Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:
Question: Find the smallest number by which the given number must be divided so that the resulting number is a perfect square: (i) 14283 (ii) 1800 (iii) 2904 Solution: For each question, factorise the number into its prime factors.(i) 14283 = 3 x 3 x 3 x 23 x 23Grouping the factors into pairs: 14283 = (3 x 3) x (23 x 23) x 3 Here, the factor 3 does not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 14283 must be divided for it to ...
Read More →sin (45° + θ) – cos (45° – θ) = ?
Question: sin (45 +) cos (45 ) = ?(a) 0(b) 1(c) 2(d) 2 Solution: $\sin \left(45^{\circ}+\theta\right)-\cos \left(45^{\circ}-\theta\right)$ $=\cos \left(90^{\circ}-\left(45^{\circ}+\theta\right)\right)-\cos \left(45^{\circ}-\theta\right) \quad\left(\because \sin \theta=\cos \left(90^{\circ}-\theta\right)\right)$ $=\cos \left(90^{\circ}-45^{\circ}-\theta\right)-\cos \left(45^{\circ}-\theta\right)$ $=\cos \left(45^{\circ}-\theta\right)-\cos \left(45^{\circ}-\theta\right)$ $=0$ Hence, the correct op...
Read More →Find the smallest number by which the given number must bew multiplied so that the product is a perfect square:
Question: Find the smallest number by which the given number must bew multiplied so that the product is a perfect square: (i) 23805 (ii) 12150 (iii) 7688 Solution: Factorise each number into its prime factors. (i) 23805 = 3 x 3 x 5 x 23 x 23Grouping 23805 into pairs of equal factors: 23805 = (3 x 3) x (23 x 23) x 5 Here, the factor 5 does not occur in pairs. To be a perfect square, every prime factor has to be in pairs. Hence, the smallest number by which 23805 must be multiplied is 5.(ii) 12150...
Read More →If cos (α + β) = 0 then sin (α – β) = ?
Question: If cos ( + ) = 0 then sin ( ) = ? (a) sin 2(b) cos 2(c) sin (d) cos Solution: Given:cos(+) = 0 As we know that, $\cos 90^{\circ}=0$ Since, $\cos (\alpha+\beta)=0$ $\Rightarrow \alpha+\beta=90^{\circ}$ $\Rightarrow \alpha=90^{\circ}-\beta \quad \ldots(1)$ Now, $\sin (\alpha-\beta)=\sin \left(90^{\circ}-\beta-\beta\right)$ $=\sin \left(90^{\circ}-2 \beta\right)$ $=\cos 2 \beta \quad\left(\because \sin \left(90^{\circ}-\theta\right)=\cos \theta\right)$ Hence, the correct option is (b)....
Read More →Show that each of the following numbers is a perfect square.
Question: Show that each of the following numbers is a perfect square. Also, find the number whose square is the given number in each case: (i) 1156 (ii) 2025 (iii) 14641 (iv) 4761 Solution: In each problem, factorise the number into its prime factors. (i) 1156 = 2 x 2 x 17 x 17 Grouping the factors into pairs of equal factors, we obtain: 1156 = (2 x 2) x (17 x 17) No factors are left over. Hence, 1156 is a perfect square. Moreover, by grouping 1156 into equal factors: 1156 = (2 x 17) x (2 x 17)...
Read More →If A and B are acute angles such that sin A = cos B then (A + B) = ?
Question: IfAandBare acute angles such that sinA= cosBthen (A+B) = ?(a) 30(b) 45(c) 60(d) 90 Solution: Given:sinA= cosB $\sin A=\cos B$ $\Rightarrow \cos \left(90^{\circ}-A\right)=\cos B \quad\left(\because \sin \theta=\cos \left(90^{\circ}-\theta\right)\right)$ $\Rightarrow 90^{\circ}-A=B$ $\Rightarrow 90^{\circ}=B+A$ $\Rightarrow A+B=90^{\circ}$ Hence, the correct option is $(\mathrm{d})$....
Read More →Water sprinkler used for grass lawns begins
Question: Water sprinkler used for grass lawns begins to rotate as soon as the water is supplied. Explain the principle on which it works. Solution: Water sprinkler works on Newton's third law of motion....
Read More →Two friends on roller-skates are standing 5 metres
Question: Two friends on roller-skates are standing 5 metres apart facing each other. One of them throws a ball of 2 $\mathrm{kg}$ towards the other, who catches it. How will this activity affect the position of the two ? Explain. Solution: When one boy throws a ball towards the other boy, he moves in the backward direction to conserve the linear momentum. On the other hand, the second boy will move away from the first boy after receiving momentum from the ball. Therefore, the distance between t...
Read More →A truck of mass M is moved under a force F.
