विविध प्रश्नावली-7
1. योगफल:–
(a)
3a2-7b2+2c2, 4c2-9b2+3a2, b2-a2+c2
=3a2-7b2+2c2+4c2-9b2+3a2+b2-a2+c2
=3a2+3a2-a2-7b2-9b2+b2+2c2+4c2+c2
=5a2-15b2+7c2
(b)4xy+3yz+4zx, 5×2-4zx+2xy, y2-3yz+4z2
4xy+3yz+4zx+5×2-4zx+2xy+y2-3yz+4z2
4xy+2xy+3yz-3yz+4zx-4zx+5×2+y2+4z2
=5×2+y2+4z2+6xy-6yz
(c)x2+y2,y2+z2, x2+y2+z2, 5xy-4yz+7zx
=x2+y2++y2+z2+x2+y2+z2+5xy-4yz+7zx
=2×2+3y2+2z2+5xy-4yz+7zx
2.(a)12a4-5a3+4a2-7, 9+4a-3a3+15a2
=(12a4-5a3+4a2-7)-(9+4a-3a3+15a2)
=12a4-2a3-11a2-16-4a
(b)(15×3-3xyz+15y2z-7yz2+8)-(8y3+7yz2-6x+5yz-11)
=15×3-3xyz+15y2z-7yz2+8-8y3-7yz2+6x-5yz+11
=15×3-3xyz-8y3+15y2z-14yz2+6x-5yz+11
(c)(13y4-6y2+7,8-4y+6y3-y4
=(13y4-6y2+7)-(8-4y+6y3-y4)
=13y4-6y2+7-8+4y-6y3+y4
=14y4-6y3-6y2+4y-1
3. ऐसे घनाभ का आयतन घन इकाई में ज्ञात कीजिए जिनकी लंबाई, चौड़ाई और ऊंचाई क्रमशः निम्नलिखित हैं|
(a)4abc, 6ab2c, 9a3bc2
=4abc×6ab2c×9a3bc2
=216a5b4c4
(b)a2b2, abc2, bc
=a2b2×abc2×bc
=a3b4c3
4. योगफल ज्ञात कीजिए-
(a)4x(5×2-6x+7), 4(x3+9×2-6x+1)
=4x(5×2-6x+7)+4(x3+9×2-6x+1)
=20×3-24×2+28x+4×3+36×2-24x+4
=24×3+12×2+4x+4
(b)5a(-a+4b+7c), 3b(-5a+2b-9c), 8c(a-2b+7c)
=5a(-a+4b+7c)+3b(-5a+2b-9c)+8c(a-2b+7c)
=-5a2+20ab+35ac-15ab+6b2-27bc+8ac-16bc+56c2
=-5a2+6b2+56c2+5ab+43ac-43bc
5. घटाइए—-
(a)6m(l-2m+4p), 6p(8l-p+m)
=6ml-12m2-24mp-(48pl-6p2+6pm)
=6ml-12m2-24mp-48pl+6p2-6pm
=-12m2+6p2+6ml+18mp-48pl
(b)9a(2b+3c)-4(a+3b)c, 6b(a+b-2c)-3a(a+2b-c)
=(18ab+27ac-4ac-12bc)-(6ab+6b2-12bc-3a2-6ab+3ac)
=18ab+27ac-4ac-12bc-6ab-6b2+12bc+3a2+6ab-3ac
=3a2+18ab+20ac-6b2
6. गुणा कीजिए—-
(a)(3l-5m)×(-l+m-3n)
=-3l2+3ml-9nl+5ml-5m2+15mn
=-3l2+8ml-9nl+15mn-5m2
(b)(3x+2y-7z)×(5x-4y+6z)
=15×2-12xy+18zx+10xy-8y2+12yz-35zx+28yz-42z2
=15×2-8y2-42z2-2xy+11zx+40yz
7. सरल कीजिए—-
(a)(p+3q)(4p+q)+(3p-5q)(2p-7q)
=4p2+pq+12pq+3q2+6p2-21pq-10pq+35q2
=4p2+6p2+3q2+35q2+pq+12pq-21pq-10pq
=10p2-18pq+38q2
(b)(4.3a-5b)(3a+b)-5(a+1.3b)b
=12.9a2+4.3ab-15ab-5ab-5ab-6.5b2
=12.9a2-15.7ab-11.5b2
(c)(3x-5y)(9×2+15xy+25y2)
