Test 1: Calculus

Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom of the page

Total Marks: 100

1) Define a sequence $s_n$ of real numbers by $s_n=\sum _{i=1}^n{\displaystyle \frac{(\log (n+i)-\log n{)}^2}{n+i}}$. Then, the value of $\underset{n\to \mathrm{\infty }}{lim}{s}_n$ is equal to:

(a) $0$

(b) ${\displaystyle \frac{(\log 2{)}^3}{4}}$

(c) ${\displaystyle \frac{(\log 2{)}^3}{3}}$

(d) Limit does not exist

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2) Let

$L=\underset{n\to \mathrm{\infty }}{lim}{(1+{\displaystyle \frac{1}{n}})}^n$. Then, $L$ lies in the interval

(a) $(0,1)$

(b) $(1,2)$

(c) $(2,3)$

(d) $(3,4)$

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3) If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then ${\displaystyle \frac{dy}{dx}}$ is equal to

(a) 1

(b) ${\displaystyle \frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}}$

(c) 0

(d) ${\displaystyle \frac{\sqrt{1-x^2}}{\sqrt{1-y^2}}}$

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4) If ${\mathrm{I}}_n={\displaystyle \frac{d^n}{d{x}^n}}(x^n\log x)$, then $I_n-n{I}_{n-1}$ is equal to:

(a) 0

(b) 1

(c) $n!$

(d) $(n-1)!$

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5) The maximum distance of the normal from the centre of the ellipse

is given by

(a) $\sqrt{a^2+b^2}$

(b) $(a-b)$

(c) ${\displaystyle \frac{1}{2}}\sqrt{a^2+b^2}$

(d) ${\displaystyle \frac{1}{2}}(a-b)$

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6) Find the values of $a$ and $b$ in order that

may be equal to 1.

(a) $a={\displaystyle \frac{-5}{2}}$, $b={\displaystyle \frac{-3}{2}}$

(b) $a={\displaystyle \frac{-5}{8}}$, $b={\displaystyle \frac{-3}{8}}$

(c) $a={\displaystyle \frac{-5}{4}}$, $b={\displaystyle \frac{-3}{4}}$

(d) $a={\displaystyle \frac{5}{2}}$, $b={\displaystyle \frac{3}{2}}$

7) If $z$ is a homogeneous function of $x$, $y$ of order $n$. Consilder the following statements:

(i): $x{\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}+y{\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }y}}=nf$

(ii): $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}}+2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}}+y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2}}=n^{2}f$

Which of the following options is correct?

(a) (i) is correct but (ii) is incorrect

(b) (ii) is correct but (i) is incorrect

(c) Both (i) and (ii) are correct

(d) Both (i) nor (ii) are incorrect

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8) The extreme value of $xy(a-x-y)$ is given by:

(a) $(0,0)$

(b) $(0,a)$

(c) $({\displaystyle \frac{a}{3}},{\displaystyle \frac{a}{3}})$

(d) The given function has no extreme values

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9) Let If $f^{\mathrm{\prime }}(x)={\displaystyle \frac{1}{1+x^2}}$ for all $x$ and $f(0)=0$. Then, $f(2)$ lies in the interval:

(a) $(0.4,2)$

(b) $(3,4)$

(b) $(4,8)$

(b) $(0,1)$

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10) If $u={\displaystyle \frac{x+y}{1-xy}}$ and $v={\tan }^{-1}x+{\tan }^{-1}y$, then,

(a) $u=\tan v$

(b) $v=\tan u$

(c) $u=\cot v$

(d) $v=\cot u$

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11) Let $f(x,y)=\{\begin{array}{ll}{x}^2{\tan }^{-1}{\displaystyle \frac{y}{x}}-y^2{\tan }^{-1}{\displaystyle \frac{x}{y}},& x\ne 0,y\ne 0\\ 0,& x=y=0\end{array}$

Then,

(a) $f(x,y)$ is continuous and differentiable at $(0,0)$

(b) $f(x,y)$ is continuous but not differentiable at $(0,0)$

(c) $f(x,y)$ is neither continuous nor differentiable at $(0,0)$

(d) $f(x,y)$ is diffrentiable but not continuous at $(0,0)$

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12) The volume generated by revolving $y^2=4ax$ about the latus rectum is given by:

(a) ${\displaystyle \frac{32\pi a^3}{15}}$

(b) ${\displaystyle \frac{16\pi a^3}{15}}$

(c) ${\displaystyle \frac{8\pi a^3}{15}}$

(d) ${\displaystyle \frac{4\pi a^3}{15}}$

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13) The value of ${\int }_1^x{\displaystyle \frac{tdt}{1+t^2}}+{\int }_1^{{\displaystyle \frac{1}{x}}}{\displaystyle \frac{dt}{t(1+t^2)}}$ is given by:

(a) $1$

(b) $\tan x$

(c) ${\tan }^{-1}x$

(d) $0$

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14) The maximum and minimum values of $f(x,y)=7x^2+8xy+y^2$, where $x$, $y$ are constrained by the relation $x^2+y^2=1$, are given by:

(a) Max= 9, Min=1

(b) Max= 9, Min= -1

(c) Max= 8, Min=2

(d) Max= 8, Min=-1

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15) Let $f(x)=x$ if $x$ is rational, and $1-x$ if $x$ is irrational. Then,

(a) $f(x)$ is continuous at all points

(b) $f(x)$ is discontinuous at all points

(c) $f(x)$ has a removable discontinuity at $x={\displaystyle \frac{1}{2}}$

(d) $f(x)$ is continuous at $x={\displaystyle \frac{1}{2}}$

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16) The value of

over the area of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ is given by:

(a) $\pi ab$

(b) ${\displaystyle \frac{\pi ab}{5}}$

(c) ${\displaystyle \frac{2\pi ab}{5}}$

(d) ${\displaystyle \frac{4\pi ab}{5}}$

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17) Let the volume of a right circular cylinder of greatest volume which can be inscribed in a sphere of volume $V$ is given by $v$. Then, $V/v$ is given by:

(a) $2$

(b) $\sqrt{2}$

(c) $3$

(d) $\sqrt{3}$

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18) The value of the integral ${\int }_0^{{\displaystyle \frac{\pi }{2}}}\log (\sin x)dx$ is given by:

(a) Integral is divergent

(b) ${\displaystyle \frac{-\pi }{2}}\log 2$

(c) ${\displaystyle \frac{\pi }{2}}\log 2$

(d) 0

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19) The percentage error in the volume of a right circular cone, when an error of $1\mathrm{\%}$ is made in measuring the height and an error of $0.5\mathrm{\%}$ is made in measuring the base radius, is given by:

(a) $2$

(b) $0.5$

(c) $1$

(d) ${\displaystyle \frac{\pi }{3}}$

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20) The value of ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\log (1+a^2{x}^2)}{1+b^2{x}^2}}dx$ is given by:

(a) ${\displaystyle \frac{\pi }{b}}\log \left({\displaystyle \frac{b+a}{b}}\right)$

(b) ${\displaystyle \frac{\pi }{a}}\log \left({\displaystyle \frac{b+a}{b}}\right)$

(c) ${\displaystyle \frac{2\pi }{b}}\log \left({\displaystyle \frac{b+a}{b}}\right)$

(d) ${\displaystyle \frac{\pi }{2b}}\log \left({\displaystyle \frac{b+a}{b}}\right)$