For some θ ∈ (0,π/2). if the eccentricity of the hyperbola,

Question: For some $\theta \in\left(0, \frac{\pi}{2}\right)$, if the eccentricity of the hyperbola, $x^{2}-y^{2} \sec ^{2} \theta=10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^{2} \sec ^{2} \theta+y^{2}=5$, then the length of the latus rectum of the ellipse, is:$\sqrt{30}$$\frac{4 \sqrt{5}}{3}$$2 \sqrt{6}$$\frac{2 \sqrt{5}}{3}$Correct Option: , 2 Solution: Given $\theta \in\left(0, \frac{\pi}{2}\right)$ equation of hyperbola $\Rightarrow x^{2}-y^{2} \sec ^{2} \theta=10$ $\Rightarro...

Solve this following

Question: Let $\lambda \neq 0$ be in $\mathrm{R}$. If $\alpha$ and $\beta$ are the roots of the equation, $x^{2}-x+2 \lambda=0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3 x^{2}-10 x+27 \lambda=0$, then $\frac{\beta \gamma}{\lambda}$ is equal to : 3627 918Correct Option: , 4 Solution: $\alpha+\beta=1, \quad \alpha \beta=2 \lambda$ $\alpha+\beta=\frac{10}{3}, \quad \alpha \gamma=\frac{27 \lambda}{3}=9 \lambda$ $\gamma-\beta=\frac{7}{3}$ $\frac{\gamma}{\beta}=\frac{9}{2} \Rightarro...

Let the latus ractum of the parabola

Question: Let the latus ractum of the parabola $y^{2}=4 x$ be the common chord to the circles $C_{1}$ and $C_{2}$ each of them having radius $2 \sqrt{5}$. Then, the distance between the centres of the circles $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ is:8$4 \sqrt{5}$12$8 \sqrt{5}$Correct Option: 1, Solution: Length of latus rectum $=4$ $\mathrm{DB}=2$ $C_{1} B=\sqrt{\left(C_{1} D\right)^{2}-(D B)^{2}}=4$ $\mathrm{C}_{1} \mathrm{C}_{2}=8$...

Solve this following

Question: Let $\bigcup_{\mathrm{i}=1}^{50} \mathrm{X}_{\mathrm{i}}=\mathrm{U}_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{Y}_{\mathrm{i}}=\mathrm{T}$, where each $\mathrm{X}_{\mathrm{i}}$ contains 10 elements and each $Y_{i}$ contains 5 elements. If each element of the set $\mathrm{T}$ is an element of exactly 20 of sets $X_{i}{ }^{\prime} s$ and exactly 6 of sets $Y_{i}{ }^{\prime} s$, then $\mathrm{n}$ is equal to : 45155030Correct Option: , 4 Solution: $\mathrm{n}\left(\mathrm{X}_{\mathrm{i}}\right)=...

The equation of the normal to the curve

Question: The equation of the normal to the curve $y=(1+x)^{2 y}+\cos ^{2}\left(\sin ^{-1} x\right)$ at $x=0$ is :$y=4 x+2$$x+4 y=8$$y+4 x=2$$2 \mathrm{y}+\mathrm{x}=4$Correct Option: , 2 Solution: Given equation of curve $y=(1+x)^{2 y}+\cos ^{2}\left(\sin ^{-1} x\right)$ at $x=0$ $y=(1+0)^{2 y}+\cos ^{2}\left(\sin ^{-1} 0\right)$ $y=1+1$ $y=2$ So we have to find the normal at $(0,2)$ Now $y=e^{2 y \ln (1+x)}+\cos ^{2}\left(\cos ^{-1} \sqrt{1-x^{2}}\right)$ $y=e^{2 y \ln (1+x)}+\left(\sqrt{1-x^{...

Let the function

Question: Let $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ be two relations defined as follows: $\mathrm{R}_{1}=\left\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R}^{2}: \mathrm{a}^{2}+\mathrm{b}^{2} \in \mathrm{Q}\right\}$ and $\mathrm{R}_{2}=\left\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R}^{2}: \mathrm{a}^{2}+\mathrm{b}^{2} \notin \mathrm{Q}\right\}$ where $Q$ is the set of all rational numbers. Then:$R_{2}$ is transitive but $R_{1}$ is not transitive$R_{1}$ is transitive but $R_{2}$ is not transitive$\mathr...

