Consider an arithmetic series and a geometric
Question: Consider an arithmetic series and a geometric series having four initial terms from the set $\{11,8,21,16,26,32,4\}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to_________. Solution: GP : $4,8,16,32,64,128,256,512,1024,2048$, 4096,8192 AP : $11,16,21,26,31,36$ Common terms : $16,256,4096$ only...
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Question: For the four circles $\mathrm{M}, \mathrm{N}, \mathrm{O}$ and $\mathrm{P}$, following four equations are given : Circle $\mathrm{M}: \mathrm{x}^{2}+\mathrm{y}^{2}=1$ Circle $\mathrm{N}: \mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}=0$ Circle $\mathrm{O}: \mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-2 \mathrm{y}+1=0$ Circle $P: x^{2}+y^{2}-2 y=0$ If the centre of circle $\mathrm{M}$ is joined with centre of the circle N, further centre of circle $\mathrm{N}$ is joined with centre of the circ...
Read More →If A =
Question: If $\mathrm{A}=\left[\begin{array}{cc}2 3 \\ 0 -1\end{array}\right]$, then the value of $\operatorname{det}\left(\mathrm{A}^{4}\right)+\operatorname{det}\left(\mathrm{A}^{10}-(\operatorname{Adj}(2 \mathrm{~A}))^{10}\right)$ is equal to Solution: $2 \mathrm{~A}$ adj $(2 \mathrm{~A})=|2 \mathrm{~A}| \mathrm{I}$ $\Rightarrow \mathrm{A}$ adj $(2 \mathrm{~A})=-4 \mathrm{I} \quad \ldots . .(\mathrm{i})$ Now, E = $\left|\mathrm{A}^{4}\right|+\left|\mathrm{A}^{10}-(\operatorname{adj}(2 \mathrm...
Read More →Prove the following
Question: Let $S_{k}=\sum_{r=1}^{k} \tan ^{-1}\left(\frac{6^{r}}{2^{2 r+1}+3^{2 r+1}}\right) .$ Then $\lim _{k \rightarrow \infty} S_{k}$ is equal to :$\tan ^{-1}\left(\frac{3}{2}\right)$$\frac{\pi}{2}$$\cot ^{-1}\left(\frac{3}{2}\right)$$\tan ^{-1}(3)$Correct Option: , 3 Solution: $\mathrm{S}_{\mathrm{k}}=\sum_{\mathrm{r}=1}^{\mathrm{k}} \tan ^{-1}\left(\frac{6^{\mathrm{r}}}{2^{2 \mathrm{r}+1}+3^{2 \mathrm{r}+1}}\right)$ Divide by $3^{2 \mathrm{r}}$ $\sum_{r=1}^{k} \tan ^{-1}\left(\frac{\left(\...
Read More →Solve the Following Questions
Question: If $\quad \overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{b}}=-\beta \hat{\mathrm{i}}-\alpha \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ such that $\vec{a} \cdot \vec{b}=1$ and $\vec{b} \cdot \vec{c}=-3$, then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to Solution: $\vec{a} \cdot \vec{b}=1 \Rightarrow-\alpha \beta-\alp...
Read More →The number of roots of the equation,
Question: The number of roots of the equation, $(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$ in the interval $[0, \pi]$ is equal to :3482Correct Option: , 2 Solution: $(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$ $(81)^{\sin ^{2} x}+\frac{(81)^{1}}{(18)^{\sin ^{2} x}}=30$ $(81)^{\sin ^{2} x}=t$ $t+\frac{81}{t}=30$ $\mathrm{t}^{2}-30 \mathrm{t}+81=0$ $(\mathrm{t}-3)(\mathrm{t}-27)=0$ $(81)^{\sin ^{2} x}=3^{1} \quad$ or $(81)^{\sin ^{2} x}=3^{3}$ $3^{4 \sin ^{2} x}=3^{1} \quad$ or $\quad 3^{4 \sin ^{2} ...
