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 147 must be multiplied so that it becomes a perfect square.
Question: Find the smallest number by which 147 must be multiplied so that it becomes a perfect square. Also, find the square root of the number so obtained. Solution: The prime factorisation of 147: 147 = 3 x 7 x 7 Grouping the factors into pairs of equal factors, we get: 147 = 3 x (7 x 7) The factor, 3 does not have a pair. Therefore, we must multiply 147 by 3 to make a perfect square. The new number is: (3 x 3) x (7 x 7) = 441 Taking one factor from each pair on the LHS, the square root of th...
Read More →Two stones are thrown vertically upwards simultaneously
Question: Two stones are thrown vertically upwards simultaneously with their initial velocities $u_{1}$ and $u_{2}$ respectively. Prove that the heights reached by them would be in the ratio of $u_{1}{ }^{2}: u_{2}{ }^{2}$ (Assume upward acceleration is $-g$ and downward acceleration to be $+g$ ). Solution: For Ist Stone $u=u_{1}, \quad a=-g, v=0$ (Velocity at the highest point $\left.=0\right)$ $\mathrm{S}=h_{1}$ Using $v^{2}-u^{2}=2 a S$, we get $0-u_{1}^{2}=-2 g h_{1}$ or $b_{1}=\frac{u_{1}^{...
Read More →Find the smallest number by which 180 must be multiplied so that it becomes a perfect square.
Question: Find the smallest number by which 180 must be multiplied so that it becomes a perfect square. Also, find the square root of the perfect square so obtained. Solution: The prime factorisation of 180: 180 = 2 x 2 x 3 x 3 x 5 Grouping the factors into pairs of equal factors, we get: 180 = (2 x 2) x (3 x 3) x 5 The factor, 5 does not have a pair. Therefore, we must multiply 180 by 5 to make a perfect square. The new number is: (2 x 2) x (3 x 3) x (5 x 5) = 900 Taking one factor from each pa...
Read More →Obtain the values of a, b, c, x, y and z.
Question: If $\left[\begin{array}{rcc}x+3 z+4 2 y-7 \\ 4 x+6 a-1 0 \\ b-3 3 b z+2 c\end{array}\right]=\left[\begin{array}{rrr}0 6 3 y-2 \\ 2 x -3 2 c-2 \\ 2 b+4 -21 0\end{array}\right]$ Obtain the values ofa,b,c,x,yandz. Solution: Since all the corresponding element of a matrix are equal, $x+3=0$ $\Rightarrow x=-3$ Also, $2 y-7=3 y-2$ $\Rightarrow 2 y-3 y=-2+7$ $\Rightarrow-y=5$ $\Rightarrow y=-5$ $z+4=6$ $\Rightarrow z=6-4$ $\Rightarrow z=2$ $a-1=-3$ $\Rightarrow a=-3+1$ $\Rightarrow a=-2$ $3 b...
Read More →cosec257° – tan233° = ?
Question: cosec257 tan233 = ?(a) 1(b) 0(c) 1(d) 2 Solution: $\operatorname{cosec}^{2} 57^{\circ}-\tan ^{2} 33^{\circ}$ $=\left(\operatorname{cosec}\left(90^{\circ}-33^{\circ}\right)\right)^{2}-\tan ^{2} 33^{\circ}$ $=\sec ^{2} 33^{\circ}-\tan ^{2} 33^{\circ} \quad\left(\because \operatorname{cosec}\left(90^{\circ}-\theta\right)=\sec \theta\right)$ $=1$ (using the identity : $\sec ^{2} \theta-\tan ^{2} \theta=1$ ) Hence, the correct option is (a)....
Read More →Obtain the values of a, b, c, x, y and z.
Question: If $\left[\begin{array}{rcc}x+3 z+4 2 y-7 \\ 4 x+6 a-1 0 \\ b-3 3 b z+2 c\end{array}\right]=\left[\begin{array}{rrr}0 6 3 y-2 \\ 2 x -3 2 c-2 \\ 2 b+4 -21 0\end{array}\right]$ Obtain the values ofa,b,c,x,yandz. Solution: Since all the corresponding element of a matrix are equal, $x+3=0$ $\Rightarrow x=-3$ Also, $2 y-7=3 y-2$ $\Rightarrow 2 y-3 y=-2+7$ $\Rightarrow-y=5$ $\Rightarrow y=-5$ $z+4=6$ $\Rightarrow z=6-4$ $\Rightarrow z=2$ $a-1=-3$ $\Rightarrow a=-3+1$ $\Rightarrow a=-2$ $3 b...
