प्रश्नावली-7C
1. सर्वसमिका के उपयोग से निम्नलिखित वर्गों के मान ज्ञात कीजिए—-
[(a+b)2=a2+2ab+b2]
(i)(2x+7y)2
=(2x)2+2×2x×7y+(7y)2
=4×2+28xy+49y2
(ii)(ab+3c)2
=(ab)2+2×ab×3c+(3c)2
=a2b2+6abc+9c2
(iii)( 2 x+ 5 y)2
5 2
=( 2 x)2+2× 2 x× 5 y+( 5 y)2
5 5 2 2
= 4 x2 +2xy+ 25 y2
25 4
(iv) (2ab+5b)2
=(2ab)2+(5b)2+2×2ab×5b
=4a2b2+25b2+20ab2
(v)(0.3x+0.8y)2
=(0.3x)2+2×0.3x×0.8y+(0.8y)2
=0.09×2+0.48xy+0.64y2
(vi) (3xy+7yz)2
=(3xy)2+2×3xy×7yz+(7yz)2
=9x2y2+42xy2z+49y2z2
2. उपयुक्त सर्वसमिका के प्रयोग से निम्नलिखित के मान ज्ञात कीजिए—–
(i)(6x-5y)2
=(6x)2-2×6x×5y+(5y)2
=36×2-60xy+25y2
(ii)(2a2-b)2
=(2a2)2-2×2a2×b+(b)2
=4a4-4a2b+b2
(iii) ( 4 x – 7 y)2
7 4
=( 4 x)2-2× 4 x× 7 y+( 7 y)2
7 7 4 4
= 16×2 – 2yx2 + 49y2
49 16
(iv) (3ab-4bd)2
=(3ab)2+(4bd)2-2×3ab×4bd
=9a2b2+16b2d2-24ab2d
(v)(0.3x-0.8y)2
=(0.3x)2-2×0.3x×0.8y+(0.8y)2
=0.09×2-0.48xy+0.64y2
(vi)(3a2-2b2)2
=(3a2)2+(2b2)2-2×3a2×2b2
=9a4+4b4-12a2b2
3. सरल कीजिए-
[(a+b)(a-b)=a2-b2]
(i)(x-4)(x+4)
=(x)2-(4)2=x2-16
(ii)(2a-3b)(2a+3b)
=(2a)2-(3b)2=4a2-9b2
(iii)(4a2+2ab)(4a2-2ab)
=(4a2)2-(2ab)2=4a4-4a2b2
(iv)(0.3a+0.9b)(0.3a-0.9b)
=(0.3a)2-(0.9b)2=0.09a2-0.81b2
(v)( 2 a – 3 b)( 2 a+ 3 b)
11 7 11 7
=( 2a )2 – ( 3b )2= 4a2 – 9b2
11 7 121 49
4. सर्वसमिकाओं के प्रयोग से निम्नलिखित में प्रत्येक का गुणनफल प्राप्त कीजिए—
[(x+a)(x+b)=x2+(a+b)x+ab]
(i)(a+3)(a+8)
=(a)2+(3+8)a+3×8
=a2+11a+24
(ii)(2p+7)(2p+11)
=(2p)2+(7+11)2p+7×11
=4p2+36p+77
(iii)(3x-4y)(3x+2y)
=(3x)2+(-4y+2y)×3x +2y×(-4y)
=9×2-6xy-8y2
(iv)(4a-7b)(4a-6b)
=(4a)2+(-7b-6b)×4a -7b×(-6b)
=16a2-52ab+42b2
(v)(2×2+9)(2×2-7)
=(2×2)2+(9-7)2×2+9×(-7)
=4×4-18×2-63
(vi)(a-8b2)(a-9b2)
=(a)2-(-8b2-9b2)a+(-8b2)×(-9b2)
=a4 -17ab2 +72b4
(vii)(8xy+z)(8xy-3z)
=(8xy)2+(z-3z)8xy+z×(-3z)
=64x2y2 -16xyz-3z2
5. बीजीय तादात्म्य के प्रयोग से निम्नलिखित के मान ज्ञात कीजिए-
(i)(104)2
=(100+4)2=(100)2+(4)2+2×100×4
=10000+16+800=10816
(ii)(9.8)2
=(10-0.2)2=(10)2-2×10×0.2+(0.2)2
=100-4+0.04=96.04
(iii)(110)2
=(100+10)2=(100)2+2×100×10+(10)2
