If the four complex numbers
Question: If the four complex numbers $z, \bar{z}, \bar{z}-2 \operatorname{Re}(\bar{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to :42$4 \sqrt{2}$$2 \sqrt{2}$Correct Option: , 4 Solution: Let $z=x+i y$ Length of side $=4$ $\mathrm{AB}=4$ $|z-\overline{2}|=4$ $|2 y|=4 ;|y|=2$ $B C=4$ $\mid \bar{z}-(\bar{z}-2 \operatorname{Re}(\bar{z}) \mid=4$ $|2 x|=4 ;|x|=2$ $|z|=\sqrt{x^{2}+y^{2}}=\sqrt{4+4}=2 \sqrt{2}$...
Read More →Solve this following
Question: If $\{p\}$ denotes the fractional part of the number $\mathrm{p}$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to $\frac{1}{8}$$\frac{5}{8}$ $\frac{3}{8}$$\frac{7}{8}$Correct Option: 1 Solution: $\left\{\frac{3^{200}}{8}\right\}=\left\{\frac{\left(3^{2}\right)^{100}}{8}\right\}$ $=\left\{\frac{(1+8)^{100}}{8}\right\}$ $=\left\{\frac{1+{ }^{100} C_{1} \cdot 8+{ }^{100} C_{2} \cdot 8^{2}+\ldots+{ }^{100} C_{100} 8^{100}}{8}\right\}$ $=\left\{\frac{1+8 m}{8}\right\}$ $=\frac{1}{8}$...
Read More →If S is the sum of the first 10 terms of the series
Question: If $\mathrm{S}$ is the sum of the first 10 terms of the series $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots$ then $\tan (S)$ is equal to :$\frac{5}{11}$$-\frac{6}{5}$$\frac{10}{11}$$\frac{5}{6}$Correct Option: , 4 Solution: $\mathrm{S}=\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\ldots$ $\mathrm{S}=\tan ^{-1}\left(\frac{2...
Read More →Solve this following
Question: $\lim _{x \rightarrow 1}\left(\frac{\int_{0}^{(x-1)^{2}} t \cos \left(t^{2}\right) d t}{(x-1) \sin (x-1)}\right)$does not existis equal to $\frac{1}{2}$is equal to 1is equal to $-\frac{1}{2}$Correct Option: 1 Solution: $\lim _{x \rightarrow 1} \frac{\int_{0}^{(x-1)^{2}} t \cos \left(t^{2}\right) d t}{(x-1) \sin (x-1)}\left(\frac{0}{0}\right)$. Apply L Hopital Rule $=\lim _{x \rightarrow 1} \frac{2(x-1) \cdot(x-1)^{2} \cos (x-1)^{4}-0}{(x-1) \cdot \cos (x-1)+\sin (x-1)}\left(\frac{0}{0}...
Read More →Let λ ∈ R. the system of linear equations
Question: Let $\lambda \in \mathrm{R}$. The system of linear equations $2 x_{1}-4 x_{2}+\lambda x_{3}=1$ $x_{1}-6 x_{2}+x_{3}=2$ $\lambda x_{1}-10 x_{2}+4 x_{3}=3$ is inconsistent for :exactly one negative value of $\lambda$exactly one positive value of $\lambda$.every value of $\lambda$.exactly two values of $\lambda$.Correct Option: 1 Solution: $D=\left|\begin{array}{ccc}2 -4 \lambda \\ 1 -6 1 \\ \lambda -10 4\end{array}\right|$ $=2(3 \lambda+2)(\lambda-3)$ $\mathrm{D}_{1}=-2(\lambda-3)$ $\mat...
Read More →Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ?
Question: Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ? $2 ! 3 ! 4 !$$(3 !)^{3} \cdot(4 !)$$(3 !)^{2} \cdot(4 !)$$3 !(4 !)^{3}$Correct Option: , 2 Solution: Total numbers in three familes $=3+3+4=10$ so total arrangement $=10$ ! Favourable cases $=$ $3 !$ $3 ! \times 3 ! \times 4 !$ $\therefore$ Probability of same family memebers are together $=\frac{3 ! 3 !...
