Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The area of a rhombus is 480 cm2and the length of one of its diagonalsis 20 cm. The length of each side of the rhombus is(a) 24 cm (b) 30 cm (c) 26 cm (d) 28 cm Solution: We have, $\mathrm{BD}=20 \mathrm{~cm}$ $\Rightarrow \mathrm{BO}=\frac{\mathrm{BD}}{2}=\frac{20}{2}=10 \mathrm{~cm}$ As, area of the rhombus $A B C D=480 \mathrm{~cm}^{2}$ $\Rightarrow \frac{1}{2} \times \mathrm{AC} \times \mathrm{BD}=480$ $\Rightarrow \frac{1}{2} \ti...
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Question: Factorize each of the following expression:p2q2p4q4 Solution: $p^{2} q^{2}-p^{4} q^{4}$ $=p^{2} q^{2}\left(1-p^{2} q^{2}\right)$ $=p^{2} q^{2}\left[1-(p q)^{2}\right]$ $=p^{2} q^{2}(1-p q)(1+p q)$...
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Question: If $\left|\begin{array}{cc}2 x 5 \\ 8 x\end{array}\right|=\left|\begin{array}{cc}6 -2 \\ 7 3\end{array}\right|$, then $x=$ (a) 3 (b) $\pm 3$ (c) $\pm 6$ (d) 6 Solution: $\left|\begin{array}{cc}2 x 5 \\ 8 x\end{array}\right|=\left|\begin{array}{cc}6 -2 \\ 7 3\end{array}\right|$ $\Rightarrow 2 x^{2}-40=18+14$ $\Rightarrow 2 x^{2}-40=32$ $\Rightarrow 2 x^{2}=72$ $\Rightarrow x^{2}=36$ $\Rightarrow x=\pm 6$ Hence, the correct option is (c)....
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Question: Factorize each of the following expression:(3x+ 4y)4x4 Solution: $(3 x+4 y)^{4}-x^{4}$ $=\left[(3 x+4 y)^{2}\right]^{2}-\left(x^{2}\right)^{2}$ $=\left[(3 x+4 y)^{2}+x^{2}\right]\left[(3 x+4 y)^{2}-x^{2}\right]$ $=\left[(3 x+4 y)^{2}+x^{2}\right][(3 x+4 y)+x][(3 x+4 y)-x]$ $=\left\{(3 \mathrm{x}+4 \mathrm{y})^{2}+\mathrm{x}^{2}\right\}(3 \mathrm{x}+4 \mathrm{y}+\mathrm{x})(3 \mathrm{x}+4 \mathrm{y}-\mathrm{x})$ $=\left\{(3 \mathrm{x}+4 \mathrm{y})^{2}+\mathrm{x}^{2}\right\}(4 \mathrm{x...
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Question: Let $A=\left[\begin{array}{ccc}1 \sin \theta 1 \\ -\sin \theta 1 \sin \theta \\ -1 -\sin \theta 1\end{array}\right]$, where $0 \leq \theta \leq 2 \pi$. Then, (a) Det $(A)=0$ (b) Det $(A) \in(2, \infty)$ (c) Det $(A) \in(2,4)$ (d) Det $(A) \in[2,4]$ Solution: (d) $\operatorname{Det}(A) \in[2,4]$ $\mid \begin{array}{lll}1 \sin \theta 1\end{array}$ $-\sin \theta \quad 1 \quad \sin \theta$ $\begin{array}{lll}-1 -\sin \theta 1 \mid\end{array}$ $=\mid \begin{array}{lll}1 \sin \theta 2\end{ar...
Read More →The side of an equilateral triangle is equal to the radius of a circle whose area is 154 cm2.
Question: The side of an equilateral triangle is equal to the radius of a circle whose area is 154 cm2. The area of the triangle is(a) 49cm2 (b) $\frac{49 \sqrt{3}}{4} \mathrm{~cm}^{2}$ (c) $\frac{7 \sqrt{3}}{4} \mathrm{~cm}^{2}$ (d) $77 \mathrm{~cm}^{2}$ Solution: (b) $\frac{49 \sqrt{3}}{4} \mathrm{~cm}^{2}$ Area of a circle $=\pi r^{2}$ $\Rightarrow 154=\pi r^{2}$ $\Rightarrow r=\sqrt{\frac{154 \times 7}{22}}$ $=\sqrt{7 \times 7}$ $=7 \mathrm{~cm}$ The radius of the circle is equal to the side...
