eSaral - JEE | NEET | Class 8-10 Exam Prep App

The vapour pressure of water is 12.3 kPa at 300 K.

Question: The vapour pressure of water is 12.3 kPa at 300 K. Calculate vapour pressure of 1 molal solution of a non-volatile solute in it. Solution: 1 molal solution means 1 mol of the solute is present in 1000 g of the solvent (water). Molar mass of water = 18 g mol1 $\therefore$ Number of moles present in $1000 \mathrm{~g}$ of water $=\frac{1000}{18}$ = 55.56 mol Therefore, mole fraction of the solute in the solution is $x_{2}=\frac{1}{1+55.56}=0.0177$ It is given that, Vapour pressure of wate...

A body of mass m=10 kg is attached to one end of a wire of length 0.3 m.

Question: A body of mass $m=10 \mathrm{~kg}$ is attached to one end of a wire of length $0.3 \mathrm{~m}$. The maximum angular speed (in $\mathrm{rad} \mathrm{s}^{-1}$ ) with which it can be rotated about its other end in space station is (Breaking stress of wire $=4.8 \times 10^{7} \mathrm{Nm}^{-2}$ and area of cross-section of the wire $=10^{-2} \mathrm{~cm}^{2}$ ) is________ Solution: (4) Given : Wire length, $l=0.3 \mathrm{~m}$ Mass of the body, $\mathrm{m}=10 \mathrm{~kg}$ Breaking stress, ...

Heptane and octane form an ideal solution. At 373 K,

Question: Heptane and octane form an ideal solution. At 373 K, the vapour pressures of the two liquid components are 105.2 kPa and 46.8 kPa respectively. What will be the vapour pressure of a mixture of 26.0 g of heptane and 35 g of octane? Solution: Vapour pressure of heptane $\left(p_{1}^{0}\right)=105.2 \mathrm{kPa}$ Vapour pressure of octane $\left(p_{2}^{0}\right)=46.8 \mathrm{kPa}$ We know that, Molar mass of heptane (C7H16) = 7 12 + 16 1 = 100 g mol1 $\therefore$ Number of moles of heptan...

A chord of a circle subtends an angle of θ at the centre of the circle.

Question: A chord of a circle subtends an angle ofat the centre of the circle. The area of the minor segment cut off by the chord is one eighth of the area of the circle. Prove that $8 \sin \frac{\theta}{2} \cos \frac{\theta}{2}+\pi=\frac{\pi \theta}{45}$ Solution: We know that the area of circle and area of minor segment of angle $\theta$ in a circle of radius $r$ is given by, $A^{\prime}=\pi r^{2}$ and $A=\left\{\frac{\pi \theta}{360^{\circ}}-\sin \frac{\theta}{2} \cos \frac{\theta}{2}\right\}...

Three solid spheres each of mass m and diameter d are stuck together

Question: Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$. The ratio $\frac{\mathrm{I}_{0}}{\mathrm{I}_{\mathrm{A}}}$ of moment of inertia $\mathrm{I}_{0}$ of the system about an axis passing the centroid and about center of any of the spheres $\mathrm{I}_{\mathrm{A}}$ and perpendicular to the plane of the triangle is: $\frac{13}{23}$$\frac{15}{13}$$\frac{23}{13}$$\frac{13}{15}$...

An aqueous solution of 2% non-volatile solute exerts a pressure of 1.004 bar at the normal boiling point of the solvent.

Question: An aqueous solution of 2% non-volatile solute exerts a pressure of 1.004 bar at the normal boiling point of the solvent. What is the molar mass of the solute? Solution: Here, Vapour pressure of the solution at normal boiling point (p1) = 1.004 bar Vapour pressure of pure water at normal boiling point $\left(p_{1}^{0}\right)=1.013$ bar Mass of solute, (w2) = 2 g Mass of solvent (water), (w1) = 98 g Molar mass of solvent (water), (M1) = 18 g mol1 According to Raoults law, $\frac{p_{1}^{0...

