The distantce of the point
Question: The distantce of the point $(1,3,-7)$ from the plane passing through the point $(1,-1,-1)$, having normal perpendicular to both the lines $\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$, is :-$\frac{10}{\sqrt{74}}$$\frac{20}{\sqrt{74}}$$\frac{10}{\sqrt{83}}$$\frac{5}{\sqrt{83}}$Correct Option: 1 Solution:...
Read More →Let y(x) be the solution of the differential equation
Question: Let $y(x)$ be the solution of the differential equation $(x \log x) \frac{d y}{d x}+y=2 x \log x,(x \geq 1)$. Then $y(e)$ is equal to:2$2 \mathrm{e}$$\mathrm{e}$0Correct Option: 1 Solution:...
Read More →Solve this following
Question: If $\mathrm{A}=\left[\begin{array}{ccc}1 2 2 \\ 2 1 -2 \\ \mathrm{a} 2 \mathrm{~b}\end{array}\right]$ is a matrix satisfying the equation $\mathrm{AA}^{\mathrm{T}}=9 \mathrm{I}$, where $\mathrm{I}$ is $3 \times 3$ identity matrix, then the ordered pair $(\mathrm{a}, \mathrm{b})$ is equal to:$(2,1)$$(-2,-1)$$(2,-1)$$(-2,1)$Correct Option: , 2 Solution:...
Read More →If the image of the point
Question: If the image of the point $\mathrm{P}(1,-2,3)$ in the plane, $2 \mathrm{x}+3 \mathrm{y}-4 \mathrm{z}+22=0$ measured parallel to line, $\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{4}=\frac{\mathrm{z}}{5}$ is $\mathrm{Q}$, then $\mathrm{PQ}$ is equal to :-$6 \sqrt{5}$$3 \sqrt{5}$$2 \sqrt{42}$$\sqrt{42}$Correct Option: , 3 Solution:...
Read More →Let the population of rabbits surviving at
Question: Let the population of rabbits surviving at a time t be governed by the differential equation $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}=\frac{1}{2} \mathrm{p}(\mathrm{t})-200$. If $\mathrm{p}(0)=100$, then $\mathrm{p}(\mathrm{t})$ equals :$400-300 \mathrm{e}^{\mathrm{t} / 2}$$300-200 \mathrm{e}^{-1 / 2}$$600-500 \mathrm{e}^{\mathrm{t} / 2}$$400-300 \mathrm{e}^{-1 / 2}$Correct Option: 1 Solution:...
Read More →If the line,
Question: If the line, $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}+2}{-1}=\frac{\mathrm{z}+4}{3}$ lies in the plane, $\mathrm{xx}+\mathrm{my}-\mathrm{z}=9$, then $\mathrm{l}^{2}+\mathrm{m}^{2}$ is equal to :-226185Correct Option: 1 Solution:...
Read More →The equation of the curve passing through
Question: The equation of the curve passing through the origin and satisfying the differential equation $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=4 x^{2}$ is :$\left(1+x^{2}\right) y=x^{3}$$3\left(1+x^{2}\right) y=4 x^{3}$$3\left(1+x^{2}\right) y=2 x^{3}$$\left(1+x^{2}\right) y=3 x^{3}$Correct Option: , 2 Solution:...
Read More →Solve this following
Question: If $\mathrm{A}$ is an $3 \times 3$ non-singular matrix such that $\mathrm{AA}^{\prime}=\mathrm{A}^{\prime} \mathrm{A}$ and $\mathrm{B}=\mathrm{A}^{-1} \mathrm{~A}^{\prime}$, the $\mathrm{BB}^{\prime}$ equals :$\mathrm{I}+\mathrm{B}$I$\mathrm{B}^{-1}$$\left(\mathrm{B}^{-1}\right)^{\prime}$Correct Option: , 2 Solution:...
Read More →If a curve passes through the point
Question: If a curve passes through the point $\left(2, \frac{7}{2}\right)$ and has slope $\left(1-\frac{1}{x^{2}}\right)$ at any point $(x, y)$ on it, then the ordinate of the point on the curve whose abscissa is $-2$ is :$-\frac{5}{2}$$\frac{5}{2}$$-\frac{3}{2}$$\frac{3}{2}$Correct Option: , 3 Solution:...
Read More →The distance of the point (1,-5,9) from the plane
Question: The distance of the point $(1,-5,9)$ from the plane $x-y+z=5$ measured along the line $\mathrm{x}=\mathrm{y}=\mathrm{z}$ is :$\frac{20}{3}$$3 \sqrt{10}$$10 \sqrt{3}$$\frac{10}{\sqrt{3}}$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: If $P=\left[\begin{array}{lll}1 \alpha 3 \\ 1 3 3 \\ 2 4 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to41150Correct Option: , 2 Solution:...
Read More →Consider the differential equation
Question: Consider the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}^{3}}{2\left(\mathrm{xy}^{2}-\mathrm{x}^{2}\right)}$ : Statement 1 : The substitution $\mathrm{z}=\mathrm{y}^{2}$ transforms the above equation into a first order homogenous differential equation. Statement 2 : The solution of this differential equation is $y^{2} e^{-\frac{y^{2}}{x}}=C$.Statement 1 is false and statement 2 is true.Both statements are true.Statement 1 is true and statement 2 is false.Bot...
