A light and a heavy object have the same momentum.
Question: A light and a heavy object have the same momentum. Find out the ratio of their kinetic energies. Which one has a larger kinetic energy ? Solution: K.E. $=\frac{p^{2}}{2 m}$ For same momentum $(p), K . E . \alpha \frac{1}{m}$ $\therefore \frac{(\text { K.E }) \text { light object }}{(\text { K.E. }) \text { heavy object }}=\frac{\text { mass of heavy object }}{\text { mass of light object }}1$ Thus, K.E. of light object K.E. of heavy object....
Read More →Solve the following equations
Question: If $\left[\begin{array}{ccc}9 -1 4 \\ -2 1 3\end{array}\right]=A+\left[\begin{array}{ccc}1 2 -1 \\ 0 4 9\end{array}\right]$, then find matrix $A$. Solution: $\left[\begin{array}{ccc}9 -1 4 \\ -2 1 3\end{array}\right]=A+\left[\begin{array}{ccc}1 2 -1 \\ 0 4 9\end{array}\right]$ $\Rightarrow A=\left[\begin{array}{ccc}9 -1 4 \\ -2 1 3\end{array}\right]-\left[\begin{array}{ccc}1 2 -1 \\ 0 4 9\end{array}\right]$ $=\left[\begin{array}{ccc}9-1 -1-2 4+1 \\ -2-0 1-4 3-9\end{array}\right]$ $=\le...
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 number whose square is the resulting number. Solution: Prime factorisation of 1152: 1152 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3Grouping them into pairs of equal factors: 1152 = (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x 2 The factor, 2 at the end is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 1152 must be divided by 2 for it to be a perfec...
Read More →If an electric iron of 1200W is used for 30 minutes everyday,
Question: If an electric iron of $1200 \mathrm{~W}$ is used for 30 minutes everyday, find electric energy consumed in the month of April. Solution: Energy consumed in one day $=P \times t=1200 \mathrm{~W} \times 1 / 2 \mathrm{~h}=600 \mathrm{Wh}$ Energy consumed in 30 days $=600 \mathrm{Wh} \times 30=1800 \mathrm{Wh}=18 \mathrm{kWh}$....
Read More →Find the smallest number by which 28812 must be
Question: Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square. Solution: Prime factorisation of 28812: 28812 = 2 x 2 x 3 x 7 x 7 x 7 x 7Grouping them into pairs of equal factors: 28812 = (2 x 2) x (7 x 7) x (7 x 7) x 3 The factor, 3 is not paired. Hence, the smallest number by which 28812 must be divided such that the resulting number is a perfect square is 3....
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Question: If $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$, find the value of $x$ Solution: $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$ $\Rightarrow\left[\begin{array}{l}2 x-y \\ 3 x+y\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$ Corresponding elements of equal matrices are...
Read More →Find the smallest number by which 4851 must be multiplied
Question: Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect suqare. Solution: Prime factorisation of 4851: 4851 = 3 x 3 x 7 x 7 x 11Grouping them into pairs of equal factors: 4851 = (3 x 3) x (7 x 7) x 11 The factor, 11 is not paired. The smallest number by which 4851 must be multiplied such that the resulting number is a perfect square is 11....
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Question: If $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$, find the value of $x$ Solution: $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$ $\Rightarrow\left[\begin{array}{l}2 x-y \\ 3 x+y\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$\ Corresponding elements of equal matrices ar...
Read More →A ball is dropped from a height of
Question: A ball is dropped from a height of $10 \mathrm{~m}$. If the energy of the ball reduces by $40 \%$ after striking the ground, how much high can the ball bounce back? $\left(g=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$. (CBSE 2013) Solution: P.E. $=m g h$, Reduced energy $=\frac{60}{100} m g h=\frac{6}{10} m g h$ Let $h^{\prime}=$ height upto which body goes after bounce. $\therefore \frac{6}{10} m g h=m g h^{\prime}$ $\therefore h^{\prime}=\frac{6}{10} \times 10=6 \mathrm{~m}$...
Read More →Find the length of a side of a square playground whose area is equal to
Question: Find the length of a side of a square playground whose area is equal to the area of a rectangular field of diamensions 72 m and 338 m. Solution: The area of the playground = 72338 = 24336 m2 The length of one side of a square is equal to the square root of its area. Hence, we just need to find the square root of 24336.Hence, the length of one side of the playground is 156 metres....
