The area enclosed between the concentric circles is
Question: The area enclosed between the concentric circles is $770 \mathrm{~cm}^{2}$. If the radius of the outer circle is $21 \mathrm{~cm}$, find the radius of the inner circle. Solution: Let the radius of outer and inner two circles ber1 andr2respectively. Area enclosed between concentric circles $=\pi r_{1}{ }^{2}-\pi r_{2}{ }^{2}$ $\Rightarrow 770=\frac{22}{7}\left(21^{2}-r_{2}^{2}\right)$ $\Rightarrow 245=21^{2}-r_{2}^{2}$ $\Rightarrow r_{2}^{2}=441-245$ $\Rightarrow r_{2}^{2}=196$ $\Righta...
Read More →A block of mass m=1 kg slides with velocity v=6 m/s
Question: A block of mass $m=1 \mathrm{~kg}$ slides with velocity $v=6 \mathrm{~m} / \mathrm{s}$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about $O$ and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. If the rod has mass $M=2 \mathrm{~kg}$, and length $l=1 \mathrm{~m}$, the value of $\theta$ is approximately: (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ ) $63^{\circ}$$55^{\...
Read More →Explain
Question: Explain (i)The basis of similarities and differences between metallic and ionic crystals. (ii)Ionic solids are hard and brittle. Solution: (i)The basis of similarities between metallic and ionic crystals is that both these crystal types are held by the electrostatic force of attraction. In metallic crystals, the electrostatic force acts between the positive ions and the electrons. In ionic crystals, it acts between the oppositely-charged ions. Hence, both have high melting points. The ...
Read More →The point on the x-axis which is equidistant
Question: The point on thex-axis which is equidistant from points (1, 0) and (5, 0) is(a) (0, 2) (b) (2, 0) (c) (3, 0) (d) (0, 3) [CBSE 2013] Solution: Let A(1, 0) and B(5, 0) be the given points. Suppose the required point on thex-axis be P(x, 0).It is given that P(x, 0) is equidistant from A(1, 0) and B(5, 0). PA = PB $\Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2}$ $\Rightarrow[x-(-1)]^{2}+(0-0)^{2}=(x-5)^{2}+(0-0)^{2}$ (Using distance formula) $\Rightarrow(x+1)^{2}=(x-5)^{2}$ $\Rightarrow x^{2}...
Read More →How many lattice points are there in one unit cell of each of the following lattice?
Question: How many lattice points are there in one unit cell of each of the following lattice? (i)Face-centred cubic (ii)Face-centred tetragonal (iii)Body-centred Solution: (i)There are 14 (8 from the corners + 6 from the faces) lattice points in face-centred cubic. (ii)There are 14 (8 from the corners + 6 from the faces) lattice points in face-centred tetragonal. (iii)There are 9 (1 from the centre + 8 from the corners) lattice points in body-centred cubic....
Read More →How will you distinguish between the following pairs of terms:
Question: How will you distinguish between the following pairs of terms: (i)Hexagonal close-packing and cubic close-packing? (ii)Crystal lattice and unit cell? (iii)Tetrahedral void and octahedral void? Solution: i. A 2-D hexagonal close-packing contains two types of triangular voids (a and b) as shown in figure 1. Let us call this 2-D structure as layer A. Now, particles are kept in the voids present in layer A (it can be easily observed from figures 2 and 3 that only one of the voids will be o...
Read More →If two adjacent vertices of a parallelogram are (3, 2) and (–1, 0) and the diagonals intersect at (2, –5)
Question: If two adjacent vertices of a parallelogram are (3, 2) and (1, 0) and the diagonals intersect at (2, 5) then find the coordinates of the other two vertices Solution: Let ABCD be the parallelogram with two adjacent vertices A(3, 2) and B(1, 0). Suppose O(2, 5) be the point of intersection of the diagonals AC and BD.Let C(x1,y1) and D(x2,y2) be the coordinates of the other vertices of the parallelogram. We know that the diagonals of the parallelogram bisect each other. Therefore, O is th...
Read More →In Fig. 14.46, the area of ΔABC (in square units) is
Question: In Fig. 14.46, the area of ΔABC(in square units) is [CBSE 2013] (a) 15 (b) 10 (c) $7.5$ (d) $2.5$ Solution: The coordinates of A are (1, 3). Distance of A from thex-axis, AD =y-coordinate of A = 3 unitsThe number of units between B and C on thex-axis are 5. BC = 5 unitsNow, Area of $\triangle \mathrm{ABC}=\frac{1}{2} \times \mathrm{BC} \times \mathrm{AD}=\frac{1}{2} \times 5 \times 3=\frac{15}{2}=7.5$ square units Thus, the area of ∆ABC is 7.5 square units.Hence, the correct answer is ...
Read More →The midpoints of the sides BC, CA and AB of a ΔABC are D(3, 4), E(8, 9) and F(6, 7) respectively.
