The equation of the line passing through
Question: The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2 y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is:$\frac{x+4}{-1}=\frac{y-3}{1}=\frac{z-1}{1}$$\frac{x+4}{3}=\frac{y-3}{-1}=\frac{z-1}{1}$$\frac{x+4}{1}=\frac{y-3}{1}=\frac{z-1}{3}$$\frac{x-4}{2}=\frac{y+3}{1}=\frac{z+1}{4}$Correct Option: , 2 Solution: Normal vector of plane containing two intersecting lines is parallel to vector. $\left(\overrightarrow{\mathrm{V}}_{1}\rig...
Read More →The straight line x + 2y = 1 meets the coordinate axes at A and B.
Question: The straight line $x+2 y=1$ meets the coordinate axes at A and B. A circle is drawn through A, $B$ and the origin. Then the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is :$\frac{\sqrt{5}}{4}$$\frac{\sqrt{5}}{2}$$2 \sqrt{5}$$4 \sqrt{5}$Correct Option: , 2 Solution: Equation of circle $(x-1)(x-0)+(y-0)\left(y-\frac{1}{2}\right)=0$ $\Rightarrow x^{2}+y^{2}-x-\frac{y}{2}=0$ Equation of tangent of origin is $2 \mathrm{x}+\mathrm{y}=0$ $\ell_{1...
Read More →Solve this following
Question: If the system of equations $x+y+z=5$ $x+2 y+3 z=9$ $x+3 y+\alpha z=\beta$ has infinitely many solutions, then $\beta-\alpha$ equals:518218Correct Option: , 4 Solution: $\mathrm{D}=\left|\begin{array}{lll}1 1 1 \\ 1 2 3 \\ 1 3 \alpha\end{array}\right|=\left|\begin{array}{ccc}1 1 1 \\ 0 1 2 \\ 0 2 \alpha-1\end{array}\right|=(\alpha-1)-4=(\alpha-5)$ for infinite solutions $\mathrm{D}=0 \Rightarrow \alpha=5$ $D_{x}=0 \Rightarrow\left|\begin{array}{lll}5 1 1 \\ 9 2 3 \\ \beta 3 5\end{array}...
Read More →Let 0 < θ < π/2 . If the eccentricity of the
Question: Let $0\theta\frac{\pi}{2}$. If the eccentricity of the hyperbola $\frac{x^{2}}{\cos ^{2} \theta}-\frac{y^{2}}{\sin ^{2} \theta}=1$ is greater than 2, then the length of its latus rectum lies in the interval:$(2,3]$$(3, \infty)$$(3 / 2,2]$$(1,3 / 2]$Correct Option: , 2 Solution: $\mathrm{e}=\sqrt{1+\tan ^{2} \theta}=\sec \theta$ As, $\sec \theta2 \Rightarrow \cos \theta\frac{1}{2}$ $\Rightarrow \theta \in\left(60^{\circ}, 90^{\circ}\right)$ Now, $\ell(\mathrm{L} \cdot \mathrm{R})=\frac{...
Read More →Prove the following
Question: If $\mathrm{A}=\left[\begin{array}{cc}\cos \theta -\sin \theta \\ \sin \theta \cos \theta\end{array}\right]$, then the matrix $\mathrm{A}^{-50}$ when $\theta=\frac{\pi}{12}$, is equal to :$\left[\begin{array}{cc}\frac{\sqrt{3}}{2} \frac{1}{2} \\ -\frac{1}{2} \frac{\sqrt{3}}{2}\end{array}\right]$$\left[\begin{array}{cc}\frac{1}{2} \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} \frac{1}{2}\end{array}\right]$$\left[\begin{array}{cc}\frac{1}{2} -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} \frac{1}...
Read More →If θ denotes the acute angle between the curves,
Question: If $\theta$ denotes the acute angle between the curves, $y=10-x^{2}$ and $y=2+x^{2}$ at a point of their intersection, then $|\tan \theta|$ is equal to :4/9$7 / 17$$8 / 17$$8 / 15$Correct Option: , 4 Solution: Point of intersection is $\mathrm{P}(2,6)$. Also, $\mathrm{m}_{1}=\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{P}(2,6)}=-2 \mathrm{x}=-4$ $\mathrm{m}_{2}=\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{P}(2,6)}=2 \mathrm{x}=4$ $\therefore|\tan \theta|=\left|\frac{m...
