प्रश्नावली-3
1. यदि a3+b3+c3-3abc=p(a2+b2+c2-ab-bc-ca) तो p का मान बतावें
Answer:—-
p=(a+b+c)
Details:—-
a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)
2. यदि a3+b3+c3 -3abc=(a+b+c)(a2+b2+c2+k)(ab+bc+ca) तो k का मान क्या होगा?
Answer:—-
k=-1
a3+b3+c3-3abc=(a
3.(i)यदि k(a3+b3+c3-3abc)={(a-b)2+(b-c)2+(c-a)2 तो k का मान ज्ञात करें-
Answer:—
k=2
Details:—-
k(a3+b3+c3-3abc)=(a+b+c)(a2+b2-2ab+b2+c2-2bc+c2+a2-2ca)
=k(a3+b3+c3-3abc)(2a2+2b2+2c2-2ab-2bc-2ca)
=k(a3+b3+c3-3abc)=2(a+b+c)(a2+b2+c2-ab-bc-ca)
=k=2
(ii)यदि a3+b3+c3-3ab-(a+b+c)(m-ab-bc-ca) तो M का मान निकाले-
Answer:—-
M=a2+b2+c2
(iii)a3+b3+c3-z=(a+b+c)(a2+b2+c2-ab-bc-ca)
z=3abc
4.गुणनखण्ड निकाले-
(a)27×3+108x2y+144xy2+64y3
(3x)3+3(3x)2(4y)+3(3x)(4y)2+(4y)3
(3x+4y)3=(3x+4y)(3x+4y)(3x+4y)
Formula:-
(a+b)3=a3+b3+3a2b+3ab2
(b)a3+27b3+9ab(a+3b)
(a)3+(3b)3+3×a×3b(a+3b)
(a+3b)3=(a+3b)(a+3b)(a+3b)
(c) 27×3-135x2y+225xy2-125y3
(3x)3-3×(3x)2×5y+3×3x×(5y)2-(5y)3
=(3x-5y)3=(3x-5y)(3x-5y)(3x-5y)
(d)8a3+b3+12a2b+6ab2
(2a)3+(b)3+3×(2a)2×b+3×2a×(b)2
(2a+3b)3=(2a+3b)(2a+3b)(2a+3b)
(e)27-125a3-135a+225a2
(3)3-3×(3)2×5a+3×3×(5a)2-(5a)3
(3-5a)3=(3-5a)(3-5a)(3-5a)
(f)8×3+27y3+36x2y+54xy2
(2x)3+(3y)3+3×(2x)2×3y+3×2x×(3y)2
(2x+3y)3=(2x+3y)(2x+3y)(2x+3y)
5. गुणनखण्ड निकाले-
(a)a3+1
(a)3+(1)3=(a+1){(a)2-a×1+(1)2}
(a+1)(a2-a+1)
(b)m6+n6
(m2)3+(n2)3
(m2+n2){(m2)2-m2n2+(n2)2}
(m2+n2)(m4-m2n2+n4)
(c)27a6+125b3
(3a2)3+(5b)3
(3a2+5b){(3a2)2-3a2×5b+(5b)2}
(3a2+5b)(9a4-15a2b+25b2)
(d)8a3x4+64b6x
x(8a3x3+64b6)=x{(2ax)3+(4b2)3}
x(2ax+4b2){(2ax)2-2ax×4b2+(4b2)2}
x(2ax+4b2)(4a2x2-8axb2+16b4)
x×2×4(ax+2b2)(a2x2-2axb2+4b4)
8x(ax+2b2)(a2x2-2axb2+4b4)
6. गुणनखण्ड निकाले-
(a)54×3-16y3
2(27×3-8y3)
2{(3y)3-(2y)3}=2×(3x-2y){(3x)2+3x×2y+(2y)2}
2(3x-2y)(9×2+6xy+4y2)
(b)x6y9-1
(x2y3)3-(1)3=(x2y3-1){(x2y3)2+x2×y3+(1)2}
(x2y3-1)(x4y6+x2y3+1)
(c)(x+y)3-(z)3
(x+y-z){(x+y)2+(x+y)×z+z2}
