Solve this
Question: $\frac{1-\sin \theta}{1+\sin \theta}=(\sec \theta-\tan \theta)^{2}$ Solution: $\frac{1-\sin \theta}{1+\sin \theta}$ $=\frac{1-\sin \theta}{1+\sin \theta} \times \frac{1-\sin \theta}{1-\sin \theta}$ $=\frac{(1-\sin \theta)^{2}}{1-\sin ^{2} \theta} \quad\left[(a+b)(a-b)=a^{2}-b^{2}\right]$ $=\frac{(1-\sin \theta)^{2}}{\cos ^{2} \theta} \quad\left(\sin ^{2} \theta+\cos ^{2} \theta=1\right)$ $=\left(\frac{1-\sin \theta}{\cos \theta}\right)^{2}$ $=\left(\frac{1}{\cos \theta}-\frac{\sin \the...
Read More →Find the least number of six digits which is a perfect square.
Question: Find the least number of six digits which is a perfect square. Solution: The least number with six digits is 100000. To find the least square number with six digits, we must find the smallest number that must be added to 100000 in order to make a perfect square. For that, we have to find the square root of 100000 by the long division method as follows: 100000 is 489 (4389 3900) less than 3172. Hence, to be a perfect square, 489 should be added to 100000. 100000 + 489 = 100489 Hence, th...
Read More →Identical packets are dropped from two aeroplanes.
Question: Identical packets are dropped from two aeroplanes. One above the equator and the other above the north pole both at height h. Assuming all conditions are identical will those packets take same time to reach the surface of earth. Justify your answer. (CBSE Sample Paper) Solution: Time taken by an object to fall through height h at a place is given by $t=\sqrt{\frac{2 h}{g}}$ Since, value of ' $g$ ' at poles is greater than at the equator, therefore, packet dropped above the north pole w...
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{ll}1 2 \\ 3 4\end{array}\right]$, find $A+A^{\top}$ Solution: Given : $A=\left[\begin{array}{ll}1 2 \\ 3 4\end{array}\right]$ $A^{T}=\left[\begin{array}{ll}1 3 \\ 2 4\end{array}\right]$ $A+A^{T}=\left[\begin{array}{ll}1 2 \\ 3 4\end{array}\right]+\left[\begin{array}{ll}1 3 \\ 2 4\end{array}\right]$ $\Rightarrow A+A^{T}=\left[\begin{array}{ll}1+1 2+3 \\ 3+2 4+4\end{array}\right]$ $\Rightarrow A+A^{T}=\left[\begin{array}{ll}2 5 \\ 5 8\end{array}\right]$...
Read More →Suppose gravity of earth suddenly becomes zero,
Question: Suppose gravity of earth suddenly becomes zero, then in which direction will the moon begin to move if no other celestial body affects it? Solution: Gravity of earth provides necessary centripetal force to the moon to move in a circular path around the earth. If gravity becomes zero, there is no centripetal force and hence, the moon will begin to move in a straight line along to the tangent at the point on the circular path due to inertia of direction....
Read More →Find the least number of 4 digits which is a perfect square.
Question: Find the least number of 4 digits which is a perfect square. Solution: The least number with four digits is 1000. To find the least square number with four digits, we must find the smallest number that must be added to 1000 in order to make a perfect square. For that, we have to find the square root of 1000 by the long division method as shown below:1000 is 24 (124 100) less than the nearest square number 322. Thus, 24 must be added to 1000 to be a perfect square. 1000 + 24 = 1024 Henc...
Read More →(sec A – cos A) (cot A + tan A) = sec A tan A
Question: (secA cosA) (cotA+ tanA) = secAtanA Solution: $(\sec A-\cos A)(\cot A+\tan A)$ $=\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{\cos A}{\sin A}+\frac{\sin A}{\cos A}\right)$ $=\left(\frac{1-\cos ^{2} A}{\cos A}\right)\left(\frac{\cos ^{2} A+\sin ^{2} A}{\sin A \cos A}\right)$ $=\frac{\sin ^{2} A}{\cos A} \times \frac{1}{\sin A \cos A} \quad\left(\sin ^{2} \theta+\cos ^{2} \theta=1\right)$ $=\frac{\sin A}{\cos A} \times \frac{1}{\cos A}$ $=\tan A \sec A$...
Read More →Solve this
Question: If $\left[\begin{array}{cc}2 x+y 3 y \\ 0 4\end{array}\right]=\left[\begin{array}{ll}6 0 \\ 6 4\end{array}\right]$, then find $x$ Solution: The corresponding elements of two equal matrices are equal. Given : $\left[\begin{array}{cc}2 x+y 3 y \\ 0 4\end{array}\right]=\left[\begin{array}{ll}6 0 \\ 6 4\end{array}\right]$ $2 x+y=6 \quad \ldots(1)$ $3 y=0$ $\Rightarrow y=0$ Putting the value of $y$ in eq. (1) $2 x+0=6$ $\Rightarrow 2 x=6$ $\Rightarrow x=\frac{6}{2}$ $\therefore x=3$...
