If the greatest value of the term independent of ' x '
Question: If the greatest value of the term independent of ' $x$ ' in the expansion of $\left(x \sin \alpha+a \frac{\cos \alpha}{x}\right)^{10}$ is $\frac{10 !}{(5 !)^{2}}$, then the value of 'a' is equal to:$-1$1$-2$2Correct Option: , 4 Solution: $\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}}(\mathrm{x} \sin \alpha)^{10-\mathrm{r}}\left(\frac{\mathrm{a} \cos \alpha}{\mathrm{x}}\right)^{\mathrm{r}}$ $\mathrm{r}=0,1,2, \ldots, 10$ $\mathrm{T}_{r+1}$ will be independent of $\mathrm{x}...
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Question: Let $\mathrm{A}=\left[\begin{array}{cc}1 2 \\ -1 4\end{array}\right] .$ If $\mathrm{A}^{-1}=\alpha \mathrm{I}+\beta \mathrm{A}, \alpha, \beta \in \mathbf{R}, \mathrm{I}$ is a $2 \times 2$ identity matrix, then $4(\alpha-\beta)$ is equal to:5$\frac{8}{3}$24Correct Option: , 4 Solution: $\mathrm{A}=\left[\begin{array}{cc}1 2 \\ -1 4\end{array}\right],|\mathrm{A}|=6$ $\mathrm{A}^{-1}=\frac{\mathrm{adj} \mathrm{A}}{|\mathrm{A}|}=\frac{1}{6}\left[\begin{array}{cc}4 -2 \\ 1 1\end{array}\righ...
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Question: If $f(x)=\left\{\begin{array}{ll}\int_{0}^{x}(5+|1-t|) d t, x2 \\ 5 x+1, x \leq 2\end{array}\right.$, then$f(x)$ is not continuous at $x=2$$f(x)$ is everywhere differentiable$f(x)$ is continuous but not differentiable at $x=2$$f(x)$ is not differentiable at $x=1$Correct Option: , 3 Solution: $f(\mathrm{x})=\int_{0}^{1}(5+(1-\mathrm{t})) \mathrm{dt}+\int_{1}^{\mathrm{x}}(5+(\mathrm{t}-1)) \mathrm{dt}$ $=6-\frac{1}{2}+\left.\left(4 \mathrm{t}+\frac{\mathrm{t}^{2}}{2}\right)\right|_{1} ^{...
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Question: Let $A=\left[\begin{array}{lll}0 1 0 \\ 1 0 0 \\ 0 0 1\end{array}\right]$ Then the number of $3 \times 3$ matrices B with entries from the set $\{1,2,3,4,5\}$ and satisfying $\mathrm{AB}=\mathrm{BA}$ is Solution: Let matrix $B=\left[\begin{array}{lll}a b c \\ d e f \\ g n i\end{array}\right]$ $\because \quad \mathrm{AB}=\mathrm{BA}$ $\left[\begin{array}{lll}0 1 0 \\ 1 0 0 \\ 0 0 1\end{array}\right]\left[\begin{array}{lll}a b c \\ d e f \\ g h i\end{array}\right]=\left[\begin{array}{lll...
Read More →The first of the two samples in a group has 100
Question: The first of the two samples in a group has 100 items with mean 15 and standard deviation 3 . If the whole group has 250 items with mean $15.6$ and standard deviation $\sqrt{13.44}$, then the standard deviation of the second sample is :8645Correct Option: , 3 Solution: $\sigma^{2}=\frac{\mathrm{n}_{1} \sigma_{1}^{2}+\mathrm{n}_{2} \sigma_{2}^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}+\frac{\mathrm{n}_{1} \mathrm{n}_{2}}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{2}}\left(\overline{\mathrm{x}...
Read More →If the digits are not allowed to repeat in any number formed
Question: If the digits are not allowed to repeat in any number formed by using the digits $0,2,4,6,8$, then the number of all numbers greater than 10,000 is equal to Solution: $=4 \times 4 \times 3 \times 2=96$...
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Question: If $\sin \theta+\cos \theta=\frac{1}{2}$, then $16(\sin (2 \theta)+\cos (4 \theta)+\sin (6 \theta))$ is equal to:23272327Correct Option: , 3 Solution: $\sin \theta+\cos \theta=\frac{1}{2}$ $\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=\frac{1}{4}$ $\sin 2 \theta=-\frac{3}{4}$ Now : $\cos 4 \theta=1-2 \sin ^{2} 2 \theta$ $=1-2\left(-\frac{3}{4}\right)^{2}$ $=1-2 \times \frac{9}{16}=-\frac{1}{8}$ $\sin 6 \theta=3 \sin 2 \theta-4 \sin ^{3} 2 \theta$ $=\left(3-4 \sin ^{2} 2 ...