Read More →Find the square root of each of the following by prime factorization.
Question: Find the square root of each of the following by prime factorization. (i) 441 (ii) 196 (iii) 529 (iv) 1764 (v) 1156 (vi) 4096 (vii) 7056 (viii) 8281 (ix) 11664 (x) 47089 (xi) 24336 (xii) 190969 (xiii) 586756 (xiv) 27225 (xv) 3013696 Solution: (i) Resolving 441 into prime factors: 441 = 3 x 3 x 7 x 7 Grouping the factors into pairs of equal factors: 441 = (3 x 3) x (7 x 7) Taking one factor for each pair, we get the square root of 441: 3 x 7 = 21(ii) Resolving 196 into prime factors: 19...
Read More →Obtain a relation for the distance travelled
Question: Obtain a relation for the distance travelled by an object moving with a uniform acceleration in the interval between 4 th and 5 th seconds. Solution: We know, distance travelled by a uniformly accelerated object in time t is given by $\mathrm{S}=u t+\frac{1}{2} a t^{2}$ Distance travelled after 4 th second. $\mathrm{S}_{4}=4 u+8 a$ Distance travelled after 5 th second, $S_{5}=5 u+\frac{25}{2} a$ $\therefore$ Distance travelled between 4 th and 5 th seconds $=\mathrm{S}_{5}-\mathrm{S}_{...
Read More →sec 70° sin 20° + cos 20° cosec 70° = ?
Question: sec 70 sin 20 + cos 20 cosec 70 = ?(a) 0(b) 1(c) 2(d) 2 Solution: $\sec 70^{\circ} \sin 20^{\circ}+\cos 20^{\circ} \operatorname{cosec} 70^{\circ}$ $=\sec \left(90^{\circ}-20^{\circ}\right) \sin 20^{\circ}+\cos 20^{\circ} \operatorname{cosec}\left(90^{\circ}-20^{\circ}\right)$ $=\operatorname{cosec} 20^{\circ} \sin 20^{\circ}+\cos 20^{\circ} \sec 20^{\circ} \quad\left(\because \sec \left(90^{\circ}-\theta\right)=\operatorname{cosec} \theta\right.$ and $\left.\operatorname{cosec}\left(9...
Read More →Solve the following equations
Question: If $\left[\begin{array}{cc}x-y z \\ 2 x-y \omega\end{array}\right]=\left[\begin{array}{rr}-1 4 \\ 0 5\end{array}\right]$, find $x, y, z$ 우$\omega$ Solution: Given : $\left[\begin{array}{cc}x-y z \\ 2 x-y w\end{array}\right]=\left[\begin{array}{cc}-1 4 \\ 0 5\end{array}\right]$ Since all the corresponding elements of a matrix are equal, $x-y=-1$ $\Rightarrow x=-1+y$ ....(1) $2 x-y=0$ .....(2) $z=4$ $w=5$ Putting the value of $x$ in eq. (2), we get $2(-1+y)-y=0$ $\Rightarrow-2+2 y-y=0$ $...
Read More →An electron moving with a velocity
Question: An electron moving with a velocity of $5 \times 10^{4} \mathrm{~ms}^{-1}$ enters into a uniform electric field and acquires a uniform acceleration of $104 \mathrm{~ms}^{-2}$ in the direction of its initial motion. (i) Calculate the time in which the electron would acquire a velocity double of its initial velocity. (ii) How much distance the electron would cover in this time? Solution: Here, $u=5 \times 10^{4} \mathrm{~ms}^{-1}, a=10^{4} \mathrm{~ms}^{-2}$ (i) Let after time $t, v=2 u=1...
Read More →sin 43° cos 47° + cos 43° sin 47° = ?
Question: sin 43 cos 47 + cos 43 sin 47 = ?(a) 0(b) 1(c) sin 4(d) cos 4 Solution: $\sin 43^{\circ} \cos 47^{\circ}+\cos 43^{\circ} \sin 47^{\circ}$ $=\sin \left(90^{\circ}-47^{\circ}\right) \cos 47^{\circ}+\cos \left(90^{\circ}-47^{\circ}\right) \sin 47^{\circ}$ $=\cos 47^{\circ} \cos 47^{\circ}+\sin 47^{\circ} \sin 47^{\circ} \quad\left(\because \sin \left(90^{\circ}-\theta\right)=\cos \theta\right.$ and $\left.\cos \left(90^{\circ}-\theta\right)=\sin \theta\right)$ $=\cos ^{2} 47^{\circ}+\sin ...