Read More →If the minimum and the maximum values of the function
Question: If the minimum and the maximum values of the function $\mathrm{f}:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, defined by : function $\mathrm{f}:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, defined by : $f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta -1-\sin ^{2} \theta 1 \\ -\cos ^{2} \theta -1-\cos ^{2} \theta 1 \\ 12 10 -2\end{array}\right|$ are $m$ and $M$ respectively, then the ordered pair $(\mathrm{m}, \mathrm{M})$ is equal to :$(...
Read More →Which of the following points lies on the locus of the foot of perpendicular drawn upon any
Question: Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ from any of its foci ?$(-1, \sqrt{3})$$(-1, \sqrt{2})$$(-2, \sqrt{3})$(1,2)Correct Option: 1 Solution: Let foot of perpendicular is $(\mathrm{h}, \mathrm{k})$ $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ (Given) $a=2, b=\sqrt{2}, e=\sqrt{1-\frac{2}{4}}=\frac{1}{\sqrt{2}}$ $\therefore$ Focus $(\mathrm{ae}, 0)=(\sqrt{2}, 0)$ Equation of tangent $y...
Read More →Prove the following
Question: If $\int\left(e^{2 x}+2 e^{x}-e^{-x}-1\right) e^{\left(e^{2}+e^{-t}\right)} d x$ $=g(x) e^{\left(e^{x}+e^{-x}\right)}+c$, where $c$ is a constant of integration, then $g(0)$ is equal to :2$\mathrm{e}^{2}$$\mathrm{e}$1Correct Option: 1 Solution: $e^{2 x}+2 e^{x}-e^{-x}-1$ $=e^{x}\left(e^{x}+1\right)-e^{-x}\left(e^{x}+1\right)+e^{x}$ $=\left[\left(e^{x}+1\right)\left(e^{x}-e^{-x}\right)+e^{x}\right]$ so $I=\int\left(e^{x}+1\right)\left(e^{x}-e^{-x}\right) e^{e^{x}+e^{-x}}+\int e^{x} \cdo...
Read More →The product of the roots of the equation
Question: The product of the roots of the equation $9 x^{2}-18|x|+5=0$, is$\frac{25}{9}$$\frac{25}{81}$$\frac{5}{27}$$\frac{5}{9}$Correct Option: , 2 Solution: $9 x^{2}-18|x|+5=0$ $9|x|^{2}-15|x|-3|x|+5=0\left(\because x^{2}=|x|^{2}\right)$ $3|x|(3|x|-5)-(3|x|-5)=0$ $|x|=\frac{1}{3}, \frac{5}{3}$ $x=\pm \frac{1}{3}, \pm \frac{5}{3}$ Product of roots $=\frac{25}{81}$...
Read More →A survey shows that 73% of the persons working in an office like coffee,
Question: A survey shows that $73 \%$ of the persons working in an office like coffee, whereas $65 \%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be:63385436Correct Option: , 4 Solution: $\mathrm{C} \rightarrow$ person like coffee $\mathrm{T} \rightarrow$ person like Tea $\mathrm{n}(\mathrm{C})=73$ $\mathrm{n}(\mathrm{T})=65$ $n(C \cup T) \leq 100$ $n(C)+n(T)-n(C \cap T) \leq 100$ $73+65-x \leq 100$ $x \geq 38$ $73-x \geq 0 \Rightarrow x \leq 7...
Read More →A satellite is in an elliptical orbit around a planet P.
Question: A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is :$1: 6$$3: 4$$1: 3$$1: 2$Correct Option: 1, Solution: By angular momentum conservation $\mathrm{r}_{\min } \mathrm{V}_{\max }=\mathrm{r}_{\max } \mathrm{V}_{\min }$ ......(1) Given $\mathrm{V}_...
Read More →If y=y(x) is the solution of the differential
Question: If $y=y(x)$ is the solution of the differential equation $\frac{5+\mathrm{e}^{\mathrm{x}}}{2+\mathrm{y}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{e}^{\mathrm{x}}=0$ satisfying $y(0)=1$, then a value of $y\left(\log _{e} 13\right)$ is : 1$-1$20Correct Option: , 2 Solution: $\frac{\left(5+\mathrm{e}^{\mathrm{x}}\right)}{2+\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{e}^{\mathrm{x}}$ $\int \frac{d y}{2+y}=\int \frac{-e^{x}}{e^{x}+5} d x$ $\ln (y+2)=-\ln \left(e^{x}+5\right)+k...
