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Continuity and Differentiability Class 12 MCQs Questions with Answers
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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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