Factorize each of the following expression:
Question: Factorize each of the following expression:36l2 (m + n)2 Solution: $36 l^{2}-(m+n)^{2}$ $=(6 l)^{2}-(m+n)^{2}$ $=[6 l-(m+n)][6 l+(m+n)]$ $=(6 l-m-n)(6 l+m+n)$...
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Question: Choose the correct answer of the following question: The height of an equilateral triangle is $3 \sqrt{3} \mathrm{~cm}$. Its area is (a) $6 \sqrt{3} \mathrm{~cm}^{2}$ (b) $27 \mathrm{~cm}^{2}$ (c) $9 \sqrt{3} \mathrm{~cm}^{2}$ (d) $27 \sqrt{3} \mathrm{~cm}^{2}$ Solution: Let the side of the equilateral triangle be $x$. As, area of the equilateral triangle $=\frac{\sqrt{3}}{4} \times(\text { side })^{2}$ $\Rightarrow \frac{1}{2} \times$ Base $\times$ Height $=\frac{\sqrt{3}}{4} \times x...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:64 (a+ 1)2 Solution: $64-(a+1)^{2}$ $=(8)^{2}-(a+1)^{2}$ $=[8-(a+1)][8+(a+1)]$ $=(8-a-1)(8+a+1)$ $=(7-a)(9+a)$...
Read More →If a, b, c are in A.P., then the determinant
Question: If $a, b, c$ are in A.P., then the determinant $\left|\begin{array}{lll}x+2 x+3 x+2 a \\ x+3 x+4 x+2 b \\ x+4 x+5 x+2 c\end{array}\right|$ (a) 0 (b) 1 (c) $x$ (d) $2 x$ Solution: (a) 0 $\mid x+2 x+3 \quad x+2 a$ $x+3 x+4 \quad x+2 b$ $x+4 x+5 \quad x+2 c \mid$ $=\mid \begin{array}{lll}0 0 2(a+c-2 b)\end{array}$ $x+3 \quad x+4 \quad x+2 b$ $\begin{array}{llll}x+4 x+5 x+2 c \mid {\left[\text { Applying } \mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{3}-\mathrm{R}_{2}, \mathrm{R}_...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x8 1 Solution: $x^{8}-1$ $=\left(x^{4}\right)^{2}-1^{2}$ $=\left(x^{4}-1\right)\left(x^{4}+1\right)$ $=\left[\left(x^{2}\right)^{2}-1^{2}\right]\left(x^{4}+1\right)$ $=\left(x^{2}-1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$ $=\left(x^{2}-1^{2}\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$ $=(x-1)(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)$...
Read More →In following figure if A D
Question: In following figure if $A D$ is the bisector of $\angle B A C$, then prove that $A BB D$. Solution: Given ABC is a triangle such that AD is the bisector of BAC. To prove AB BD. Proof Since, AD is the bisector of BAC. But BAD = CAD (i) ADB CAD [exterior angle of a triangle is greater than each of the opposite interior angle] ADB BAD [from Eq. (i)] AB BD [side opposite to greater angle is longer] Hence proved....
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question: Each side of an equilateral triangle is $6 \sqrt{3} \mathrm{~cm}$. The altitude of the triangle is (a) $8 \mathrm{~cm}$ (b) $9 \mathrm{~cm}$ (c) $3 \sqrt{3} \mathrm{~cm}$ (d) $6 \mathrm{~cm}$ Solution: As, area of an equilateral triangle $=\frac{\sqrt{3}}{4} \times(\text { side })^{2}$ $\Rightarrow \frac{1}{2} \times$ Base $\times$ Height $=\frac{\sqrt{3}}{4} \times(6 \sqrt{3})^{2}$ $\Rightarrow \frac{1}{2} \times 6 \sqrt{3} \times$ ...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a4 16b4 Solution: $a^{4}-16 b^{4}=a^{4}-2^{4} b^{4}=\left(a^{2}\right)^{2}-\left(2^{2} b^{2}\right)^{2}$ $=\left(a^{2}-2^{2} b^{2}\right)\left(a^{2}+2^{2} b^{2}\right)$ $=\left[a^{2}-(2 b)^{2}\right]\left(a^{2}+4 b^{2}\right)$ $=(a-2 b)(a+2 b)\left(a^{2}+4 b^{2}\right)$...
