Let x, y be positive real numbers and m, n positive integers.

Question: Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x^{m} y^{n}}{\left(1+x^{2 m}\right)\left(1+y^{2 n}\right)}$ is :-$\frac{1}{2}$$\frac{1}{4}$$\frac{m+n}{6 m n}$1Correct Option: , 2 Solution: $\frac{x^{m} y^{n}}{\left(1+x^{2 m}\right)\left(1+y^{2 n}\right)}=\frac{1}{\left(x^{m}+\frac{1}{x^{m}}\right)\left(y^{n}+\frac{1}{y^{n}}\right)} \leq \frac{1}{4}$ using $\mathrm{AM} \geq \mathrm{GM}$...

Solve this following

Question: The length of the chord of the parabola $x^{2}=4 y$ having equation $x-\sqrt{2} y+4 \sqrt{2}=0$ is : $2 \sqrt{11}$$3 \sqrt{2}$$6 \sqrt{3}$$8 \sqrt{2}$Correct Option: , 3 Solution: $x^{2}=4 y$ $x-\sqrt{2} y+4 \sqrt{2}=0$ Solving together we get $x^{2}=4\left(\frac{x+4 \sqrt{2}}{\sqrt{2}}\right)$ $\sqrt{2} x^{2}+4 x+16 \sqrt{2}$ $\sqrt{2} x^{2}-4 x-16 \sqrt{2}=0$ $\mathrm{x}_{1}+\mathrm{x}_{2}=2 \sqrt{2} ; \quad \mathrm{x}_{1} \mathrm{x}_{2}=\frac{-16 \sqrt{2}}{\sqrt{2}}=-16$ Similarly, ...

If the solve the problem

Question: Given $\frac{\mathrm{b}+\mathrm{c}}{11}=\frac{\mathrm{c}+\mathrm{a}}{12}=\frac{\mathrm{a}+\mathrm{b}}{13}$ for a $\triangle \mathrm{ABC}$ with usual notation. If $\frac{\cos \mathrm{A}}{\alpha}=\frac{\cos \mathrm{B}}{\beta}=\frac{\cos \mathrm{C}}{\gamma}$, then the ordered triad $(\alpha, \beta, \gamma)$ has a value :-$(3,4,5)$$(19,7,25)$$(7,19,25)$$(5,12,13)$Correct Option: , 3 Solution: $\mathrm{b}+\mathrm{c}=11 \lambda, \mathrm{c}+\mathrm{a}=12 \lambda, \mathrm{a}+\mathrm{b}=13 \lam...

All x satisfying the inequality

Question: All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+100$, lie in the interval:-$(-\infty, \cot 5) \cup(\cot 4, \cot 2)$$(\cot 5, \cot 4)$$(\cot 2, \infty)$$(-\infty, \cot 5) \cup(\cot 2, \infty)$Correct Option: , 4 Solution: $\cot ^{-1} x5, \quad \cot ^{-1} x2$ $\Rightarrow x\cot 5, x\cot 2$...

Solve this following

Question: If mean and standard deviation of 5 observations $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}, \mathrm{x}_{5}$ are 10 and 3 , respectively, then the variance of 6 observations $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{5}$ and $-50$ is equal to :$582.5$$507.5$$586.5$$509.5$Correct Option: , 2 Solution: $\bar{x}=10 \Rightarrow \sum_{i=1}^{5} x_{i}=50$ S.D. $=\sqrt{\frac{\sum_{i=1}^{5} \mathrm{x}_{\mathrm{i}}^{2}}{5}-(\overline{\mathrm{x}})^{2}}=8$ $\Rightarr...

Question: Let $z$ be a complex number such that $|z|+z=3+i($ where $i=\sqrt{-1})$. Then $|z|$ is equal to :-$\frac{5}{4}$$\frac{\sqrt{41}}{4}$$\frac{\sqrt{34}}{3}$$\frac{5}{3}$Correct Option: , 4 Solution: $|z|+z=3+i$ $z=3-|z|+i$ Let $3-|z|=a \Rightarrow|z|=(3-a)$ $\Rightarrow \mathrm{z}=\mathrm{a}+\mathrm{i} \Rightarrow|\mathrm{z}|=\sqrt{\mathrm{a}^{2}+1}$ $\Rightarrow 9+a^{2}-6 a=a^{2}+1 \Rightarrow a=\frac{8}{6}=\frac{4}{3}$ $\Rightarrow|z|=3-\frac{4}{3}=\frac{5}{3}$...

