Prove the following
Question: If $\int \frac{\mathrm{d} \theta}{\cos ^{2} \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|\mathrm{f}(\theta)|+\mathrm{C}$ where $C$ is a constant of integration, then the ordered pair $(\lambda, f(\theta))$ is equal to :$(-1,1+\tan \theta)$$(-1,1-\tan \theta)$$(1,1-\tan \theta)$$(1,1+\tan \theta)$Correct Option: 1 Solution:...
Read More →Solve this following
Question: The $\operatorname{sum} \sum_{\mathrm{k}=1}^{20}(1+2+3+\ldots+\mathrm{k})$ is $\ldots$ Solution:...
Read More →Solve the following equations:
Question: If $\frac{d y}{d x}=\frac{x y}{x^{2}+y^{2}} ; y(1)=1$; then a value of $x$ satisfying $y(x)=e$ is :$\sqrt{2} \mathrm{e}$$\frac{\mathrm{e}}{\sqrt{2}}$$\frac{1}{2} \sqrt{3} \mathrm{e}$$\sqrt{3} \mathrm{e}$Correct Option: , 4 Solution:...
Read More →Let the normal at a point P on the curve
Question: Let the normal at a point $P$ on the curve $y^{2}-3 x^{2}+y+10=0$ intersect the $y$-axis at $\left(0, \frac{3}{2}\right)$. If $\mathrm{m}$ is the slope of the tangent at $\mathrm{P}$ to the curve, then $|\mathrm{m}|$ is equal to Solution:...
Read More →Solve the following equations:
Question: If $A=\{x \in \mathbf{R}:|x|2\}$ and $B=\{x \in \mathbf{R}:|x-2| \geq 3\}$ then :$\mathrm{A} \cup \mathrm{B}=\mathbf{R}-(2,5)$$\mathrm{A} \cap \mathrm{B}=(-2,-1)$$\mathrm{B}-\mathrm{A}=\mathbf{R}-(-2,5)$$\mathrm{A}-\mathrm{B}=[-1,2)$Correct Option: , 3 Solution:...
Read More →The least positive value of 'a' for which the equation
Question: The least positive value of 'a' for which the equation $2 \mathrm{x}^{2}+(\mathrm{a}-10) \mathrm{x}+\frac{33}{2}=2 \mathrm{a}$ has real roots is Solution:...
Read More →If 10 different balls are to be placed in 4 distinct boxes at random,
Question: If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is :$\frac{945}{2^{11}}$$\frac{965}{2^{11}}$$\frac{945}{2^{10}}$$\frac{965}{2^{10}}$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: The number of all $3 \times 3$ matrices A, with enteries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $\mathrm{AA}^{\mathrm{T}}$ is 3 , is Solution:...
Read More →If one end of a focal chord
Question: If one end of a focal chord $\mathrm{AB}$ of the parabola $y^{2}=8 x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of the tangent to it at B is :$2 x+y-24=0$$x-2 y+8=0$$2 x-y-24=0$$x+2 y+8=0$Correct Option: , 2 Solution:...
Read More →Solve this following
Question: Let $f(x)=x \cos ^{-1}(-\sin |x|), x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, then which of the following is true ?$f^{\prime}$ is decreasing in $\left(-\frac{\pi}{2}, 0\right)$ and increasing in $\left(0, \frac{\pi}{2}\right)$$f$ is not differentiable at $x=0$$f^{\prime}(0)=-\frac{\pi}{2}$$f^{\prime}$ is increasing in $\left(-\frac{\pi}{2}, 0\right)$ and decreasing in $\left(0, \frac{\pi}{2}\right)$Correct Option: 1 Solution:...
Read More →Let a function
Question: Let a function $f:[0,5] \rightarrow \mathbf{R}$ be continuous, $f(1)=3$ and $F$ be defined as : $F(x)=\int_{1}^{x} t^{2} g(t) d t$, where $g(t)=\int_{1}^{t} f(u) d u$. Then for the function $F$, the point $x=1$ is:a point of local minima.not a critical point.a point of inflection.a point of local maxima.Correct Option: 1 Solution:...
Read More →The shortest distance between the lines
Question: The shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$ is$\frac{7}{2} \sqrt{30}$$3 \sqrt{30}$3$2 \sqrt{30}$Correct Option: , 2 Solution:...
Read More →Prove the following
Question: If $x=\sum_{n=0}^{\infty}(-1)^{n} \tan ^{2 n} \theta$ and $y=\sum_{n=0}^{\infty} \cos ^{2 n} \theta$, for $0\theta\frac{\pi}{4}$, then : $y(1+x)=1$$x(1+y)=1$$y(1-x)=1$$x(1-y)=1$Correct Option: , 3 Solution:...
Read More →A random variable X has the following probability distribution:
Question: A random variable $X$ has the following probability distribution: $\begin{array}{lllllll}X : 1 2 3 4 5\end{array}$ $\mathrm{P}(\mathrm{X}) \quad: \quad \mathrm{K}^{2} \quad 2 \mathrm{~K} \quad \mathrm{~K} \quad 2 \mathrm{~K} \quad 5 \mathrm{~K}^{2}$ Then $\mathrm{P}(\mathrm{X}2)$ is equal to :$\frac{7}{12}$$\frac{23}{36}$$\frac{1}{36}$$\frac{1}{6}$Correct Option: , 2 Solution:...
