Factorize each of the following expression:
Question: Factorize each of the following expression:a4b4 81c4 Solution: $a^{4} b^{4}-81 c^{4}$ $=\left(a^{2} b^{2}\right)^{2}-\left(9 c^{2}\right)^{2}$ $=\left(a^{2} b^{2}+9 c^{2}\right)\left(a^{2} b^{2}-9 c^{2}\right)$ $=\left(a^{2} b^{2}+9 c^{2}\right)\left[(a b)^{2}-(3 c)^{2}\right]$ $=\left(a^{2} b^{2}+9 c^{2}\right)(a b+3 c)(a b-3 c)$...
Read More →In the given figure ABCD is a quadrilateral in which ∠ABC = 90°
Question: In the given figureABCDis a quadrilateral in which ABC= 90, BDC= 90,AC= 17 cm,BC= 15 cm,BD= 12 cm andCD= 9 cm. The area of quad.ABCDis (a) 102 cm2(b) 114 cm2(c) 95 cm2(d) 57 cm2 Solution: (b) 114 sq cm Using Pythagoras' theorem in $\triangle A B C$, we get: $A C^{2}=A B^{2}+B C^{2}$ $\Rightarrow A B=\sqrt{A C^{2}-B C^{2}}$ $=\sqrt{17^{2}-15^{2}}$ $=8 \mathrm{~cm}$ Area of $\Delta A B C=\frac{1}{2} \times A B \times B C$ $=\frac{1}{2} \times 8 \times 15$ $=60 \mathrm{~cm}^{2}$ Area of $...
Read More →ABC is a right triangle with AB = AC.
Question: ABC is a right triangle with AB = AC.If bisector of A meets BC at D,then prove that BC = 2AD. Solution: Given $\triangle A B C$ is a right angled triangle with $A B=A C, A D$ is the bisector of $\angle A$. To prove $B C=2 A D$ [given] Proof $\ln \triangle A B C$, $A B=A C$ [given] $\Rightarrow$ $\angle C=\angle B$ $\ldots(1)$ [angles opposite to equal sides are equal] Now, in right angled $\triangle A B C, \angle A+\angle B+\angle C=180^{\circ}$ [by angle sum property of a triangle in ...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:2a5 32a Solution: $2 a^{5}-32 a$ $=2 a\left(a^{4}-16\right)$ $=2 a\left[\left(a^{2}\right)^{2}-4^{2}\right]$ $=2 a\left(a^{2}+4\right)\left(a^{2}-4\right)$ $=2 a\left(a^{2}+4\right)\left(a^{2}-2^{2}\right)$ $=2 a\left(a^{2}+4\right)(a+2)(a-2)$ $=2 a(a-2)(a+2)\left(a^{2}+4\right)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a4 16(b c)4 Solution: $a^{4}-16(b-c)^{4}$ $=\left(a^{2}\right)^{2}-\left[4(b-c)^{2}\right]^{2}$ $=\left[a^{2}+4(b-c)^{2}\right]\left[a^{2}-4(b-c)^{2}\right]$ $=\left[a^{2}+4(b-c)^{2}\right]\left\{a^{2}-[2(b-c)]^{2}\right\}$ $=\left[a^{2}+4(b-c)^{2}\right][a+2(b-c)][a-2(b-c)]$ $=\left[a^{2}+4(b-c)^{2}\right](a+2 b-2 c)(a-2 b+2 c)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:16a4b4 Solution: $16 a^{4}-b^{4}$ $=\left(4 a^{2}\right)^{2}-\left(b^{2}\right)^{2}$ $=\left(4 a^{2}+b^{2}\right)\left(4 a^{2}-b^{2}\right)$ $=\left(4 a^{2}+b^{2}\right)\left[(2 a)^{2}-b^{2}\right]$ $=\left(4 a^{2}+b^{2}\right)(2 a+b)(2 a-b)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a2b2+a b Solution: $a^{2}-b^{2}+a-b=\left(a^{2}-b^{2}\right)+(a-b) \quad$ [Grouping the terms] $=(a+b)(a-b)+(a-b)$ $=(a-b)(a+b+1) \quad[$ Taking out the common factor $(a-b)]$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x4 (2y 3z)2 Solution: $\mathrm{x}^{4}-(2 \mathrm{y}-3 \mathrm{z})^{2}$ $=\left(\mathrm{x}^{2}\right)^{2}-(2 \mathrm{y}-3 \mathrm{z})^{2}$ $=\left[\mathrm{x}^{2}-(2 \mathrm{y}-3 \mathrm{z})\right]\left[\mathrm{x}^{2}+(2 \mathrm{y}-3 \mathrm{z})\right]$ $=\left(\mathrm{x}^{2}-2 \mathrm{y}+3 \mathrm{z}\right)\left(\mathrm{x}^{2}+2 \mathrm{y}-3 \mathrm{z}\right)$...
