Prove the following
Question: Let $f(\mathrm{x})=\mathrm{a}^{\mathrm{x}}(\mathrm{a}0)$ be written as $f(\mathrm{x})=f_{1}(\mathrm{x})+f_{2}(\mathrm{x})$, where $f_{1}(\mathrm{x})$ is an even function of $f_{2}(x)$ is an odd function. Then $f_{1}(\mathrm{x}+\mathrm{y})+f_{1}(\mathrm{x}-\mathrm{y})$ equals$2 f_{1}(\mathrm{x}) f_{1}(\mathrm{y})$$2 f_{1}(\mathrm{x}) f_{2}(\mathrm{y})$$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{2}(\mathrm{x}-\mathrm{y})$$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{1}(\mathrm{x}-\mathrm{y})$Correct Option:...
Read More →If the lengths of the sides of a triangle are in A.P.
Question: If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is :$5: 9: 13$$5: 6: 7$$4: 5: 6$$3: 4: 5$Correct Option: , 3 Solution: $\mathrm{a}\mathrm{b}\mathrm{c}$ are in A.P. $\angle \mathrm{C}=2 \angle \mathrm{A}$ (Given) $\Rightarrow \sin C=\sin 2 A$ $\Rightarrow \sin C=2 \sin A \cdot \cos A$ $\Rightarrow \frac{\sin C}{\sin A}=2 \cos A$ $\Rightarrow \frac{\mathrm{c}}{\mathrm{a}}=2 \frac{\m...
Read More →If the eccentricity of the standard hyperbola passing
Question: If the eccentricity of the standard hyperbola passing through the point $(4,6)$ is 2 , then the equation of the tangent to the hyperbola at $(4,6)$ is-$2 x-y-2=0$$3 x-2 y=0$$2 x-3 y+10=0$$x-2 y+8=0$Correct Option: 1 Solution: Let us Suppose equation of hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $\mathrm{e}=2 \Rightarrow \mathrm{b}^{2}=3 \mathrm{a}^{2}$ passing through $(4,6) \Rightarrow \mathrm{a}^{2}=4, \mathrm{~b}^{2}=12$ $\Rightarrow$ equaiton of tangent $x-\frac{y}{2}...
Read More →If the system of linear equations
Question: If the system of linear equations $x-2 y+k z=1$ $2 x+y+z=2$ $3 x-y-k z=3$ has a solution $(x, y, z), z \neq 0$, then $(x, y)$ lies on the straight line whose equation is :$3 x-4 y-1=0$$3 x-4 y-4=0$$4 x-3 y-4=0$$4 x-3 y-1=0$Correct Option: , 3 Solution: $x-2 y+k z=1$ ............(1) $2 x+y+z=2$ ..............(2) $3 x-y-k z=3$ .............(3) (1) $+(3)$ $\Rightarrow 4 x-3 y=4$...
Read More →The value of
Question: Let $\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\mathrm{x} \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$, for some real $x$. Then $|\vec{a} \times \vec{b}|=r$ is possible if :$3 \sqrt{\frac{3}{2}}r5 \sqrt{\frac{3}{2}}$$0\mathrm{r} \leq \sqrt{\frac{3}{2}}$$\sqrt{\frac{3}{2}}$r \geq 5 \sqrt{\frac{3}{2}}$Correct Option: 4, Solution: $\vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 3 2...
Read More →Solve that equation
Question: The sum $\sum_{k=1}^{20} k \frac{1}{2^{k}}$ is equal to-$2-\frac{3}{2^{17}}$$2-\frac{11}{2^{19}}$$1-\frac{11}{2^{20}}$$2-\frac{21}{2^{20}}$Correct Option: , 2 Solution: $\mathrm{S}=\sum_{\mathrm{k}=1}^{20} \frac{1}{2^{\mathrm{k}}}$ $\mathrm{S}=\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{3^{2}}+\ldots+\frac{20}{2^{20}}$ $\mathrm{S} \times \frac{1}{2}=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{19}{2^{20}}+\frac{20}{2^{21}}$ $\Rightarrow\left(1-\frac{1}{2}\right) \mathrm{S}=\frac{1}{2}+\frac{1...
