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MATHS 2 Marks Questions

Factorisation · UP Board (UPMSP) STD 8 (UPMSP - English Medium). 110 questions.

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Question 22 Marks
Factorise:
$=a b\left(x^2+y^2\right)-x y\left(a^2-b^2\right)$
Answer
$=a b\left(x^2+y^2\right)-x y\left(a^2-b^2\right)$
$=a b x^2+a b y^2+x y a^2-x y b^2$
$=a b x^2-x y a^2-x y b^2+a b y^2$
$=a x(b x-a y)-b y(b x-a y)$
$=(b x-a y)(a x-b y)$
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Question 32 Marks
Factorise:
$16 x^5-144 x^3$
Answer
$16 x^5-144 x^3$
$=16 x^3\left[x^2-9\right]$
$=16 x^3\left[(x)^2-(3)^2\right]$
$=16 x^3(x+3)(x-3)$
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Question 42 Marks
Factorise: $16(2 p-3 q)^2-4(2 p-3 q)$
Answer
$16(2 p-3 q)^2-4(2 p-3 q)$
$=(2 p-3 q)\{16(2 p-3 q)-4\}$
$=(2 p-3 q)(32 p-48 q-4)$
$=4(2 p-3 q)(8 p-12 q-1)$
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Question 72 Marks
Factorise:
$3m^2+ 24m + 36$
Answer
$ 3 m^2+24 m+36 $
$ =3\left(m^2+8 m+12\right) $
$ =3\left(m^2+6 m+2 m+12\right) $
$ \{12=6 \times 2,8=6+2\} $
$ =33(m+6)+(m+2) $
$ =(2 x-3)(3 x+2) $
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Question 82 Marks
Factorise:
$a b\left(x^2+y^2\right)-x y\left(a^2+b^2\right)$
Answer
$a b\left(x^2+y^2\right)-x y\left(a^2+b^2\right)$
$=a b x^2+a b y^2+x y a^2-x y b^2$
$=a b x^2-x y a^2-x y b^2+a b y^2$
$=a x(b x-a y)-b y(b x-a y)$
$=(b x-a y)(a x-b y)$
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Question 112 Marks
Factorise:
$x^2-17 x+16$
Answer
$x^2-17 x+16$
$ =x^2-x-16 x+16 $
$ \{16=(-1) \times(-16), 17=-1-16\} $
$ =x(x-1)-16(x-1) $
$ =(x-1)(x-16) $
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Question 122 Marks
Factorise:
$x^2-x(a+2 b)+2 a b$
Answer
$x^2-x(a+2 b)+2 a b$
$=x^2-x a-2 b x+2 a b$
$= x(x - a) - 2bx + 2ab$
$ = x(x - a) - 2b(x - a) $
$= (x - a)(x - 2b)$
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Question 132 Marks
Factorise:
$x^2-5 x-24$
Answer
$x^2-5 x-24$
$=x^2-8 x+3 x-24$
${-24 = -8 × 3, -5 = -8 + 3}$
$= x(x - 8) + 3(x - 8)$
$= (x - 8)(x + 3)$
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Question 142 Marks
Factorise:$49 a^2+84 a b+36 b^2 $
Answer
$49 a^2+84 a b+36 b^2 $
$ =(7 a)^2+2 \times 7 a \times 6 b+(6 b)^2$
$\{\because$ $a^2+ 2ab + b^2= (a + b)^2\}$
$= (7a + 6b)^2$
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Question 152 Marks
Factorise:
$4 x^2-9 y^2$
Answer
$4 x^2-9 y^2$
$=(2 x)^2-(3 y)^2$
$=(2 x+3 y)(2 x-3 y)$
$\{\because a^2- b^2= (a + b)(a - b)\}$
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Question 162 Marks
Factorise: $(x + y)(2x + 5) - (x + y)(x + 3)$
Answer
$(x + y)(2x + 5) - (x + y)(x + 3) $
$= (x + y)(2 + 5 - x - 3) $
$= (x + y)(x + 2)$
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Question 172 Marks
Factorise:
$ 4 y^2+20 y+25 $
Answer
$ 4 y^2+20 y+25 $
$=(2 y)^2+2 \times 2 y \times 5+(5)^2 $
$=\left(2 y^2+5\right)^2 $
$\left\{\therefore a^2+2 a b+b=(a+b)^2\right\} $
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Question 182 Marks
Factorise: $20 a^2-45 b^2$
Answer
Using: $a^2-b^2=(a+b)(a-b) $
$20 a^2-45 b^2$
$=5\left(4 a^2-9 b^2\right)$
$=5(2 a)^2-(3 b)^2$
$=5(2 a+3 b)(2 a-3 b)$
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Question 192 Marks
$(2x + 5y)2 - 1$