(x+y-z)(x2+y2+2xy+xz+yz+z2)
(d)27y3-125z3
(3y)3-(5z)3=(3y-5z){(3y)2+3y×5z+(5z)2}
(3y-5z)(9y2+15yz+25z2)
7. प्रत्येक व्यंजक का गुणनखण्ड करें-
(a)x3-12x(x-4)-63
x3-12×2+48x-63
x3-3×x2×4+3×x×(4)2-(4)3+1
(x-4)3+(1)3
(x-4+1){(x-4)2-(x-4)×1+(1)2)}
(x-3)(x2-2×x×4+(4)2-x+4+1)
(x-3)(x2-9x+21)
(b)8×3+12x2y+6xy2-7y3
(2x)3+3×(2x)2×y+3×2x×(y)2+y3-(8y3)
(2x+y)3-(2y)3=
(2x+y-2y){(2x+y)2+(2x+y)×2y+(2y)2}
(2x-y)(4×2+2×2x×y+y2+4xy+2y2+4y2)
(2x-y)(4×2+8xy+7y2)
(c)x3+6x2y+12xy2+7y3
x3+3×x2×y+3×x×y2-y3+8y3
(x-y)3+(2y)3
(x-y+2y){(x-y)2-(x-y)2y+(2y)2}
(x+y)(x2-2xy+y2-2xy+2y2+4y2)
(x+y)(x2-4xy+7y2)
(d)8p3+36p2q+54pq2+26q3
(2p)3+3×(2p)2×3q+3×2p×(3q)2+(3q)3-q3
(2p+3q)3-q3
(2p+3q-q){(2p+3q)2+(2p+3q)×q+q2}
(2p+2q){(2p)2+2×2p×3q+(3q)2+2pq+3q2+q2}
(2p+2q)(4p2+12pq+9q2+2pq+3q2+q2)
(2p+2q)(4p2+14pq+13q2)
2(p+q)(4p2+14pq+13q2)
8.(i) a3+b3-c3+3abc
a3+b3+(-c)3-(-3abc)
(a+b-c)(a2+b2+c2-ab+bc+ca)
(ii) a3-b3+c3+3abc
a3+(-b)3+c3-(-3abc)
(a-b+c)(a2+b2+c2+ab+bc-ca)
(iii) -x3+y3+z3+3xyz
(-x)3+y3+z3-(-3xyz)
(-x+y+z)(x2+y2+z2+xy-yz+zx)
(iv) -x3-y3+z3-3xyz
(-x)3+(-y)3+z3-3xyz
(-x-y+z)(x2+y2+z2-xy+yz+zx)
9. गुणनखण्ड निकाले-
(i)a3+b3-3ab+1
a3+b3+(1)3-3ab×1
(a+b+1)(a2+b2+(1)2-ab-b×1-1×a)
(a+b+1)(a2+b2+1-ab-b-a)
(ii)x3+y3-1+3xy
x3+y3+(-1)3-(-3×x×y×1)
(x+y-1)(x2+y2+1-xy+y+x)
(x+y-1)(x2+y2+1-xy+y+x)
(iii)m3-n3-1-3mn
(m)3+(-n)3+(-1)3-3×m×n×1
(m-n-1)(m2+n2+(1)2+mn-n+m)
(m-n-1)(m2+n2+1+mn-n+m)
10. गुणनखण्ड कीजिये—-
(i)m3+8n3+27p3-18mnp
(m)3+(2n)3+(3p)3-3×m×2n×3p
(m+2n+3p)(m2+(2n)2+(3p)2-m×2n-2n×3p-3p×m)
(m+2n+3p)(m2+4n2+9p2-2mn-6np-3mp)
(ii) 1+8×3+18xy-27y3
(1)3+(2x)3+(-3y)3+3×1×2x×3y
(1+2x-3y){(1)2+(2x)2+(-3y)2-1×2x-2xy-3y-(-3y×1)}
(1+2x-3y)(1+4×2+9y2-2x+6xy+3y)
(iii) 8×3+y3+27-18xy
(2x)3+(y)3+(3)3-3×2x×y×3
(2x+y+3){(2x)2+(y)2+(3)2-2xy-y×3-3×2x)
(2x+y+3)(4×2+y2+9-2xy-3y-6x)
(iv)x3-8y3+z3+6xyz
x3+(-2y)3+z3-3×x×(-2y)×z
(x-2y+z){x2+4y2+z2-x×(-2y)-(-2y×z)-z×x)
