Let the function
Question: Let $S=\{1,2, \ldots \ldots, 20\}$. A subset $B$ of $S$ is said to be "nice", if the sum of the elements of B is 203 . Then the probability that a randomly chosen subset of $S$ is "nice" is :-$\frac{6}{2^{20}}$$\frac{5}{2^{20}}$$\frac{4}{2^{20}}$$\frac{7}{2^{20}}$Correct Option: , 2 Solution:...
Read More →Solve the following systems of equations:
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\mathrm{b}_{1} \hat{\mathrm{i}}+\mathrm{b}_{2} \hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}} \quad$ and $\overrightarrow{\mathrm{c}}=5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $\vec{a}$. If $\vec{a}+\vec{b}$ is perpendicular to $\overrightarrow{\mathrm{c}}$, then $|\overri...
Read More →An unbiased coin is tossed. If the outcome is a head then a pair
Question: An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1,2,3, \ldots, 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is :$\frac{13}{36}$$\frac{19}{36}$$\frac{19}{72}$$\frac{15}{72}$Correct Option: , 3 Solution: $\mathrm{P}(\mathrm...
Read More →Let a function
Question: Let a function $f:(0, \infty) \rightarrow(0, \infty)$ be defined by $f(x)=\left|1-\frac{1}{x}\right| .$ Then $f$ is :-Injective onlyNot injective but it is surjectiveBoth injective as well as surjectiveNeither injective nor surjectiveCorrect Option: , 2 Solution: $f(x)=\left|1-\frac{1}{x}\right|=\frac{|x-1|}{x}=\left\{\begin{array}{cc}\frac{1-x}{x} 0x \leq 1 \\ \frac{x-1}{x} x \geq 1\end{array}\right.$ $\Rightarrow f(x)$ is not injective but range of function is $[0, \infty)$ Remark: I...
Read More →An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled
Question: An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1,2,3, \ldots, 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is :$\frac{13}{36}$$\frac{19}{36}$$\frac{19}{72}$$\frac{15}{72}$Correct Option: , 3 Solution: $\mathrm{P}(\mathrm...
Read More →Let z_0 be a root of the quadratic equation,
Question: Let $z_{0}$ be a root of the quadratic equation, $x^{2}+x+1=0$. If $z=3+6 i z_{0}^{81}-3 i z_{0}^{93}$, then arg $z$ is equal to:$\frac{\pi}{4}$$\frac{\pi}{3}$0$\frac{\pi}{6}$Correct Option: 1 Solution: $z_{0}=\omega$ or $\omega^{2}$ (where $\omega$ is a non-real cube root of unity) $z=3+6 i(\omega)^{81}-3 i(\omega)^{93}$ $z=3+3 i$ $\Rightarrow \arg z=\frac{\pi}{4}$...
Read More →The area of the region
Question: The area of the region $A=[(x, y): 0 \leq y \leq x|x|+1$ and $-1 \leq x \leq 1]$ in sq. units, is :$\frac{2}{3}$$\frac{1}{3}$2$\frac{4}{3}$Correct Option: , 3 Solution: The graph is a follows $\int_{-1}^{0}\left(-x^{2}+1\right) d x+\int_{0}^{1}\left(x^{2}+1\right) d x=2$...
Read More →solve this
Question: Let $(x+10)^{50}+(x-10)^{50}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots \ldots$ $+\mathrm{a}_{50} \mathrm{x}^{50}$, for all $\mathrm{x} \in \mathrm{R}$, then $\frac{\mathrm{a}_{2}}{\mathrm{a}_{0}}$ is equal to:-$12.50$$12.00$$12.75$$12.25$Correct Option: , 4 Solution: $(10+x)^{50}+(10-x)^{50}$ $\Rightarrow a_{2}=2 .{ }^{50} C_{2} 10^{48}, a_{0}=2.10^{50}$ $\frac{\mathrm{a}_{2}}{\mathrm{a}_{0}}=\frac{{ }^{50} \mathrm{C}_{2}}{10^{2}}=12.25$...
