Find the greatest number of two digits which is a perfect square.
Question: Find the greatest number of two digits which is a perfect square. Solution: We know that102is equal to 100 and 92is equal to 81. Since 10 and 9 are consecutive numbers, there is no perfect square between 100 and 81. Since 100 is the first perfect square that has more than two digits, 81 is the greatest two-digit perfect square....
Read More →The atmosphere is held to the earth by
Question: The atmosphere is held to the earth by (a) gravity (b) wind (c) clouds (d) earths magnetic field. Solution: (a) gravity...
Read More →Two particles are placed at some distance.
Question: Two particles are placed at some distance. If the mass of each of the two particles is doubled, keeping the distance between them unchanged, the value of gravitational force between them will be (a) $1 / 4$ times (b) 4 times (c) $1 / 2$ times (d) unchanged. Solution: (b) Explanation: $\mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}, \mathrm{~F}^{\prime}=\frac{\mathrm{G}\left(2 m_{1}\right)\left(2 m_{2}\right)}{r^{2}}=4 \mathrm{~F}$....
Read More →Law of gravitation gives the gravitational force between
Question: Law of gravitation gives the gravitational force between (a) the earth and a point mass only (b) the earth and sun only (c) any two bodies having some mass (d) two charged bodies only. Solution: (c), $\mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}$...
Read More →By what numbers should each of the following be divided to get a perfect square in each case?
Question: By what numbers should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number. (i) 16562 (ii) 3698 (iii) 5103 (iv) 3174 (v) 1575 Solution: Factorising each number.(i) 16562 = 2 x 7 x 7 x 13 x 13Grouping them into pairs of equal factors: 16562 = 2 x (7 x 7) x (13 x 13) The factor, 2 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 16562 must be divided by 2 for it to be a p...
Read More →Find the value of x from the following:
Question: Find the value of $x$ from the following: $\left[\begin{array}{cc}2 x-y 5 \\ 3 y\end{array}\right]=\left[\begin{array}{cc}6 5 \\ 3 -2\end{array}\right]$ Solution: The corresponding elements of two equal matrices are equal. Given : $\left[\begin{array}{cc}2 x-y 5 \\ 3 y\end{array}\right]=\left[\begin{array}{cc}6 5 \\ 3 -2\end{array}\right]$ $2 x-y=6$ ...(1) $y=-2$ Putting the value of $y$ in eq. (1) $2 x-(-2)=6$ $\Rightarrow 2 x+2=6$ $\Rightarrow 2 x=6-2$ $\Rightarrow 2 x=4$ $\Rightarro...
Read More →Find the value of x from the following:
Question: Find the value of $x$ from the following: $\left[\begin{array}{cc}2 x-y 5 \\ 3 y\end{array}\right]=\left[\begin{array}{cc}6 5 \\ 3 -2\end{array}\right]$ Solution: The corresponding elements of two equal matrices are equal. Given : $\left[\begin{array}{cc}2 x-y 5 \\ 3 y\end{array}\right]=\left[\begin{array}{cc}6 5 \\ 3 -2\end{array}\right]$ $2 x-y=6$ ...(1) $y=-2$ Putting the value of $y$ in eq. (1) $2 x-(-2)=6$ $\Rightarrow 2 x+2=6$ $\Rightarrow 2 x=6-2$ $\Rightarrow 2 x=4$ $\Rightarro...
Read More →In the relation
Question: In the relation $F=G M \mathrm{~m} / \mathrm{d}^{2}$, the quantity $G$ (a) depends on the value of $g$ at the place of observation (b) is used only when the earth is one of the two masses (c) is greatest at the surface of the earth (d) is universal constant of nature. Solution: (d) Explanation: The value of ' $G$ ' is same throughout the universe....
Read More →Fill in the blanks.
Question: Fill in the blanks.(i) cot 34 tan 56 = .......(ii) cosec 31 sec 59 = .......(iii) cos267 + cos223 = ..........(iv) cosec254 tan236 = .........(v) sec240 cot250 = .......... Solution: (i) cot34 tan56 = ....... $\cot 34^{\circ}-\tan 56^{\circ}$ $=\cot \left(90^{\circ}-56^{\circ}\right)-\tan 56^{\circ}$ $=\tan 56^{\circ}-\tan 56^{\circ} \quad\left(\because \cot \left(90^{\circ}-\theta\right)=\tan \theta\right)$ $=0$ Hence, $\cot 34^{\circ}-\tan 56^{\circ}=\underline{0} .$ (ii) cosec31 sec...
Read More →A boy is whirling a stone tied with a string in an horizontal circular path the string breaks,
Question: A boy is whirling a stone tied with a string in an horizontal circular path the string breaks, the stone (a) will continue to move in the circular path (b) will move along a straight line towards the centre of the circular path (c) will move along a straight line tangential to the circular path (d) will move along a straight line perpendicular to the circular path away from the boy. Solution: (c) Explanation : Due to inertia of directions....