AB is the diameter of a circle, centre O,

Question: $A B$ is the diameter of a circle, centre $O, C$ is a point on the circumference such that $\angle C O B=\theta$. The area of the minor segment cut off by $A C$ is equal to twice the area of the sector $B O C$. Prove that $\sin \frac{\theta}{2} \cos \frac{\theta}{2}=\pi\left(\frac{1}{2}-\frac{\theta}{120}\right)$ Solution: We know that the area of minor segment of anglein a circle of radiusris, $A=\left\{\frac{\pi \theta}{360^{\circ}}-\sin \frac{\theta}{2} \cos \frac{\theta}{2}\right\}...

What is meant by positive and negative deviations from

Question: What is meant by positive and negative deviations from Raoult's law and how is the sign ofΔsolHrelated to positive and negative deviations from Raoult's law? Solution: According to Raoults law, the partial vapour pressure of each volatile component in any solution is directly proportional to its mole fraction. The solutions which obey Raoults law over the entire range of concentration are known as ideal solutions.The solutions that do not obey Raoults law (non-ideal solutions) have vap...

If the points P(−3, 9), Q(a, b) and R(4, −5) are collinear and a + b = 1, find the values of a and b.

Question: If the pointsP(3, 9),Q(a,b) andR(4,5) are collinear anda+b= 1, find the values ofaandb. Solution: LetA(x1= 3,y1= 9),B(x2=a,y2=b) andC(x3= 4,y3=5) be the given points.The given points are collinear if $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow(-3)(b+5)+a(-5-9)+4(9-b)=0$ $\Rightarrow-3 b-15-14 a+36-4 b=0$ $\Rightarrow 2 a+b=3$ Now, solvinga+b= 1 and 2a+b= 3, we geta= 2 andb= 1.Hence,a= 2 andb= 1....

The partial pressure of ethane over a solution containing

Question: The partial pressure of ethane over a solution containing 6.56 103g of ethane is 1 bar. If the solution contains 5.00 102g of ethane, then what shall be the partial pressure of the gas? Solution: Molar mass of ethane (C2H6) = 2 12 + 6 1 = 30 g mol1 $\therefore$ Number of moles present in $6.56 \times 10^{-3} \mathrm{~g}$ of ethane $=\frac{6.56 \times 10^{-3}}{30}$ = 2.187 104mol Let the number of moles of the solvent bex. According to Henrys law, p=KHx $\Rightarrow 1$ bar $=K_{H} \cdot...

A uniform sphere of mass 500 g rolls

Question: A uniform sphere of mass $500 \mathrm{~g}$ rolls without slipping on a plane horizontal surface with its centre moving at a speed of $5.00 \mathrm{~cm} / \mathrm{s}$. Its kinetic energy is:$8.75 \times 10^{-4} \mathrm{~J}$$8.75 \times 10^{-3} \mathrm{~J}$$6.25 \times 10^{-4} \mathrm{~J}$$1.13 \times 10^{-3} \mathrm{~J}$Correct Option: 1 Solution: (1) $K . E$ of the sphere $=$ translational $K . E+$ rotational $K . E$ $=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}$ Where, $I=$ moment of...

Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear if

Question: Prove that the points $A(a, 0), B(0, b)$ and $C(1,1)$ are collinear ii $\left(\frac{1}{a}+\frac{1}{b}\right)=1$. Solution: Consider the pointsA(a, 0),B(0,b) andC(1, 1).Here, (x1=a,y1= 0), (x2= 0,y2=b) and (x3= 1,y3= 1).It is given that the points are collinear. So, $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow a(b-1)+0(1-0)+1(0-b)=0$ $\Rightarrow a b-a-b=o$ Divid ing the equation by $a b$ : $\Rightarrow 1-\frac{1}{b}-\frac{1}...

State Henry’s law and mention some important applications?

