The shortest distance between the line
Question: The shortest distance between the line $x-y=1$ and the curve $x^{2}=2 y$ is :$\frac{1}{2}$$\frac{1}{2 \sqrt{2}}$$\frac{1}{\sqrt{2}}$0Correct Option: , 2 Solution: Shortest distance between curves is always along common normal. $\left.\frac{d y}{d x}\right|_{P}=$ slope of line $=1$ $\Rightarrow x_{0}=1$ $\therefore \mathrm{y}_{0}=\frac{1}{2}$ $\Rightarrow \mathrm{P}\left(1, \frac{1}{2}\right)$ $\therefore$ Shortest distance $=\left|\frac{1-\frac{1}{2}-1}{\sqrt{1^{2}+1^{2}}}\right|=\frac...
Read More →Let (x) =
Question: Let $f(x)=\sin ^{-1} x$ and $g(x)=\frac{x^{2}-x-2}{2 x^{2}-x-6}$. If $g(2)=\lim _{x \rightarrow 2} g(x)$, then the domain of the function $f \circ g$ is :$(-\infty,-2] \cup\left[-\frac{3}{2}, \infty\right)$$(-\infty,-2] \cup[-1, \infty)$$(-\infty,-2] \cup\left[-\frac{4}{3}, \infty\right)$$(-\infty,-1] \cup[2, \infty)$Correct Option: , 3 Solution: Domain of $f \circ g(x)=\sin ^{-1}(g(x))$ $\Rightarrow|g(x)| \leq 1 \quad, \quad g(2)=\frac{3}{7}$ $\left|\frac{x^{2}-x-2}{2 x^{2}-x-6}\right...
Read More →Solve the following
Question: The integral $\int \frac{\mathrm{e}^{3 \log _{e} 2 x}+5 \mathrm{e}^{2 \log _{e} 2 x}}{\mathrm{e}^{4 \log _{e} x}+5 \mathrm{e}^{3 \log _{e} x}-7 \mathrm{e}^{2 \log _{e} x}} \mathrm{dx}, x0$, is equal to : (where $c$ is a constant of integration)$\log _{e}\left|x^{2}+5 x-7\right|+c$$4 \log _{e}\left|x^{2}+5 x-7\right|+c$$\frac{1}{4} \log _{e}\left|x^{2}+5 x-7\right|+c$$\log _{e} \sqrt{x^{2}+5 x-7}+c$Correct Option: , 2 Solution: $\int \frac{\mathrm{e}^{3 \log _{\mathrm{e}} 2 \mathrm{x}}+...
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Question: Let $\mathrm{M}$ be any $3 \times 3$ matrix with entries from the set $\{0,1,2\}$. The maximum number of such matrices, for which the sum of diagonal elements of $\mathrm{M}^{\mathrm{T}} \mathrm{M}$ is seven, is Solution: $\left[\begin{array}{lll}a b c \\ d e f \\ g h i\end{array}\right]\left[\begin{array}{lll}a d g \\ b e h \\ c f i\end{array}\right]$ $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}+g^{2}+h^{2}+i^{2}=7$ Case-I : Seven (1's) and two ( 0 's) ${ }^{9} \mathrm{C}_{2}=36$ Case-II : On...
Read More →A natural number has
Question: A natural number has prime factorization given by $n=2^{x} 3^{y} 5^{z}$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6}, yz$. Then the number of odd divisors of $\mathrm{n}$, including 1 , is :1166x12Correct Option: , 4 Solution: $y+z=5$ $\frac{1}{y}+\frac{1}{z}=\frac{5}{6}$$yz$ $\Rightarrow y=3, z=2$ $\Rightarrow \mathrm{n}=2^{\mathrm{x}} \cdot 3^{3} \cdot 5^{2}=(2.2 .2 \ldots)$ Number of odd divisors $=4 \times 3=12$...