If the probability of hitting a target by a shooter,

Question: If the probability of hitting a target by a shooter, in any shot, is $1 / 3$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6}$, is :6543Correct Option: , 2 Solution: $1-{ }^{n} C_{0}\left(\frac{1}{3}\right)^{0}\left(\frac{2}{3}\right)^{n}\frac{5}{6}$ $\frac{1}{6}\left(\frac{2}{3}\right)^{\mathrm{n}} \Rightarrow 0.1666\left(\frac{2}{3}\right)^{\mathrm{n}}$ $\mathrm{n}_{\mi...

The number of natural numbers less than 7,000

Question: The number of natural numbers less than 7,000 which can be formed by using the digits $0,1,3,7,9$ (repitition of digits allowed) is equal to :250374372375Correct Option: , 2 Solution: 2 ways for $\mathrm{a}_{4}$ Number of numbers $=2 \times 5^{3}$ Required number $=5^{3}+2 \times 5^{3}-1$ $=374$...

If the solve the problem

Question: Let $f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}-\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}, \quad x \in R$, where a, b and d are non-zero real constants. Then :-$\mathrm{f}$ is a decreasing function of $x$$\mathrm{f}$ is neither increasing nor decreasing function of $x$$f^{\prime}$ is not a continuous function of $x$$\mathrm{f}$ is an increasing function of $x$Correct Option: , 4 Solution: $f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}-\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}$ $f^{\prime}(x)=\frac{a^{2}}{\left(a^{2}+x^{...

A data consists of n observations:

Question: A data consists of $\mathrm{n}$ observations: $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \ldots, \mathrm{x}_{\mathrm{n}}$. If $\sum_{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{x}_{\mathrm{i}}+1\right)^{2}=9 \mathrm{n} \quad$ and $\sum_{i=1}^{n}\left(x_{i}-1\right)^{2}=5 n$, then the standard deviation of this data is :5$\sqrt{5}$$\sqrt{7}$2Correct Option: , 2 Solution: $\sum\left(x_{i}+1\right)^{2}=9 n$ ..............(1) $\sum\left(x_{i}-1\right)^{2}=5 n$ ................(2) $(1)+(2) \Righta...

The value of

Question: The value of $\int_{-\pi / 2}^{\pi / 2} \frac{d x}{[x]+[\sin x]+4}$, where $[t]$ denotes the greatest integer less than or equal to $\mathrm{t}$, is :$\frac{1}{12}(7 \pi+5)$$\frac{3}{10}(4 \pi-3)$$\frac{1}{12}(7 \pi-5)$$\frac{3}{20}(4 \pi-3)$Correct Option: , 4 Solution: $I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{d x}{[x]+[\sin x]+4}$ $=\int_{\frac{-\pi}{2}}^{-1} \frac{d x}{-2-1+4}+\int_{-1}^{0} \frac{d x}{-1-1+4}$ $+\int_{0}^{1} \frac{d x}{0+0+4}+\int_{1}^{\frac{\pi}{2}} \frac{d x...

If 19 th term of a non-zero A.P. is zero,

Question: If 19 th term of a non-zero A.P. is zero, then its $(49$ th term $)$ :(29th term) is :-$3: 1$$4: 1$$2: 1$$1: 3$Correct Option: 1 Solution: $a+18 d=0$ ......(1) $\frac{a+48 d}{a+28 d}=\frac{-18 d+48 d}{-18 d+28 d}=\frac{3}{1}$...

Let f : [0 , 1]→ R be such that

Question: Let $f:[0,1] \rightarrow \mathrm{R}$ be such that $f(\mathrm{xy})=f(\mathrm{x}) . f(\mathrm{y})$ for all $\mathrm{x}, \mathrm{y}, \varepsilon[0,1]$, and $f(0) \neq 0$. If $\mathrm{y}=\mathrm{y}(\mathrm{x})$ satisfies the differential equation, $\frac{\mathrm{dy}}{\mathrm{dx}}=f(\mathrm{x})$ with $y(0)=1$, then $y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to4352Correct Option: , 2 Solution: $f(x y)=f(x) . f(y)$ $f(0)=1$ as $f(0) \neq 0$ $\Rightarrow f(x)=1$ $\frac{\mat...

