The minimum value of
Question: The minimum value of $f(x)=a^{a^{x}}+a^{1-a^{x}}$ where $\mathrm{a}, \mathrm{x} \in \mathrm{R}$ and $\mathrm{a}0$, is equal to :$2 \mathrm{a}$$2 \sqrt{\mathrm{a}}$$a+\frac{1}{a}$$a+1$Correct Option: , 2 Solution: A.M. $\geq$ G.M. $f(x)=a^{a^{x}}+a^{1-a^{x}}=a^{a^{x}}+\frac{a}{a^{a^{x}}} \geq 2 \sqrt{a}$...
Read More →Solve this following
Question: For the statements $\mathrm{p}$ and $\mathrm{q}$, consider the following compound statements : (a) $(\sim \mathrm{q} \wedge(\mathrm{p} \rightarrow \mathrm{q})) \rightarrow \sim \mathrm{p}$ (b) $((\mathrm{p} \vee \mathrm{q}) \wedge \sim \mathrm{p}) \rightarrow \mathrm{q}$ Then which of the following statements is correct? (a) and (b) both are not tautologies.(a) and (b) both are tautologies.(a) is a tautology but not (b).(b) is a tautology but not (a).Correct Option: , 2 Solution: (A) B...
Read More →Solve the Following Questions
Question: If $0\mathrm{a}, \mathrm{b}1$, and $\tan ^{-1} \mathrm{a}+\tan ^{-1} \mathrm{~b}=\frac{\pi}{4}$, then the value of $(a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots$ is:$\log _{e} 2$$e^{2}-1$$\mathrm{e}$$\log _{e}\left(\frac{e}{2}\right)$Correct Option: 1 Solution: $\tan ^{-1} \mathrm{a}+\tan ^{-1} \mathrm{~b}=\frac{\pi}{4} \quad 0\mathrm{a}, \mathrm{b}1$ $\Rightarrow \frac{a+b}{1-a b}=1$ $a+b=1-a b$ $(a+1)(b+1)=2$ No...
Read More →If for the matrix,
Question: If for the matrix, $\mathrm{A}=\left[\begin{array}{cc}1 -\alpha \\ \alpha \beta\end{array}\right], \mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{2}$, then the value of $\alpha^{4}+\beta^{4}$ is :4231Correct Option: , 4 Solution: $\mathrm{A}=\left[\begin{array}{cc}1 -\alpha \\ \alpha \beta\end{array}\right] \quad \mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{2}$ $\Rightarrow\left[\begin{array}{cc}1 -\alpha \\ \alpha \beta\end{array}\right]\left[\begin{array}{cc}1 \alpha \\ -\alpha \beta\end{array}\right]...
Read More →A function f(x) is given by
Question: A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series $\mathrm{f}\left(\frac{1}{20}\right)+\mathrm{f}\left(\frac{2}{20}\right)+\mathrm{f}\left(\frac{3}{20}\right)+\ldots \ldots+\mathrm{f}\left(\frac{39}{20}\right)$ is equal to :$\frac{19}{2}$$\frac{49}{2}$$\frac{29}{2}$$\frac{39}{2}$Correct Option: , 4 Solution: $f(x)=\frac{5^{x}}{5^{x}+5} \quad f(2-x)=\frac{5}{5^{x}+5}$ $f(x)+f(2-x)=1$ $\Rightarrow f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+...
Read More →Consider the following system of equations :
Question: Consider the following system of equations : $x+2 y-3 z=a$ $2 x+6 y-11 z=b$ $x-2 y+7 z=c$ where $a, b$ and $c$ are real constants. Then the system of equations :has a unique solution when $5 a=2 b+c$has infinite number of solutions when $5 \mathrm{a}=2 \mathrm{~b}+\mathrm{c}$has no solution for all $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$has a unique solution for all a, b and cCorrect Option: , 2 Solution: $P_{1}: x+2 y-3 z=a$ $P_{2}: 2 x+6 y-11 z=b$ $P_{3}: x-2 y+7 z=c$ Clearly $5 P_...