Solve this

Question: The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|1\end{array}\right.$ is :continuous on $\mathrm{R}-\{1\}$ and differentiable on $\mathrm{R}-\{-1,1\}$.both continuous and differentiable on $R-\{-1\}$.continuous on $R-\{-1\}$ and differentiable on $\mathrm{R}-\{-1,1\}$.both continuous and differentiable on $\mathrm{R}-\{1\}$Correct Option: 1 Solution: for continuity at $\mathrm{x}=-1$ L.H.L. $=\frac{\pi}{4}-\frac{\pi}{4}=0$ R.H.L. ...

Solve this

Question: The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|1\end{array}\right.$ is :continuous on $\mathrm{R}-\{1\}$ and differentiable on $\mathrm{R}-\{-1,1\}$.both continuous and differentiable on $R-\{-1\}$.continuous on $R-\{-1\}$ and differentiable on $\mathrm{R}-\{-1,1\}$.both continuous and differentiable on $\mathrm{R}-\{1\}$Correct Option: Solution: for continuity at $\mathrm{x}=-1$ L.H.L. $=\frac{\pi}{4}-\frac{\pi}{4}=0$ R.H.L. $=...

limite X rightarrow 0 {tan(pi/4+x)} 1/x

Question: $\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x}$ is equal to :2$\mathrm{e}$1$\mathrm{e}^{2}$Correct Option: , 4 Solution: $\lim _{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}$ $=\lim _{x \rightarrow 0} \frac{1}{x}\left\{\tan \left(\frac{x}{4}+x\right)-1\right\}$ $=e^{\lim _{x \rightarrow 9}\left(\frac{1+\tan x-1+\tan x}{x(1-\tan x)}\right)}$ $=\lim _{x \rightarrow x(1-\tan x)}$ $=\mathrm{e}^{2}$...

The imaginary part of

Question: The imaginary part of $(3+2 \sqrt{-54})^{1 / 2}-(3-2 \sqrt{-54})^{1 / 2}$ can be : $-2 \sqrt{6}$6$\sqrt{6}$$-\sqrt{6}$Correct Option: 1 Solution: $(3+2 \sqrt{-54})=3+2 \times 3 \times \sqrt{6} \mathrm{i}$ $=(3+\sqrt{6} i)^{2}$ $(3-2 \sqrt{54})=(3-\sqrt{6} i)^{2}$ $(3+2 \sqrt{-54})^{1 / 2}+(3-2 \sqrt{-54})^{1 / 2}$ $=\pm(3+\sqrt{6} \mathrm{i}) \pm(3-\sqrt{6} \mathrm{i})$ $=6,-6,2 \sqrt{6} i,-2 \sqrt{6} i$...

If the value of the integral

Question: If the value of the integral $\int_{0}^{1 / 2} \frac{x^{2}}{\left(1-x^{2}\right)^{3 / 2}} d x$ is $\frac{\mathrm{k}}{6}$, then $\mathrm{k}$ is equal to :$2 \sqrt{3}-\pi$$3 \sqrt{2}+\pi$$3 \sqrt{2}-\pi$$2 \sqrt{3}+\pi$Correct Option: 1 Solution: $\int_{0}^{1 / 2} \frac{\left(\left(x^{2}-1\right)+1\right)}{\left(1-x^{2}\right)^{3 / 2}} d x$ $\int_{0}^{1 / 2} \frac{d x}{\left(1-x^{2}\right)^{3 / 2}}-\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$ $\int_{0}^{1 / 2} \frac{x^{-3}}{\left(x^{-2}-...

If the sum of first 11 terms of an A.P.,

Question: If the sum of first 11 terms of an A.P., $a_{1} a_{2}, a_{3}, \ldots$ is $0\left(\mathrm{a}_{1} \neq 0\right)$, then the sum of the A.P., $\mathrm{a}_{1}, \mathrm{a}_{3}, \mathrm{a}_{5}, \ldots, \mathrm{a}_{23}$ is $\mathrm{ka}_{1}$, where $\mathrm{k}$ is equal to :$\frac{121}{10}$$-\frac{72}{5}$$\frac{72}{5}$$-\frac{121}{10}$Correct Option: , 2 Solution: $a_{1}+a_{2}+a_{3}+\ldots+a_{11}=0$ $\Rightarrow\left(a_{1}+a_{11}\right) \times \frac{11}{2}=0$ $\Rightarrow a_{1}+a_{11}=0$ $\Righ...

