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

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

Q1.

$\underset{x \to \infty}{L t} {(1 + \frac{3}{x})}^{x}$ is equal to

A. none of these.

B. 3 e

C. e1/3

D. e3

Answer : Option D
Explaination / Solution:

lim_{x \to \infty} {(1 + \frac{3}{x})}^{x}

= lim_{t \to 0^{+}} {(1 + 3 t)}^{\frac{1}{t}} = lim_{3 t \to 0^{+}} {[{(1 + 3 t)}^{\frac{1}{3 t}}]}^{3} = e^{3}

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

Q2. If

y = e^{2 x + 3 \log x}

, then

\frac{d y}{d x}

is equal to

A. x2e2x(2+3x)

B. x2e2x(2x+3)

C. None of these

D. x3e2x

Answer : Option B
Explaination / Solution:

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

Q3. If

y = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}},

then

\frac{d y}{d x}

is equal to

A. y2+1

B. 1−y2

C. 1+y2

D. none of these.

Answer : Option C
Explaination / Solution:

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

Q4.

\frac{d}{d x} (\log | x |)

is equal to

(x \neq 0)

A. ±1/x

B. 1/x or −1/x

C. 1/x

D. 1/|x|

Answer : Option C
Explaination / Solution:

ddx(log|x|)=1|x|x|x|=x|x|2=xx2=1x

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

Q5. If f(x) be any function which assumes only positive values and f’ (x) exists then f’ (x) is equal to

A. f(x)ddx{elog(f(x))}

B. f(x)ddx{logf(x))}

C. none of these

D. f(x)ddx(ef(x))

Answer : Option B
Explaination / Solution:

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

Q6.

\frac{d}{d x} (\log | \tan \frac{x}{2} |)

is equal to

A. cosec x

B. none of these

C. 2sinx

D. 1tanx2

Answer : Option A
Explaination / Solution:

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

Q7. The differential coefficient of log (| log x |) w.r.t. log x is

A. 1x|logx|

B. none of these

C. 1logx

D. 1xlogx

Answer : Option C
Explaination / Solution:

Let y =

\log (| l o g x |) a n d z = l o g x,

then , we have ;

\frac{d y}{d z} = \frac{\frac{d y}{d x}}{\frac{d z}{d x}} = (\frac{1}{\log x} . \frac{1}{x}) / (\frac{1}{x})

\Rightarrow \frac{1}{\log x}

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

Q8. If x = a

c o s^{3} t, y = a s i n^{3}

then

\frac{d y}{d x}

is equal to

A. cos t

B. – tan t.

C. cot t

D. cosec t

Answer : Option B
Explaination / Solution:

dydx=dydtdxdt=3asin2tcost3acos2t(−sint)=−tant

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

Q9.

Derivative of $\sin^{3} x w . r . t . \cos^{3} x i s$

\sin^{3} x w . r . t . \cos^{3} x i s

A. – tan x

B. tan x

C. cot x.

D. tan3x

Answer : Option A
Explaination / Solution:

L e t y = s i n^{3} x a n d z = c o s^{3} x,

then ,

\frac{d y}{d z} = \frac{\frac{d y}{d x}}{\frac{d z}{d x}} = \frac{3 \sin^{2} x \cos x}{3 \cos^{2} x (- \sin x)} = - \tan x

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

Q10. Derivative of cos (sin x) w.r.t. sin x is

A. sec2(xx2+1√)

B. none of these

C. – sin (sin x)cosx

D. – sin (sin x)

Answer : Option D
Explaination / Solution:

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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