Question: A colloidal system consisting of a gas dispersed in a solid is called a/an: solid solgelaerosolfoamCorrect Option: 1 Solution: Colloid of gas dispersed in solid is called solid sol....

Ga (atomic mass 70 u ) crystallizes in a hexagonal close packed structure.

Question: Ga (atomic mass $70 \mathrm{u}$ ) crystallizes in a hexagonal close packed structure. The total number of voids in $0.581 \mathrm{~g}$ of $\mathrm{Ga}$ is_______________ $\times 10^{21}$. (Round off to the Nearest Integer). Solution: (15) HCP structure: Per atom, there will be one octahedral void (OV) and two tetrahedral voids (TV). Therefore total three voids per atom are present in HCP structure. $\rightarrow$ therefore total no of atoms of Ga will be- $=\frac{\text { Mass }}{\text {...

Ga (atomic mass 70 u ) crystallizes in a hexagonal close packed structure.

Question: Ga (atomic mass $70 \mathrm{u}$ ) crystallizes in a hexagonal close packed structure. The total number of voids in $0.581 \mathrm{~g}$ of $\mathrm{Ga}$ is_______________ $\times 10^{21}$. (Round off to the Nearest Integer). Solution: (15) HCP structure: Per atom, there will be one octahedral void (OV) and two tetrahedral voids (TV). Therefore total three voids per atom are present in HCP structure. $\rightarrow$ therefore total no of atoms of Ga will be- $=\frac{\text { Mass }}{\text {...

A certain element crystallises in a bce lattice of unit cell

Question: A certain element crystallises in a bce lattice of unit cell edge length $27 \backslash \mathrm{AA}$. If the same element under the same conditions crystallises in the fcc lattice, the edge length of the unit cell in $\backslash \mathrm{AA}$ will be________________ (Round off to the Nearest Integer). [Assume each lattice point has a single atom] [Assume $\sqrt{3}=1.73, \sqrt{2}=1.41$ ] Solution: (33) For $\mathrm{BCC} \sqrt{3} \mathrm{a}=4 \mathrm{r}$ so $\quad \mathrm{r}=\frac{\sqrt{3...

Question: A certain element crystallises in a bce lattice of unit cell edge length $27 \backslash \mathrm{AA}$. If the same element under the same conditions crystallises in the fcc lattice, the edge length of the unit cell in $\backslash \mathrm{AA}$ will be________________ (Round off to the Nearest Integer). [Assume each lattice point has a single atom] [Assume $\sqrt{3}=1.73, \sqrt{2}=1.41$ ] Solution: (33) For $\mathrm{BCC} \sqrt{3} \mathrm{a}=4 \mathrm{r}$ so $\quad \mathrm{r}=\frac{\sqrt{3...

A point source of light S, placed at a distance 60 cm

Question: A point source of light $S$, placed at a distance $60 \mathrm{~cm}$ infront of the centre of plane mirror of width $50 \mathrm{~cm}$, hangs vertically on a wall. A man walks infront of the mirror along a line parallel to the mirror at a distance $1.2 \mathrm{~m}$ from it (see in the figure). The distance between the extreme points where he can see the image of the light source in the mirror is $\mathrm{cm}$ Solution: (150) from similar triangle IMP and IQR $\frac{Q R}{25}=\frac{180}{60...

Given below are two statements: One is labeled as Assertion

Question: Given below are two statements: One is labeled as Assertion $A$ and the other is labeled as Reason $\mathrm{R}$. Assertion A: For a simple microscope, the angular size of the object equals the angular size of the image. Reason $\mathrm{R}$ : Magnification is achieved as the small object can be kept much closer to the eye than $25 \mathrm{~cm}$ and hence it subtends a large angle. In the light of the above statements, choose the most appropriate answer from the options given below:Both ...

Prove the following

Question: If $\vec{x}$ and $\vec{y}$ be two non-zero vectors such that $|\vec{x}+\vec{y}|=|\vec{x}|$ and $2 \vec{x}+\lambda \vec{y}$ is perpendicular to $\vec{y}$, then the value of $\lambda$ is_________. Solution: $\because|\vec{x}+\vec{y}|=|\vec{x}|$ Squaring both sides we get $|\vec{x}|^{2}+2 \vec{x} \cdot \vec{y}+|\vec{y}|^{2}=|\vec{x}|^{2}$ $\Rightarrow 2 \vec{x} \cdot \vec{y}+\vec{y} \cdot \vec{y}=0$ $\ldots$ (i) Also $2 \vec{x}+\lambda \vec{y}$ and $\vec{y}$ are perpendicular $\therefore ...

Prove the following

Question: If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is__________. Solution: Let angle between $\vec{a}$ and $\vec{b}$ be $\theta$. $|\vec{a}+\vec{b}|=\sqrt{1+1+2 \cos \theta}=2\left|\cos \frac{\theta}{2}\right| \quad[\because|a|=|b|=1]$ Similarly, $|\vec{a}-\vec{b}|=2\left|\sin \frac{\theta}{2}\right|$ So, $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|=2\left[\sqrt{3}\left|\cos \frac{\theta}{2}\right|+\left|\sin \frac{\the...