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Question: Let $A=\{n \in N: n$ is a 3 -digit number $\}$ $\mathrm{B}=\{9 \mathrm{k}+2: \mathrm{k} \in \mathrm{N}\}$ and $\mathrm{C}=\{9 \mathrm{k}+l: \mathrm{k} \in \mathrm{N}\}$ for some $l(0l9)$ If the sum of all the elements of the set $\mathrm{A} \cap(\mathrm{B} \cup \mathrm{C})$ is $274 \times 400$, then $l$ is equal to Solution: $\mathrm{B}$ and $\mathrm{C}$ will contain three digit numbers of the form $9 \mathrm{k}+2$ and $9 \mathrm{k}+\ell$ respectively. We need to find sum of all elemen...
Read More →Let A be a 3x3 matrix with that det(A) = 4,
Question: Let $\mathrm{A}$ be a $3 \times 3$ matrix with $\operatorname{det}(\mathrm{A})=4$. Let $\mathrm{R}_{\mathrm{i}}$ denote the $\mathrm{i}^{\text {th }}$ row of $\mathrm{A}$. If a matrix $\mathrm{B}$ is obtained by performing the operation $\mathrm{R}_{2} \rightarrow 2 \mathrm{R}_{2}+5 \mathrm{R}_{3}$ on $2 \mathrm{~A}$, then $\operatorname{det}(\mathrm{B})$ is equal to :168012864Correct Option: , 4 Solution: $|\mathrm{A}|=4$ $\Rightarrow|2 \mathrm{~A}|=2^{3} \times 4=32$ $\because \mathr...
Read More →For x > 0,
Question: For $x0$, if $f(x)=\int_{i}^{x} \frac{\log _{e} t}{(1+t)} d t$, then $f(e)+f\left(\frac{1}{e}\right)$ is equal to1-1$\frac{1}{2}$0Correct Option: , 3 Solution: $f(x)=\int_{i}^{x} \frac{\log _{e} t}{(1+t)} d t$ $f\left(\frac{1}{x}\right)=\int_{1}^{1 / x} \frac{\ell \mathrm{nt}}{1+t} \mathrm{dt}$, let $\mathrm{t}=\frac{1}{\mathrm{y}}$ $=+\int_{1}^{x} \frac{\ell \text { ny }}{1+y} \cdot \frac{y}{y^{2}} d y$ $=\int_{1}^{x} \frac{\ell n y}{y(1+y)} d y$ hence $f(\mathrm{x})+f\left(\frac{1}{\...
Read More →Solve the Following Questions
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(x)=\left\{\begin{array}{cc}2 \sin \left(-\frac{\pi x}{2}\right), \text { if } x-1 \\ \left|a x^{2}+x+b\right|, \text { if }-1 \leq x \leq 1 \\ \sin (\pi x), \text { if } x1\end{array}\right.$ If $f(\mathrm{x})$ is continuous on $\mathrm{R}$, then $\mathrm{a}+\mathrm{b}$ equals:-3-131Correct Option: , 2 Solution: $f(\mathrm{x})$ is continuous on $\mathrm{R}$ $\Rightarrow f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$ $|a+1+b|=\l...
Read More →If the least and the largest real values of
Question: If the least and the largest real values of $\alpha$, for which the equation $\mathrm{z}+\alpha|\mathrm{z}-1|+2 i=0$ ( $\mathrm{z} \in \mathrm{C}$ and $i=\sqrt{-1}$ ) has a solution, are $\mathrm{p}$ and $\mathrm{q}$ respectively; then $4\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right)$ is equal to Solution: Put $z=x+i y$ $x+i y+\alpha|x+i y-1|+2 i=0$ $\Rightarrow \quad x+\alpha \sqrt{(x-1)^{2}+y^{2}}+i(y+2)=0+0 i$ $\Rightarrow \quad y+2=0$ and $x+\alpha \sqrt{(x-1)^{2}+y^{2}}=0$ $\Rightarro...
Read More →Let A = {1, 2, 3, ..., 10}
Question: Let $\mathrm{A}=\{1,2,3, \ldots, 10\}$ and $f: \mathrm{A} \rightarrow \mathrm{A}$ be defined as $f(\mathrm{k})=\left\{\begin{array}{cl}\mathrm{k}+1 \text { if } \mathrm{k} \text { is odd } \\ \mathrm{k} \text { if } \mathrm{k} \text { is even }\end{array}\right.$ Then the number of possible functions $\mathrm{g}: \mathrm{A} \rightarrow \mathrm{A}$ such that go $f=f$ is$10^{5}$${ }^{10} \mathrm{C}_{5}$$5^{5}$$5 !$Correct Option: 1 Solution: $f(\mathrm{x})=\left\{\begin{array}{cl}\mathrm...
