Prove the following

Question: Let $\mathrm{O}$ be the origin. Let $\overrightarrow{\mathrm{OP}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OQ}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \mathrm{x} \hat{\mathrm{k}}, \mathrm{x}, \mathrm{y} \in \mathrm{R}, \mathrm{x}0$, be such that $|\overrightarrow{\mathrm{PQ}}|=\sqrt{20}$ and the vector $\overrightarrow{\mathrm{OP}}$ is perpendicular to $\overrightarrow{\mathrm{OQ}}$. If $\overrightarrow{\mathrm{OR}}=3 \hat{\...

The secondary valency and the number of hydrogen bonded water molecule(s) in

Question: The secondary valency and the number of hydrogen bonded water molecule(s) in $\mathrm{CuSO}_{4} \cdot 5 \mathrm{H}_{2} \mathrm{O}$, respectively, are6 and 44 and 16 and 55 and 1Correct Option: , 2 Solution: Hydrogen bonded water molecule $=1$ Secondary valency $=4$...

Question: The secondary valency and the number of hydrogen bonded water molecule(s) in $\mathrm{CuSO}_{4} \cdot 5 \mathrm{H}_{2} \mathrm{O}$, respectively, are6 and 44 and 16 and 55 and 1Correct Option: , 2 Solution: Hydrogen bonded water molecule $=1$ Secondary valency $=4$...

The angle of deviation through a prism is minimum when

Question: The angle of deviation through a prism is minimum when (A) Incident ray and emergent ray are symmetric to the prism (B) The refracted ray inside the prism becomes parallel to its base (C) Angle of incidence is equal to that of the angle of emergence (D) When angle of emergence is double the angle of incidence Choose the correct answer from the options given below:Statements $(\mathrm{A}),(\mathrm{B})$ and $(\mathrm{C})$ are trueOnly statement (D) is trueOnly statements (A) and (B) are ...

Prove the following

Question: If $\quad \overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{b}}=-\beta \hat{\mathrm{i}}-\alpha \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ such that $\vec{a} \cdot \vec{b}=1$ and $\vec{b} \cdot \vec{c}=-3$, then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to______. Solution: $\overrightarrow{\mathrm{a}} \cdot \overrightar...

Given below are two statements:

Question: Given below are two statements: One is labelled as Assertion A and the other labelled as reason $\mathrm{R}$ Assertion $\mathbf{A}$ : During the boiling of water having temporary hardness, $\mathrm{Mg}\left(\mathrm{HCO}_{3}\right)_{2}$ is converted to $\mathrm{MgCO}_{3}$ Reason $\mathbf{R}$ : The solubility product of $\mathrm{Mg}(\mathrm{OH})_{2}$ is greater than that of $\mathrm{MgCO}_{3}$. In the light of the above statements, choose the most appropriate answer from the options give...

Prove the following

Question: Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=7 \hat{\mathrm{i}}+\hat{\mathrm{j}}-6 \hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=-3$, then $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\m...

Match List-I with List-II

Question: Match List-I with List-II Choose the most appropriate answer from the options given below:$\mathrm{a}-\mathrm{i}, \mathrm{b}-\mathrm{iv}, \mathrm{c}-\mathrm{iii}, \mathrm{d}-\mathrm{ii}$$\mathrm{a}-\mathrm{iii}, \mathrm{b}-\mathrm{ii}, \mathrm{c}-\mathrm{iv}, \mathrm{d}-\mathrm{i}$$\mathrm{a}-\mathrm{iii}, \mathrm{b}-\mathrm{iv}, \mathrm{c}-\mathrm{i}, \mathrm{d}-\mathrm{i}$$\mathrm{a}-\mathrm{iii}, \mathrm{b}-\mathrm{ii}, \mathrm{c}-\mathrm{i}, \mathrm{d}-\mathrm{iv}$Correct Option: ,...

Prove the following

Question: Let $\vec{c}$ be a vector perpendicular to the vectors $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ If $\overrightarrow{\text { c. }}(\hat{i}+\hat{j}+3 \hat{k})=8$ then the value of $\vec{c} \cdot(\vec{a} \times \vec{b})$ is equal to_________. Solution: $\vec{c}=\lambda(\vec{a} \times \vec{b})$ $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\left|\...

Prove the following

Question: Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}$. If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{r}}, \quad \overrightarrow{\mathrm{r}} \cdot(\alpha \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=3$ and $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-\alpha \hat{\mathrm{k}})=-1, \alpha \in \mathrm{R}$, then the value of $\alpha+|\overrighta...

One of the by-products formed during the recovery of

Question: One of the by-products formed during the recovery of $\mathrm{NH}_{3}$ from Solvay process is :$\mathrm{Ca}(\mathrm{OH})_{2}$$\mathrm{NaHCO}_{3}$$\mathrm{CaCl}_{2}$$\mathrm{NH}_{4} \mathrm{Cl}$Correct Option: , 3 Solution: (3)...

