Prove the following

Question: Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $1+m-n=0$ and $\mathrm{I}^{2}+\mathrm{m}^{2}-\mathrm{n}^{2}=0$. Then the value of $\sin ^{4} \alpha+\cos ^{4} \alpha$ is :(1) $\frac{3}{4}$(2) $\frac{1}{2}$(3) $\frac{5}{8}$(4) $\frac{3}{8}$Correct Option: , 3 Solution: $\left.\right|^{2}+m^{2}+n^{2}=1$ $\therefore 2 n^{2}=1 \Rightarrow n=\pm \frac{1}{\sqrt{2}}$ $\left.\therefore\right|^{2}+m^{2}=\frac{1}{2} \ \mid+m=\frac{1}{\sqrt{2}}$ $\Rightarr...

A particle is projected with velocity

Question: A particle is projected with velocity $\mathrm{v}_{0}$ along $\mathrm{x}$-axis. A damping forceis acting on the particle which is proportional to the square of the distance from the origin i.e. $m a=-\alpha X^{2}$. The distance at which the particle stops :(1) $\left(\frac{2 v_{0}}{3 \alpha}\right)^{\frac{1}{3}}$(2) $\left(\frac{3 v_{0}^{2}}{2 \alpha}\right)^{\frac{1}{2}}$(3) $\left(\frac{3 v_{0}^{2}}{2 \alpha}\right)^{\frac{1}{3}}$(4) $\left(\frac{2 v_{0}^{2}}{3 \alpha}\right)^{\frac{...

Glycerol is separated in soap industries by :

Question: Glycerol is separated in soap industries by :Fractional distillationDifferential extractionSteam distillationDistillation under reduced pressureCorrect Option: , 4 Solution: Glycerol can be separated from spent-lye in soap industry by using reduce pressure distillation technique....

Question: Glycerol is separated in soap industries by :Fractional distillationDifferential extractionSteam distillationDistillation under reduced pressureCorrect Option: , 4 Solution: Glycerol can be separated from spent-lye in soap industry by using reduce pressure distillation technique....

Prove the following

Question: Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda=2 y-1=-2 z$ and $x=y+2 \lambda=z-\lambda$ is $\frac{\sqrt{7}}{2 \sqrt{2}}$, then the value of $|\lambda|$ is Solution: $\frac{x-\lambda}{1}=\frac{y-\frac{1}{2}}{\frac{1}{2}}=\frac{z}{-\frac{1}{2}}$ $\frac{x-\lambda}{2}=\frac{y-\frac{1}{2}}{1}=\frac{2}{-1} \quad \ldots \ldots(1) \quad$ Point on line $=\left(\lambda, \frac{1}{2}, 0\right)$ $\frac{x}{1}=\frac{y+2 \lambda}{1}=\frac{z-\lambda}{1}$ Point on li...

In Carius method of estimation of halogen,

Question: In Carius method of estimation of halogen, $0.172 \mathrm{~g}$ of an organic compound showed presence of $0.08 \mathrm{~g}$ of bromine. Which of these is the correct structure of the compound?Correct Option: , 2 Solution: Mole of bromine $=\frac{0.08}{80}=10^{-3} \mathrm{~mole}$ Molar mass of compound is given by the following equation, $\frac{0.172}{M}=10^{-3}$ $\Rightarrow M=\frac{0.172}{10^{-2}}=172 \mathrm{~g}$ $\because$ Molar mass of $\mathrm{C}_{6} \mathrm{H}_{6} \mathrm{NBr}$ $...

The vector equation of the plane passing

Question: The vector equation of the plane passing through the intersection of the planes $\overrightarrow{\mathrm{r}} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{\mathrm{r}} \cdot(\hat{i}-2 \hat{j})=-2$, and the point $(1,0,2)$ is : (1) $\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})=\frac{7}{3}$(2) $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=7$(3) $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\math...

In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane.

Question: In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $\mathrm{k}$, the frequency of oscillation of given body is : (1) $\frac{1}{2 \pi} \sqrt{\frac{2 k}{\operatorname{Mg} \sin \alpha}}$(2) $\frac{1}{2 \pi} \sqrt{\frac{\mathrm{k}}{\mathrm{Mg} \sin \alpha}}$(3) $\frac{1}{2 \pi} \sqrt{\frac{2 \mathrm{k}}{\mathrm{M}}}$(4) $\frac{1}{2 \pi} \sqrt...

Given below are two statements :

Question: Given below are two statements : Statement I : A mixture of chloroform and aniline can be separated by simple distillation Statement II : When separating aniline from a mixture of aniline and water by steam distillation aniline boils below its boiling point In the light of the above statements, choose the most appropriate answer from the options given belowStatement I is true, statement II is falseBoth Statement I and Statement II are trueBoth Statement I and Statement II are falseStat...

