If the mean and variance of the following data: 6, 10, 7, 13, a, 12, b, 12
Question: If the mean and variance of the following data: 6, 10, 7, 13, a, 12, b, 12 are 9 and $\frac{37}{4}$ respectively, then $(a-b)^{2}$ is equal to:24123216Correct Option: , 4 Solution: Mean $=\frac{6+10+7+13+a+12+b+12}{8}=9$ $60+a+b=72$ $a+b=12$ ......(1) variance $=\frac{\sum x_{i}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}=\frac{37}{4}$ $\Sigma x_{i}^{2}=6^{2}+10^{2}+7^{2}+13^{2}+a^{2}+b^{2}+12^{2}+12^{2}$ $=a^{2}+b^{2}+642$ $\frac{\mathrm{a}^{2}+\mathrm{b}^{2}+642}{8}-(9)^{2}=\frac{3...
Read More →Solve the Following Questions
Question: $\cos ^{-1}(\cos (-5))+\sin ^{-1}(\sin (6))-\tan ^{-1}(\tan (12))$ is equal to : (The inverse trigonometric functions take the principal values)$3 \pi-11$$4 \pi-9$$4 \pi-11$$3 \pi+1$Correct Option: , 3 Solution: $\cos ^{-1}(\cos (-5))+\sin ^{-1}(\sin (6))-\tan ^{-1}(\tan (12))$ $\Rightarrow(2 \pi-5)+(6-2 \pi)-(12-4 \pi)$ $\Rightarrow 4 \pi-11$...
Read More →Solve the Following Questions
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function. Then $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\frac{\pi}{4} \int_{2}^{\sec ^{2} x} f(x) d x}{x^{2}-\frac{\pi^{2}}{16}}$ is equal to :$f(2)$$2 f(2)$$2 f(\sqrt{2})$$4 f(2)$Correct Option: , 2 Solution: $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\frac{\pi}{4} \int_{2}^{\sec ^{2} x} f(x) d x}{x^{2}-\frac{\pi^{2}}{16}}$ $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\pi}{4} \cdot \frac{\left[f\left(\sec ^{2} x\right) \cdot 2...
Read More →Let [x] denote the greatest integer less than or equal to x.
Question: Let $[\mathrm{x}]$ denote the greatest integer less than or equal to $x$. Then, the values of $x \in \mathbf{R}$ satisfying the equation $\left[e^{x}\right]^{2}+\left[e^{x}+1\right]-3=0$ lie in the interval:$\left[0, \frac{1}{\mathrm{e}}\right)$$\left[\log _{\mathrm{e}} 2, \log _{\mathrm{e}} 3\right)$$[1, \mathrm{e})$$\left[0, \log _{\mathrm{e}} 2\right)$Correct Option: , 4 Solution: $\left[e^{x}\right]^{2}+\left[e^{x}+1\right]-3=0$ $\Rightarrow\left[e^{x}\right]^{2}+\left[e^{x}\right]...
Read More →Let [x] denote the greatest integer less than or equal
Question: Let $[x]$ denote the greatest integer less than or equal to $x$. Then, the values of $x \in \mathbf{R}$ satisfying the equation $\left[e^{x}\right]^{2}+\left[e^{x}+1\right]-3=0$ lie in the interval:$\left[0, \frac{1}{\mathrm{e}}\right)$$\left[\log _{e} 2, \log _{e} 3\right)$$[1, \mathrm{e})$$\left[0, \log _{\mathrm{e}} 2\right)$Correct Option: , 4 Solution: $\left[e^{x}\right]^{2}+\left[e^{x}+1\right]-3=0$ $\Rightarrow\left[e^{x}\right]^{2}+\left[e^{x}\right]+1-3=0$ Let $\left[e^{x}\ri...
Read More →Let A=[a_ij] be a real matrix of order
Question: Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a real matrix of order $3 \times 3$, such that $\mathrm{a}_{\mathrm{il}}+\mathrm{a}_{\mathrm{i} 2}+\mathrm{a}_{\mathrm{i} 3}=1$, for $\mathrm{i}=1,2,3 .$ Then, the sum of all the entries of the matrix $\mathrm{A}^{3}$ is equal to :2139Correct Option: , 3 Solution: $A=\left[\begin{array}{lll}a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33}\end{array}\right]$ Let $x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\...
