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Question: If $\sqrt{3} \tan \theta=3 \sin \theta$, find the value of $\sin \theta$. Solution: Given : $\sqrt{3} \tan \theta=3 \sin \theta$ $\sqrt{3} \tan \theta=3 \sin \theta$ $\Rightarrow \sqrt{3} \frac{\sin \theta}{\cos \theta}=3 \sin \theta$ $\Rightarrow \frac{\sqrt{3} \sin \theta}{\cos \theta}-3 \sin \theta=0$ $\Rightarrow \frac{\sqrt{3} \sin \theta-3 \sin \theta \cos \theta}{\cos \theta}=0$ $\Rightarrow \sqrt{3} \sin \theta-3 \sin \theta \cos \theta=0$ $\Rightarrow \sqrt{3} \sin \theta(1-\s...
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Question: If $\cot A+\frac{1}{\cot A}=2$, find the value of $\left(\cot ^{2} A+\frac{1}{\cot ^{2} A}\right)$ Solution: Given : $\cot A+\frac{1}{\cot A}=2$ $\cot A+\frac{1}{\cot A}=2$ Squaring both sides, we get $\Rightarrow\left(\cot A+\frac{1}{\cot A}\right)^{2}=2^{2}$ $\Rightarrow \cot ^{2} A+\left(\frac{1}{\cot A}\right)^{2}+2(\cot A)\left(\frac{1}{\cot A}\right)=4$ $\Rightarrow \cot ^{2} A+\frac{1}{\cot ^{2} A}+2=4$ $\Rightarrow \cot ^{2} A+\frac{1}{\cot ^{2} A}=4-2$ $\Rightarrow \cot ^{2} A...
Read More →If A is a square matrix such that
Question: If $A$ is a square matrix such that $A^{2}=A$, then $(I+A)^{3}-7 A$ is equal to (a) $A$ (b) $I-A$ (c) $I$ (d) $3 \mathrm{~A}$ Solution: (c) $I$ Here, $A^{2}=A \quad \ldots(1)$ $A^{3}=A^{2} A$ $=A^{2} \quad$ [From eq. (1) $]$ $=A$ $\therefore A^{3}=A \quad \ldots(2)$ We know that $(I+A)^{3}=I^{3}+3(I)^{2} A+3(I) A^{2}+A^{3}$ $\Rightarrow(I+A)^{3}=I+3 A+3 A+A \quad$ [From eqs. (1) and (2)] $\Rightarrow(I+A)^{3}=I+7 A$ $\Rightarrow(I+A)^{3}-7 A=I$...
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Question: In the figure of $\triangle \mathrm{PQR}, \angle \mathrm{P}=\theta^{\circ}$ and $\angle \mathrm{R}=\phi^{\circ}$. Find (i) $(\sqrt{x+1}) \cot \phi$ (ii) $\left(\sqrt{x^{3}+x^{2}}\right) \tan \theta$ (iii) $\cos \theta$ Solution: In $\triangle \mathrm{PQR}, \angle \mathrm{Q}=90^{\circ}$ Using Pythagoras theorem, we get $\mathrm{PQ}=\sqrt{\mathrm{PR}^{2}-\mathrm{QR}^{2}}$ $=\sqrt{(x+2)^{2}-x^{2}}$ $=\sqrt{x^{2}+4 x+4-x^{2}}$ $=\sqrt{4(x+1)}$ $=2 \sqrt{x+1}$ Now, (i) $(\sqrt{x+1}) \cot \p...
Read More →If S = [Sij] is a scalar matrix such that sij = k and A is a square matrix of the same order, then AS = SA = ?
Question: If $S=\left[S_{i j}\right]$ is a scalar matrix such that $s_{i j}=k$ and $A$ is a square matrix of the same order, then $A S=S A=?$ (a) $A^{k}$ (b) $k+A$ (c) $k A$ (d) $k S$ Solution: (c) $k A$ Here, $S=\left[S_{i j}\right]$ $\Rightarrow S=\left[\begin{array}{ll}k 0 \\ 0 k\end{array}\right] \quad\left[\because S_{i j}=k\right]$ Let $A=\left[\begin{array}{ll}a_{11} a_{12} \\ a_{21} a_{22}\end{array}\right] \quad[\because A$ is square matrix $]$' Now, $A S=\left[\begin{array}{ll}a_{11} a...
