Solve the Following Questions
Question: For $\mathrm{p}0$, a vector $\overrightarrow{\mathrm{v}}_{2}=2 \hat{\mathrm{i}}+(\mathrm{p}+1) \hat{\mathrm{j}}$ is obtained by rotating the vector $\overrightarrow{\mathrm{v}}_{1}=\sqrt{3}$ pi $+\hat{\mathrm{j}}$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta=\frac{(\alpha \sqrt{3}-2)}{(4 \sqrt{3}+3)}$, then the value of $\alpha$ is equal to Solution: $\left|\overrightarrow{\mathrm{V}}_{1}\right|=\left|\overrightarrow{\mathrm{V}_{2}}\right|$ $3 \math...
Read More →The values of a and b, for which the system of equations
Question: The values of $a$ and $b$, for which the system of equations $2 x+3 y+6 z=8$ $x+2 y+a z=5$ $3 x+5 y+9 z=b$ has no solution, are :$a=3, b \neq 13$$a \neq 3, b \neq 13$$a \neq 3, b=3$$a=3, b=13$Correct Option: 1 Solution: $D=\left|\begin{array}{lll}2 3 6 \\ 1 2 a \\ 3 5 9\end{array}\right|=3-a$ $D=\left|\begin{array}{lll}2 3 8 \\ 1 2 5 \\ 3 5 b\end{array}\right|=b-13$ If $a=3, b \neq 13$, no solution....
Read More →Let Q be the foot of the perpendicular from the point P(7,-2,13)
Question: Let $Q$ be the foot of the perpendicular from the point $\mathrm{P}(7,-2,13)$ on the plane containing the $\operatorname{lines} \frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and $\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}$. Then $(\mathrm{PQ})^{2}$, is equal to___________ Solution: Containing the line $\left|\begin{array}{ccc}\mathrm{x}+1 \mathrm{y}-1 \mathrm{z}-3 \\ 6 7 8 \\ 3 5 7\end{array}\right|=0$ $9(x+1)-18(y-1)+9(z-3)=0$ $x-2 y+z=0$ $\mathrm{PQ}=\left|\frac{7+4+13}{\sqrt{6}}\right|=...
Read More →Let three vectors
Question: Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$ and $|\overrightarrow{\mathrm{a}}|=2$. Then which one of the following is not true ?$\overrightarrow{\mathrm{a}} \times((\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}) \times(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}})...
Read More →Let a curve
Question: Let a curve $y=y(x)$ be given by the solution of the differential equation $\cos \left(\frac{1}{2} \cos ^{-1}\left(\mathrm{e}^{-\mathrm{x}}\right)\right) \mathrm{dx}=\sqrt{\mathrm{e}^{2 \mathrm{x}}-1} \mathrm{dy}$ If it intersects $\mathrm{y}$-axis at $\mathrm{y}=-1$, and the intersection point of the curve with $x$-axis is $(\alpha, 0)$, then $e^{\alpha}$ is equal to Solution: $\cos \left(\frac{1}{2} \cos ^{-1}\left(\mathrm{e}^{-\mathrm{x}}\right)\right) \mathrm{d} \mathrm{x}=\sqrt{\m...
Read More →Let f : [ 0 , ∞ ) → [ 0 , ∞ ) be defined as
Question: Let $f:[0, \infty) \rightarrow[0, \infty)$ be defined as $\mathrm{f}(\mathrm{x})=\int_{0}^{\mathrm{x}}[\mathrm{y}] \mathrm{dy}$ where $[\mathrm{x}]$ is the greatest integer less than or equal to $\mathrm{x}$. Which of the following is true?$f$ is continuous at every point in $[0, \infty)$ and differentiable except at the integer points.$\mathrm{f}$ is both continuous and differentiable except at the integer points in $[0, \infty)$.f is continuous everywhere except at the integer points...
Read More →The number of solutions of the equation
Question: The number of solutions of the equation $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0$ $x0$, is Solution: $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0$ $\log _{(x+1)}(2 x+5)(x+1)+2 \log _{(2 x+5)}(x+1)=4$ $\log _{(x+1)}(2 x+5)+1+2 \log _{(2 x+5)}(x+1)=4$ Put $\log _{(x+1)}(2 x+5)=t$ $\mathrm{t}+\frac{2}{\mathrm{t}}=3 \Rightarrow \mathrm{t}^{2}-3 \mathrm{t}+2=0$ $t=1,2$ $\log _{(x+1)}(2 x+5)=1 \ \quad \log _{(x+1)}(2 x+5)=2$ $x+1=2 x+3 \q...