Prove the following

Question: Let $a, b \in R$. If the mirror image of the point $P(a, 6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20, b,-a-9)$, then $|a+b|$ is equal to:(1) 86(2) 88(3) 84(4) 90Correct Option: , 2 Solution: $P(a, 6,9), Q(20, b,-a-9)$ mid point of $\mathrm{PQ}=\left(\frac{\mathrm{a}+20}{2}, \frac{\mathrm{b}+6}{2},-\frac{\mathrm{a}}{2}\right)$ lie on line $\frac{\frac{a+20}{2}-3}{7}=\frac{\frac{b+6}{2}-2}{5}=\frac{-\frac{a}{2}-1}{-9}$ $\frac{a+20-6}{14}=\frac{b+6-...

In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane.

Question: In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $\mathrm{k}$, the frequency of oscillation of given body is : (1) $\frac{1}{2 \pi} \sqrt{\frac{2 k}{\operatorname{Mg} \sin \alpha}}$(2) $\frac{1}{2 \pi} \sqrt{\frac{\mathrm{k}}{\mathrm{Mg} \sin \alpha}}$(3) $\frac{1}{2 \pi} \sqrt{\frac{2 \mathrm{k}}{\mathrm{M}}}$(4) $\frac{1}{2 \pi} \sqrt...

Which of the following is 'a' FALSE statement?

Question: Which of the following is 'a' FALSE statement?Carius tube used in the estimation of sulphur in an organic compoundKjedahl's method is used for the estimation of nitrogen in an organic compoundPhosphoric acid produced on oxidation of phosphorus present in an organic compound is precipitated as $\mathrm{Mg}_{2} \mathrm{P}_{2} \mathrm{O}_{7}$ by adding magnesia mixtureCarius method is used for the estimation of nitrogen in an organic compoundCorrect Option: , 4 Solution: (4) Fact...

The distance of the point

Question: The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane $x+y+z=17$ is:(1) $\sqrt{38}$(2) $19 \sqrt{2}$(3) $2 \sqrt{19}$(4) 38Correct Option: 1 Solution: $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}=\lambda$ $\Rightarrow x=\lambda+3, y=2 \lambda+4, z=2 \lambda+5$ Which lines on given plane hence $\Rightarrow \lambda+3+2 \lambda+4+2 \lambda+5=17$ $\Rightarrow \lambda=\frac{5}{5}=1$ Hence, point of intersection...

When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is :

Question: When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is :(1) elliptical(2) parabolic(3) straight line(4) circularCorrect Option: 1 Solution: (1) We know that is SHM: $V=\omega \sqrt{A^{2}-x^{2}}$...

The equation of the plane passing

Question: The equation of the plane passing through the point $(1,2,-3)$ and perpendicular to the planes $3 x+y-2 z=5$ and $2 x-5 y-z=7$, is:(1) $3 x-10 y-2 z+11=0$(2) $6 x-5 y-2 z-2=0$(3) $11 x+y+17 z+38=0$(4) $6 x-5 y+2 z+10=0$Correct Option: , 3 Solution: Normal vector of required plane is $\vec{n}=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 3 1 -2 \\ 2 -5 -1\end{array}\right|=-11 \hat{i}-\hat{j}-17 \hat{k}$ $\therefore 11(x-1)+(y-2)+17(z+3)=0$ $11 x+y+17 z+38=0$...

Using the provided information in the following paper chromatogram:

Question: Using the provided information in the following paper chromatogram: Fig: Paper chromatography for compounds A and B The calculated $R_{f}$ value of $A$ ________________ $\times 10^{-1}$ Solution: (4) $\mathrm{R}_{\mathrm{f}}=\frac{\text { Dis tan ce travelledby compound }}{\text { Dis tan ce travelled by solvent }}$ On chromatogram distance travelled by compound is $\rightarrow 2 \mathrm{~cm}$ Distance travelled by solvent $=5 \mathrm{~cm}$ So $\mathrm{R}_{\mathrm{f}}=\frac{2}{5}=4 \ti...

The equation of the planes parallel to the plane

Question: The equation of the planes parallel to the plane $x-2 y+2 z-3=0$ which are at unit distance from the point $(1,2,3)$ is $a x+b y+c z+d=0$. If $(b-d)=K(c-a)$, then the positive value of $K$ is Solution: Let plane is $x-2 y+2 z+\lambda=0$ distance from $(1,2,3)=1$ $\Rightarrow \frac{|\lambda+3|}{5}=1 \Rightarrow \lambda=0,-6$ $\Rightarrow a=1, b=-2, c=2, d=-6$ or 0 $\mathrm{b}-\mathrm{d}=4$ or $-2, \mathrm{c}-\mathrm{a}=1$ $\Rightarrow \mathrm{k}=4$ or $-2$...

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support.