Question: State Henrys law and mention some important applications? Solution: Henrys law states that partial pressure of a gas in the vapour phase is proportional to the mole fraction of the gas in the solution. Ifpis the partial pressure of the gas in the vapour phase andxis the mole fraction of the gas, then Henrys law can be expressed as: p=KHx Where, KHis Henrys law constant Some important applications of Henrys law are mentioned below. (i)Bottles are sealed under high pressure to increase t...

Find a relation between x and y, if the points A(x, y), B(−5, 7) and C(−4, 5) are collinear.

Question: Find a relation betweenxandy, if the pointsA(x,y),B(5, 7) andC(4, 5) are collinear. Solution: LetA(x1=x,y1=y),B(x2= 5,y2= 7) andC(x3= 4,y3= 5) be the given points.The given points are collinear if $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow x(7-5)+(-5)(5-y)+(-4)(y-7)=0$ $\Rightarrow 7 x-5 x-25+5 y-4 y+28=0$ $\Rightarrow 2 x+y+3=0$ Hence, the required relation is 2x+y+3 = 0....

A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length l.

Question: A particle of mass $m$ is fixed to one end of a light spring having force constant $k$ and unstretched length $l$. The other end is fixed. The system is given an angular speed $\omega$ about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is:$\frac{m l \omega^{2}}{k-\omega m}$$\frac{m l \omega^{2}}{k-m \omega^{2}}$$\frac{m l \omega^{2}}{k+m \omega^{2}}$$\frac{m l \omega^{2}}{k+m \omega}$Correct Option: , 2 Solution: (2)...

A chord AB of a circle, of radius 14 cm makes an angle

Question: A chord AB of a circle, of radius 14 cm makes an angle of 60 at the centre of the circle. Find the area of the minor segment of the circle. (Use = 22/7) Solution: We know that the area of minor segment of anglein a circle of radiusris, $A=\left\{\frac{\pi \theta}{360^{\circ}}-\sin \frac{\theta}{2} \cos \frac{\theta}{2}\right\} r^{2}$ It is given that, $r=14 \mathrm{~cm}$ $\theta=60^{\circ}$ Substituting these values in above formula $A=\left\{\frac{3.14 \times 60^{\circ}}{360^{\circ}}-...

Find a relation between x and y, if the points

Question: Find a relation betweenxandy, if the pointsA(2, 1),B(x,y) andC(7, 5) are collinear. Solution: LetA(x1= 2,y1= 1),B(x2=x,y2=y) andC(x3= 7,y3= 5) be the given points.The given points are collinear if $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow 2(y-5)+x(5-1)+7(1-y)=0$ $\Rightarrow 2 y-10+4 x+7-7 y=0$ $\Rightarrow 4 x-5 y-3=0$ Hence, the required relation is 4x 5y 3 = 0....

Why do gases always tend to be less soluble in liquids

Question: Why do gases always tend to be less soluble in liquids as the temperature is raised? Solution: Solubility of gases in liquids decreases with an increase in temperature. This is because dissolution of gases in liquids is an exothermic process. Gas $+$ Liquid $\longrightarrow$ Solution $+$ Heat Therefore, when the temperature is increased, heat is supplied and the equilibrium shifts backwards, thereby decreasing the solubility of gases....

What role does the molecular interaction play in a solution of alcohol and water?

Question: What role does the molecular interaction play in a solution of alcohol and water? Solution: In pure alcohol and water, the molecules are held tightly by a strong hydrogen bonding. The interaction between the molecules of alcohol and water is weaker than alcoholalcohol and waterwater interactions. As a result, when alcohol and water are mixed, the intermolecular interactions become weaker and the molecules can easily escape. This increases the vapour pressure of the solution, which in t...