Read More →Is it possible that an object is in the state
Question: Is it possible that an object is in the state of accelerated motion due to external force acting on it, but no work is being done by the force. Explain it with an example. Solution: Yes. When body moves in a circular path....
Read More →The area of a square field is
Question: The area of a square field is $30 \frac{1}{4} \mathrm{~m}^{2}$. Calculate the length of the side of the square. Solution: The length of one side is equal to the square root of the area of the field. Hence, we just need to calculate the value of$\sqrt{30 \frac{1}{4}}$ We have; $\sqrt{30 \frac{1}{4}}=\frac{\sqrt{121}}{\sqrt{4}}$ Now, calculating the square root of the numerator and the denominator: $\sqrt{121}=\sqrt{11 \times 11}=11$ $\sqrt{4}=2$ Therefore, the length of the side of the ...
Read More →For a 2 × 2 matrix A = [aij] whose elements are given by
Question: For a $2 \times 2$ matrix $A=\left[a_{i j}\right]$ whose elements are given by $a_{i j}=\frac{i}{j}$, write the value of $a_{12}$. Solution: Here, $a_{i j}=\frac{i}{j}$ $1 \leq i \leq 2$ $1 \leq j \leq 2$ $\Rightarrow a_{12}=\frac{1}{2}$ Therefore, the value of $a_{12}$ is $\frac{1}{2}$....
Read More →The velocity of a body moving in a straight
Question: The velocity of a body moving in a straight line is increased by applying a constant force $F$, for some distance in the direction of the motion. Prove that the increase in the kinetic energy of the body is equal to the work done by the force on the body. Solution: Consider a body or an object of mass $m$ moving with velocity $u$. Let a force $F$ be applied on the body so that the velocity attained by the body after travelling a distance $S$ is $v$ (Figure 5 ). Work done by the force o...
Read More →The area of a square field is
Question: The area of a square field is $80 \frac{244}{729}$ square metres. Find the length of each side of the field. Solution: The length of one side is the square root of the area of the field. Hence, we need to calculate the value of $\sqrt{80 \frac{244}{729}}$ We have $\sqrt{80 \frac{244}{729}}=\sqrt{\frac{58564}{729}}=\frac{\sqrt{58564}}{\sqrt{729}}$ Now, to calculate the square root of the numerator and the denominator:We know that: $\sqrt{729}=27$ Therefore, length of one side of the fie...
Read More →If a matrix has 5 elements, write all possible orders it can have.
Question: If a matrix has 5 elements, write all possible orders it can have. Solution: We know that if a matrix is of order $m \times n$, then it has $m n$ elements. If the matrix has 5 elements, then the number of elements will be $1 \times 5$ or $5 \times 1$, i.e. there will be 2 possible orders of the matrix....
Read More →Find the value of:
Question: Find the value of: (i) $\frac{\sqrt{80}}{\sqrt{405}}$ (ii) $\frac{\sqrt{441}}{\sqrt{625}}$ (iii) $\frac{\sqrt{1587}}{\sqrt{1728}}$ (iv) $\sqrt{72} \times \sqrt{338}$ (v) $\sqrt{45} \times \sqrt{20}$ Solution: (i)We have: $\frac{\sqrt{80}}{\sqrt{405}}=\sqrt{\frac{80}{405}}=\sqrt{\frac{16}{81}}=\frac{\sqrt{16}}{\sqrt{81}}=\frac{4}{9}$ (ii) Computing the square roots: $\sqrt{441}=\sqrt{(3 \times 3) \times(7 \times 7)}=3 \times 7=21$ $\sqrt{625}=\sqrt{(5 \times 5) \times(5 \times 5)}=5 \ti...
