KC Sinha Mathematics Solution Class 9 Chapter 4 Algebraic Identities(बीजीय सर्वसमिकाएँ) exercise 4.2

Exercise 4.2

Question 1

(i) x²+4x+4
Sol:
Using identity:
a²+b²+2ab=(a+b)²=(a+b)(a+b)
⇒x²+2²+2(x)(2)
⇒(x+2)²
⇒(x+2)(x+2)

(ii) x²-18x+81
Sol:
Using identity:
a²+b²-2ab=(a-b)²=(a-b)(a-b)
⇒x²+9²-2(x)(9)
⇒(x-9)²
⇒(x-9)(x-9)

(iii) 4x²+4x+1
Sol:
Using identity:
a²+b²+2ab=(a+b)²=(a+b)(a+b)
⇒(2x)²+1²+2(2x)(1)
⇒(2x+1)²
⇒(2x+1)(2x+1)

(iv) $\dfrac{4}{9}a^2+b^2+\dfrac{4}{3}ab$
Sol:
Using identity:
a²+b²+2ab=(a+b)²=(a+b)(a+b)
⇒$\left(\dfrac{2}{3}a\right)^2+b^2+2\left(\dfrac{2}{3}a\right)(b)$
⇒$\left(\dfrac{2}{3}a+b\right)^2$
⇒$\left(\dfrac{2}{3}a+b\right)$ $\left(\dfrac{2}{3}a+b\right)$

(v) 25a²-60ab+36b²
Sol:
Using identity:
a²+b²-2ab=(a-b)²=(a-b)(a-b)
⇒(5a)²+(6b)²-2(5a)(6b)
⇒(5a-6b)²
⇒(5a-6b)(5a-6b)

(vi) 9a²-30ab+25b²
Sol:
Using identity:
a²+b²-2ab=(a-b)²=(a-b)(a-b)
⇒(3a)²+(5b)²-2(3a)(5b)
⇒(3a-5b)²
⇒(3a-5b)(3a-5b)

(vii) 4(x-y)²-12(x-y)(x+y)+9(x+y)²
Sol:
Using identity:
(a²-b²)=(a+b)(a-b)
⇒4(x²+y²-2xy)-12(x²-y²)+9(x²+y²+2xy)
⇒4x²+4y²-8xy-12x²+12y²+9x²+9y²+18xy
⇒4x²-12x²+9x²+4y²+12y²+9y²-8xy+18xy
⇒13x²-12x²+13y²+12y²+10xy
⇒x²+25y²+10xy
⇒x²+(5y)²+2(x)(5y)
Using identity:
a²+b²+2ab=(a+b)²
⇒(x+5y)²
⇒(x+5y)(x+5y)

Question 2

निम्नलिखित का गुणनखंड निकाले
(i) p²+q²+9r²+2pq+6pr+6qr
Sol:
Using identity:
a²+b²+c²+2ab+2bc+2ca=(a+b+c)²
⇒p²+q²+(3r)²+2pq+2(q)(3r)+2(3r)(p)
⇒(p+q+3r)²
⇒(p+q+3r)(p+q+3r)

(ii) 4a²+9b²+c²+12ab+4ac+6bc
Sol:
Using identity:
a²+b²+c²+2ab+2bc+2ca=(a+b+c)²
⇒(2a)²+(3b)²+c²+2(2a)(3b)+2(3b)(c)+2(c)(2a)
⇒(2a+3b+c)²
⇒(2a+3b+c)(2a+3b+c)

(iii) x²+4+9z²+4x-6rz-12z
Sol:
Using identity:
a²+b²+c²+2ab+2bc+2ca=(a+b+c)²
⇒(x)²+(2)²+(-3z)²+2(x)(2)+2(2)(-3z)+2(-3z)(x)
⇒(x+2-3z)²
⇒(x+2-3z)(x+2-3z)

(iv) 4x²+9y²+z²-12xy-4xz+6yz
Sol:
Using identity:
a²+b²+c²+2ab+2bc+2ca=(a+b+c)²
⇒(-2x)²+(3y)²+(z)²+2(-2x)(3y)+2(3y)(z)+2(z)(-2x)
⇒(-2x+3y+z)²
[Taking common (-)]
⇒(2x-3y-z)(2x-3y-z)