Read More →Solve the Following Questions
Question: If $\lim _{x \rightarrow 0} \frac{\alpha x e^{x}-\beta \log _{e}(1+x)+\gamma x^{2} e^{-x}}{x \sin ^{2} x}=10, \alpha, \beta, \gamma \in \mathbf{R}$ then the value of $\alpha+\beta+\gamma$ is Solution: $\lim _{x \rightarrow 0} \frac{\alpha x\left(1+x+\frac{x^{2}}{2}\right)-\beta\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{3}\right)+\gamma x^{2}(1-x)}{x^{3}}$ $\lim _{x \rightarrow 0} \frac{x(\alpha-\beta)+x^{2}\left(\alpha+\frac{\beta}{2}+\gamma\right)+x^{3}\left(\frac{\alpha}{2}-\frac{\beta}{3}...
Read More →The least positive integer n such that
Question: The least positive integer n such that $\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}, \mathrm{i}=\sqrt{-1}$ is a positive integer, is________ Solution: $\frac{(2 i)^{n}}{(1-i)^{n-2}}=\frac{(2 i)^{n}}{(-2 i)^{\frac{n-2}{2}}}$ $=\frac{(2 i)^{\frac{n+2}{2}}}{(-1)^{\frac{n-2}{2}}}=\frac{2^{\frac{n+2}{2}} i^{\frac{n+2}{2}}}{(-1)^{\frac{n-2}{2}}}$ This is positive integer for $n=6$...
Read More →Let the foot of perpendicular from a point
Question: Let the foot of perpendicular from a point $\mathrm{P}(1,2,-1)$ to the straight line $\mathrm{L}: \frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{0}=\frac{\mathrm{z}}{-1}$ be $\mathrm{N}$. Let a line be drawn from $P$ parallel to the plane $x+y+2 z=0$ which meets $L$ at point $Q$. If $\alpha$ is the acute angle between the lines $\mathrm{PN}$ and $\mathrm{PQ}$, then $\cos \alpha$ is equal to_________.$\frac{1}{\sqrt{5}}$$\frac{\sqrt{3}}{2}$$\frac{1}{\sqrt{3}}$$\frac{1}{2 \sqrt{3}}$Correct Optio...
Read More →Solve the Following Questions
Question: Let $\left\{a_{n}\right\}_{n=1}^{\infty}$ be a sequence such that $a_{1}=1, a_{2}=1$ and $a_{n+2}=2 a_{n+1}+a_{n}$ for all $n \geq 1$. Then the value of $47 \sum_{n=1}^{\infty} \frac{a_{n}}{2^{3 n}}$ is equal to Solution: $a_{n+2}=2 a_{n+1}+a_{n}$, let $\sum_{n=1}^{\infty} \frac{a_{n}}{8^{n}}=P$ Divide by $8^{\mathrm{n}}$ we get $\frac{a_{n+2}}{8^{n}}=\frac{2 a_{n+1}}{8^{n}}+\frac{a_{n}}{8^{n}}$ $\Rightarrow 64 \frac{a_{n+2}}{8^{n+2}}=\frac{16 a_{n+1}}{8^{n+1}}+\frac{a_{n}}{8^{n}}$ $64...
Read More →Let A be a 3 × 3 real matrix.
Question: Let A be a 3 3 real matrix. If $\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 \mathrm{~A}))))=2^{41}$, then the value of $\operatorname{det}\left(\mathrm{A}^{2}\right)$ equal_________ Solution: $\operatorname{adj}(2 \mathrm{~A})=2^{2} \operatorname{adj} \mathrm{A}$ $\Rightarrow \operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))=\operatorname{adj}(4 \operatorname{adj} \mathrm{A})=16 \operatorname{adj}(\operatorname{adj} \mathrm{A})$ $=16|\mathrm{~A...
Read More →Solve the Following Questions
Question: For $\mathrm{k} \in \mathrm{N}$, let $\frac{1}{\alpha(\alpha+1)(\alpha+2) \ldots \ldots . .(\alpha+20)}=\sum_{\mathrm{K}=0}^{20} \frac{\mathrm{A}_{\mathrm{k}}}{\alpha+\mathrm{k}}$ where $\alpha0$. Then the value of $100\left(\frac{\mathrm{A}_{14}+\mathrm{A}_{15}}{\mathrm{~A}_{13}}\right)^{2}$ is equal to Solution: $\frac{1}{\alpha(\alpha+1) \ldots . .(\alpha+20)}=\sum_{\mathrm{k}=0}^{20} \frac{\mathrm{A}_{\mathrm{k}}}{\alpha+\mathrm{k}}$ $\mathrm{A}_{14}=\frac{1}{(-14)(-13) \ldots . .(...