=27×3+45x2y+75xy2-45x2y-75xy2-125y3
=27×3-125y3
(d)(2a-5b+7c)(a+b+c)-6ab+12bc-14ca
=2a2+2ab+2ac-5ab-5b2-5bc+7ac+7bc+7c2-6ab+12bc-14ca
=2a2-5b2+7c2-9ab+14bc-5ac
8. सर्वसमिकाओं के प्रयोग से निम्नलिखित वर्गों के मान ज्ञात कीजिए-
[(a+b)2=a2+2ab+b2]
(a)(4ab+6bc)2
=(4ab)2+2×4ab×6bc+(6bc)2
=16a2b2+48ab2c+36b2c2
(b)(0.5x+0.3y)2
=(0.5x)2+2×0.5x×0.3y+(0.3y)2
=0.25×2+0.09y2+0.03xy
(c)(6x-5yz)2
=(6x)2-2×6x×5yz+(5yz)2
=36×2-60xyz+25y2z2
(d)(0.6a – 0.8b)2
=(0.6a)2-2×0.6a×0.8b+(0.8b)2
=0.36a2 – 0.96ab +0.64b2
9.सर्वसमिकाओं के प्रयोग से सरल कीजिए-
(a)( 2 x- 5 y)( 2 x + 5y )
13 11 13 11
=( 2x )2 – ( 5y )2
13 11
= 4×2 – 25y2
169 121
(b)(x+8)(x+15)
=x2+(8+15)x+8×15
=x2+23x+120
(c)(2m2-7)(2m2+9)
=(2m2)2+(-7+9)2m2+9×(-7)
=4m4+4m2-63
(d)(5×2-7y2)2+70x2y2
=(5×2)2-2×5×2×7y2+(7y2)2+70x2y2
=25×4-70x2y2+49y4+70x2y2
=25×4+49y4
(e)(7a-5b)2+(7a+5b)2
=(7a)2-2×7a×5b+(5b)2+(7a)2+2×7a×5b+(5b)2
=49a2-70ab+25b2+49a2+70ab+25b2
=98a2+50b2
(f)(6a2b-3ab2)2-(4a2b+5ab2)2
=[(6a2b)2-2×6a2b×3ab2+(3ab2)2]-[(4a2b)2+2×4a2b×5ab2+(5ab2)2]
=36a4b2+9a2b4-36a3b3-16a4b2+25a2b4+40a2b3
=20a4b2-76a3b3-16a2b4
10. बीजीय तादात्म्य के प्रयोग से निम्नलिखित में प्रत्येक का मान ज्ञात कीजिए-
(a)(107)2=(100+7)2
=(100)2+2×100×7+(7)2
=10000+1400+49=11449
(b)(7.3)2=(10-2.7)2
=(10)2-2×10×2.7+(2.7)2
=100-54+7.29=107.29-54=53.29
(c)102×98=(100+2)(100-2)
=(100)2-(2)2=10000-4=9996
(d)99×98=(100-1)(100-2)
=(100)2+(-1-2)×100+(-1)×(-2)
=10000-300+2=9702
(e)(176)2-(24)2
=(176+24)(176-24)
=200×152=30400
11. सिद्ध कीजिए—–
(a)(7xy+4y)2-(7xy-4y)2=112xy2
LHS=(7xy+4y)2-(7xy-4y)2
=(7xy)2+2×7xy×4y+(4y)2-[(7xy)2-2×7xy×4y+(4y)2]
=49x2y2+16y2+56xy2-49x2y2-16y2+56xy2
=112xy2=RHS
(b)a3(b-c)(b2+bc+c2)+b3(c-a)(c2+ca+a2)+c3(a-b)(a2+ab+b2)=0
=(a3b-a3c)(b2+bc+c2)+(b3c-ab3)(c2+ca+a2)+(ac3-bc3)(a2+ab+b2)
=a3b3+a3b2c+a3bc2-a3b2c-a3bc2-a3c3+b3c3+b3c2a+a2b3c-ab3c2-a2b3c-a3b3+a3c3+a2bc3+ab2c3-a2bc3-ab2c3-b3c3
0=RHS
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