Let n>2 be an integer. Suppose that there are n Metro stations
Question: Let $n2$ be an integer. Suppose that there are $\mathrm{n}$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of $\mathrm{n}$ is :-199101201200Correct Option: , 3 Solution: Number of blue lines $=$ Number of si...
Read More →If the solve the problem
Question: If $\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$ then the value of $\mathrm{k}$ is_______________ Solution: $\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\left(1-\cos \frac{x^{2}}{2}\right)}{4\left(\frac{x^{2}}{2}\rig...
Read More →The probability of a man hitting a target is
Question: The probability of a man hitting a target is $\frac{1}{10}$. The least number of shots required, so that the probability of his hitting the target at least once is greater than $\frac{1}{4}$, is Solution: We have, $1-$ (probability of all shots result in failure) $\frac{1}{4}$ $\Rightarrow 1-\left(\frac{9}{10}\right)^{\mathrm{n}}\frac{1}{4}$ $\Rightarrow \frac{3}{4}\left(\frac{9}{10}\right)^{\mathrm{n}} \Rightarrow \mathrm{n} \geq 3$...
Read More →Let the function
Question: Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{x} 1 \\ 1 0\end{array}\right], \mathrm{x} \in \mathrm{R}$ and $\mathrm{A}^{4}=\left[\mathrm{a}_{\mathrm{ij}}\right] .$ If $a_{11}=109$, then $a_{22}$ is equal to____________ Solution: $A=\left[\begin{array}{ll}x 1 \\ 1 0\end{array}\right]$ $A^{2}=\left[\begin{array}{ll}x 1 \\ 1 0\end{array}\right]\left[\begin{array}{ll}x 1 \\ 1 0\end{array}\right]=\left[\begin{array}{cc}x^{2}+1 x \\ x 1\end{array}\right]$ $A^{4}=\left[\begin{array}{cc}x^{2...
Read More →If the system of equations
Question: If the system of equations $x-2 y+3 z=9$ $2 x+y+z=b$ $x-7 y+a z=24$ has infinitely many solutions, then $a-b$ is equal to Solution: $D=\left|\begin{array}{ccc}1 -2 3 \\ 2 1 1 \\ 1 -7 a\end{array}\right|=0 \Rightarrow a=8$ also, $D_{1}=\left|\begin{array}{ccc}9 -2 3 \\ b 1 1 \\ 24 -7 8\end{array}\right|=0 \Rightarrow b=3$ hence, $a-b=8-3=5$...
Read More →The area (in sq. units) of an equilateral triangle inscribed in the parabola
Question: The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^{2}=8 x$, with one of its vertices on the vertex of this parabola, is:$64 \sqrt{3}$$256 \sqrt{3}$$192 \sqrt{3}$$128 \sqrt{3}$Correct Option: , 3 Solution: $\tan 30^{\circ}=\frac{4 \mathrm{t}}{2 \mathrm{t}^{2}}=\frac{2}{\mathrm{t}} \Rightarrow \mathrm{t}=2 \sqrt{3}$ $\mathrm{AB}=8 \mathrm{t}=16 \sqrt{3}$ Area $=256.3 \cdot \frac{\sqrt{3}}{4}=192 \sqrt{3}$...
Read More →The number of integral values of k for
Question: The number of integral values of $\mathrm{k}$ for which the line, $3 x+4 y=k$ intersects the circle, $x^{2}+y^{2}-2 x-4 y+4=0$ at two distinct points is Solution: Circle $x^{2}+y^{2}-2 x-4 y+4=0$ $\Rightarrow(x-1)^{2}+(y-2)^{2}=1$ Centre: $(1,2)$ radius $=1$ line $3 x+4 y-k=0$ intersects the circle at two distinct points. $\Rightarrow$ distance of centre from the line $$ radius $\Rightarrow\left|\frac{3 \times 1+4 \times 2-\mathrm{k}}{\sqrt{3^{2}+4^{2}}}\right|1$ $\Rightarrow|11-\mathr...
