Prove the following

Question: If vectors $\overrightarrow{a_{1}}=x \hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{a_{2}}=\hat{i}+y \hat{j}+z \hat{k}$ are collinear, then a possible unit vector parallel to the vector $x \hat{i}+y \hat{j}+z \hat{k}$ is :$\frac{1}{\sqrt{2}}(-\hat{j}+\hat{k})$$\frac{1}{\sqrt{2}}(i-j)$$\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$$\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$Correct Option: , 3 Solution: $\frac{x}{1}=-\frac{1}{y}=\frac{1}{z}=\lambda($ let $)$ Unit vector parallel to $x \ha...

Among the sulphates of alkaline earth metals,

Question: Among the sulphates of alkaline earth metals, the solubilities of $\mathrm{BeSO}_{4}$ and $\mathrm{MgSO}_{4}$ in water, respectively, are :poor and poorhigh and poorhigh and highpoor and highCorrect Option: Solution: $\mathrm{BeSO}_{4}$ and $\mathrm{MgSO}_{4}$ are readily soluble in water due to greater hydration enthalpies of $\mathrm{Be}^{2+}$ and $\mathrm{Mg}^{2+}$ ions, dominate over their lattice enthalpies and therefore their sulphates are highly soluble....

Prove the following

Question: If $\vec{a} \ \vec{b}$ are perpendicular vactors, then $\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})))$ is equal to$\frac{1}{2}|\vec{a}|^{4} \vec{b}$$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$$|\vec{a}|^{4} \vec{b}$\overrightarrow{0}Correct Option: , 3 Solution: $\bar{a} \times\left(\bar{a} \times\left((\bar{a} \cdot \bar{b})^{-}-\mid \vec{a}^...

An alkaline earth metal 'M' readily forms water soluble sulphate and water insoluble hydroxide.

Question: An alkaline earth metal 'M' readily forms water soluble sulphate and water insoluble hydroxide. Its oxide $\mathrm{MO}$ is very stable to heat and does not have rock-salt structure. $\mathrm{M}$ is :SrCaMgBeCorrect Option: , 4 Solution: Solubility of $\mathrm{BeSO}_{4}$ is highest among the given met sulphates $\mathrm{BeSO}_{4}\mathrm{MgSO}_{4}\mathrm{CaSO}_{4}\mathrm{SrSO}_{4}\mathrm{BaSO}_{4}$ Solubility order for hydroxide is $\mathrm{Be}(\mathrm{OH})_{2}\mathrm{Mg}(\mathrm{OH})_{2...

Prove the following

Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8 \sqrt{3}$ square units, then $\vec{a} \cdot \vec{b}$ is equal to Solution: $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ $\overrightarrow{\mat...

On combustion of LI ,

Question: On combustion of $\mathrm{Li}, \mathrm{Na}$ and $\mathrm{K}$ in excess of air, the major oxides formed, respectively, are :$\mathrm{Li}_{2} \mathrm{O}_{2}, \mathrm{Na}_{2} \mathrm{O}_{2}$ and $\mathrm{K}_{2} \mathrm{O}_{2}$ (2)$\mathrm{Li}_{2} \mathrm{O}, \mathrm{Na}_{2} \mathrm{O}_{2}$ and $\mathrm{KO}_{2}$$\mathrm{Li}_{2} \mathrm{O}, \mathrm{Na}_{2} \mathrm{O}$ and $\mathrm{K}_{2} \mathrm{O}_{2}$$\mathrm{Li}_{2} \mathrm{O}, \mathrm{Na}_{2} \mathrm{O}_{2}$ and $\mathrm{K}_{2} \mathrm{...

Question: On combustion of $\mathrm{Li}, \mathrm{Na}$ and $\mathrm{K}$ in excess of air, the major oxides formed, respectively, are :$\mathrm{Li}_{2} \mathrm{O}_{2}, \mathrm{Na}_{2} \mathrm{O}_{2}$ and $\mathrm{K}_{2} \mathrm{O}_{2}$ (2)$\mathrm{Li}_{2} \mathrm{O}, \mathrm{Na}_{2} \mathrm{O}_{2}$ and $\mathrm{KO}_{2}$$\mathrm{Li}_{2} \mathrm{O}, \mathrm{Na}_{2} \mathrm{O}$ and $\mathrm{K}_{2} \mathrm{O}_{2}$$\mathrm{Li}_{2} \mathrm{O}, \mathrm{Na}_{2} \mathrm{O}_{2}$ and $\mathrm{K}_{2} \mathrm{...