Read More →Let a1 , a2 ,....,a10 be an AP with common difference –3 and b1 , b2 ,...., b10 be a GP with common ratio 2.
Question: Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{10}$ be an AP with common difference $-3$ and $b_{1}, b_{2}, \ldots, b_{10}$ be a GP with common ratio 2 . Let $c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10$. If $c_{2}=12$ and $c_{3}=13$, then $\sum_{\mathrm{k}=1}^{10} c_{\mathrm{k}}$ is equal to______ Solution: $c_{2}=a_{2}+b_{2}=a_{1}-3+2 b_{1}=12$ $a_{1}+2 b_{1}=15$ .....(1) $\alpha^{2}-\alpha+2 \lambda=0$ ......(2) from (1) \ (2) $b_{1}=2, a_{1}=11$ $\sum_{\mathrm{k}=1}^{10} \mathrm...
Read More →Solve the Following Questions
Question: Let $A=\left\{a_{i j}\right\}$ be a $3 \times 3$ matrix, where $\mathrm{a}_{\mathrm{ij}}=\left\{\begin{array}{c}(-1)^{\mathrm{j}-\mathrm{i}} \text { if } \mathrm{i}\mathrm{j} \\ 2 \quad \text { if } \mathrm{i}=\mathrm{j} \\ (-1)^{\mathrm{i}+\mathrm{j}} \text { if } \mathrm{i}\mathrm{j}\end{array}\right.$ then $\operatorname{det}\left(3 \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right)$ is equal to Solution: $A=\left[\begin{array}{rrr}2 -1 1 \\ -1 2 -1 \\ 1 -1 2\end{array}\right]...
Read More →Let g : N → N be defined as
Question: Let $g: \mathbf{N} \rightarrow \mathbf{N}$ be defined as $g(3 n+1)=3 n+2$ $g(3 n+2)=3 n+3$ $\mathrm{g}(3 \mathrm{n}+3)=3 \mathrm{n}+1$, for all $\mathrm{n} \geq 0$ Then which of the following statements is true?There exists an onto function $f: \mathbf{N} \rightarrow \mathbf{N}$ such that fog $=\mathrm{f}$There exists a one-one function $\mathrm{f}: \mathbf{N} \rightarrow \mathbf{N}$ such that fog $=\mathrm{f}$gogog $=\mathrm{g}$There exists a function $\mathrm{f} / \mathrm{N} \rightar...
Read More →In a triangle ABC,
Question: In a triangle $\mathrm{ABC}$, if $|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{CA}}|=5$ and $|\overrightarrow{\mathrm{BA}}|=7$, then the projection of the vector $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal to$\frac{19}{2}$$\frac{13}{2}$$\frac{11}{2}$$\frac{15}{2}$Correct Option: , 3 Solution: Projection of $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal to $=|\mathrm{BA} \overrightarrow{\mathrm{A}}| \cos \angle \mathr...
Read More →Solve this
Question: If $\int_{0}^{100 \pi} \frac{\sin ^{2} x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}} d x=\frac{\alpha \pi^{3}}{1+4 \pi^{2}}, \alpha \in \mathbf{R}$ where $[\mathrm{x}]$ is the greatest integer less than or equal to $x$, then the value of $\alpha$ is :$200\left(1-\mathrm{e}^{-1}\right)$$100(1-\mathrm{e})$$50(\mathrm{e}-1)$$150\left(\mathrm{e}^{-1}-1\right)$Correct Option: 1 Solution: $I=\int_{0}^{100 \pi} \frac{\sin ^{2} x}{e^{\{x / \pi\}}} d x=100 \int_{0}^{\pi} \frac{\...
Read More →If the projection of the vector
Question: If the projection of the vector $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ on the sum of the two vectors $2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$ and $-\lambda \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ is 1 , then $\lambda$ is equal to Solution: $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ $\overrightarrow{\mathrm{b}}=(2-\lambda) \hat{\mathrm{i}}+6 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ $\frac{\overrightarr...
