8th Maths 9.5

Q1.What is an Identity ?
Answer
An equality, true for every value of the variable in it, is called an identity.
1. (a + b)2=a2+2ab+b2
2. (a - b)2=a2-2ab+b2
3. (a + b)(a - b) = a2-b2
4. (x + a)(x + b) = x2+ (a + b)x + ab

Exercise 9.5

Q1. Use a suitable identity to get each of the following products.
i) (x+3)(x+3) 
ii) (2y+5)(2y+5)
iii) (2a-7)(2a-7)
iv) (3a-12)(3a-12)
v) (1.1m-0.4)(1.1m+0.4)
vi) (a2+b2)(-a2+b2)
vii) (6x-7)(6x+7)
viii) (-a+c)(-a+c)
ix) (x2+3y4)(x2+3y4)
x) (7a-9b)(7a-9b)


i) (x+3)(x+3)
Sol.
=(x+3)(x+3)
=(x+3)2
[ (a + b)2=a2+2ab+b2]
where a = x, b = 3
=x2+2×x×3+(3)2
=x2+6x+9
Answer
=x2+6x+9

ii) (2y+5)(2y+5)
Sol.
=(2y+5)(2y+5)
=(2y+5)2
[ (a + b)2=a2+2ab+b2]
where a = 2y,b = 5
=(2y)2+2×2y×5+(5)2
=4y2+20y+25
Answer
=4y2+20y+25

iii) (2a-7)(2a-7)
Sol.
=(2a-7)(2a-7)
=(2a-7)2
[ (a - b)2=a2-2ab+b2]
where a =2a,b =7
=(2a)2+2×2a×7+(7)2
=4a2+28a+49
Answer
=4a2+28a+49

iv) (3a-12)(3a-12)
Sol.
=(3a-12)(3a-12)
=(3a-12)2
[ (a - b)2=a2-2ab+b2]
where a =3a,b =12
=(3a)2-2×3a×12+(12)2
=9a2-3a+14
Answer
=9a2-3a+14

v) (1.1m-0.4)(1.1m+0.4)
Sol.
=(1.1m-0.4)(1.1m+0.4)
[ (a + b)(a - b) = a2-b2]
Where a = 1.1m, b = 0.4
=(1.1m)2-(0.4)2
=1.21m2-1.6
Answer
=1.21m2-1.6

vi) (a2+b2)(-a2+b2)
Sol.
=(a2+b2)(-a2+b2)
=(b2+a2)(b2-a2)
[ (a + b)(a - b) = a2-b2]
Where a=b2,b=a2
=(b2)2-(a2)2
=b4-a4
Answer
=b4-a4

vii) (6x-7)(6x+7)
Sol.
=(6x-7)(6x+7)
[ (a + b)(a - b) = a2-b2]
Where a = 6x, b = 7
=(6x)2-(7)2
=36x2-49
Answer
=36x2-49

viii) (-a+c)(-a+c)
Sol.
=(-a+c)(-a+c)
=(c-a)(c-a)
=(c-a)2
[ (a - b)2=a2-2ab+b2]
where a = c, b = a
=c2-2×c×a+a2
=c2-2ac+a2
Answer
=a2+2ac+c2

ix) (x2+3y4)(x2+3y4)
Sol.
=(x2+3y4)(x2+3y4)
=(x2+3y4)2
[ (a + b)2=a2+2ab+b2]
where a=x2,b=3y4
=(x2)2+2×x2×3y4+(3y4)2
=x24+3xy4+9y216
Answer
=x24+3xy4+9y216

x) (7a-9b)(7a-9b)
Sol.
=(7a-9b)(7a-9b)
=(7a-9b)2
[ (a - b)2=a2-2ab+b2]
where a = 7a, b = 9b
=(7a)2-2×7a×9b+(9b)2
=49a2-126ab+81b2
Answer
=49a2-126ab+81b2

Q2. Use the identity (x+a)(x+b)=x2+(a+b)x+ab to find the following products.
i) (x+3)(x+7)
ii) (4x+5)(4x+1)
iii) (4x-5)(4x-1)
iv) (4x+5)(4x-1)
v) (2x+5y)(2x+3y)
vi) (2a2+9)(2a2+5)
vii) (xyz-4)(xyz-2)

i) (x+3)(x+7)
Sol.
=(x+3)(x+7)
[ (x + a)(x + b) = x2+ (a + b)x + ab ]
where x = x, a = 3, b = 7
=x2+(3+7)x+3×7
=x2+10x+21
Answer
=x2+10x+21

ii) (4x+5)(4x+1)
Sol.
 =(4x+5)(4x+1)
[ (x + a)(x + b) = x2+ (a + b)x + ab ]
where x = 4x, a = 5, b = 1
=(4x)2+(5+1)4x+5×1
=16x2+24x+5
Answer
=16x2+24x+5