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Question: Factorize each of the following expression:a4 (2b+c)4 Solution: $a^{4}-(2 b+c)^{4}$ $=\left(a^{2}\right)^{2}-\left[(2 b+c)^{2}\right]^{2}$ $=\left[a^{2}+(2 b+c)^{2}\right]\left[a^{2}-(2 b+c)^{2}\right]$ $=\left[a^{2}+(2 b+c)^{2}\right]\{[a+(2 b+c)][a-(2 b+c)]\}$ $=\left[a^{2}+(2 b+c)^{2}\right](a+2 b+c)(a-2 b-c)$...
Read More →The image of an object placed at
Question: The image of an object placed at a point A before a plane mirror LM is seen at the point B by an observer at D as shown in figure. Prove that the image is as far behind the mirror as the object is in front of the mirror. Solution: Given An object $O A$ placed at a point $A, L M$ be a plane mirror, $D$ be an observer and $O B$ is the image. To prove The image is as far behind the mirror as the object is in front of the mirror i.e., $O B=O A$ Proof $\because C N \perp L M$ and $A B \perp...
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Question: Factorize each of the following expression:256x5 81x Solution: $256 x^{5}-81 x$ $=x\left(256 x^{4}-81\right)$ $=x\left[\left(16 x^{2}\right)^{2}-9^{2}\right]$ $=x\left(16 x^{2}+9\right)\left(16 x^{2}-9\right)$ $=x\left(16 x^{2}+9\right)\left[(4 x)^{2}-3^{2}\right]$ $=x\left(16 x^{2}+9\right)(4 x+3)(4 x-3)$...
Read More →The side of a square is equal to the side of an equilateral triangle.
Question: The side of a square is equal to the side of an equilateral triangle. The ratio of their areas is(a) 4 : 3 (b) $2: \sqrt{3}$ (c) $4: \sqrt{3}$ (d) none of these Solution: (c) $4: \sqrt{3}$ Let:Length of the side of the square = Length of the side of the equilateral triangle =aunitNow, Area of the square $=a \times a=a^{2}$ unit $^{2}$ Area of the equilateral triangle $=\frac{\sqrt{3}}{4} a^{2} \mathrm{unit}^{2}$ Ratio of areas $=\frac{\text { Area of the square }}{\text { Area of the e...
Read More →The number of distinct real roots
Question: The number of distinct real roots of $\left|\begin{array}{ccc}\operatorname{cosec} x \sec x \sec x \\ \sec x \operatorname{cosec} x \sec x \\ \sec x \sec x \operatorname{cosec} x\end{array}\right|=0$ lies in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is (a) 1 (b) 2 (C) 3 (d) 0 Solution: (b) 2 Let $\Delta=\mid \operatorname{cosec} x \quad \sec x \quad \sec x$ $\begin{array}{ccc}\sec x \operatorname{cosec} x \sec x \\ \sec x \sec x \operatorname{cosec} x\end{array}$ $\left[\...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x5 16x3 Solution: $x^{5}-16 x^{3}$ $=x^{3}\left(x^{2}-16\right)$ $=x^{3}\left(x^{2}-4^{2}\right)$ $=x^{3}(x-4)(x+4)$...
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Question: Factorize each of the following expression:75a3b2- 108ab4 Solution: $75 a^{3} b^{2}-108 a b^{4}$ $=3 a b^{2}\left(25 a^{2}-36 b^{2}\right)$ $=3 a b^{2}\left[(5 a)^{2}-(6 b)^{2}\right]$ $=3 a b^{2}(5 a-6 b)(5 a+6 b)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression: $\frac{1}{16} x^{2} y^{2}-\frac{4}{49} y^{2} z^{2}$ Solution: $\frac{1}{16} x^{2} y^{2}-\frac{4}{49} y^{2} z^{2}$ $=y^{2}\left(\frac{1}{16} x^{2}-\frac{4}{49} z^{2}\right)$ $=y^{2}\left[\left(\frac{1}{4} x\right)^{2}-\left(\frac{2}{7} z\right)^{2}\right]$ $=y^{2}\left(\frac{1}{4} x-\frac{2}{7} z\right)\left(\frac{1}{4} x+\frac{2}{7} z\right)$ $=y^{2}\left(\frac{x}{4}-\frac{2}{7} z\right)\left(\frac{x}{4}+\frac{2}{7} z\right)$...
