The locus of a point which divides the line segment joining the point
Question: The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $\mathrm{x}^{2}=4 \mathrm{y}$, internally in the ratio $1: 2$, is-$9 x^{2}-3 y=2$$9 x^{2}-12 y=8$$x^{2}-3 y=2$$4 x^{2}-3 y=2$Correct Option: , 2 Solution:...
Read More →Which one of the following is a tautology ?
Question: Which one of the following is a tautology ? $P \wedge(P \vee Q)$$P \vee(P \wedge Q)$$\mathrm{Q} \rightarrow(\mathrm{P} \wedge(\mathrm{P} \rightarrow \mathrm{Q}))$$(\mathrm{P} \wedge(\mathrm{P} \rightarrow \mathrm{Q})) \rightarrow \mathrm{Q}$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: For a 0 , let the curves $C_{1}: y^{2}=a x$ and $C_{2}: x^{2}=$ ay intersect at origin $O$ and a point $P$. Let the line $\mathrm{x}=\mathrm{b}(0\mathrm{b}\mathrm{a})$ intersect the chord OP and the $x$-axis at points $Q$ and $R$, respectively. If the line $x=b$ bisects the area bounded by the curves, $C_{1}$ and $C_{2}$, and the area of $\triangle \mathrm{OQR}=\frac{1}{2}$, then 'a' satisfies the equation$x^{6}-12 x^{3}+4=0$$x^{6}-12 x^{3}-4=0$$x^{6}+6 x^{3}-4=0$$x^{6}-6 x^{3}+4=0$Cor...
Read More →Solve this following
Question: If the equation, $x^{2}+b x+45=0(b \in R)$ has conjugate complex roots and they satisfy $|z+1|=2 \sqrt{10}$, then$b^{2}-b=42$$b^{2}+b=12$$b^{2}+b=72$$b^{2}-b=30$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: Let $y=y(x)$ be a solution of the differential equation, $\sqrt{1-x^{2}} \frac{d y}{d x}+\sqrt{1-y^{2}}=0,|x|1$. If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to $-\frac{\sqrt{3}}{2}$$\frac{1}{\sqrt{2}}$$\frac{\sqrt{3}}{2}$$-\frac{1}{\sqrt{2}}$Correct Option: , 2 Solution:...
Read More →The mean and the standard deviation
Question: The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 resepectively. Each of these 10 observations is multiplied by $\mathrm{p}$ and then reduced by $\mathrm{q}$, where $\mathrm{p} \neq 0$ and $\mathrm{q} \neq 0$. If the new mean and new s.d. become half of their original values, then $\mathrm{q}$ is equal to$-20$10$-10$$-5$Correct Option: 1 Solution:...
Read More →Solve this following
Question: Let two points be $\mathrm{A}(1,-1)$ and $\mathrm{B}(0,2)$. If a point $\mathrm{P}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)$ be such that the area of $\triangle \mathrm{PAB}=5 \mathrm{sq}$. units and it lies on the line, $3 x+y-4 \lambda=0$, then a value of $\lambda$ is143$-3$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: $\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}$ is equal to$\frac{1}{\mathrm{e}}$$e^{2}$e$\frac{1}{\mathrm{e}^{2}}$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: Let $f(x)=\left(\sin \left(\tan ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right)^{2}-1,|x|1$. If $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1}(f(\mathrm{x}))\right)$ and $\mathrm{y}(\sqrt{3})=\frac{\pi}{6}$, then $\mathrm{y}(-\sqrt{3})$ is equal to$\frac{5 \pi}{6}$$-\frac{\pi}{6}$$\frac{\pi}{3}$$\frac{2 \pi}{3}$Correct Option: 1 Solution:...
Read More →Solve this following
Question: If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are the greatest value of ${ }^{19} \mathrm{C}_{\mathrm{p}},{ }^{20} \mathrm{C}_{\mathrm{q}}$ and ${ }^{21} C_{r}$ respectively, then$\frac{a}{11}=\frac{b}{22}=\frac{c}{21}$$\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{21}$$\frac{a}{10}=\frac{b}{11}=\frac{c}{42}$$\frac{a}{11}=\frac{b}{22}=\frac{c}{42}$Correct Option: , 4 Solution:...
Read More →Solve this following
Question: Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be $1 \mathrm{cu}$. unit. If $\theta$ be the angle between the edges $\overrightarrow{\mathrm{u}}$ and $\overrightarrow{\mathrm{w}}$, then $\cos \theta$ can be$...
Read More →Solve this following
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be such that for all $x \in R\left(2^{1+x}+2^{1-x}\right), f(x)$ and $\left(3^{x}+3^{-x}\right)$ are in A.P., then the minimum value of $f(x)$ is 0324Correct Option: , 2 Solution: $f(x)=\frac{2\left(2^{x}+2^{-x}\right)+\left(3^{x}+3^{-x}\right)}{2} \geq 3$ (A.M $\geq$ G.M)...