Read More →Solve this following
Question: If the equation $a|z|^{2}+\overline{\bar{\alpha} z+\alpha \bar{z}}+d=0$ represents a circle where a,d are real constants then which of the following condition is correct? $|\alpha|^{2}-\mathrm{ad} \neq 0$$|\alpha|^{2}-a d0$ and $a \in R-\{0\}$$|\alpha|^{2}-a d \geq 0$ and $a \in R$$\alpha=0, a, d \in R^{+}$Correct Option: , 2 Solution: az $\bar{z}+\alpha \bar{z}+\bar{\alpha} z+d=0 \rightarrow$ Circle centre $=\frac{-\alpha}{a} \quad 2=\sqrt{\frac{\alpha \bar{\alpha}}{a^{2}}-\frac{d}{a}...
Read More →Let y=y(x) be a solution curve of the differential equation
Question: Let $y=y(x)$ be a solution curve of the differential equation $(y+1) \tan ^{2} x d x+\tan x d y+y d x=0$ $x \in\left(0, \frac{\pi}{2}\right) .$ If $\lim _{x \rightarrow 0+} x y(x)=1$, then the value of $y\left(\frac{\pi}{4}\right)$ is :$-\frac{\pi}{4}$$\frac{\pi}{4}-1$$\frac{\pi}{4}+1$$\frac{\pi}{4}$Correct Option: , 4 Solution: $(y+1) \tan ^{2} x d x+\tan x d y+y d x=0$ or $\frac{d y}{d x}+\frac{\sec ^{2} x}{\tan x} \cdot y=-\tan x$ $I F=e^{\int \frac{\sec ^{2} x}{\tan x} d x}=e^{\ln ...
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Question: A vector $\vec{a}$ has components $3 \mathrm{p}$ and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $\vec{a}$ has components $p+1$ and $\sqrt{10}$, then a value of $p$ is equal to:1$-\frac{5}{4}$$\frac{4}{5}$$-1$Correct Option: , 4 Solution: $\vec{a}_{\text {Old }}=3 p \hat{i}+\hat{j}$ $\vec{a}_{\text {New }}=(p+1) \hat{i}+\sqrt{10} \hat{j}$ $\Rightarrow\le...
Read More →Solve this following
Question: If $\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{3 x^{3}}$ is equal to $L$, then the value of $(6 L+1)$ is $\frac{1}{6}$$\frac{1}{2}$62Correct Option: , 4 Solution: $\lim _{x \rightarrow 0} \frac{\left(x+\frac{x^{3}}{3 !} \ldots\right)-\left(x-\frac{x^{3}}{3} \ldots\right)}{3 x^{3}}=\frac{1}{6}$ So $6 L+1=2$...
Read More →The locus of the midpoints of the chord of the circle,
Question: The locus of the midpoints of the chord of the circle, $x^{2}+y^{2}=25$ which is tangent to the hyperbola, $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ is :$\left(x^{2}+y^{2}\right)^{2}-16 x^{2}+9 y^{2}=0$$\left(x^{2}+y^{2}\right)^{2}-9 x^{2}+144 y^{2}=0$$\left(x^{2}+y^{2}\right)^{2}-9 x^{2}-16 y^{2}=0$$\left(x^{2}+y^{2}\right)^{2}-9 x^{2}+16 y^{2}=0$Correct Option: , 4 Solution: $y-k=-\frac{h}{k}(x-h)$ $\mathrm{ky}-\mathrm{k}^{2}=-\mathrm{hx}+\mathrm{h}^{2}$ $\mathrm{hx}+\mathrm{ky}=\mathrm{h...
Read More →Let there be three independent events
Question: Let there be three independent events $E_{1}, E_{2}$ and $\mathrm{E}_{3}$. The probability that only $\mathrm{E}_{1}$ occurs is $\alpha$, only $\mathrm{E}_{2}$ occurs is $\beta$ and only $\mathrm{E}_{3}$ occurs is $\gamma$. Let ' $p$ ' denote the probability of none of events occurs that satisfies the equations $(\alpha-2 \beta) \mathrm{p}=\alpha \beta$ and $(\beta-3 \gamma) \mathrm{p}=2 \beta \gamma .$ All the given probabilities are assumed to lie in the interval $(0,1)$. Then, $\fra...