Read More →A student scores the following marks in five tests :
Question: A student scores the following marks in five tests : $45,54,41,57,43$. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is$\frac{10}{\sqrt{3}}$$\frac{100}{\sqrt{3}}$$\frac{100}{3}$$\frac{10}{3}$Correct Option: 1 Solution: Let $x$ be the $6^{\text {th }}$ observation $\Rightarrow 45+54+41+57+43+x=48 \times 6=288$ $\Rightarrow x=48$ variance $=\left(\frac{\sum \mathrm{x}_{\mathrm{i}}^{2}}{6}-(\overl...
Read More →The minimum number of times one has to toss a fair coin
Question: The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least $90 \%$ is :5324Correct Option: , 4 Solution: Probability of observing at least one head out of $n$ tosses $=1-\left(\frac{1}{2}\right)^{\mathrm{n}} \geq 0.9$ $\Rightarrow\left(\frac{1}{2}\right)^{\mathrm{n}} \leq 0.1$ $\Rightarrow n \geq 4$ $\Rightarrow$ minimum number of tosses $=4$...
Read More →The negation of the boolean expression
Question: The negation of the boolean expression $\sim \mathrm{s} \vee(\sim \mathrm{r} \wedge \mathrm{s})$ is equivalent to:$\mathrm{r}$$\mathrm{s} \wedge \mathrm{r}$$\mathrm{s} \vee \mathrm{r}$$\sim \mathrm{S} \wedge \sim \mathrm{r}$Correct Option: , 2 Solution: $\sim(\sim \mathrm{S} \vee(\sim \mathrm{r} \wedge \mathrm{S}))$ $\mathrm{s} \wedge(\mathrm{r} \vee \sim \mathrm{s})$ $(s \wedge r) \vee(s \wedge \sim s)$ $(s \wedge r) \vee(c)$ $(s \wedge r)$...
Read More →Prove the following
Question: If $\lim _{x \rightarrow 1} \frac{x^{2}-a x+b}{x-1}=5$, then $a+b$ is equal to :-$-7$$-4$51Correct Option: 1 Solution: $\lim _{x \rightarrow 1} \frac{x^{2}-a x+b}{x-1}=5$ $1-a+b=0$ .............(i) $2-a=5$ ..............(ii) $\Rightarrow a+b=-7$...
Read More →The number of real roots of the equation
Question: The number of real roots of the equation $5+\left|2^{x}-1\right|=2^{x}\left(2^{x}-2\right)$ is :2341Correct Option: , 4 Solution: Let $2^{\mathrm{x}}=\mathrm{t}$ $5+|t-1|=t^{2}-2 t$ So, number of real root is 1 ....
Read More →If the tangent to the curve
Question: If the tangent to the curve $\mathrm{y}=\frac{\mathrm{x}}{\mathrm{x}^{2}-3}, \mathrm{x} \in \mathrm{R}$, $(x \neq \pm \sqrt{3})$, at a point $(\alpha, \beta) \neq(0,0)$ on it is parallel to the line $2 x+6 y-11=0$, then :$|6 \alpha+2 \beta|=19$$|2 \alpha+6 \beta|=11$$|6 \alpha+2 \beta|=9$$|2 \alpha+6 \beta|=19$Correct Option: 1 Solution: $\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{(\alpha, \beta)}=\frac{-\alpha^{2}-3}{\left(\alpha^{2}-3\right)^{2}}$ Given that : $\frac{-\alpha^{2}-3...
Read More →Prove the following
Question: Let $\mathrm{f}(\mathrm{x})=\log _{\mathrm{e}}(\sin \mathrm{x}),(0\mathrm{x}\pi)$ and $g(x)=\sin ^{-1}\left(e^{-x}\right),(x \geq 0)$. If $\alpha$ is a positive real number such that $\mathrm{a}=(\mathrm{fog})^{\prime}(\alpha)$ and $\mathrm{b}=(\mathrm{fog})(\alpha)$, then :$a \alpha^{2}-b \alpha-a=0$$a \alpha^{2}+b \alpha-a=-2 \alpha^{2}$$a \alpha^{2}+b \alpha+a=0$$a \alpha^{2}-b \alpha-a=1$Correct Option: , 4 Solution: fog $(x)=(-x) \Rightarrow(f g(\alpha))=-\alpha=b$ $(f g(x))^{\pri...
