Prove the following
Question: Let $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ has the magnitude 12 then one such vector is$4(2 \hat{i}+2 \hat{j}-\hat{k})$$4(-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$$4(2 \hat{i}-2 \hat{j}-\hat{k})$$4(2 \hat{i}+2 \hat{j}+\hat{k})$Correct Option: , 3 Solution: $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$ $=2(\vec{b} \times \vec{a})$...
Read More →If a is A symmetric matrix and
Question: If $\mathrm{a}$ is $\mathrm{A}$ symmetric matrix and $\mathrm{B}$ is a skewsymmetrix matrix such that $\mathrm{A}+\mathrm{B}=\left[\begin{array}{cc}2 3 \\ 5 -1\end{array}\right]$, then $\mathrm{AB}$ is equal to :$\left[\begin{array}{cc}-4 2 \\ 1 4\end{array}\right]$$\left[\begin{array}{cc}-4 -2 \\ -1 4\end{array}\right]$$\left[\begin{array}{cc}4 -2 \\ -1 -4\end{array}\right]$$\left[\begin{array}{ll}4 -2 \\ 1 -4\end{array}\right]$Correct Option: , 3 Solution: $\mathrm{A}=\mathrm{A}^{\pr...
Read More →A 2 m ladder leans against a vertical wall.
Question: A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate $25 \mathrm{~cm} / \mathrm{sec}$., then the rate (in $\mathrm{cm} / \mathrm{sec}$.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is $1 \mathrm{~m}$ above the ground is :$25 \sqrt{3}$25$\frac{25}{\sqrt{3}}$$\frac{25}{3}$Correct Option: , 3 Solution: $x^{2}+y^{2}=4\left(\frac{d y}{d t}=-25\right)$ $x \frac{d x}{d...
Read More →Let P be the point of intersection of the common tangents
Question: Let $P$ be the point of intersection of the common tangents to the parabola $\mathrm{y}^{2}=12 \mathrm{x}$ and the hyperbola $8 x^{2}-y^{2}=8$. If $S$ and $S^{\prime}$ denote the foci of the hyperbola where $S$ lies on the positive $x$-axis then $P$ divides $\mathrm{SS}^{\prime}$ in a ratio:$5: 4$$14: 13$$2: 1$$13: 11$Correct Option: 1 Solution: Equation of tangents $\mathrm{y}^{2}=12 \mathrm{x} \quad \Rightarrow \mathrm{y}=2 \mathrm{x}+\frac{3}{\mathrm{~m}}$ $\frac{x^{2}}{1}-\frac{y^{...
Read More →The equation
Question: The equation $y=\sin x \sin (x+2)-\sin ^{2}(x+1)$ represents a straight line lying in :second and third quadrants onlythird and fourth quadrants onlyfirst, third and fourth quadrantsfirst, second and fourth quadrantsCorrect Option: , 2 Solution: $2 y=2 \sin x \sin (x+2)-2 \sin ^{2}(x+1)$ $2 y=\cos 2-\cos (2 x+2)-(1-\cos (2 x+2))$ $=\cos 2-1$ $2 y=-2 \sin ^{2} \frac{1}{2}$ $y=-\sin ^{2} \frac{1}{2} \leq 0$...
Read More →Prove the following
Question: For $\mathrm{x} \in \mathrm{R}$, let $[\mathrm{x}]$ denote the greatest integer $\leq \mathrm{x}$, then the sum of the series $\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\ldots \ldots+\left[-\frac{1}{3}-\frac{99}{100}\right]$ is$-153$$-133$$-131$$-135$Correct Option: 2, Solution: $\underbrace{\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\ldots+\left[-\frac{1}{3}-\frac{66}{100}\right]}_{(-1) 67}$ $+\unde...
Read More →If the volume of parallelopiped formed by
Question: If the volume of parallelopiped formed by the vectors $\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}$ and $\lambda \hat{\mathrm{i}}+\hat{\mathrm{k}}$ is minimum, then $\lambda$ is equal to :$\sqrt{3}$$-\frac{1}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$$-\sqrt{3}$Correct Option: , 3 Solution: $f(\lambda)=\left|\lambda^{3}-\lambda+1\right|$ Its graph as follows where $\alpha \approx-1.32$ $\because \quad$ Question is asking minimum value of vo...