Question: A truck of mass $M$ is moved under a force $F$. If the truck is then loaded with an object equal to the mass of the truck and the driving force is halved, then how does the acceleration change? Solution: Initial acceleration, $a_{1}=\frac{F}{M}$; Final acceleration, $a_{2}=\frac{\mathrm{F} / 2}{2 \mathrm{M}}=\frac{\mathrm{F}}{4 \mathrm{M}}=\frac{a_{1}}{4}$....
Read More →sin (60° + θ) – cos (30° – θ) = ?
Question: sin (60 + ) cos (30 ) = ?(a) 2sin (b) 2cos (c) 0(d) 1 Solution: $\sin \left(60^{\circ}+\theta\right)-\cos \left(30^{\circ}-\theta\right)$ $=\cos \left(90^{\circ}-\left(60^{\circ}+\theta\right)\right)-\cos \left(30^{\circ}-\theta\right) \quad\left(\because \sin \theta=\cos \left(90^{\circ}-\theta\right)\right)$ $=\cos \left(90^{\circ}-60^{\circ}-\theta\right)-\cos \left(30^{\circ}-\theta\right)$ $=\cos \left(30^{\circ}-\theta\right)-\cos \left(30^{\circ}-\theta\right)$ $=0$ Hence, the c...
Read More →Which of the following numbers are perfect squares?
Question: Which of the following numbers are perfect squares? (i) 484 (ii) 625 (iii) 576 (iv) 941 (v) 961 (vi) 2500 Solution: (i) 484 = 222 (ii) 625 = 252 (iii) 576 = 242 (iv) Perfect squares closest to 941 are 900 (302) and 961 (312). Since 30 and 31 are consecutive numbers, there are no perfect squares between 900 and 961. Hence, 941 is not a perfect square. (v) 961 = 312 (vi) 2500 = 502 Hence, all numbers except that in (iv), i.e. 941, are perfect squares....
Read More →Velocity versus time graph of a ball of mass
Question: Velocity versus time graph of a ball of mass $50 \mathrm{~g}$ rolling on a concrete floor is shown in Fig. 1 Calculate the acceleration and frictional force of the floor on the ball. Solution: Here, $\mathrm{m}=50 \mathrm{~g}=5 \times 10^{-2} \mathrm{~kg}$ $u=80 \mathrm{~m} \mathrm{~s}^{-1}, v=0, t=8 \mathrm{~s}$ $\therefore$ Acceleration, $a=\frac{v-u}{t}=\frac{-80}{8}=-10 \mathrm{~m} \mathrm{~s}^{-2}$ Force, $\mathrm{F}=m a=-5 \times 10^{-2} \times 10=-0.5 \mathrm{~N}$ This force is ...
Read More →Solve this
Question: $\left(\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}}\right)=?$ (a) $\frac{1}{3}$ (b) $\frac{1}{\sqrt{3}}$ (c) $\sqrt{3}$ (d) $\frac{2}{\sqrt{3}}$ Solution: $\left(\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}}\right)$ $=\left(\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan \left(90^{\circ}-72^{\cir...
Read More →If A and B are symmetric matrices of the same order,
Question: IfAandBare symmetric matrices of the same order, write whetherABBAis symmetric or skew-symmetric or neither of the two. Solution: Since $A$ and $B$ are symmetric matrices, $A^{T}=A$ and $B^{T}=B$. Here, $(A B-B A)^{T}=(A B)^{T}-(B A)^{T}$ $\Rightarrow(A B-B A)^{T}=B^{T} A^{T}-A^{T} B^{T} \quad\left[\because(A B)^{T}=B^{T} A^{T}\right]$ $\Rightarrow(A B-B A)^{T}=B A-A B \quad\left[\because B^{T}=B\right.$ and $\left.A^{T}=A\right]$ $\Rightarrow(A B-B A)^{T}=-(A B-B A)$ Therefore, $A B-B...
Read More →A ball of mass $m$ is thrown vertically upward
Question: A ball of mass $m$ is thrown vertically upward with an initial speed Its speed decreases continuously till it becomes zero. Thereafter, the ball begins to fell downward and attains the speed $v$ again before striking the ground. It implies that the magnitude of initial and final momentum of the ball are same. Yet, it is not example of conservation of momentum. Explain why? (CBSE Sample Paper) Solution: Momentum of the system is conserved only if no external force acts on the system. Ho...
Read More →A horse continues to apply a force in
Question: A horse continues to apply a force in order to move a cart with a constant velocity. Explain why? Solution: The cart will move with a constant velocity if no net external force acts on it. When horse applies a force on the cart, frictional force, also acts between the tyres of the cart and the road to oppose its motion. The cart will move with constant velocity only if the force applied by the horse is equal to the frictional force....