Read More →Solve this following
Question: Let $\mathrm{A}=\left(\begin{array}{lll}1 0 0 \\ 2 1 0 \\ 3 2 1\end{array}\right)$. If $\mathrm{u}_{1}$ and $\mathrm{u}_{2}$ are column matrices such that $\mathrm{Au}_{1}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ and $A u_{2}=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$, then $u_{1}+u_{2}$ is equal to : $\left(\begin{array}{c}1 \\ -1 \\ -1\end{array}\right)$$\left(\begin{array}{c}-1 \\ 1 \\ 0\end{array}\right)$$\left(\begin{array}{c}-1 \\ 1 \\ -1\end{array}\right)$$...
Read More →If the surface area of a sphere of radius
Question: If the surface area of a sphere of radius $\mathrm{r}$ is increasing uniformly at the rate $8 \mathrm{~cm}^{2} / \mathrm{s}$, then the rate of change of its volume is:proportional to $\mathrm{r}^{2}$constantproportional to $\mathrm{r}$proportional to $\sqrt{r}$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: Statement-1 : Determinant of a skew-symmetric matrix of order 3 is zero. Statement-1 : For any matrix $A$, $\operatorname{det}\left(A^{T}\right)=\operatorname{det}(A)$ and $\operatorname{det}(-A)=-\operatorname{det}(A)$. Where $\operatorname{det}(B)$ denotes the determinant of matrix $B$. Then :Statement- 1 is true and statement- 2 is falseBoth statements are trueBoth statements are falseStatement- 1 is false and statement- 2 is true.Correct Option: 1 Solution:...
Read More →The distance of the point (1,0,2) from the point
Question: The distance of the point $(1,0,2)$ from the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane $x-y+z=16$, is :$3 \sqrt{21}$13$2 \sqrt{14}$8Correct Option: , 2 Solution:...
Read More →At present a firm is manufacturing 2000 items.
Question: At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{\mathrm{dP}}{\mathrm{dx}}=100-12 \sqrt{\mathrm{x}}$. If the firm employs 25 more workers, then the new level of production of items is :2500300035004500Correct Option: , 3 Solution:...
Read More →Solve this following
Question: Let $A$ and $B$ be two symmetric matrices of order 3 . Statement-1 : A(BA) and (AB)A are symmetric matrices. Statement- $2: \mathrm{AB}$ is symmetric matrix if matrix multiplication of A with B is commutative.Statement- 1 is true, Statement- 2 is false.Statement-1 is false, Statement- 2 is trueStatement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statemen...
Read More →The population
Question: The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}$ $=0.5 \mathrm{p}(\mathrm{t})-450$. If $\mathrm{p}(0)=850$, then the time at which the population becomes zero is :$\ln 18$$2 \ln 18$$\ln 9$$\frac{1}{2} \ln 18$Correct Option: , 2 Solution:...
Read More →The equation of the plane containing the line
Question: The equation of the plane containing the line $2 x-5 y+z=3 ; x+y+4 z=5$, and parallel to the plane, $x+3 y+6 z=1$, is :$x+3 y+6 z=7$$2 x+6 y+12 z=-13$$2 x+6 y+12 z=13$$x+3 y+6 z=-7$Correct Option: 1 Solution:...
Read More →Consider the differential equation
Question: Consider the differential equation $\mathrm{y}^{2} \mathrm{dx}+\left(\mathrm{x}-\frac{1}{\mathrm{y}}\right) \mathrm{dy}=0$. It $\mathrm{y}(1)=1$, then $\mathrm{x}$ is given by :$1-\frac{1}{y}+\frac{\frac{1}{e^{y}}}{e}$$4-\frac{2}{y}-\frac{\frac{1}{e^{y}}}{e}$$3-\frac{1}{y}+\frac{\frac{1}{e^{y}}}{e}$$1+\frac{1}{y}-\frac{\frac{1}{e^{y}}}{e}$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: Let A be a $2 \times 2$ matrix with non-zero entries and let $\mathrm{A}^{2}=\mathrm{I}$, where $\mathrm{I}$ is $2 \times 2$ identity matrix. Define $\operatorname{Tr}(\mathrm{A})=$ sum of diagonal elements of $\mathrm{A}$ and $|\mathrm{A}|=$ determinant of matrix $\mathrm{A}$. Statement-1: $\operatorname{Tr}(\mathrm{A})=0$. Statement-2: $|\mathrm{A}|=1$.Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.Statement- 1 is true, Statement $-2$ i...
Read More →The image of the line
Question: The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane $2 x-y+z+3=0$ is the line:$\frac{\mathrm{x}+3}{3}=\frac{\mathrm{y}-5}{1}=\frac{\mathrm{z}-2}{-5}$$\frac{x+3}{-3}=\frac{y-5}{-1}=\frac{z+2}{5}$$\frac{x-3}{3}=\frac{y+5}{1}=\frac{z-2}{-5}$$\frac{x-3}{-3}=\frac{y+5}{-1}=\frac{z-2}{5}$Correct Option: 1 Solution:...
Read More →The curve that passes through the point (2, 3),
Question: The curve that passes through the point (2, 3), and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact, is given by :(1) $\left(\frac{x}{2}\right)^{2}+\left(\frac{y}{3}\right)^{2}=2$ (2) (3) (4)$2 y-3 x=0$$y=\frac{6}{x}$$x^{2}+y^{2}=13$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: The number of $3 \times 3$ non-singular matrices, with four entries as 1 and all other entries as 0 , isLess than 456At least 7Correct Option: , 4 Solution:...
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