If the foot of the perpendicular from point
Question: If the foot of the perpendicular from point $(4,3,8)$ on the line $\mathrm{L}_{1}: \frac{\mathrm{x}-\mathrm{a}}{l}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-\mathrm{b}}{4}$, $l \neq 0$ is $(3,5,7)$, then the shortest distance between the line $L_{1}$ and line $\mathrm{L}_{2}: \frac{\mathrm{x}-2}{3}=\frac{\mathrm{y}-4}{4}=\frac{\mathrm{z}-5}{5}$ is equal to :$\frac{1}{2}$$\frac{1}{\sqrt{6}}$$\sqrt{\frac{2}{3}}$$\frac{1}{\sqrt{3}}$Correct Option: , 2 Solution: $(3,5,7)$ satisfy the line $L...
Read More →Solve the Following Questions
Question: Let $\overrightarrow{\mathrm{X}}$ be a vector in the plane containing vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. If the vector $\overrightarrow{\mathrm{x}}$ is perpendicular to $(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and its projection on $\overrightarrow{\mathrm{a}}$ is $\frac{17 \sqrt{6}}{2}$, then the value of $|\vec{x}|^{2}$ is equal to Solution: Let $\overrightarrow{\mathrm{x}}=\lambda \overrightarrow{\mathrm{a}}+\mu \ov...
Read More →Solve this following
Question: Let $\mathrm{g}(\mathrm{x})=\int_{0}^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3} \leq f(\mathrm{t}) \leq 1$ for all $\mathrm{t} \in[0,1]$ and $0 \leq f(\mathrm{t}) \leq \frac{1}{2}$ for all $\mathrm{t} \in(1,3]$ The largest possible interval in which $\mathrm{g}(3)$ lies is : $\left[-1,-\frac{1}{2}\right]$$\left[-\frac{3}{2},-1\right]$$\left[\frac{1}{3}, 2\right]$$[1,3]$Correct Option: 3, Solution: $\frac{1}{3} \leq f(\mat...
Read More →Let P(x) = x squre + bx + c be a quadratic polynomial
Question: Let $\mathrm{P}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{bx}+\mathrm{c}$ be a quadratic polynomial with real coefficients such that $\int_{0}^{1} \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}=1$ and $\mathrm{P}(\mathrm{x})$ leaves remainder 5 when it is divided by $(x-2)$. Then the value of $9(b+c)$ is equal to:915711Correct Option: , 3 Solution: $\int_{0}^{1}\left(x^{2}+b x+c\right) d x=1$ $\frac{1}{3}+\frac{b}{2}+c=1 \quad \Rightarrow \quad \frac{b}{2}+c=\frac{2}{3}$ $3 b+6 c=4$ ......(1) $...
Read More →Solve the Following Questions
Question: Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} \mathrm{b} \\ \mathrm{c} \mathrm{d}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]$ such that $\mathrm{AB}=\mathrm{B}$ and $\mathrm{a}+\mathrm{d}=2021$, then the value of $\mathrm{ad}-\mathrm{bc}$ is equal to Solution: $A=\left[\begin{array}{ll}a b \\ c d\end{array}\right], \quad B=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right]$ $\mat...
Read More →Let ABC be a triangle with
Question: Let $\mathrm{ABC}$ be a triangle with $\mathrm{A}(-3,1)$ and $\angle \mathrm{ACB}=\theta, 0\theta\frac{\pi}{2}$. If the equation of the median through $\mathrm{B}$ is $2 \mathrm{x}+\mathrm{y}-3=0$ and the equation of angle bisector of $C$ is $7 x-4 y-1=0$, then $\tan \theta$ is equal to:$\frac{1}{2}$$\frac{3}{4}$$\frac{4}{3}$2Correct Option: , 3 Solution: $\therefore \quad \mathrm{M}\left(\frac{\mathrm{a}-3}{2}, \frac{\mathrm{b}+1}{2}\right)$ lies on $2 \mathrm{x}+\mathrm{y}-3=0$ $\Rig...
Read More →Let the coefficients of third
Question: Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0$, be in the ratio $12: 8: 3$. Then the term independent of $x$ in the expansion, is equal to Solution: $\mathrm{T}_{\mathrm{r}+1}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}(\mathrm{x})^{\mathrm{n}-\mathrm{r}}\left(\frac{\mathrm{a}}{\mathrm{x}^{2}}\right)^{\mathrm{r}}$ $={ }^{n} C_{r} a^{r} x^{n-3 r}$ ${ }^{n} C_{2} a^{2}:{ }^{n} C_{3} a^{3}:{ }^{n} C_{4} a^{4}=12: 8: 3$...