Read More →Bisectors of the angles B and C of
Question: Bisectors of the angles B and C of an isosceles ΔABC with AB = AC intersect each other at O. Show that external angle adjacent to ABC is equal to BOC. Solution: Given $\triangle A B C$ is an isosceles triangle in which $A B=A C, B O$ and $C O$ are the bisectors of $\angle A B C$ and $\angle A C B$ respectively intersect at $O$. To show $\angle D B A=\angle B O C$ Construction Line CB produced to $D$. Proof In $\triangle A B C$, $A B=A C$ [given] $\angle A C B=\angle A B C$ [angles oppo...
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Question: Factorize each of the following expression:3a5 48a3 Solution: $3 a^{5}-48 a^{3}$ $=3 a^{3}\left(a^{2}-16\right)$ $=3 a^{3}\left(a^{2}-4^{2}\right)$ $=3 a^{3}(a-4)(a+4)$...
Read More →Solve the following equations
Question: $\left|\begin{array}{cccc}\log _{3} 512 \log _{4} 3 \\ \log _{3} 8 \log _{4} 9\end{array}\right| \times\left|\begin{array}{llll}\log _{2} 3 \log _{8} 3 \\ \log _{3} 4 \log _{3} 4\end{array}\right|$ (a) 7 (b) 10 (c) 1 (d) 17 Solution: (b) 10 $\mid \log _{3} 512 \quad \log _{4} 3$ $\log _{3} 8 \quad \log _{4} 9|\times| \log _{2} 3 \quad \log _{8} 3$ $\log _{3} 4 \quad \log _{3} 4 \mid$ $=\left|\log _{3} 2^{9} \quad \log _{2^{2}} 3\right|$ $\log _{3} 2^{3} \quad \log _{2^{2}} 3^{3}|\times...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:(x+ 2y)2 4(2x y)2 Solution: $(x+2 y)^{2}-4(2 x-y)^{2}=(x+2 y)^{2}-[2(2 x-y)]^{2}$ $=[(x+2 y)-2(2 x-y)][(x+2 y)+2(2 x-y)]$ $=(x+2 y-4 x+2 y)(x+2 y+4 x-2 y)$ $=5 x(4 y-3 x)$...
Read More →Each side of an equilateral triangle is 8 cm. Its area is
Question: Each side of an equilateral triangle is 8 cm. Its area is(a) 24 cm2 (b) $04 \sqrt{9} \mathrm{~m}^{2}$ (c) $16 \sqrt{3} \mathrm{~cm}^{2}$ (d) $8 \sqrt{3} \mathrm{~cm}^{2}$ Solution: (c) $16 \sqrt{3} \mathrm{~cm}^{2}$ Let the side of the equilateral triangle bea.Given:a= 8 cmNow, Area of the equilateral triangle $=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 8 \times 8=16 \sqrt{3} \mathrm{~cm}^{2}$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:(2a b)2 16c2 Solution: $(2 a-b)^{2}-16 c^{2}$ $=(2 a-b)^{2}-(4 c)^{2}$ $=[(2 a-b)-4 c][(2 a-b)+4 c]$ $=(2 a-b-4 c)(2 a-b+4 c)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:144a2 169b2 Solution: $144 a^{2}-169 b^{2}$ $=(12 a)^{2}-(13 b)^{2}$ $=(12 a-13 b)(12 a+13 b)$...
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Question: Factorize each of the following expression:125x2 45y2 Solution: $125 x^{2}-45 y^{2}$ $=5\left(25 x^{2}-9 y^{2}\right)$ $=5\left[(5 x)^{2}-(3 y)^{2}\right]$ $=5(5 x-3 y)(5 x+3 y)$...
