Let A be a fixed point
Question: Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2 \mathrm{t}, 0)$. Let $\mathrm{M}$ be the mid-point of $\mathrm{AB}$ and the perpendicular bisector of $\mathrm{AB}$ meets the $\mathrm{y}$-axis at $\mathrm{C}$. The locus of the mid-point $\mathrm{P}$ of $\mathrm{MC}$ is:$3 x^{2}-2 y-6=0$$3 x^{2}+2 y-6=0$$2 x^{2}+3 y-9=0$$2 x^{2}-3 y+9=0$Correct Option: , 3 Solution: $\mathrm{A}(0,6)$ and $\mathrm{B}(2 \mathrm{t}, 0)$ Perpendicular bisector of $\mathrm{AB}$ is $(y-3)=\frac{...
Read More →Let f : R → R and g : R → R be
Question: Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{cc}x+a, x0 \\ |x-1|, x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x+1, x0 \\ (x-1)^{2}+b, x \geq 0\end{array}\right.$ where $a, b$ are non-negative real numbers. If (gof) $(\mathrm{x})$ is continuous for all $\mathrm{x} \in \mathrm{R}$, then $\mathrm{a}+\mathrm{b}$ is equal to_________. Solution: $g[f(x)]=\left[\begin{array}{cc}f(x)+1 f(x)0 \\ (f(x)-1)^{2}+b f(x) \geq 0\end{array...
Read More →A wire of length 36 m is cut into two pieces,
Question: A wire of length $36 \mathrm{~m}$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $\mathrm{k}$ (meter), then $\left(\frac{4}{\pi}+1\right) \mathrm{k}$ is equal to _______. Solution: Let $x+y=36$ $x$ is perimeter of square and $y$ is perimeter of circle side of square $=\mathrm{x} / 4$ radius of circle $=\frac{\mathrm{y}}{2 \pi}$ Sum Ar...
Read More →Let a tangent be drawn to the ellipse
Question: Let a tangent be drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in\left(0, \frac{\pi}{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to : $\frac{\pi}{8}$$\frac{\pi}{4}$$\frac{\pi}{6}$$\frac{\pi}{3}$Correct Option: , 3 Solution: Equation of tangent be $\frac{x \cos \theta}{3 \sqrt{3}}+\frac{y \cdot \sin \theta}{1}=1, \quad \theta \in\left(0, \frac{\pi}{2}\r...
Read More →Solve the Following Questions
Question: If for $x, y \in \mathbf{R}, x0$ $y=\log _{10} x+\log _{10} x^{1 / 3}+\log _{10} x^{1 / 9}+\ldots .$ upto $\infty$ terms and $\frac{2+4+6+\ldots+2 y}{3+6+9+\ldots+3 y}=\frac{4}{\log _{10} x}$, then the ordered pair $(\mathrm{x}, \mathrm{y})$ is equal to :$\left(10^{6}, 6\right)$$\left(10^{4}, 6\right)$$\left(10^{2}, 3\right)$$\left(10^{6}, 9\right)$Correct Option: , 4 Solution: $\frac{2(1+2+3+\ldots+y)}{3(1+2+3+\ldots+y)}=\frac{4}{\log _{10} x}$ $\Rightarrow \log _{10} x=6 \Rightarrow ...
Read More →For real numbers α, β, γ and
Question: For real numbers $\alpha, \beta, \gamma$ and $\delta$, if $\int \frac{\left(x^{2}-1\right)+\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)} d x$ $=\alpha \log _{e}\left(\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)\right)$ $+\beta \tan ^{-1}\left(\frac{\gamma\left(x^{2}-1\right)}{x}\right)+\delta \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)+C$ where $\mathrm{C}$ is an arbitrary constant, then the value of $10(\alpha+\beta \gamma+\d...
Read More →Solve the Following Questions
Question: If $0x1$, then $\frac{3}{2} x^{2}+\frac{5}{3} x^{3}+\frac{7}{4} x^{4}+\ldots .$, is equal to :$x\left(\frac{1+x}{1-x}\right)+\log _{e}(1-x)$$x\left(\frac{1-x}{1+x}\right)+\log _{e}(1-x)$$\frac{1-x}{1+x}+\log _{e}(1-x)$$\frac{1+x}{1-x}+\log _{e}(1-x)$Correct Option: 1 Solution: Let $\mathrm{t}=\frac{3}{2} \mathrm{x}^{2}+\frac{5}{3} \mathrm{x}^{3}+\frac{7}{4} \mathrm{x}^{4}+\ldots . \infty$ $=\left(2-\frac{1}{2}\right) x^{2}+\left(2-\frac{1}{3}\right) x^{3}+\left(2-\frac{1}{4}\right) x^{...