Read More →If the volume of a parallelopiped,
Question: If the volume of a parallelopiped, whose coterminus edges are given by the vectors $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\mathrm{n} \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+n \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \quad(\mathrm{n} \geq 0)$, is $158 \mathrm{cu}$. units, then :$\vec{a} \cdot \vec{c}=17$$\vec{b} \cdot \vec{c}=10$$n=7$$\mathr...
Read More →The negation of the Boolean expression
Question: The negation of the Boolean expression $\mathrm{x} \leftrightarrow \sim \mathrm{y}$ is equivalent to:$(\sim \mathrm{x} \wedge \mathrm{y}) \vee(\sim \mathrm{x} \wedge \sim \mathrm{y})$$(x \wedge \sim y) \vee(\sim x \wedge y)$$(x \wedge y) \vee(\sim x \wedge \sim y)$$(x \wedge y) \wedge(\sim x \vee \sim y)$Correct Option: , 3 Solution: $p \leftrightarrow q \equiv(p \rightarrow q) \wedge(q \rightarrow p)$ $x \leftrightarrow \sim y \equiv(x \rightarrow \sim y) \wedge(-y \rightarrow x)$ $\b...
Read More →If the common tangent to the parabolas,
Question: If the common tangent to the parabolas, $y^{2}=4 x$ and $x^{2}=4 y$ also touches the circle, $x^{2}+y^{2}=c^{2}$, then $c$ is equal to :$\frac{1}{2}$$\frac{1}{2 \sqrt{2}}$$\frac{1}{\sqrt{2}}$$\frac{1}{4}$Correct Option: , 3 Solution: $y=m x+\frac{1}{m}\left(\operatorname{tangent}\right.$ at $\left.y^{2}=4 x\right)$ $y=m x-m^{2}\left(\operatorname{tangent}\right.$ at $\left.x^{2}=4 y\right)$ $\frac{1}{m}=-m^{2}$ (for common tangent) $m^{3}=-1$ $\mathrm{m}=-1$ $y=-x-1$ $x+y+1=0$ This lin...
Read More →If the variance of the following frequency distribution :
Question: If the variance of the following frequency distribution : Class $: \begin{array}{lll}10-20 20-30 30-40\end{array}$ Frequency: $\begin{array}{lll}2 x 2\end{array}$ is 50 , then $\mathrm{x}$ is equal to Solution: $\because$ Variance is independent of shifting of origin $\begin{array}{rlrrrrrr}\Rightarrow \mathrm{x}_{\mathrm{i}}: 15 25 35 \text { or } -10 0 10 \\ \mathrm{f}_{\mathrm{i}}: 2 \mathrm{x} 2 2 \mathrm{x} 2\end{array}$ $\Rightarrow \quad$ Variance $\left(\sigma^{2}\right)=\frac{...
Read More →If the function f(x) = { k1 (x-π)^2 - 1 ,
Question: If the function $\mathrm{f}(\mathrm{x})= \begin{cases}\mathrm{k}_{1}(\mathrm{x}-\pi)^{2}-1, \mathrm{x} \leq \pi \\ \mathrm{k}_{2} \cos \mathrm{x}, \mathrm{x}\pi\end{cases}$ is twice differentiable, then the ordered pair $\left(\mathrm{k}_{1}, \mathrm{k}_{2}\right)$ is equal to :$\left(\frac{1}{2}, 1\right)$$(1,1)$$\left(\frac{1}{2},-1\right)$$(1,0)$Correct Option: 1 Solution: $f(x)$ is continuous and differentiable $f(\pi)=f(\pi)=f\left(\pi^{+}\right)$ $-1=-k_{2}$ $\mathrm{k}_{2}=1$ $f...
Read More →The total number of 3 -digit numbers, whose sum of digits is 10 , is
Question: The total number of 3 -digit numbers, whose sum of digits is 10 , is Solution: Let three digit number is $x y z$ $x+y+z=10 ; \quad x \geq 1, y \geq 0 \quad z \geq 0$ ...(1) Let $\mathrm{T}=\mathrm{x}-1 \Rightarrow \mathrm{x}=\mathrm{T}+1$ where $\mathrm{T} \geq 0$ Put in (1) $\mathrm{T}+\mathrm{y}+\mathrm{z}=9 ; \quad 0 \leq \mathrm{T} \leq 8,0 \leq \mathrm{y}, \mathrm{z} \leq 9$ No. of non negative integral solution $={ }^{9+3-1} \mathrm{C}_{3-1}-1($ when $\mathrm{T}=9)$ $=55-1=54$...