Read More →Let f : S → S where S = (0,∞) be a twice
Question: Let $\mathrm{f}: \mathrm{S} \rightarrow \mathrm{S}$ where $\mathrm{S}=(0, \infty)$ be a twice differentiable function such that $\mathrm{f}(\mathrm{x}+1)=\mathrm{xf}(\mathrm{x})$. If $g: S \rightarrow R$ be defined as $g(x)=\log _{e} f(x)$, then the value of $\left|g^{\prime \prime}(5)-g^{\prime \prime}(1)\right|$ is equal to :$\frac{205}{144}$$\frac{197}{144}$$\frac{187}{144}$1Correct Option: 1 Solution: $\operatorname{lnf}(x+1)=\ln (x f(x))$ $\operatorname{lnf}(x+1)=\ln x+\operatorna...
Read More →Solve this following
Question: If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology ? $((\mathrm{P} \Rightarrow \mathrm{Q}) \wedge \sim \mathrm{Q}) \Rightarrow \mathrm{Q}$$((\mathrm{P} \Rightarrow \mathrm{Q}) \wedge \sim \mathrm{Q}) \Rightarrow \sim \mathrm{P}$$((\mathrm{P} \Rightarrow \mathrm{Q}) \wedge \sim \mathrm{Q}) \Rightarrow \mathrm{P}$$((P \Rightarrow Q) \wedge \sim Q) \Rightarrow(P \wedge Q)$Correct Option: , 2 Solution: LHS of all the options are some i.e. $((\m...
Read More →Consider a set of
Question: Consider a set of $3 \mathrm{n}$ numbers having variance 4. In this set, the mean of first $2 n$ numbers is 6 and the mean of the remaining $n$ numbers is 3. A new set is constructed by adding 1 into each of first $2 \mathrm{n}$ numbers, and subtracting 1 from each of the remaining $n$ numbers. If the variance of the new set is $\mathrm{k}$, then $9 \mathrm{k}$ is equal to Solution: Let number be $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots \ldots \mathrm{a}_{2 \mathrm{n}}, ...
Read More →The value of
Question: The value of $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2 n-1} \frac{n^{2}}{n^{2}+4 r^{2}}$ is:$\frac{1}{2} \tan ^{-1}(2)$$\frac{1}{2} \tan ^{-1}(4)$$\tan ^{-1}(4)$$\frac{1}{4} \tan ^{-1}(4)$Correct Option: 2, Solution: $L=\lim _{n \rightarrow \infty} \frac{1}{n} \cdot \sum_{r=0}^{2 n-1} \frac{1}{1+4\left(\frac{r}{n}\right)^{2}}$ $\Rightarrow \mathrm{L}=\int_{0}^{2} \frac{1}{1+4 \mathrm{x}^{2}} \mathrm{dx}$ $\Rightarrow \mathrm{L}=\left.\frac{1}{2} \tan ^{-1}(2 \mathrm{x})\r...
Read More →Solve the Following Questions
Question: Let $\tan \alpha, \tan \beta$ and $\tan \gamma ; \alpha, \beta, \gamma \neq \frac{(2 \mathrm{n}-1) \pi}{2}$, $\mathrm{n} \in \mathrm{N}$ be the slopes of three line segments $\mathrm{OA}$, $\mathrm{OB}$ and $\mathrm{OC}$, respectively, where $\mathrm{O}$ is origin. If circumcentre of $\triangle \mathrm{ABC}$ coincides with origin and its orthocentre lies on $y$-axis, then the value of $\left(\frac{\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^{2}$...
Read More →Solve this following
Question: Consider a hyperbola $\mathrm{H}: \mathrm{x}^{2}-2 \mathrm{y}^{2}=4 .$ Let the tangent at a point $\mathrm{P}(4, \sqrt{6})$ meet the $\mathrm{x}$-axis at $\mathrm{Q}$ and latus rectum at $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{x}_{1}0 .$ If $\mathrm{F}$ is a focus of $\mathrm{H}$ which is nearer to the point $\mathrm{P}$, then the area of $\triangle Q F R$ is equal to $4 \sqrt{6}$$\sqrt{6}-1$$\frac{7}{\sqrt{6}}-2$$4 \sqrt{6}-1$Correct Option: , 3 Solution: $\fra...