Read More →Solve this following
Question: If $\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c$ where $\mathrm{c}$ is a constant of integration, then $\lambda f\left(\frac{\pi}{3}\right)$ is equal to$-2$$-\frac{9}{8}$2$\frac{9}{8}$Correct Option: 1 Solution:...
Read More →Prove the following
Question: Given $: f(x)=\left\{\begin{array}{cc}x 0 \leq x\frac{1}{2} \\ \frac{1}{2} x=\frac{1}{2} \\ 1-x , \quad \frac{1}{2}x \leq 1\end{array}\right.$ and $g(x)=\left(x-\frac{1}{2}\right)^{2}, x \in R$. Then the area (in sq. units) of the region bounded by the curves, $\mathrm{y}=f(\mathrm{x})$ and $\mathrm{y}=\mathrm{g}(\mathrm{x})$ between the lines, $2 \mathrm{x}=1$ and $2 \mathrm{x}=\sqrt{3}$, is :$\frac{1}{3}+\frac{\sqrt{3}}{4}$$\frac{\sqrt{3}}{4}-\frac{1}{3}$$\frac{1}{2}+\frac{\sqrt{3}}{...
Read More →The inverse function of
Question: The inverse function of $f(\mathrm{x})=\frac{8^{2 \mathrm{x}}-8^{-2 \mathrm{x}}}{8^{2 \mathrm{x}}+8^{-2 \mathrm{x}}}, \mathrm{x} \in(-1,1)$, is$\frac{1}{4}\left(\log _{8} \mathrm{e}\right) \log _{\mathrm{e}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}}\right)$$\frac{1}{4} \log _{\mathrm{e}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}}\right)$$\frac{1}{4}\left(\log _{8} e\right) \log _{e}\left(\frac{1+x}{1-x}\right)$$\frac{1}{4} \log _{\mathrm{e}}\left(\frac{1+x}{1-x}\right)$Correct Option: , 3 Sol...
Read More →Let a, b in R, a neq 0 be such that the equation,
Question: Let $a, b \in R$, $a \neq 0$ be such that the equation, $a x^{2}-2 b x+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation, then $\alpha^{2}+\beta^{2}$ is equal to :26252824Correct Option: , 2 Solution:...
Read More →Solve this following
Question: Let $\mathrm{A}$ and $\mathrm{B}$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE?$\mathrm{P}(\mathrm{A} / \mathrm{B})=\frac{2}{3}$$\mathrm{P}(\mathrm{A} /(\mathrm{A} \cup \mathrm{B}))=\frac{1}{4}$$\mathrm{P}\left(\mathrm{A} / \mathrm{B}^{\prime}\right)=\frac{1}{3}$$P\left(A^{\prime} / B^{\prime}\right)=\frac{1}{3}$Correct Option: , 3 Solution:...
Read More →The length of the minor axis (along y-axis) of an ellipse
Question: The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x+6 y=8$; then its eccentricity is:$\sqrt{\frac{5}{6}}$$\frac{1}{2} \sqrt{\frac{11}{3}}$$\frac{1}{3} \sqrt{\frac{11}{3}}$$\frac{1}{2} \sqrt{\frac{5}{3}}$Correct Option: , 2 Solution:...
Read More →For which of the following ordered pairs
Question: For which of the following ordered pairs $(\mu, \delta)$, the system of linear equations $x+2 y+3 z=1$ $3 x+4 y+5 z=\mu$ $4 x+4 y+4 z=\delta$ is inconsistent?$(1,0)$$(4,6)$$(3,4)$$(4,3)$Correct Option: , 4 Solution:...
Read More →Prove the following
Question: If $x=2 \sin \theta-\sin 2 \theta$ and $y=2 \cos \theta-\cos 2 \theta$, $\theta \in[0,2 \pi]$, then $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ at $\theta=\pi$ is :$\frac{3}{2}$$-\frac{3}{4}$$\frac{3}{4}$$-\frac{3}{8}$Correct Option: , 4 Solution:...
Read More →The following system of linear equations
Question: The following system of linear equations $7 x+6 y-2 z=0$ $3 x+4 y+2 z=0$ $x-2 y-6 z=0$, hasinfinitely many solutions, $(x, y, z)$ satisfying $\mathrm{x}=2 \mathrm{z}$no solutiononly the trivial solutioninfinitely many solutions, (x,y, z) satisfying $y=2 z$Correct Option: 1 Solution:...
Read More →Let [t] denote the greatest integer
Question: Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A$. Then the function, $\mathrm{f}(\mathrm{x})=\left[\mathrm{x}^{2}\right] \sin (\pi \mathrm{x})$ is discontinuous, when $\mathrm{x}$ is equal to :$\sqrt{\mathrm{A}+5}$$\sqrt{\mathrm{A}+1}$$\sqrt{\mathrm{A}}$$\sqrt{\mathrm{A}+21}$Correct Option: , 2 Solution:...
Read More →Solve this following
Question: If $\mathrm{c}$ is a point at which Rolle's theorem holds for the function, $f(\mathrm{x})=\log _{\mathrm{e}}\left(\frac{\mathrm{x}^{2}+\alpha}{7 \mathrm{x}}\right)$ in the interval $[3,4]$, where $\alpha \in \mathrm{R}$, then $f^{\prime \prime}(\mathrm{c})$ is equal to$\frac{\sqrt{3}}{7}$$\frac{1}{12}$$-\frac{1}{24}$$-\frac{1}{12}$Correct Option: , 2 Solution:...
Read More →