Read More →If the solve the problem
Question: If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y0)$, then $d y / d x$ at $x=e$ is equal to :$\frac{e}{\sqrt{4+e^{2}}}$$\frac{(1+2 \mathrm{e})}{2 \sqrt{4+\mathrm{e}^{2}}}$$\frac{(2 e-1)}{2 \sqrt{4+e^{2}}}$$\frac{(1+2 e)}{\sqrt{4+e^{2}}}$Correct Option: , 3 Solution: Differentiating with respect to x, $x \cdot \frac{1}{\ln x} \cdot \frac{1}{x}+\ln (\ln x)-2 x+2 y \cdot \frac{d y}{d x}=0$ at $x=$ e we get $1-2 e+2 y \frac{d y}{d x}=0 \Rightarrow \frac{d y}{d x}=\frac{2 e-1}{2 y}$ $...
Read More →Let A = { 0 ∈ ( -π/2 , π) : 3 + 2isinθ/1 - 2isinθ is purely imaginary}
Question: Let $A=\left\{0 \in\left(-\frac{\pi}{2}, \pi\right): \frac{3+2 i \sin \theta}{1-2 i \sin \theta}\right.$ is purely imaginary $\}$ Then the sum of the elements in $\mathrm{A}$ is :$\frac{5 \pi}{6}$$\frac{2 \pi}{3}$$\frac{3 \pi}{4}$$\pi$Correct Option: 2, Solution: Given $\mathrm{z}=\frac{3+2 \mathrm{i} \sin \theta}{1-2 \mathrm{i} \sin \theta}$ is purely img so real part becomes zero. $\mathrm{z}=\left(\frac{3+2 \mathrm{i} \sin \theta}{1-2 \mathrm{i} \sin \theta}\right) \times\left(\frac...
Read More →The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and
Question: The direction ratios of normal to the plane through the points (0, 1, 0) and (0, 0, 1) andmaking an anlge $\frac{\pi}{4}$ with the plane $y-z+5=0$ are:$2 \sqrt{3}, 1,-1$$2, \sqrt{2},-\sqrt{2}$$2,-1,1$$\sqrt{2}, 1,-1$Correct Option: , 4 Solution: Let the equation of plane be $a(x-0)+b(y+1)+c(z-0)=0$ It passes through $(0,0,1)$ then $b+c=0$ ......(1) Now $\cos \frac{\pi}{4}=\frac{a(0)+b(1)+c(-1)}{\sqrt{2} \sqrt{a^{2}+b^{2}+c^{2}}}$ $\Rightarrow a^{2}=-2 b c$ and $b=-c$ we get $a^{2}=2 c^...
Read More →Solve this following
Question: Let $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ be any two non-zero complex numbers such that $3\left|z_{1}\right|=4\left|z_{2}\right|$. If $\mathrm{z}=\frac{3 \mathrm{z}_{1}}{2 \mathrm{z}_{2}}+\frac{2 \mathrm{z}_{2}}{3 \mathrm{z}_{1}}$ then :$|\mathrm{z}|=\frac{1}{2} \sqrt{\frac{17}{2}}$$\operatorname{Re}(z)=0$$|\mathrm{z}|=\sqrt{\frac{5}{2}}$$\operatorname{Im}(z)=0$Correct Option: , 4 Solution: $3\left|z_{1}\right|=4\left|z_{2}\right|$ $\Rightarrow \frac{\left|z_{1}\right|}{\left|z_{2}\rig...
Read More →For x ∈ R - { 0 , 1} , Let f1 (x) = 1/x ,
Question: For $\mathrm{x} \in \mathrm{R}-\{0,1\}$, let $\mathrm{f}_{1}(\mathrm{x})=\frac{1}{\mathrm{x}}$ $\mathrm{f}_{2}(\mathrm{x})=1-\mathrm{x}$ and $\mathrm{f}_{3}(\mathrm{x})=\frac{1}{1-\mathrm{x}}$ be three given functions. If a function, $\mathrm{J}(\mathrm{x})$ satisfies $\left(f_{2}{ }^{\circ}{ }^{\circ} \mathrm{f}_{1}\right)(\mathrm{x})=\mathrm{f}_{3}(\mathrm{x})$ then $\mathrm{J}(\mathrm{x})$ is equal to :-$\mathrm{f}_{3}(\mathrm{x})$$f_{1}(x)$$\mathrm{f}_{2}(\mathrm{x})$$\frac{1}{x} f...
Read More →Let the function
Question: Let $[\mathrm{x}]$ denote the greatest integer less than or equal to $x$. Then :- $\lim _{x \rightarrow 0} \frac{\tan \left(\pi \sin ^{2} x\right)+(|x|-\sin (x[x]))^{2}}{x^{2}}$equals $\pi$equals 0equals $\pi+1$does not existCorrect Option: , 4 Solution: R.H.L. $=\lim _{x \rightarrow 0^{+}} \frac{\tan \left(\pi \sin ^{2} x\right)+(|x|-\sin (x[x]))^{2}}{x^{2}}$ $\left(\right.$ as $\left.x \rightarrow 0^{+} \Rightarrow[x]=0\right)$ $=\lim _{x \rightarrow 0^{+}} \frac{\tan \left(\pi \sin ...