Read More →Solve the following equations
Question: If $\left[\begin{array}{cc}x-y z \\ 2 x-y \omega\end{array}\right]=\left[\begin{array}{rr}-1 4 \\ 0 5\end{array}\right]$, find $x, y, z$ 우$\omega$ Solution: Given : $\left[\begin{array}{cc}x-y z \\ 2 x-y w\end{array}\right]=\left[\begin{array}{cc}-1 4 \\ 0 5\end{array}\right]$ Since all the corresponding elements of a matrix are equal, $x-y=-1$ $\Rightarrow x=-1+y$ ....(1) $2 x-y=0$ .....(2) Putting the value of $x$ in eq. (2), we get $2(-1+y)-y=0$ $\Rightarrow-2+2 y-y=0$ $\Rightarrow-...
Read More →Using following data, draw time-displacement graph for a moving object:
Question: Using following data, draw time-displacement graph for a moving object: Use this graph to find average velocity for first $4 \mathrm{~s}$, for next $4 \mathrm{~s}$ and for last $6 \mathrm{~s}$. Solution: Displacement-time graph is shown in figure 8 . Average velocity for first $4 \mathrm{~s}$ = Slope of displacement-time graph (i.e. OA part) $=\frac{4 m}{4 s}=1 \mathrm{~ms}^{-1}$ Average velocity for next $4 s$ (i.e. in the interval of $4 s$ to $8 s$ ) = zero. Average velocity for last...
Read More →Write the possible unit's digits of the square root of the following numbers.
Question: Write the possible unit's digits of the square root of the following numbers. Which of these numbers are odd square roots? (i) 9801 (ii) 99856 (iii) 998001 (iv) 657666025 Solution: (i) The unit digit of the number 9801 is 1. So, the possible unit digits are 1 or 9 (Table 3.4). Note that 9801 is equal to 992. Hence, the square root is an odd number. (ii) The unit digit of the number 99856 is 6. So, the possible unit digits are 4 or 6 (Table 3.4). Since its last digit is 6 (an even numbe...
Read More →cos 1° cos 2° cos 3° ... cos 180° = ?
Question: cos 1 cos 2 cos 3 ... cos 180 = ? (a) 0(b) 1(c) 1 (d) $\frac{1}{2}$ Solution: $\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \ldots \cos 180^{\circ}$ $=\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \ldots \cos 90^{\circ} \ldots \ldots \cos 180^{\circ}$ $=\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \ldots \times 0 \times \ldots \ldots \cos 180^{\circ} \quad\left(\because \cos 90^{\circ}=0\right)$ $=0$ Hence, the correct option is (a)....
Read More →Find the squares of the following numbers by visual method:
Question: Find the squares of the following numbers by visual method: (i) 52 (ii) 95 (iii) 505 (iv) 702 (v) 99 Solution: (i) We have: 52 = 50 + 2 Let us draw a square having side 52 units. Let us split it into 50 units and 2 units. The sum of the areas of these four parts is the square of 52. Thus, the square of 52 is 2704. (ii) We have: 95 = 90 + 5 Let us draw a square having side 95 units. Let us split it into 90 units and 5 units. The sum of the areas of these four parts is the square of 95. ...
Read More →Solve the following equations
Question: If $\left[\begin{array}{cc}x-y z \\ 2 x-y \omega\end{array}\right]=\left[\begin{array}{rr}-1 4 \\ 0 5\end{array}\right]$, find $x, y, z$ 우$\omega$ Solution: Given : $\left[\begin{array}{cc}x-y z \\ 2 x-y w\end{array}\right]=\left[\begin{array}{cc}-1 4 \\ 0 5\end{array}\right]$ Since all the corresponding elements of a matrix are equal, $x-y=-1$ $\Rightarrow x=-1+y$ ....(1) $2 x-y=0$ .....(2) Putting the value of $x$ in eq. (2), we get $2(-1+y)-y=0$ $\Rightarrow-2+2 y-y=0$ $\Rightarrow-...
Read More →An object starting from rest travels
Question: An object starting from rest travels $20 \mathrm{~m}$ in first $2 \mathrm{~s}$ and $160 \mathrm{~m}$ in next $4 \mathrm{~s}$. What will be the velocity after $7 \mathrm{~s}$ from the start? Solution: Ist Case : Here $u=0, \mathrm{~S}_{1}=20 \mathrm{~m}, t_{1}=2 \mathrm{~s}$ Using $S_{1}=u t_{1}+\frac{1}{2} a t_{1}^{2}$, we get $20=0+\frac{1}{2} a \times 4=2 a$ or $a=10 \mathrm{~ms}^{-2}$ 2nd Case : $\mathrm{S}_{2}=160 \mathrm{~m}, t_{2}=4 \mathrm{~s}$ $\therefore S_{1}+S_{2}=20+160=180...