Read More →The sum of all those terms which are rational numbers
Question: The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:89273543Correct Option: , 4 Solution: $T_{r+1}={ }^{12} C_{r}\left(2^{1 / 3}\right)^{r} \cdot\left(3^{1 / 4}\right)^{12-r}$ $\mathrm{T}_{s+1}$ will be rational number when $r=0,3,6,9,12$ $\ \mathrm{r}=0,4,8,12$ $\Rightarrow \mathrm{r}=0,12$ $\mathrm{T}_{1}+\mathrm{T}_{13}=1 \times 3^{3}+1 \times 2^{4} \times 1$ $=24+16=43$...
Read More →Solve the Following Questions
Question: Let $a_{1}, a_{2}, \ldots \ldots, a_{21}$ be an AP such that $\sum_{n=1}^{20} \frac{1}{a_{n} a_{n+1}}=\frac{4}{9}$. If the sum of this $A P$ is 189 , then a a is equal to :57724836Correct Option: , 2 Solution: $\sum_{n=1}^{20} \frac{1}{a_{n} a_{n+1}}=\sum_{n=1}^{20} \frac{1}{a_{n}\left(a_{n}+d\right)}$ $=\frac{1}{d} \sum_{n=1}^{20}\left(\frac{1}{a_{n}}-\frac{1}{a_{n}+d}\right)$ $\Rightarrow \frac{1}{d}\left(\frac{1}{a_{1}}-\frac{1}{a_{21}}\right)=\frac{4}{9}$ (Given) $\Rightarrow \frac...
Read More →Let A={0,1,2,3,4,5,6,7}. Then the number of bijective functions
Question: Let $A=\{0,1,2,3,4,5,6,7\} .$ Then the number of bijective functions $f: \mathrm{A} \rightarrow \mathrm{A}$ such that $f(1)+f(2)=3-f(3)$ is equal to Solution: $f(1)+f(2)=3-f(3)$ $\Rightarrow \mathrm{f}(1)+\mathrm{f}(2)=3+\mathrm{f}(3)=3$ The only possibility is : $0+1+2=3$ $\Rightarrow$ Elements $1,2,3$ in the domain can be mapped with $0,1,2$ only. So number of bijective functions. $=\lfloor 3 \times\lfloor 5=720$...
Read More →The compound statement
Question: The compound statement $(P \vee Q) \wedge(\sim P) \Rightarrow Q$ is equivalent to:$P \vee Q$$P \wedge \sim Q$$\sim(\mathrm{P} \Rightarrow \mathrm{Q})$$\sim(\mathrm{P} \Rightarrow \mathrm{Q}) \Leftrightarrow \mathrm{P} \wedge \sim \mathrm{Q}$Correct Option: 1 Solution: Using Truth Table...
Read More →The range of the function,
Question: The range of the function, $f(x)=\log _{\sqrt{5}}\left(3+\cos \left(\frac{3 \pi}{4}+x\right)+\cos \left(\frac{\pi}{4}+x\right)+\cos \left(\frac{\pi}{4}-x\right)-\cos \left(\frac{3 \pi}{4}-x\right)\right)$ is: $(0, \sqrt{5})$$[-2,2]$$\left[\frac{1}{\sqrt{5}}, \sqrt{5}\right]$$[0,2]$Correct Option: , 4 Solution: $\mathrm{f}(\mathrm{x})=\log _{\sqrt{5}}$ $\left(3+\cos \left(\frac{3 \pi}{4}+x\right)+\cos \left(\frac{\pi}{4}+x\right)+\cos \left(\frac{\pi}{4}-x\right)-\cos \left(\frac{3 \pi}...
Read More →Solve the Following Questions
Question: Let $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{15}$ be 15 points on a circle. The number of distinct triangles formed by points $P_{i}, P_{j}, P_{k}$ such that $i+j+k \neq 15$, is :12419443455Correct Option: , 3 Solution: Total Number of Triangles $={ }^{15} \mathrm{C}_{3}$ $\mathrm{i}+\mathrm{j}+\mathrm{k}=15$ (Given) Number of Possible triangles using the vertices $\mathrm{P}_{\mathrm{i}}, \mathrm{P}_{\mathrm{j}}$, $P_{k}$ such that $i+j+k \neq 15$ is equal to ${ }^{15} C_{...