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Question: If $x=\cot \mathrm{A}+\cos \mathrm{A}$ and $y=\cot \mathrm{A}-\cos \mathrm{A}$, prove that $\left(\frac{x-y}{x+y}\right)^{2}+\left(\frac{x-y}{2}\right)^{2}=1$. Solution: LHS $=\left(\frac{x-y}{x+y}\right)^{2}+\left(\frac{x-y}{2}\right)^{2}$ $=\left[\frac{(\cot \mathrm{A}+\cos \mathrm{A})-(\cot \mathrm{A}-\cos \mathrm{A})}{(\cot \mathrm{A}+\cos \mathrm{A})+(\cot \mathrm{A}-\cos \mathrm{A})}\right]^{2}+\left[\frac{(\cot \mathrm{A}+\cos \mathrm{A})-(\cot \mathrm{A}-\cos \mathrm{A})}{2}\ri...
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Question: If $A=\left[\begin{array}{cc}\alpha \beta \\ \gamma -\alpha\end{array}\right]$ is such that $A^{2}=1$, then (a) $1+\alpha^{2}+\beta y=0$ (b) $1-\alpha^{2}+\beta y=0$ (c) $1-\alpha^{2}-\beta y=0$ (d) $1+\alpha^{2}-\beta y=0$ Solution: (c) $1-\alpha^{2}-\beta y=0$ Here, $A^{2}=I$ $\Rightarrow\left[\begin{array}{cc}\alpha \beta \\ \gamma -\alpha\end{array}\right]\left[\begin{array}{cc}\alpha \beta \\ \gamma -\alpha\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ $\R...
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Question: If $x=\operatorname{cosec} \mathrm{A}+\cos \mathrm{A}$ and $y=\operatorname{cosec} \mathrm{A}-\cos \mathrm{A}$, then prove that $\left(\frac{2}{x+y}\right)^{2}+\left(\frac{x-y}{2}\right)^{2}-1=0$ Solution: LHS $=\left(\frac{2}{x+y}\right)^{2}+\left(\frac{x-y}{2}\right)^{2}-1$ $=\left[\frac{2}{(\operatorname{cosec} A+\cos A)+(\operatorname{cosec} A-\cos A)}\right]^{2}+\left[\frac{(\operatorname{cosec} A+\cos A)-(\operatorname{cosec} A-\cos A)}{2}\right]^{2}-1$ $=\left[\frac{2}{\operator...
Read More →Solve the following equations for
Question: If $A=\left[\begin{array}{ll}1 -1 \\ 2 -1\end{array}\right], B=\left[\begin{array}{rr}a 1 \\ b -1\end{array}\right]$ and $(A+B)^{2}=A^{2}+B^{2}$, values of a and b are (a) $a=4, b=1$ (b) $a=1, b=4$ (c) $a=0, b=4$ (d) $a=2, b=4$ Solution: (b) $a=1, b=4$ Here, $(A+B)^{2}=A^{2}+B^{2}$ $\Rightarrow A^{2}+A B+B A+B^{2}=A^{2}+B^{2}$ $\Rightarrow A B+B A=O$ $\Rightarrow A B=-B A$ $\Rightarrow\left[\begin{array}{ll}1 -1 \\ 2 -1\end{array}\right]\left[\begin{array}{cc}a 1 \\ b -1\end{array}\rig...
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Question: If $A=\left[\begin{array}{lll}1 2 x \\ 0 1 0 \\ 0 0 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}1 -2 y \\ 0 1 0 \\ 0 0 1\end{array}\right]$ and $A B=I_{3}$, then $x+y$ equals (a) 0 (b) $-1$ (c) 2 (d) none of these Solution: (a) 0 Given : $A B=I_{3}$ $\Rightarrow\left[\begin{array}{lll}1 2 x \\ 0 1 0 \\ 0 0 1\end{array}\right]\left[\begin{array}{ccc}1 -2 y \\ 0 1 0 \\ 0 0 1\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right]$ $\Rightarrow \quad\le...