Read More →The equation of one of the straight lines which passes through the point
Question: The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angles $\tan ^{-1}(\sqrt{2})$ with the straight line, $y+1=3 \sqrt{2} x$ is $4 \sqrt{2} x+5 y-(15+4 \sqrt{2})=0$$5 \sqrt{2} x+4 y-(15+4 \sqrt{2})=0$$4 \sqrt{2} x+5 y-4 \sqrt{2}=0$$4 \sqrt{2} x-5 y-(5+4 \sqrt{2})=0$Correct Option: 1 Solution: $y=m x+c$ $3=m+c$ $\sqrt{2}=\left|\frac{m-3 \sqrt{2}}{1+3 \sqrt{2} m}\right|$ $=6 m+\sqrt{2}=m-3 \sqrt{2}$ $=\sin =-4 \sqrt{2} \rightarrow m=\frac{-4 \sqr...
Read More →If y=y(x) is the solution of the differential
Question: If $y=y(x)$ is the solution of the differential equation, $\frac{d y}{d x}+2 y \tan x=\sin x, y\left(\frac{\pi}{3}\right)=0$, then the maximum value of the function $\mathrm{y}(\mathrm{x})$ over $\mathbb{R}$ is equal to :8$\frac{1}{2}$$-\frac{15}{4}$$\frac{1}{8}$Correct Option: , 4 Solution: $\frac{d y}{d x}+2 y \tan x=\sin x$ $\mathrm{I} \mathrm{F}=\mathrm{e}^{\int 2 \tan x \mathrm{dx}}=\mathrm{e}^{2 \ln \sec x}$ I.F. $=\sec ^{2} x$ $y \cdot\left(\sec ^{2} x\right)=\int \sin x \cdot \...
Read More →On the ellipse
Question: On the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$ let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x+2 y=0 .$ Let $S$ and $S^{\prime}$ be the foci of the ellipse and e be its eccentricity. If $\mathrm{A}$ is the area of the triangle $\mathrm{SPS}^{\prime}$ then, the value of $\left(5-\mathrm{e}^{2}\right) . \mathrm{A}$ is :6121424Correct Option: 1 Solution: Equation of tangent: $y=2 x+6$ at $\mathrm{P}$ $\therefore \ma...
Read More →Solve the Following Questions
Question: If $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ and its first derivative with respect to $x$ is $-\frac{b}{a} \log _{c} 2$ when $x=1$, where a and b are integers, then the minimum value of $\left|a^{2}-b^{2}\right|$ is Solution: $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ at $x=1 ; 2^{2 x}=4$ for $\sin \left(\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right)$ Let $\tan ^{-1} \mathrm{x}=\theta ; \theta \in\left(-\frac{\pi...
Read More →Let [x] denote greatest integer less than or
Question: Let $[x]$ denote greatest integer less than or equal to $x$. If for $n \in \mathbb{N},\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}$ then $\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j}+1$ is equal to :2$2^{n-1}$1$\mathrm{n}$Correct Option: , 3 Solution: $\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}$ $\left(1-x+x^{3}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2} \ldots \ldots+a_{3 n} x^{3 n}$ $\sum_{j=0}^{\left[\frac{3 n}...
Read More →The integral
Question: The integral$\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{4 x^{2}-4 x+6}} d x$ is equal to (where $\mathrm{c}$ is a constant of integration) $\frac{1}{2} \sin \sqrt{(2 x-1)^{2}+5}+c$$\frac{1}{2} \cos \sqrt{(2 x+1)^{2}+5}+c$$\frac{1}{2} \cos \sqrt{(2 x-1)^{2}+5}+c$$\frac{1}{2} \sin \sqrt{(2 x+1)^{2}+5}+c$Correct Option: 1 Solution: $\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{(2 x-1)^{2}+5}} d x$ $(2 \mathrm{x}-1)^{2}+5=\mathrm{t}^{2}$ $2(2 \mathrm{x}-1) 2 \mathrm{dx}=2 \...
Read More →If the function
Question: If the function $f(x)=\frac{\cos (\sin x)-\cos x}{x^{4}}$ is continuous at each point in its domain and $\mathrm{f}(0)=\frac{1}{\mathrm{k}}$, then $\mathrm{k}$ is Solution: $\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^{4}}=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{2 \sin \left(\frac{\sin x+x}{2}\right) \sin \left(\frac{x-\sin x}{2}\right)}{x^{4}}=\frac{1}{K}$ $\Rightarrow \lim _{x \rightarrow 0} 2\left(\frac{\sin x+x}{2 x}\right)\left(\frac{x-\sin x}{2 x^{3}}\right)=\f...
Read More →A pack of cards has one card missing.
