Rocket works on the principle of conservation of
Question: Rocket works on the principle of conservation of (a) mass (b) energy (c) momentum (d) velocity. Solution: (c) momentum...
Read More →An object of mass 2kg is sliding with a constant velocity of
Question: An object of mass $2 \mathrm{~kg}$ is sliding with a constant velocity of $4 \mathrm{~m} \mathrm{~s}^{-1}$ on a frictionless horizontal table. The force required to keep the object moving with the same velocity is (a) $32 \mathrm{~N}$ (b) $0 \mathrm{~N}$ (c) $2 \mathrm{~N}$ (d) $8 \mathrm{~N}$. Solution: (b) Explanation : No force is needed to keep the object moving with constant velocity....
Read More →If A is a skew-symmetric matrix and n is an even natural number,
Question: If $A$ is a skew-symmetric matrix and $n$ is an even natural number, write whether $A^{n}$ is symmetric or skew-symmetric or neither of these 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 $...
Read More →Solve this
Question: $\frac{\left(\sin ^{2} 23^{\circ}+\sin ^{2} 67^{\circ}\right)}{\left(\cos ^{2} 13^{\circ}+\cos ^{2} 77^{\circ}\right)}+\sin ^{2} 59^{\circ}+\cos 59^{\circ} \sin 21^{\circ}=?$ (a) 3(b) 2(c) 1(d) 0 Solution: $\frac{\left(\sin ^{2} 23^{\circ}+\sin ^{2} 67^{\circ}\right)}{\left(\cos ^{2} 13^{\circ}+\cos ^{2} 77^{\circ}\right)}+\sin ^{2} 59^{\circ}+\cos 59^{\circ} \sin 31^{\circ}$ $=\left[\frac{\sin ^{2} 23^{*}+\left(\sin \left(90^{\circ}-23^{\circ}\right)\right)^{2}}{\cos ^{2} 13^{*}+\left...
Read More →Find the square roots of 121 and 169 by
Question: Find the square roots of 121 and 169 by the method of repeated subtraction. Solution: To find the square root of 121: 121 1 = 120 120 3 = 117 117 5 = 112 112 7 = 105 105 9 = 96 96 11 = 85 85 13 = 72 72 15 = 57 57 17 = 40 40 19 = 21 21 21 = 0 In total, there are 11 numbers to subtract from 121. Hence, the square root of 121 is 11 To find the square root of 169: 169 1 = 168 168 3 = 165 165 5 = 160 160 7 = 153 153 9 = 144 144 11 = 133 133 13 = 120 120 15 = 105 105 17 = 88 88 19 = 69 69 21...
Read More →A passenger in a moving train tosses a coin which falls behind him,
Question: A passenger in a moving train tosses a coin which falls behind him, it means that motion of the train is (a) accelerated (b) uniform (c) retarded (d) along circular tracks. Solution: (a) accelerated...
Read More →Find the least square number, exactly divisible by each one of the numbers:
Question: Find the least square number, exactly divisible by each one of the numbers: (i) 6, 9, 15 and 20 (ii) 8, 12, 15 and 20 Solution: (i) The smallest number divisible by 6, 9, 15 and 20 is their L.C.M., which is equal to 60. Factorising 60 into its prime factors: 60 = 2 x 2 x 3 x 5 Grouping them into pairs of equal factors: 60 = (2 x 2) x 3 x 5 The factors 3 and 5 are not paired. To make 60 a perfect square, we have to multiply it by 3 x 5, i.e . by15. The perfect square is 60 x 15, which i...
Read More →By pulling his hands backwards,
Question: By pulling his hands backwards, he increases the time to reduce the momentum of the ball. Hence, less force is exerted on his hands. The inertia of an object tends to cause the object (a) to increase its speed (b) to decrease its speed (c) to resist any change in its state of motion (d) to decelerate due to friction. Solution: (c) to resist any change in its state of motion...
Read More →The area of a square field is 5184 cm
Question: The area of a square field is 5184 cm2. A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area of the rectangular field. Solution: First, we have to find the perimeter of the square. The area of the square isr2, whereris the side of the square. Then, we have the equation as follows: r2= 5184 = (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x (3 x 3) Taking the square root, we getr= 2 x 2 x 2 x 3 x 3 = 72 Hence the perim...