Read More →Solve the Following Questions
Question: Let $S_{n}=1 \cdot(n-1)+2 \cdot(n-2)+3 \cdot(n-3)+\ldots+$ $(\mathrm{n}-1) \cdot 1, \mathrm{n} \geq 4$ The sum $\sum_{n=4}^{\infty}\left(\frac{2 S_{n}}{n !}-\frac{1}{(n-2) !}\right)$ is equal to :$\frac{\mathrm{e}-1}{3}$$\frac{\mathrm{e}-2}{6}$$\frac{\mathrm{e}}{3}$$\frac{\mathrm{e}}{6}$Correct Option: 1 Solution: Let $\mathrm{T}_{\mathrm{r}}=\mathrm{r}(\mathrm{n}-\mathrm{r})$ $\mathrm{T}_{\mathrm{r}}=\mathrm{nr}-\mathrm{r}^{2}$ $\Rightarrow S_{n}=\sum_{r=1}^{n} T_{r}=\sum_{r=1}^{n}\le...
Read More →The numbers of pairs
Question: The numbers of pairs $(a, b)$ of real numbers, such that whenever $\alpha$ is a root of the equation $\mathrm{x}^{2}+\mathrm{ax}+\mathrm{b}=0, \alpha^{2}-2$ is also a root of this equation, is :6248Correct Option: 1 Solution: Consider the equation $x^{2}+a x+b=0$ If has two roots (not necessarily real $\alpha \ \beta$ ) Either $\alpha=\beta$ or $\alpha \neq \beta$ Case (1) If $\alpha=\beta$, then it is repeated root. Given that $\alpha^{2}-2$ is also a root So, $\alpha=\alpha^{2}-2 \Ri...
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Question: Let $\mathrm{E}_{1}: \frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1, \mathrm{a}\mathrm{b}$. Let $\mathrm{E}_{2}$ be another ellipse such that it touches the end points of major axis of $E_{1}$ and the foci of $E_{2}$ are the end points of minor axis of $E_{1}$. If $E_{1}$ and $E_{2}$ have same eccentricities, then its value is :$\frac{-1+\sqrt{5}}{2}$$\frac{-1+\sqrt{8}}{2}$$\frac{-1+\sqrt{3}}{2}$$\frac{-1+\sqrt{6}}{2}$Correct Option: 1, Solution: $e^{2}=...
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Question: If the coefficients of $x^{7}$ in $\left(x^{2}+\frac{1}{b x}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac{1}{b x^{2}}\right)^{11}, b \neq 0$, are equal, then the value of $b$ is equal to:2112Correct Option: , 3 Solution: Coefficient of $x^{7}$ in $\left(x^{2}+\frac{1}{b x}\right)^{11}$ ${ }^{11} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{2}\right)^{11-\mathrm{r}} \cdot\left(\frac{1}{\mathrm{bx}}\right)^{\mathrm{r}}$ ${ }^{11} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{22-3 \mathrm{r}} \cdot \fra...
Read More →solve the following equation
Question: Let $\mathrm{S}=\left\{\mathrm{n} \in \mathbf{N} \mid\left(\begin{array}{ll}0 \mathrm{i} \\ 1 0\end{array}\right)^{\mathrm{n}}\left(\begin{array}{ll}\mathrm{a} \mathrm{b} \\ \mathrm{c} \mathrm{d}\end{array}\right)=\left(\begin{array}{ll}\mathrm{a} \mathrm{b} \\ \mathrm{c} \mathrm{d}\end{array}\right) \forall \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in \mathbf{R}\right\}$ where $\mathrm{i}=\sqrt{-1}$. Then the number of 2 -digit numbers in the set $\mathrm{S}$ is_________. Soluti...
Read More →Consider the parabola with
Question: Consider the parabola with vertex $\left(\frac{1}{2}, \frac{3}{4}\right)$ and the $\operatorname{directrix} y=\frac{1}{2} .$ Let $P$ be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $\mathrm{P}$ intersects the parabola again at the point $Q$, then $(P Q)^{2}$ is equal to:$\frac{75}{8}$$\frac{125}{16}$$\frac{25}{2}$$\frac{15}{2}$Correct Option: , 2 Solution: $\left(y-\frac{3}{4}\right)=\left(x-\frac{1}{2}\right)^{2}$ For $x=-\frac{1}{2}$ ...