Read More →If sum of the first 21 terms of the series
Question: If sum of the first 21 terms of the series $\log _{9} 1 / 2 x+\log _{9}^{1 / 3} x+\log _{9} 1 / 4 x+\ldots .$, where $x0$ is 504 , then $x$ is equal to2439781Correct Option: , 4 Solution: $\mathrm{s}=2 \log _{9} \mathrm{x}+3 \log _{9} \mathrm{x}+\ldots \ldots+22 \log _{9} \mathrm{x}$ $\mathrm{s}=\log _{9} \mathrm{x}(2+3+\ldots . .+22)$ $\mathrm{s}=\log _{9} \mathrm{x}\left\{\frac{21}{2}(2+22)\right\}$ Given $252 \log _{9} x=504$ $\Rightarrow \log _{9} \mathrm{x}=2 \Rightarrow \mathrm{x...
Read More →Let a and b respectively be the points of local maximum and local minimum of the function
Question: Let a and b respectively be the points of local maximum and local minimum of the function $f(x)=2 x^{3}-3 x^{2}-12 x$. If $A$ is the total area of the region bounded by $y=f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, then $4 A$ is equal to___________ Solution: $f^{\prime}(x)=6 x^{2}-6 x-12=6(x-2)(x+1)$ Point $=(2,-20) \(-1,7)$ $A=\int_{-1}^{0}\left(2 x^{3}-3 x^{2}-12 x\right) d x+\int_{0}^{2}\left(12 x+3 x^{2}-2 x^{3}\right) d x$ $A=\left(\frac{x^{4}}{2}-x^{3}-6 x^{2}\right)_{-1}...
Read More →The value of
Question: The value of $\mathrm{k} \in \mathbf{R}$, for which the following system of linear equations $3 x-y+4 z=3$ $x+2 y-3 z=-2$ $6 x+5 y+k z=-3$ has infinitely many solutions, is :3-553Correct Option: , 2 Solution: $\left|\begin{array}{ccc}3 -1 4 \\ 1 2 -3 \\ 6 5 \mathrm{~K}\end{array}\right|=0$ $\Rightarrow 3(2 \mathrm{~K}+15)+\mathrm{K}+18-28=0$ $\Rightarrow 7 \mathrm{~K}+35=0 \Rightarrow \mathrm{K}=-5$...
Read More →The sum of all values of x in
Question: The sum of all values of $x$ in $[0,2 \pi]$, for which $\sin x+\sin 2 x+\sin 3 x+\sin 4 x=0$, is equal to :$8 \pi$$11 \pi$$12 \pi$$9 \pi$Correct Option: , 4 Solution: $(\sin x+\sin 4 x)+(\sin 2 x+\sin 3 x)=0$ $\Rightarrow 2 \sin \frac{5 x}{2}\left\{\cos \frac{3 x}{2}+\cos \frac{x}{2}\right\}=0$ $\Rightarrow 2 \sin \frac{5 x}{2}\left\{2 \cos x \cos \frac{x}{2}\right\}=0$ $2 \sin \frac{5 x}{2}=0 \Rightarrow \frac{5 x}{2}=0, \pi, 2 \pi, 3 \pi, 4 \pi, 5 \pi$ $\Rightarrow x=0, \frac{2 \pi}{...
Read More →Let P be a variable point
Question: Let $\mathrm{P}$ be a variable point on the parabola $y=4 x^{2}+1 .$ Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $\mathrm{P}$ to the line $\mathrm{y}=\mathrm{x}$ is :$(3 x-y)^{2}+(x-3 y)+2=0$$2(3 x-y)^{2}+(x-3 y)+2=0$$(3 x-y)^{2}+2(x-3 y)+2=0$$2(x-3 y)^{2}+(3 x-y)+2=0$Correct Option: , 2 Solution: $\frac{\mathrm{K}-\mathrm{C}}{\mathrm{h}-\mathrm{C}}=-1$ $\mathrm{C}=\frac{\mathrm{h}+\mathrm{K}}{2} \quad \mathrm{P}(\mathrm{x}, ...
Read More →The sum of all 3-digit numbers less than or equal to 500 , that are formed without using the digit "1" and they all are multiple of 11 , is
Question: The sum of all 3-digit numbers less than or equal to 500 , that are formed without using the digit "1" and they all are multiple of 11 , is Solution: $209,220,231, \ldots, 495$ Sum $=\frac{27}{2}(209+495)=9504$ Required $=9501-(231+341+451+319+418)$ 7744...