The functional groups that are responsible for the ion-exchange property of cation and anion exchange resins,

Question: The functional groups that are responsible for the ion-exchange property of cation and anion exchange resins, respectively, are :$-\mathrm{SO}_{3} \mathrm{H}$ and $-\mathrm{NH}_{2}$$-\mathrm{SO}_{3} \mathrm{H}$ and $-\mathrm{COOH}$$-\mathrm{NH}_{2}$ and $-\mathrm{COOH}$$-\mathrm{NH}_{2}$ and $-\mathrm{SO}_{3} \mathrm{H}$Correct Option: 1 Solution: Cation exchanger contains $-\mathrm{SO}_{3} \mathrm{H}$ or $-\mathrm{COOH}$ groups while anion exchanger contains basic groups like $-\mathr...

Young's moduli of two wires A and B are in the ratio 7: 4.

Question: Young's moduli of two wires $\mathrm{A}$ and $\mathrm{B}$ are in the ratio $7: 4$. Wire $\mathrm{A}$ is $2 \mathrm{~m}$ long and has radius $\mathrm{R}$. Wire $\mathrm{B}$ is $1.5 \mathrm{~m}$ long and has radius $2 \mathrm{~mm}$. If the two wires stretch by the same length for a given load, then the value of $R$ is close to:$1.5 \mathrm{~mm}$$1.9 \mathrm{~mm}$$1.7 \mathrm{~mm}$$1.3 \mathrm{~mm}$Correct Option: , 3 Solution: (3) $\Delta_{1}=\Delta_{2}$ or $\frac{F l_{1}}{\pi r_{1}^{2} ...

The correct order of conductivity of ions in water is :

Question: The correct order of conductivity of ions in water is :$\mathrm{Na}^{+}\mathrm{K}^{+}\mathrm{Rb}^{+}\mathrm{Cs}^{+}$$\mathrm{Cs}^{+}\mathrm{Rb}^{+}\mathrm{K}^{+}\mathrm{Na}^{+}$$\mathrm{K}^{+}\mathrm{Na}^{+}\mathrm{Cs}^{+}\mathrm{Rb}^{+}$$\mathrm{Rb}^{+}\mathrm{Na}^{+}\mathrm{K}^{+}\mathrm{Li}^{+}$Correct Option: , 2 Solution: As the size of gaseous ion decreases, it get more hydrated in water and hence, the size of aqueous ion increases. When this bulky ion move in solution, it experi...

Prove the following

Question: Let a vector $\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}$ be obtained by rotating the vector $\sqrt{3} \hat{\mathrm{i}}+\hat{\mathrm{j}}$ by an angle $45^{\circ}$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha, \beta),(0, \beta)$ and $(0,0)$ is equal to$\frac{1}{2}$1$\frac{1}{\sqrt{2}}$$2 \sqrt{2}$Correct Option: 1 Solution: Area of $\Delta\left(\mathrm{OA}^{\prime} \mathrm{B}\right)=\frac{1}{2} \mathrm{OA}^{...

Choose the CORRECT answer from the options given below:

Question: Statement I : Sodium hydride can be used as an oxidising agent. Statement II : The lone pair of electrons on nitrogen in pyridine makes it basic. Choose the CORRECT answer from the options given below:Both statement I and statement II are falseStatement $I$ is true but statement II is false Statement $\mathrm{I}$ is false but statement $\mathrm{II}$ is trueBoth statement I and statement II are trueCorrect Option: , 3 Solution: (1) $\mathrm{NaH}$ (sodium Hydride) is used as a reducing r...

A boy's catapult is made of rubber cord which is 42 cm long,

Question: A boy's catapult is made of rubber cord which is $42 \mathrm{~cm}$ long, with $6 \mathrm{~mm}$ diameter of cross-section and of negligible mass. The boy keeps a stone weighing $0.02 \mathrm{~kg}$ on it and stretches the cord by $20 \mathrm{~cm}$ by applying a constant force. When released, the stone flies off with a velocity of $20 \mathrm{~ms}^{-1}$. Neglect the change in the area of cross-section of the cord while stretched. The Young's modulus of rubber is closest to :$10^{6} \mathr...

Given below are two statements:

Question: Given below are two statements: Statement $\mathrm{I}:$ Both $\mathrm{CaCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}$ and $\mathrm{MgCl}_{2} \cdot 8 \mathrm{H}_{2} \mathrm{O}$ undergo dehydration on heating. Statement II : $\mathrm{BeO}$ is amphoteric whereas the oxides of other elements in the same group are acidic. In the light of the above statements, choose the correct answer from the options given below :Statement I is false but statement II is trueBoth statement I and statement II a...

