Find the value of y for which the points
Question: Find the value ofyfor which the pointsA(3, 9),B(2,y) andC(4, 5) are collinear. Solution: LetA(x1= 3,y1= 9),B(x2= 2,y2=y) andC(x3= 4,y3= 5) be the given points.The given points are collinear if $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow(-3)(y+5)+2(-5-9)+4(9-y)=0$ $\Rightarrow-3 y-15-28+36-4 y=0$ $\Rightarrow 7 y=36-43$ $\Rightarrow y=-1$...
Read More →A solution is obtained by mixing 300 g of 25% solution and 400 g of 40% solution by mass.
Question: A solution is obtained by mixing 300 g of 25% solution and 400 g of 40% solution by mass. Calculate the mass percentage of the resulting solution. Solution: Total amount of solute present in the mixture is given by, $300 \times \frac{25}{100}+400 \times \frac{40}{100}$ = 75 + 160 = 235 g Total amount of solution = 300 + 400 = 700 g Therefore, mass percentage $(w / w)$ of the solute in the resulting solution, $=\frac{235}{700} \times 100 \%$ = 33.57% And, mass percentage (w/w) of the so...
Read More →As shown in the figure, a bob of mass m
Question: As shown in the figure, a bob of mass $m$ is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius $\mathrm{r}$ and mass $\mathrm{m}$. When released from rest the bob starts falling vertically. When it has covered a distance of $\mathrm{h}$, the angular speed of the wheel will be: $\frac{1}{r} \sqrt{\frac{4 g h}{3}}$$r \sqrt{\frac{3}{2 g h}}$$\frac{1}{r} \sqrt{\frac{2 g h}{3}}$$r \sqrt{\frac{3}{4 g h}}$Correct Option: 1 Solution: (1) When the bob co...
Read More →How many mL of 0.1 M HCl are required to react completely with 1 g mixture of
Question: How many mL of 0.1 M HCl are required to react completely with 1 g mixture of Na2CO3and NaHCO3containing equimolar amounts of both? Solution: Let the amount of Na2CO3in the mixture bexg. Then, the amount of NaHCO3in the mixture is (1 x) g. Molar mass of Na2CO3= 2 23 + 1 12 + 3 16 = 106 g mol1 $\therefore$ Number of moles $\mathrm{Na}_{2} \mathrm{CO}_{3}=\frac{x}{106} \mathrm{~mol}$ Molar mass of NaHCO3= 1 23 + 1 1 12 + 3 16 = 84 g mol1 $\therefore$ Number of moles of $\mathrm{NaHCO}_{3...
Read More →Find the value of p for which the points
Question: Find the value ofpfor which the pointsA(5, 1),B(1,p) andC(4, 2) are collinear. Solution: If the area of the triangle formed by three points is equal to zero, then the points are collinear Area of the triangle formed by the vertices $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ is $\frac{1}{2}\left|x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right|$. Now, the given pointsA(5, 1),B(1,p) andC(4, 2) are c...
Read More →A chord of a circle of radius 14 cm makes a right angle at the centre.
Question: A chord of a circle of radius 14 cm makes a right angle at the centre. Find the areas of the minor and major segments of the circle. 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 of the circle of radius $r=14 \mathrm{~cm}$ makes right angle at the centre. So, $\theta=90^{\circ}$ Substituting the value of r and anglein above ...
Read More →For what value of x are the points A
Question: For what value ofxare the pointsA(3, 12),B(7, 6) andC(x, 9) collinear? Solution: A(3, 12),B(7, 6) andC(x, 9) are the given points. Then:(x1= 3,y1= 12), (x2= 7,y2= 6) and (x3=x,y3= 9)It is given that the pointsA,BandCare collinear. Therefore, $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)(6-9)+7(9-12)+x(12-6)=0$ $\Rightarrow(-3)(-3)+7(-3)+x(6)=0$ $\Rightarrow 9-21+6 x=0$ $\Rightarrow 6 x-12=0$ $\Rightarrow 6 x=12$ $\Rightar...
Read More →The linear mass density of a thin rod A B of length L varies
Question: The linear mass density of a thin rod $A B$ of length $L$ varies from A to B as $\lambda(x)=\lambda_{0}\left(1+\frac{x}{\text { L }}\right)$, where $x$ is the distance from $\mathrm{A}$. If $\mathrm{M}$ is the mass of the rod then its moment of inertia about an axis passing through $\mathrm{A}$ and perpendicular to the rod is :$\frac{5}{12} \mathrm{ML}^{2}$$\frac{7}{18} \mathrm{ML}^{2}$$\frac{2}{5} \mathrm{ML}^{2}$$\frac{3}{7} \mathrm{ML}^{2}$Correct Option: , 2 Solution: Mass of the s...
Read More →A solution of glucose in water is labelled as 10% w/w, what would be the molality and mole fraction of each component in the solution?