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Question: If $A=\left[\begin{array}{ll}1 a \\ 0 1\end{array}\right]$, then $A^{n}($ where $n \in N)$ equals (a) $\left[\begin{array}{cc}1 n a \\ 0 1\end{array}\right]$ (b) $\left[\begin{array}{cc}1 n^{2} a \\ 0 1\end{array}\right]$ (c) $\left[\begin{array}{cc}1 n a \\ 0 0\end{array}\right]$ (d) $\left[\begin{array}{cc}n n a \\ 0 n\end{array}\right]$ Solution: (a) $\left[\begin{array}{cc}1 n a \\ 0 1\end{array}\right]$ Here, $A=\left[\begin{array}{ll}1 a \\ 0 1\end{array}\right]$ $\Rightarrow A^{...
Read More →From a point on the ground,
Question: From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are 45 and 60 respectively. Find height of the tower. Solution: It is given that tower is placed at a 20 m high building. The top and bottom of the tower makes an angle of respectively with the ground. We have to find the height of the tower. LetDBis the tower LetADis the building Height of the building = 20 m Height of the tower =xm According to th...
Read More →Solve the following equations for
Question: If $A=\left[\begin{array}{lll}n 0 0 \\ 0 n 0 \\ 0 0 n\end{array}\right]$ and $B=\left[\begin{array}{lll}a_{1} a_{2} a_{3} \\ b_{1} b_{2} b_{3} \\ c^{1} c_{2} c_{3}\end{array}\right]$, then $A B$ is equal to (a) $B$ (b) $n B$ (c) $B^{n}$ (d) $A+B$ Solution: (b) $n B$ Here, $A=\left[\begin{array}{lll}n 0 0 \\ 0 n 0 \\ 0 0 n\end{array}\right]$ and $B=\left[\begin{array}{lll}a_{1} a_{2} a_{3} \\ b_{1} b_{2} b_{3} \\ c_{1} c_{2} c_{3}\end{array}\right]$ $\therefore A B=\left[\begin{array}{c...
Read More →The slant height of the frustum of a cone is 4 cm
Question: The slant height of the frustum of a cone is 4 cm and the circumferences of its circular ends are 18 cm and 6 cm. Find curved surface area of the frustum. Solution: It is given that slant height of frustum of a cone is 4 cm. Circumferences of its ends are 18 cm and 6 cm. We have to find the curved surface area of the frustum. Letlbe the slant height Letbe the radii of two circular ends of the cone Circumference of one end Circumference of other end Now, $2 \pi r=6$ $\pi r=3$ $2 \pi R=1...
Read More →A well of diameter 3 m ad 14 m deep is dug.
Question: A well of diameter 3 m ad 14 m deep is dug. The earth, taken out of it, has been evenly spread all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment. Solution: Given a well with diameter 3 m and height 14 m. The earth dug out from well is used to make a circular embankment of 4m width. We have to find the height of the embankment. LetRbe the radius of well LetHbe the height of well Letrbe the radius of embankment Lethbe the ...
Read More →If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
Question: If $A, B$ are square matrices of order $3, A$ is non-singular and $A B=0$, then $B$ is a (a) : matrix (b) singular matrix (c) unit-matrix (d) non-singular matrix Solution: (a) : matrixSinceAis non-singular matrix and the determinant of a non-singular matrix is non-zero,Bshould be a : matrix....
Read More →If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
Question: If $A, B$ are square matrices of order $3, A$ is non-singular and $A B=0$, then $B$ is a (a) : matrix (b) singular matrix (c) unit-matrix (d) non-singular matrix Solution: (a)a: matrixSinceAis non-singular matrix and the determinant of a non-singular matrix is non-zero,Bshould be a : matrix....
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Question: Let $A=\left[\begin{array}{lll}a 0 0 \\ 0 a 0 \\ 0 0 a\end{array}\right]$, then $A^{n}$ is equal to (a) $\left[\begin{array}{ccc}a^{n} 0 0 \\ 0 a^{n} 0 \\ 0 0 a\end{array}\right]$ (b) $\left[\begin{array}{ccc}a^{n} 0 0 \\ 0 a 0 \\ 0 0 a\end{array}\right]$ (c) $\left[\begin{array}{ccc}a^{n} 0 0 \\ 0 a^{n} 0 \\ 0 0 a^{n}\end{array}\right]$ (d) $\left[\begin{array}{ccc}n a 0 0 \\ 0 n a 0 \\ 0 0 n a\end{array}\right]$ Solution: (c) $\left[\begin{array}{ccc}a^{n} 0 0 \\ 0 a^{n} 0 \\ 0 0 a^{...