The incident ray, reflected ray and the outward drawn normal are denoted by

Question: The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors $\bar{a}, \vec{b}$ and $\vec{c}$ respectively. Then choose the correct relation for these vectors.(1) $\vec{b}=2 \vec{a}+\vec{c}$(2) $\vec{b}=\vec{a}-\vec{c}$(3) $\vec{b}=\vec{a}+2 \vec{c}$(4) $\vec{b}=\vec{a}-2(\vec{a} \cdot \vec{c}) \vec{c}$Correct Option: , 4 Solution: We see from the diagram that because of the law of reflection, the component of the unit vector $\vec{a}$ along $\vec{b}$ ch...

Prove the following

Question: Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the projection of $\vec{c}$ on $\vec{a}$ and $\vec{b}$ is perpendicular to $\vec{c}$, then the value of $|\vec{a}+\vec{b}-\vec{c}|$ is_________. Solution: $\because$ Projection of $\vec{b}$ on $\vec{a}=$ Projection of $\vec{c}$ on $\vec{a}$ $\therefore \vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}$ Given, $\vec{b} \cdot \vec{c}=0$ $...

If the volume of a parallelopiped,

Question: If the volume of a parallelopiped, whose coterminus edges are given by the vectors $\vec{a}=\hat{i}+\hat{j}+n \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-n \hat{k}$ and $\vec{c}=\hat{i}+n \hat{j}+3 \hat{k}(n \geq 0)$, is 158 cu.units, then:$\vec{a} \cdot \vec{c}=17$$\vec{b} \cdot \vec{c}=10$$n=7$$n=9$Correct Option: , 2 Solution: We know that the volume of parallelopiped $=\left[\begin{array}{lll}\bar{a} \bar{b} \bar{c}\end{array}\right]$ $\left|\begin{array}{ccc}1 1 n \\ 2 4 -n \\ 1 n 3\end{...

Prove the following

Question: If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $|\hat{i} \times(\vec{a} \times \hat{i})|^{2}+|\hat{j} \times(\vec{a} \times \hat{j})|^{2}+|\hat{k} \times(\vec{a} \times \hat{k})|^{2}$ is equal to___________. Solution: $\hat{i} \times(\bar{a} \times \hat{i})=(\hat{i} \cdot \hat{i}) \bar{a}-(\hat{i} \cdot \bar{a}) \hat{i}=\hat{j}+2 \hat{k}$ Similarly, $\hat{j} \times(\bar{a} \times \hat{j})=2 \hat{i}+2 \hat{k}$ $\hat{k} \times(\bar{a} \times \hat{k})=2 \hat{i}+\hat{j}$ $\the...

Prove the following

Question: Let $x_{0}$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})$, where $\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}, \quad \vec{b}=-2 \hat{i}+x \hat{j}-\hat{k}$ and $\vec{c}=7 \hat{i}-2 \hat{j}+x \hat{k}$. Then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ at $x=x_{0}$ is :$-4$$-30$14$-22$Correct Option: , 4 Solution: It is given that $f(x)=\bar{a} \cdot(\bar{b} \times \bar{c})=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ x -2...

A short straight object of height 100 cm

Question: A short straight object of height $100 \mathrm{~cm}$ lies before the central axis of a spherical mirror whose focal length has absolute value $|f|=40 \mathrm{~cm}$. The image of object produced by the mirror is of height $25 \mathrm{~cm}$ and has the same orientation of the object. One may conclude from the information :Image is real, same side of concave mirror.Image is virtual, opposite side of convex mirror.Image is virtual, opposite side of concave mirror.Image is real, same side o...

Prove the following

Question: Let $a, b, c \in \mathbf{R}$ be such that $a^{2}+b^{2}+c^{2}=1$. If $a \cos \theta-b \cos \left(\theta+\frac{2 \pi}{3}\right)=c \cos \left(\theta+\frac{4 \pi}{3}\right)$, where $\theta=\frac{\pi}{9}$, then the angle between the vectors $a \hat{i}+b \hat{j}+c \hat{k}$ and $b \hat{i}+c \hat{j}+a \hat{k}$ is :$\frac{\pi}{2}$$\frac{2 \pi}{3}$$\frac{\pi}{9}$0Correct Option: 1 Solution: $a \cos \theta=b \cos \left(\theta+\frac{2 \pi}{3}\right)=c \cos \left(\theta+\frac{4 \pi}{3}\right)=k$ $a...