Question: The midpoints of the sidesBC,CAandABof a ΔABCareD(3, 4),E(8, 9) andF(6, 7) respectively. Find the coordinates of the vertices of the triangle. Solution: Let the coordinates of A, B, C be $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ respectively. Because D is the mid-point of BC, using mid-point formula, we have $\frac{x_{2}+x_{3}}{2}=3 \quad$ and $\quad \frac{y_{2}+y_{3}}{2}=4$ $\Rightarrow x_{2}+x_{3}=6 \quad$ and $\quad y_{2}+y_{3}=8 \quad \ld...
Read More →Moment of inertia of a cylinder of mass M, length L and radius R about
Question: Moment of inertia of a cylinder of mass $M$, length $L$ and radius $R$ about an axis passing through its centre and perpendicular to the axis of the cylinder is $I=M\left(\frac{R^{2}}{4}+\frac{L^{2}}{12}\right) .$ If such a cylinder is to be made for a given mass of a material, the ratio $L / R$ for it to have minimum possible $I$ is :$\frac{2}{3}$$\frac{3}{2}$$\sqrt{\frac{3}{2}}$$\sqrt{\frac{2}{3}}$Correct Option: , 3 Solution: (3) Let there be a cylinder of mass $m$ length $L$ and ra...
Read More →The coordinates of the point P dividing the line
Question: The coordinates of the point P dividing the line segment joining the pointsA(1, 3) andB(4, 6) in the ratio 2 : 1 are (a) $(2,4)$ (b) $(3,5)$ (c) $(4,2)$ (d) $(5,3)$ [CBSE 2012] Solution: It is given that P divides the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2 : 1.Using section formula, we get Coordinates of $\mathrm{P}=\left(\frac{2 \times 4+1 \times 1}{2+1}, \frac{2 \times 6+1 \times 3}{2+1}\right)=\left(\frac{9}{3}, \frac{15}{3}\right)=(3,5)$ Thus, the coordi...
Read More →How will you distinguish between the following pairs of terms:
Question: How will you distinguish between the following pairs of terms: (i)Hexagonal close-packing and cubic close-packing? (ii)Crystal lattice and unit cell? (iii)Tetrahedral void and octahedral void? Solution: i. A 2-D hexagonal close-packing contains two types of triangular voids (a and b) as shown in figure 1. Let us call this 2-D structure as layer A. Now, particles are kept in the voids present in layer A (it can be easily observed from figures 2 and 3 that only one of the voids will be o...
Read More →If the coordinates of one end of a diameter of a circle are (2, 3)
Question: If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (2, 5), then the coordinates of the other end of the diameter are (a) (6, 7) (b) $(6,-7)$ (c) $(6,7)$ (d) $(-6,-7)$ [CBSE 2012] Solution: Let O(2, 5) be the centre of the given circle and A(2, 3) and B(x,y) be the end points of a diameter of the circle.Then, O is the mid-point of AB.Using mid-point formula, we have $\therefore \frac{2+x}{2}=-2$ and $\frac{3+y}{2}=5$ $\Rightarrow 2+x...
Read More →Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres.
Question: Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 \mathrm{~kg}-\mathrm{m}^{2}$ and $10 \mathrm{rad} \mathrm{s}^{-1}$ respectively while those for the second one are $0.2 \mathrm{~kg}-\mathrm{m}^{2}$ and $5 \mathrm{rad} \mathrm{s}^{-1}$ respectively. At some instant they get stuck together and start rotating as a single system about thei...
Read More →In what ratio does the point
Question: In what ratio does the point $\left(\frac{24}{11}, y\right)$ divide the line segment joining the points $\mathrm{P}(2,-2)$ and $\mathrm{Q}(3,7) ?$ Also, find the value of $y .$ Solution: Let the point $\mathrm{P}\left(\frac{24}{11}, y\right)$ divides the line $\mathrm{PQ}$ in the ratio $k: 1$ Then, by the section formula: $x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}$ The coordinates of $\mathrm{R}$ are $\left(\frac{24}{11}, y\right)$. $\frac{24}{11}=\frac{3 k+2}{k+1}, ...
Read More →If the line segment joining the points (3, −4),
Question: If the line segment joining the points $(3,-4)$, and $(1,2)$ is trisected at points $P(a,-2)$ and $Q\left(\frac{5}{3}, b\right)$, Then, (a) $a=\frac{8}{3}, b=\frac{2}{3}$ (b) $a=\frac{7}{3}, b=0$ (c) $a=\frac{1}{3}, b=1$ (d) $a=\frac{2}{3}, b=\frac{1}{3}$ Solution: We have two points $A(3,-4)$ and $B(1,2)$. There are two points $P(a,-2)$ and $Q\left(\frac{5}{3}, b\right)$ which trisect the line segment joining $A$ and $B$. Now according to the section formula if any point $P$ divides a...
Read More →A uniform cylinder of mass M and radius R
Question: A uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height $a(aR)$ by applying a force $F$ at its centre ' $O$ ' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of $F$ required is : $M g \sqrt{1-\left(\frac{R-a}{R}\right)^{2}}$$M g \sqrt{\left(\frac{R}{R-a}\right)^{2}-1}$$\mathrm{Mg} \frac{a}{R}$$M g \sqrt{1-\frac{a^{2}}{R^{2}}}$Correct Option: 1 Solution: For step up, $F \times R \geq M g \tim...