Read More →The contrapositive of the statement "If you will work,
Question: The contrapositive of the statement "If you will work, you will earn money" is :You will earn money, if you will not workIf you will earn money, you will workIf you will not earn money, you will not workTo earn money, you need to workCorrect Option: , 3 Solution: Constrapositive of $\mathrm{p} \rightarrow \mathrm{q}$ is $\sim \mathrm{q} \rightarrow \sim \mathrm{p}$ $\Rightarrow$ If you will not earn money, you will not work. option (3)...
Read More →If the locus of the mid-point
Question: If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, $x^{2}+y^{2}=1$ is a circle of radius $r$, then $r$ is equal to :1$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{4}$Correct Option: , 2 Solution: $h=\frac{\cos \theta+3}{2}$c $\mathrm{k}=\frac{\sin \theta+2}{2}$ $\Rightarrow\left(\mathrm{h}-\frac{3}{2}\right)^{2}+(\mathrm{k}-1)^{2}=\frac{1}{4}$. $\Rightarrow \mathrm{r}=\frac{1}{2}$...
Read More →If the curve
Question: If the curve $x^{2}+2 y^{2}=2$ intersects the line $\mathrm{x}+\mathrm{y}=1$ at two points $\mathrm{P}$ and $\mathrm{Q}$, then the angle subtended by the line segment PQ at the origin is :$\frac{\pi}{2}+\tan ^{-1}\left(\frac{1}{3}\right)$$\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{3}\right)$$\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{4}\right)$$\frac{\pi}{2}+\tan ^{-1}\left(\frac{1}{4}\right)$Correct Option: , 4 Solution: Homogenising $x^{2}+2 y^{2}-2(x+y)^{2}=0$ $\Rightarrow-x^{2}-4 x y=0 \Ri...
Read More →Solve this following
Question: Let $\mathrm{P}=\left[\begin{array}{ccc}3 -1 -2 \\ 2 0 \alpha \\ 3 -5 0\end{array}\right]$, where $\alpha \in \mathrm{R}$. Suppose $\mathrm{Q}=\left[\mathrm{q}_{\mathrm{ij}}\right]$ is a matrix satisfying $\mathrm{PQ}=\mathrm{kI}_{3}$ for some non-zero $\mathrm{k} \in \mathrm{R} .$ If $\mathrm{q}_{23}=-\frac{\mathrm{k}}{8}$ and $|\mathrm{Q}|=\frac{\mathrm{k}^{2}}{2}$, then $\alpha^{2}+\mathrm{k}^{2}$ is equal to Solution: $\mathrm{PQ}=\mathrm{kI}$ $|\mathrm{P}| .|\mathrm{Q}|=\mathrm{k}...
Read More →Prove the following
Question: $\operatorname{cosec}\left[2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :$\frac{56}{33}$$\frac{65}{56}$$\frac{65}{33}$$\frac{75}{56}$Correct Option: , 2 Solution: $\operatorname{cosec}\left[2 \tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{3}{4}\right)\right]$ $\operatorname{cosec}\left[\tan ^{-1}\left(\frac{5}{12}\right)+\tan ^{-1}\left(\frac{3}{4}\right)\right]$ $=\operatorname{cosec}\left[\tan ^{-1}\left(\frac{56}{33}\right)\right]=\frac{65}{56}$ o...
Read More →Let slope of the tangent
Question: Let slope of the tangent line to a curve at any point $\mathrm{P}(\mathrm{x}, \mathrm{y})$ be given by $\frac{\mathrm{xy}^{2}+\mathrm{y}}{\mathrm{x}}$. If the curve intersects the line $x+2 y=4$ at $x=-2$, then the value of $y$, for which the point $(3, y)$ lies on the curve, is :$\frac{18}{35}$$-\frac{4}{3}$$-\frac{18}{19}$$-\frac{18}{11}$Correct Option: , 3 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{xy}^{2}+\mathrm{y}}{\mathrm{x}}$ $\frac{x d y-y d x}{y^{2}}=x d x$ $-\m...