Solve this following

Question: The value of $\cos \frac{\pi}{2^{2}} \cdot \cos \frac{\pi}{2^{3}} \cdot \ldots \cdot \cdot \cdot \cos \frac{\pi}{2^{10}} \cdot \sin \frac{\pi}{2^{10}}$ is : $\frac{1}{256}$$\frac{1}{2}$$\frac{1}{512}$$\frac{1}{1024}$Correct Option: , 3 Solution: $2 \sin \frac{\pi}{2^{10}} \cos \frac{\pi}{2^{10}} \ldots \ldots \cdot \cos \frac{\pi}{2^{2}}$ $\frac{1}{2^{9}} \sin \frac{\pi}{2}=\frac{1}{512}$ Option (3)...

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis,

Question: A circle cuts a chord of length $4 \mathrm{a}$ on the $x$-axis and passes through a point on the $y$-axis, distant $2 b$ from the origin. Then the locus of the centre of this circle, is :-A hyperbolaA parabolaA straight lineAn ellipseCorrect Option: , 2 Solution: Let equation of circle is $x^{2}+y^{2}+2 f x+2 f y+e=0$, it passes through $(0,2 b)$ $\Rightarrow 0+4 b^{2}+2 g \times 0+4 f+c=0$ $\Rightarrow 4 b^{2}+4 f+c=0$ ........(1) $2 \sqrt{\mathrm{g}^{2}-\mathrm{c}}=4 \mathrm{a}$ .......

If the system of linear equations

Question: If the system of linear equations $x-4 y+7 z=g$ $3 y-5 z=h$ $-2 x+5 y-9 z=k$ is consistent, then :$\mathrm{g}+\mathrm{h}+\mathrm{k}=0$$2 \mathrm{~g}+\mathrm{h}+\mathrm{k}=0$$\mathrm{g}+\mathrm{h}+2 \mathrm{k}=0$$\mathrm{g}+2 \mathrm{~h}+\mathrm{k}=0$Correct Option: , 2 Solution: $\mathrm{P}_{1} \equiv \mathrm{x}-4 \mathrm{y}+7 \mathrm{z}-\mathrm{g}=0$ $\mathrm{P}_{2} \equiv 3 \mathrm{x}-5 \mathrm{y}-\mathrm{h}=0$ $\mathrm{P}_{3} \equiv-2 \mathrm{x}+5 \mathrm{y}-9 \mathrm{z}-\mathrm{k}=...

If the solve the problem

Question: Let $\mathrm{S}_{\mathrm{n}}=1+\mathrm{q}+\mathrm{q}^{2}+\ldots \ldots .+\mathrm{q}^{\mathrm{n}}$ and $\mathrm{T}_{\mathrm{n}}=1+\left(\frac{\mathrm{q}+1}{2}\right)+\left(\frac{\mathrm{q}+1}{2}\right)^{2}+\ldots \ldots+\left(\frac{\mathrm{q}+1}{2}\right)^{\mathrm{n}}$ where $\mathrm{q}$ is a real number and $\mathrm{q} \neq 1$ If ${ }^{101} \mathrm{C}_{1}+{ }^{101} \mathrm{C}_{2} \cdot \mathrm{S}_{1}+\ldots \ldots+{ }^{101} \mathrm{C}_{101} \cdot \mathrm{S}_{100}=\alpha \mathrm{T}_{100...

Solve this following

Question: The positive value of $\lambda$ for which the co-efficient of $x^{2}$ in the expression $x^{2}\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)^{10}$ is 720, is :$\sqrt{5}$4$2 \sqrt{2}$3Correct Option: , 2 Solution: $x^{2}\left({ }^{10} C_{r}(\sqrt{x})^{10-r}\left(\frac{\lambda}{x^{2}}\right)^{r}\right)$ $x^{2}\left[{ }^{10} C_{r}(x)^{\frac{10-r}{2}}(\lambda)^{\mathrm{r}}(x)^{-2 r}\right]$ $\mathrm{X}^{2}\left[{ }^{10} \mathrm{C}_{\mathrm{r}} \lambda^{\mathrm{r}} \mathrm{X}^{\frac{10-5 \math...

Let a, b, and c be the 7th , 11th and 13th terms respectively of

Question: Let $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ be the $7^{\text {th }}, 11^{\text {th }}$ and $13^{\text {th }}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{\mathrm{a}}{\mathrm{c}}$ is equal to:$\frac{1}{2}$42$\frac{7}{13}$Correct Option: , 2 Solution: $a=A+6 d$ $b=A+10 d$ $c=A+12 d$ $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in G.P. $\Rightarrow(\mathrm{A}+10 \mathrm{~d})^{2}=(\mathrm{A}+6 \mathrm{~d})(\mathrm{a}+12 \math...

