If α and β are the roots of the equation,
Question: If $\alpha$ and $\beta$ are the roots of the equation, $7 x^{2}-3 x-2=0$, then the value of $\frac{\alpha}{1-\alpha^{2}}+\frac{\beta}{1-\beta^{2}}$ is equal to:$\frac{27}{16}$$\frac{1}{24}$$\frac{27}{32}$$\frac{3}{8}$Correct Option: 1 Solution: $7 x^{2}-3 x-2=0$ $\alpha+\beta=\frac{3}{7} \quad \alpha \beta=\frac{-2}{7}$ $\frac{\alpha}{1-\alpha^{2}}+\frac{\beta}{1-\beta^{2}}=\frac{\alpha+\beta-\alpha \beta(\alpha+\beta)}{1-\alpha^{2}-\beta^{2}+\alpha^{2} \beta^{2}}$ $=\frac{\frac{3}{7}+...
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Question: If $\mathrm{I}_{1}=\int_{0}^{1}\left(1-\mathrm{x}^{50}\right)^{100} \mathrm{dx}$ and $\mathrm{I}_{2}=\int_{0}^{1}\left(1-\mathrm{x}^{50}\right)^{101} \mathrm{dx}$ such that $\mathrm{I}_{2}=\alpha \mathrm{I}_{1}$ then $\alpha$ equals to$\frac{5050}{5051}$$\frac{5050}{5049}$$\frac{5049}{5050}$$\frac{5051}{5050}$Correct Option: 1, Solution: $\mathrm{I}_{1}=\int_{0}^{1}\left(1-\mathrm{x}^{50}\right)^{100} \mathrm{dx}$ and $\mathrm{I}_{2}=\int_{0}^{1}\left(1-\mathrm{x}^{50}\right)^{101} \ma...
Read More →If the system of linear equations
Question: If the system of linear equations $x+y+3 z=0$ $x+3 y+k^{2} z=0$ $3 x+y+3 z=0$ has a non-zero solution $(x, y, z)$ for some $k \in R$, then $x+\left(\frac{y}{z}\right)$ is equal to :9$-3$$-9$3Correct Option: , 2 Solution: $x+y+3 z=0$ $\ldots \ldots(\mathrm{i})$ $x+3 y+k^{2} z=0$ $\ldots . .(\mathrm{ii})$ $3 x+y+3 z=0$ ...........(iii) $\left|\begin{array}{ccc}1 1 3 \\ 1 3 k^{2} \\ 3 1 3\end{array}\right|=0$ $\Rightarrow 9+3+3 k^{2}-27-k^{2}-3=0$ $\Rightarrow \mathrm{k}^{2}=9$ (i) $-$ (i...
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Question: Let $a, b, c, d$ and $p$ be any non zero distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c$ $+\mathrm{cd}) \mathrm{p}+\left(\mathrm{b}^{2}+\mathrm{c}^{2}+\mathrm{d}^{2}\right)=0 .$ Then : a,c,p are in G.P.$\mathrm{a}, \mathrm{c}, \mathrm{p}$ are in A.P.$\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in G.P.$a, b, c, d$ are in A.P.Correct Option: , 3 Solution: $\left(a^{2}+b^{2}+c^{2}\right) p^{2}+2(a b+b c+c d) p+b^{2}+c^{2}+d^{2}$ $=0$ $\Rightarrow\l...
Read More →A ray of light coming from the point
Question: A ray of light coming from the point $(2,2 \sqrt{3})$ is incident at an angle $30^{\circ}$ on the line $x=1$ at the point A. The ray gets reflected on the line $x=1$ and meets $x$-axis at the point $B$. Then, the line $\mathrm{AB}$ passes through the point:$\left(3,-\frac{1}{\sqrt{3}}\right)$$(3,-\sqrt{3})$$\left(4,-\frac{\sqrt{3}}{2}\right)$$(4,-\sqrt{3})$Correct Option: , 2 Solution: For point $\mathrm{A}$ $\tan 60^{\circ}=\frac{2 \sqrt{3}-\mathrm{k}}{2-1}$ $\sqrt{3}=2 \sqrt{3}-\math...
Read More →Let f(x) = x . [x/2], for -10 < x < 10, where [t] denotes the greatest integer function.
Question: Let $\mathrm{f}(\mathrm{x})=\mathrm{x} \cdot\left[\frac{\mathrm{x}}{2}\right]$, for $-10\mathrm{x}10$, where $[\mathrm{t}]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to_____. Solution: $x \in(-10,10)$ $\frac{x}{2} \in(-5,5) \rightarrow 9$ integers check continuity at $x=0$ $\left.\begin{array}{l}\mathrm{f}(0)=0 \\ \mathrm{f}\left(0^{+}\right)=0 \\ \mathrm{f}\left(0^{-}\right)=0\end{array}\right\} \quad$ continuous at $\mathrm{x}=0...