Answer
$(2x + 5y)2 - 1$
$= (2x + 5y + 1)(2x + 5y - 1)$
$\{\because  a2 - b2 = (a + b)(a - b) \}$
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Question 202 Marks
Factorise:
$2 a+6 b-3(a+3 b)^2$
Answer
$2 a+6 b-3(a+3 b)^2$
$=2(a+3 b)-3(a+3 b)^2$
$=(a+3 b)\{2-3(a+3 b)\}$
$=(a+3 b)(2-3 a-9 b)$
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Question 212 Marks
Evalute: $ \left\{(405)^2-(395)^2\right\} $
Answer
$ \left\{(405)^2-(395)^2\right\} $
$ (405)^2-(395)^2 $
$ =(405+395)(405-395) $
$ \left\{\because a^2-b^2(a+b)(a-b)\right\} $
$ =800 \times 10=8000 $
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Question 222 Marks
Factorise:
$6 a b-b^2+12 a c-2 b c$
Answer
$6 a b-b^2+12 a c-2 b c$
$=6 a b+12 a c-b^2-2 b c$
$=6 a(b+2 c)-b(b+2 c)$
$= (b + 2c)(6a - b)$
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Question 232 Marks
Factorise: $y^2+y-72$
Answer
$y^2+y-72$
$=y^2+9 y-8 y-72$
$\{-72=9 \times(-8), 1=9-8\}$
$= y(y + 9) - 8(y + 9) $
$= (y + 9)(y - 8)$
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Question 242 Marks
Factorise: $ (2 a+3 b)^2-16 c^2$
Answer
$ (2 a+3 b)^2-16 c^2$
$ =(2 a+3 b) 2-(4 c)^2 $
$ =(2 a+3 b+4 c)(2 a+3 b-4 c) $
$ \{\because a 2-b 2=(a+b)(a-b)\} $
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Question 252 Marks
Factorise:
$p^2-6 p-16$
Answer
$p^2-6 p-16$
$=p^2+8 p-2 p-16$
$\{-16=8 \times(-2), 6=8-2\}$
$= p(p + 8) - 2(p + 8)$
$ = (p + 8)(p - 2)$
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Question 262 Marks
Factorise:
$x^2-23 x+42$
Answer
$x^2-23 x+42$
$=x^2-2 x-21 x+42$
$\{42=(-2) \times(-21),-23=-2-21\}$
$= x(x - 2) - 21(x - 2) $
$= (x - 2)(x - 21)$
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Question 272 Marks
Factorise: $ x^2+6 a x+9 a^2$
Answer
$ x^2+6 a x+9 a^2$
$=(x)^2+2 \times x \times 3 a+(3 a)^2 $
$ =(x+3 a)^2 $
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Question 282 Marks
Factorise: $x^2+x-132$
Answer
$x^2+x-132$
$=x^2+12 x-11 x-132$
${-132 = 12 × (-11), 1 = 12 - 11}$
$= x(x + 12) - 11(x + 12) $
$= (x + 12)(x - 11)$
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Question 292 Marks
Factorise: $1 - (b - c)^2$
Answer
$1 - (b - c)^2$
$= (1)^2 - (b - c)^2$
$= (1 + b + c)(1 - b + c)$
$\{\because a^2- b^2= (a + b)(a - b)\}$
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Question 312 Marks
Factorise: $x^2- 7x - 30$
Answer
$x^2-7 x-30$
$=x^2-10 x+3 x-30$
${-30 = -10 × 3, -7 = -10 + 3}$
$= x(x - 10) + 3(x - 10) $
$= (x - 10)(x + 3)$
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Question 322 Marks
Factorise: $4a^2- 9$
Answer
$4a^2- 9$
$= (2a)^2- (3)^2$
$= (2a + 3)(2a - 3)$ $\{\therefore a^2- b^2= (a + b)(a - b)\}$
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Question 332 Marks
Factorise: $y^2+19 y+60$
Answer
$y^2+19 y+60$
$=y^2+19 y+60$
${60 = 15 × 4, 19 = 15 + 4}$
$= y(y + 15) + 4(y + 15)$
$= (x + 15)(x + 4)$
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Question 342 Marks
Factorise: $63 a^2-112 b^2=7$
Answer
Using: $a^2-b^2=(a+b)(a-b) $
$63 a^2-112 b^2=7$
$\left(9 a^2-16 b^2\right)$
$=7 [(3 a)^2-(4 b)^2]$
$=7(3 a+4 b)(3 a-4 b).$
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Question 352 Marks
Factorise: $1-6 x+9 x^2$
Answer
$1-6 x+9 x^2$
$=(1)^2-2 \times 1 \times 3 x+(3 x)^2$
$=(1-3 x)^2$
$\{\because a2 - 2ab + b2 = (a - b)2\}$
