Let I be the purchase value of an equipment and
Question: Let I be the purchase value of an equipment and $\mathrm{V}(\mathrm{t})$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by differential equation $\frac{\mathrm{dV}(\mathrm{t})}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-\mathrm{t})$, where $\mathrm{k}0$ is a constant and $\mathrm{T}$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is :-$I-\frac{k(T-t)^{2}}{2}$$e^{-k T}$$T^{2}-\frac{I}{k}$$\mathrm{I}-\fr...
Read More →Prove the following
Question: If $\frac{d y}{d x}=y+30$ and $y(0)=2$, then $y(\ln 2)$ is equal to :-13-275Correct Option: , 3 Solution:...
Read More →The angle between the lines whose direction cosines satisfy the equations
Question: The angle between the lines whose direction cosines satisfy the equations $\ell+\mathrm{m}+\mathrm{n}=0$ and $\ell^{2}=\mathrm{m}^{2}+\mathrm{n}^{2}$ is :$\frac{\pi}{3}$$\frac{\pi}{4}$$\frac{\pi}{6}$$\frac{\pi}{2}$Correct Option: 1 Solution:...
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Question: Let $A$ be a $2 \times 2$ matrix Statement-1 : adj $(\operatorname{adj} A)=A$ Statement-2 : $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|$Statement $-1$ is true, Statement $-2$ is false.Statement-1 is false, Statement-2 is true.Statement-1 is true, Statement $-2$ is true;Statement $-2$ is a correct explanation for Statement-1.Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.Correct Option: , 4 Solution:...
Read More →Solution of the differential equation
Question: Solution of the differential equation $\cos x d y=y(\sin x-y) d x, 0x\frac{\pi}{2}$ is :$\sec \mathrm{x}=(\tan \mathrm{x}+\mathrm{c}) \mathrm{y}$$y \sec x=\tan x+c$$y \tan x=\sec x+c$$\tan x=(\sec x+c) y$Correct Option: 1 Solution:...
Read More →Let ABC be a triangle with vertices at points
Question: Let $\mathrm{ABC}$ be a triangle with vertices at points $\mathrm{A}(2,3,5), \mathrm{B}(-1,3,2)$ and $\mathrm{C}(\lambda, 5, \mu)$ in three dimensional space. If the median through $\mathrm{A}$ is equally inclined with the axes, then $(\lambda, \mu$.) is equal to:(10,7)$$(7.5)$$(7,10)$$(5,7)$Correct Option: , 3 Solution:...
Read More →The equation of a plane through the line of intersection
Question: The equation of a plane through the line of intersection of the planes $x+2 y=3, y-2 z+1=0$, and perpendicular to the first plane is :$2 x-y+7 z=11$$2 \mathrm{x}-\mathrm{y}+10 \mathrm{z}=11$$2 x-y-9 z=10$$2 \mathrm{x}-\mathrm{y}-10 \mathrm{z}=9$Correct Option: , 2 Solution:...
Read More →The differential equation which represents the family of curves
Question: The differential equation which represents the family of curves $y=c_{1} e^{c_{2} x}$, where $c_{1}$ and $c_{2}$ are arbitrary constants, is :-$y^{\prime \prime}=y^{\prime}$$y y^{\prime \prime}=\left(y^{\prime}\right)^{2}$$y^{\prime}=y^{2}$$y^{\prime \prime}=y^{\prime} y$Correct Option: , 2 Solution:...
Read More →The integral
Question: The integral $\int \frac{\sin ^{2} x \cos ^{2} x}{\left(\sin ^{5} x+\cos ^{3} x \sin ^{2} x+\sin ^{3} x \cos ^{2} x+\cos ^{5} x\right)^{2}} d x$ is equal to (where C is a constant of integration)$\frac{-1}{3\left(1+\tan ^{3} x\right)}+C$$\frac{1}{1+\cot ^{3} x}+C$$\frac{-1}{1+\cot ^{3} x}+C$$\frac{1}{3\left(1+\tan ^{3} x\right)}+C$Correct Option: 1 Solution:...