The same size images are formed by a convex lens when the object is placed

Question: The same size images are formed by a convex lens when the object is placed at $20 \mathrm{~cm}$ or at $10 \mathrm{~cm}$ from the lens. The focal length of convex lens is______ cm. Solution: (15) $\frac{1}{v}-\frac{1}{v}=\frac{1}{f} \ldots(1)$ $m=\frac{v}{w} \ldots(2)$ from (1) and (2) we get $m=\frac{f}{f+u}$ given conditions $m_{1}=-m_{2}$ $\frac{f}{f-10}=\frac{-f}{f-20}$ $f-20=-f+10$ $2 f=30$ $\mathrm{f}=15 \mathrm{~cm}$...

Among the statements (A)-(D),

Question: Among the statements (A)-(D), the correct ones are: (A) Lithium has the highest hydration enthalpy among the alkali metals. (B) Lithium chloride is insoluble in pyridine. (C) Lithium cannot form ethynide upon its reaction with ethyne. (D) Both lithium and magnesium react slowly with $\mathrm{H}_{2} \mathrm{O}$.(A), (B) and (D) only(A), (C) and (D) only(B) and (D) only(A) and (D) onlyCorrect Option: , 2 Solution: $\mathrm{LiCl}$ is soluble in pyridine....

A metal (A) on heating in nitrogen gas gives compound B.

Question: A metal (A) on heating in nitrogen gas gives compound B. B on treatment with $\mathrm{H}_{2} \mathrm{O}$ gives a colourless gas which when passed through $\mathrm{CuSO}_{4}$ solution gives a dark blueviolet coloured solution. A and B respectively, are:$\mathrm{Na}$ and $\mathrm{NaNO}_{3}$$\mathrm{Na}$ and $\mathrm{Na}_{3} \mathrm{~N}$$\mathrm{Mg}$ and $\mathrm{Mg}_{3} \mathrm{~N}_{2}$$\mathrm{Mg}$ and $\mathrm{Mg}\left(\mathrm{NO}_{3}\right)_{2}$Correct Option: , 3 Solution:...

Let the position vectors of points 'A' and 'B' be

Question: Let the position vectors of points 'A' and 'B' be $\hat{i}+\hat{j}+k$ and $2 \hat{i}+\hat{j}+3 \hat{k}$, respectively. A point 'P' divides the line segment $\mathrm{AB}$ internally in the ratio $\lambda: 1(\lambda0)$. If $O$ is the region and $\overrightarrow{O B} \cdot \overrightarrow{O P}-3|\overrightarrow{O A} \times \overrightarrow{O P}|^{2}=6$, then $\lambda$ is equal to_________. Solution: Let position vector of $A$ and $B$ be $\vec{a}$ and $\vec{b}$ respectively. $\therefore$ Po...

The focal length f is related to the radius of curvature r of the spherical convex mirror by -

Question: The focal length $\mathrm{f}$ is related to the radius of curvature $\mathrm{r}$ of the spherical convex mirror by -$f=r$$f=-\frac{1}{2} r$$\mathrm{f}=+\frac{1}{2} \mathrm{r}$$f=-r$Correct Option: , 3 Solution: So, $\frac{R}{2}=f$ $F=+\frac{1}{2} R$...

The focal length f is related to the radius of curvature r of the spherical convex mirror by -

Question: The focal length $\mathrm{f}$ is related to the radius of curvature $\mathrm{r}$ of the spherical convex mirror by -$f=r$$f=-\frac{1}{2} r$$\mathrm{f}=+\frac{1}{2} \mathrm{r}$$f=-r$Correct Option: , 3 Solution: So, $\frac{R}{2}=f$ $F=+\frac{1}{2} R$...

Match the following compounds (Column-I) with their uses (Column-II) :

Question: Match the following compounds (Column-I) with their uses (Column-II) : (I)-(D), (II)-(A), (III)-(C), (IV)-(B)(I)-(B), (II)-(D), (III)-(A), (IV)-(C)(I)-(B), (II)-(C), (III)-(D), (IV)-(A)(I)-(C), (II)-(D), (III)-(B), (IV)-(A)Correct Option: , 2 Solution: (I) $\mathrm{Ca}(\mathrm{OH})_{2}$ is used in white wash. (II) $\mathrm{NaCl}$ is used in preparation of washing soda $\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)$. (III) $\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2} \mathrm{O}$ i...

Let a, b and c be three unit vectors such that

Question: Let $a, b$ and $c$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ is equal to__________. Solution: $|\vec{a}|=|\vec{b}|=|\vec{c}|=1$ $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$ $\Rightarrow \vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}=-2$ Now, $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ $=2|\vec{a}|^{2}+4|\vec{b}|^{2}+4|\vec{c}|^{2}+4(\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c})=2$...

Three rays of light, namely red R,

Question: Three rays of light, namely red $(R)$, green $(G)$ and blue $(B)$ are incident on the face PQ of a right angled prism PQR as shown in figure. The refractive indices of the material of the prism for red, green and blue wavelength are $1.27,1.42$ and $1.49$ respectively. The colour of the ray(s) emerging out of the face $\mathrm{PR}$ is :greenredblue and greenblueCorrect Option: , 2 Solution: (2) Assuming that the right angled prism is an isoceles prism, so the other angles will be $45^{...