Read More →Solve this following
Question: For each $t \in R$, let [t] be the greatest integer less than or equal to $\mathrm{t}$. Then, $\lim _{x \rightarrow 1+} \frac{(1-|x|+\sin |1-x|) \sin \left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}$equals $-1$equals 1does not existequals 0Correct Option: , 4 Solution: $\lim _{x \rightarrow 1^{+}} \frac{(1-|x|+\sin |1-x|) \sin \left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}$ $=\lim _{x \rightarrow 1^{+}} \frac{(1-x)+\sin (x-1)}{(x-1)(-1)} \sin \left(\frac{\pi}{2}(-1)\right)$ $=\lim _{x \right...
Read More →Prove the following
Question: If $A=\left[\begin{array}{ccc}e^{t} e^{-t} \cos t e^{-t} \sin t \\ e^{t} -e^{-t} \cos t-e^{-t} \sin t -e^{-t} \sin t+e^{-t} \cos t \\ e^{t} 2 e^{-t} \sin t -2 e^{-t} \cos t\end{array}\right]$ Then $\mathrm{A}$ is-Invertible only if $t=\frac{\pi}{2}$not invertible for any $t \varepsilon R$invertible for all $t \varepsilon R$invertible only if $t=\pi$Correct Option: , 3 Solution: $|A|=e^{-t}\left|\begin{array}{ccc}1 \cos t \sin t \\ 1 -\cos t-\sin t -\sin t+\cos t \\ 1 2 \sin t -2 \cos t...
Read More →Evaluate the following integrals:
Question: The integral $\int_{\pi / 6}^{\pi / 4} \frac{d x}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\right)}$ equals :-$\frac{1}{10}\left(\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{9 \sqrt{3}}\right)\right)$$\frac{1}{5}\left(\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{3 \sqrt{3}}\right)\right)$$\frac{\pi}{10}$$\frac{1}{20} \tan ^{-1}\left(\frac{1}{9 \sqrt{3}}\right)$Correct Option: 1 Solution: $\mathrm{I}=\int_{\pi / 6}^{\pi / 4} \frac{\mathrm{dx}}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\right)}$ $I=\frac{1}{...
Read More →Let A be a point on the line
Question: Let $A$ be a point on the line $\overrightarrow{\mathrm{r}}=(1-3 \mu) \hat{\mathrm{i}}+(\mu-1) \hat{\mathrm{j}}+(2+5 \mu) \hat{\mathrm{k}}$ and $\mathrm{B}(3,2,6)$ be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow{\mathrm{AB}}$ is parallel to the plane $x-4 y+3 z=1$ is :$\frac{1}{2}$$-\frac{1}{4}$$\frac{1}{4}$$\frac{1}{8}$Correct Option: , 3 Solution: Let point $\mathrm{A}$ is $(1-3 \mu) \hat{\mathrm{i}}+(\mu-1) \hat{\mathrm{j}}+(2+5 \mu) \hat{\math...
Read More →If both the roots of the quadratic equation
Question: If both the roots of the quadratic equation $x^{2}-m x+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $\mathrm{m}$ lies in the interval:$(4,5)$$(3,4)$$(5,6)$$(-5,-4)$Correct Option: 1 Solution: $x^{2}-m x+4=0$ $\alpha, \beta \in[1,5]$ (1) $\begin{aligned} \mathrm{D} 0 \Rightarrow \mathrm{m}^{2}-160 \\ \Rightarrow \mathrm{m} \in(-\infty,-4) \cup(4, \infty) \end{aligned}$ (2) $f(1) \geq 0 \Rightarrow 5-\mathrm{m} \geq 0 \Rightarrow \mathrm{m} \in(-\infty, 5]$ (3) $...