Read More →The area (in sq. units) of the region, given by the
Question: The area (in sq. units) of the region, given by the set $\left\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid x \geq 0,2 x^{2} \leq y \leq 4-2 x\right\}$ is :$\frac{8}{3}$$\frac{17}{3}$$\frac{13}{3}$$\frac{7}{3}$Correct Option: , 4 Solution: Required area $=\int_{0}^{1}\left(4-2 x-2 x^{2}\right) d x=4 x-x^{2}-\left.\frac{2 x^{3}}{3}\right|_{0} ^{1}$ $=4-1-\frac{2}{3}=\frac{7}{3}$...
Read More →Let a vector
Question: Let a vector $\overrightarrow{\mathrm{a}}$ be coplanar with vectors $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{d}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$, and $|\overrightarrow{\mathrm{a}}|=\sqrt{10}$. Then a possible value of $\left[\begin{array}{lll}\vec{a} \vec{b}...
Read More →Solve the Following Questions
Question: Let $\mathrm{g}(\mathrm{t})=\int_{-\pi / 2}^{\pi / 2} \cos \left(\frac{\pi}{4} \mathrm{t}+f(\mathrm{x})\right) \mathrm{dx}$, where $f(x)=\log _{e}\left(x+\sqrt{x^{2}+1}\right), x \in \mathbf{R}$. Then which one of the following is correct?$g(1)=g(0)$$\sqrt{2} g(1)=g(0)$$g(1)=\sqrt{2} g(0)$$\mathrm{g}(1)+\mathrm{g}(0)=0$Correct Option: , 2 Solution: $\mathrm{g}(\mathrm{t})=\int_{-\pi / 2}^{\pi / 2}\left(\cos \frac{\pi}{4} \mathrm{t}+\mathrm{f}(\mathrm{x})\right) \mathrm{d} \mathrm{x}$ $...
Read More →The Boolean expression
Question: The Boolean expression $(\mathrm{p} \Rightarrow \mathrm{q}) \wedge(\mathrm{q} \Rightarrow \sim \mathrm{p})$ is equivalent to :$\sim \mathrm{q}$$\mathrm{q}$$\mathrm{p}$pCorrect Option: , 4 Solution: $(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \sim \mathrm{p})$ $\equiv(\sim \mathrm{p} \vee \mathrm{q}) \wedge(\sim \mathrm{q} \vee \sim \mathrm{p}) \quad\{\mathrm{p} \rightarrow \mathrm{q} \equiv \sim \mathrm{p} \vee \mathrm{q}\}$ $\equiv(\sim \mathrm{p} \vee \mathrm{q...
Read More →If the mean and variance of
Question: If the mean and variance of six observations $7,10,11,15, \mathrm{a}, \mathrm{b}$ are 10 and $\frac{20}{3}$, respectively, then the value of $|a-b|$ is equal to :91171Correct Option: , 4 Solution: $10=\frac{7+10+11+15+a+b}{6}$ $\Rightarrow a+b=17$..(1) $\frac{20}{3}=\frac{7^{2}+10^{2}+11^{2}+15^{2}+a^{2}+b^{2}}{6}-10^{2}$ $a^{2}+b^{2}=145$..(2) Solve (i) and (ii) $a=9, b=8$ or $a=8, b=9$ $|a-b|=1$...
Read More →Solve this
Question: $\lim _{x \rightarrow 2}\left(\sum_{n=1}^{9} \frac{x}{n(n+1) x^{2}+2(2 n+1) x+4}\right)$ is equal to :$\frac{9}{44}$$\frac{5}{24}$$\frac{1}{5}$$\frac{7}{36}$Correct Option: 1 Solution: $S=\lim _{x \rightarrow 2} \sum_{n=1}^{9} \frac{x}{n(n+1) x^{2}+2(2 n+1) x+4}$ $\mathrm{S}=\sum_{\mathrm{n}=1}^{9} \frac{2}{4\left(\mathrm{n}^{2}+3 \mathrm{n}+2\right)}=\frac{1}{2} \sum_{\mathrm{n}=1}^{9}\left(\frac{1}{\mathrm{n}+1}-\frac{1}{\mathrm{n}+2}\right)$ $\mathrm{S}=\frac{1}{2}\left(\frac{1}{2}-...
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