Read More →tan 5° tan 25° tan 30° tan 65° tan 85° = ?
Question: tan 5 tan 25 tan 30 tan 65 tan 85 = ? (a) 1 (b) $\sqrt{3}$ (c) $\frac{1}{\sqrt{3}}$ (d) $\frac{1}{2}$ Solution: $\tan 5^{\circ} \tan 25^{\circ} \tan 30^{\circ} \tan 65^{\circ} \tan 85^{\circ}$ $=\tan \left(90^{\circ}-85^{\circ}\right) \tan 25^{\circ} \tan 30^{\circ} \tan \left(90^{\circ}-25^{\circ}\right) \tan 85^{\circ}$ $=\cot 85^{\circ} \tan 25^{\circ} \tan 30^{\circ} \cot 25^{\circ} \tan 85^{\circ} \quad\left(\because \tan \left(90^{\circ}-\theta\right)=\cot \theta\right)$ $=\left(...
Read More →Find the squares of the following numbers using the identity
Question: Find the squares of the following numbers using the identity (ab)2=a2 2ab+b2: (i) 395 (ii) 995 (iii) 495 (iv) 498 (v) 99 (vi) 999 (vii) 599 Solution: (i)Decomposing: 395 = 400 5 Here,a= 400 andb= 5 Using the identity (ab)2=a2 2ab+b2: 3952= (4005)2= 40022(400)(5) + 52= 160000 4000 + 25 = 156025 (ii) Decomposing: 995 = 1000 5 Here,a= 1000 andb= 5 Using the identity (ab)2=a2 2ab+b2: 9952= (10005)2= 100022(1000)(5) + 52= 1000000 10000 + 25 = 990025 (iii) Decomposing: 495 = 500 5 Here,a= 50...
Read More →Find the squares of the following numbers using the identity
Question: Find the squares of the following numbers using the identity (a+b)2=a2+ 2ab+b2:(i) 405 (ii) 510 (iii) 1001 (iv) 209 (v) 605 Solution: (i) On decomposing: 405 = 400 + 5 Here,a= 400 andb= 5 Using the identity (a+b)2=a2+ 2ab+b2: 4052= (400 + 5)2= 4002+ 2(400)(5) + 52 = 160000 + 4000 + 25 = 164025 (ii) On decomposing: 510 = 500 + 10 Here,a= 500 andb= 10 Using the identity (a+b)2=a2+ 2ab+b2: 5102= (500 + 10)2= 5002+ 2(500)(10) + 102 = 250000 + 10000 + 100 = 260100 (iii) On decomposing: 1001...
Read More →Solve this
Question: If $\left[\begin{array}{cc}x 3 x-y \\ 2 x+z 3 y-\omega\end{array}\right]=\left[\begin{array}{ll}3 2 \\ 4 7\end{array}\right]$, find $x, y, z, \omega$. Solution: Since all the corresponding elements of a matrix are equal, $\left[\begin{array}{cc}x 3 x-y \\ 2 x+z 3 y-w\end{array}\right]=\left[\begin{array}{ll}3 2 \\ 4 7\end{array}\right]$ $x=3$ .....(1) $3 x-y=2$ ....(2) Putting the value of $x$ in eq. (2), we get $3(3)-y=2$ $\Rightarrow 9-y=2$ $\Rightarrow-y=-7$ $\Rightarrow y=7$ $2 x+z...
Read More →tan 10° tan 15° tan 75° tan 80° = ?
Question: tan 10 tan 15 tan 75 tan 80 = ? (a) $\sqrt{3}$ (b) 1 (c) $\frac{1}{\sqrt{3}}$ (d) 1 Solution: $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$ $=\tan \left(90^{\circ}-80^{\circ}\right) \tan 15^{\circ} \tan \left(90^{\circ}-15^{\circ}\right) \tan 80^{\circ}$ $=\cot 80^{\circ} \tan 15^{\circ} \cot 15^{\circ} \tan 80^{\circ} \quad\left(\because \tan \left(90^{\circ}-\theta\right)=\cot \theta\right)$ $=\left(\frac{1}{\tan 80^{\circ}} \tan 80^{\circ}\right)\left(\tan 15^{\c...
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