Read More →If the area of the triangle whose one vertex is at the vertex of the parabola,
Question: If the area of the triangle whose one vertex is at the vertex of the parabola, $y^{2}+4\left(x-a^{2}\right)=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is 250 sq. units, then a value of ' $a$ ' is :-$5 \sqrt{5}$$(10)^{2 / 3}$$5\left(2^{1 / 3}\right)$5Correct Option: , 4 Solution: Vertex is $\left(a^{2}, 0\right)$ $\mathrm{y}^{2}=-\left(\mathrm{x}-\mathrm{a}^{2}\right)$ and $\mathrm{x}=0 \Rightarrow(0, \pm 2 \mathrm{a})$ Area of triangle is...
Read More →Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin,
Question: Let the length of the latus rectum of an ellipse with its major axis along $x$-axis and centre at the origin, be 8 . If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?$(4 \sqrt{3}, 2 \sqrt{3})$$(4 \sqrt{3}, 2 \sqrt{2})$$(4 \sqrt{2}, 2 \sqrt{2})$$(4 \sqrt{2}, 2 \sqrt{3})$Correct Option: , 2 Solution: $\frac{2 b^{2}}{a}=8$ and $2 a e=2 b$ $\Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\mathrm{e}$ and ...
Read More →For each x ε R, let [x] be the greatest integer less
Question: For each $\mathrm{x} \varepsilon \mathrm{R}$, let $[\mathrm{x}]$ be the greatest integer less than or equal to $x$. Then $\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}$ is equal to$-\sin 1$01$\sin 1$Correct Option: 1 Solution: $\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}$ $\mathrm{x} \rightarrow 0^{-}$ $[x]=-1 \Rightarrow \lim _{x \rightarrow 0^{-}} \frac{x(-x-1) \sin (-1)}{-x}=-\sin 1$ $|x|=-x$...
Read More →Solve this following
Question: Consider the statement: "P(n): $n^{2}-n+41$ is prime." Then which one of the following is true? $P(5)$ is false but $P(3)$ is trueBoth $\mathrm{P}(3)$ and $\mathrm{P}(5)$ are false$P(3)$ is false but $P(5)$ is trueBoth $P(3)$ and $P(5)$ are trueCorrect Option: , 4 Solution: $P(n): n^{2}-n+41$ is prime $P(5)=61$ which is prime $P(3)=47$ which is also prime...
Read More →Solve this following
Question: If $5,5 \mathrm{r}, 5 \mathrm{r}^{2}$ are the lengths of the sides of a triangle, then $\mathrm{r}$ cannot be equal to: $\frac{3}{2}$$\frac{3}{4}$$\frac{5}{4}$$\frac{7}{4}$Correct Option: , 4 Solution: $\mathrm{r}=1$ is obviously true. Let $0\mathrm{r}1$ $\Rightarrow \quad r+r^{2}1$ $\Rightarrow r^{2}+r-10$ $\left(r-\frac{-1-\sqrt{5}}{2}\right)\left(r-\left(\frac{-1+\sqrt{5}}{2}\right)\right)$ $\Rightarrow r-\frac{-1-\sqrt{5}}{2}$ or $r\frac{-1+\sqrt{5}}{2}$ $r \in\left(\frac{\sqrt{5}-...
Read More →Let K be the set of all real values of x where the function
Question: Let $\mathrm{K}$ be the set of all real values of $x$ where the function $\mathrm{f}(\mathrm{x})=\sin |\mathrm{x}|-|\mathrm{x}|+2(\mathrm{x}-\pi) \cos |\mathrm{x}|$ is not differentiable. Then the set $K$ is equal to:-$\{\pi\}$$\{0\}$$\phi$ (an empty set)$\{0, \pi\}$Correct Option: , 3 Solution: $f(x)=\sin |x|-|x|+2(x-\pi) \cos x$ $\because \sin |\mathrm{x}|-|\mathrm{x}|$ is differentiable function at $\mathrm{x}=0$ $\therefore \mathrm{k}=\phi$...