Read More →Solve the Following Questions
Question: The proposition $\mathrm{p} \rightarrow \sim(\mathrm{p} \wedge \sim \mathrm{q})$ is equivalent to:$(\sim \mathrm{p}) \vee \mathrm{q}$q$(\sim \mathrm{p}) \wedge \mathrm{q}$$(\sim p) \vee(\sim q)$Correct Option: 1 Solution: $\mathrm{p} \rightarrow \sim(\mathrm{p} \wedge \sim \mathrm{q})$ $=\sim p \vee \sim(p \wedge \sim q)$ $=\sim p \vee \sim p \vee q$ $=\sim(p \wedge q) \vee q$ $=\sim p \vee q$...
Read More →If the letters of the word 'MOTHER' be permuted and
Question: If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is Solution: MOTHER $1 \rightarrow \mathrm{E}$ $2 \rightarrow \mathrm{H}$ $3 \rightarrow \mathrm{M}$ $4 \rightarrow \mathrm{O}$ $5 \rightarrow \mathrm{R}$ $6 \rightarrow \mathrm{T}$ So position of word MOTHER in dictionary $2 \times 5 !+2 \times 4 !+3 \times 3 !+2 !+1$ $=240+48+18+2+1$ $=309$...
Read More →Solve this
Question: If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}$ where $ab0$, then $\frac{d x}{d y}$ at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is : $\frac{a-b}{a+b}$$\frac{a+b}{a-b}$$\frac{2 a+b}{2 a-b}$$\frac{a-2 b}{a+2 b}$Correct Option: , 2 Solution: $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}$ $\Rightarrow a^{2}-\sqrt{2} a b \cos y+\sqrt{2} a b \cos x$ $-2 b^{2} \cos x \cos y=a^{2}-b^{2}$ Differentiating both sides : $0-\sqrt{2} \mathrm{ab}\left(-\sin \mathrm{y} \frac{\...
Read More →Solve the Following Questions
Question: $2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to:$\frac{7 \pi}{4}$$\frac{5 \pi}{4}$$\frac{3 \pi}{2}$$\frac{\pi}{2}$Correct Option: , 3 Solution: $2 \pi-\left(\sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{16}{65}\right)\right)$ $=2 \pi-\left(\tan ^{-1}\left(\frac{4}{3}\right)+\tan ^{-1}\left(\frac{5}{12}\right)+\tan ^{-1}\left(\frac{16}{63}\right)\right)$ $=2 \pi-\left(\tan ^{-1}\left(...
Read More →if lim x rightarrow 1 x+x sequre + x qube +........+xn - n
Question: If $\lim _{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{n}-n}{x-1}=820,(n \in N)$ then the value of $\mathrm{n}$ is equal to Solution: $\lim _{x \rightarrow 1} \frac{x+x^{2}+\ldots \ldots+x^{2}-n}{x-1}=820$ $\Rightarrow \lim _{x \rightarrow 1}\left(\frac{x-1}{x-1}+\frac{x^{2}-1}{x-1}+\ldots . . \frac{x^{n}-1}{x-1}\right)=820$ $\Rightarrow 1+2+\ldots . .+\mathrm{n}=820$ $\Rightarrow \mathrm{n}(\mathrm{n}+1)=2 \times 820$ $\Rightarrow \mathrm{n}(\mathrm{n}+1)=40 \times 41$ Since $n \in...
Read More →Solve the Following Questions
Question: $2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to:$\frac{7 \pi}{4}$$\frac{5 \pi}{4}$$\frac{3 \pi}{2}$$\frac{\pi}{2}$Correct Option: , 3 Solution: $2 \pi-\left(\sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{16}{65}\right)\right)$...
Read More →Let $mathrm{f}$ be a twice differentiable function on
Question: Let $\mathrm{f}$ be a twice differentiable function on $(1,6)$. If $f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$, for all $x \in(1,6)$, then :$f(5) \leq 10$$\mathrm{f}^{\prime}(5)+\mathrm{f}^{\prime \prime}(5) \leq 20$$\mathrm{f}(5)+\mathrm{f}^{\prime}(5) \geq 28$$f(5)+f^{\prime}(5) \leq 26$Correct Option: , 3 Solution: $f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1, f^{\prime \prime}(x) \geq 4, \forall x \in(1,6)$ $f^{\prime \prime}(x)=\frac{f^{\pri...
