RS Aggarwal 2020 2021 Solutions for Class 9 Maths Chapter 3 Factorisation Of Polynomials are provided here with simple step-by-step explanations. These solutions for Factorisation Of Polynomials are extremely popular among class 9 students for Maths Factorisation Of Polynomials Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RS Aggarwal 2020 2021 Book of class 9 Maths Chapter 3 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RS Aggarwal 2020 2021 Solutions. All RS Aggarwal 2020 2021 Solutions for class 9 Maths are prepared by experts and are 100% accurate.

Page No 99:

Question 1:

Answer:

We have:
9x2+12xy=3x3x+4y

Page No 99:

Question 2:

We have:
9x2+12xy=3x3x+4y

Answer:

We have:
18x2y-24xyz=6xy3y-4z

Page No 99:

Question 3:

We have:
18x2y-24xyz=6xy3y-4z

Answer:

We have:
27a3b3-45a4b2=9a3b23b-5a

Page No 99:

Question 4:

We have:
27a3b3-45a4b2=9a3b23b-5a

Answer:

We have:
2ax+y-3bx+y=x+y2a-3b

Page No 99:

Question 5:

We have:
2ax+y-3bx+y=x+y2a-3b

Answer:

We have:
2xp2+q2+4yp2+q2=2xp2+q2+2yp2+q2=2p2+q2x+2y
                            

Page No 99:

Question 6:

We have:
2xp2+q2+4yp2+q2=2xp2+q2+2yp2+q2=2p2+q2x+2y
                            

Answer:

We have:
xa-5+y5-a=xa-5-ya-5
                    =a-5x-y

Page No 99:

Question 7:

We have:
xa-5+y5-a=xa-5-ya-5
                    =a-5x-y

Answer:

We have:
4a+b-6a+b2=2a+b2-3a+b
                     =2a+b2-3a-3b

Page No 99:

Question 8:

We have:
4a+b-6a+b2=2a+b2-3a+b
                     =2a+b2-3a-3b

Answer:

We have:
83a-2b2-103a-2b=23a-2b43a-2b-5
                           =23a-2b12a-8b-5

Page No 99:

Question 9:

We have:
83a-2b2-103a-2b=23a-2b43a-2b-5
                           =23a-2b12a-8b-5

Answer:

We have:
xx+y3-3x2yx+y=xx+yx+y2-3xy
                        =xx+yx2+y2+2xy-3xy=xx+yx2+y2-xy

Page No 99:

Question 10:

We have:
xx+y3-3x2yx+y=xx+yx+y2-3xy
                        =xx+yx2+y2+2xy-3xy=xx+yx2+y2-xy

Answer:

We have:
x3+2x2+5x+10=x3+2x2+5x+10
                   =x2x+2+5x+2=x+2x2+5

Page No 99:

Question 11:

We have:
x3+2x2+5x+10=x3+2x2+5x+10
                   =x2x+2+5x+2=x+2x2+5

Answer:

We have:
x2+xy-2xz-2yz=x2+xy-2xz+2yz                           =xx+y-2zx+y                           =x+yx-2z

Page No 99:

Question 12:

We have:
x2+xy-2xz-2yz=x2+xy-2xz+2yz                           =xx+y-2zx+y                           =x+yx-2z

Answer:

We have:
a3b-a2b+5ab-5b=ba3-a2+5a-5                              =ba3-a2+5a-5

                      =ba2a-1+5a-1=ba-1a2+5

Page No 99:

Question 13:

We have:
a3b-a2b+5ab-5b=ba3-a2+5a-5                              =ba3-a2+5a-5

                      =ba2a-1+5a-1=ba-1a2+5

Answer:

We have:
8-4a-2a3+a4= 8-4a-2a3-a4                         = 42-a- a32-a                         = 2-a 4 - a3
                     

Page No 99:

Question 14:

We have:
8-4a-2a3+a4= 8-4a-2a3-a4                         = 42-a- a32-a                         = 2-a 4 - a3
                     

Answer:

We have:
x3-2x2y+3xy2-6y3=x3-2x2y+3xy2-6y3
                       =x2x-2y+3y2x-2y=x-2yx2+3y2

Page No 99:

Question 15:

We have:
x3-2x2y+3xy2-6y3=x3-2x2y+3xy2-6y3
                       =x2x-2y+3y2x-2y=x-2yx2+3y2

Answer:

We have:
px-5q+pq-5x=px-5x+pq-5q
                   =xp-5+qp-5=p-5x+q

Page No 99:

Question 16:

We have:
px-5q+pq-5x=px-5x+pq-5q
                   =xp-5+qp-5=p-5x+q

Answer:

We have:
x2+y-xy-x=x2-xy-x-y
               =xx-y-1x-y=x-yx-1

Page No 99:

Question 17:

We have:
x2+y-xy-x=x2-xy-x-y
               =xx-y-1x-y=x-yx-1

Answer:

We have:
3a-12-6a+2=3a-12-23a-1
                   =3a-13a-1-2=3a-13a-1-2=3a-13a-3=33a-1a-1

Page No 99:

Question 18:

We have:
3a-12-6a+2=3a-12-23a-1
                   =3a-13a-1-2=3a-13a-1-2=3a-13a-3=33a-1a-1

Answer:

We have:
2x-32-8x+12=2x-32-42x-3
                    =2x-32x-3-4=2x-32x-3-4=2x-32x-7

Page No 99:

Question 19:

We have:
2x-32-8x+12=2x-32-42x-3
                    =2x-32x-3-4=2x-32x-3-4=2x-32x-7

Answer:

We have:
a3+a-3a2-3=a3-3a2+a-3
                =a2a-3+1a-3=a-3a2+1

Page No 99:

Question 20:

We have:
a3+a-3a2-3=a3-3a2+a-3
                =a2a-3+1a-3=a-3a2+1

Answer:

We have:
3ax-6ay-8by+4bx=3ax-6ay+4bx-8by
                       =3ax-2y+4bx-2y=x-2y3a+4b

Page No 99:

Question 21:

We have:
3ax-6ay-8by+4bx=3ax-6ay+4bx-8by
                       =3ax-2y+4bx-2y=x-2y3a+4b

Answer:

We have:
abx2+a2x+b2x+ab=abx2+b2x+a2x+ab
                       =bxax+b+aax+b=ax+bbx+a

Page No 99:

Question 22:

We have:
abx2+a2x+b2x+ab=abx2+b2x+a2x+ab
                       =bxax+b+aax+b=ax+bbx+a

Answer:

We have:
x3-x2+ax+x-a-1=x3-x2+ax-a+x-1
                        =x2x-1+ax-1+1x-1=x-1x2+a+1



Page No 100:

Question 23:

We have:
x3-x2+ax+x-a-1=x3-x2+ax-a+x-1
                        =x2x-1+ax-1+1x-1=x-1x2+a+1

Answer:

We have:
2x+4y-8xy-1=2x-8xy-1-4y
                  =2x1-4y-11-4y=1-4y2x-1

Page No 100:

Question 24:

We have:
2x+4y-8xy-1=2x-8xy-1-4y
                  =2x1-4y-11-4y=1-4y2x-1

Answer:

We have:
abx2+y2-xya2+b2=abx2+aby2-a2xy-b2xy
                          =abx2-a2xy-b2xy-aby2=axbx-ay-bybx-ay=bx-ayax-by

Page No 100:

Question 25:

We have:
abx2+y2-xya2+b2=abx2+aby2-a2xy-b2xy
                          =abx2-a2xy-b2xy-aby2=axbx-ay-bybx-ay=bx-ayax-by

Answer:

We have:
a2+abb+1+b3=a2+ab2+ab+b3
                    =a2+ab2+ab+b3=aa+b2+ba+b2=a+b2a+b

Page No 100:

Question 26:

We have:
a2+abb+1+b3=a2+ab2+ab+b3
                    =a2+ab2+ab+b3=aa+b2+ba+b2=a+b2a+b

Answer:

We have:
a3+ab1-2a-2b2=a3+ab-2a2b-2b2
                      =a3-2a2b+ab-2b2=a2a-2b+ba-2b=a-2ba2+b

Page No 100:

Question 27:

We have:
a3+ab1-2a-2b2=a3+ab-2a2b-2b2
                      =a3-2a2b+ab-2b2=a2a-2b+ba-2b=a-2ba2+b

Answer:

We have:
2a2+bc-2ab-ac=2a2-2ab-ac-bc
                    =2aa-b-ca-b=a-b2a-c

Page No 100:

Question 28:

We have:
2a2+bc-2ab-ac=2a2-2ab-ac-bc
                    =2aa-b-ca-b=a-b2a-c

Answer:

We have:
ax+by2+bx-ay2=ax2+2×ax×by+by2+bx2-2×bx×ay+ay2
                        =a2x2+2abxy+b2y2+b2x2-2abxy+a2y2=a2x2+b2y2+b2x2+a2y2=a2x2+b2x2+a2y2+b2y2=x2a2+b2+y2a2+b2=a2+b2x2+y2

Page No 100:

Question 29:

We have:
ax+by2+bx-ay2=ax2+2×ax×by+by2+bx2-2×bx×ay+ay2
                        =a2x2+2abxy+b2y2+b2x2-2abxy+a2y2=a2x2+b2y2+b2x2+a2y2=a2x2+b2x2+a2y2+b2y2=x2a2+b2+y2a2+b2=a2+b2x2+y2

Answer:

We have:
aa+b-c-bc=a2+ab-ac-bc
                  =a2-ac+ab-bc=aa-c+ba-c=a-ca+b

Page No 100:

Question 30:

We have:
aa+b-c-bc=a2+ab-ac-bc
                  =a2-ac+ab-bc=aa-c+ba-c=a-ca+b

Answer:

We have:
aa-2b-c+2bc=a2-2ab-ac+2bc
                    =a2-2ab-ac-2bc=aa-2b-ca-2b=a-2ba-c

Page No 100:

Question 31:

We have:
aa-2b-c+2bc=a2-2ab-ac+2bc
                    =a2-2ab-ac-2bc=aa-2b-ca-2b=a-2ba-c

Answer:

We have:
a2x2+ax2+1x+a=ax2+1x+a2x2+a
                      =xax2+1+aax2+1=ax2+1x+a

Page No 100:

Question 32:

We have:
a2x2+ax2+1x+a=ax2+1x+a2x2+a
                      =xax2+1+aax2+1=ax2+1x+a

Answer:

We have:
abx2+1+xa2+b2=abx2+ab+a2x+b2x
                        =abx2+a2x+b2x+ab=axbx+a+bbx+a=bx+aax+b

Page No 100:

Question 33:

We have:
abx2+1+xa2+b2=abx2+ab+a2x+b2x
                        =abx2+a2x+b2x+ab=axbx+a+bbx+a=bx+aax+b

Answer:

We have:
x2-a+bx+ab=x2-ax-bx+ab
                   =x2-ax-bx-ab=xx-a-bx-a=x-ax-b

Page No 100:

Question 34:

We have:
x2-a+bx+ab=x2-ax-bx+ab
                   =x2-ax-bx-ab=xx-a-bx-a=x-ax-b

Answer:

We have: x2+1x2-2-3x+3x= x2-2+1x2-3x+3x=x2-2×x×1x+1x2-3x-1x=x-1x2-3x-1x=x-1xx-1x-3



Page No 105:

Question 1:

We have: x2+1x2-2-3x+3x= x2-2+1x2-3x+3x=x2-2×x×1x+1x2-3x-1x=x-1x2-3x-1x=x-1xx-1x-3

Answer:

9x2-16y2=3x2-4y2=3x+4y3x-3y                  a2-b2=a+ba-b

Page No 105:

Question 2:

9x2-16y2=3x2-4y2=3x+4y3x-3y                  a2-b2=a+ba-b

Answer:

254x2-19y2=52x2-13y2=52x+13y52x-13y                  a2-b2=a+ba-b

Page No 105:

Question 3:

254x2-19y2=52x2-13y2=52x+13y52x-13y                  a2-b2=a+ba-b

Answer:

81-16x2=92-4x2=9+4x9-4x                  a2-b2=a+ba-b

Page No 105:

Question 4:

81-16x2=92-4x2=9+4x9-4x                  a2-b2=a+ba-b

Answer:

5-20x2=51-4x2=512-2x2=51+2x1-2x                  a2-b2=a+ba-b

Page No 105:

Question 5:

5-20x2=51-4x2=512-2x2=51+2x1-2x                  a2-b2=a+ba-b

Answer:

2x4-32=2x4-16=2x22-42=2x2+4x2-4                  a2-b2=a+ba-b

=2x2+4x2-22=2x2+4x+2x-2                  a2-b2=a+ba-b

Page No 105:

Question 6:

2x4-32=2x4-16=2x22-42=2x2+4x2-4                  a2-b2=a+ba-b

=2x2+4x2-22=2x2+4x+2x-2                  a2-b2=a+ba-b

Answer:

3a3b-243ab3=3aba2-81b2                         =3aba2-9b2                         =3aba-9ba+9b                         

Page No 105:

Question 7:

3a3b-243ab3=3aba2-81b2                         =3aba2-9b2                         =3aba-9ba+9b                         

Answer:

3x3-48x=3xx2-16                =3xx2-42                =3xx-4 x+4

Page No 105:

Question 8:

3x3-48x=3xx2-16                =3xx2-42                =3xx-4 x+4

Answer:

27a2-48b2=39a2-16b2                    =33a2-4b2                    =33a-4b3a+4b

Page No 105:

Question 9:

27a2-48b2=39a2-16b2                    =33a2-4b2                    =33a-4b3a+4b

Answer:

x-64x3=x1-64x2              =x1-8x2              =x1-8x 1+8x

Page No 105:

Question 10:

x-64x3=x1-64x2              =x1-8x2              =x1-8x 1+8x

Answer:

8ab2-18a3=2a4b2-9a2                    =2a2b2-3a2                    =2a2b-3a2b+3a

Page No 105:

Question 11:

8ab2-18a3=2a4b2-9a2                    =2a2b2-3a2                    =2a2b-3a2b+3a