Read More →The coefficient of t^4 in the expansion of
Question: The coefficient of $\mathrm{t}^{4}$ in the expansion of $\left(\frac{1-t^{6}}{1-t}\right)^{3}$ is12151014Correct Option: , 2 Solution: $\left(1-t^{6}\right)^{3}(1-t)^{-3}$ $\left(1-t^{18}-3 t^{6}+3 t^{12}\right)(1-t)^{-3}$ $\Rightarrow$ cofficient of $t^{4}$ in $(1-t)^{-3}$ is ${ }^{3+4-1} C_{4}=6 C_{2}=15$...
Read More →Let a and b be the roots of the quadratic equation
Question: Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2} \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0$ $\left(0\theta45^{\circ}\right)$, and $\alpha\beta$. Then $\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)$ is equal to :-$\frac{1}{1-\cos \theta}+\frac{1}{1+\sin \theta}$$\frac{1}{1+\cos \theta}+\frac{1}{1-\sin \theta}$$\frac{1}{1-\cos \theta}-\frac{1}{1+\sin \theta}$$\frac{1}{1+\cos \theta}-\frac{1}{1-\sin \theta}$Correct Option: 1 Solut...
Read More →Solve this following
Question: Let $I=\int_{a}^{b}\left(x^{4}-2 x^{2}\right) d x$. If I is minimum then the ordered pair (a, b) is : $(-\sqrt{2}, 0)$$(-\sqrt{2}, \sqrt{2})$$(0, \sqrt{2})$$(\sqrt{2},-\sqrt{2})$Correct Option: , 2 Solution: Let $f(x)=x^{2}\left(x^{2}-2\right)$ As long as $f(x)$ lie below the $x$-axis, definite integral will remain negative, so correct value of $(a, b)$ is $(-\sqrt{2}, \sqrt{2})$ for minimum of I...
Read More →Solve that equation
Question: If $\int_{0}^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} \mathrm{d} \theta=1-\frac{1}{\sqrt{2}},(\mathrm{k}0)$, then the value of $\mathrm{k}$ is :2$\frac{1}{2}$41Correct Option: 1 Solution: $\frac{1}{\sqrt{2 k}} \int_{0}^{\pi / 3} \frac{\tan \theta}{\sqrt{\sec \theta}} \mathrm{d} \theta=\frac{1}{\sqrt{2 \mathrm{k}}} \int_{0}^{\pi / 3} \frac{\sin \theta}{\sqrt{\cos \theta}} \mathrm{d} \theta$ $=-\left.\frac{1}{\sqrt{2 k}} 2 \sqrt{\cos \theta}\right|_{0} ^{\pi / 3}=-\frac...
Read More →The plane passing through the point
Question: The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point: $(-1,-1,-1)$$(-1,-1,1)$$(1,1,-1)$$(1,1,1)$Correct Option: , 4 Solution: Let $\overrightarrow{\mathrm{n}}$ be the normal vector to the plane passing through $(4,-1,2)$ and parallel to the lines $\mathrm{L}_{1} \ \mathrm{~L}_{2}$ then $\overrightarrow{\mathrm{n}}=\left|\begin{array}{ccc}\hat{\m...
Read More →If the point (2, a ,b ) lies on the plane which passes through the points (3,4,2) and (7,0,6) and
Question: If the point $(2, \alpha, \beta)$ lies on the plane which passes through the points $(3,4,2)$ and $(7,0,6)$ and is perpendicular to the plane $2 x-5 y=15$, then $2 \alpha$ $-3 \beta$ is equal to :-517127Correct Option: , 4 Solution: Normal vector of plane $=\left|\begin{array}{ccc}\mathrm{i} \mathrm{j} \mathrm{k} \\ 2 -5 0 \\ 4 -4 4\end{array}\right|=-4(5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ equation of plane is $5(x-7)+2 y-3(z-6)=0$ $5 x+2 y-3 z=17$...
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