Read More →Solve this
Question: The determinant $\left|\begin{array}{lll}b^{2}-a b b-c b c-a c \\ a b-a^{2} a-b b^{2}-a b \\ b c-c a c-a a b-a^{2}\end{array}\right|$ equals (a) $a b c(b-c)(c-a)(a-b)$ (b) $(b-c)(c-a)(a-b)$ (c) $(a+b+c)(b-c)(c-a)(a-b)$ (d) none of these Solution: $\left|\begin{array}{lll}b^{2}-a b b-c b c-a c \\ a b-a^{2} a-b b^{2}-a b \\ b c-c a c-a a b-a^{2}\end{array}\right|$ $=\left|\begin{array}{lll}b(b-a) b-c c(b-a) \\ a(b-a) a-b b(b-a) \\ c(b-a) c-a a(b-a)\end{array}\right|$ $=(b-a)^{2}\left|\be...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:(2x+ 1)2 9x4 Solution: $(2 \mathrm{x}+1)^{2}-9 \mathrm{x}^{4}$ $=(2 \mathrm{x}+1)^{2}-\left(3 \mathrm{x}^{2}\right)^{2}$ $=\left[(2 \mathrm{x}+1)-3 \mathrm{x}^{2}\right]\left[(2 \mathrm{x}+1)+3 \mathrm{x}^{2}\right]$ $=\left(-3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)$ We can factorise the quadratic expressions in the curved brackets as: $\left(-3 \mathrm{x}^{2}+3 \mathrm{x}-\mathrm{x}+1\right)\left(3 \m...
Read More →ABCD is a quadrilateral in which AB = BC
Question: ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC. Thinking Process Firstly, use the property that if two sides of a triangle are equal, then their opposite angles are equal. Further, show that ΔBAD and ΔBCD are congruent by SAS rule. Solution: Given $A B C D$ is a quadrilateral in which $A B=B C$ and $A D=C D$. To show $B D$ bisects both the angles $A B C$ and $A D C$. Proof Since, $A B=B C$(given) $\therefore$ $\angle 2=\angle 1$ $...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:4(xy+ 1)2 9(x 1)2 Solution: $4(x y+1)^{2}-9(x-1)^{2}$ $=[2(x y+1)]^{2}-[3(x-1)]^{2}$ $=[2(x y+1)-3(x-1)][2(x y+1)+3(x-1)]$ $=(2 x y+2-3 x+3)(2 x y+2+3 x-3)$ $=(2 x y-3 x+5)(2 x y+3 x-1)$...
Read More →If x, y, z are different from zero
Question: If $x, y, z$ are different from zero and $\left|\begin{array}{ccc}1+x 1 1 \\ 1 1+y 1 \\ 1 1 1+z\end{array}\right|=0$, then the value $x^{-1}+y^{-1}+z^{-1}$ is (a) $x y z$ (b) $x^{-1} y^{-1} z^{-1}$ (c) $-x-y-z$ (d) $-1$ Solution: $\left|\begin{array}{ccc}1+x 1 1 \\ 1 1+y 1 \\ 1 1 1+z\end{array}\right|=0$ $\Rightarrow\left|\begin{array}{ccc}x 0 -z \\ 0 y -z \\ 1 1 1+z\end{array}\right|=0$ [Applying $R_{2} \rightarrow R_{2}-R_{3}$ and $\left.R_{1} \rightarrow R_{1}-R_{3}\right]$ $\Righta...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:16(2x 1)2 25y2 Solution: $16(2 x-1)^{2}-25 y^{2}$ $=[4(2 x-1)]^{2}-(5 y)^{2}$ $=[4(2 x-1)-5 y][4(2 x-1)+5 y]$ $=(8 x-4-5 y)(8 x-4+5 y)$ $=(8 x-5 y-4)(8 x+5 y-4)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x yx2+y2 Solution: $x-y-x^{2}+y^{2}$ $=(x-y)+\left(y^{2}-x^{2}\right)$ [Regrouping the terms] $=(x-y)+(y+x)(y-x)$ $=(x-y)-(y+x)(x-y)$ $[\because(y-x)=-(x-y)]$ $=(x-y)[1-(y+x)]$ $=(x-y)(1-x-y)$...
Read More →The value of the determinant
Question: The value of the determinant $\left|\begin{array}{lll}a-b b+c a \\ b-c c+a b \\ c-a a+b c\end{array}\right|$ is (a) $a^{3}+b^{3}+c^{3}$ (b) $3 b c$ (c) $a^{3}+b^{3}+c^{3}-3 a b c$ (d) none of these Solution: $\left|\begin{array}{lll}a-b b+c a \\ b-c c+a b \\ c-a a+b c\end{array}\right|$ $=\left|\begin{array}{lll}-b b+c+a a \\ -c c+a+b b \\ -a a+b+c c\end{array}\right|$ $\left[\right.$ Applying $C_{1} \rightarrow C_{1}-C_{3}$ and $\left.C_{2} \rightarrow C_{2}+C_{3}\right]$ $=(-1)(a+b+c...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:49(a b)2 25(a + b)2 Solution: $49(a-b)^{2}-25(a+b)^{2}$ $=[7(a-b)]^{2}-[5(a+b)]^{2}$ $=[7(a-b)-5(a+b)][7(a-b)+5(a+b)]$ $=(7 a-7 b-5 a-5 b)(7 a-7 b+5 a+5 b)$ $=(2 a-12 b)(12 a-2 b)$ $=2(a-6 b) 2(6 a-b)$ $=4(a-6 b)(6 a-b)$...