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Question: Factorize each of the following expression:(x + y)2 (a b)2 Solution: $(x+y)^{2}-(a-b)^{2}$ $=[(x+y)-(a-b)][(x+y)+(a-b)]$ $=(x+y-a+b)(x+y+a-b)$...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The lengths of the sides of a triangular field are 20 m, 21 m and 29 m.The cost of cultivating the field at₹9 per m2is(a)₹2610 (b)₹3780 (c)₹1890 (d)₹1800 Solution: As, the sides of the triangle are $20 \mathrm{~m}, 21 \mathrm{~m}$ and $29 \mathrm{~m}$ So, the semi $-$ perimeter $=\frac{20+21+29}{2}=35 \mathrm{~m}$ Now, the area of the triangular field $=\sqrt{35(35-20)(35-21)(35-29)}$ $=\sqrt{35 \times 15 \times 14 \times 6}$ $=\sqrt{...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:(3 + 2a)2 25a2 Solution: $(3+2 a)^{2}-25 a^{2}$ $=(3+2 a)^{2}-(5 a)^{2}$ $=[(3+2 a)-5 a][(3+2 a)+5 a]$ $=(3+2 a-5 a)(3+2 a+5 a)$ $=(3-3 a)(3+7 a)$ $=3(1-a)(3+7 a)$...
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Question: Factorize each of the following expression:9(a b)2 100(x y)2 Solution: $9(a-b)^{2}-100(x-y)^{2}$ $=[3(a-b)]^{2}-[10(x-y)]^{2}$ $=[3(a-b)-10(x-y)][3(a-b)+10(x-y)]$ $=(3 a-3 b-10 x+10 y)(3 a-3 b+10 x-10 y)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:(x- 4y)2 625 Solution: $(x-4 y)^{2}-625$ $=(x-4 y)^{2}-25^{2}$ $=[(x-4 y)-25][(x-4 y)+25]$ $=(x-4 y-25)(x-4 y+25)$...
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Question: Factorize each of the following expression:x3 144x Solution: $x^{3}-144 x$ $=x\left(x^{2}-144\right)$ $=x\left(x^{2}-12^{2}\right)$ $=x(x-12)(x+12)$...
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Question: If $A+B+C=\pi$, then the value of $\left|\begin{array}{ccc}\sin (A+B+C) \sin (A+C) \cos C \\ -\sin B 0 \tan A \\ \cos (A+B) \tan (B+C) 0\end{array}\right|$ is equal to (a) 0 (b) 1 (c) $2 \sin B \tan A \cos C$ (d) none of these Solution: (a) 0 $A+B+C=\pi$ $\Rightarrow A+C=\pi-B, A+B=\pi-C$ and $B+C=\pi-A$ Thus the determinant becomes $\mid \begin{array}{lll}\sin \pi \sin (\pi-B) \cos C\end{array}$ $\begin{array}{lll}-\sin B 0 \tan A\end{array}$ $\cos (\pi-C) \quad \tan (\pi-A) \quad 0$ ...
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Question: Factorize each of the following expression: $a^{4}-\frac{1}{b^{4}}$ Solution: $a^{4}-1 / b^{4}$ =(a2)2-1/(b2)2 =a2-1/b2a2+1/b2 =a-1/ba+1/ba2+1/b2...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The base and height of a triangle are in the ratio 3:4 and its area is216 cm2. The height of the triangle is(a) 18 cm (b) 24 cm (c) 21 cm (d) 28 cm Solution: Let the base of the triangle be $3 x$ and its height be $4 x$. As, the area of the triangle $=216 \mathrm{~cm}^{2}$ $\Rightarrow \frac{1}{2} \times$ Base $\times$ Height $=216$ $\Rightarrow \frac{1}{2} \times 3 x \times 4 x=216$ $\Rightarrow 6 x^{2}=216$ $\Rightarrow x^{2}=\frac{...
Read More →Find all the angles of an equilateral triangle.
Question: Find all the angles of an equilateral triangle. Solution: Let $A B C$ be an equilateral triangle such that $A B=B C=C A$ We have, $\quad A B=A C \Rightarrow \angle C=\angle B$ [angles opposite to equal sides are equal] Let $\angle C=\angle B=x^{\circ}$ $\ldots(i)$ Now, $B C=B A$ $\Rightarrow$ $\angle A=\angle C$ ... (ii) [angles opposite to equal sides are equal] From Eqs. (i) and (ii), $\angle A=\angle B=\angle C=x$ Now, in $\triangle A B C$, $\angle A+\angle B+\angle C=180^{\circ}$ [...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:25x4y4 1 Solution: $25 x^{4} y^{4}-1$ $=\left(5 x^{2} y^{2}\right)^{2}-1$ $=\left(5 x^{2} y^{2}-1\right)\left(5 x^{2} y^{2}+1\right)$...
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