Read More →The sum of the real roots of the equatuion
Question: The sum of the real roots of the equatuion $\left|\begin{array}{ccc}x -6 -1 \\ 2 -3 x x-3 \\ -3 2 x x+2\end{array}\right|=0$, is equal to 610$-4$Correct Option: , 3 Solution: By expansion, we get $-5 x^{3}+30 x-30+5 x=0$ $\Rightarrow-5 x^{3}+35 x-30=0$ $\Rightarrow x^{3}-7 x+6=0$, All roots So, sum of roots $=0$...
Read More →Suppose that 20 pillars of the same height have been erected
Question: Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number beams is:210190170180Correct Option: , 3 Solution: Total cases $=$ number of diagonals $={ }^{20} C_{2}-20=170$...
Read More →Minimum number of times a fair coin must be tossed
Question: Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99 \%$ is :5678Correct Option: , 3 Solution: $1-\left(\frac{1}{2}\right)^{n}\frac{99}{100}$ $\Rightarrow\left(\frac{1}{2}\right)^{\mathrm{n}}\frac{1}{100}$ $\Rightarrow \mathrm{n}=7$...
Read More →The angles A, B and C of a triangal ABC are
Question: The angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of a triangle $\mathrm{ABC}$ are in A.P. and $\mathrm{a}: \mathrm{b}=1: \sqrt{3}$. If $\mathrm{c}=4 \mathrm{~cm}$, then the area (in sq. cm) of this triangle is :$4 \sqrt{3}$$\frac{2}{\sqrt{3}}$$2 \sqrt{3}$$\frac{4}{\sqrt{3}}$Correct Option: , 3 Solution: $\angle \mathrm{B}=\frac{\pi}{3}$, by sine Rule $\sin \mathrm{A}=\frac{1}{2}$ $\Rightarrow \mathrm{A}=30^{\circ}, \mathrm{a}=2, \mathrm{~b}=2 \sqrt{3}, \mathrm{c}=4$ $\Delta=\frac{1...
Read More →Let a1, a2, a3,.......... be an A.P. with
Question: Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots . .$ be an A. P. with $\mathrm{a}_{6}=2$. Then the common difference of this A. P., which maximises the produce $\mathrm{a}_{1} \mathrm{a}_{4} \mathrm{a}_{5}$, is :$\frac{6}{5}$$\frac{8}{5}$$\frac{2}{3}$$\frac{3}{2}$Correct Option: , 2 Solution: Let a is first term and $\mathrm{d}$ is common difference then, a $+5 \mathrm{~d}=2$ (given) ...(1) $f(d)=(2-5 d)(2-2 d)(2-d)$ $\mathrm{f}^{\prime}(\mathrm{d})=0 \quad \Rightarrow \mat...
Read More →Let y=y(x) be the solution of the differential equation,
Question: Let $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}+y \tan x=2 x+x^{2} \tan x$ $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $y(0)=1$. Then :$\mathrm{y}^{\prime}\left(\frac{\pi}{4}\right)+\mathrm{y}^{\prime}\left(\frac{-\pi}{4}\right)=-\sqrt{2}$$y^{\prime}\left(\frac{\pi}{4}\right)-y^{\prime}\left(\frac{-\pi}{4}\right)=\pi-\sqrt{2}$$y\left(\frac{\pi}{4}\right)-y\left(-\frac{\pi}{4}\right)=\sqrt{2}$$\mathrm{y}\left(\frac{\pi}{4}\right)+\mathrm{y}\l...
Read More →The integral
Question: The integral $\int_{\pi / 6}^{\pi / 3} \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3} x d x$ equal to$3^{7 / 6}-3^{5 / 6}$$3^{5 / 3}-3^{1 / 3}$$3^{4 / 3}-3^{1 / 3}$$3^{5 / 6}-3^{2 / 3}$Correct Option: 1 Solution: $I=\int \frac{1}{\cos ^{2 / 3} x \sin ^{1 / 3} x \cdot \sin x} d x$ $=\int \frac{\tan ^{2 / 3} x}{\tan ^{2} x} \cdot \sec ^{2} x \cdot d x$ $=\int \frac{\sec ^{2} x}{\tan ^{4 / 3} x} \cdot d x \quad\left\{\tan x=t, \sec ^{2} x d x=d t\right\}$ $=\int \frac{d t}{t^{4 / 3}}=\frac{...