The sum of the follwing series

Question: The sum of the follwing series $1+6+\frac{9\left(1^{2}+2^{2}+3^{2}\right)}{7}+\frac{12\left(1^{2}+2^{2}+3^{2}+4^{2}\right)}{9}$ $+\frac{15\left(1^{2}+2^{2}+\ldots+5^{2}\right)}{11}+\ldots$ up to 15 terms, is:7820783075207510Correct Option: 1 Solution: $T_{n}=\frac{(3+(n-1) \times 3)\left(1^{2}+2^{2}+\ldots .+n^{2}\right)}{(2 n+1)}$ $T_{n}=\frac{3 . \frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}}{2 \mathrm{n}+1}=\frac{\mathrm{n}^{2}(\mathrm{n}+1)}{2}$ $\mathrm{S}_{15}=\frac{1}{2} \s...

If the solve the problem

Question: If $\left|\begin{array}{ccc}a-b-c 2 a 2 a \\ 2 b b-c-a 2 b \\ 2 c 2 c c-a-b\end{array}\right|$ $=(a+b+c)(x+a+b+c)^{2}, x \neq 0$ and $a+b+c \neq 0$, then $x$ is equal to :-$-(a+b+c)$$2(a+b+c)$$a b c$$-2(a+b+c)$Correct Option: , 4 Solution: $\left|\begin{array}{ccc}a-b-c 2 a 2 a \\ 2 b b-c-a 2 b \\ 2 c 2 c c-a-b\end{array}\right|$ $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}$ $=\left|\begin{array}{ccc}a+b+c a+b+c a+b+c \\ 2 b b-c-a 2 b \\ 2 c 2 c c-a-b\end{ar...

Let S be the set of all triangles in the xy-plane,

Question: Let $S$ be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $\mathrm{S}$ has area 50 sq. units, then the number of elements in the set $S$ is:9183236Correct Option: , 4 Solution: Let $A(\alpha, 0)$ and $B(0, \beta)$ be the vectors of the given triangle $A O B$ $\Rightarrow|\alpha \beta|=100$ $\Rightarrow$ Number of triangles $=4 \times$ (number of divisors of 1...

Solve this following

Question: Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots . \mathrm{a}_{10}$ be in G.P. with $\mathrm{a}_{\mathrm{i}}0$ for $\mathrm{i}=1,2, \ldots ., 10$ and $\mathrm{S}$ be the set of pairs $(\mathrm{r}, \mathrm{k})$, $\mathrm{r} \mathrm{k} \in \mathrm{N}$ (the set of natural numbers) for which $\left|\begin{array}{lll}\log _{e} a_{1}^{r} a_{2}^{k} \log _{e} a_{2}^{r} a_{3}^{k} \log _{e} a_{3}^{r} a_{4}^{k} \\ \log _{e} a_{4}^{r} a_{5}^{k} \log _{e} a_{5}^{r} a_{6}^{k} \log _{e} a_...

If the circles

Question: If the circles $x^{2}+y^{2}-16 x-20 y+164=r^{2}$ and $(x-4)^{2}+(y-7)^{2}=36$ intersect at two distinct points, then:$0\mathrm{r}1$$1r11$$r11$$r=11$Correct Option: , 2 Solution: $x^{2}+y^{2}-16 x-20 y+164=r^{2}$ $\mathrm{A}(8,10), \mathrm{R}_{1}=\mathrm{r}$ $(x-4)^{2}+(y-7)^{2}=36$ $\mathrm{B}(4,7), \mathrm{R}_{2}=6$ $\left|\mathrm{R}_{1}-\mathrm{R}_{2}\right|\mathrm{AB}\mathrm{R}_{1}+\mathrm{R}_{2}$ $\Rightarrow 1\mathrm{r}11$...

If the solve the problem

Question: Let $\sqrt{3} \hat{\mathrm{i}}+\hat{\mathrm{j}}, \hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}$ and $\beta \hat{\mathrm{i}}+(1-\beta) \hat{\mathrm{j}}$ respectively be the position vectors of the points $A, B$ and $C$ with respect to the origin $\mathrm{O}$. If the distance of $\mathrm{C}$ from the bisector of the acute angle between OA and $\mathrm{OB}$ is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :-2134Correct Option: , 2 Solution: Angle bisector is $x-y=0$...