Read More →The general solution of the differential equation
Question: The general solution of the differential equation $\sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+x y \frac{d y}{d x}=0$ is : (where $\mathrm{C}$ is a constant of integration) $\sqrt{1+y^{2}}+\sqrt{1+x^{2}}=\frac{1}{2} \log _{e}\left(\frac{\sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}-1}\right)+C$$\sqrt{1+y^{2}}-\sqrt{1+x^{2}}=\frac{1}{2} \log _{e}\left(\frac{\sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}-1}\right)+C$$\sqrt{1+y^{2}}+\sqrt{1+x^{2}}=\frac{1}{2} \log _{c}\left(\frac{\sqrt{1+x^{2}}-1}{\sqrt{1+x^{2}}+1}\right)+C$$...
Read More →The number of words, with or without meaning,
Question: The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is_________. Solution: $\mathrm{S}_{2} \mathrm{YL}_{2} \mathrm{ABU}$ ABCC type words $=240$...
Read More →The natural number m, for which the coefficient
Question: The natural number $\mathrm{m}$, for which the coefficient of $x$ in the binomial expansion of $\left(x^{m}+\frac{1}{x^{2}}\right)^{22}$ is 1540 , is___________. Solution: $\mathrm{T}_{\mathrm{r}+1}={ }^{22} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{\mathrm{m}}\right)^{22-\mathrm{r}}\left(\frac{1}{\mathrm{x}^{2}}\right)^{\mathrm{r}}={ }^{22} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{22 \mathrm{~m}-\mathrm{mr}-2 \mathrm{r}}$ $={ }^{22} \mathrm{C}_{\mathrm{r}} \mathrm{x}$ $\because{ }^{22} \mat...
Read More →If the line, 2x - y + 3 = 0 is at a distance 1/spart5 and
Question: If the line, $2 x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4 x-2 y+\alpha=0$ and $6 x-3 y+\beta=0$, respectively, then the sum of all possible values of $\alpha$ and $\beta$ is________ Solution: Apply distance between parallel line formula $4 x-2 y+\alpha=0$ $4 x-2 y+6=0$ $\left|\frac{\alpha-6}{255}\right|=\frac{1}{55}$ $|\alpha-6|=2 \Rightarrow \alpha=8,4$ sum $=12$ again $6 x-3 y+\beta=0$ $6 x-3 y+9=0$ $\left|\frac{\beta-9}{3 \sqrt{5}}\rig...
Read More →Four fair dice are thrown independently 27 times.
Question: Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is Solution: 4 dice are independently thrown. Each die has probability to show 3 or 5 is $\mathrm{p}=\frac{2}{6}=\frac{1}{3}$ $\therefore \quad \mathrm{q}=1-\frac{1}{3}=\frac{2}{3}$ (not showing 3 or 5 ) Experiment is performed with 4 dices independently. $\therefore$ Their binomial distribution is $(q+p)^{4}=(q)^{4}+{ }^{4} C_{1} q^{3} p+{ }^{4} C_{2} q^{2}...
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Question: If $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y})$ and $\sum_{\mathrm{x}=1}^{\infty} f(\mathrm{x})=2, \mathrm{x}, \mathrm{y} \in \mathrm{N}$ where $\mathrm{N}$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is $\frac{1}{9}$$\frac{4}{9}$$\frac{1}{3}$$\frac{2}{3}$Correct Option: , 2 Solution: $f(x+y)=f(x) . f(y)$ $\sum_{x=1}^{\infty} f(x)=2$ where $x, y \in N$ $f(1)+f(2)+f(3)+\ldots . \infty=2 \ldots(1)($ Given $)$ Now for $f(2)$ put $\mathrm{x}=\mathrm{y...
Read More →If α is the positive root of the equation,
Question: If $\alpha$ is the positive root of the equation, $p(x)=x^{2}-x-2=0$, then $\lim _{x \rightarrow a^{+}} \frac{\sqrt{1-\cos (p(x))}}{x+\alpha-4}$ is equal to$\frac{3}{\sqrt{2}}$$\frac{3}{2}$$\frac{1}{\sqrt{2}}$$\frac{1}{2}$Correct Option: 1 Solution: $x^{2}-x-2=0$ roots are $2 \-1$ $\Rightarrow \lim _{x \rightarrow 2^{+}} \frac{\sqrt{1-\cos \left(x^{2}-x-2\right)}}{(x-2)}$ $=\lim _{x \rightarrow 2^{+}} \frac{\sqrt{2 \sin ^{2} \frac{\left(x^{2}-x-2\right)}{2}}}{(x-2)}$ $=\lim _{x \righta...
