M.C.Q (1 Marks)

MCQ 11 Mark

$\frac{d}{d x}\left(\log _a x\right)$ is equal to :

✓
$\frac{1}{x \cdot \log _e a}$
B
$\frac{\log _e a}{x}$
C
$\frac{1}{x}$
D
$\frac{x}{\log _e a}$

Answer

Correct option: A.

$\frac{1}{x \cdot \log _e a}$

MCQ 21 Mark

If $\lim _{x \rightarrow 0}\left(1+\frac{a}{x}+\frac{b}{x^2}\right)^{2 x}=e^2$, then values of $a$ and $b$ are equal to :

A
$a \in R , b=2$
✓
$a=1, b \in R$
C
$a \in R , b \in R$
D
$a=1, b=2$

Answer

Correct option: B.

$a=1, b \in R$

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MCQ 31 Mark

Evaluate $\lim _{x \rightarrow \infty} \frac{5 x^2+3 x+2}{x^2+x+1} :$

A
1
B
2
C
3
✓
5

Answer

Correct option: D.

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MCQ 41 Mark

$\lim _{x \rightarrow 0} \frac{\sqrt{1-\cos 2 x}}{\sqrt{2} x}$ is equal to :

✓
$1$
B
$-1$
C
$0$
D
does not exist

Answer

Correct option: A.

$1$

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MCQ 51 Mark

If $f(x)=\left\{\begin{array}{cc}2 x+b, & x<\alpha \\ x+d, & x \geq \alpha\end{array}\right.$ is such that $\lim _{x \rightarrow \alpha} f(x)=l$, then $l$ is equal to :

✓
$2 d-b$
B
$2 b-d$
C
$2 d+b$
D
$b-2 d$

Answer

Correct option: A.

$2 d-b$

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MCQ 61 Mark

If $\lim _{x \rightarrow 0} \frac{\log (3+x)-\log (3-x)}{x}= K$, then value of K is equal to :

A
$0$
B
$-\frac{1}{3}$
✓
$\frac{2}{3}$
D
$-\frac{2}{3}$

Answer

Correct option: C.

$\frac{2}{3}$

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MCQ 71 Mark

If $\lim _{x \rightarrow 0} \frac{[(a-n) n x-\tan x] \sin x}{x^2}=0$, where $n$ is non -zero real number, then $a$ is equal to :

A
$0$
B
$\frac{n+1}{n}$
C
$n$
✓
$n+\frac{1}{n}$

Answer

Correct option: D.

$n+\frac{1}{n}$

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MCQ 81 Mark

If $f(x)=\frac{2 x+c}{x-1}$ and $f^{\prime}(0)=0$, then value of $c$ is equal to :

A
$0$
B
1
C
2
✓
-2

Answer

Correct option: D.

-2

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MCQ 91 Mark

Derivative of $\sec x^{o}$ is equal to :

A
$\sec x^{\circ} \tan x^{\circ}$
B
$\frac{\pi}{180} \sec x \tan x$
✓
$\frac{\pi}{180} \sec x^{\circ} \tan x^{\circ}$
D
$\sec x \tan x$

Answer

Correct option: C.

$\frac{\pi}{180} \sec x^{\circ} \tan x^{\circ}$

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MCQ 101 Mark

If $f(x)=x^3+6 x^2-5$, then $f^{\prime}(1)$ is equal to :

A
$0$
B
9
C
4
✓
15

Answer

Correct option: D.

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MCQ 111 Mark

Derivative of $\log _e \sqrt{x}$ is equal to :

✓
$\frac{1}{2 x}$
B
$\frac{1}{2 \sqrt{x}}$
C
$2 \sqrt{x}$
D
$\frac{1}{2} \sqrt{x}$

Answer

Correct option: A.

$\frac{1}{2 x}$

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MCQ 121 Mark

$\frac{d}{d x}\left(5^x\right)$ is equal to :

A
$5^x$
B
$10^x$
C
$10^x \log _e 5$
✓
$5^x \log _e 5$

Answer

Correct option: D.