Read More →Let t denote the greatest integer < t. If for some
Question: Let $[t]$ denote the greatest integer $\leq t$. If for some $\lambda \in \mathrm{R}-\{0,1\}, \lim _{\mathrm{x} \rightarrow 0}\left|\frac{1-\mathrm{x}+|\mathrm{x}|}{\lambda-\mathrm{x}+[\mathrm{x}]}\right|=\mathrm{L}$, then $\mathrm{L}$ is equal to :12$\frac{1}{2}$0Correct Option: , 2 Solution: $\mathrm{LHL}: \lim _{\mathrm{x} \rightarrow 0^{-}}\left|\frac{1-\mathrm{x}-\mathrm{x}}{\lambda-\mathrm{x}-1}\right|=\left|\frac{1}{\lambda-1}\right|$ $\mathrm{RHL}: \lim _{x \rightarrow 0^{+}}\le...
Read More →let vector a, vector b and vector c be three unite vectors such that
Question: Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$ Then $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ is equal to Solution: $|\vec{a}|=|\vec{b}|=|\vec{c}|=1$ $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{b}|^{2}=8$ $\Rightarrow|\vec{a}|^{2}+|\vec{b}|^{2}-2 \vec{a} \cdot \vec{b}+|\vec{a}|^{2}+|\vec{c}|^{2}-2 \vec{a} \cdot \vec{c}=8$ $\Rightarrow \quad 4-2(\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c})=8$ $\Rightarrow \vec{a} \cd...
Read More →If a and b are the roots of the equation
Question: If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+p x+2=0$ and $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ are the roots of the equation $2 x^{2}+2 q x+1=0$, then $\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)$ is equal to:$\frac{9}{4}\left(9+\mathrm{p}^{2}\right)$$\frac{9}{4}\left(9-\mathrm{q}^{2}\right)$$\frac{9}{4}\left(9-\mathrm{p}^{2}\right)$$\frac{9}{4}\left(9+q^{2}\right)$Corre...
Read More →The integral
Question: The integral $\int^{2} \| x-1|-x| d x$ is equal to_________. Solution: $\int_{0}^{2}|x-1|-x \mid d x$ Let $f(x) \| x-1|-x|$ $= \begin{cases}1, x \geq 1 \\ |1-2 x|, x \leq 1\end{cases}$ $A=\frac{1}{2}+1=\frac{3}{2}$ or $\int_{0}^{1 / 2}(1-2 x) d x+\int_{1 / 2}^{1}(2 x-1)+\int_{0}^{2} 1 d x$ $=\left[x-x^{2}\right]_{0}^{1}+\left[x^{2}-x\right]_{1 / 2}^{1}+[x]_{1}^{2}$ $=3 / 2$...
Read More →The value of
Question: The value of $\sum_{r=0}^{20}{ }^{50-r} C_{6}$ is equal to : ${ }^{51} \mathrm{C}_{7}+{ }^{30} \mathrm{C}_{7}$${ }^{51} C_{7}-{ }^{30} C_{7}$${ }^{50} \mathrm{C}_{7}-{ }^{30} \mathrm{C}_{7}$${ }^{50} \mathrm{C}_{6}-{ }^{30} \mathrm{C}_{6}$Correct Option: , 2 Solution: $\sum_{\mathrm{r}=0}^{20}{ }^{50-\mathrm{r}} \mathrm{C}_{6}={ }^{50} \mathrm{C}_{6}+{ }^{49} \mathrm{C}_{6}+{ }^{48} \mathrm{C}_{6}+\ldots . .+{ }^{30} \mathrm{C}_{6}$ $={ }^{50} \mathrm{C}_{6}+{ }^{49} \mathrm{C}_{6}+\ld...
Read More →If the number of integral terms in the expansion
Question: If the number of integral terms in the expansion of $\left(3^{1 / 2}+5^{1 / 8}\right)^{n}$ is exactly 33 , then the least value of $n$ is :264256128248Correct Option: , 2 Solution: $\mathrm{T}_{\mathrm{r}+1}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}(3)^{\frac{\mathrm{n}-\mathrm{r}}{2}}(5)^{\frac{\mathrm{r}}{8}} \quad(\mathrm{n} \geq \mathrm{r})$ Clearly $\mathrm{r}$ should be a multiple of $8 .$ $\because$ there are exactly 33 integral terms Possible values of $\mathrm{r}$ can be $0,8,1...