Read More →P is a point on the bisector of ∠ABC.
Question: P is a point on the bisector of ABC. If the line through P, parallel to BA meet BC at Q, prove that BPQ is an isosceles triangle. Solution: Given we have P is a point on the bisector of ABC and draw the line through P parallel to BA and meet BC at Q. To prove $\triangle B P Q$ is an isosceles triangle. Proof$\angle 1=\angle 2$$[\because B P$ is bisector of $\angle B$ (given)] Now, $\angle 1=\angle 3$ [alternate interior angles as $P Q \| A B$ ] $\therefore$ $\angle 2=\angle 3$ $\Righta...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x4 1 Solution: $x^{4}-1$ $=\left(x^{2}\right)^{2}-1$ $=\left(x^{2}+1\right)\left(x^{2}-1\right)$ $=\left(x^{2}+1\right)(x+1)(x-1)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x4 625 Solution: $x^{4}-625$ $=\left(x^{2}\right)^{2}-25^{2}$ $=\left(x^{2}+25\right)\left(x^{2}-25\right)$ $=\left(x^{2}+25\right)\left(x^{2}-5^{2}\right)$ $=\left(x^{2}+25\right)(x+5)(x-5)$...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:One side of a rhombus is 20 cm long and one of its diagonals measures24 cm. The area of the rhombus is(a) 192 cm2 (b) 480 cm2 (c) 240 cm2 (d) 384 cm2 Solution: WE have, $\mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DA}=20 \mathrm{~cm}$ and $\mathrm{BD}=24 \mathrm{~cm}$ Also, $\mathrm{BO}=\frac{\mathrm{BD}}{2}=\frac{24}{2}=12 \mathrm{~cm}$ In $\Delta \mathrm{AOB}$, Using Pythagoras theorem $\mathrm{AO}^{2}=\mathrm{AB}^{2}-\mathrm{BO}^{2...
Read More →Solve the following equations
Question: If $f(x)=\left|\begin{array}{ccc}0 x-a x-b \\ x+a 0 x-c \\ x+b x+c 0\end{array}\right|$, then (a) $f(a)=0$ (b) $f(b)=0$ (c) $f(0)=0$ (d) $f(1)=0$ Solution: Let $f(x)=\left|\begin{array}{ccc}0 x-a x-b \\ x+a 0 x-c \\ x+b x+c 0\end{array}\right|$. Now, $f(a)=\left|\begin{array}{ccc}0 a-a a-b \\ a+a 0 a-c \\ a+b a+c 0\end{array}\right|$ $=\left|\begin{array}{ccc}0 0 a-b \\ 2 a 0 a-c \\ a+b a+c 0\end{array}\right|$ $=(a-b)\left(2 a^{2}+2 a c\right) \neq 0$ $f(b)=\left|\begin{array}{ccc}0 b...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a4b4 16c4 Solution: $a^{4} b^{4}-16 c^{4}$ $=\left[\left(a^{2} b^{2}\right)^{2}-\left(4 c^{2}\right)^{2}\right]$ $=\left(a^{2} b^{2}+4 c^{2}\right)\left(a^{2} b^{2}-4 c^{2}\right)$ $=\left(a^{2} b^{2}+4 c^{2}\right)\left[(a b)^{2}-(2 c)^{2}\right]$ $=\left(a^{2} b^{2}+4 c^{2}\right)(a b+2 c)(a b-2 c)$...
Read More →ABC is an isosceles triangle with AB = AC
Question: ABC is an isosceles triangle with AB = AC and D is a point on BC such that AD BC (see figure). To prove that BAD = CAD, a student proceeded as follows In $\triangle A B D$ and $\triangle A C D$, we have $A B=A C$ [given] $\angle B=\angle C$ [because $A B=A C$ ] and $\angle A D B=\angle A D C$ Therefore, $\triangle A B D \cong \triangle A C D$ [by AAS congruence rule] So, $\angle B A D=\angle C A D$ [by CPCT] What is the defect in the above arguments? Solution: In $\triangle A B C$, $A ...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:3x3y 243xy3 Solution: $3 x^{3} y-243 x y^{3}$ $=3 x y\left(x^{2}-81 y^{2}\right)$ $=3 x y\left[x^{2}-(9 y)^{2}\right]$ $=3 x y(x-9 y)(x+9 y)$...
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