Read More →The sum of all values of
Question: The sum of all values of $\theta \in\left(0, \frac{\pi}{2}\right)$ satisfying $\sin ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}$ is :$\frac{\pi}{2}$$\pi$$\frac{3 \pi}{8}$$\frac{5 \pi}{4}$Correct Option: 1 Solution: $\sin ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}, \theta \in\left(0, \frac{\pi}{2}\right)$ $\Rightarrow 1-\cos ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}$ $\Rightarrow 4 \cos ^{4} 2 \theta-4 \cos ^{2} 2 \theta+1=0$ $\Rightarrow\left(2 \cos ^{2} 2 \theta-1\right)^{2}=0$ ...
Read More →Two cards are drawn successively with replacement
Question: Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let $X$ denote the random variable of number of aces obtained in the two drawn cards. Then $\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)$ equals :$52 / 169$$25 / 169$$49 / 169$$24 / 169$Correct Option: , 2 Solution: Two cards are drawn successively with replacement 4 Aces 48 Non Aces $\mathrm{P}(\mathrm{x}=1)=\frac{{ }^{4} \mathrm{C}_{1}}{{ }^{52} \mathrm{C}_{1}} \times \frac{48 \mathrm{C}_{1}...
Read More →Let the function
Question: Let $f_{k}(x)=\frac{1}{k}\left(\sin ^{k} x+\cos ^{k} x\right)$ for $k=1,2$, 3 ,.... Then for all $x \in R$, the value of $\mathrm{f}_{4}(\mathrm{x})-\mathrm{f}_{6}(\mathrm{x})$ is equal to :-$\frac{5}{12}$$\frac{-1}{12}$$\frac{1}{4}$$\frac{1}{12}$Correct Option: , 4 Solution: $f_{4}(x)-f_{6}(x)$ $=\frac{1}{4}\left(\sin ^{4} x+\cos ^{4} x\right)-\frac{1}{6}\left(\sin ^{6} x+\cos ^{6} x\right)$ $=\frac{1}{4}\left(1-\frac{1}{2} \sin ^{2} 2 x\right)-\frac{1}{6}\left(1-\frac{3}{4} \sin ^{2}...
Read More →5 students of a class have an average height
Question: 5 students of a class have an average height $150 \mathrm{~cm}$ and variance $18 \mathrm{~cm}^{2}$. A new student, whose height is $156 \mathrm{~cm}$, joined them. The variance (in $\mathrm{cm}^{2}$ ) of the height of these six students is:22201618Correct Option: , 2 Solution: Given $\vec{x}=\frac{\Sigma x_{i}}{5}=150$ $\Rightarrow \sum_{i=1}^{5} x_{i}=750$ $\ldots \ldots(\mathrm{i})$ $\frac{\sum \mathrm{x}_{\mathrm{i}}^{2}}{5}-(\overrightarrow{\mathrm{x}})^{2}=18$ $\frac{\sum x_{i}^{2...
Read More →A square is inscribed inthe circle
Question: A square is inscribed inthe circle $x^{2}+y^{2}-6 x+8 y-103=0$ with its sides parallel to the corrdinate axes. Then the distance of the vertex of this square which is nearest to the origin is :-13$\sqrt{137}$6$\sqrt{41}$Correct Option: , 4 Solution: $\mathrm{R}=\sqrt{9+16+103}=8 \sqrt{2}$ $\mathrm{OA}=13$ $\mathrm{OB}=\sqrt{265}$ $\mathrm{OC}=\sqrt{137}$ $\mathrm{OD}=\sqrt{41}$...
Read More →Solve this following
Question: If the parabolas $y^{2}=4 b(x-c)$ and $y^{2}=8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$$(1,1,0)$$\left(\frac{1}{2}, 2,3\right)$$\left(\frac{1}{2}, 2,0\right)$$(1,1,3)$Correct Option: 1, 2, 3, 4 Solution: Normal to these two curves are $y=m(x-c)-2 b m-b m^{3}$ $y=m x-4 a m-2 a m^{3}$ If they have a common normal $(c+2 b) m+b m^{3}=4 a m+2 a m^{3}$ Now $(4 a-c-2 b) m=(b-2 a) m^{3}$ We get all options are correct for $m...
Read More →Let a1, a2, a3,.......... a30, be an A.P.