Read More →The coefficient of
Question: The coefficient of $\mathrm{x}^{18}$ in the product $(1+x)(1-x)^{10}\left(1+x+x^{2}\right)^{9}$ is :$-84$84126-126Correct Option: 1, Solution: $(1+x)(1-x)^{10}\left(1+x+x^{2}\right)^{9}$ $\left(1-x^{2}\right)\left(1-x^{3}\right)^{9}$ ${ }^{9} C_{6}=84$...
Read More →Let f : R → R be a continuously differentiable
Question: Let $f: R \rightarrow R$ be a continuously differentiable function wuch that $f(2)=6$ and $f(2)=\frac{1}{48}$. If $\int_{6}^{\mathrm{f}(x)} 4 \mathrm{t}^{3} \mathrm{dt}=(\mathrm{x}-2) \mathrm{g}(\mathrm{x})$, then $\lim _{x \rightarrow 2} \mathrm{~g}(\mathrm{x})$ is equal to :24361218Correct Option: , 4 Solution: $\lim _{x \rightarrow 2} g(x)=\lim _{x \rightarrow 2} \frac{\int_{6}^{f(x)} 4 t^{3} d t}{x-2}$ $=\lim _{x \rightarrow 2} \frac{4 . f^{3}(x) \cdot f^{\prime}(x)}{1}$ $=4 \mathr...
Read More →Consider the differential equation,
Question: Consider the differential equation, $y^{2} d x+\left(x-\frac{1}{y}\right) d y=0$. If value of $y$ is 1 when $x=1$, the the value of $x$ for which $y=2$, is :$\frac{1}{2}+\frac{1}{\sqrt{\mathrm{e}}}$$\frac{3}{2}-\sqrt{\mathrm{e}}$$\frac{5}{2}+\frac{1}{\sqrt{\mathrm{e}}}$$\frac{3}{2}-\frac{1}{\sqrt{\mathrm{e}}}$Correct Option: , 4 Solution: $y^{2} d x+x d y=\frac{d y}{y}$ $\frac{d x}{d y}+\frac{x}{y^{2}}=\frac{1}{y^{3}}$ $I F=e^{\int \frac{1}{y^{2}} d y}=e^{-\frac{1}{y}}$ $e^{-\frac{1}{y...
Read More →If the truth value of the statement
Question: If the truth value of the statement $\mathrm{P} \rightarrow(\sim \mathrm{p} \vee \mathrm{r})$ is false $(\mathrm{F})$, then the truth values of the statements $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are respectively :$\mathrm{F}, \mathrm{T}, \mathrm{T}$T, F, FT, T, FT, F, TCorrect Option: , 3 Solution: $\mathrm{P} \rightarrow(\sim \mathrm{q} \vee \mathrm{r})$ $\sim \mathrm{p} \vee(\sim \mathrm{q} \vee \mathrm{r})$ $\left.\left.\begin{array}{l}\sim \mathrm{p} \rightarrow \mathrm{F} \\ \sim...
Read More →Let a random variable
Question: Let a random variable $\mathrm{X}$ have a binomial distribution with mean 8 and variance 4 . If $\mathrm{P}(\mathrm{x} \leq 2)=\frac{\mathrm{k}}{2^{16}}$, then $\mathrm{k}$ is equal to :171121137Correct Option: , 4 Solution: $\mathrm{np}=8$ $\mathrm{npq}=4$ $\mathrm{q}=\frac{1}{2} \Rightarrow \mathrm{p}=\frac{1}{2}$ $\mathrm{n}=16$ $\mathrm{p}(\mathrm{x}=\mathrm{r})=16 \mathrm{C}_{\mathrm{r}}\left(\frac{1}{2}\right)^{16}$ $\mathrm{p}(\mathrm{x} \leq 2)=\frac{{ }^{16} \mathrm{C}_{0}+{ }...
Read More →Prove the following
Question: If $B=\left[\begin{array}{ccc}5 2 \alpha 1 \\ 0 2 1 \\ \alpha 3 -1\end{array}\right]$ is the inverse of a $3 \times 3$ matrix A, then the sum of all values of $\alpha$ for which det $(\mathrm{A})+1=0$, is :021-1Correct Option: , 3 Solution: $|B|=5(-5)-2 \alpha(-\alpha)-2 \alpha$ $=2 \alpha^{2}-2 \alpha-25$ $1+|\mathrm{A}|=0$ $\alpha^{2}-\alpha-12=0$ Sum $=1$...