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Question 362 Marks
Factorise: $q^2-10 q+21$
Answer
$q^2-10 q+21$
$=q^2+7 q-3 q+21$
${21 = (-7) × (-3), -10 = -7 - 3}$
$= q(q - 7) - 3(q - 7)$
$ = (q - 7)(q - 3)$
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Question 372 Marks
Factorise: $6 p^2+11 p-10$
Answer
$6 p^2+11 p-10$
$=6 x^2+15 x-4 x-10$
${6 × (-10) = 60, -60 = 15 × (-4), 11 = 15 - 4}$
$= 3x(2x + 5) - 2(2x + 5) $
$= (3x - 2)(2x + 5)$
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Question 382 Marks
Factorise: $3 x^2+10 x+8$
Answer
$3 x^2+10 x+8$
$=3 x^2+6 x+4 x+8$
${24 = 3 × 8, 24 = 6 × 4, 10 = 6 + 4}$
$= 3x(x + 2) + 4(x + 2) $
$= (x + 2)(3x + 4)$
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Question 392 Marks
Factorise: $28-31 x-5 x^2$
Answer
$28-31 x-5 x^2$
$=28-35 x+4 x-5 x^2$
${28 × (-5) = -140, -140 = -35 × 4, -31 = -35 + 4}$
$= 7(4 - 5x) + x(4 - 5x)$
$= (4 - 5x)(7 + x)$
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Question 412 Marks
Factorise: $a^2 b^2-6 a b c+9 c^2$
Answer
$a^2 b^2-6 a b c+9 c^2$
$=(a b)^2-2 \times a b \times 3 c+(3 c)^2$
$=(a b-3 c)^2$
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Question 422 Marks
Factorise: $y^2+10 y+24$
Answer
$y^2+10 y+24$
$=y^2+6 y+4 y+24$
$ {24 = 6 × 4, 10 = 6 + 4} $
$= y(y + 6) + 4(y + 6) $
$= (y + 6)(y + 4)$
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Question 432 Marks
Factorise: $m^2+2 m^2 n^2+n^4$
Answer
$m^2+2 m^2 n^2+n^4$
$=\left(m^2\right)^2+2 m^2 n^2+\left(n^2\right)^2$
$=\left(m^2+n^2\right)^2$
$\{\because a^2+ 2ab + b^2= (a + b)^2\}$
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Question 442 Marks
Factorise: $2 x^2+x-45$
Answer
$2 x^2+x-45$
$=2 x^2+10 x-9 x-45$
${2 × (-45) = -90, -90 = (10) × (-9), 1 = 10 - 9}$
$= 2x(x - 5) + 9(x - 5) $
$= (x - 5)(2x + 9)$
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Question 452 Marks
Factorise: $9 a^2-b^2+4 b-4$
Answer
$9 a^2-b^2+4 b-4$
$=9 a^2-\left(b^2-4 b+4\right)$
$=(3 a)^2-[(b)^2-2 \times b \times 2+(2)^2$
$=(3 a)^2-(b-2)^2.$
$\{\because a^2- 2ab + b^2= (a - b)^2\}$
$= (3a + b - 2)(3a - b + 2)$
$\{\because a^2- b^2= (a + b)(a - b)\}$
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Question 462 Marks
Factorise: $7 x^2-19 x-6$
Answer
$7 x^2-19 x-6$
$=7 x^2-21 x+2 x-6$
${7 × (-6) = -42, -42 = -21 × 2, -19 = -21 + 2}$
$= 7x(x - 3) + 2(x - 3) $
$= (x - 3)(7x + 2)$
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Question 472 Marks
Factorise: $p^2-4 p-11$
Answer
$p^2-4 p-11$
$=p^2+11 p-7 p-77$
${-77 = -11 × 7, -4 = -11 + 7}$
$= p(p - 11) + 7(p - 11) $
$= (p - 11)(p + 7)$
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Question 482 Marks
Factorise: $(l+3)^2-(l-m)^2$​​​​​​​
Answer
$(l+3)^2-(l-m)^2$
$= (l + m + l - m)(l + m - l + m)$
$\{\because a2 - b2 = (a + b)(a - b)\}$
$= 2l × 2m = 4lm$
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Question 492 Marks
Factorise: $x^2-10 x+24$
Answer
$x^2-10 x+24$
$=x^2-6 x-4 x+24$
${24 = (-6) × (-4), -10 = -6 - 4}$
$= x(x - 6) - 4(x - 6) $
$= (x - 6)(x - 4)$
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Question 502 Marks
Factorise: $81-49 \mathrm{x}^2$
Answer
$81-49 \mathrm{x}^2$
$=(9)^2-(7 \mathrm{x})^2=(9+7 \mathrm{x})(9-7 \mathrm{x})$
 $\{\because a^2- b^2= (a + b)(a - b)\}$
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