Read More →By what number should each of the following numbers be multiplied to get a perfect square in each case?
Question: By what number should each of the following numbers be multiplied to get a perfect square in each case? Also, find the number whose square is the new number. (i) 8820 (ii) 3675 (iii) 605 (iv) 2880 (v) 4056 (vi) 3468 (vii) 7776 Solution: Factorising each number. (i) 8820 = 2 x 2 x 3 x 3 x 5 x 7 x 7 Grouping them into pairs of equal factors: 8820 = (2 x 2) x (3 x 3) x (7 x 7) x 5 The factor, 5 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, ...
Read More →The gravitational force between two objects is F.
Question: The gravitational force between two objects is F. If masses of both objects are halved without changing distance between them, then the gravitation force would become (a) $F / 4$ (b) $F / 2$ (c) $F$ (d) $2 \mathrm{~F}$ Solution: (a) $\mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}, \mathrm{~F}^{\prime}=\frac{\mathrm{G}\left(m_{1} / 2\right)\left(m_{2} / 2\right)}{r^{2}}=\frac{\mathrm{F}}{4} .$...
Read More →Solve this
Question: If $\left[\begin{array}{rr}x+3 4 \\ y-4 x+y\end{array}\right]=\left[\begin{array}{ll}5 4 \\ 3 9\end{array}\right]$, find $x$ and $y$ Solution: The corresponding elements of two equal matrices are equal. Given : $\left[\begin{array}{cc}x+3 4 \\ y-4 x+y\end{array}\right]=\left[\begin{array}{ll}5 4 \\ 3 9\end{array}\right]$ $x+3=5$ and $y-4=3$ $\Rightarrow x=5-3$ and $y=3+4$ $\Rightarrow x=2$ and $y=7$ $\therefore x=2$ and $y=7$...
Read More →Solve this
Question: If $\left[\begin{array}{rr}x+3 4 \\ y-4 x+y\end{array}\right]=\left[\begin{array}{ll}5 4 \\ 3 9\end{array}\right]$, find $x$ and $y$ Solution: The corresponding elements of two equal matrices are equal. Given : $\left[\begin{array}{cc}x+3 4 \\ y-4 x+y\end{array}\right]=\left[\begin{array}{ll}5 4 \\ 3 9\end{array}\right]$ $x+3=5$ and $y-4=3$ $\Rightarrow x=5-3$ and $y=3+4$ $\Rightarrow x=2$ and $y=7$ $\therefore x=2$ and $y=7$...
Read More →The value of acceleration due to gravity
Question: The value of acceleration due to gravity (a) is same on equator and poles (b) is least on poles (c) is least on equator (d) increases from pole to equator. Solution: (c) Explanation: $g=\frac{\mathrm{GM}}{\mathrm{R}^{2}}$ Equatorial radius ( $R$ ) is more than the polar radius....
Read More →Find the values of x and y,
Question: Find the values of $x$ and $y$, if $2\left[\begin{array}{ll}1 3 \\ 0 x\end{array}\right]+\left[\begin{array}{ll}y 0 \\ 1 2\end{array}\right]=\left[\begin{array}{ll}5 6 \\ 1 8\end{array}\right]$ Solution: Given : $2\left[\begin{array}{ll}1 3 \\ 0 x\end{array}\right]+\left[\begin{array}{ll}y 0 \\ 1 2\end{array}\right]=\left[\begin{array}{ll}5 6 \\ 1 8\end{array}\right]$ $\Rightarrow\left[\begin{array}{cc}2 6 \\ 0 2 x\end{array}\right]+\left[\begin{array}{ll}y 0 \\ 1 2\end{array}\right]=\...
Read More →Two objects of different masses falling freely near the surface of moon would
Question: Two objects of different masses falling freely near the surface of moon would (a) have same velocities at any instant (b) have different accelerations (c) experience forces of same magnitude (d) undergo a change in their inertia. Solution: (a) Explanation : During free fall, acceleration remains the same irrespective of the mass of the object. Force is directly proportional to the mass of a freely falling object....
Read More →Write a square matrix which is both symmetric as well as skew-symmetric.
Question: Write a square matrix which is both symmetric as well as skew-symmetric. Solution: Let $A=\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\right]$ $A^{T}=\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\right]$ Since $A^{T}=A, A$ is a symmmetric matrix. Now, $-A=-\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\right]$ $\Rightarrow-A=\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\right]$ Since $A^{T}=-A, A$ is $a$ skew $-$ symmetric matrix. Thus, $A=\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\right...
Read More →If sin 3A = cos (A – 10°) and 3A is acute then A = ?