Question: In the given figure, a mass $M$ is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is $\mathrm{k}$. The mass oscillates on a frictionless surface with time period $T$ and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it. The new amplitude of oscillation will be - (1) $A \sqrt{\frac{M}{M+m}}$(2) $A \sqrt{\frac{M}{M-m}}$(3) $A \sqrt{\frac{M-m}{M}}$(4) $A...

Let the plane ax+by+cz+d=0 bisect the

Question: Let the plane $a x+b y+c z+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles. If $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are integers, then the minimum value of $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)$ is Solution: Plane is $1(x-3)-3(y-0)+3(z+2)=0$ $x-3 y+3 z+3=0$ $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)_{\min }=28$...

Let P be an arbitrary point having sum

Question: Let $P$ be an arbitrary point having sum of the squares of the distance from the planes $x+y+z=0, l x-n z=0$ and $x-2 y+z=0$ equal to 9 . If the locus of the point $\mathrm{P}$ is $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=9$, then the value of $l-\mathrm{n}$ is equal to________. Solution: Let point $\mathrm{P}$ is $(\alpha, \beta, \gamma)$ $\left(\frac{\alpha+\beta+\gamma}{\sqrt{3}}\right)^{2}+\left(\frac{\ell \alpha-\mathrm{n} \gamma}{\sqrt{\ell^{2}+\mathrm{n}^{2}}}\right)^{2}+\le...

If the equation of plane passing through

Question: If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}$ is $\alpha x+\beta y+\gamma z=24$ then $\alpha+\beta+\gamma$ is equal to :(1) 20(2) 19(3) 18(4) 21Correct Option: , 2 Solution: $\therefore$ Reflection $(-2,4,-6)$ Plane : $\left|\begin{array}{ccc}\mathrm{x}-2 \mathrm{y}-1 \mathrm{z}+1 \\ 3 -2 1 \\ 4 -3 5\end{array}\right|=0$ ...

The function of time representing a simple harmonic motion

Question: The function of time representing a simple harmonic motion with a period of $\frac{\pi}{\omega}$ is :(1) $\sin (\omega t)+\cos (\omega t)$(2) $\cos (\omega t)+\cos (2 \omega t)+\cos (3 \omega t)$(3) $\sin ^{2}(\omega t)$(4) $3 \cos \left(\frac{\pi}{4}-2 \omega t\right)$Correct Option: , 4 Solution: (4) Time period $\mathrm{T}=\frac{2 \pi}{\omega^{\prime}}$ $\frac{\pi}{\omega}=\frac{2 \pi}{\omega^{\prime}}$ $\omega^{\prime}=2 \omega \rightarrow$ Angular frequency of SHM Option (c) $\sin...

The function of time representing a simple harmonic motion

Question: The function of time representing a simple harmonic motion with a period of $\frac{\pi}{\omega}$ is :(1) $\sin (\omega t)+\cos (\omega t)$(2) $\cos (\omega t)+\cos (2 \omega t)+\cos (3 \omega t)$(3) $\sin ^{2}(\omega t)$(4) $3 \cos \left(\frac{\pi}{4}-2 \omega t\right)$Correct Option: , 4 Solution: (4) Time period $\mathrm{T}=\frac{2 \pi}{\omega^{\prime}}$ $\frac{\pi}{\omega}=\frac{2 \pi}{\omega^{\prime}}$ $\omega^{\prime}=2 \omega \rightarrow$ Angular frequency of SHM Option (c) $\sin...

If the equation of the plane passing

Question: If the equation of the plane passing through the line of intersection of the planes $2 x-7 y+4 z-3=0,3 x-5 y+4 z+11=0$ and the point $(-2,1,3)$ is $a x+b y+c z-7=0$, then the value of $2 \mathrm{a}+\mathrm{b}+\mathrm{c}-7$ is Solution: Required plane is $\mathrm{p}_{1}+\lambda \mathrm{p}_{2}=(2+3 \lambda) \mathrm{x}-(7+5 \lambda) \mathrm{y}$ $+(4+4 \lambda) \mathrm{z}-3+11 \lambda=0$ which is satisfied by $(-2,1,3)$. Hence, $\lambda=\frac{1}{6}$ Thus, plane is $15 x-47 y+28 z-7=0$ So, ...

Solve the following

Question: Diborane $\left(\mathrm{B}_{2} \mathrm{H}_{6}\right)$ reacts independently with $\mathrm{O}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$ to produce, respectively;$\mathrm{B}_{2} \mathrm{O}_{3}$ and $\mathrm{H}_{3} \mathrm{BO}_{3}$$\mathrm{B}_{2} \mathrm{O}_{3}$ and $\left[\mathrm{BH}_{4}\right]^{-}$$\mathrm{H}_{3} \mathrm{BO}_{3}$ and $\mathrm{B}_{2} \mathrm{O}_{3}$$\mathrm{HBO}_{2}$ and $\mathrm{H}_{3} \mathrm{BO}_{3}$Correct Option: 1 Solution:...