Read More →Find the value of:
Question: Find the value of: (i) $\frac{\sqrt{80}}{\sqrt{405}}$ (ii) $\frac{\sqrt{441}}{\sqrt{625}}$ (iii) $\frac{\sqrt{1587}}{\sqrt{1728}}$ (iv) $\sqrt{72} \times \sqrt{338}$ (v) $\sqrt{45} \times \sqrt{20}$ Solution: (i)We have: $\frac{\sqrt{80}}{\sqrt{405}}=\sqrt{\frac{80}{405}}=\sqrt{\frac{16}{81}}=\frac{\sqrt{16}}{\sqrt{81}}=\frac{4}{9}$ (ii) Computing the square roots: $\sqrt{441}=\sqrt{(3 \times 3) \times(7 \times 7)}=3 \times 7=21$ $\sqrt{625}=\sqrt{(5 \times 5) \times(5 \times 5)}=5 \ti...
Read More →Solve this
Question: If $\left[\begin{array}{cc}x x-y \\ 2 x+y 7\end{array}\right]=\left[\begin{array}{ll}3 1 \\ 8 7\end{array}\right]$, then find the value of $y$. Solution: We have, $\left[\begin{array}{cc}x x-y \\ 2 x+y 7\end{array}\right]=\left[\begin{array}{ll}3 1 \\ 8 7\end{array}\right]$ The corresponding elements of two equal matrices are equal. $\therefore x=3$ $x-y=1 \quad \ldots(1)$ Putting the value of $x$ in eq. (1) $3-y=1$ $\Rightarrow 3-1=y$ $\therefore y=2$...
Read More →The weight of a person on a planet
Question: The weight of a person on a planet $\mathrm{A}$ is about half that on the earth. He can jump upto $0.4 \mathrm{~m}$ height on the surface of the earth. How high he can jump on the planet $A$ ? Solution: P.E. of person at height $h_{1}$ on earth $=$ P.E. at height $h_{2}$ on planet i.e. $m g_{e} h_{1}=m g_{p} h_{2}$ or $h_{2}=\left(g / g_{p}\right) h_{1}=\left(\frac{g_{e}}{g / 2}\right) \times 0.4 \mathrm{~m}=0.8 \mathrm{~m}$....
Read More →The power of a motor pump is
Question: The power of a motor pump is $2 \mathrm{~kW}$. How much water per minute the pump can raise to a height of $10 \mathrm{~m}$ ? (Given $g=10 \mathrm{~ms}^{-2}$ ). Solution: $P=2 \mathrm{~kW}=2000 \mathrm{~W}, h=10 \mathrm{~m}, t=80 \mathrm{~s}$ $\mathrm{P}=\frac{m g h}{t}$ $\therefore m=\frac{\mathrm{P} t}{g h}=\frac{2000 \times 60}{10 \times 10}=1200 \mathrm{~kg}$...
Read More →Can any object have momentum even
Question: Can any object have momentum even if its mechanical energy is zero? Explain. Solution: Mechanical energy $=\mathrm{K} . \mathrm{E} .+\mathrm{P} . \mathrm{E} .=0 .$ Since, $\mathrm{K} . \mathrm{E} .=0$, so momentum $=0\left(\because \mathrm{K} . \mathrm{E}=\frac{p^{2}}{2 m}\right)$...
Read More →What is the total number of 2 × 2 matrices with each entry 0 or 1?
Question: What is the total number of $2 \times 2$ matrices with each entry 0 or $1 ?$ Solution: In a $2 \times 2$ matrix, the total number of elements are 4 and each entry can be written in 2 ways. Number of ways in which 4 entries can be written $=4^{2}=16$ [Applying the above property]...
Read More →Can any object have mechanical
Question: Can any object have mechanical energy even if its momentum is zero? Explain. Solution: Yes. Mechanical energy $=$ K.E. $+$ P.E. $=\frac{p^{2}}{2 m}+$ P.E. When, momentum $(p)=0$ Mechanical energy $=$ P.E....
Read More →If A is 2 × 3 matrix and B is a matrix
Question: If $A$ is $2 \times 3$ matrix and $B$ is a matrix such that $A^{\top} B$ and $B A^{\top}$ both are defined, then what is the order of $B$ ? Solution: Order of $A=2 \times 3$ Order of $A^{T}=3 \times 2$ Let order of $B=m \times n$ Given : $A^{T} B$ and $B A^{T}$ are defined If $A^{T}{ }_{3 \times 2} B_{m \times n}$ exists, then the number of columns in $A^{T}$ must be equal to number of rows in $B$. $\Rightarrow m=2$ If $B_{m \times n} A^{T} 3 \times 2$ exists, then the number of column...
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