Let f : (-1 , infinite) rightarrow R be defined by f(0) = 1 and

Question: Let $f:(-1, \infty) \rightarrow R$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x} \log _{e}(1+x), x \neq 0$. Then the function $f$ :decreases in $(-1, \infty)$decreases in $(-1,0)$ and increases in $(0, \infty)$increases in $(-1, \infty)$increases in $(-1,0)$ and decreases in $(0, \infty)$Correct Option: 1 Solution: $f^{\prime}(x)=\frac{\frac{x}{1+x}-\ell n(1+x)}{x^{2}}$ $=\frac{x-(1+x) \ell n(1+x)}{x^{2}(1+x)}$ Suppose $h(x)=x-(1+x) \ell n(1+x)$ $\Rightarrow h^{\prime}(x)=1-\ell n(1+x)-...

If the surface area of a cube is increasing at a rate

Question: If the surface area of a cube is increasing at a rate of $3.6 \mathrm{~cm}^{2} / \mathrm{sec}$, retaining its shape; then the rate of change of its volume (in $\mathrm{cm}^{3} / \mathrm{sec}$ ), when the length of a side of the cube is $10 \mathrm{~cm}$, is:9181020Correct Option: 1 Solution: $\frac{\mathrm{d}}{\mathrm{dt}}\left(6 \mathrm{a}^{2}\right)=3.6 \Rightarrow 12 \mathrm{a} \frac{\mathrm{da}}{\mathrm{dt}}=3.6$ $\mathrm{a} \frac{\mathrm{da}}{\mathrm{dt}}=0.3$ $\frac{\mathrm{dv}}{...

Let a, b, c ∈ R be all non - zero and satisfy

Question: Let $a, b, c \in R$ be all non-zero and satisfy $\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}=2$. If the matrix $A=\left(\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right)$ satisfies $\mathrm{A}^{\mathrm{T}} \mathrm{A}=\mathrm{I}$, then a value of abc can be :$\frac{2}{3}$$-\frac{1}{3}$3$\frac{1}{3}$Correct Option: , 4 Solution: $\mathrm{A}^{\mathrm{T}} \mathrm{A}=\mathrm{I}$ $\Rightarrow a^{2}+b^{2}+c^{2}=1$ and $a b+b c+c a=0$ Now, $(a+b+c)^{2}=1$ $\Rightarrow a+b+c=\pm 1$ S...

If the equation of a plane $mathrm{P}$, passing through the

Question: If the equation of a plane $\mathrm{P}$, passing through the intesection of the planes, $x+4 y-z+7=0$ and $3 x+y+5 z=8$ is $a x+b y+6 z=15$ for some $a, b \in R$, then the distance of the point $(3,2,$, -1) from the plane $\mathrm{P}$ is Solution: $\mathrm{D}_{1}=\left|\begin{array}{ccc}-7 4 -1 \\ 8 1 5 \\ 15 \mathrm{~h} 6\end{array}\right|=0 \Rightarrow \mathrm{b}=-3$ $\mathrm{D}=\left|\begin{array}{ccc}1 4 -1 \\ 3 1 5 \\ \mathrm{a} \mathrm{b} 6\end{array}\right|=0 \Rightarrow 21 \mat...

Let f: R rightarrow R be a function which satisfies

Question: Let $f: R \rightarrow R$ be a function which satisfies $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R}$. If $\mathrm{f}(1)=2$ and $g(n)=\sum_{k=1}^{(n-1)} f(k), n \in N$ then the value of $n$, for which $g(n)=20$, is :59204Correct Option: 1 Solution: $f(x+y)=f(x)+f(y)$ $\Rightarrow \mathrm{f}(\mathrm{n})=\mathrm{nf}(1)$ $\mathrm{f}(\mathrm{n})=2 \mathrm{n}$ $g(n)=\sum_{k=1}^{n-1} 2 n=2\left(\frac{(n-1) n}{2}...

If the solve the problem

Question: If $\left(\frac{1+i}{1-i}\right)^{\frac{m}{2}}=\left(\frac{1+i}{i-1}\right)^{\frac{n}{3}}=1,(m, n \in N)$ then the greatest common divisor of the least values of $\mathrm{m}$ and $\mathrm{n}$ is___________ Solution: $\left(\frac{1+\mathrm{i}}{1-\mathrm{i}}\right)^{\mathrm{m} / 2}=\left(\frac{1+\mathrm{i}}{\mathrm{i}-1}\right)^{\mathrm{n} / 3}=1$ $\Rightarrow\left(\frac{(1+i)^{2}}{2}\right)^{m / 2}=\left(\frac{(1+i)^{2}}{-2}\right)^{n / 3}=1$ $\Rightarrow(\mathrm{i})^{\mathrm{m} / 2}=(-...

