Prove the following
Question: A vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in R)$ lies in the plane of the vectros $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4 \hat{k}$. If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c}$, then:$\vec{a} \cdot \hat{i}+1=0$$\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{i}}+3=0$$\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{k}}+4=0$$\vec{a} \cdot \hat{k}+2=0$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: If the function $f(x)=\left\{\begin{array}{l}a|\pi-x|+1, x \leq 5 \\ b|x-\pi|+3, x5\end{array}\right.$ is continuous at $x=5$, then the value of $a-b$ is :- $\frac{2}{5-\pi}$$\frac{2}{\pi-5}$$\frac{2}{\pi+5}$$\frac{-2}{\pi+5}$Correct Option: 1 Solution:...
Read More →Let P be a plane passing through the points (2,1,0),(4,1,1)
Question: Let $P$ be a plane passing through the points $(2,1,0),(4,1,1)$ and $(5,0,1)$ and $R$ be any point $(2,1,6)$. Then the image of $R$ in the plane $P$ is :$(6,5,-2)$$(4,3,2)$$(3,4,-2)$$(6,5,2)$Correct Option: 1 Solution:...
Read More →The area (in sq. units) of the smaller of the two circles that touch the parabola
Question: The area (in sq. units) of the smaller of the two circles that touch the parabola, $\mathrm{y}^{2}=4 \mathrm{x}$ at the point $(1,2)$ and the $x$-axis is :-$4 \pi(2-\sqrt{2})$$8 \pi(3-2 \sqrt{2})$$4 \pi(3+\sqrt{2})$$8 \pi(2-\sqrt{2})$Correct Option: , 2 Solution:...
Read More →Total number of 6-digit numbers in
Question: Total number of 6-digit numbers in which only and all the five digits $1,3,5,7$ and 9 appear, is :$\frac{5}{2}(6 !)$$5^{6}$$\frac{1}{2}(6 !)$6 !Correct Option: 1 Solution: Total number of 6-digit numbers in which only and all the five digits $1,3,5,7$ and 9 is ${ }^{5} \mathrm{C}_{1} \times \frac{6 !}{2 !}$...
Read More →The total number of matrices
Question: The total number of matrices $A=\left(\begin{array}{ccc}0 2 y 1 \\ 2 x y -1 \\ 2 x -y 1\end{array}\right),(x, y \in R, x \neq y)$ for which $\mathrm{A}^{\mathrm{T}} \mathrm{A}=3 \mathrm{I}_{3}$ is :- 6234Correct Option: , 4 Solution:...
Read More →If y=y(x) is the solution of the differential
Question: If $y=y(x)$ is the solution of the differential equation, $e^{y}\left(\frac{d y}{d x}-1\right)=e^{x}$ such that $y(0)=0$, then $\mathrm{y}(1)$ is equal to :$2+\log _{e} 2$$2 \mathrm{e}$$\log _{e} 2$$1+\log _{\mathrm{e}} 2$Correct Option: , 4 Solution: $e^{y} \frac{d y}{d x}-e^{y}=e^{x}$, Let $e^{y}=t$ $\Rightarrow \mathrm{e}^{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dt}}{\mathrm{dx}}$ $\frac{d t}{d x}-t=e^{x}$ I.F. $=e^{\int-d x}=e^{-x}$ $t e^{-x}=x+c \Rightarrow e^{y-...
Read More →Solve this following
Question: The vertices $\mathrm{B}$ and $\mathrm{C}$ of a $\triangle \mathrm{ABC}$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $B C=5$ units. Then the area (in sq. units) of this triangle, given that the point $\mathrm{A}(1,-1,2)$, is :- $2 \sqrt{34}$$\sqrt{34}$6$5 \sqrt{17}$Correct Option: , 2 Solution:...
Read More →Two poles standing on a horizontal ground are of heights
Question: Two poles standing on a horizontal ground are of heights $5 \mathrm{~m}$ and $10 \mathrm{~m}$ respectively. The line joining their tops makes an angle of $15^{\circ}$ with ground. Then the distance (in m) between the poles, is :- $\frac{5}{2}(2+\sqrt{3})$$5(\sqrt{3}+1)$$5(2+\sqrt{3})$$10(\sqrt{3}-1)$Correct Option: , 3 Solution:...
Read More →A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is
Question: A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan ^{-1}\left(\frac{1}{2}\right) .$ Water is poured into it at a constant rage of 5 cubic meter per minute. The the rate (in $\mathrm{m} / \mathrm{min}$.), at which the level of water is rising at the instant when the depth of water in the tank is $10 \mathrm{~m}$; is :- $2 / \pi$$1 / 5 \pi$$1 / 10 \pi$$1 / 15 \pi$Correct Option: , 2 Solution:...
Read More →Prove the following
Question: Let $x^{k}+y^{k}=a^{k},(a, K0)$ and $\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$, then $\mathrm{k}$ is :$\frac{3}{2}$$\frac{1}{3}$$\frac{2}{3}$$\frac{4}{3}$Correct Option: , 3 Solution: $x^{k}+y^{k}=a^{k}(a, k0)$ $k x^{k-1}+k y^{k-1} \frac{d y}{d x}=0$ $\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{\mathrm{x}}{\mathrm{y}}\right)^{\mathrm{k}-1}=0 \Rightarrow \mathrm{k}-1=-\frac{1}{3} \Rightarrow \mathrm{k}=2 / 3$...