(x-2y+z)(x2+4y2+z2+2xy+2yz-zx)
11. गुणनखण्ड ज्ञात करें-
(i)(a-b)3-(b-c)3+(c-a)3+3(a-b)(b-c)(c-a)
(a-b)3+{-(b-c}3+(c-a)3-(-3(a-b)(b-c)(c-a)
-(a-b+b+c+c-a){(a-b)2+{-(b-c)}2+(c-a)2
-{(a-b)×-(b-c)-(-b-c)×(c-a)-(c-a)×(a-b)
(-2b+2c){a2-2ab+b2+b2+c2-2bc+c2+a2-2ca+ab-ac(-b2+bc+bc-ab-c2+ac-ac+b-c+a2-ab)
2(c-a)(3a2+b2+c2-3ab-3ac+bc)
(ii)(x+y)3+(y+z)3+(z+x)3-3(x+y)(y+z)(z+x)
(x+y+y+z+z+x){(x+y)2+(y+z)2+(z+x)2-(x+y)(y+z)-(y+z)(z+x)-(z+x)(x+y)}
(2x+2y+2z)(x2+y2+2xy+y2+z2+2yz+z2+x2+2zx-xy-yz-zx-y2-yz-xy-z2-zx-zx+zy-x2-xy)
(2x+2y+2z)(x2+y2+z2-xy-yz-zx)
2(x+y+z)(x2+y2+z2-xy-yz-zx)
12. गुणनखण्ड ज्ञात करें-
(i)(y-z)3+(z-x)3+(x-y)3
3(x-y)(y-z)(z-x)
(ii) (2x-y)3-(x+y)3+(2y-x)3
=3(x+y)(x-2y)(2x-y)
(iii)a3(b-c)3+b3(c-a)3-c3(b-a)3
{c(b-a)}3+{a(b-c)}3+{b(c-a)3}
3abc(a-b)(b-c)(c-a)
13. यदि x=265, y=267, z=268 तो x3+y3+z3-3xyz का मान बताओ|
Answer:—–
x3+y3+z3-3xyz
x+y+z=265+267+268=800
x-y=(265-267)=2
y-z=(267-268)=-1
z-x=(268-265)=3
Formula:—-
x3+y3+z3-3xyz=
1 (x+y+z)[(x-y)2+(y-z)2+(z-x)]
2
1 ×800[(-2)2+(-1)2+(3)2]
2
400×(4+1+9)
400×14=5600
14. यदि x=2y+6 हो तो, x3-8y3-36xy-216 का मान ज्ञात करें-
Answer:—-
x=2y+6
x-2y=6
दोनों ओर घन करने पर,
(x-2y)3=(6)3
x3-(2y)3-3×x×2y(x-2y)=216
x3-8y3-6xy×6=216
x3-8y3-36xy-216=0
15. यदि x+y+z=0, तो सिद्ध करें कि (x3+y3+z3)=3xyz
Answer:—-
x+y+z=0
x+y=-z
(x+y)3=(-z)3
x3+y3+3xy(x+y)=(-z)3
x3+y3+3xy(-z)=(-z)3
x3+y3+z3=3xyz
16.(i) यदि x+y=z तो सिद्ध करें कि x3+y3+z3+3xyz=z3
Answer:—-
a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)
x3+y3+z3+3xyz=z3
x3+y3+(-z)3-3xy(-z)=(x+y-z)(x2+y2+z2-xy+yz+zx)
(z-x)(x2+y2+z2-xy+yz+zx) (x+y=z)
0×(x2+y2+z2-xy+yz+zx)
0 Proved
(ii)यदि x-y=z तो सिद्ध करें कि x3-y3-z3-3xyz=0
Answer:—–
a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)
x3+(-y)3+(-z)3-3×x×(-y)×(-z)
(x-y-z)(x2+y2+z2+xy-yz+zx)
(z-x)(x2+y2+z2+xy-yz+zx)
0×(x2+y2+z2+xy-yz+zx)
0 Proved
0 टिप्पणियाँ