Read More →If two lines L_1 and L_{2 in space, are defined by
Question: If two lines $L_{1}$ and $L_{2}$ in space, are defined by $\mathrm{L}_{1}=\{\mathrm{x}=\sqrt{\lambda} \mathrm{y}+(\sqrt{\lambda}-1)$ $\mathrm{z}=(\sqrt{\lambda}-1) \mathrm{y}+\sqrt{\lambda}\}$ and $\mathrm{L}_{2}=\{\mathrm{x}=\sqrt{\mu} \mathrm{y}+(1-\sqrt{\mu})$ $z=(1-\sqrt{\mu}) y+\sqrt{\mu}\}$ then $L_{1}$ is perpendicular to $L$, for all non-negative reals $\lambda$ and $\mu$, such that :$\lambda=\mu$$\lambda \neq \mu$$\sqrt{\lambda}+\sqrt{\mu}=1$$\lambda+\mu=0$Correct Option: 1, 4...
Read More →Solve the equation
Question: Let $I_{n}=\int \tan ^{n} x d x,(n1) . I_{4}+I_{6}=a \tan ^{5} x+b x^{5}+C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :-$\left(-\frac{1}{5}, 0\right)$$\left(-\frac{1}{5}, 1\right)$$\left(\frac{1}{5}, 0\right)$$\left(\frac{1}{5},-1\right)$Correct Option: , 3 Solution:...
Read More →Solve the equation
Question: Let $I_{n}=\int \tan ^{n} x d x,(n1) . I_{4}+I_{6}=a \tan ^{5} x+b x^{5}+C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :-$\left(-\frac{1}{5}, 0\right)$$\left(-\frac{1}{5}, 1\right)$$\left(\frac{1}{5}, 0\right)$$\left(\frac{1}{5},-1\right)$Correct Option: Solution:...
Read More →The integral
Question: The integral $\int \frac{2 x^{12}+5 x^{9}}{\left(x^{5}+x^{3}+1\right)^{3}} d x$ is equal to :-$\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$$\frac{-x^{5}}{\left(x^{5}+x^{3}+1\right)^{2}}+C$$\frac{x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$$\frac{x^{5}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: If $x^{2}+y^{2}+\sin y=4$, then the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(-2,0)$ is $-34$$-32$$-2$4Correct Option: 1 Solution: Solution not required...
Read More →If the projections of a line segment on thex,
Question: If the projections of a line segment on thex, $y$ and $z$-axes in 3-dimensional space are 2,3 and 6 respectively, then the length ofthe line segment is :79126Correct Option: 1 Solution:...
Read More →The integral
Question: The integral $\int \frac{\mathrm{dx}}{\mathrm{x}^{2}\left(\mathrm{x}^{4}+1\right)^{\frac{3}{4}}}$ equals :$-\left(x^{4}+1\right)^{\frac{1}{4}}+c$$-\left(\frac{x^{4}+1}{x^{4}}\right)^{\frac{1}{4}}+c$$\left(\frac{x^{4}+1}{x^{4}}\right)^{\frac{1}{4}}+c$$\left(x^{4}+1\right)^{\frac{1}{4}}+c$Correct Option: Solution:...
Read More →Solve this following
Question: If $f(x)=\left|\begin{array}{ccc}\cos x x 1 \\ 2 \sin x x^{2} 2 x \\ \tan x x 1\end{array}\right|$, then $\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{x}$ exists and is equal to 0exists and is equal to $-2$exists and is equal to 2does not existCorrect Option: , 2 Solution: Solution not required...
Read More →The integral
Question: The integral $\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x$ is equal to :$(x-1) e^{x+\frac{1}{x}}+c$$x e^{x+\frac{1}{x}}+c$$(x+1) e^{x+\frac{1}{x}}+c$$-x e^{x+\frac{1}{x}}+c$Correct Option: Solution:...