Question: A solution of glucose in water is labelled as 10% w/w, what would be the molality and mole fraction of each component in the solution? If the density of solution is 1.2 g mL1, then what shall be the molarity of the solution? Solution: 10% w/w solution of glucose in water means that 10 g of glucose in present in 100 g of the solution i.e., 10 g of glucose is present in (100 10) g = 90 g of water. Molar mass of glucose (C6H12O6) = 6 12 + 12 1 + 6 16 = 180 g mol1 Then, number of moles of ...
Read More →Find the value of x for which the points
Question: Find the value of $x$ for which the points $A(x, 2), B(-3,-4)$ and $C(7,-5)$ are collinear. Solution: Let $A\left(x_{1}, y_{1}\right)=A(x, 2), B\left(x_{2}, y_{2}\right)=B(-3,-4)$ and $C\left(x_{3}, y_{3}\right)=C(7,-5)$. So, the condition for three collinear points is $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ $\Rightarrow x(-4+5)-3(-5-2)+7(2+4)=0$ $\Rightarrow x+21+42=0$ $\Rightarrow x=-63$ Hence,x= 63....
Read More →A chord PQ of length 12 cm subtends an angle
Question: A chordPQof length 12 cm subtends an angle of 120 at the centre of a circle. Find the area of the minor segment cut off by the chordPQ. 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 chordPQdivides the circle in two segments. We have $\angle P O Q=120^{\circ}$ and $P Q=12 \mathrm{~cm}$. So, $P L=\frac{P Q}{2} \mathrm{~cm}$ $=\frac{...
Read More →Concentrated nitric acid used in laboratory work is 68% nitric acid by mass in aqueous solution.
Question: Concentrated nitric acid used in laboratory work is 68% nitric acid by mass in aqueous solution. What should be the molarity of such a sample of the acid if the density of the solution is 1.504 g mL1? Solution: Concentrated nitric acid used in laboratory work is 68% nitric acid by mass in an aqueous solution. This means that 68 g of nitric acid is dissolved in 100 g of the solution. Molar mass of nitric acid (HNO3) = 1 1 + 1 14 + 3 16 = 63 g mol1 Then, number of moles of $\mathrm{HNO}_...
Read More →Define the following terms:
Question: Define the following terms: (i)Mole fraction (ii)Molality (iii)Molarity (iv)Mass percentage. Solution: (i) Mole fraction: The mole fraction of a component in a mixture is defined as the ratio of the number of moles of the component to the total number of moles of all the components in the mixture. i.e., Mole fraction of a component $=\frac{\text { Number of moles of the component }}{\text { Total number of moles of all components }}$ Mole fraction is denoted by x. If in a binary soluti...
Read More →AB is a chord of a circle with centre O and radius 4 cm.
Question: ABis a chord of a circle with centreOand radius 4 cm.ABis of length 4 cm and divides the circle into two segments. Find the area of the minor segment. 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 chordABdivides the circle in two segments. We have $O A=4 \mathrm{~cm}$ and $A B=4 \mathrm{~cm}$. So, $A L=\frac{A B}{2} \mathrm{~cm}$ ...
Read More →Show that the following points are collinear:
Question: Show that the following points are collinear:(i)A(2, 2),B(3, 8) andC(1, 4)(ii)A(5, 1),B(5, 5) andC(10, 7)(iii)A(5, 1),B(1, 1) andC(11, 4)(iv)A(8, 1),B(3, 4) andC(2, 5) Solution: (i)LetA(x1= 2,y1= 2),B(x2= 3,y2= 8) andC(x3= 1,y3 = 4) be the given points. Now $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)$ $=2(8-4)+(-3)(4+2)+(-1)(-2-8)$ $=8-18+10$ $=0$ Hence, the given points are collinear. (ii)LetA(x1= 5,y1= 1),B(x2= 5,y2= 5) andC(x3= 10,y3 = ...
Read More →Define the following terms:
Question: Define the following terms: (i)Mole fraction (ii)Molality (iii)Molarity (iv)Mass percentage. Solution: (i) Mole fraction: The mole fraction of a component in a mixture is defined as the ratio of the number of moles of the component to the total number of moles of all the components in the mixture. i.e., Mole fraction of a component $=\frac{\text { Number of moles of the component }}{\text { Total number of moles of all components }}$ Mole fraction is denoted by x. If in a binary soluti...
Read More →Four point masses, each of mass m,
Question: Four point masses, each of mass $m$, are fixed at the corners of a square of side $l$. The square is rotating with angular frequency $\omega$, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is: $\mathrm{m} l^{2} \omega$$4 \mathrm{~m} l^{2} \omega$$3 \mathrm{~m} l^{2} \omega$$2 \mathrm{~m} R^{2} \omega$Correct Option: , 3 Solution: (3) Angular momentum, $L=I \omega$ $...
Read More →Give an example of solid solution in which the solute is a gas.
Question: Give an example of solid solution in which the solute is a gas. Solution: In case a solid solution is formed between two substances (one having very large particles and the other having very small particles), an interstitial solid solution will be formed. For example, a solution of hydrogen in palladium is a solid solution in which the solute is a gas....