iii) (4x-5)(4x-1)
Sol.
=(4x-5)(4x-1)
[ (x - a)(x - b) = x2- (a + b)x + ab ]
where x = 4x, a = 5, b = 1
=(4x)2-(5+1)4x+5×1
=16x2-24x+5
Answer
=16x2-24x+5

iv) (4x+5)(4x-1)
Sol.
=(4x+5)(4x-1)
[ (x + a)(x - b) = x2+ (a - b)x - ab ]
where x = 4x, a = 5, b = 1
=(4x)2+(5-1)4x-5×1
=16x2+16x-5
Answer
=16x2+16x-5

v) (2x+5y)(2x+3y)
Sol.
=(2x+5y)(2x+3y)
[ (x + a)(x + b) = x2+ (a + b)x + ab ]
where x = 2x, a = 5y, b = 3y
=(2x)2+(5y+3y)2x+5y×3y
=4x2+16xy+15y2
Answer
=4x2+16xy+15y2

vi) (2a2+9)(2a2+5)
Sol.
=(2a2+9)(2a2+5)
[ (x + a)(x + b) = x2+ (a + b)x + ab ]
where x=2a2,a =9,b =5
=(2a2)2+(9+5)2a2+9×5
=4a4+28a2+45
Answer
=4a4+28a2+45

vii) (xyz-4)(xyz-2)
Sol.
=(xyz-4)(xyz-2)
[ (x - a)(x - b) = x2- (a + b)x + ab ]
where x = xyz, a = 4, b = 2
=(xyz)2-(4+2)xyz+4×2
=x2y2z2-6xyz+8
Answer
=x2y2z2-6xyz+8


Q3. Find the following squares by using the identities.
i) (b-7)2
ii) (xy-3z)2
iii) (6x2-5y)2
iv) (23m+32n)2
v) (0.4p-0.5q)2
vi) (2xy+5y)2

i) (b-7)2
Sol.
=(b-7)2
[ (a - b)2=a2-2ab+b2]
where a =b,b =7
=(b)2-2×b×7+(7)2
=b2-14b+49
Answer
=b2-14b+49

ii) (xy-3z)2
Sol.
=(xy-3z)2
[ (a - b)2=a2-2ab+b2]
where a = xy, b = 3z
=(xy)2-2×xy×3z+(3z)2
=x2y2-6xyz+9z2
Answer
=x2y2-6xyz+9z2

iii) (6x2-5y)2
Sol.
=(6x2-5y)2
[ (a - b)2=a2-2ab+b2]
where a=6x2,b =5y
=(6x2)2-2×6x2×5y+(5y)2
=36x4-60x2y+25y2
Answer
=36x4-60x2y+25y2

iv) (23m+32n)2
Sol.
=(23m+32n)2
[ (a + b)2=a2+2ab+b2]
where a = 23m,b = 32n
=(23m)2+2×23m×32n+(32n)2
=49m2+2mn+94n2
Answer
=49m2+2mn+94n2

v) (0.4p-0.5q)2
Sol.
=(0.4p-0.5q)2
[ (a - b)2=a2-2ab+b2]
where a = 0.4p, b = 0.5q
=(0.4p)2-2×0.4p×0.5q+(0.5q)2
=0.16p2-0.04pq+0.25q2
Answer
=0.16p2-0.04pq+0.25q2

vi) (2xy+5y)2
Sol.
=(2xy+5y)2
[ (a + b)2=a2+2ab+b2]
where a = 2xy,b = 5y
=(2xy)2+2×2xy×5y+(5y)2
=4x2y2+20xy+25y2
Answer
=4x2y2+20xy+25y2

Q4. Simplify.
i) (a2-b2)2
ii) (2x+5)2-(2x-5)2
iii) (7m-8n)2+(7m+8n)2
iv) (4m+5n)2+(5m+4n)2
v) (2.5p-1.5q)2-(1.5p-2.5q)2
vi) (ab+bc)2-2ab2c
vii) (m2-n2m)2+2m3n2

i) (a2-b2)2
Sol.
=(a2-b2)2
[ (a - b)2=a2-2ab+b2]
where a =a2,b =b2
=(a2)2-2×a2×b2+(b2)2
=a4-2a2b2+b4
Answer
=a4-2a2b2+b4

ii) (2x+5)2-(2x-5)2
Sol.
=(2x+5)2-(2x-5)2
[  a2-b2=(a+b)(a-b)  ]
Where a = (2x+5), b = (2x-5)
=(2x+5+2x-5)(2x+5-2x+5)
=4x×10
=40x
Answer
=40x