Read More →The area of the region,
Question: The area of the region, enclosed by the circle $x^{2}+y^{2}=2$ which is not common to the region bounded by the parabola $\mathrm{y}^{2}=\mathrm{x}$ and the straight line $\mathrm{y}=\mathrm{x}$, is :$\frac{1}{3}(12 \pi-1)$$\frac{1}{6}(12 \pi-1)$$\frac{1}{6}(24 \pi-1)$$\frac{1}{3}(6 \pi-1)$Correct Option: , 2 Solution:...
Read More →Solve this following
Question: If the two lines $\mathrm{x}+(\mathrm{a}-1) \mathrm{y}=1$ and $2 x+a^{2} y=1(a \in R-\{0,1\})$ are perpendicular, then the distance of their point of intersection from the origin is :-$\frac{2}{5}$$\frac{2}{\sqrt{5}}$$\frac{\sqrt{2}}{5}$$\sqrt{\frac{2}{5}}$Correct Option: , 4 Solution:...
Read More →An unbiased coin is tossed 5 times.
Question: An unbiased coin is tossed 5 times. Suppose that a variable $\mathrm{X}$ is assigned the value $k$ when $k$ consecutive heads are obtained for $\mathrm{k}=3,4,5$ otherwise $X$ takes the value $-1$. Then the expected value of $X$, is :$\frac{3}{16}$$-\frac{3}{16}$$\frac{1}{8}$$-\frac{1}{8}$Correct Option: , 3 Solution:...
Read More →If the distance between the foci of an ellipse is 6 and
Question: If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12 , then the length of its latus rectum is :$\sqrt{3}$$2 \sqrt{3}$$3 \sqrt{2}$$\frac{3}{\sqrt{2}}$Correct Option: , 3 Solution: Given $2 \mathrm{ae}=6 \Rightarrow \mathrm{ae}=3$ ................(1) and $\frac{2 a}{e}=12 \Rightarrow a=6 e ...............(2) from (1) and (2) $6 \mathrm{e}^{2}=3 \Rightarrow \mathrm{e}=\frac{1}{\sqrt{2}}$ $\Rightarrow a=3 \sqrt{2}$ Now, $b^{2}=a^{2}\left(1-e^{2...
Read More →If y=m x+4 is a tangent to both the parabolas,
Question: If $y=m x+4$ is a tangent to both the parabolas, $y^{2}=4 x$ and $x^{2}=2 b y$, then $b$ is equal to :128$-64$$-128$$-32$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: If $\cos x \frac{d y}{d x}-y \sin x=6 x,\left(0x\frac{\pi}{2}\right)$ and $\mathrm{y}\left(\frac{\pi}{3}\right)=0$, then $\mathrm{y}\left(\frac{\pi}{6}\right)$ is equal to :-$-\frac{\pi^{2}}{4 \sqrt{3}}$$-\frac{\pi^{2}}{2}$$-\frac{\pi^{2}}{2 \sqrt{3}}$$\frac{\pi^{2}}{2 \sqrt{3}}$Correct Option: , 3 Solution:...
Read More →Let α be a root of the equation
Question: Let $\alpha$ be a root of the equation $x^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1 1 1 \\ 1 \alpha \alpha^{2} \\ 1 \alpha^{2} \alpha^{4}\end{array}\right]$, then the matrix $A^{31}$ is equal to:$\mathrm{A}^{3}$$\mathrm{A}$$\mathrm{A}^{2}$$\mathrm{I}_{3}$Correct Option: 1 Solution:...
Read More →Solve the following
Question: If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)$ then $\frac{d y}{d \alpha}$ at $\alpha=\frac{5 \pi}{6}$ is :4$-\frac{1}{4}$$\frac{4}{3}$$-4$Correct Option: 1 Solution:...
Read More →If some three consecutive in the binomial expansion of
Question: If some three consecutive in the binomial expansion of $(x+1)^{n}$ is powers of $x$ are in the ratio $2: 15: 70$, then the average of these three coefficient is :-964625227232Correct Option: , 4 Solution:...
Read More →Solve this following
Question: Let $z \in C$ be such that $|z|1$. If $\omega=\frac{5+3 z}{5(1-z)}$, then:-$5 \operatorname{Im}(\omega)1$$4 \operatorname{Im}(\omega)5$$5 \operatorname{Re}(\omega)1$$5 \operatorname{Re}(\omega)4$Correct Option: , 3 Solution:...
Read More →Five numbers are in A.P., whose sum is 25 and product is 2520 .
Question: Five numbers are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers is $-\frac{1}{2}$, then the grea est number amongst them is :$\frac{21}{2}$27167Correct Option: , 3 Solution:...
Read More →Solve this following
Question: The value of $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}$ is :-$\frac{1}{36}$$\frac{1}{32}$$\frac{1}{18}$$\frac{1}{16}$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: The value of the integral $\int_{0}^{1} x \cot ^{-1}\left(1-x^{2}+x^{4}\right) d x$ is :-$\frac{\pi}{4}-\frac{1}{2} \log _{\mathrm{e}} 2$$\frac{\pi}{2}-\log _{\mathrm{e}} 2$$\frac{\pi}{2}-\frac{1}{2} \log _{\mathrm{e}} 2$$\frac{\pi}{4}-\log _{\mathrm{e}} 2$Correct Option: 1 Solution:...
Read More →The value of
Question: If $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)=1$, where $z=x+i y$, then the point $(x, y)$ lies on a :circle whose centre is at $\left(-\frac{1}{2},-\frac{3}{2}\right)$circle whose diameter is $\frac{\sqrt{5}}{2}$straight line whose slope is $\frac{3}{2}$straight line whose slope is $-\frac{2}{3}$Correct Option: , 2 Solution:...
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