Read More →Which of the following Boolean expressions
Question: Which of the following Boolean expressions is not a tautology?$(\mathrm{p} \Rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$$(\mathrm{q} \Rightarrow \mathrm{p}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$$(\mathrm{p} \Rightarrow \sim \mathrm{q}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$$(\sim \mathrm{p} \Rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$Correct Option: , 4 Solution: (1) $(\mathrm{p} \rightarrow \mathrm{q}) \vee(\sim \mathrm{q...
Read More →Let an ellipse
Question: Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}b^{2}$, passes through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus $\mathrm{F}(\alpha, 0), \alpha0$, of $\mathrm{E}$ and radius $\frac{2}{\sqrt{3}}$, intersects $\mathrm{E}$ at two points $\mathrm{P}$ and $\mathrm{Q}$, then $\mathrm{PQ}^{2}$ is equal to :$\frac{8}{3}$$\frac{4}{3}$$\frac{16}{3}$3Correct Option: , 3 Solution: $\frac{3}{2 a^{2}}+\frac{1}{b^...
Read More →Let a function
Question: Let a function $\mathrm{g}:[0,4] \rightarrow \mathbf{R}$ be defined as $g(x)= \begin{cases}\max _{0 \leq t \leq x}\left\{t^{3}-6 t^{2}+9 t-3\right\}, 0 \leq x \leq 3 \\ 4-x , 3x \leq 4\end{cases}$ then the number of points in the interval $(0,4)$ where $g(x)$ is NOT differentiable, is Solution: $f(x)=x^{3}-6 x^{2}+9 x-3$ $f^{\prime}(x)=3 x^{2}-12 x+9=3(x-1)(x-3)$ $f(1)=1 \mathrm{f}(3)=-3$ $g(x)=\left[\begin{array}{rr}f(x) 0 \leq x \leq 1 \\ 0 1 \leq x \leq 3 \\ -1 3x \leq 4\end{array}\...
Read More →Let the mean and variance of four numbers
Question: Let the mean and variance of four numbers $3,7, x$ and $y(xy)$ be 5 and 10 respectively. Then the mean of four numbers $3+2 x, 7+2 y, x+y$ and $x-y$ is_________ Solution: $5=\frac{3+7+x+y}{4} \Rightarrow x+y=10$ $\operatorname{Var}(x)=10=\frac{3^{2}+7^{2}+x^{2}+y^{2}}{4}-25$ $140=49+9+x^{2}+y^{2}$ $x^{2}+y^{2}=82$ $x+y=10$ $\Rightarrow(\mathrm{x}, \mathrm{y})=(9,1)$ Four numbers are $21,9,10,8$ Mean $=\frac{48}{4}=12$...
Read More →If the point on the curve
Question: If the point on the curve $\mathrm{y}^{2}=6 \mathrm{x}$, nearest to the point $\left(3, \frac{3}{2}\right)$ is $(\alpha, \beta)$, then $2(\alpha+\beta)$ is equal to Solution: $\mathrm{P} \equiv\left(\frac{3}{2} \mathrm{t}^{2}, 3 \mathrm{t}\right)$ Normal at point $P$ $\mathrm{tx}+\mathrm{y}=3 \mathrm{t}+\frac{3}{2} \mathrm{t}^{3}$ Passes through $\left(3, \frac{3}{2}\right)$ $\Rightarrow 3 \mathrm{t}+\frac{3}{2}=3 \mathrm{t}+\frac{3}{2} \mathrm{t}^{3}$ $\mathrm{P} \equiv\left(\frac{3}{...
Read More →The number of real roots of the equation
Question: The number of real roots of the equation $e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0$ is :2461Correct Option: 1 Solution: $e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0$ $\Rightarrow\left(e^{3 x}-1\right)^{2}-e^{x}\left(e^{3 x}-1\right)=12 e^{2 x}$ $\left(e^{3 x}-1\right)^{2}\left(e^{x}-e^{-x}-e^{-2 x}\right)=12$ $\Rightarrow \underbrace{\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{-2 \mathrm{x}}}_{\text {increasing }(\operatorname{let} \mathrm{f}(\mathrm{x}))}=\un...
Read More →If the shortest distance between the straight lines
Question: If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(\mathrm{x}-2)=2(\mathrm{y}-\lambda)=(\mathrm{z}-3), \lambda \in \mathbf{R}$ is $\frac{1}{\sqrt{38}}$, then the integral value of $\lambda$ is equal to :325$-1$Correct Option: 1 Solution: $\mathrm{L}_{1}: \frac{(\mathrm{x}-1)}{2}=\frac{(\mathrm{y}-2)}{1}=\frac{(\mathrm{z}-1)}{3} \quad \overrightarrow{\mathrm{r}}_{1}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \mathrm{k}$ $\mathrm{L}_{2}: \frac{(\mathrm{x}-2)}{1}...