Question: A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :$\frac{3}{4}$$\frac{52}{867}$$\frac{39}{50}$$\frac{22}{425}$Correct Option: , 3 Solution: $\mathrm{E}_{1}$ : Event denotes spade is missing $\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{1}{4} ; \mathrm{P}\left(\overline{\mathrm{E}}_{1}\right)=\frac{3}{4}$ A : Event drawn two cards are spade $P(A)=\frac{\frac{1}{4} \times\left(\frac{{ }^...
Read More →The mean and standard deviation of 20 observations were calculated
Question: The mean and standard deviation of 20 observations were calculated as 10 and $2.5$ respectively. It was found that by mistake one data value was taken as 25 instead of $35 .$ If $\alpha$ and $\sqrt{\beta}$ are the mean and standard deviation respectively for correct data, then $(\alpha, \beta)$ is :$(11,26)$$(10.5,25)$$(11,25)$$(10.5,26)$Correct Option: 4, Solution: Given : $\operatorname{Mean}(\overline{\mathrm{x}})=\frac{\sum \mathrm{x}_{\mathrm{i}}}{20}=10$ or $\Sigma x_{i}=200$ (in...
Read More →The range of a in R for which the function
Question: The range of a $\in \mathbb{R}$ for which the function $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$ $\mathrm{x} \neq 2 \mathrm{n} \pi, \mathrm{n} \in \mathbb{N}$, has critical points, is :$(-3,1)$$\left[-\frac{4}{3}, 2\right]$$[1, \infty)$$(-\infty,-1]$Correct Option: , 2 Solution: $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+(a-7) \sin x$ $f(x)=(4 a-3)(1)+(a-7) \cos x=0$ $\Rightarrow \quad \cos x=\frac{3-4 a}{a-7}$ $-1 \leq...
Read More →The maximum value of
Question: The maximum value of $\mathrm{z}$ in the following equation $z=6 x y+y^{2}$, where $3 x+4 y \leq 100$ and $4 x+3 y \leq 75$ for $x \geq 0$ and $y \geq 0$ is Solution: $z=6 x y+y^{2}=y(6 x+y)$ $3 x+4 y \leq 100$...(1) $4 x+3 y \leq 75$...(2). $x \geq 0$ $y \geq 0$ $x \leq \frac{75-3 y}{4}$ $Z=y(6 x+y)$ $z \leq y\left(6 \cdot\left(\frac{75-3 y}{4}\right)+y\right)$ $z \leq \frac{1}{2}\left(225 y-7 y^{2}\right) \leq \frac{(225)^{2}}{2 \times 4 \times 7}$ $=\frac{50625}{56}$ $\approx 904.01...
Read More →Solve this following
Question: Let $\alpha, \beta, \gamma$ be the real roots of the equation, $x^{3}+a x^{2}+b x+c=0,(a, b, c \in R$ and $a, b \neq 0)$. If the system of equations (in, $\mathrm{u}, \mathrm{v}, \mathrm{w}$ ) given by $\alpha u+\beta v+\gamma w=0, \beta u+\gamma v+\alpha w=0 ;$ $\gamma \mathrm{u}+\alpha \mathrm{v}+\beta \mathrm{w}=0$ has non-trivial solution, then the value of $\frac{\mathrm{a}^{2}}{\mathrm{~b}}$ is 5310Correct Option: , 2 Solution: $\left|\begin{array}{ccc}\alpha \beta \gamma \\ \bet...
Read More →Let P be a plane l x+m y+n z=0 containing
Question: Let P be a plane $l x+m y+n z=0$ containing the line, $\frac{1-\mathrm{x}}{1}=\frac{\mathrm{y}+4}{2}=\frac{\mathrm{z}+2}{3} .$ If plane $\mathrm{P}$ divides the line segment $A B$ joining points $\mathrm{A}(-3,-6,1)$ and $\mathrm{B}(2,4,-3)$ in ratio $\mathrm{k}: 1$ then the value of $\mathrm{k}$ is equal to :1.5324Correct Option: , 3 Solution: $\left(\frac{2 k-3}{k+1}, \frac{4 k-6}{k+1}, \frac{-3 k+1}{k+1}\right)$ $\frac{x-1}{-1}=\frac{y+4}{2}=\frac{z+2}{3}$ Plane $l x+m y+n z=0$ $l(-...
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