Read More →Let a, b and c be in G.P. with common ratior,
Question: Let $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ be in G. P. with common ratio r, where $a \neq 0$ and $0r \leq \frac{1}{2}$. If $3 a, 7 b$ and $15 c$ are the first three terms of an A. P., then the $4^{\text {th }}$ term of this A. P. is :$\frac{7}{3} a$$a$$\frac{2}{3} a$$5 \mathrm{a}$Correct Option: , 2 Solution: $\mathrm{b}=\mathrm{ar}$ $\mathrm{c}=\mathrm{ar}{ }^{2}$ $3 \mathrm{a}, 7 \mathrm{~b}$ and $15 \mathrm{c}$ are in A.P. $\Rightarrow 14 b=3 a+15 c$ $\Rightarrow 14(a r)=3 a+15 a...
Read More →If z and w are two complex numbers such that
Question: If $\mathrm{z}$ and $\mathrm{w}$ are two complex numbers such that $|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$, then :$\overline{Z W}=i$$\bar{Z} W=-1$$z \bar{w}=\frac{1-i}{\sqrt{2}}$$z \bar{w}=\frac{-1+i}{\sqrt{2}}$Correct Option: , 2 Solution: $|\mathrm{z}| .|\mathrm{w}|=1 \quad \mathrm{z}=\mathrm{re}^{\mathrm{i}(\theta+\pi / 2)}$ and $\mathrm{w}=\frac{1}{\mathrm{r}} \mathrm{e}^{\mathrm{i} \theta}$ $\bar{z} . w=e^{-i(\theta+\pi / 2)} \cdot e^{i \theta}=e^{-i(\pi / 2)}=-i$ $z \cdot...
Read More →If the plane 2 x-y+2 z+3=0 has the distances
Question: If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x-2 y+4 z+\lambda=0$ and $2 x-y+2 z+\mu=0$, respectively, then the maximum value of $\lambda+\mu$ is equal to :155139Correct Option: , 3 Solution: $4 x-2 y+4 z+6=0$ $\frac{|\lambda-6|}{\sqrt{16+4+16}}=\left|\frac{\lambda-6}{6}\right|=\frac{1}{3}$ $|\lambda-6|=2$ $\lambda=8,4$ $\frac{|\mu-3|}{\sqrt{4+4+1}}=\frac{2}{3}$ $|\mu-3|=2$ $\mu=5,1$ $\therefore$ Maximum value of $(\mu+\lambda)...
Read More →The area (in sq. units) of the region bounded
Question: The area (in sq. units) of the region bounded by the curves $y=2^{x}$ and $y=|x+1|$, in the first quadrant is:$\frac{3}{2}-\frac{1}{\log _{\mathrm{e}} 2}$$\frac{1}{2}$$\log _{\mathrm{e}} 2+\frac{3}{2}$$\frac{3}{2}$Correct Option: 1 Solution: Required Area $\int_{0}^{1}\left((x+1)-2^{x}\right) d x$ $=\left(\frac{x^{2}}{2}+x-\frac{2^{x}}{\ln 2}\right)_{0}^{1}$ $=\left(\frac{1}{2}+1-\frac{2}{\ln 2}\right)-\left(0+0-\frac{1}{\ln 2}\right)$ $=\frac{3}{2}-\frac{1}{\ln 2}$...
Read More →Lines are drawn parallel to the line 4 x-3 y+2=0,
Question: Lines are drawn parallel to the line $4 x-3 y+2=0$, at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?$\left(-\frac{1}{4}, \frac{2}{3}\right)$$\left(\frac{1}{4}, \frac{1}{3}\right)$$\left(-\frac{1}{4},-\frac{2}{3}\right)$$\left(\frac{1}{4},-\frac{1}{3}\right)$Correct Option: 1 Solution: Required line is $4 x-3 y+\lambda=0$ So, required equation of line is $4 x-3 y+3=0$ and $4 x-3 y-3=0$ (1) $4\left(-\frac{1}{4}\right)-3\left(...
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