Read More →Find the greatest number of 5 digits which is a perfect square.
Question: Find the greatest number of 5 digits which is a perfect square. Solution: The greatest number with five digits is 99999. To find the greatest square number with five digits, we must find the smallest number that must be subtracted from 99999 in order to make a perfect square. For that, we have to find the square root of 99999 by the long division method as follows:Hence, we must subtract 143 from 99999 to get a perfect square: 99999 143 = 99856...
Read More →Solve this
Question: If $\left[\begin{array}{cc}2 x+y 3 y \\ 0 4\end{array}\right]=\left[\begin{array}{ll}6 0 \\ 6 4\end{array}\right]$, then find $x$ Solution: The corresponding elements of two equal matrices are equal. Given : $\left[\begin{array}{cc}2 x+y 3 y \\ 0 4\end{array}\right]=\left[\begin{array}{ll}6 0 \\ 6 4\end{array}\right]$ $2 x+y=6 \quad \ldots(1)$ $3 y=0$ $\Rightarrow y=0$ Putting the value of $y$ in eq. (1) $2 x+0=6$ $\Rightarrow 2 x=6$ $\Rightarrow x=\frac{6}{2}$ $\therefore x=3$...
Read More →Find the least number which must be added to the following numbers to make them a perfect square:
Question: Find the least number which must be added to the following numbers to make them a perfect square: (i) 5607 (ii) 4931 (iii) 4515600 (iv) 37460 (v) 506900 Solution: (i) Using the long division method:We can see that 5607 is 18 more than 752. Hence, we have to add 18 to 5607 to get a perfect square.(ii)Using the long division method:We can see that 4931 is 110 more than 712. Hence, we have to add 110 to 4931 to get a perfect square.(iii)Using the long division method:We can see that 45156...
Read More →On the earth, a stone is thrown from a height in
Question: On the earth, a stone is thrown from a height in a direction parallel to the earths surface while another stone is simultaneously dropped from the same height. Which stone would reach the ground first and why ? Solution: Both stones will reach the ground simultaneously. Initial velocity of both the stones in the downward direction is zero and the acceleration of both the stones in the downward direction is same and equal to $\mathrm{g}$. Using $h=u t+\frac{1}{2} a t^{2}$, we get $b=0+\...
Read More →Solve this
Question: $\frac{\sin ^{4} \theta+\cos ^{4} \theta}{1-2 \sin ^{2} \theta \cos ^{2} \theta}=1$ Solution: We know $\sin ^{2} \theta+\cos ^{2} \theta=1$ Squaring on both sides, we get $\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}=1$ $\Rightarrow\left(\sin ^{2} \theta\right)^{2}+\left(\cos ^{2} \theta\right)^{2}+2 \sin ^{2} \theta \cos ^{2} \theta=1 \quad\left[(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $\Rightarrow \sin ^{4} \theta+\cos ^{4} \theta=1-2 \sin ^{2} \theta \cos ^{2} \theta$ $\therefore \...
Read More →What is the source of centripetal force
Question: What is the source of centripetal force that a planet requires to revolve around the sun ? On what factors does that force depend? Solution: The source of centripetal force that a planet requires to revolve around the sun is the gravitational force between the sun and the planet. Thus, $\mathrm{F}=\frac{\mathrm{GM}_{s} \mathrm{M}_{p}}{r^{2}}$ where $m$ is the mass of the sun, is the mass of the planet and $r$ is the distance between the sun and the planet. Thus, the force depends upon ...
Read More →Solve this
Question: If matrix $A=\left[\begin{array}{lll}1 2 3\end{array}\right]$, write $A A^{\top}$. Solution: Given: $A=\left[\begin{array}{lll}1 2 3\end{array}\right]$ $A^{T}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$ $A A^{T}=\left[\begin{array}{lll}1 2 3\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$ $\Rightarrow A A^{T}=1+4+9$ $\Rightarrow A A^{T}=14$...
Read More →Find the least number which must be subtracted from the following numbers to make them a perfect square:
Question: Find the least number which must be subtracted from the following numbers to make them a perfect square: (i) 2361 (ii) 194491 (iii) 26535 (iv) 16160 (v) 4401624 Solution: (i) Using the long division method:We can see that 2361 is 57 more than 472. Hence, 57 must be subtracted from 2361 to get a perfect square.(ii) Using the long division method:We can see that 194491 is 10 more than 4412. Hence, 10 must be subtracted from 194491 to get a perfect square.(iii)Using the long division meth...