Read More →The negation of the Boolean expression
Question: The negation of the Boolean expression $p \vee(\sim p \wedge q)$ is equivalent to :$\sim \mathrm{p} \vee \sim \mathrm{q}$$\sim \mathrm{p} \vee \mathrm{q}$$\sim \mathrm{p} \wedge \sim \mathrm{q}$$\mathrm{p} \wedge \sim \mathrm{q}$Correct Option: , 3 Solution: Negation of $\phi \vee(\sim p \wedge q)$ $\mathrm{p} \vee(\sim \mathrm{p} \wedge \mathrm{q})=(\mathrm{p} \vee \sim \mathrm{p}) \wedge(\mathrm{p} \vee \mathrm{q})$ $=(\mathrm{T}) \wedge(\mathrm{p} \vee \mathrm{q})$ $=(\mathrm{p} \ve...
Read More →If the co-ordinates of two points A and B are
Question: If the co-ordinates of two points $\mathrm{A}$ and $\mathrm{B}$ are $(\sqrt{7}, 0)$ and $(-\sqrt{7}, 0)$ respectively and $\mathrm{P}$ is any point on the conic, $9 x^{2}+16 y^{2}=144$, then $\mathrm{PA}+\mathrm{PB}$ is equal to : 86169Correct Option: 1 Solution: $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ $a=4 ; b=3 ; e=\sqrt{\frac{16-9}{16}}=\frac{\sqrt{7}}{4}$ $\mathrm{A}$ and $\mathrm{B}$ are foci $\Rightarrow \mathrm{PA}+\mathrm{PB}=2 \mathrm{a}=2 \times 4=8$...
Read More →If 2 to the power 10 + 2 to the power 9.31 + .........
Question: If $2^{10}+2^{9} \cdot 3^{1}+2^{8} \cdot 3^{2}+\ldots .+2 \cdot 3^{9}+3^{10}=S-2^{11}$, then $S$ is equal to:$\frac{3^{11}}{2}+2^{10}$$3^{11}-2^{12}$$3^{11}$$2 \cdot 3^{11}$Correct Option: , 3 Solution: $\mathrm{a}=2^{10} ; \mathrm{r}=\frac{3}{2} ; \mathrm{n}=11$ (G.P.) $\mathrm{S}^{\prime}=\left(2^{10}\right) \frac{\left(\left(\frac{3}{2}\right)^{11}-1\right)}{\frac{3}{2}-1}=2^{11}\left(\frac{3^{11}}{2^{11}}-1\right)$ $\mathrm{S}^{\prime}=3^{11}-2^{11}=\mathrm{S}-2^{11}$ (Given) $\the...
Read More →The value of
Question: The value of $\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+e^{\sin x}} d x$ is$\pi$$\frac{3 \pi}{2}$$\frac{\pi}{4}$$\frac{\pi}{2}$Correct Option: , 4 Solution: $I=\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+e^{\sin x}} d x$ $\ldots(1)$ Apply King property $I=\int_{-z / 2}^{\pi / 2} \frac{1}{1+e^{-\sin x}} d x=\int_{-\pi / 2}^{\pi / 2} \frac{e^{\sin x}}{1+e^{\sin x} x} d x \ldots(2)$ Add (1) (2) $2 \mathbf{I}=\int_{-\pi / 2}^{\pi / 2} \mathrm{~d} \mathrm{x}=\pi$ $I=\frac{\pi}{2}$...
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Question: Let $L_{1}$ be a tangent to the parabola $y^{2}=4(x+1)$ and $L_{2}$ be a tangent to the parabola $y^{2}=8(x+2)$ such that $L_{1}$ and $L_{2}$ intersect at right angles. Then $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ meet on the straight line :$x+3=0$$x+2 y=0$$2 x+1=0$$x+2=0$Correct Option: 1 Solution: $y^{2}=4(x+1)$ equation of tangent $\mathrm{y}=\mathrm{m}(\mathrm{x}+1)+\frac{1}{\mathrm{~m}}$ $y=m x+m+\frac{1}{m}$ $y^{2}=8(x+2)$ equation of tangent $\mathrm{y}=\mathrm{m}^{\prime}(\mathrm...