Question: Let $a_{1}, a_{2}, \ldots \ldots . ., a_{30}$ be an A. P., $S=\sum_{i=1}^{30} a_{i}$ and $\mathrm{T}=\sum_{\mathrm{i}=1}^{15} \mathrm{a}_{(2 \mathrm{i}-1)} .$ If $\mathrm{a}_{5}=27$ and $\mathrm{S}-2 \mathrm{~T}=75$, then $\mathrm{a}_{10}$ is equal to :57474252Correct Option: , 4 Solution: $S=a_{1}+a_{2}+\ldots \ldots+a_{30}$ $S=\frac{30}{2}\left[a_{1}+a_{30}\right]$ $S=15\left(a_{1}+a_{30}\right)=15\left(a_{1}+a_{1}+29 d\right)$ $\mathrm{T}=\mathrm{a}_{1}+\mathrm{a}_{3}+\ldots .+\math...
Read More →If the system of linear equations
Question: If the system of linear equations 2x + 2y + 3z = a 3x y + 5z = b x 3y + 2z = c where a, b, c are non-zero real numbers, has more then one solution, then :b c a = 0a + b + c = 0b + c a = 0b c + a = 0Correct Option: 1 Solution: $P_{1}: 2 x+2 y+3 z=a$ $P_{2}: 3 x-y+5 z=b$ $P_{3}: x-3 y+2 z=c$ We find $\mathrm{P}_{1}+\mathrm{P}_{3}=\mathrm{P}_{2} \Rightarrow \mathrm{a}+\mathrm{c}=\mathrm{b}$...
Read More →The value of
Question: Let $\vec{a}=\hat{i}-\hat{j}, \vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c}+\vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{c}=4$, then $|\vec{c}|^{2}$ is equal to:-$\frac{19}{2}$8$\frac{17}{2}$9Correct Option: 1 Solution: $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=-\overrightarrow{\mathrm{b}}$ $(\vec{a} \times \vec{c}) \times \vec{a}=-\vec{b} \times \vec{a}$ $\Rightarrow(\vec{a} \times \vec{c}) \times \vec{a}=\vec...
Read More →Solve this following
Question: Let $f(x)= \begin{cases}\max \left\{|x|, x^{2}\right\}, |x| \leq 2 \\ 8-2|x|, 2x \mid \leq 4\end{cases}$ Let $S$ be the set of points in the interval $(-4,4)$ at which $\mathrm{f}$ is not differentiable. Then $S$ : is an empty setequals $\{-2,-1,1,2\}$equals $\{-2,-1,0,1,2\}$equals $\{-2,2\}$Correct Option: , 3 Solution: $f(x)= \begin{cases}8+2 x, -4 \leq x-2 \\ x^{2}, -2 \leq x \leq-1 \\ |x|, -1x1 \\ x^{2}, 1 \leq x \leq 2 \\ 8-2 x, 2x \leq 4\end{cases}$ $f(x)$ is not differentiable a...
Read More →The value of the integral
Question: The value of the integral $\int_{-2}^{2} \frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$ (where $[\mathrm{x}]$ denotes the greatest integer less than ${ }^{20} \mathrm{Cr}$ or equal to $\mathrm{x}$ ) is :44 sin4sin 40Correct Option: , 4 Solution: $I=\int_{-2}^{2} \frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$ $I=\int_{0}^{2}\left(\frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}+\frac{\sin ^{2}(-x)}{\left[-\frac{x}{\pi}\right]+\frac{1}{2}}\righ...
Read More →The system of linear equations.
Question: The system of linear equations. $x+y+z=2$ $2 x+3 y+2 z=5$ $2 x+3 y+\left(a^{2}-1\right) z=a+1$has infinitely many solutions for $\mathrm{a}=4$is inconsistent when $|\mathrm{a}|=\sqrt{3}$is inconsistent when $\mathrm{a}=4$has a unique solution for $|a|=\sqrt{3}$Correct Option: , 2 Solution: $\mathrm{D}=\left|\begin{array}{ccc}1 1 1 \\ 2 3 2 \\ 2 3 \mathrm{a}^{2}-1\end{array}\right|=\mathrm{a}^{2}-3$ $D_{1}=\left|\begin{array}{ccc}2 1 1 \\ 5 3 2 \\ a+1 3 a^{2}-1\end{array}\right|=a^{2}-a...
Read More →In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y.
Question: In a triangle, the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $\mathrm{y}$. If $\mathrm{x}^{2}-\mathrm{c}^{2}=\mathrm{y}$, where $\mathrm{c}$ is the length of the third side of the triangle, then the circumradius of the triangle is :$\frac{y}{\sqrt{3}}$$\frac{c}{\sqrt{3}}$$\frac{c}{3}$$\frac{3}{2} y$Correct Option: , 2 Solution: Given $a+b=x$ and $a b=y$ If $x^{2}-c^{2}=y \Rightarrow(a+b)^{2}-c^{2}=a b$ $\Rightarrow a^{2}+b^{2}-c^{2}=-a b...
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