Read More →In a group of 400 people, 160 are smokers and non-vegetarian;
Question: In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35 \%, 20 \%$ and $10 \%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :$\frac{7}{45}$$\frac{14}{45}$$\frac{28}{45}$$\frac{8}{45}$Correct Optio...
Read More →Solve this following
Question: Let $\mathrm{B}_{i}(i=1,2,3)$ be three independent events in a sample space. The probability that only $\mathrm{B}_{1}$ occur is $\alpha$, only $B_{2}$ occurs is $\beta$ and only $B_{3}$ occurs is $\gamma$. Let p be the probability that none of the events $\mathrm{B}_{i}$ occurs and these 4 probabilities satisfy the equations $(\alpha-2 \beta) \mathrm{p}=\alpha \beta$ and $(\beta-3 \gamma) p=2 \beta \gamma($ All the probabilities are assumed to lie in the interval $(0,1))$. Then $\frac...
Read More →Solve the Following Questions
Question: Let $\mathrm{F}_{1}(\mathrm{~A}, \mathrm{~B}, \mathrm{C})=(\mathrm{A} \wedge \sim \mathrm{B}) \vee[\sim \mathrm{C} \wedge(\mathrm{A} \vee \mathrm{B})] \vee \sim \mathrm{A}$ and $\mathrm{F}_{2}(\mathrm{~A}, \mathrm{~B})=(\mathrm{A} \vee \mathrm{B}) \vee(\mathrm{B} \rightarrow \sim \mathrm{A})$ be two logical expressions. Then :$\mathrm{F}_{1}$ and $\mathrm{F}_{2}$ both are tautologies$F_{1}$ is a tautology but $F_{2}$ is not a tautology$F_{1}$ is not tautology but $F_{2}$ is a tautology...
Read More →A plane passes through the points A (1,2,3),
Question: A plane passes through the points $\mathrm{A}(1,2,3), \mathrm{B}(2,3,1)$ and $\mathrm{C}(2,4,2)$. If $\mathrm{O}$ is the origin and $\mathrm{P}$ is $(2,-1,1)$,$\sqrt{\frac{2}{7}}$$\sqrt{\frac{2}{3}}$$\sqrt{\frac{2}{11}}$$\sqrt{\frac{2}{5}}$Correct Option: , 3 Solution: Normal to plane $\overrightarrow{\mathrm{n}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} \hat{\mathrm{j}} \hat{\mathrm{k}} \\ 1 1 -2 \\ 0 1 1\end{array}\right|$ $=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ $\over...
Read More →Solve this following
Question: Let three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be such that $\overrightarrow{\mathrm{c}}$ is coplanar with $\vec{a}$ and $\vec{b}, \vec{a} \cdot \vec{c}=7$ and $\vec{b}$ is perpendicular to $\overrightarrow{\mathrm{c}}$, where $\overrightarrow{\mathrm{a}}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{k}}$, then the value of $2|\overrightarrow{\mathrm{a...
Read More →If 0 < x,y < π and cosx - cos(x+y) = 3/4,
Question: If $0x, y\pi$ and $\cos x+\cos y-\cos (x+y)=\frac{3}{2}$, then $\sin x+\cos y$ is equal to :$\frac{1}{2}$$\frac{1+\sqrt{3}}{2}$$\frac{\sqrt{3}}{2}$$\frac{1-\sqrt{3}}{2}$Correct Option: , 2 Solution: $\cos x+\cos y-\cos (x+y)=\frac{3}{2}$ $\cos ^{2}\left(\frac{x+y}{2}\right)-\cos \left(\frac{x+y}{2}\right) \cdot \cos \left(\frac{x-y}{2}\right)$ $+\frac{1}{4} \cdot \cos ^{2}\left(\frac{x-y}{2}\right)+\frac{1}{4} \sin ^{2}\left(\frac{x-y}{2}\right)=0$ $\Rightarrow\left(\cos \left(\frac{x+...