Read More →If (a, b, c) is the image of the point (1,2,-3) in the line,
Question: If $(a, b, c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}$, then $a+b+c$ is equal to$-1$231Correct Option: , 2 Solution: Line is $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}=\lambda:$ Let point $R$ is $(2 \lambda-1,-2 \lambda+3,-\lambda)$ Direction ratio of $\mathrm{PQ}=(2 \lambda-2,-2 \lambda+1,3-\lambda)$ $P Q$ is $\perp^{r}$ to line $\Rightarrow 2(2 \lambda-2)-2(-2 \lambda+1)-1(3-\lambda)=0$ $4 \lambda-4+4 \lambda-2-3+\lambda=0$ $9 \la...
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Question: If $\sum_{i=1}^{n}\left(x_{i}-a\right)=n$ and $\sum_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a,(n, a1)$ then the standard deviation of $\mathrm{n}$ observations $\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots, \mathrm{x}_{\mathrm{n}}$ is $\mathrm{n} \sqrt{\mathrm{a}-1}$$\sqrt{a-1}$$a-1$$\sqrt{\mathrm{n}(\mathrm{a}-1)}$Correct Option: , 2 Solution: S.D $=\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-a\right)}{n}-\left(\frac{\sum_{i=1}^{n}\left(x_{i}-a\right)}{n}\right)^{2}}$ $=\sqrt{\frac{n a}{n}-\left(\fra...
Read More →Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition),
Question: Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is : $\frac{15}{101}$$\frac{5}{101}$$\frac{5}{33}$$\frac{10}{99}$Correct Option: , 3 Solution: Out of 11 consecutive natural numbers either 6 even and 5 odd numbers or 5 even and 6 odd numbers when 3 numbers are selected at random then total cases $={ }^{11} \mathrm{C}_{3}$ Since these 3 numbers are in A.P. Let n...
Read More →The mean and variance of 7 observations are 8 and 16 , respectively.
Question: The mean and variance of 7 observations are 8 and 16 , respectively. If five observations are $2,4,10,12,14$, then the absolute difference of the remaining two observations is:2431Correct Option: 1 Solution: $\bar{x}=\frac{2+4+10+12+14+x+y}{7}=8$ $x+y=14$......(i) $(\sigma)^{2}=\frac{\sum\left(\mathrm{x}_{\mathrm{i}}\right)^{2}}{\mathrm{n}}-\left(\frac{\sum \mathrm{x}_{\mathrm{i}}}{\mathrm{n}}\right)^{2}$ $16=\frac{4+16+100+144+196+x^{2}+y^{2}}{7}-8^{2}$ $16+64=\frac{460+x^{2}+y^{2}}{7...
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Question: The area (in sq. units) of the region $A=\{(x, y)$ $\left.:|x|+|y| \leq 1,2 y^{2} \geq|x|\right\}$ is : $\frac{1}{6}$$\frac{1}{3}$$\frac{7}{6}$$\frac{5}{6}$Correct Option: 4, Solution: $|x|+|y| \leq 1$ $2 y^{2} \geq|x|$ For point of intersection $x+y=1 \Rightarrow x=1-y$ $\mathrm{y}^{2}=\frac{\mathrm{x}}{2} \Rightarrow 2 \mathrm{y}^{2}=\mathrm{x}$ $2 \mathrm{y}^{2}=1-\mathrm{y} \Rightarrow 2 \mathrm{y}^{2}+\mathrm{y}-1=0$ $(2 \mathrm{y}-1)(\mathrm{y}+1)=0$ $y=\frac{1}{2}$ or $-1$ Now A...
Read More →If the point P on the curve,
Question: If the point $P$ on the curve, $4 x^{2}+5 y^{2}=20$ is farthest from the point $Q(0,-4)$, then $P Q^{2}$ is equal to :21364829Correct Option: , 2 Solution: Given ellipse is $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ Let point $\mathrm{P}$ is $(\sqrt{5} \cos \theta, 2 \sin \theta)$ $(\mathrm{PQ})^{2}=5 \cos ^{2} \theta+4(\sin \theta+2)^{2}$ $(\mathrm{PQ})^{2}=\cos ^{2} \theta+16 \sin \theta+20$ $(\mathrm{PQ})^{2}=-\sin ^{2} \theta+16 \sin \theta+21$ $=85-(\sin \theta-8)^{2}$ will be maximum wh...
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Question: The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$ $x+2 y+3 z=5$ $x+3 y+\lambda z=\mu$ has infinitely many solutions are, respectively 5 and 76 and 84 and 95 and 8Correct Option: Solution: For infinite many solutions $\mathrm{D}=\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$ Now $\mathrm{D}=\left|\begin{array}{lll}1 1 1 \\ 1 2 3 \\ 1 3 \lambda\end{array}\right|=0$ 1. $(2 \lambda-9)-1 .(\lambda-3)+1 .(3-2)=0$ $\therefore \lambda=5$ Now $D_{1}=\left|\be...
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