Read More →Let S_n denote the sum of the first n terms of an A.P.
Question: Let $S_{n}$ denote the sum of the first $n$ terms of an A.P. If $\mathrm{S}_{4}=16$ and $\mathrm{S}_{6}=-48$, then $\mathrm{S}_{10}$ is equal to:$-320$$-260$$-380$$-410$Correct Option: 1 Solution: $2\{2 \mathrm{a}+3 \mathrm{~d}\}=16$ $3(2 \mathrm{a}+5 \mathrm{~d})=-48$ $2 \mathrm{a}+3 \mathrm{~d}=8$ $2 \mathrm{a}+5 \mathrm{~d}=-16$ $\mathrm{d}=-12$ $\mathrm{S}_{10}=5\{44-9 \times 12\}$ $=-320$...
Read More →The number of solutions of the equation
Question: The number of solutions of the equation $1+\sin ^{4} x=\cos ^{2} 3 x, x \in\left[-\frac{5 \pi}{2}, \frac{5 \pi}{2}\right]$ is :5473Correct Option: 1 Solution: $1+\sin ^{4} x=\cos ^{2} 3 x$ $\sin x=0 \ \cos 3 x=1$ $0,2 \pi,-2 \pi,-\pi, \pi$...
Read More →If the data
Question: If the data $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots . \mathrm{x}_{10}$ is such that the mean of first four of these is 11 , the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is :42$\sqrt{2}$$2 \sqrt{2}$Correct Option: , 2 Solution: $\mathrm{x}_{1}+\ldots+\mathrm{x}_{4}=44$ $\mathrm{x}_{5}+\ldots+\mathrm{x}_{10}=96$ $\overline{\mathrm{x}}=14, \Sigma \mathrm{x}_{\mathrm{i}}=140$ Variance $=\frac{\sum \mathrm{x}_...
Read More →The number of ways of choosing 10 objects out of 31 objects
Question: The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :$2^{20}$$2^{20}-1$$2^{20}+1$$2^{21}$Correct Option: 1 Solution:...
Read More →Prove the following
Question: If $\int_{0}^{\frac{\pi}{2}} \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=m(\pi+n)$, then $m \cdot n$ is equal to :$-1$1$\frac{1}{2}$$-\frac{1}{2}$Correct Option: 1 Solution: $\int_{0}^{\pi / 2} \frac{\cot x d x}{\cot x+\cos e c x}$ $\int_{0}^{\pi / 2} \frac{\cos x}{\cos x+1}=\int \frac{2 \cos ^{2} \frac{x}{2}-1}{2 \cos ^{2} \frac{x}{2}}$ $\int_{0}^{\pi / 2}\left(1-\frac{1}{2} \sec ^{2} \frac{x}{2}\right) d x$ $\left[x-\tan \frac{x}{2}\right]_{0}^{\frac{\pi}{2}}$ $\frac{1}{2}[\pi-2...
Read More →If α and β are the roots of the equation
Question: If $\alpha$ and $\beta$ are the roots of the equation $375 x^{2}-25 x-2=0$, then $\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \alpha^{r}+\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \beta^{r}$ is equal to :$\frac{21}{346}$$\frac{29}{358}$$\frac{1}{12}$$\frac{7}{116}$Correct Option: , 3 Solution: $375 x^{2}-25 x-2=0$ $\alpha+\beta=\frac{25}{375}, \alpha \beta=\frac{-2}{375}$ $\Rightarrow\left(\alpha+\alpha^{2}+\ldots\right.$ upto infinite terms $)+\left(\beta+\beta^{2}\right.$ $+\ldots$ ...
Read More →The value of
Question: The value of $\sin ^{-1}\left(\frac{12}{13}\right)-\sin ^{-1}\left(\frac{3}{5}\right)$ is equal to:$\pi-\sin ^{-1}\left(\frac{63}{65}\right)$$\pi-\cos ^{-1}\left(\frac{33}{65}\right)$$\frac{\pi}{2}-\sin ^{-1}\left(\frac{56}{65}\right)$$\frac{\pi}{2}-\cos ^{-1}\left(\frac{9}{65}\right)$Correct Option: , 3 Solution: $\sin ^{-1}\left(\frac{12}{13}\right)-\sin ^{-1}\left(\frac{3}{5}\right)$ $\sin ^{-1}\left(x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right)$ $=\sin ^{1}\left(\frac{33}{65}\right)=\co...