Read More →Solve the Following Questions
Question: Let $\mathrm{f}:[-3,1] \rightarrow \mathrm{R}$ be given as $f(x)= \begin{cases}\min \left\{(x+6), x^{2}\right\}, -3 \leq x \leq 0 \\ \max \left\{\sqrt{x}, x^{2}\right\}, 0 \leq x \leq 1\end{cases}$ If the area bounded by $y=f(x)$ and $x$-axis is $\mathrm{A}$, then the value of $6 \mathrm{~A}$ is equal to Solution: $\mathrm{f}:[-3,1] \rightarrow \mathrm{R}$ $f(x)= \begin{cases}\min \left\{(x+6), x^{2}\right\}, -3 \leq x \leq 0 \\ \max \left\{\sqrt{x}, x^{2}\right\} , 0 \leq x \leq 1\end...
Read More →Let f be a real valued function,
Question: Let $\mathrm{f}$ be a real valued function, defined on $\mathrm{R}-\{-1,1\}$ and given by $f(x)=3 \log _{e}\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}$ Then in which of the following intervals, function $f(x)$ is increasing?$(-\infty,-1) \cup\left(\left[\frac{1}{2}, \infty\right)-\{1\}\right)$$(-\infty, \infty)-\{-1,1\}$$\left(-1, \frac{1}{2}\right]$$\left(-\infty, \frac{1}{2}\right]-\{-1\}$Correct Option: 1 Solution: $f(x)=3 \ell n(x-1)-3 \ell n(x+1)-\frac{2}{x-1}$ $f^{\prime}(x)=\frac...
Read More →If the sum of an infinite GP a,
Question: If the sum of an infinite GP a, ar, ar $^{2}, a r^{3}, \ldots$ is 15 and the sum of the squares of its each term is 150 , then the sum of $\operatorname{ar}^{2}, \operatorname{ar}^{4}, \operatorname{ar}^{6}, \ldots$ is :$\frac{5}{2}$$\frac{1}{2}$$\frac{25}{2}$$\frac{9}{2}$Correct Option: , 2 Solution: Sum of infinite terms : $\frac{a}{1-r}=15$ ............(i) Series formed by square of terms: $a^{2}, a^{2} r^{2}, a^{2} r^{4}, a^{2} r^{6} \quad \ldots .$ $\operatorname{Sum}=\frac{\mathr...
Read More →Let A (-1,1), B (3,4) and C (2,0) be given three points.
Question: Let $\mathrm{A}(-1,1), \mathrm{B}(3,4)$ and $\mathrm{C}(2,0)$ be given three points. A line $\mathrm{y}=\mathrm{mx}, \mathrm{m}0$, intersects lines $\mathrm{AC}$ and $\mathrm{BC}$ at point $\mathrm{P}$ and $\mathrm{Q}$ respectively. Let $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ be the areas of $\triangle \mathrm{ABC}$ and $\triangle \mathrm{PQC}$ respectively, such that $A_{1}=3 A_{2}$, then the value of $m$ is equal to :$\frac{4}{15}$123Correct Option: , 2 Solution: $\mathrm{P} \equiv\lef...
Read More →Solve the Following Questions
Question: Let $\mathrm{f}:[-1,1] \rightarrow \mathrm{R}$ be defined as $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ for all $x \in[-1,1]$, where $a, b, c \in R$ such that $f(-1)=2, f^{\prime}(-1)=1$ and for $x \in(-1,1)$ the maximum value of $\mathrm{f}^{\prime \prime}(\mathrm{x})$ is $\frac{1}{2}$. If $\mathrm{f}(\mathrm{x}) \leq \alpha$, $x \in[-1,1]$, then the least value of $\alpha$ is equal to Solution: $f:[-1,1] \rightarrow R$ $f(x)=a x^{2}+b x+c$ $f(-1)=a-b+c=2$..(1) $f...
Read More →Let the centroid of an equilateral triangle
Question: Let the centroid of an equilateral triangle $\mathrm{ABC}$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3 .$ If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\triangle \mathrm{ABC}$, then $(\mathrm{R}+\mathrm{r})$ is equal to : $\frac{9}{\sqrt{2}}$$7 \sqrt{2}$$2 \sqrt{2}$$3 \sqrt{2}$Correct Option: 1 Solution: $\mathrm{r}=\mathrm{OM}=\frac{3}{\sqrt{2}}$ $\ \sin 30^{\circ}=\frac{1}{2}=\frac{\mathrm{r}}{\math...