Read More →Solve this
Question: Let $y=y(x)$ be the solution of the differential equation, $x y^{\prime}-y=x^{2}(x \cos x+\sin x), x0$. If $y(\pi)=\pi$, then $y^{\prime \prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right)$ is equal to :. $2+\frac{\pi}{2}$$1+\frac{\pi}{2}$$1+\frac{\pi}{2}+\frac{\pi^{2}}{4}$$2+\frac{\pi}{2}+\frac{\pi^{2}}{4}$Correct Option: 1 Solution: $x \frac{d y}{d x}-y=x^{2}(x \cos x+\sin x), x0$ $\frac{d y}{d x}-\frac{y}{x}=x(x \cos x+\sin x) \Rightarrow \frac{d y}{d x}+P y=Q$ so, I.F. $=...
Read More →If p(x) be a polynomial of degree three that has a local maximum value 8
Question: If $\mathrm{p}(\mathrm{x})$ be a polynomial of degree three that has a local maximum value 8 at $x=1$ and a local minimum value 4 at $x=2$; then $p(0)$ is equal to:12$-24$6$-12$Correct Option: , 4 Solution: Since $p(x)$ has realtive extreme at $x=1 \ 2$ so $\mathrm{p}^{\prime}(\mathrm{x})=0$ at $\mathrm{x}=1 \ 2$ so $p^{\prime}(x)=0$ at $x=1 \ 2$ $\Rightarrow \mathrm{p}^{\prime}(\mathrm{x})=\mathrm{A}(\mathrm{x}-1)(\mathrm{x}-2)$ $\Rightarrow \mathrm{p}(\mathrm{x})=\int \mathrm{A}\left...
Read More →The solution curve of the differential equation,
Question: The solution curve of the differential equation, $\left(1+\mathrm{e}^{-\mathrm{x}}\right)\left(1+\mathrm{y}^{2}\right) \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}^{2}$, which passes through the point $(0,1)$, is :$y^{2}=1+y \log _{e}\left(\frac{1+e^{x}}{2}\right)$$y^{2}+1=y\left(\log _{\mathrm{e}}\left(\frac{1+\mathrm{e}^{\mathrm{x}}}{2}\right)+2\right)$$\mathrm{y}^{2}=1+\mathrm{y} \log _{\mathrm{e}}\left(\frac{1+\mathrm{e}^{-\mathrm{x}}}{2}\right)$$y^{2}+1=y\left(\log _{e}\left(\frac{1...
Read More →Solve this
Question: If $1+\left(1-2^{2} .1\right)+\left(1-4^{2} .3\right)+\left(1-6^{2} .5\right)+\ldots . .+\left(1-20^{2} .19\right)$ $=\alpha-220 \beta$, then an ordered pair $(\alpha, \beta)$ is equal to : $(10,97)$$(11,103)$$(10,103)$$(11,97)$Correct Option: , 2 Solution: $1+\left(1-2^{2} .1\right)+\left(1-4^{2} .3\right)+\ldots \ldots+\left(1-20^{2} .19\right)$ $=\alpha-220 \beta$ $=11-\left(2^{2} .1+4^{2} .3+\ldots \ldots .+20^{2} .19\right)$ $=11-2^{2} \cdot \sum_{\mathrm{r}=1}^{10} \mathrm{r}^{2}...
Read More →The value of
Question: The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^{3}$ is :$\frac{1}{2}(\sqrt{3}-\mathrm{i})$$-\frac{1}{2}(\sqrt{3}-\mathrm{i})$$-\frac{1}{2}(1-\mathrm{i} \sqrt{3})$$\frac{1}{2}(1-\mathrm{i} \sqrt{3})$Correct Option: , 2 Solution: The value of $\left(\frac{1+\sin 2 \pi / 9+i \cos 2 \pi / 9}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)$ $=\left(\frac{1+\sin \left(\frac{\pi}{2}-\frac{5 \pi}{18}\right)+i...
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