Read More →'Stability of a crystal is reflected in the magnitude of its melting point'.
Question: 'Stability of a crystal is reflected in themagnitude of its melting point'. Comment. Collect melting points of solid water, ethyl alcohol, diethyl ether and methane from a data book. What can you say about the intermolecular forces between these molecules? Solution: Higher the melting point, greater is the intermolecular force of attraction and greater is the stability. A substance with higher melting point is more stable than a substance with lower melting point. The melting points of...
Read More →A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5)
Question: A line intersects they-axis andx-axis at the pointsPandQrespectively. If (2, 5) is the midpoint ofPQthen find the coordinates ofPandQ. Solution: Suppose the line intersects they-axis at P(0,y) and thex-axis at Q(x, 0). It is given that (2, 5) is the mid-point of PQ. Using mid-point formula, we have $\left(\frac{x+0}{2}, \frac{0+y}{2}\right)=(2,-5)$ $\Rightarrow\left(\frac{x}{2}, \frac{y}{2}\right)=(2,-5)$ $\Rightarrow \frac{x}{2}=2$ and $\frac{y}{2}=-5$ $\Rightarrow x=4, y=-10$ Thus, t...
Read More →The ratio in which the line segment joining points A (a1, b1)
Question: The ratio in which the line segment joining pointsA(a1,b1) andB(a2,b2) is divided byy-axis is(a) a1:a2(b)a1:a2(c)b1:b2(d) b1:b2 Solution: Let $\mathrm{P}(0, y)$ be the point of intersection of $y$-axis with the line segment joining $\mathrm{A}\left(a_{1}, b_{1}\right)$ and $\mathrm{B}\left(a_{2}, b_{2}\right)$ which divides the line segment AB in the ratio $\lambda: 1$. Now according to the section formula if point a point $P$ divides a line segment joining $A\left(x_{1}, y_{1}\right)$...
Read More →How can you determine the atomic mass of an unknown metal if you know its density and the dimension of its unit cell?
Question: How can you determine the atomic mass of an unknown metal if you know its density and the dimension of its unit cell? Explain. Solution: By knowing the density of an unknown metal and the dimension of its unit cell, the atomic mass of the metal can be determined. Leta be the edge length of a unit cell of a crystal, d be the density of the metal,mbe the mass of one atom of the metal andzbe the number of atoms in the unit cell. Now, density of the unit cell $=\frac{\text { Mass of the un...
Read More →The ratio in which the line segment joining P (x1, y1)
Question: The ratio in which the line segment joiningP(x1,y1) andQ(x2,y2) is divided by x-axis is(a) y1: y2(b) y1: y2(c) x1: x2(d) x1: x2 Solution: Let $C(x, 0)$ be the point of intersection of $x$-axis with the line segment joining $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ which divides the line segment $P Q$ in the ratio $\lambda: 1$. Now according to the section formula if point a point $\mathrm{P}$ divides a line segment joining $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $...
Read More →The midpoint P of the line segment joining the points A(−10, 4) and B(−2, 0) lies on the line segment joining the points C(−9, −4) and D(−4, y).
Question: The midpointPof the line segment joining the pointsA(10, 4) andB(2, 0) lies on the line segment joining the pointsC(9, 4) andD(4,y). Find the ratio in whichPdividesCD. Also find the value ofy. Solution: The midpoint of $A B$ is $\left(\frac{-10-2}{2}, \frac{4+0}{2}\right)=P(-6,2)$. Let $k$ be the ratio in which $P$ divides $C D$. So $(-6,2)=\left(\frac{k(-4)-9}{k+1}, \frac{k(y)-4}{k+1}\right)$ $\Rightarrow \frac{k(-4)-9}{k+1}=-6$ and $\frac{k(y)-4}{k+1}=2$ $\Rightarrow k=\frac{3}{2}$ N...
Read More →Shown in the figure is rigid and uniform one meter long rod $A B$ held in horizontal position
Question: Shown in the figure is rigid and uniform one meter long rod $A B$ held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass ' $m$ ' and has another weight of mass $2 \mathrm{~m}$ hung at a distance of 75 $\mathrm{cm}$ from $A$. The tension in the string at $A$ is :$0.5 \mathrm{mg}$$2 \mathrm{mg}$$0.75 \mathrm{mg}$$1 \mathrm{mg}$Correct Option: , 4 Solution: (4) Net torque, $\tau_{\text {net }}$ about $B$ is zero at equilibrium $\therefo...
Read More →What is meant by the term 'coordination number'?
Question: (i)What is meant by the term 'coordination number'? (ii)What is the coordination number of atoms: (a)in a cubic close-packed structure? (b)in a body-centred cubic structure? Solution: (i)The number of nearest neighbours of any constituent particle present in the crystal lattice is called its coordination number. (ii)The coordination number of atoms (a)in a cubic close-packed structure is 12, and (b)in a body-centred cubic structure is 8...
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