Read More →The locus of the point of intersection of the lines
Question: The locus of the point of intersection of the lines $(\sqrt{3}) \mathrm{kx}+\mathrm{ky}-4 \sqrt{3}=0$ and $\sqrt{3} x-y-4(\sqrt{3}) k=0$ is a conic, whose eccentricity is_________. Solution: $K=\frac{4 \sqrt{3}}{\sqrt{3} x+y}=\frac{\sqrt{3} x-y}{4 \sqrt{3}}$ $\Rightarrow 3 x^{2}-y^{2}=48$ $\Rightarrow \frac{x^{2}}{16}-\frac{y^{2}}{48}=1$ Now, $48=16\left(\mathrm{e}^{2}-1\right)$ $\Rightarrow \mathrm{e}=\sqrt{4}=2$...
Read More →The locus of the mid-point of the line segment joining the focus of the parabola
Question: The locus of the mid-point of the line segment joining the focus of the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ to a moving point of the parabola, is another parabola whose directrix is : $x=-\frac{a}{2}$$x=\frac{a}{2}$$x=0$$\mathrm{X}=\mathrm{a}$Correct Option: 3 Solution: $\mathrm{h}=\frac{\mathrm{at}^{2}+\mathrm{a}}{2}, \mathrm{k}=\frac{2 \mathrm{a} t+0}{2}$ $\Rightarrow \quad \mathrm{t}^{2}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}$ and $\mathrm{t}=\frac{\mathrm{k}}{\mathrm{a}}$ ...
Read More →If the system of equations
Question: If the system of equations $k x+y+2 z=1$ $3 x-y-2 z=2$ $-2 x-2 y-4 z=3$ has infinitely many solutions, then $\mathrm{k}$ is equal to_______. Solution: We observe $5 \mathrm{P}_{2}-\mathrm{P}_{1}=3 \mathrm{P}_{3}$ So, $15-\mathrm{K}=-6$ $\Rightarrow \mathrm{K}=21$...
Read More →let vector a, vector i + vector 2j - vector k,
Question: Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector such that $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{b}}=0$, then $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}$ is equal to_________. Solution: $(\overr...
Read More →If vectors
Question: If vectors $\overrightarrow{\mathrm{a}}_{1}=\mathrm{x} \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{a}}_{2}=\hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$ are collinear, then a possible unit vector parallel to the vector $x \hat{i}+y \hat{j}+z \hat{k}$ is$\frac{1}{\sqrt{2}}(-\hat{\mathrm{j}}+\hat{\mathrm{k}})$$\frac{1}{\sqrt{2}}(\hat{i}-\hat{j})$$\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$$\frac{1}{\sqrt{3}}(\hat{i}-\hat{j...
Read More →Solve this following
Question: If $e^{\left(\cos ^{2} x+\cos ^{4} x+\cos ^{6} x+\ldots \infty\right) \log _{c} 2}$ satisfies the equation $\mathrm{t}^{2}-9 \mathrm{t}+8=0$, then the value of $\frac{2 \sin x}{\sin x+\sqrt{3} \cos x}\left(0x\frac{\pi}{2}\right)$ is $2 \sqrt{3}$$\frac{3}{2}$$\sqrt{3}$$\frac{1}{2}$Correct Option: , 4 Solution: $e^{\left(\cos ^{2} \theta+\cos ^{4} \theta+\ldots \ldots\right)\left(n^{2}\right.}=2^{\cos ^{2} \theta+\cos ^{4} \theta+\ldots \infty}$ $=2^{\cot ^{2} \theta}$ Now $t^{2}-9 t+9=0...
Read More →The total number of numbers,
Question: The total number of numbers, lying between 100 and 1000 that can be formed with the digits $1,2,3,4,5$, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 , is_______. Solution: We need three digits numbers. Since $1+2+3+4+5=15$ So, number of possible triplets for multiple of 15 is $1 \times 2 \times 2$ so Ans. $=4 \times\lfloor 3+4 \times 3-1 \times 2 \times\lfloor 2=32$...