iii) (7m-8n)2+(7m+8n)2
Sol.
=(7m-8n)2+(7m+8n)2
=(7m)2-2×7m×8n+(8n)2+(7m)2+2×7m×8n+(8n)2
=49m2-112mn+64n2+49m2+112mn+64n2
=98m2+128n2
Answer
=98m2+128n2

iv) (4m+5n)2+(5m+4n)2
Sol.
=(4m+5n)2+(5m+4n)2
=(4m)2+2×4m×5n+(5n)2+(5m)2+2×5m×4n+(4n)2
=16m2+40mn+25n2+25m2+40mn+16n2
=41m2+80mn+41n2
Answer
=41m2+80mn+41n2

v) (2.5p-1.5q)2-(1.5p-2.5q)2
Sol.
=(2.5p-1.5q)2-(1.5p-2.5q)2
=(2.5p)2-2×2.5p×1.5q+(1.5q)2-(1.5p)2-2×1.5p×2.5q+(2.5q)2
=6.25p2-7.50pq+2.25q2-2.25p2+7.50pq-6.25q2
=4p2-4q2
Answer
=4p2-4q2

vi) (ab+bc)2-2ab2c
Sol.
=(ab+bc)2-2ab2c
=(ab)2+2×ab×bc+(bc)2-2ab2c
=a2b2+2ab2c+b2c2-2ab2c
=a2b2+b2c2
Answer
=a2b2+b2c2

vii) (m2-n2m)2+2m3n2
Sol.
 =(m2-n2m)2+2m3n2
=(m2)2-2×m2×n2m+(n2m)2+2m3n2
=m4-2m3n2+n4m2+2m3n2
=m4+n4m2
Answer
=m4+n4m2

Q5. Show that.
i) (3x+7)2-84x=(3x-7)2
ii) (9p-5q)2+180pq=(9p+5q)2
iii) (43m-34n)2+2mn=169m2+916n2
iv) (4pq+3q)2-(4pq-3q)2=48pq2
v) (a-b)(a+b)+(b-c)(b+c)+(c-a)(c+a)=0

i) (3x+7)2-84x=(3x-7)2
Sol.
=LHS.
=(3x+7)2-84x
=(3x)2+2×3x×7+(7)2-84x
=9x2+42x+49-84x
=9x2-42x+49
=(3x)2-2×3x×7+(7)2
=(3x-7)2
=RHS.
LHS.=RHS.
Proved

ii) (9p-5q)2+180pq=(9p+5q)2
Sol.
LHS.
=(9p-5q)2+180pq
=(9p)2-2×9p×5q+(5q)2+180pq
=81p2-90pq+25q2+180pq
=81p2+90pq+25q2
=(9p)2+2×9p×5q+(5q)2
=(9p+5q)2
=RHS.
LHS.=RHS.
Proved

iii) (43m-34n)2+2mn=169m2+916n2
Sol.
LHS.
=(43m-34n)2+2mn
=(43m)2-2×43m×34n+(34n)2+2mn
=169m2-2mn+916n2+2mn
=169m2+916n2
RHS.
LHS.=RHS.
Proved

iv) (4pq+3q)2-(4pq-3q)2=48pq2
Sol.
=(4pq+3q)2-(4pq-3q)2
=(4pq+3q+4pq-3q)(4pq+3q-4pq+3q)
=(8pq)(6q)
=48pq2
LHS.
LHS.=RHS.
Proved

v) (a-b)(a+b)+(b-c)(b+c)+(c-a)(c+a)=0
Sol.
LHS.
=(a-b)(a+b)+(b-c)(b+c)+(c-a)(c+a)
=a2-b2+b2-c2+c2-a2
=0
RHS.
LHS.=RHS.
Proved

Q6. Using identities, evaluate
i) 712
ii) 992
iii) 1022
iv) 9982
v) 5.22
vi) 297×303
vii) 78×82
viii) 8.92
ix) 1.05×9.5

i) 712
Sol.
=712
=(70+1)2
[ (a + b)2=a2+2ab+b2]
where a = 70,b = 1
=(70)2+2×70×1+(1)2
=4900+140+1
=5041
Answer
=5041

ii) 992
Sol.
=992
=(100-1)2
[ (a + b)2=a2-2ab+b2]
where a = 100,b = 1
=(100)2-2×100×1+(1)2
=10000-200+1
=9801
Answer
=9801

iii) 1022
Sol.
=1022
=(100+2)2
[ (a + b)2=a2+2ab+b2]
where a = 100,b = 2
=(100)2+2×100×2+(2)2
=10000+400+4
=10404
Answer
=10404

iv) 9982
Sol.
=9982
=(1000-2)2
[ (a + b)2=a2-2ab+b2]
where a = 1000,b = 2
=(1000)2-2×1000×2+(2)2
=1000000-4000+4
=996004
Answer
=996004