Read More →Let L be a
Question: Let $L$ be a line obtained from the intersection of two planes $x+2 y+z=6$ and $y+2 z=4$. If point $\mathrm{P}(\alpha, \beta, \gamma)$ is the foot of perpendicular from $(3,2,1)$ on $\mathrm{L}$, then the value of $21(\alpha+\beta+\gamma)$ equals :14268136102Correct Option: , 4 Solution: $x+2 y+z=6$ (y+2 z=4) \times 2 $x-3 z=-2 \Rightarrow x=3 z-2 \Rightarrow y=4-2 z$ $\frac{x+2}{3}=z$$\frac{y-4}{-2}=z$ $\Rightarrow$ line of intersection of two planes is $\frac{x+2}{3}=\frac{y-4}{-2}=z...
Read More →Solve this following
Question: $\lim _{n \rightarrow \infty} \tan \left\{\sum_{r=1}^{n} \tan ^{-1}\left(\frac{1}{1+r+r^{2}}\right)\right\}$ is equal to Solution: $\lim _{n \rightarrow \alpha} \tan \left(\sum_{r=1}^{n} \tan ^{-1}\left(\frac{1}{1+r(r+1)}\right)\right)$ $=\lim _{n \rightarrow \alpha} \tan \left(\sum_{r=1}^{n} \tan ^{-1}\left(\frac{r+1-r}{1+r(r+1)}\right)\right)$ $=\tan \left(\lim _{n \rightarrow \alpha} \sum_{r=1}^{n}\left[\tan ^{-1}(r+1)-\tan ^{-1}(r)\right]\right)$ $=\tan \left(\lim _{n \rightarrow \...
Read More →A hyperbola passes through the foci of the ellipse
Question: A hyperbola passes through the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities in one, then the equation of the hyperbola is :$\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$$\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$$x^{2}-y^{2}=9$$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$Correct Option: , 2 Solution: For ellipse $e_{1}=\sqrt{1-\frac{b^{2}}{a^{2}}}=\frac{3}{5}$...
Read More →The minimum value of
Question: The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $\left(0, \frac{\pi}{2}\right)$ is Solution: Let $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ $\Rightarrow f^{\prime}(x)=0 \Rightarrow \sin x=2 / 3$ $\therefore f(x)_{\min }=\frac{4}{2 / 3}+\frac{1}{1-2 / 3}=9$ $f(\mathrm{x}) \max \rightarrow \infty$ $f(x)$ is continuous function $\therefore \alpha_{\min }=9$...
Read More →The triangle of maximum
Question: The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is :An isosceles triangle with base equal to $2 \mathrm{r}$.An equilateral triangle of height $\frac{2 r}{3}$.An equilateral triangle having each of its side of length $\sqrt{3} \mathrm{r}$.A right angle triangle having two of its sides of length $2 \mathrm{r}$ and $\mathrm{r}$.Correct Option: , 3 Solution: $h=r \sin \theta+r$ base $=\mathrm{BC}=2 \mathrm{r} \cos \theta$ $\theta \in\left[0, \frac{\pi...
Read More →If α , β ∈ R are such that 1 - 2i
Question: If $\alpha, \beta \in R$ are such that $1-2 i$ (here $i^{2}=-1$ ) is a root of $z^{2}+\alpha z+\beta=0$, then $(\alpha-\beta)$ is equal to :$-3$-773Correct Option: , 2 Solution: $\because \alpha, \beta \in \mathrm{R} \Rightarrow$ other root is $1+2 \mathrm{i}$ $\alpha=-($ sum of roots $)=-(1-2 \mathrm{i}+1+2 \mathrm{i})=-2$ $\beta=$ product of roots $=(1-2 \mathrm{i})(1+2 \mathrm{i})=5$ $\therefore \alpha-\beta=-7$ option (2)...
Read More →Solve this following
Question: If one of the diameters of the circle $x^{2}+y^{2}-2 x-6 y+6=0$ is a chord of another circle ' $C$ ', whose center is at $(2,1)$, then its radius is Solution: $x^{2}+y^{2}+2 x-6 y+6=0$ center $(1,3)$ radius $=2$ distance between $(1,3)$ and $(2,1)$ is $\sqrt{5}$ $\therefore(\sqrt{5})^{2}+(2)^{2}=r^{2}$ $\Rightarrow \mathrm{r}=3$...
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