Read More →Define the term solution.
Question: Define the term solution. How many types of solutions are formed? Write briefly about each type with an example. Solution: Homogeneous mixtures of two or more than two components are known as solutions. There are three types of solutions. (i) Gaseous solution: The solution in which the solvent is a gas is called a gaseous solution. In these solutions, the solute may be liquid, solid, or gas. For example, a mixture of oxygen and nitrogen gas is a gaseous solution. (ii) Liquid solution: ...
Read More →Figure 15.18 shows a sector of a circle of radius r cm containing an angle θ°.
Question: Figure 15.18 shows a sector of a circle of radius r cm containing an angle. The area of the sector is A cm2and perimeter of the sector is 50 cm. Prove that (i) $\theta=\frac{360}{\theta}\left(\frac{25}{r}-1\right)$ (ii) $A=25 r-r^{2}$ Solution: It is given that the radius of circle is $r \mathrm{~cm}$ and angle $\angle A O B=\theta^{\circ}$. (i) We know that the arc lengthlof a sector of an anglein a circle of radius r is $l=\frac{\theta}{360^{\circ}} \times 2 \pi r$ Perimeter of secto...
Read More →Shown in the figure is a hollow icecream cone(it is open at the top).
Question: Shown in the figure is a hollow icecream cone(it is open at the top). If its mass is $M$, radius of its top, $R$ and height, $H$, then its moment of inertia about its axis is : $\frac{M R^{2}}{2}$$\frac{M\left(R^{2}+H^{2}\right)}{4}$$\frac{M H^{2}}{3}$$\frac{M R^{2}}{3}$Correct Option: , 4 Solution: (4) Hollow ice-cream cone can be assume as several parts of discs having different radius, so $I=\int d I=\int d m\left(r^{2}\right)$ ...(i) From diagram, $\frac{r}{h}=\tan \theta=\frac{R}{...
Read More →Calculate the osmotic pressure in pascals exerted by a solution prepared by
Question: Calculate the osmotic pressure in pascals exerted by a solution prepared by dissolving 1.0 g of polymer of molar mass 185,000 in 450 mL of water at 37C. Solution: It is given that: Volume of water,V= 450 mL = 0.45 L Temperature,T= (37 + 273)K = 310 K Number of moles of the polymer, $n=\frac{1}{185000} \mathrm{~mol}$ We know that: Osmotic pressure, $\pi=\frac{n}{V} \mathrm{R} T$ $=\frac{1}{185000} \mathrm{~mol} \times \frac{1}{0.45 \mathrm{~L}} \times 8.314 \times 10^{3} \mathrm{~Pa} \m...
Read More →For what value of k (k > 0) is the area of the triangle with vertices (−2, 5), (k, −4) and
Question: For what value ofk(k 0) is the area of the triangle with vertices (2, 5), (k, 4) and(2k+ 1, 10) equal to 53 square units? Solution: Let $A\left(x_{1}=-2, y_{1}=5\right), B\left(x_{2}=k, y_{2}=-4\right)$ and $C\left(x_{3}=2 k+1, y_{3}=10\right)$ be the vertices of the triangle. So Area $(\Delta A B C)=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$ $\Rightarrow 53=\frac{1}{2}[(-2)(-4-10)+k(10-5)+(2 k+1)(5+4)]$ $\Rightarr...
Read More →Calculate the mass of ascorbic acid
Question: Calculate the mass of ascorbic acid (Vitamin C, C6H8O6) to be dissolved in 75 g of acetic acid to lower its melting point by 1.5C.Kf= 3.9 K kg mol1. Solution: Mass of acetic acid,w1= 75 g Molar mass of ascorbic acid (C6H8O6),M2= 6 12 + 8 1 + 6 16 = 176 g mol1 Lowering of melting point, ΔTf= 1.5 K We know that: $\Delta T_{f}=\frac{K_{f} \times w_{2} \times 1000}{M_{2} \times w_{1}}$ $\Rightarrow w_{2}=\frac{\Delta T_{f} \times M_{2} \times w_{1}}{K_{f} \times 1000}$ $=\frac{1.5 \times 1...
Read More →Find the value of k so that the area of the triangle with vertices A
Question: Find the value ofkso that the area of the triangle with verticesA(k+ 1, 1),B(4, 3) andC(7, k) is 6 square units. Solution: Let $A\left(x_{1}, y_{1}\right)=A(k+1,1), B\left(x_{2}, y_{2}\right)=B(4,-3)$ and $C\left(x_{3}, y_{3}\right)=C(7,-k)$. Now Area $(\Delta A B C)=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$ $\Rightarrow 6=\frac{1}{2}[(k+1)(-3+k)+4(-k-1)+7(1+3)]$ $\Rightarrow 6=\frac{1}{2}\left[k^{2}-2 k-3-4 k-4+2...
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