Read More →Solve this following
Question: If $f(x)=\sin ^{-1}\left(\frac{2 \times 3^{x}}{1+9^{x}}\right)$, then $f^{\prime}\left(-\frac{1}{2}\right)$ equals : $\sqrt{3} \log _{e} \sqrt{3}$$-\sqrt{3} \log _{e} 3$$-\sqrt{3} \log _{e} \sqrt{3}$$\sqrt{3} \log _{\mathrm{c}} 3$Correct Option: 1 Solution: Solution not required...
Read More →Let Q be the foot of perpendicular from the origin to the plane
Question: Let $Q$ be the foot of perpendicular from the origin to the plane $4 x-3 y+z+13=0$ and $R$ be a point $(-1,1,-6)$ on the plane. Then length $Q R$ is :$3 \sqrt{\frac{7}{2}}$$\sqrt{14}$$\sqrt{\frac{19}{2}}$$\frac{3}{\sqrt{2}}$Correct Option: 1 Solution:...
Read More →Solve the equation
Question: If $\int f(\mathrm{x}) \mathrm{dx}=\Psi(\mathrm{x})$, then $\int \mathrm{x}^{5} f\left(\mathrm{x}^{3}\right) \mathrm{dx}$ is equal to :$\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x\right]+C$$\frac{1}{3} \mathrm{x}^{3} \Psi\left(\mathrm{x}^{3}\right)-3 \int \mathrm{x}^{3} \Psi\left(\mathrm{x}^{3}\right) \mathrm{dx}+\mathrm{C}$$\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x+C$$\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\ri...
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Question: If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}}(|t| \geq 1)$, then $\frac{d y}{d x}$ is equal to : $-\frac{y}{x}$$\frac{9}{1+9 x^{3}}$$-\frac{x}{y}$$\frac{y}{x}$Correct Option: 1 Solution: Solution not required...
Read More →The acute angle between two lines such that the direction
Question: The acute angle between two lines such that the direction cosines $\ell, \mathrm{m}, \mathrm{n}$ of each of them satisfy the equations $\ell+\mathrm{m}+\mathrm{n}=0$ and $\ell^{2}+\mathrm{m}^{2}-\mathrm{n}^{2}=0$ is :-$30^{\circ}$$45^{\circ}$$60^{\circ}$$15^{\circ}$Correct Option: 3, Solution:...
Read More →Solve this following Question
Question: A vector $\overrightarrow{\mathrm{n}}$ is inclined to $\mathrm{x}$-axis at $45^{\circ}$, to $\mathrm{y}$-axis at $60^{\circ}$ and at an acute angle to $\mathrm{z}$-axis. If $\overrightarrow{\mathrm{n}}$ is a normal to a plane passing through the point $(\sqrt{2},-1,1)$, then the equation of the plane is :$\sqrt{2} \mathrm{x}-\mathrm{y}-\mathrm{z}=2$$\sqrt{2} \mathrm{x}+\mathrm{y}+\mathrm{z}=2$$3 \sqrt{2} x-4 y-3 z=7$$4 \sqrt{2} \mathrm{x}+7 \mathrm{y}+\mathrm{z}=2$Correct Option: , 2 S...
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Question: Let $g(x)=\cos x^{2}, f(x)=\sqrt{x}$ and $\alpha, \beta(\alpha\beta)$ be the roots of the quadratic equation $18 x^{2}-9 \pi x+\pi^{2}=0$. Then the area (in sq. units) bounded by the curve $y=(g o f)(x)$ and the lines $x=\alpha, x=\beta$ and $y=0$ is-$\frac{1}{2}(\sqrt{3}+1)$$\frac{1}{2}(\sqrt{3}-\sqrt{2})$$\frac{1}{2}(\sqrt{2}-1)$$\frac{1}{2}(\sqrt{3}-1)$Correct Option: , 4 Solution:...
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