Read More →Prove the following
Question: If $A=\left[\begin{array}{cc}0 -\tan \left(\frac{\theta}{2}\right) \\ \tan \left(\frac{\theta}{2}\right) 0\end{array}\right]$ and $\left(\mathrm{I}_{2}+\mathrm{A}\right)\left(\mathrm{I}_{2}-\mathrm{A}\right)^{-1}=\left[\begin{array}{cc}\mathrm{a} -\mathrm{b} \\ \mathrm{b} \mathrm{a}\end{array}\right]$, then $13\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)$ is equal to________. Solution: $\mathrm{a}^{2}+\mathrm{b}^{2}=\left|\mathrm{I}_{2}+\mathrm{A} \| \mathrm{I}_{2}-\mathrm{A}\right|^{-1}...
Read More →Solve this following
Question: $\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}}(\sin \sqrt{t}) d t}{x^{3}}$ is equal to : $\frac{2}{3}$$\frac{3}{2}$0$\frac{1}{15}$Correct Option: 1 Solution: $\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}=\lim _{x \rightarrow 0^{+}} \frac{(\sin x) 2 x}{3 x^{2}}$ $=\lim _{x \rightarrow 0^{+}}\left(\frac{\sin x}{x}\right) \times \frac{2}{3}=\frac{2}{3}$...
Read More →Solve the Following Questions
Question: If $\sqrt{3}\left(\cos ^{2} x\right)=(\sqrt{3}-1) \cos x+1$, the number of solutions of the given equation when $x \in\left[0, \frac{\pi}{2}\right]$ is Solution: $\sqrt{3}(\cos x)^{2}-\sqrt{3} \cos x+\cos x-1=0$ $\Rightarrow(\sqrt{3} \cos x+1)(\cos x-1)=0$ $\Rightarrow \cos x=1$ or $\cos x=-\frac{1}{\sqrt{3}}$ (reject) $\Rightarrow x=0$ only...
Read More →The value of the integral
Question: The value of the integral $\int_{0}^{\pi}|\sin 2 \mathrm{x}| \mathrm{dx}$ is Solution: Put $2 \mathrm{x}=\mathrm{t} \Rightarrow 2 \mathrm{dx}=\mathrm{dt}$ $\Rightarrow \mathrm{I}=\frac{1}{2} \int_{0}^{2 \pi}|\sin \mathrm{t}| \mathrm{dt}$ $=\int_{0}^{\pi}|\sin t| d t$ $=2$...
Read More →Prove the following
Question: Let $A=\left[\begin{array}{lll}x y z \\ y z x \\ z x y\end{array}\right]$, where $x, y$ and $z$ are real numbers such that $x+y+z0$ and $x y z=2$. If $\mathrm{A}^{2}=\mathrm{I}_{3}$, then the value of $\mathrm{x}^{3}+\mathrm{y}^{3}+\mathrm{z}^{3}$ is Solution: $\mathrm{A}^{2}=\mathrm{I}$ $\Rightarrow \mathrm{AA}^{\prime}=\mathrm{I} \quad\left(\right.$ as $\left.\mathrm{A}^{\prime}=\mathrm{A}\right)$ $\Rightarrow \mathrm{A}$ is orthogonal So, $x^{2}+y^{2}+z^{2}=1$ and $x y+y z+z x=0$ $\...
Read More →Solve this following
Question: Two vertical poles are $150 \mathrm{~m}$ apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is : $20 \sqrt{3}$$25 \sqrt{3}$3025Correct Option: , 2 Solution: $\tan \theta=\frac{\mathrm{h}}{75}=\frac{75}{3 \mathrm{~h}}$ $\Rightarrow \mathrm{h}^{2}=\frac{(75)^{2}}{3}$ $\mathrm{h}=25 \sqrt{3} \mathrm...
Read More →The area bounded by the lines
Question: The area bounded by the lines $y=\| x-1|-2|$ is Solution: Remark : Question is incomplete it should be area bounded by $\mathrm{y}=|\mathrm{x}-1|-2 \mid$ and $\mathrm{y}=2$ Area $=2\left(\frac{1}{2} .4 .2\right)$...
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