Read More →Solve this following
Question: Let the line $y=m x$ and the ellipse $2 x^{2}+y^{2}=1$ intersect at a ponit $\mathrm{P}$ in the first quadrant. If the normal to this ellipse at $\mathrm{P}$ meets the co-ordinate axes at $\left(-\frac{1}{3 \sqrt{2}}, 0\right)$ and $(0, \beta)$, then $\beta$ is equal to$\frac{2}{\sqrt{3}}$$\frac{2 \sqrt{2}}{3}$$\frac{2}{3}$$\frac{\sqrt{2}}{3}$Correct Option: , 4 Solution:...
Read More →The projection of the line segment joining
Question: The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10$ ) is________. Solution:...
Read More →If the vectors,
Question: If the vectors, $\vec{p}=(a+1) \hat{i}+a \hat{j}+a \hat{k}$, $\overrightarrow{\mathrm{q}}=a \hat{\mathrm{i}}+(\mathrm{a}+1) \hat{\mathrm{j}}+\mathrm{a} \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{r}}=a \hat{\mathrm{i}}+\mathrm{aj}+(\mathrm{a}+1) \hat{\mathrm{k}} \quad(\mathrm{a} \in \mathrm{R}) \quad$ are coplanar and $3(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{q}})^{2}-\lambda|\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{q}}|^{2}=0$, then the value of...
Read More →Prove the following
Question: If for $x \geq 0, y=y(x)$ is the solution of the differential equation $(x+1) d y=\left((x+1)^{2}+y-3\right) d x, y(2)=0$ then $y(3)$ is equal to______. Solution:...
Read More →The number of distinct solutions of the equation
Question: The number of distinct solutions of the equation $\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|$ in the interval $[0,2 \pi]$, is___________. Solution:...
Read More →The coefficient of
Question: The coefficient of $x^{4}$ is the expansion of $\left(1+x+x^{2}\right)^{10}$ is___________. Solution:...
Read More →If for all real triplets
Question: If for all real triplets $(a, b, c), f(x)=a+b x+c x^{2}$ then $\int_{0}^{1} f(x) d x$ is equal to :$\frac{1}{2}\left\{f(1)+3 f\left(\frac{1}{2}\right)\right\}$$2\left\{3 f(1)+2 f\left(\frac{1}{2}\right)\right\}$$\frac{1}{6}\left\{f(0)+f(1)+4 f\left(\frac{1}{2}\right)\right\}$$\frac{1}{3}\left\{f(0)+f\left(\frac{1}{2}\right)\right\}$Correct Option: , 3 Solution:...
Read More →Consider the following reactions
Question: Consider the following reactions 2-butene The mass percentage of carbon in $\mathrm{A}$ is Solution:...
Read More →The value of
Question: The value of $\cos ^{3}\left(\frac{\pi}{8}\right) \cdot \cos \left(\frac{3 \pi}{8}\right)+\sin ^{3}\left(\frac{\pi}{8}\right) \cdot \sin \left(\frac{3 \pi}{8}\right)$ is:$\frac{1}{4}$$\frac{1}{\sqrt{2}}$$\frac{1}{2 \sqrt{2}}$$\frac{1}{2}$Correct Option: , 3 Solution:...
Read More →Prove the following
Question: If $f(x)= \begin{cases}\frac{\sin (a+2) x+\sin x}{x} ; x0 \\ b ; x=0 \\ \frac{\left(x+3 x^{2}\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}} ; x0\end{cases}$ is continuous at $x=0$, then $a+2 b$ is equal to :$-1$1$-2$0Correct Option: , 4 Solution:...
Read More →The sum of the total number of bonds between chromium and oxygen atoms in chromate and dichromate ions is
Question: The sum of the total number of bonds between chromium and oxygen atoms in chromate and dichromate ions is Solution: separately....
Read More →Let C be the centroid of the triangle
Question: Let $\mathrm{C}$ be the centroid of the triangle with vertices $(3,-1),(1,3)$ and $(2,4)$. Let $P$ be the point of intersection of the lines $x+3 y-1=0$ and $3 x-y+1=0$. Then the line passing through the points $\mathrm{C}$ and $\mathrm{P}$ also passes through the point :$(7,6)$$(-9,-6)$$(-9,-7)$$(9,7)$Correct Option: , 2 Solution:...
Read More →A sample of milk splits after
Question: A sample of milk splits after $60 \mathrm{~min}$. at $300 \mathrm{~K}$ and after $40 \mathrm{~min}$. at $400 \mathrm{~K}$ when the population of lactobacillus acidophilus in it doubles. The activa tion energy (in $\mathrm{kJ} / \mathrm{mol}$ ) for this process is closest to (Given, $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \ln \left(\frac{2}{3}\right)=0.4$, $\left.e^{-3}=4.0\right)$ Solution:...
Read More →