Read More →Solve this
Question: Let $\lambda \neq 0$ be in $\mathbf{R}$. If $\alpha$ and $\beta$ are the roots of the equation $\mathrm{x}^{2}-\mathrm{x}+2 \lambda=0$, and $\alpha$ and $\gamma$ are the roots of equation $3 x^{2}-10 x+27 \lambda=0$, then $\frac{\beta \gamma}{\lambda}$ is equal to__________ Solution: $3 \alpha^{2}-10 \alpha+27 \lambda=0$ .....(1) $\alpha^{2}-\alpha+2 \lambda=0$ ......(2) $(1)-3(2)$ gives $-7 \alpha+21 \lambda=0 \Rightarrow \alpha=3 \lambda$ Put $\alpha=3 \lambda$ in equation (1) we get...
Read More →Let a parabola P be such that its vertex
Question: Let a parabola P be such that its vertex and focus lie on the positive $x$-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $\mathrm{O}(0,0)$ to the parabola $\mathrm{P}$ which meet $\mathrm{P}$ at $\mathrm{S}$ and $R$, then the area (in sq. units) of $\Delta S O R$ is equal to :$16 \sqrt{2}$1632$8 \sqrt{2}$Correct Option: , 2 Solution: Clearly RS is latus-rectum $\because \mathrm{VF}=2=\mathrm{a}$ $\therefore \mathrm{RS}=4 \mathrm{a}=8$ Now $\...
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Question: Let $\left(\begin{array}{l}\mathrm{n} \\ \mathrm{k}\end{array}\right)$ denotes ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}$ and $\left[\begin{array}{l}\mathrm{n} \\ \mathrm{k}\end{array}\right]=\left\{\begin{array}{cl}\left(\begin{array}{l}\mathrm{n} \\ \mathrm{k}\end{array}\right), \text { if } 0 \leq \mathrm{k} \leq \mathrm{n} \\ 0, \text { otherwise }\end{array}\right.$ If $A_{k}=\sum_{i=0}^{9}\left(\begin{array}{l}9 \\ i\end{array}\right)\left[\begin{array}{c}12 \\ 12-k+i\end{array}\...
Read More →The values of lambda and
Question: The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3 x+5 y+5 z=26$, $x+2 y+\lambda z=\mu$ has no solution, are :$\lambda=3, \mu=5$$\lambda=3, \mu \neq 10$$\lambda \neq 2, \mu=10$$\lambda=2, \mu \neq 10$Correct Option: , 4 Solution: $x+y+z=6$ .........(i) $3 x+5 y+5 z=26$ .........(ii) $x+2 y+\lambda z=\mu$ .........(ii) $5 \times(\mathrm{i})-(\mathrm{ii}) \Rightarrow 2 \mathrm{x}=4 \Rightarrow \mathrm{x}=2$ $\therefore$ from (i) and (iii) $y+z=4$ ............(...
Read More →Let 9 distinct balls be distributed among 4 boxes,
Question: Let 9 distinct balls be distributed among 4 boxes, $B_{1}, B_{2}, B_{3}$ and $B_{4}$. If the probability than $B_{3}$ contains exactly 3 balls is $\mathrm{k}\left(\frac{3}{4}\right)^{9}$ then $\mathrm{k}$ lies in the set:$\{x \in \mathbf{R}:|x-3|1\}$$\{x \in \mathbf{R}:|x-2| \leq 1\}$$\{x \in \mathbf{R}:|x-1|1\}$$\{x \in \mathbf{R}:|x-5| \leq 1\}$Correct Option: , 2 Solution: required probability $=\frac{{ }^{9} \mathrm{C}_{3} \cdot 3^{6}}{4^{9}}$ $=\frac{{ }^{9} \mathrm{C}_{3}}{27} \c...
Read More →Consider a triangle having vertices
Question: Consider a triangle having vertices $\mathrm{A}(-2,3), \mathrm{B}(1,9)$ and $\mathrm{C}(3,8)$. If a line $\mathrm{L}$ passing through the circum-centre of triangle $A B C$, bisects line $B C$, and intersects $\mathrm{y}$-axis at point $\left(0, \frac{\alpha}{2}\right)$, then the value of real number $\alpha$ is Solution: $(\sqrt{50})^{2}=(\sqrt{45})^{2}+(\sqrt{5})^{2}$ $\angle \mathrm{B}=90^{\circ}$ Circum-center $=\left(\frac{1}{2}, \frac{11}{2}\right)$ Mid point of $\mathrm{BC}=\left...
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