Read More →If the line
Question: If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the palne $2 x+3 y-z+13=0$ at a point $P$ and the plane $3 x+y+4 z=16$ at a point $Q$, then PQ is equal to :$2 \sqrt{14}$$\sqrt{14}$$2 \sqrt{7}$14Correct Option: 1 Solution: $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}=\lambda$ $\mathrm{x}=3 \lambda+2, \mathrm{y}=2 \lambda-1, \mathrm{z}=-\lambda+1$ Intersection with plane $2 x+3 y-z+13=0$ $2(3 \lambda+2)+3(2 \lambda-1)-(-\lambda+1)+13=0$ $13 \lambda+13=0 \quad \lambda=-...
Read More →Solve the following quadratic equations
Question: If $\mathrm{e}^{\mathrm{y}}+\mathrm{xy}=\mathrm{e}$, the ordered pair $\left(\frac{\mathrm{dy}}{\mathrm{dx}}, \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)$ at $x=0$ is equal to :$\left(-\frac{1}{\mathrm{e}}, \frac{1}{\mathrm{e}^{2}}\right)$$\left(\frac{1}{\mathrm{e}}, \frac{1}{\mathrm{e}^{2}}\right)$$\left(\frac{1}{\mathrm{e}},-\frac{1}{\mathrm{e}^{2}}\right)$$\left(-\frac{1}{\mathrm{e}},-\frac{1}{\mathrm{e}^{2}}\right)$Correct Option: 1 Solution: $e^{y}=x y=e$ differentiat...
Read More →If the normal to the ellipse
Question: If the normal to the ellipse $3 x^{2}+4 y^{2}=12$ at a point $P$ on it is parallel to the line, $2 x+y=4$ and the tangent to the ellipse at $P$ passes through $Q(4,4)$ then $P Q$ is equal to :$\frac{\sqrt{221}}{2}$$\frac{\sqrt{157}}{2}$$\frac{\sqrt{61}}{2}$$\frac{5 \sqrt{5}}{2}$Correct Option: , 4 Solution: $3 x^{2}+4 y^{2}=12$ $x=2 \cos \theta, y=\sqrt{3} \sin \theta$ Let $\mathrm{P}(2 \cos \theta, \sqrt{3 \sin \theta})$ Equation of normal is $\frac{\mathrm{a}^{2} \mathrm{x}}{\mathrm{...
Read More →If three of the six vertices of a regular hexagon are chosen
Question: If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is :$\frac{3}{10}$$\frac{1}{10}$$\frac{3}{20}$$\frac{1}{5}$Correct Option: , 2 Solution: Only two equilateral tringles are possible $\mathrm{A}_{1} \mathrm{~A}_{3}$ $\mathrm{A}_{5}$ and $\mathrm{A}_{2} \mathrm{~A}_{5} \mathrm{~A}_{6}$ $\frac{2}{6_{C_{2}}}=\frac{2}{20}=\frac{1}{10}$...
Read More →For x ∈ ( 0 , 3/2 ), let f(x) =
Question: For $x \in\left(0, \frac{3}{2}\right)$, let $f(x)=\sqrt{x}, g(x)=\tan x$ and $h(x)=\frac{1-x^{2}}{1+x^{2}} .$ If $\phi(x)=(($ hof $) \circ g)(x)$, then $\phi=\left(\frac{\pi}{3}\right)$ is equal to :$\tan \frac{\pi}{12}$$\tan \frac{7 \pi}{12}$$\tan \frac{11 \pi}{12}$$\tan \frac{5 \pi}{12}$Correct Option: , 3 Solution: $f(x)=\sqrt{x}, g(x)=\tan x, h(x)=\frac{1-x^{2}}{1+x^{2}}$ $f o g(x)=\sqrt{\tan x}$ $\operatorname{hofog}(x)=h(\sqrt{\tan x})=\frac{1-\tan x}{1+\tan x}$ $=-\tan \left(\fr...
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