Read More →Solve the Following Questions
Question: If $1, \log _{10}\left(4^{x}-2\right)$ and $\log _{10}\left(4^{x}+\frac{18}{5}\right)$ are in arithmetic progression for a real number $x$, then the value of the determinant $\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) x-1 x^{2} \\ 1 0 x \\ x 1 0\end{array}\right|$ is equal to : Solution: $2 \log _{10}\left(4^{x}-2\right)=1+\log _{10}\left(4^{x}+\frac{18}{5}\right)$ $\left(4^{x}-2\right)^{2}=10\left(4^{x}+\frac{18}{5}\right)$ $\left(4^{x}\right)^{2}+4-4\left(4^{x}\right)-32=0$ ...
Read More →Solve this
Question: If $\mathrm{A}=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} \frac{1}{\sqrt{5}}\end{array}\right), \mathrm{B}=\left(\begin{array}{ll}1 0 \\ i 1\end{array}\right), i=\sqrt{-1}$, and $Q=A^{\mathrm{T}} B A$, then the inverse of the matrix $\mathrm{A} \mathrm{Q}^{2021} \mathrm{~A}^{\mathrm{T}}$ is equal to :$\left(\begin{array}{cc}\frac{1}{\sqrt{5}} -2021 \\ 2021 \frac{1}{\sqrt{5}}\end{array}\right)$$\left(\begin{array}{cc}1 0 \\ -2021 i 1\end{array}\r...
Read More →Given that the inverse trigonometric functions take principal values only.
Question: Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy $\sin ^{-1}\left(\frac{3 x}{5}\right)+\sin ^{-1}\left(\frac{4 x}{5}\right)=\sin ^{-1} x$ is equal to:2130Correct Option: , 3 Solution: $\sin ^{-1} \frac{3 x}{5}+\sin ^{-1} \frac{4 x}{5}=\sin ^{-1} x$ $\sin ^{-1}\left(\frac{3 x}{5} \sqrt{1-\frac{16 x^{2}}{25}}+\frac{4 x}{5} \sqrt{1-\frac{9 x^{2}}{25}}\right)=\sin ^{-1} x$ $\frac{3 x}{5} \sqrt{1-\frac{16 x^{2}}{...
Read More →Solve this following
Question: Let $f: \mathbb{R}-\{3\} \rightarrow \mathrm{R}-\{1\}$ be defined by $f(\mathrm{x})=\frac{\mathrm{x}-2}{\mathrm{x}-3}$. Let $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be given as $g(x)=2 x-3$. Then, the sum of all the values of $x$ for which $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$ is equal to 7253Correct Option: , 3 Solution: $f(x)=y=\frac{x-2}{x-3}$ $\therefore x=\frac{3 y-2}{y-1}$ $\therefore f^{-1}(x)=\frac{3 x-2}{x-1}$ $\ g(x)=y=2 x-3$ $\therefore x=\frac{y+3}{2}$ $\therefore g^{-1...
Read More →If the point of intersections of the ellipse
Question: If the point of intersections of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ and the circle $x^{2}+y^{2}=4 b, b4$ lie on the curve $y^{2}=3 x^{2}$, then $b$ is equal to:125610Correct Option: 1 Solution: $y^{2}=3 x^{2}$ and $x^{2}+y^{2}=4 b$ Solve both we get so $\quad x^{2}=b$ $\frac{x^{2}}{16}+\frac{3 x^{2}}{b^{2}}=1$ $\frac{b}{16}+\frac{3}{b}=1$ $b^{2}-16 b+48=0$ $(b-12)(b-4)=0$ $\mathrm{b}=12, \mathrm{~b}4$...
Read More →Let the system of linear equations
Question: Let the system of linear equations $4 x+\lambda y+2 z=0$ $2 x-y+z=0$ $\mu x+2 y+3 z=0, \lambda, \mu \in R$ has a non-trivial solution. Then which of the following is true? $\mu=6, \lambda \in \mathrm{R}$$\lambda=2, \mu \in \mathrm{R}$$\lambda=3, \mu \in \mathrm{R}$$\mu=-6, \lambda \in \mathrm{R}$Correct Option: 1 Solution: For non-trivial solution $\left|\begin{array}{ccc}4 \lambda 2 \\ 2 -1 1 \\ \mu 2 3\end{array}\right|=0$ $\Rightarrow 2 \mu-6 \lambda+\lambda \mu=12$ when $\mu=6, \qu...
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