Read More →If ^1P_1+2.^2 P_2+3.
Question: If ${ }^{1} \mathrm{P}_{1}+2 \cdot{ }^{2} \mathrm{P}_{2}+3 \cdot{ }^{3} \mathrm{P}_{3}+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}={ }^{\mathrm{q}} \mathrm{P}_{\mathrm{r}}-\mathrm{s}, 0 \leq \mathrm{s} \leq 1$ then ${ }^{q+s} \mathrm{C}_{r-s}$ is equal to _______. Solution: ${ }^{1} \mathrm{P}_{1}+2 \cdot{ }^{2} \mathrm{P}_{2}+3 \cdot{ }^{3} \mathrm{P}_{3}+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}$ $=1 !+2.2 !+3.3 !+\ldots .15 \times 15 !$ $=\sum_{r=1}^{15}(r+1-1 r !)$ $=\sum_{r=1}^{15}(r+1)...
Read More →Let A = [a1 a2] and B = [b1 b2] be two 2 x 1 matrices
Question: Let $A=\left[\begin{array}{l}a_{1} \\ a_{2}\end{array}\right]$ and $B=\left[\begin{array}{l}b_{1} \\ b_{2}\end{array}\right]$ be two $2 \times 1$ matrices with real entries such that $\mathrm{A}=\mathrm{XB}$, where $\mathrm{X}=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 -1 \\ 1 \mathrm{k}\end{array}\right]$, and $\mathrm{k} \in \mathrm{R} .$ If $\mathrm{a}_{1}^{2}+\mathrm{a}_{2}^{2}=\frac{2}{3}\left(\mathrm{~b}_{1}^{2}+\mathrm{b}_{2}^{2}\right)$ and $\left(\mathrm{k}^{2}+1\right) \mathr...
Read More →Solve this following
Question: Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be $0.4096$ and $0.2048$ respectively. Then the probability of getting exactly 3 successes is equal to : $\frac{32}{625}$$\frac{80}{243}$$\frac{40}{243}$$\frac{128}{625}$Correct Option: 1 Solution: $\mathrm{P}(\mathrm{X}=1)={ }^{5} \mathrm{C}_{1} \cdot \mathrm{p} \cdot \mathrm{q}^{4}=0.4096$ $\mathrm{P}(\mathrm{X}=2)={ }^{5} \mathrm{C}_{2} \cdot \mathrm{p}^{2} \cdot \mathrm{q}...
Read More →Solve this following
Question: Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$. If $|\vec{a} \times \vec{b}|=|\vec{a}|$, then the angle between the vectors $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}))$ and $\overrightarrow{\mathrm{a}}$ is equal to : $\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$$\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$$\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$ $\si...
Read More →Consider the statistics of two sets of observations as follows :
Question: Consider the statistics of two sets of observations as follows : If the variance of the combined set of these two observations is $\frac{17}{9}$, then the value of $n$ is equal to__________. Solution: $\sigma^{2}=\frac{\mathrm{n}_{1} \sigma_{1}^{2}+\mathrm{n}_{2} \sigma_{2}^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}+\frac{\mathrm{n}_{1} \mathrm{n}_{2}}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)}\left(\overline{\mathrm{x}}_{1}-\overline{\mathrm{x}}_{2}\right)^{2}$ $\mathrm{n}_{1}=10, \mathrm{n...
Read More →Let the line L be the projection of the line
Question: Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2 y-z=3 .$ If $d$ is the distance of the point $(0,0,6)$ from $\mathrm{L}$, then $\mathrm{d}^{2}$ is equal to _________. Solution: $\mathrm{L}_{1}: \frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-3}{1}=\frac{\mathrm{z}-4}{2}$ for foot of $\perp \mathrm{r}$ of $(1,3,4)$ on $\mathrm{x}-2 \mathrm{y}-\mathrm{z}-3=0$ $(1+t)-2(3-2 t)-(4-t)-3=0$ $\Rightarrow \mathrm{t}=2$ So foot of $\perp \math...