A steel wire having a radius of 2.0 mm, carrying a load of 4 kg,

Question: A steel wire having a radius of $2.0 \mathrm{~mm}$, carrying a load of $4 \mathrm{~kg}$, is hanging from a ceiling. Given that $\mathrm{g}=3.1 \pi \mathrm{ms}^{-2}$, what will be the tensile stress that would be developed in the wire?$6.2 \times 10^{6} \mathrm{Nm}^{-2}$$5.2 \times 10^{6} \mathrm{Nm}^{-2}$$3.1 \times 10^{6} \mathrm{Nm}^{-2}$$4.8 \times 10^{6} \mathrm{Nm}^{-2}$Correct Option: , 3 Solution: (3) Given, Radius of wire, $r=2 \mathrm{~mm}$ Mass of the load $m=4 \mathrm{~kg}$ ...

Two steel wires having same length are suspended from a ceiling under the same load.

Question: Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is $1: 4$, the ratio of their diameters is:$\sqrt{2}: 1$$1: 2$$2: 1$$1: \sqrt{2}$Correct Option: 1 Solution: (1) If force $F$ acts along the length $\mathrm{L}$ of the wire of crosssection $A$, then energy stored in unit volume of wire is given by Energy density $=\frac{1}{2}$ stress $\times$ strain $=\frac{1}{2} \times \frac{F}{A} \times \frac{F}{A Y...

The length of the perpendicular

Question: The length of the perpendicular from the point $(2,-1,4)$ on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is :greater than 3 but less than 4less than 2greater than 2 but less than 3greater than 4Correct Option: 1, Solution: Let $P$ be the foot of perpendicular from point $T(2,-1,4)$ on the given line. So $P$ can be assumed as $P(10 \lambda-3,-7 \lambda+2, \lambda)$ DR's of $T P$ is proportional to $10 \lambda-5,-7 \lambda+3, \lambda-4$ $\because \quad T P$ and given l...

The redox reaction among the following is:

Question: The redox reaction among the following is:formation of ozone from atmospheric oxygen in the presence of sunlightreaction of $\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right] \mathrm{Cl}_{3}$ with $\mathrm{AgNO}_{3}$reaction of $\mathrm{H}_{2} \mathrm{SO}_{4}$ with $\mathrm{NaOH}$combination of dinitrogen with dioxygen at $2000 \mathrm{~K}$Correct Option: , 4 Solution:...

Two different wires having lengths

Question: Two different wires having lengths $L_{1}$ and $L_{2}$, and respective temperature coefficient of linear expansion $\alpha_{1}$ and $\alpha_{2}$, are joined end-to-end. Then the effective temperature coefficient of linear expansion is :$\frac{\alpha_{1} L_{1}+\alpha_{2} L_{2}}{L_{1}+L_{2}}$$2 \sqrt{\alpha_{1} \alpha_{2}}$$\frac{\alpha_{1}+\alpha_{2}}{2}$$4 \frac{\alpha_{1} \alpha_{2}}{\alpha_{1}+\alpha_{2}} \frac{L_{2} L_{1}}{\left(L_{2}+L_{1}\right)^{2}}$Correct Option: 1 Solution: (1...

Oxidation number of potassium in

Question: Oxidation number of potassium in $\mathrm{K}_{2} \mathrm{O}, \mathrm{K}_{2} \mathrm{O}_{2}$ and $\mathrm{KO}_{2}$, respectively, is:$+2,+1$ and $+\frac{1}{2}$$+1,+1$ and $+1$$+1,+4$ and $+2$$+1,+2$ and $+4$Correct Option: , 2 Solution: $\mathrm{K}_{2} \mathrm{O}: 2 x-2=0 \Rightarrow x=+1$ $\mathrm{~K}_{2} \mathrm{O}_{2}: 2 x-2=0 \Rightarrow x=+1$ $\mathrm{KO}_{2}: x-1=0 \Rightarrow x=+1$ Thus, potassium shows $+1$ state in all its oxides, superoxides and peroxides....

The equation of a plane containing

Question: The equation of a plane containing the line of intersection of the planes $2 x-y-4=0$ and $y+2 z-4=0$ and passing through the point $(1,1,0)$ is :$x-3 y-2 z=-2$$2 x-z=2$$x-y-z=0$$x+3 y+z=4$Correct Option: , 3 Solution: Let the equation of required plane be; $(2 x-y-4)+\lambda(y+2 z-4)=0$ $\because$ This plane passes through the point $(1,1,0)$ then $(2-1-4)+\lambda(1+0-4)=0$ $\Rightarrow \lambda=-1$ Then, equation of required plane is, $(2 x-y-4)-(y+2 z-4)=0$ $\Rightarrow 2 x-2 y-2 z=0...