=10000+2000+100=12100
(iv)105×95
=(100+5)(100-5)
=(100)2-(5)2=10000-25=9975
(v)304×306
=(300+4)(300+6)
=(300)2+(4+6)×100+6×4
=90000+1000+24=93024
(vi)96×97
=(100-4)(100-3)
=(100)2+(-4-3)×100+(-4)×(-3)
=10000-700+12=9312
(vii)(107)2-(93)2
=(107+93)(107-93)
=200×14=2800
(viii)(175)2-(25)2
=(175+25)(175-25)
=200×150=30000
6. सरल कीजिए-
(i)(3x-2y)2 + (3x+2y)2
=[(3x)2 -2×3x×2y+(2y)2+(3x)2+2×3x×2y+(2y)2]
=9×2 -6xy+4y2+9×2+12xy+4y2
=18×2+8y2
(ii)(4a+2b)2-(4a-2b)2
=[(4a)2+2×4a×2b+(2b)2-{(4a)2-2×4a×2b+(2b)2]
=16a2+16ab+4b2-16a2+16ab-4b2
=32ab
(iii)(8a-7b)2+112ab
=(8a)2-2×8a×7b+(7b)2+112ab
=64a2-112ab+49b2+112ab
=64a2+49b2
(iv)(2×2 – 3y2)2+12x2y2
=(2×2)2-2×2×2×3y2+(3y2)2+12x2y2
=4×4-12x2y2+9y4+12x2y2
=4×4+9×4
(v)(8×2+7y2)2-112x2y2
=(8×2)2+(7y2)2+2×8×2×7y2-112x2y2
=64×4+49y4+112x2y2-112x2y2
=64×4+49y4
(vi)(4a2x-2ax2)2-(2a2x+3ax2)2
=[(4a2)2-2×4a2x×2ax2+(2ax2)2-{(2a2x)2+(3ax2)2+2×2a2x×3ax2]
=16a4-16a3x3+4a2x4-4a4x4-9a2x4-12a3x3
=12a4x2-28a3x3-5a2x4
7. सिद्ध कीजिए—-
(i)(4p-7)2+112p=(4p+7)2
LHS=(4p-7)2+112p
=(4p)2-2×4p×7+(7)2+112p
=16p2-56p+49+112p
=16p2+56p+49
RHS=(4p+7)2
=(4p)2+2×4p×7+(7)2
=16p2+56p+49
LHS=RHS
(ii)( 2a + 3b )2 -2ab= 4a2 + 9b2
3 2 9 4
LHS=( 2a )2+2× 2a × 3b + ( 3b )2-2ab
3 3 2 2
= 4a2 +2ab+ 9a2 -2ab
9 4
= 4a2 + 9a2 =RHS
9 4
(iii)(m2-n2m)2+4m3n2=(m2+n2m)2
LHS=(m2-n2m)2+4m3n2
=(m2)2-2×m2×n2m+(n2m)2+4m3n2
=m4-2m3n2+n4m2+4m3n2
=m4+2m3n2+n4m2
RHS=(m2+n2m)2
=(m2)2+(n2m)2+2×m2×n2m
=m4+n4m2+2m3n2
(iv)(4ab+3b)2-(4ab-3b)2=48ab2
LHS=(4ab+3b)2-(4ab-3b)2
=[(4ab)2+2×4ab×3b+(3b)2-{(4ab)2-2×4ab×3b+(3b)2]
=16a2b2+24ab2+9b2
-16a2b2+24ab2-9b4
=48ab2=RHS
(v)(8×3-4)2-(8×3+4)2=-128×3
LHS=(8×3-4)2-(8×3+4)2
=(8×3)2-2×8×3×4+(4)2-[(8×3)2+2×8×3×4+(4)2]
=64×6-64×3+16-64-64×6-64×3-16
=-128×3
(vi)a2(b+c)(b-c)+b2(c+a)(c-a)+c2(a+b)(a-b)=0
LHS=a2(b+c)(b-c)+b2(c+a)(c-a)+c2(a+b)(a-b)
=a2(b2-c2)+b2(c2-a2)+c2(a2-b2)
=a2b2-a2c2+b2c2-b2a2+c2a2-c2b2
=0=RHS
Proved
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