Read More →In the given figure, ABCD is a rectangle in which diag.
Question: In the given figure, $A B C D$ is a rectangle in which diag. $A C=17 \mathrm{~cm}, \angle B C A=\theta$ and $\sin \theta=\frac{8}{17}$. Find (i) the area of rect.ABCD, (ii) the perimeter of rect.ABCD. Solution: Given: In $\Delta A B C$, $A C=17 \mathrm{~cm}$ $\sin \theta=\frac{8}{17}$ Since, $\sin \theta=\frac{P}{H}$ $\Rightarrow P=8$ and $H=17$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 8^{2}+B^{2}=17^{2}$ $\Rightarrow B^{2}=289-64$ $\Rightarrow B^{2}=225$ $\Rightarrow...
Read More →Prove that the lengths of tangents drawn
Question: Prove that the lengths of tangents drawn from an external point to a circle are equal. Solution: We have to prove that the lengths of tangents drawn from an external point to a circle are equal. Draw a circle with centre O and tangentsPAandPB, wherePis the external point andAandBare the points of contact of the tangents.JoinOA,OBandOP. Now in $\triangle O A P$ and $\triangle O B P$ $\angle O A P=\angle O B P \quad\left[\right.$ Both $90^{\circ}$, because radius is perpendicular to the ...
Read More →If the matrix A B is zero, then
Question: If the matrix $A B$ is zero, then (a) It is not necessary that either $A=O$ or, $B=0$ (b) $A=O$ or $B=0$ (c) $A=O$ and $B=O$ (d) all the above statements are wrong Solution: (a) It is not necessary that either $A=O$ or, $B=0$ Let $A=\left[\begin{array}{ll}0 2 \\ 0 0\end{array}\right]$ and $B=\left[\begin{array}{ll}1 0 \\ 0 0\end{array}\right]$ $\therefore A B=\left[\begin{array}{ll}0 2 \\ 0 0\end{array}\right]\left[\begin{array}{ll}1 0 \\ 0 0\end{array}\right]=\left[\begin{array}{ll}0 ...
Read More →A sum of Rs. 1400 is to be used to give seven cash prizes
Question: A sum of Rs. 1400 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 40 less than the preceding price, find the value of each of the prizes. Solution: It is given that total prize money is Rs 1400 /-. There are a total of 7 prizes distributed in a way that each prize is less than the previous prize by Rs 40/- We have to find the value of the prizes. Letais the value of a prize Then the value of consecutive prizes...
Read More →A train travels at a certain average speed for a distance of 63 km
Question: A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed? Solution: It is given that a train travels a distance of 63 km with a certain average speed and 72 km with a speed which is 6 km/hr more than the original average speed. Time taken for the whole journey is 3hr. We have to find out th...
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Question: If $\left[\begin{array}{cc}\cos \frac{2 \pi}{7} -\sin \frac{2 \pi}{7} \\ \sin \frac{2 \pi}{7} \cos \frac{2 \pi}{7}\end{array}\right]^{k}=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$, then the least positive integral value of $k$ is (a) 3 (b) 4 (c) 6 (d) 7 Solution: (d) 7 Here, $A=\left[\begin{array}{cc}\cos \frac{2 \pi}{7} -\sin \frac{2 \pi}{7} \\ \sin \frac{2 \pi}{7} \cos \frac{2 \pi}{7}\end{array}\right]$ $\Rightarrow A^{2}=A \times A$ $\Rightarrow A^{2}=\left[\begin{array}{c...
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Question: In a right $\Delta \mathrm{ABC}$, right-angled at $\mathrm{B}$, if $\tan \mathrm{A}=1$, then verify that $2 \sin \mathrm{A} \cdot \cos \mathrm{A}=1$. Solution: We have, $\tan \mathrm{A}=1$ $\Rightarrow \frac{\sin \mathrm{A}}{\cos \mathrm{A}}=1$ $\Rightarrow \sin A=\cos A$ $\Rightarrow \sin \mathrm{A}-\cos \mathrm{A}=0$ Squaring both sides, we get $(\sin \mathrm{A}-\cos \mathrm{A})^{2}=0$ $\Rightarrow \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}-2 \sin \mathrm{A} \cdot \cos \mathrm{A}=0$...
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