Question 4

निम्नलिखित का गुणनखंड ज्ञात करे ।
(i) x²-1
Sol:
Using identity:
a²-b²=(a+b)(a-b)
⇒x²-1²
⇒(x-1)(x+1)

(ii) x²-4
Sol:
Using identity:
a²-b²=(a+b)(a-b)
⇒x²-2²
⇒(x-2)(x+2)

(iii) 49-64x²
Sol:
Using identity:
a²-b²=(a+b)(a-b)
⇒(7)²-(8x)²
⇒(7-8x)(7+8x)

(iv) 4x²-81
Sol:
Using identity:
a²-b²=(a+b)(a-b)
⇒(2x)²-(9)²
⇒(2x-9)(2x+9)

(v) 4a²-(2b-c)²
Sol:
Using identity:
a²-b²=(a+b)(a-b)
⇒(2a)²-(2b-c)²
⇒[2a+2b-c][2a-(2b-c)]
⇒(2a+2b-c)(2a-2b+c)

(vi) 16(2x-1)²-25z²
Sol:
Using identity:
a²-b²=(a+b)(a-b)
⇒[4(2x-1)]²-(5z)²
⇒[4(2x-1)+5z][4(2x-1)-5z]
⇒[8x-4+5z][8x-4-5z]
⇒(8a-5z-4)(8x+5z-4)

(vii) $\frac{x^2}{4}-\frac{y^2}{4}$
Sol :

(viii) 25(x-y)²-4(x+4)²
Sol :

Question 5

निमनलिखित का गुणनखंड ज्ञात करे ।
(i) x²+y²+2xy-z²
Sol:
Using identity:
a²+b²+2ab=(a+b)²
⇒(x+y)²-(z)²
Using identity:
a²-b²=(a+b)(a-b)
⇒(x+y+z)(x+y-z)

(ii) x²+y²-z²-2xy
Sol:
Using identity:
a²+b^2-2ab=(a-b)²
⇒x²+y²-2xy-z²
⇒(x-y)²-(z)²
Using identity:
a²-b²=(a+b)(a-b)
⇒[(x-y)+(z)][(x-y)-(z)]
⇒(x-y+z)(x-y-z)

(iii) a²-b²-c²-2bc
Sol:
⇒(a)²-(b²+c²+2bc)
Using identity:
x²+y²+2xy=(x+y)²
⇒(a)²-(b+c)²
Using identity:
x²-y²=(x+y)(x-y)
⇒[(a)+(b+c)][(a)-(b+c)]
⇒(a+b+c)(a-b-c)

(iv) x²-1-2a-a²
Sol:
⇒x²-[a²+1²+2(a)(1)]
Using identity:
a²+b²+2ab=(a+b)²
⇒(x)²-(a+1)²
Using identity:
a²-b²=(a+b)(a-b)
⇒[x+a+1][x-(a+1)]
⇒(x+a+1)(x-a-1)

(v) x⁴-14x²y²+y⁴
Sol:
⇒x⁴+y⁴-14x²y²
⇒[(x²)²+(y²)²+2(x²)(y²)]-[16(x²)(y²)]
Using identity:
a²-b²=(a+b)(a-b)
⇒[(x²+y²)²]-[16(x²)(y²)]
⇒(x²+y²)²-(4xy)²
⇒(x²+y²-4xy)(x²+y²+4xy)

(vi) x⁴-7x²y²+y⁴
Sol:
⇒x⁴+y⁴-7x²y²
⇒[(x²)²+(y²)²+2(x²)(y²)]-[9x²y²]
Using identity:
a²+b²+2ab=(a+b)²
⇒[(x²+y²)²]-[9x²y²]
⇒(x²+y²)²-(3xy)²
Using identity:
a²-b²=(a+b)(a-b)
⇒[(x²+y²)+(3xy)][(x²+y²)-(3xy)]
⇒(x²+y²+3xy)(x²+y²-3xy)

(vii) 4x⁴+3x²+9
Sol :
=(2x²)²+2.2x²+3²-9x²
=(2x²+3)²-(3x)²
=(2x²+3+3x)(2x²+3-3x)