Read More →An apple falls from a tree because of gravitational
Question: An apple falls from a tree because of gravitational attraction between the earth and apple. If $F_{1}$ is the magnitude of force exerted by the earth on the apple and $F_{2}$ is the magnitude of force exerted by apple on earth, then (a) $F_{1}$ is very much greater than $F_{2}$ (b) $F_{2}$ is very much greater than $F_{1}$ (c) $F_{1}$ is only a little greater than $F_{2}$ (d) $F_{1}$ and $F_{2}$ are equal. Solution: (d) Explanation: $\overrightarrow{\mathrm{F}}_{1}=-\overrightarrow{\ma...
Read More →sin4θ – cos4θ = 1 – 2cos2θ
Question: sin4 cos4 = 1 2cos2 Solution: $\sin ^{4} \theta-\cos ^{4} \theta$ $=\left(\sin ^{2} \theta\right)^{2}-\left(\cos ^{2} \theta\right)^{2}$ $=\left(\sin ^{2} \theta+\cos ^{2} \theta\right)\left(\sin ^{2} \theta-\cos ^{2} \theta\right) \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$ $=1 \times\left(1-\cos ^{2} \theta-\cos ^{2} \theta\right) \quad\left(\sin ^{2} \theta+\cos ^{2} \theta=1\right)$ $=1-2 \cos ^{2} \theta$...
Read More →Find the square root of each of the following by long division method:
Question: Find the square root of each of the following by long division method: (i) 12544 (ii) 97344 (iii) 286225 (iv) 390625 (v) 363609 (vi) 974169 (vii) 120409 (viii) 1471369 (ix) 291600 (x) 9653449 (xi) 1745041 (xii) 4008004 (xiii) 20657025 (xiv) 152547201 (xv) 20421361 (xvi) 62504836 (xvii) 82264900 (xviii) 3226694416 (xix) 6407522209 (xx) 3915380329 Solution: (i)Hence, the square root of 12544 is 112.(ii)Hence, the square root of 97344 is 312.(iii)Hence, the square root of 286225 is 535.(i...
Read More →Find the value of x,
Question: Find the value of $x$, if $\left[\begin{array}{rr}3 x+y -y \\ 2 y-x 3\end{array}\right]=\left[\begin{array}{rr}1 2 \\ -5 3\end{array}\right]$ Solution: The corresponding elements of two equal matrices are equal. $3 x+y=1 \quad \ldots(1)$ $-y=2$ $\Rightarrow y=-2$ Putting the value of $y$ in eq. (1) $3 x+(-2)=1$ $\Rightarrow 3 x-2=1$ $\Rightarrow 3 x=1+2$ $\Rightarrow 3 x=3$ $\Rightarrow x=\frac{3}{3}=1$ $\therefore x=1$...
Read More →The weight of an object at the centre of the earth of radius R is
Question: The weight of an object at the centre of the earth of radius $R$ is (a) zero (b) infinite (c) $R$ times the weight at the surface of the earth (d) $1 / R^{2}$ times the weight at surface of the earth. Solution: (a) Explanation: $W=m g$. The value of ' $g$ ' at the centre of earth is zero....
Read More →Find the value of x,
Question: Find the value of $x$, if $\left[\begin{array}{rr}3 x+y -y \\ 2 y-x 3\end{array}\right]=\left[\begin{array}{rr}1 2 \\ -5 3\end{array}\right]$ Solution: The corresponding elements of two equal matrices are equal. $3 x+y=1 \quad \ldots(1)$ $-y=2$ $\Rightarrow y=-2$ Putting the value of $y$ in eq. (1) $3 x+(-2)=1$ $\Rightarrow 3 x-2=1$ $\Rightarrow 3 x=1+2$ $\Rightarrow 3 x=3$ $\Rightarrow x=\frac{3}{3}=1$ $\therefore x=1$...
Read More →The force of attraction between two unit point
Question: The force of attraction between two unit point masses separated by a unit distance is called (a) gravitational potential (b) acceleration due to gravity (c) gravitational field (d) universal gravitational constant. Solution: (d) Explanation: $\mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}=\frac{\mathrm{G} \times 1 \times 1}{1}=\mathrm{G}$....
Read More →Find the least number of three digits which is perfect square.
Question: Find the least number of three digits which is perfect square. Solution: Let us make a list of the squares starting from 1. 12= 1 22= 4 32= 9 42= 16 52= 25 62= 36 72= 49 82= 64 92= 81 102= 100 The square of 10 has three digits. Hence, the least three-digit perfect square is 100....
Read More →Find the value of y,
Question: Find the value of $y$, if $\left[\begin{array}{ll}x-y 2 \\ x 5\end{array}\right]=\left[\begin{array}{ll}2 2 \\ 3 5\end{array}\right]$ Solution: Given : $\left[\begin{array}{cc}x-y 2 \\ x 5\end{array}\right]=\left[\begin{array}{ll}2 2 \\ 3 5\end{array}\right]$ The corresponding elements of two equal matrices are equal. $\therefore x-y=2 \quad \ldots(1)$ and $x=3$ Putting the value of $x$ in eq. (1) $3-y=2$ $\Rightarrow 3-2=y$ $\therefore y=1$...
Read More →