Question: If sin 3A= cos (A 10) and 3Ais acute thenA= ?(a) 15(b) 20(c) 25(d) 30 Solution: Given:sin3A= cos(A 10) $\sin 3 A=\cos \left(A-10^{\circ}\right)$ $\Rightarrow \cos \left(90^{\circ}-3 A\right)=\cos \left(A-10^{\circ}\right) \quad\left(\because \sin \theta=\cos \left(90^{\circ}-\theta\right)\right)$ $\Rightarrow 90^{\circ}-3 A=A-10^{\circ}$ $\Rightarrow 90^{\circ}+10^{\circ}=A+3 A$ $\Rightarrow 4 A=100^{\circ}$ $\Rightarrow A=\frac{100^{\circ}}{4}$ $\Rightarrow A=25^{\circ}$ Hence, the co...
Read More →Using prime factorization method, find which of the following numbers are perfect squares?
Question: Using prime factorization method, find which of the following numbers are perfect squares? 189, 225, 2048, 343, 441, 2916, 11025, 3549 Solution: (i) 189 = 3 x 3 x 3 x 7 Grouping them into pairs of equal factors: 189 = (3 x 3) x 3 x 7 The factors 3 and 7 cannot be paired. Hence, 189 is not a perfect square.(ii) 225 = 3 x 3 x 5 x 5Grouping them into pairs of equal factors: 225 = (3 x 3) x (5 x 5) There are no left out of pairs. Hence, 225 is a perfect square.(iii) 2048 = 2 x 2 x2 x 2 x 2...
Read More →What is momentum ?
Question: What is momentum ? Write its SI unit. Interpret force in terms of momentum. Represent the following graphically (a) momentum versus velocity when mass is fixed. (b) momentum versus mass when velocity is constant. Solution: Momentum, $\vec{p}=$ mass $(m) \times$ velocity $(\vec{v})=m \vec{v}$ SI unit of $\vec{p}$ is $\mathrm{kg} \mathrm{m} \mathrm{s}^{-1} . \mathrm{F}=\frac{d p}{d t}$...
Read More →Which of the following numbers are perfect squares?
Question: Which of the following numbers are perfect squares? 11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121 Solution: 11: The perfect squares closest to 11 are 9 (9 = 32) and 16 (16 = 42). Since 3 and 4 are consecutive numbers, there are no perfect squares between 9 and 16, which means that 11 is not a perfect square. 12: The perfect squares closest to 12 are 9 (9 =32) and 16 (16 = 42). Since 3 and 4 are consecutive numbers, there are no perfect squares between 9 and 16, which means that 12 is no...
Read More →If sec 4A = cosec (A – 10°) and 4A is acute then A = ?
Question: If sec 4A= cosec (A 10) and 4Ais acute thenA= ?(a) 20(b) 25(c) 30(d) 40 Solution: Given:sec4A= cosec(A 10) $\sec 4 A=\operatorname{cosec}\left(A-10^{\circ}\right)$ $\Rightarrow \operatorname{cosec}\left(90^{\circ}-4 A\right)=\operatorname{cosec}\left(A-10^{\circ}\right) \quad\left(\because \sec \theta=\operatorname{cosec}\left(90^{\circ}-\theta\right)\right)$ $\Rightarrow 90^{\circ}-4 A=A-10^{\circ}$ $\Rightarrow 90^{\circ}+10^{\circ}=A+4 A$ $\Rightarrow 5 A=100^{\circ}$ $\Rightarrow A...
Read More →Derive the unit of force using the second law of motion.
Question: Derive the unit of force using the second law of motion. A force of $5 \mathrm{~N}$ produces an acceleration of $8 \mathrm{~m} \mathrm{~s}^{-2}$ on a mass $m_{1}$ and an acceleration of $24 m \mathrm{~s}^{-2}$ on a mass $m_{2}$. What acceleration would the same force provide if both the masses are tied together? Solution: $m_{1}=\frac{\mathrm{F}}{a_{1}}=\frac{5 \mathrm{~N}}{8 \mathrm{~ms}^{-2}}=\frac{5}{8} \mathrm{~kg}, m_{2}=\frac{\mathrm{F}}{a_{2}}=\frac{5}{24} \mathrm{~kg}$ $a=\frac...
Read More →Using second law of motion, derive the relation
Question: Using second law of motion, derive the relation between force and acceleration. A bullet of $10 \mathrm{~g}$ strikes a sand-bag at a speed of $10^{3} \mathrm{~ms}^{-1}$ and gets embedded after travelling $5 \mathrm{~cm}$. Calculate (i) the resistive force exerted by the sand on the bullet (ii) the time taken by the bullet to come to rest. Solution: According to this law, the change in momentum of a body per unit time (i.e. rate of change of momentum) is directly proportional to the unb...
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