Read More →Let a line L : 2x+y=k, k>0
Question: Let a line $\mathrm{L}: 2 \mathrm{x}+\mathrm{y}=\mathrm{k}, \mathrm{k}0$ be a tangent to the hyperbola $x^{2}-y^{2}=3$. If $L$ is also a tangent to the parabola $\mathrm{y}^{2}=\alpha \mathrm{x}$, then $\alpha$ is equal to :12-1224-24Correct Option: , 4 Solution: Tangent to hyperbola of Slope $m=-2$ (given) $y=-2 x \pm \sqrt{3(3)}$ $\left(y=m x \pm \sqrt{a^{2} m^{2}-b^{2}}\right)$ $\Rightarrow y+2 x=\pm 3 \Rightarrow 2 x+y=3 \quad(k0)$ For parabola $\mathrm{y}^{2}-\alpha \mathrm{x}$ $\...
Read More →The distance of line
Question: The distance of line $3 y-2 z-1=0=3 x-z+4$ from the point $(2,-1,6)$ is :$\sqrt{26}$$2 \sqrt{5}$$2 \sqrt{6}$$4 \sqrt{2}$Correct Option: , 3 Solution: $3 y-2 z-1=0=3 x-z+4$ $3 \mathrm{y}-2 \mathrm{z}-1=0 \quad$ D.R's $\Rightarrow(0,3,-2)$ $3 x-z+4=0 \quad$ D.R's $\Rightarrow(3,-1,0)$ Let DR's of given line are $\mathrm{a}, \mathrm{b}, \mathrm{c}$ Now $3 b-2 c=0 \ 3 a-c=0$ $\therefore 6 a=3 b=2 c$ $a: b: c=3: 6: 9$ Any pt on line $3 \mathrm{~K}-1,6 \mathrm{~K}+1,9 \mathrm{~K}+1$ Now $3(3...
Read More →The term independent of ' x ' in the expansion of
Question: The term independent of ' $x$ ' in the expansion of $\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}$, where $x \neq 0,1$ is equal to_________. Solution: $\left(\left(x^{1 / 3}+1\right)-\left(\frac{x^{1 / 2}+1}{x^{1 / 2}}\right)\right)^{10}$ $=\left(x^{1 / 3}-\frac{1}{x^{1 / 2}}\right)^{10}$ Now General Term $\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{1 / 3}\right)^{10-\mathrm{r}} \cdot\left(-\frac{1}{\mathrm{x}^{1 / 2}}\righ...
Read More →A ray of light through (2,1) is reflected at a point P on the y-axis and then passes through the point (5, 3).
Question: A ray of light through (2,1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}}$, then the equation of the other directrix can be:$11 x+7 y+8=0$ or $11 x+7 y-15=0$$11 x-7 y-8=0$ or $11 x+7 y+15=0$$2 x-7 y+29=0$ or $2 x-7 y-7=0$$2 x-7 y-39=0$ or $2 x-7 y-7=0$Correct Option: , 3 Solution: Equat...
Read More →The area, enclosed by the curves
Question: The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0, x=\frac{\pi}{2}$ is:$2 \sqrt{2}(\sqrt{2}-1)$$2(\sqrt{2}+1)$$4(\sqrt{2}-1)$$2 \sqrt{2}(\sqrt{2}+1)$Correct Option: 1 Solution: $A=\int_{0}^{\pi / 2}((\sin x+\cos x)-|\cos x-\sin x|) d x$ $A=\int_{0}^{\pi / 2}((\sin x+\cos x)-(\cos x-\sin x)) d x$ $+\int_{\pi / 4}^{\pi / 2}((\sin x+\cos x)-(\sin x-\cos x)) d x$ $A=2 \int_{0}^{\pi / 2} \sin x d x+2 \int_{\pi / 4}^{\pi / 2} \cos x d x$ $A=-2\left...
Read More →If α, β are roots of the equation
Question: If $\alpha, \beta$ are roots of the equation $\mathrm{x}^{2}+5(\sqrt{2}) \mathrm{x}+10=0, \alpha\beta$ and $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$ for each positive integer $\mathrm{n}$, then the value of $\left(\frac{\mathrm{P}_{17} \mathrm{P}_{20}+5 \sqrt{2} \mathrm{P}_{17} \mathrm{P}_{19}}{\mathrm{P}_{18} \mathrm{P}_{19}+5 \sqrt{2} \mathrm{P}_{18}^{2}}\right)$ is equal to__________. Solution: $x^{2}+5 \sqrt{2} x+10=0$ $\ \mathrm{p}_{\mathrm{n}}=\alpha^{\math...
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