MCQ Questions for Class 11 Commerce Applied Mathematics Limits And Continuity Quiz 13

$\displaystyle \lim _{x \rightarrow 0} \dfrac{x^{n}-\sin x^{n}}{x-\sin ^{n} x}$ is non-zero finite, then

$n$ must be equal

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4
0%
1
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2
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3

$L=\displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}}=\infty$

Equation

$a x^{2}+b x+c=0$ has

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real and equal roots
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complex roots
0%
unequal positive real roots
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unequal roots

$\displaystyle \lim _{x \rightarrow \infty} \dfrac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}}$ is equal to

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0
0%
1
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-1
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Does not exists

Value of

$L=\displaystyle\lim_{n\rightarrow \infty n}\dfrac{1}{4}\left[1.\left(\displaystyle\sum_{k=1}^{n}k\right)+2.\left(\sum_{k=1}^{n-1}k\right)+3.\left(\sum_{k=1}^{n-2}k\right)+...+n.1\right]$ is

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$1/24$
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$1/12$
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$1/6$
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$1/3$

The value of

$\lim _{n \rightarrow \infty}\left[\tan \dfrac{\pi}{2 n} \tan \dfrac{2 \pi}{2 n} \cdots \tan \dfrac{n \pi}{2 n}\right]^{1 / n}$ is

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$e$
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$e^2$
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$1$
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$e^3$

Explanation

$\operatorname{Let} A=\lim _{n \rightarrow \infty}\left[\tan \dfrac{\pi}{2 n} \tan \dfrac{2 \pi}{2 n} \cdots \tan \dfrac{n \pi}{2 n}\right]^{1 / n}\\$

$\therefore \log A=\lim _{n \rightarrow \infty} \dfrac{1}{n}\left[\log \tan \dfrac{\pi}{2 n}+\log \tan \dfrac{2 \pi}{2 n}\right.\\$

$+\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \dfrac{1}{n} \log \tan \dfrac{\pi r}{2 n}=\int_{0}^{1} \log \tan \left(\dfrac{\pi}{2} x\right) d x\\$

$=\dfrac{2}{\pi} \int_{0}^{\pi / 2} \log \tan y d y$

$I=\int_{0}^{\pi / 2} \log \tan \left(\dfrac{1}{2} \pi-y\right) d y\\$

$(\text{ by Property } \mathrm{IV})\\$

$\begin{array}{l}=\int_{0}^{\pi / 2} \log \cot y d y \\=-\int_{0}^{\pi / 2} \log \tan y d y=-I\end{array}$

$\operatorname{or} I+I=0$ or

$2 I=0$ or

$I=0\\$
from equation

$(1), \log A=0 \therefore A=e^{0}=1$

$\displaystyle \lim _{x \rightarrow \pi / 2} \dfrac{\sin (x \cos x)}{\cos (x \sin x)}$ is equal to

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0
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p/2
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p
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2p

The value of

$\displaystyle \lim _{x \rightarrow 0} \dfrac{\sqrt{\dfrac{1}{2}(1-\cos 2 x)}}{x}$ is

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1
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-1
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0
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None of these

The value of

$\displaystyle \lim_{n\infty}\dfrac{1}{n^2}\left\{ sin^3\dfrac{\pi}{4n}+2sin^3\dfrac{2\pi}{4n} + ... + nsin^3\dfrac{n\pi}{4n}\right\}$ is equal to

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$\dfrac{\sqrt{2}}{9\pi^2}(52-15 \pi )$
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$\dfrac{2}{9\pi^2}(52-15n)$
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$\dfrac{1}{9\pi^2}(15n-15)$
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None of these

CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity Quiz 13 - MCQExams.com

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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity Quiz 13 - MCQExams.com

0:0:2

Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

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