Your friend is having eye sight problem

Question: Your friend is having eye sight problem. She is not able lo see clearly a distant uniform window mesh and it appears to her as nonuniform and distorted. The doctor diagnosed the problem as:AstigmatismMyopia with AstigmatismPresbyopia with AstigmatismMyopia and hypermetropiaCorrect Option: , 2 Solution: (2) If distant objects are blurry then problem is Myopia. If objects are distorted then problem is Astigmatism...

The image of an object placed in air formed by

Question: The image of an object placed in air formed by a convex refracting surface is at a distance of $10 \mathrm{~m}$ behind the surface. The image is real and is at $\frac{2^{\mathrm{rd}}}{3}$ of the distance of the object from the surface. The wavelength of light of the distance of the object from the surface. The wavelength of light inside the surface is $\frac{2}{3}$ times the wavelength in air. The radius of the curved surface is $\frac{x}{13} m .$ the value of ' $x$ ' is Solution: $(30...

Prove the following

Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$ be three given vectors. If $\vec{r}$ is a vector such that $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{b}}=0...

The thickness at the centre of a plano convex lens

Question: The thickness at the centre of a plano convex lens is $3 \mathrm{~mm}$ and the diameter is $6 \mathrm{~cm}$. If the speed of light in the material of the lens is $2 \times 10^{8} \mathrm{~ms}^{-1}$. The focal length of the lens is $2 \times 10^{8} \mathrm{~ms}^{-1}$. The focal length of the lens is$0.30 \mathrm{~cm}$$15 \mathrm{~cm}$$1.5 \mathrm{~cm}$$30 \mathrm{~cm}$Correct Option: , 4 Solution: (4) $\mathrm{R}^{2}=\mathrm{r}^{2}+(\mathrm{R}-\mathrm{t})^{2}$ $\mathrm{R}^{2}=\mathrm{r}...

Let three vectors

Question: Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}, \vec{a} \cdot \vec{c}=7$ and $\vec{b}$ is perpendicular to $\vec{c}$, where $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{k}$, then the value of $2|\vec{a}+\vec{b}+\vec{c}|^{2}$ is Solution: $\vec{C}=\lambda(\vec{b} \times(\vec{a} \times \vec{b}))$ $=\lambda((\vec{b} \cdot \vec{b}) \vec{b}-(\vec{b} \cdot \vec{a}) \vec{b})$ $=\lambda(5(-\hat{i}+\hat{j}+\hat...

Two elements A and B have similar chemical properties.

Question: Two elements A and B have similar chemical properties. They don't form solid hydrogencarbonates, but react with nitrogen to form nitrides. A and B, respectively, are :$\mathrm{Na}$ and $\mathrm{Rb}$$\mathrm{Na}$ and $\mathrm{Ca}$$\mathrm{Cs}$ and $\mathrm{Ba}$Li and MgCorrect Option: , 4 Solution: $\mathrm{Li}$ and $\mathrm{Mg}$ do not form solid bicarbonate, but react with $\mathrm{N}_{2}$ to give nitrides. $6 \mathrm{Li}+\mathrm{N}_{2} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{Li}...

Prove the following

Question: Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$. If $|\vec{a} \times \vec{b}|=|\vec{a}|$, then the angle between the vectors $(\vec{a}+\vec{b}+(\vec{a} \times \vec{b}))$ and $\vec{a}$ is equal to:$\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$$\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$$\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$\sin ^{-1}\left(\frac{1}{\sqrt{6}}\right)$Correct Option: , 2 Solution: $|\overrightarrow{\mathrm{a}}|=|\o...

The metal mainly used in devising photoelectric cells is :

Question: The metal mainly used in devising photoelectric cells is : (1) $\mathrm{Na}$ (2) Li (3) $\mathrm{Rb}$ (4) $\mathrm{Cs}$NaLiRbCsCorrect Option: , 4 Solution: Cesium has lowest ionisation enthalpy and hence it shows photoelectric effect to the maximum extent. So, it is used in photo electric cell....

Among the following, number of metal/s which can be used as electrodes in the photoelectric cell is

Question: Among the following, number of metal/s which can be used as electrodes in the photoelectric cell is_____________ (Integer answer).LiNaRbCsCorrect Option: , 4 Solution: Cs is used in photoelectric cell due to its very low ionization potential....

A deviation

Question: A deviation of $2^{\circ}$ is produced in the yellow ray when prism of crown and flint glass are achromatically combined. Taking dispersive powers of crown and flint glass are $0.02$ and $0.03$ respectively and refractive index for yellow light for these glasses are $1.5$ and $1.6$ respectively. The refracting angles for crown glass prism will be (in degree) (Round off to the Nearest Integer) Solution: (12) $\omega_{1}=0.02 ; \mu_{1}=1.5 ; \quad \omega_{2}=0.03 ; \mu_{2}=1.6$ Achromati...