Read More →A goalkeeper in a game of football pulls his hands backwards
Question: A goalkeeper in a game of football pulls his hands backwards after holding the ball shot at the goal. This enables the goal keeper to (a) exert larger force on the ball (b) reduce the force exerted by the ball on hands (c) increase the rate of change of momentum (d) decrease the rate of change of momentum. Solution: (b) Explanation: $\mathrm{F}=\frac{\text { change in momentum }}{\text { time }}$...
Read More →sin 38° – cos 52° = ?
Question: sin 38 cos 52 = ? (a) 0(b) 1 (c) $\frac{\sqrt{3}}{2}$ (d) $\frac{2}{\sqrt{3}}$ Solution: $\sin 38^{\circ}-\cos 52^{\circ}$ $=\sin \left(90^{\circ}-52^{\circ}\right)-\cos 52^{\circ}$ $=\cos 52^{\circ}-\cos 52^{\circ} \quad\left(\because \sin \left(90^{\circ}-\theta\right)=\cos \theta\right)$ $=0$ Hence, the correct option is (a)....
Read More →According to the third law of motion, action and reaction
Question: According to the third law of motion, action and reaction (a) always act on the same body (b) always act on different bodies in opposite directions (c) have same magnitude and directions (d) act on either body at normal to each other. Solution: (b) always act on different bodies in opposite directions...
Read More →A school collected Rs 2304 as fees from its students.
Question: A school collected Rs 2304 as fees from its students. If each student paid as many paise as there were students in the school, how many students were there in the school? Solution: LetSbe the number of students. Letrbe the money donated by each student. The total contribution can be expressed by(S)(r)= Rs 2304 Since each student paid as many paise as the number of students, thenr = S.Substituting this in the first equation, we get: S x S= 2304 S2= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 ...
Read More →Which of the following statement is not correct for an
Question: Which of the following statement is not correct for an object moving along a straight path in an accelerated motion? (a) Its speed keeps changing (b) Its velocity always changes (c) It always goes away from the earth (d) A force is always acting on it. Solution: (c) It always goes away from the earth...
Read More →A society collected Rs 92.16. Each member collected as many paise as there were members.
Question: A society collected Rs 92.16. Each member collected as many paise as there were members. How many members were there and how much did each contribute? Solution: LetMbe thenumber of members. Letrbe the amount in paise donated by each member. The total contribution can be expressed as follows: M x r= Rs 92.16 = 9216 paise Since the amount received as donation is the same as the number of members: $\therefore r=M$ Substituting this in the first equation, we get: M x M= 9216 M2= 2 x 2 x 2 ...
Read More →Solve this
Question: $\frac{2 \sin ^{2} 63^{\circ}+1+2 \sin ^{2} 27^{\circ}}{3 \cos ^{2} 17^{\circ}-2+3 \cos ^{2} 73^{\circ}}=?$ (a) $\frac{2}{3}$ (b) $\frac{3}{2}$ (c) 2(d) 3 Solution: $\frac{2 \sin ^{2} 63^{\circ}+1+2 \sin ^{2} 27^{\circ}}{3 \cos ^{2} 17^{\circ}-2+3 \cos ^{2} 73^{\circ}}$ $=\frac{2\left(\sin \left(90^{\circ}-27^{\circ}\right)\right)^{2}+1+2 \sin ^{2} 27^{\circ}}{3\left(\cos \left(90^{\circ}-73^{\circ}\right)\right)^{2}-2+3 \cos ^{2} 73^{\circ}}$ $=\frac{2 \cos ^{2} 27^{\circ}+1+2 \sin ^{...
Read More →If A is a symmetric matrix and n ∈ N,
Question: IfAis a symmetric matrix andnN, write whetherAnis symmetric or skew-symmetric or neither of these two. Solution: If $A$ is a symmetric matrix, then $A^{T}=A$. Now, $\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n} \quad[$ for all $n \in N]$ $\Rightarrow\left(A^{n}\right)^{T}=(A)^{n} \quad\left[\because A^{T}=A\right]$ Hence, $A^{n}$ is a symmetric matrix....
Read More →A welfare association collected Rs 202500 as donation from the residents.
Question: A welfare association collected Rs 202500 as donation from the residents. If each paid as many rupees as there were residents, find the number of residents. Solution: LetRbe the number of residents. Letrbe the money in rupees donated by each resident. Total donation =R x r= 202500 Since the money received as donation is the same as the number of residents: $\therefore r=R$ Substituting this in the first equation, we get: R x R= 202500 R2= 202500 R2= (2 x 2) x (5 x 5) x (5 x 5) x (3 x 3...
