Continuity and Differentiability Online Objective Test | Continuity and Differentiability Online Objective Test

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q1. Let f and g be differentiable functions such that fog = I, the identity function. If g’ (a) = 2 and g (a) = b, then f ‘ (b) =

A. -2

B. 2

C. 1/2

D. none of these.

Answer : Option C
Explaination / Solution:

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q2. Differential coefficient of a function f (g (x)) w.r.t. the function g (x) is

A. none of these

B. f ‘ (g (x)) g’ (x)

C. f ‘ (g (x))

D. f ‘ (g (x))/g’ (x)

Answer : Option C
Explaination / Solution:

dd(g(x))(f(g(x))=f′(g(x))

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q3. If a function f is derivable at x = a, then

\underset{h \to 0}{L e t} \frac{f (a - h) - f (a)}{h}

is equal to

A. none of these.

B. does not exist

C. – f ‘ (a)

D. f ‘ (a)

Answer : Option C
Explaination / Solution:

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q4. If

y = \tan^{- 1}

x and

z = \cot^{- 1} x

then

\frac{d y}{d z}

is equal to

A. 1

B. none of these

C. π/2

D. -1

Answer : Option D
Explaination / Solution:

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q5.

\frac{d}{d x} (\cos^{- 1} x) = - \frac{1}{\sqrt{1 - x^{2}}}

where

A. −1

B. −1⩽x⩽1

C. −1

D. −1⩽x<1

Answer : Option A
Explaination / Solution:

\frac{d}{d x} (\cos^{- 1} x) = - \frac{1}{\sqrt{1 - x^{2}}}

is valid only if ,

1 - x^{2} > 0,

i.e. if

x^{2} < 1 i . e . i f | x | < 1

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q6. If f (x) =

x t a n^{- 1}

x then f ‘ (1 ) is equal to

A. none of these.

B. π/4 + 1/2

C. 1/2 − π/4

D. π/4 − 1/2

Answer : Option B
Explaination / Solution:

\begin{matrix} f^{'} (x) = \frac{d}{d x} (x \tan^{- 1} x) = \frac{x}{1 + x^{2}} + \tan^{- 1} x \end{matrix}

\Rightarrow f^{'} (1) = \frac{1}{1 + 1} + \tan^{- 1} 1 = \frac{1}{2} + \frac{π}{4}

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q7.

\frac{d}{d x} (\tan^{- 1} (\cot x))

is equal to

A. none of these.

- c o s s e c^{2}

C. -1

s i n^{2}

Answer : Option C
Explaination / Solution:

\frac{d}{d x} (\tan^{- 1} (\cot x))

\frac{d}{d x} (\tan^{- 1} (\tan (\frac{π}{2} - x))) = \frac{d}{d x} (\frac{π}{2} - x) = - 1

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q8.

\frac{d}{d x} (\tan^{- 1} (\sec x + \tan x)

is equal to

A. none of these.

B. -1/2

D. 1/2

Answer : Option D
Explaination / Solution:

\frac{d}{d x} (\tan^{- 1} (\sec x + \tan x))

= \frac{\sec x \tan x + \sec^{2} x}{1 + {(\sec x + \tan x)}^{2}} = \frac{\sec x (\sec x + \tan x)}{2 \sec x (\sec x + \tan x)} = \frac{1}{2}

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q9.

\underset{x \to 0}{L t} (1 + 2 x)^{\frac{x + 3}{x}}

is equal to

A. none of these.

B. e3

C. e3/2

Answer : Option D
Explaination / Solution:

# Continuity and Differentiability # CBSE 12th Mathematics Prepare / Learn

Q10.

A. is equal to non – zero real number

B. none of these

C. does not exist

D. is equal to 0

Answer : Option D
Explaination / Solution:

\begin{matrix} lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{-}} e^{\frac{1}{x}} = 0, \end{matrix}

lim_{x \to 0^{+}} f (x) = lim_{x \to 0^{+}} x = 0 ∴ lim_{x \to 0} f (x) = 0

Total Question/Mark :
Scored Mark :
Mark for Correct Answer : 1
Mark for Wrong Answer : -0.5
Mark for Left Answer : 0

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