In a triangle ABC,

Question: In a triangle $\mathrm{ABC}$, if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7$, $|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to:$\frac{25}{4}$$\frac{85}{14}$$\frac{127}{20}$$\frac{115}{16}$Correct Option: , 2 Solution: $|\vec{a}|=8,|\vec{b}|=7,|\vec{c}|=10$ $\cos \theta=\frac{|\vec{b}|^{2}+|\vec{c}|^{2}-|\vec{a}|^{2}}{2|\vec{b}||\vec{c}|}=\frac{17}{28}$ Projection o...

The Correct set from the following

Question: The Correct set from the following in which both pairs are in correct order of melting point is$\mathrm{LiF}\mathrm{LiCl} ; \mathrm{NaCl}\mathrm{MgO}$$\mathrm{LiF}\mathrm{LiCl} ; \mathrm{MgO}\mathrm{NaCl}$$\mathrm{LiCl}\mathrm{LiF} ; \mathrm{NaCl}\mathrm{MgO}$$\mathrm{LiCl}\mathrm{LiF} ; \mathrm{MgO}\mathrm{NaCl}$Correct Option: , 2 Solution: Generally M.P. $\propto$ Lattice energy $=\frac{\mathrm{KQ}_{1} \mathrm{Q}_{2}}{\mathrm{r}^{+}+\mathrm{r}^{-}}$ $\propto$ (packing efficiency)...

A vector a has components 3p and 1

Question: A vector a has components $3 \mathrm{p}$ and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components $p+1$ and $\sqrt{10}$, then a value of $p$ is equal to:1$-\frac{5}{4}$$\frac{4}{5}$$-1$Correct Option: , 4 Solution: $\overrightarrow{\mathrm{a}}$ old $=3 \mathrm{p} \hat{\mathrm{i}}+\hat{\mathrm{j}}$ $\overrightarrow{\mathrm{a}}_{\mathrm{New}}=(\mat...

Match List-I with List-II

Question: Match List-I with List-II Choose the correct answer from the options given below :$(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{i} v),(\mathrm{d})-(\mathrm{iii})$(a) $-($ ii $),(b)-($ iv $),(c)-($ iii $),(d)-($ i $)$(a) $-($ iv $),($ b $)-($ ii $),($ c $)-($ iii $),($ d $)-($ i $)$$(\mathrm{a})-(\mathrm{i}),(\mathrm{b})-(\mathrm{i} v),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{iii})$Correct Option: , 2 Solution: Range of visible region : $390 n m-760...

The refractive index of a converging lens is $1.4$.

Question: The refractive index of a converging lens is $1.4$. What will be the focal length of this lens if it is placed in a medium of same refractive index ? (Assume the radii of curvature of the faces of lens are $R_{1}$ and $R_{2}$ respectively)1Infinite$\frac{\mathrm{R}_{1} \mathrm{R}_{2}}{\mathrm{R}_{1}-\mathrm{R}_{2}}$ZeroCorrect Option: , 2 Solution: (2) $\frac{1}{\mathrm{~F}}=\left[\frac{\mu_{\mathrm{L}}}{\mu_{\mathrm{s}}}-1\right]\left[\frac{1}{\mathrm{R}_{1}}-\frac{1}{\mathrm{R}_{2}}\...

Prove the following

Question: Let $\overrightarrow{\mathrm{x}}$ be a vector in the plane containing vectors $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$. If the vector $\overrightarrow{\mathrm{x}}$ is perpendicular to $(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and its projection on $\vec{a}$ is $\frac{17 \sqrt{6}}{2}$, then the value of $|\vec{x}|^{2}$ is equal to_______. Solu...

Number of amphoteric compounds among the following is

Question: Number of amphoteric compounds among the following is$\mathrm{BeO}$$\mathrm{BaO}$$\mathrm{Be}(\mathrm{OH})_{2}$$\mathrm{Sr}(\mathrm{OH})_{2}$Correct Option: , 2 Solution: $\mathrm{BeO}$ and $\mathrm{Be}(\mathrm{OH})_{2}$ are amphoteric in nature....

Match list-I with list-II :

Question: Match list-I with list-II : Choose the most appropriate answer the option given below:$a-i v, b-i i i, c-i, d-i i$$a-i v, b-i i i, c-i i, d-i$$a-i i i, b-i v, c-v, d-i i$$a-i i i, b-i v, c-i i, d-v$Correct Option: , 2 Solution: (a) $\mathrm{Be} \rightarrow$ it is used in the Windows of $\mathrm{X}$-ray tubes (b) $\mathrm{Mg} \rightarrow$ it is used in the Incendiary bombs and signals (c) $\mathrm{Ca} \rightarrow$ it is used in the Extraction of metals (d) $\mathrm{Ra} \rightarrow$ it i...