$5^x \log _e 5$

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MCQ 131 Mark

Derivative of $\frac{1}{\sqrt{x}}$ is equal to :

A
$\frac{1}{2 x \sqrt{x}}$
✓
$-\frac{1}{2 x \sqrt{x}}$
C
$2 x \sqrt{x}$
D
$-2 x \sqrt{x}$

Answer

Correct option: B.

$-\frac{1}{2 x \sqrt{x}}$

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MCQ 141 Mark

Derivative of $x^n$ is equal to :

A
$x^{n-1}$
B
$(n-1) ~x^{n-2}$
✓
$n~ x^{n-1}$
D
$\frac{x^{n+1}}{n+1}$

Answer

Correct option: C.

$n~ x^{n-1}$

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MCQ 151 Mark

Value of $\lim _{x \rightarrow 0} \frac{e^{\frac{1}{x}}}{x}$ is :

A
$0$
B
$\infty$
C
$1$
✓
does not exist.

Answer

Correct option: D.

does not exist.

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MCQ 161 Mark

Evaluate $\lim _{x \rightarrow \pi} \frac{1+\cos x}{\tan ^2 x}:$

A
$-\frac{1}{2}$
B
$0$
✓
$\frac{1}{2}$
D
$1$

Answer

Correct option: C.

$\frac{1}{2}$

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MCQ 171 Mark

Evaluate $\lim _{x \rightarrow 0} \frac{x^2-2 x}{2 \sin x}$ :

A
$2$
B
$1$
C
$0$
✓
$-1$

Answer

Correct option: D.

$-1$

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MCQ 181 Mark

The value of $\lim _{m \rightarrow \infty}\left(1-\frac{1}{m}\right)^{-m}$ is equal to :

A
$0$
B
$\infty$
✓
$e$
D
$e^{-1}$

Answer

Correct option: C.

$e$

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MCQ 191 Mark

The value of $\lim _{x \rightarrow 0} \frac{e^x-e^{-x}}{x}$ is equal to :

A
$0$
✓
$2$
C
$1$
D
$-1$

Answer

Correct option: B.

$2$

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MCQ 201 Mark

Evaluate $\lim _{x \rightarrow 1}\left(\frac{x^n-1}{x-1}\right)$ :

A
$n^2$
✓
$n$
C
$n-1$
D
$n+1$

Answer

Correct option: B.

$n$

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MCQ 211 Mark

$\lim _{x \rightarrow \infty} \frac{\sin 2 x}{2 x}$ is equal to :

✓
$0$
B
$\frac{1}{2}$
C
1
D
2

Answer

Correct option: A.

$0$

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MCQ 221 Mark

Evaluate $\lim _{x \rightarrow 0} \frac{\sin x^{\circ}}{x}$ :

A
$0$
B
$1$
✓
$\frac{\pi}{180}$
D
$\pi$

Answer

Correct option: C.

$\frac{\pi}{180}$

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MCQ 231 Mark

Evaluate $\lim _{x \rightarrow 0}\left[\frac{\sin \alpha x}{\sin \beta x}\right]^{\frac{1}{n}}$ :

✓
$\left(\frac{\alpha}{\beta}\right)^{\frac{1}{n}}$
B
$\left(\frac{\beta}{\alpha}\right)^{\frac{1}{n}}$
C
$\left(\frac{\alpha}{\beta}\right)^n$
D
$\left(\frac{\beta}{\alpha}\right)^n$

Answer

Correct option: A.

$\left(\frac{\alpha}{\beta}\right)^{\frac{1}{n}}$

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MCQ 241 Mark

Which of the following limit does not exist?

A
$\lim _{x \rightarrow 0} \cos \frac{1}{x}$
B
$\lim _{x \rightarrow \frac{\pi}{2}} \tan x$
C
$\lim _{x \rightarrow \infty}[x]$
✓
None of these

Answer

Correct option: D.

None of these

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MCQ 251 Mark

Evaluate $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 3 x}$ :

A
$\frac{3}{4}$
✓
$\frac{4}{3}$
C
$0$
D
1

Answer

Correct option: B.

$\frac{4}{3}$

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