Read More →If A is a symmetric matrix and n ∈ N, write whether An is symmetric or skew-symmetric or neither of these two.
Question: IfAis a symmetric matrix andnN, write whetherAnis symmetric or skew-symmetric or neither of these two. Solution: If $A$ is a symmetric matrix, then $A^{T}=A$. Now, $\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n} \quad[$ for all $n \in N]$ $\Rightarrow\left(A^{n}\right)^{T}=(A)^{n} \quad\left[\because A^{T}=A\right]$ Hence, $A^{n}$ is a symmetric matrix....
Read More →If A is a symmetric matrix and n ∈ N, write whether An is symmetric or skew-symmetric or neither of these two.
Question: IfAis a symmetric matrix andnN, write whetherAnis symmetric or skew-symmetric or neither of these two. Solution: If $A$ is a symmetric matrix, then $A^{T}=A$. Now, $\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n} \quad[$ for all $n \in N]$ $\Rightarrow\left(A^{n}\right)^{T}=(A)^{n} \quad\left[\because A^{T}=A\right]$ Hence, $A^{n}$ is a symmetric matrix....
Read More →The product of two numbers is 1296.
Question: The product of two numbers is 1296. If one number is 16 times the other, find the numbers. Solution: Let the two numbers beaandb. From the first statement, we have: axb= 1296 If one number is 16 times the other, then we have: b= 16 xa. Substituting this value in the first equation, we get: ax (16 xa) = 1296 By simplifying both sides, we get: a2= 1296/16 = 81 Hence,ais the square root of 81, which is 9. To findb, use equationb= 16 xa. Sincea= 9: b= 16 x 9 = 144 So, the two numbers satis...
Read More →Find the smallest number by which 1152 must be divided so that it becomes a perfect square.
Question: Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the square root of the number so obtained. Solution: The prime factorisation of 1152: 1152 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 Grouping the factors into pairs of equal factors, we get: 1152 = (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x 2 The factor, 2, at the end, does not have a pair. Therefore, we must divide 1152 by 2 to make a perfect square. The new number is: (2 x 2) x (2 x 2) x (2 ...
Read More →sec210° – cot280° = ?
Question: sec210 cot280 = ? (a) 0(b) 1 (c) $\frac{3}{4}$ (d) $\frac{1}{2}$ Solution: $\sec ^{2} 10^{\circ}-\cot ^{2} 80^{\circ}$ $=\left(\sec \left(90^{\circ}-80^{\circ}\right)\right)^{2}-\cot ^{2} 80^{\circ}$ $=\operatorname{cosec}^{2} 80^{\circ}-\cot ^{2} 80^{\circ} \quad\left(\because \sec \left(90^{\circ}-\theta\right)=\operatorname{cosec} \theta\right)$ $=1 \quad$ (using the identity : $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$ ) Hence, the correct option is (b)....
Read More →If A is a skew-symmetric and n ∈ N such that
Question: If $A$ is a skew-symmetric and $n \in N$ such that $\left(A^{n}\right)^{T}=\lambda A^{n}$, write the value of $\lambda$. Solution: Given:Ais skew symmetric matrix. $\Rightarrow A^{T}=-A$ $\left(A^{n}\right)^{T}=\lambda A^{n}$ $\Rightarrow\left(A^{T}\right)^{n}=\lambda A^{n}$ $\Rightarrow(-A)^{n}=\lambda A^{n}$ $\Rightarrow(-1)^{n} A^{n}=\lambda A^{n}$ $\Rightarrow \lambda=(-1)^{n}$...
Read More →Find the smallest number by which 3645 must be divided so that it becomes a perfect square.
Question: Find the smallest number by which 3645 must be divided so that it becomes a perfect square. Also, find the square root of the resulting number. Solution: The prime factorisation of 3645: 3645 = 3 x 3 x 3 x 3 x 3 x 3 x 5 Grouping the factors into pairs of equal factors, we get: 3645 = (3 x 3) x (3 x 3) x (3 x 3) x 5 The factor, 5 does not have a pair. Therefore, we must divide 3645 by 5 to make a perfect square. The new number is: (3 x 3) x (3 x 3) x (3 x 3) = 729 Taking one factor from...
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