v) 5.22
Sol.
=5.22
=(5.0+0.2)2
[ (a + b)2=a2+2ab+b2]
where a = 5.0,b = 0.2
=(5.0)2+2×5.0×.2+(0.2)2
=25+2.0+0.04
=27.04
Answer
=27.04


vi) 297×303
Sol.
=297×303
=(300-3)(300+3)
[ (a + b)(a - b) = a2-b2]
Where a = 300, b = 3
=(300)2-(3)2
=90000-9
=89991
Answer
=89991

vii) 78×82
Sol.
=78×82
=(80-2)(80+2)
[ (a + b)(a - b) = a2-b2]
Where a = 80, b = 2
=(80)2-(2)2
=6400-4
=6396
Answer
=6396

viii) 8.92
Sol.
=8.92
=(8+0.9)2
[ (a + b)2=a2+2ab+b2]
where a = 8,b = 0.9
=(8)2+2×8×0.9+(0.9)2
=64+14.4+0.81
=79.21
Answer
=79.21

ix) 1.05×9.5
Sol.
=1.05×9.5
=(1+0.05)(9+0.5)
=1(9+0.5)+0.05(9+0.5)
=1×9+1t×0.5+0.05×9+0.05×0.5 
=9+0.5+0.45+0.025
=9.5+0.475
=9.975
Answer
=9.975


Q7. Using a2-b2=(a+b)(a-b), find
i) 512-482
ii) (1.02)2-(0.98)2
iii) 1532-1472
iv) 12.12-7.92

i) 512-492
Sol.
=512-492
=(51+49)(51-49)
=100×2
=200
Answer
=200

ii) (1.02)2-(0.98)2
Sol.
=(1.02)2-(0.98)2
=(1.02+0.98)(1.02-0.98)
=2×0.04
=0.08
Answer
=0.08

iii) 1532-1472
Sol.
=1532-1472
=(153+147)(153-147)
=300×6
=1800
Answer
=1800

iv) 12.12-7.92
Sol.
=12.12-7.92
=(12.1+7.9)(12.1-7.9)
=20×4.2
=84
Answer
=84

Q8. Using (x+a)(x+b)=x2+(a+b)x+ab, find
i) 103×104
ii) 5.1×5.2
iii) 103×98
iv) 9.7×9.8

i) 103×104
Sol.
=103×104
=(100+3)(100+4)
=(100)2+(3+4)100+3×4
=10000+700+12
=10712
Answer
=10712

ii) 5.1×5.2
Sol.
=5.1×5.2
=(5+0.1)(5+0.2)
=(5)2+(0.1+0.2)5+0.1×0.2
=25+0.3×5+0.02
=25+1.5+0.02
=26.52
Answer
=26.52

iii) 103×98
Sol.
=103×98
=(100+3)(100-2)
=(100)2+(3-2)100-3×2
=10000+100-6
=10000+94
=10094
Answer
=10094

iv) 9.7×9.8
Sol.
=9.7×9.8
=(9+0.7)(9+0.8)
=(9)2+(0.7+0.8)9+0.7×0.8
=81+1.5×9+0.56
=81+13.5+0.56
=95.06
Answer
=95.06

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RSKMP 5th & 8th Result

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RSKMP 5th & 8th Result राज्य शिक्षा केंद्र  [RSKMP ] भोपाल म.प्र. 5वीं और 8वीं का रिजल्ट प्राप्त करने के लिए आवश्यक दस्तावेज  कक्षा 5वीं और 8वीं का रिजल्ट प्राप्त करने के लिए आवश्यक दस्तावेज रोल नंबर, आवेदन क्रमांक, समग्र आईडी और अन्य सबमिट कर । Like, Share, and Subscribe. 5वीं और 8वीं का रिजल्ट कहाँ से प्राप्त करें कक्षा  5वीं और  8वीं का रिजल्ट उम्मीदवार नीचे दी वेबसाइट को टच करके 5th और 8th का रिजल्ट  🫲 5th और 8th का रिजल्ट  🫲 5th और 8th का रिजल्ट  🫲 5th और 8th का रिजल्ट  🫲 पर नतीजे देख सकते है । 5वीं और 8वीं के उम्मीदवारों को सलाह  यह परीक्षा जीवन का एक हिस्सा है, सारा जीवन नहीं । इस परीक्षा के परिणाम की कमियों से परेशान न होकर संयम और धैर्यपूर्वक स्वीकार करें । आपको आने वाले समय में इन कमियों को दूर करने का प्रयास करना होगा । जीवन में करने के लिए बहुत कुछ है ।  ------x---- राज्य शिक्षा केंद्र  [RSKMP ] भोपाल म.प्र. 5वीं और 8वीं का रिजल्ट प्राप्त करने के लिए आवश्यक दस्तावेज  कक्षा 5वीं और 8वीं का रिजल्ट प्राप्त करने...
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