Question: Let $\left(2 x^{2}+3 x+4\right)^{10}=\sum_{r=0}^{20} a_{r} x^{r}$. Then $\frac{a_{7}}{a_{13}}$ is equal to Solution: Given $\left(2 x^{2}+3 x+4\right)^{10}=\sum_{r=0}^{20} a_{r} x^{r}$ ........(i) replace $x$ by $\frac{2}{x}$ in above identity :- $\frac{2^{10}\left(2 x^{2}+3 x+4\right)^{10}}{x^{20}}=\sum_{r=0}^{20} \frac{a_{r} 2^{r}}{x^{r}}$ $\Rightarrow 2^{10} \sum_{\mathrm{r}=0}^{20} \mathrm{a}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}}=\sum_{\mathrm{r}=0}^{20} \mathrm{a}_{\mathrm{r}} 2^{\...

Let f(x) be a quadratic polynomial such that

Question: Let $f(x)$ be a quadratic polynomial such that $\mathrm{f}(-1)+\mathrm{f}(2)=0$. If one of the roots of $\mathrm{f}(\mathrm{x})=0$ is 3 , then its other root lies in :$(-3,-1)$$(1,3)$$(-1,0)$$(0,1)$Correct Option: , 3 Solution: $f(x)=a(x-3)(x-\alpha)$ $f(2)=a(\alpha-2)$ $f(-1)=4 a(1+\alpha)$ $f(-1)+f(2)=0 \Rightarrow a(\alpha-2+4+4 \alpha)=0$ $a \neq 0 \Rightarrow 5 \alpha=-2$ $\alpha=-\frac{2}{5}=-0.4$ $\alpha \in(-1,0)$...

Solve the Following Questions

Question: The value of $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \text { to } \infty\right)}$ is equal to______________ Solution: $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\ldots \ldots \ldots \ldots \text { to } \infty\right)}$ $=\left(\frac{4}{25}\right)^{\log \left(\frac{5}{2}\right)\left(\frac{1}{2}\right)}$ $=\left(\frac{1}{2}\right)^{\log \left(\frac{5}{2}\right)\left(\frac{4}{25}\right)}=\left(\frac{1}{2}\right)^{-2}=4$...

Solve the Following Questions

Question: The value of $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \text { to } \infty\right)}$ is equal to______________ Solution: $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\ldots \ldots \ldots \ldots \text { to } \infty\right)}$ $=\left(\frac{4}{25}\right)^{\log \left(\frac{5}{2}\right)\left(\frac{1}{2}\right)}$ $=\left(\frac{1}{2}\right)^{\log \left(\frac{5}{2}\right)\left(\frac{4}{25}\right)}=\left(\frac{1}{2}\right)^{-2}=4$...

If the equation cos4 O + sin4 O + lambda = 0 has real solutions for O.

Question: If the equation $\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval :$\left[-\frac{3}{2},-\frac{5}{4}\right]$$\left(-\frac{1}{2},-\frac{1}{4}\right]$$\left(-\frac{5}{4},-1\right)$$\left[-1,-\frac{1}{2}\right]$Correct Option: , 4 Solution: $\lambda=-\left(\sin ^{4} \theta+\cos ^{4} \theta\right)$ $\lambda=-\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cos ^{2} \theta$ $\lambda=\frac{\sin ^{2} 2 \thet...

Suppose a differentiable function

Question: Suppose a differentiable function $\mathrm{f}(\mathrm{x})$ satisfies the identity $f(x+y)=f(x)+f(y)+x y^{2}+x^{2} y$, for all real $x$ and $y$. If $\operatorname{Lim}_{x \rightarrow 0} \frac{f(x)}{x}=1$, then $f^{\prime}(3)$ is equal to Solution: Since, $\lim _{x \rightarrow 0} \frac{f(x)}{x}$ exist $\Rightarrow f(0)=0$ Now, $f^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{f(\mathrm{x}+\mathrm{h})-f(\mathrm{x})}{\mathrm{h}}$ $=\lim _{h \rightarrow 0} \frac{f(h)+x h^{2}+x^...

The diameter of the circle, whose center lies on the line x+y=2 in the first quadrant and which touches both the lines x=3 and y=2, is

Question: The diameter of the circle, whose centre lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$, is____________ Solution: $\because \quad$ center lies on $x+y=2$ and in 1 st quadrant center $=(\alpha, 2-\alpha)$ where $\alpha0$ and $2-\alpha0 \Rightarrow 0\alpha2$ $\because$ circle touches $x=3$ and $y=2$ $\Rightarrow|3-\alpha|=|2-(2-\alpha)|=$ radius $\Rightarrow|3-\alpha|=|\alpha| \Rightarrow \alpha=\frac{3}{2}$ $\therefore \quad$ radius $=\al...