Read More →Solve this following
Question: Let $S_{1}: x^{2}+y^{2}=9$ and $S_{2}:(x-2)^{2}+y^{2}=1$. Then the locus of center of a variable circle $S$ which touches $S_{1}$ internally and $S_{2}$ externally always passes through the points: $(0, \pm \sqrt{3})$$\left(\frac{1}{2}, \pm \frac{\sqrt{5}}{2}\right)$$\left(2, \pm \frac{3}{2}\right)$$(1, \pm 2)$Correct Option: , 3 Solution: $\mathrm{S}_{1}: \mathrm{x}^{2}+\mathrm{y}^{2}$ $\mathrm{S}_{2}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2}$ $\because c_{1} c_{2}=r_{1}-r_{2}$ $\therefore$ g...
Read More →If the distance of the point (1,-2,3)
Question: If the distance of the point $(1,-2,3)$ from the plane $x+2 y-3 z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is $\sqrt{\frac{7}{2}}$, then the value of $|m|$ is equal to______. Solution: $\mathrm{DC}$ of line $\equiv\left(\frac{3}{\sqrt{\mathrm{m}^{2}+10}}, \frac{-\mathrm{m}}{\sqrt{\mathrm{m}^{2}+10}}, \frac{1}{\sqrt{\mathrm{m}^{2}+10}}\right)$ $Q \equiv\left(1+\frac{3 r}{\sqrt{m^{2}+10}},-2+\frac{-m r}{\sqrt{m^{2}+10}}, 3+\frac{r}{\sqrt{m^{2}+10}}...
Read More →The sum of all integral values
Question: The sum of all integral values of $\mathrm{k}(\mathrm{k} \neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is Solution: $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ $x \in R-\{1,2\}$ $\Rightarrow \mathrm{k}(2 \mathrm{x}-4-\mathrm{x}+1)=2\left(\mathrm{x}^{2}-3 \mathrm{x}+2\right)$ $\Rightarrow \mathrm{k}(\mathrm{x}-3)=2\left(\mathrm{x}^{2}-3 \mathrm{x}+2\right)$ for $x \neq 3, \quad k=2\left(x-3+\frac{2}{x-3}+3\right)$ $x-3+\frac{2}{x-3} \g...
Read More →Solve this following
Question: Let in a series of $2 n$ observations, half of them are equal to a and remaining half are equal to $-\mathrm{a}$. Also by adding a constant $\mathrm{b}$ in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then the value of $a^{2}+b^{2}$ is equal to : 425650250925Correct Option: 1 Solution: Let observations are denoted by $\mathrm{x}_{i}$ for $1 \leq i$ $2 \mathrm{n}$ $\bar{x}=\frac{\sum x_{i}}{2 n}=\frac{(a+a+\ldots+a)-(a+a+\ldots+a...
Read More →Prove the following
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{r}} \cdot(\alpha \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=3$ and $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-\al...
Read More →Let C_1 be the curve obtained by the solution of
Question: Let $C_{1}$ be the curve obtained by the solution of differential equation $2 x y \frac{d y}{d x}=y^{2}-x^{2}, x0$. Let the curve $C_{2}$ be the solution of $\frac{2 x y}{x^{2}-y^{2}}=\frac{d y}{d x}$. If both the curves pass through $(1,1)$, then the area enclosed by the curves $C_{1}$ and $C_{2}$ is equal to :$\pi-1$$\frac{\pi}{2}-1$$\pi+1$$\frac{\pi}{4}+1$Correct Option: , 2 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}^{2}-\mathrm{x}^{2}}{2 \mathrm{xy}}, \quad \mathrm...
Read More →Solve this
Question: Let $\mathrm{z}=\frac{1-i \sqrt{3}}{2}, i=\sqrt{-1}$ Then the value of $21+\left(z+\frac{1}{z}\right)^{3}+\left(z^{2}+\frac{1}{z^{2}}\right)^{3}+\left(z^{3}+\frac{1}{z^{3}}\right)^{3}+\ldots+\left(z^{21}+\frac{1}{z^{21}}\right)^{3}$ is ________. Solution: $\mathrm{Z}=\frac{1-\sqrt{3} \mathrm{i}}{2}=\mathrm{e}^{-\mathrm{i} \frac{\pi}{3}}$ $z^{r}+\frac{1}{z^{r}}=2 \cos \left(-\frac{\pi}{3}\right) r=2 \cos \frac{r \pi}{3}$ $\Rightarrow 21+\sum_{\mathrm{r}=1}^{21}\left(\mathrm{z}^{\mathrm{...