Read More →Solve this following
Question: If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $|\hat{\mathrm{i}} \times(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}})|^{2}+|\hat{\mathrm{j}} \times(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}})|^{2}+|\hat{\mathrm{k}} \times(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}})|^{2} \quad$ is equal to Solution: $\Sigma|\vec{a}-(\vec{a} \cdot i) i|^{2}$ $\Rightarrow \quad \Sigma\left(|\mathrm{a}|^{2}+(\overrightarrow{\mathrm{a}} \cdot \mathrm{i})^{2}-2(\overrig...
Read More →Let the function
Question: Let $\mathrm{S}$ be the set of all integer solutions, $(x, y, z)$, of the system of equations $x-2 y+5 z=0$ $-2 x+4 y+z=0$ $-7 x+14 y+9 z=0$ such that $15 \leq \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2} \leq 150$. Then, the number of elements in the set $S$ is equal to__________ Solution: $\Delta=\left|\begin{array}{ccc}1 -2 5 \\ -2 4 1 \\ -7 14 9\end{array}\right|=0$ Let $\quad x=k$ $\Rightarrow \quad$ Put in (1) \ (2) $k-2 y+5 z=0$ $-2 k+4 y+z=0$' $z=0, y=\frac{k}{2}$ $\therefore \...
Read More →If 3^2 sin 2 α-1, 14 and 3^4 -2 sin 2 α are the first three terms of an A.P. for some α
Question: If $3^{2} \sin 2 \alpha-1,14$ and $3^{4-2} \sin 2 \alpha$ are the first three terms of an A.P. for some $\alpha$, then the sixth term of this A.P. is:66658178Correct Option: 1 Solution: Given that $3^{4-\sin 2 \alpha}+3^{2} \sin 2 \alpha-1=28$ Let $3^{2} \sin 2 a=\mathrm{t}$ $\frac{81}{t}+\frac{t}{3}=28$ $t=81,3$ $3^{2} \sin 2 \alpha=3^{1}, 3^{4}$ $2 \sin 2 \alpha=1,4$ $\sin 2 \alpha=\frac{1}{2}, 2$ (rejected) First term $a=3^{2} \sin 2 \alpha-1$ $a=1$ Second term $=14$ $\therefore$ co...
Read More →Solve this following
Question: Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\leq x$ respectively of a real number $x$. If $\int_{0}^{n}\{x\} d x, \int_{0}^{n}[x] d x$ and $10\left(n^{2}-n\right),(n \in N, n1)$ are three consecutive terms of a G.P., then $\mathrm{n}$ is equal to Solution: $\int_{0}^{n}\{x\} d x=n \int_{0}^{1}\{x\} d x=n \int_{0}^{1} x d x=\frac{n}{2}$ $\int_{0}^{n}[x] d x=\int_{0}^{n}(x-\{x\}) d x=\frac{n^{2}}{2}-\frac{n}{2}$ $\Rightarrow\left(\frac{\mathrm{n}^{2}...
Read More →Let a plane P contain two lines
Question: Let a plane P contain two lines $\overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}), \lambda \in \mathrm{R}$ and $\overrightarrow{\mathrm{r}}=-\hat{\mathrm{j}}+\mu(\hat{\mathrm{j}}-\hat{\mathrm{k}}), \mu \in \mathrm{R}$ If $\mathrm{Q}(\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from the point $M(1,0,1)$ to $P$, then $3(\alpha+\beta+\gamma)$ equals_______________ Solution: Dr's normal to plane $=\left|\begin{array}{ccc}\mathrm{i} \...
Read More →Solve this following
Question: Let $P Q$ be a diameter of the circle $x^{2}+y^{2}=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ respectively, then the maximum value of $\alpha \beta$ is Solution: Let $\mathrm{P}(3 \cos \theta, 3 \sin \theta)$ $Q(-3 \cos \theta,-3 \sin \theta)$ $\Rightarrow \alpha \beta=\frac{\left|(3 \cos \theta+3 \sin \theta)^{2}-4\right|}{2}$ $\Rightarrow \alpha \beta=\frac{5+9 \sin 2 \theta}{2} \leq 7$...
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