Question 6

निम्नलिखित का गुणनखंड ज्ञात करे ।
(i) (1-x²)(1-y²)+4xy
Sol:
⇒(1-x²)×(1-y²)+4xy
⇒1×(1-y²)-x²(1-y²)+4xy
⇒1-y²-x²+x²y²+4xy
⇒(1+2xy+x²y²)-(x²+y²-2xy)
Using identities:
a²+b²+2ab=(a+b)²
a²+b²-2ab=(a-b)²
⇒(1+xy)²-(x-y)²
⇒[(1+xy)+(x-y)] × [( 1+xy)- (x-y)]
⇒[(1+x-y+xy) (1-x+y+xy)]

(ii) c²+2ab-(a²+b²)
Sol:
⇒c²+2ab-a²-b²
⇒(c)²-(a²+b²-2ab)
Using identity:
a²+b²-2ab=(a-b)²
⇒c²-(a-b)²
Using identity:
a²-b²=(a+b)(a-b)
⇒(c+a-b)(c-a+b)

(iii) p-2q-p²+4q²
Sol:
⇒p-2q-p²+4q²
⇒p-2q-(p²-4q²)
Using identity:
a²-b²=(a - b)(a + b)
⇒(p-2q)-(p-2q)(p+2q)
[Taking common (p-2q)]
⇒(p-2q)(1-p-2q)

(iv) b²+c²+2(ab+bc+ca)
Sol:
adding and subtracting a²
⇒a²+b²+c²+2(ab+bc+ca)-a²
⇒a²+b²+c²+2ab+2bc+2ca-a²
Using identity:
x²+y²+z²+2xy+2yz+2zx=(x+y+z)²
⇒(a+b+c)²-a²
Using identity:
a²-b²=(a+b)(a-b)
⇒[(a+b+c)+a][(a+b+c)-a]
⇒(b+c+2a)(b+c)

Question 7

निम्नलिखित का गुणनखंड ज्ञात करे ।

(i) 27x³+54x²y+36xy²+8y³
Sol:
⇒27x³+8y³+54x²y+36xy²
⇒(3x)³+(2y)³+3(3x)²(y)+3(3x)(y)²
Using identity:
a³+b³+3a²b+3ab²=(a+b)³
⇒(3x+2y)³
⇒(3x+2y)(3x+2y)(3x+2y)

(ii) 8x³+y³+12x²y+6xy²
Sol:
⇒8x³+y³+12x²y+6xy²
⇒(2x)³+y³+3(2x)²(y)+3(2x)(y)²
Using identity:
a³+b³+3a²b+3ab²=(a+b)³
⇒(2x+y)³
⇒(2x+y)(2x+y)(2x+y)

(iii) a³x³-3a²bx²+3ab²x-b³
Sol:
⇒a³x³-b³-3a²bx²+3ab²x
⇒(ax)³-b³-3(ax)²(b)+3(ax)(b)²
Using identity:
a³-b³-3a²b+3ab²=(a-b)³
⇒(ax-b)³
⇒(ax-b)(ax-b)(ax-b)

(iv) x³-12x(x-4)-64
Sol:
⇒x³-4³-12x(x-4)
⇒x³-4³-12x²+48x
⇒x³-4³-3(x)²(4)+3(x)(4)²
Using identity:
a³-b³-3a²b+3ab²=(a-b)³
⇒(x-4)³
⇒(x-4)(x-4)(x-4)

(v) $a^3+\dfrac{3}{2}a^2+\dfrac{3}{4}a+\dfrac{1}{8}$
Sol:
⇒$a^3+\dfrac{1}{8}+\dfrac{3}{2}a^2+\dfrac{3}{4}a$
⇒$a^3+\left(\dfrac{1}{2}\right)^3+3(a)^2\left(\dfrac{1}{2}\right)+3(a)\left(\dfrac{1}{2}\right)^2$
Using identity:
a³+b³+3a²b+3ab²=(a+b)³
⇒$\left(a+\dfrac{1}{2}\right)^3$
⇒$\left(a+\dfrac{1}{2}\right)\left(a+\dfrac{1}{2}\right)\left(a+\dfrac{1}{2}\right)$

(vi) $x^3-x^2y+\frac{1}{3}xy^2-\frac{1}{27}y^3$

Sol :

=$\left(x-\frac{1}{3}y\right)^3$

=$\left(x-\frac{1}{3}y\right)\left(x-\frac{1}{3}y\right)\left(x-\frac{1}{3}y\right)$