Read More →A PT teacher wants to arrange maximum possible number of 6000 students in a field such that the number of rows is equal to the number of columns.
Question: A PT teacher wants to arrange maximum possible number of 6000 students in a field such that the number of rows is equal to the number of columns. Find the number of rows if 71 were left out after arrangement. Solution: Since 71 students were left out, there are only 5929 (6000 -71) students remaining. Hence, the number of rows or columns is simply the square root of 5929. Factorising 5929 into its prime factors: $5929=7 \times 7 \times 11 \times 11$ Grouping them into pairs of equal fa...
Read More →Two identical bullets are fired one by
Question: Two identical bullets are fired one by a light rifle and another by a heavy rifle with the same force. Which rifle will hurt the shoulder more and why? (CBSE 2012) Solution: Recoil velocity of gun $=\frac{\text { momentum of bullet }}{\text { mass of gun }}$ Recoil velocity of light rifle is more than that of heavy rifle. Therefore, light rifle will hurt the shoulder more than heavy rifle....
Read More →The students of class VIII of a school donated Rs 2401 for PM's National Relief Fund.
Question: The students of class VIII of a school donated Rs 2401 for PM's National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class. Solution: LetSbe the number of students. Letrbe the amount in rupees donated by each student. The total donation can be expressed by: $S \times r=$ Rs. 2401 Since the total amount in rupees is equal to the number of students,ris equal toS. Substituting this in the first equation: $S \t...
Read More →If A is a skew-symmetric matrix and n is an odd natural number,
Question: IfAis a skew-symmetric matrix andnis an odd natural number, write whetherAnis symmetric or skew-symmetric or neither of the two. Solution: If $A$ is a skew-symmetric matrix, then $A^{T}=-A$. $\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n} \quad[$ For all $n \in N]$ $\Rightarrow\left(A^{n}\right)^{T}=(-A)^{n} \quad\left[\because A^{T}=-A\right]$ $\Rightarrow\left(A^{n}\right)^{T}=(-1)^{n} A^{n}$ $\Rightarrow\left(A^{n}\right)^{T}=A^{n}$, if $n$ is even or $-A^{n}$, if $n$ is odd. Hence, ...
Read More →Two balls of the same size but of different materials,
Question: Two balls of the same size but of different materials, rubber and iron are kept on the smooth surface of a moving trains. The brakes are applied suddenly to stop the train. Will the balls start rolling ? If so, in which direction? Will they move with the same speed? Justify your answer. (CBSE Sample Paper) Solution: When train slows down, balls remain in motion due to inertia of motion. Hence, balls start rolling in the forward direction. Since mass of both the balls are different, so ...
Read More →Solve this
Question: $\left(\frac{2 \tan ^{2} 30^{\circ} \sec ^{2} 52^{*} \sin ^{2} 38^{*}}{\operatorname{cosec}^{2} 70^{*}-\tan ^{2} 20^{*}}\right)=?$ (a) 2(b) 1 (c) $\frac{2}{3}$ (d) $\frac{3}{2}$ Solution: $\left(\frac{2 \tan ^{2} 30^{\circ} \sec ^{2} 52^{\circ} \sin ^{2} 38^{\circ}}{\operatorname{cosec}^{2} 70^{\circ}-\tan ^{2} 20^{\circ}}\right)$ $=\left(\frac{2 \tan ^{2} 30^{\circ}\left(\sec \left(90^{\circ}-38^{\circ}\right)\right)^{2} \sin ^{2} 38^{\circ}}{\left(\operatorname{cosec}\left(90^{\circ}...
Read More →Write the prime factorization of the following numbers and hence find their square roots.
Question: Write the prime factorization of the following numbers and hence find their square roots. (i) 7744 (ii) 9604 (iii) 5929 (iv) 7056 Solution: (i) The prime factorisation of 7744: $7744=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11$ Grouping them into pairs of equal factors, we get: $7744=(2 \times 2) \times(2 \times 2) \times(2 \times 2) \times(11 \times 11)$ Taking one factor from each pair, we get: $\sqrt{7744}=2 \times 2 \times 2 \times 2 \times 11=88$ (ii) The pr...
Read More →There are three solids made up of aluminium,
Question: There are three solids made up of aluminium, steel and wood, of the same shape and same volume. Which of them would have highest inertia? (CBSE 2012) Solution: Density of steel is more than that of aluminium and wood, so its mass is greater than the solids made of aluminium and wood. Inertia depends on the mass of object. Hence, steel has the highest inertia....
Read More →A water tanker filled up to 2/3 of its height
Question: A water tanker filled up to $2 / 3$ of its height is moving with a uniform speed. On sudden application of the brake, the water in the tank would (a) move backward (b) move forward (c) be unaffected (d) rise upwards. Solution: (b) Explanation: It is due to inertia of motion....
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