Read More →Solve this following
Question: Let a complex number be $w=1-\sqrt{3} i$. Let another complex number $\mathrm{z}$ be such that $|\mathrm{zw}|=1$ and $\arg (\mathrm{z})-\arg (\mathrm{w})=\frac{\pi}{2}$. Then the area of the triangle with vertices origin, $\mathrm{z}$ and $\mathrm{w}$ is equal to : 4$\frac{1}{2}$$\frac{1}{4}$2Correct Option: , 2 Solution: $\mathrm{w}=1-\sqrt{3} \cdot i \Rightarrow|\mathrm{w}|=2$ Now, $|z|=\frac{1}{|w|} \Rightarrow|z|=\frac{1}{2}$ and $\operatorname{amp}(\mathrm{z})=\frac{\pi}{2}+\opera...
Read More →Let P be an arbitrary point
Question: Let $P$ be an arbitrary point having sum of the squares of the distance from the planes $\mathrm{x}+\mathrm{y}+\mathrm{z}=0, l \mathrm{x}-\mathrm{nz}=0$ and $\mathrm{x}-2 \mathrm{y}+\mathrm{z}=0$ equal to 9 . If the locus of the point $P$ is $x^{2}+y^{2}+z^{2}=9$, then the value of $l-n$ is equal to Solution: Let point $\mathrm{P}$ is $(\alpha, \beta, \gamma)$ $\left(\frac{\alpha+\beta+\gamma}{\sqrt{3}}\right)^{2}+\left(\frac{\ell \alpha-n \gamma}{\sqrt{\ell^{2}+n^{2}}}\right)^{2}+\lef...
Read More →Solve this following
Question: Let $S_{1}$ be the sum of first $2 n$ terms of an arithmetic progression. Let $S_{2}$ be the sum of first 4n terms of the same arithmetic progression. If $\left(S_{2}-S_{1}\right)$ is 1000 , then the sum of the first $6 n$ terms of the arithmetic progression is equal to: 1000700050003000Correct Option: , 4 Solution: $S_{2 n}=\frac{2 n}{2}[2 a+(2 n-1) d], S_{4 n}=\frac{4 n}{2}[2 a+(4 n-$ 1)d] $\Rightarrow S_{2}-S_{1}=\frac{4 n}{2}[2 a+(4 n-1) d]-\frac{2 n}{2}[2 a+(2 n-$ 1)d] $=4 \mathrm...
Read More →Solve the Following Questions
Question: Let $\mathrm{I}_{\mathrm{n}}=\int_{1}^{\mathrm{c}} \mathrm{x}^{19}(\log |\mathrm{x}|)^{\mathrm{n}} \mathrm{dx}$, where $\mathrm{n} \in \mathrm{N}$. If (20) $\mathrm{I}_{10}=\alpha \mathrm{I}_{9}+\beta \mathrm{I}_{8}$, for natural numbers $\alpha$ and $\beta$, then $\alpha-\beta$ equal to Solution: $\mathrm{I}_{\mathrm{n}}=\int_{1}^{\mathrm{e}} \mathrm{x}^{19}(\log |\mathrm{x}|)^{\mathrm{n}} \mathrm{d} \mathrm{x}$ $I_{n}=\left|(\log |x|)^{19} \frac{x^{20}}{20}\right|_{1}^{e}-\int n(\log...
Read More →If the truth value of the Boolean expression
Question: If the truth value of the Boolean expression $((\mathrm{p} \vee \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{r}) \wedge(\sim \mathrm{r})) \rightarrow(\mathrm{p} \wedge \mathrm{q}) \quad$ is $\quad$ false then the truth values of the statements $\mathrm{p}, \mathrm{q}, \mathrm{r}$ respectively can be:T F TF F TT F F$\mathrm{F} \mathrm{T} \mathrm{F}$Correct Option: , 3 Solution:...
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