Mass per unit area of a circular disc of radius a depends

Question: Mass per unit area of a circular disc of radius a depends on the distance $r$ from its centre as $\sigma(r)=A+B r$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is:$2 \pi a^{4}\left(\frac{A}{4}+\frac{a B}{5}\right)$$2 \pi a^{4}\left(\frac{a A}{4}+\frac{B}{5}\right)$$\pi a^{4}\left(\frac{A}{4}+\frac{a B}{5}\right)$$2 \pi a^{4}\left(\frac{A}{4}+\frac{B}{5}\right)$Correct Option: 1 Solution: (1) Given, mass per unit area of ci...

A sample of drinking water was found to be severely contaminated with chloroform

Question: A sample of drinking water was found to be severely contaminated with chloroform (CHCl3) supposed to be a carcinogen. The level of contamination was 15 ppm (by mass): (i)express this in percent by mass (ii)determine the molality of chloroform in the water sample. Solution: (i) $15 \mathrm{ppm}$ (by mass) means 15 parts per million $\left(10^{6}\right)$ of the solution. Therefore, percent by mass $=\frac{15}{10^{6}} \times 100 \%$ = 1.5 103% (ii)Molar mass of chloroform (CHCl3) = 1 12 +...

A chord 10 cm long is drawn in a circle whose radius is

Question: A chord $10 \mathrm{~cm}$ long is drawn in a circle whose radius is $5 \sqrt{2} \mathrm{~cm}$. Find area of both the segments. Solution: We know that the area of minor segment of anglein a circle of radiusris, $A=\left\{\frac{\pi \theta}{360^{\circ}}-\sin \frac{\theta}{2} \cos \frac{\theta}{2}\right\} r^{2}$ It is given that the chord AB divides the circle in two segments. We have $O A=5 \sqrt{2} \mathrm{~cm}$ and $A B=10 \mathrm{~cm}$. So, $A L=\frac{A B}{2} \mathrm{~cm}$ $=\frac{10}{...

For what values of k are the points A(8, 1), B(3, −2k) and C(k, −5) collinear?

Question: For what values ofkare the pointsA(8, 1),B(3, 2k) andC(k, 5) collinear? Solution: LetA(x1= 8,y1= 1),B(x2= 3,y2= 2k) andC(x3=k,y3= 5) be the given points.The given points are collinear if $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow 8(-2 k+5)+3(-5-1)+k(1+2 k)=0$ $\Rightarrow-16 k+40-18+k+2 k^{2}=0$ $\Rightarrow 2 k^{2}-15 k+22=0$ $\Rightarrow 2 k^{2}-11 k-4 k+22=0$ $\Rightarrow k(2 k-11)-2(2 k-11)=0$ $\Rightarrow(k-2)(2 k-11)...

An antifreeze solution is prepared from 222.6 g of ethylene glycol

Question: An antifreeze solution is prepared from 222.6 g of ethylene glycol (C2H6O2) and 200 g of water. Calculate the molality of the solution. If the density of the solution is 1.072 g mL1, then what shall be the molarity of the solution? Solution: Molar mass of ethylene glycol $\left[\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{OH})_{2}\right]=2 \times 12+6 \times 1+2 \times 16$ = 62 gmol1 Number of moles of ethylene glycol $=\frac{222.6 \mathrm{~g}}{62 \mathrm{gmol}^{-1}}$ = 3.59 mol Therefore, mo...

The radius of gyration of a uniform rod of length l, about

Question: The radius of gyration of a uniform rod of length $l$, about an axis passing through a point $\frac{l}{4}$ away from the centre of the rod, and perpendicular to it, is:$\frac{1}{4} l$$\frac{1}{8} l$$\sqrt{\frac{7}{48}} l$$\sqrt{\frac{3}{8}} l$Correct Option: , 3 Solution: (3) Moment inertia of the rod passing through a point away from the centre of the rod $I=I g+m \ell^{2}$ $\Rightarrow I=\frac{M I^{2}}{12}+M \times\left(\frac{I^{2}}{16}\right)=\frac{7 M I^{2}}{48}$ Using $I=M K^{2}=\...

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