Read More →The area of an equilateral triangle is
Question: The area of an equilateral triangle is $4 \sqrt{3} \mathrm{~cm}^{2}$. Its perimeter is (a) 9 cm(b) 12 cm (c) $12 \sqrt{3} \mathrm{~cm}$ (d) $6 \sqrt{3} \mathrm{~cm}$ Solution: (b) 12 cm Area of an equilateral triangle $=\frac{\sqrt{3}}{4} a^{2}$ (where $a$ is the length of the side) Thus, we have: $4 \sqrt{3}=\frac{\sqrt{3}}{4} a^{2}$ $\Rightarrow a^{2}=16$ $\Rightarrow a=4 \mathrm{~cm}$ Perimeter of the equilateral triangle $=3 a=3 \times 4=12 \mathrm{~cm}$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:12m2 27 Solution: $12 m^{2}-27$ $=3\left(4 m^{2}-9\right)$ $=3\left[(2 m)^{2}-3^{2}\right]$ $=3(2 m-3)(2 m+3)$...
Read More →Bisectors of the angles B and C
Question: Bisectors of the angles $B$ and $C$ of an isosceles triangle with $A B=A C$ intersect each other at $O$. $B O$ is produced to a point $M$. Prove that $\angle M O C=\angle A B C$. Solution: Given Lines, $O B$ and $O C$ are the anqle bisectors of $\angle B$ and $\angle C$ of an isosceles $\triangle A B C$ such that $A B=A C$ which intersect each other at $O$ and $B O$ is produced to $M$. To prove $\angle M O C=\angle A B C$. Proof $\ln \triangle A B C$, $A B=A C$[given] $\Rightarrow$ $\a...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:144a2 289b2 Solution: $144 a^{2}-289 b^{2}$ $=(12 a)^{2}-(17 b)^{2}$ $=(12 a-17 b)(12 a+17 b)$...
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Question: The value of $\left|\begin{array}{ccc}5^{2} 5^{3} 5^{4} \\ 5^{3} 5^{4} 5^{5} \\ 5^{4} 5^{5} 5^{6}\end{array}\right|$ is (a) $5^{2}$ (b) 0 (c) $5^{13}$ (d) $5^{9}$ Solution: (b) 0 $\begin{array}{ccc}\mid 5^{2} 5^{3} 5^{4} \\ 5^{3} 5^{4} 5^{5} \\ 5^{4} 5^{5} 5^{6}\end{array}$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:27x2 12y2 Solution: $27 x^{2}-12 y^{2}$ $=3\left(9 x^{2}-4 y^{2}\right)$ $=3\left[(3 x)^{2}-(2 y)^{2}\right]$ $=3(3 x-2 y)(3 x+2 y)$...
Read More →The length of a square field is 0.5 hectare.
Question: The length of a square field is 0.5 hectare. The length of its diagonal is (a) $150 \mathrm{~m}$ (b) $100 \sqrt{2}$ (c) 100 m (d) $50 \sqrt{2} \mathrm{~m}$ Solution: (c) 100 mDisclaimer :-The length cannot be in hectare So we used is as area of the square. Area of the square field $=0.5 \times 10000=5000 \mathrm{~m}^{2}$ The diagonal divides the square into two isosceles right-angled triangles.Using Pythagoras' theorem, we have: Diagonal $^{2}=a^{2}+a^{2}=2 a^{2}$ Area of a square $=a^...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:16x2 25y2 Solution: $16 x^{2}-25 y^{2}$ $=(4 x)^{2}-(5 y)^{2}$ $=(4 x-5 y)(4 x+5 y)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x2+y xy x Solution: $x^{2}+y-x y-x=\left(x^{2}-x y\right)+(y-x) \quad[$ Regrouping the expressions $]$ $=x(x-y)+(y-x)$ $=x(x-y)-(x-y)$ $[\because(y-x)=-(x-y)]$ $=(x-1)(x-y)$ $[$ Taking $(x-y)$ as the common expression $]$...
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