals
Answer
Answer: d
Hint:
Question 13.
If function f(x) is differentiable at x = a, then
\(\lim _{x \rightarrow a}\) \(\frac { x^2 f(a) – a^2 f(x) }{x-a}\) is
(a) a² f(a)
(b) af(a) – a² f'(a)
(c) 2a f(a) – a² f'(a)
(d) 2a f (a) + a² f'(a).
Answer
Answer: (c) 2a f(a) – a² f'(a)
Hint:
Question 14.
If f: R → R is a function defined by
f(x) = [x] cos (\(\frac { 2x-1 }{2}\))π, where [x] denotes the greatest integer function, then ‘f’ is
(a) continuous for every real x
(b) discontinuous only at x = 0
(c) discontinuous only at non-zero integral values of x
(d) continuous only at x = 0.
Answer
Answer: (a) continuous for every real x
Hint:
Continuous for every real x.
Question 15.
If y = sec (tan-1 x), then \(\frac { dy }{dx}\) at x = 1 is equal to
(a) \(\frac {1}{2}\)
(b) 1
(c) √2
(d) \(\frac {1}{√2}\)
Answer
Answer: (d) \(\frac {1}{√2}\)
Hint:
Here y = sec (tan-1 x).
∴ \(\frac { dy }{dx}\) = sec (tan-1 x) tan (tan-1 x). \(\frac { 1 }{1+x^2}\)
= sec (tan-1 x). x . (tan-1 x). \(\frac { 1 }{1+x}\)
\(\left.\frac{d y}{d x}\right]_{x=1}\) = sec (tan-1 1). \(\frac { 1 }{1+1}\)
= sec (\(\frac { π }{4}\)) \(\frac { 1 }{2}\) = \(\frac { √2 }{2}\) = \(\frac { 1 }{√2}\)
Question 16.
If g is the inverse of a function f and f'(x) = \(\frac { 1 }{1+x^5}\), then g'(x) is equal to
(a) 5x4
(b) \(\frac {1}{1+{g(x)}^5}\)
(c) 1 + {g(x)}5
(d) 1 + x5
Answer
Answer: (b) \(\frac {1}{1+{g(x)}^5}\)
Hint:
Here f(g(x)) = x. [∵ g is the inverse of f]
f'(g(x)) g'(x) = 1
⇒ g'(x) = \(\frac { 1 }{f'{g(x)}}\) = \(\frac { 1 }{1+{g(x)}^5}\)
Question 17.
If the function
g(x) = \(\left\{\begin{array}{ll}
k \sqrt{x+1} & ; 0 \leq x \leq 3 \\
m x+2 & ; 3 \end{array}\right.\)
is differentiable, then the value of k + m is
(a) 2
(b) \(\frac {16}{5}\)
(c) \(\frac {10}{3}\)
(d) 4
Answer
Answer: (a) 2
Hint:
We have
g(x) = \(\left\{\begin{array}{ll}
k \sqrt{x+1} & ; 0 \leq x \leq 3 \\
m x+2 & ; 3 \end{array}\right.\)
When this function is differentiable, then it is continuous
⇒ \(\lim _{x \rightarrow 3^{-}}\) g(x) = \(\lim _{x \rightarrow 3^{+}}\) g(x) = g(3)
⇒ 2k = 3m + 2 = 2k
⇒ 2k = 3m + 2 ………… (1)
Also, LHD = \(\lim _{x \rightarrow 3^{-}}\) g(x) = \(\frac {k}{4}\)
RHD = \(\lim _{x \rightarrow 3^{+}}\) g(x) = m
∴ LHD = RHD k
⇒ \(\frac {k}{4}\) = m
Solving (1) and (2),
k = \(\frac {8}{5}\) and m = \(\frac {2}{5}\)
Hence, k + m = \(\frac {8}{5}\) + \(\frac {2}{5}\) = \(\frac {10}{2}\) = 2.
Question 18.
For x ∈ R, f(x) = |log 2 – sin x| and g(x) =f(f(x)), then
(a) g is not differentiable at x = 0
(b) g'(0) = cos (log 2)
(c) g'(0) = -cos (log 2)
(d) g is differentiable at x = 0 and g'(0) = – sin (log 2).
Answer
Answer: (b) g'(0) = cos (log 2)
Hint:
We have : f(x) = log 2 – sin x
and g(x) = f(f cos x)
= log 2 – sin (log 2 – sin x).
Since ‘g’ is the sum of two differentiable functions,
∴ g is differentiable.
g'(x) = 0 – cos (log 2 – sin x) (0 – cos x)
= cos (log 2 – sin x) cos x.
Hence, g’ (x) = cos (log 2).
Fill in the blanks
Question 1.
If f(x) = \(\left\{\begin{array}{c}
\frac{x^{2}-1}{x-1}, \text { when } x \neq 1 \\
k, \text { when } x=1
\end{array}\right.\) is continuous then the value of k = …………………
Answer
Answer: 2.
Question 2.
If f(x) = x + 7, and g(x) = x – 7, x ∈R, then \(\frac { d }{dx}\) (fog) (x) = ……………….
Answer
Answer: 1.
Question 3.
If 2x + 3y = sin x, then \(\frac { dy }{dx}\) = …………………..
Answer
Answer: \(\frac { cos x-2 }{3}\)
Question 4.
\(\frac { d }{dx}\) (cosec-1 x) = …………………
Answer
Answer: \(\frac { -1 }{|x|\sqrt{x^2-1}}\)
Question 5.
\(\frac { d }{dx}\) (\(\sqrt { e^{ \sqrt{x}} }\)) = …………………
Answer
Answer: \(\frac { 1 }{4√x}\) \(\sqrt { e ^{\sqrt{x}} }\)
Question 6.
If x = at², y = 2at, then \(\frac { dy }{dx}\) = ……………….
Answer
Answer: \(\frac { 1 }{t}\)
Question 7.
The derivative of xx w.r.t. x is.
Answer
Answer: xx (1 + log x).
Question 8.
If y = x² + 3x + 2, then \(\frac { d^2y }{dx^2}\) = ………………
Answer
Answer: 2.
Question 9.
Value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in [-√3, 0] is ……………..
Answer
Answer: c = -1.
Question 10.
Value of ‘c’ in LMV Theorem for f(x) = x² in [2, 4] is …………………
Answer
Answer: c = 3.
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