Answer:

150-6x2=625-x2                 =652-x2                 =65-x5+x

Page No 105:

Question 12:

150-6x2=625-x2                 =652-x2                 =65-x5+x

Answer:

2-50x2=21-25x2=212-5x2=21+5x1-5x                  a2-b2=a+ba-b

Page No 105:

Question 13:

2-50x2=21-25x2=212-5x2=21+5x1-5x                  a2-b2=a+ba-b

Answer:

20x2-45=54x2-9=52x2-32=52x+32x-3                  a2-b2=a+ba-b

Page No 105:

Question 14:

20x2-45=54x2-9=52x2-32=52x+32x-3                  a2-b2=a+ba-b

Answer:

3a+5b2-4c2=3a+5b2-2c2=3a+5b+2c3a+5b-2c                  a2-b2=a+ba-b

Page No 105:

Question 15:

3a+5b2-4c2=3a+5b2-2c2=3a+5b+2c3a+5b-2c                  a2-b2=a+ba-b

Answer:

a2-b2-a-b=a+ba-b-1a+b                  a2-b2=a+ba-b=a+ba-b-1=a+ba-b-1

Page No 105:

Question 16:

a2-b2-a-b=a+ba-b-1a+b                  a2-b2=a+ba-b=a+ba-b-1=a+ba-b-1

Answer:

4a2-9b2-2a-3b=2a2-3b2-12a+3b=2a+3b2a-3b-12a+3b                  a2-b2=a+ba-b=2a+3b2a-3b-1=2a+3b2a-3b-1

Page No 105:

Question 17:

4a2-9b2-2a-3b=2a2-3b2-12a+3b=2a+3b2a-3b-12a+3b                  a2-b2=a+ba-b=2a+3b2a-3b-1=2a+3b2a-3b-1

Answer:

a2-b2+2bc-c2=a2-b2-2bc+c2=a2-b-c2                                     a2-2ab+b2=a-b2=a+b-ca-b-c                  a2-b2=a+ba-b=a+b-ca-b+c

Page No 105:

Question 18:

a2-b2+2bc-c2=a2-b2-2bc+c2=a2-b-c2                                     a2-2ab+b2=a-b2=a+b-ca-b-c                  a2-b2=a+ba-b=a+b-ca-b+c

Answer:

4a2-4b2+4a+1=4a2+4a+1-4b2=2a2+2×2a×1+12-4b2=2a+12-2b2                                  a2+2ab+b2=a+b2

=2a+1+2b2a+1-2b             a2-b2=a+ba-b=2a+1+2b2a+1-2b=2a+2b+12a-2b+1

Page No 105:

Question 19:

4a2-4b2+4a+1=4a2+4a+1-4b2=2a2+2×2a×1+12-4b2=2a+12-2b2                                  a2+2ab+b2=a+b2

=2a+1+2b2a+1-2b             a2-b2=a+ba-b=2a+1+2b2a+1-2b=2a+2b+12a-2b+1

Answer:

a2+2ab+b2-9c2=a+b2-3c2                               =a+b-3ca+b+3c

Page No 105:

Question 20:

a2+2ab+b2-9c2=a+b2-3c2                               =a+b-3ca+b+3c

Answer:

108a2-3b-c2=336a2-b-c2                            =36a2-b-c2                            =36a-b+c6a+b-c

Page No 105:

Question 21:

108a2-3b-c2=336a2-b-c2                            =36a2-b-c2                            =36a-b+c6a+b-c

Answer:

a+b3-a-b=a+b3-a+b                        =a+ba+b2-1                        =a+ba+b2-12                        =a+ba+b-1a+b+1

Page No 105:

Question 22:

a+b3-a-b=a+b3-a+b                        =a+ba+b2-1                        =a+ba+b2-12                        =a+ba+b-1a+b+1

Answer:

x2+y2-z2-2xy=x2+y2-2xy-z2=x-y2-z2                                  a2-2ab+b2=a-b2=x-y+zx-y-z                     a2-b2=a+ba-b

Page No 105:

Question 23:

x2+y2-z2-2xy=x2+y2-2xy-z2=x-y2-z2                                  a2-2ab+b2=a-b2=x-y+zx-y-z                     a2-b2=a+ba-b

Answer:

x2+2xy+y2-a2+2ab-b2=x2+2xy+y2-a2-2ab+b2=x+y2-a-b2             a2+2ab+b2=a+b2 and a2-2ab+b2=a-b2=x+y+a-bx+y-a-b          a2-b2=a+ba-b=x+y+a-bx+y-a+b

Page No 105:

Question 24:

x2+2xy+y2-a2+2ab-b2=x2+2xy+y2-a2-2ab+b2=x+y2-a-b2             a2+2ab+b2=a+b2 and a2-2ab+b2=a-b2=x+y+a-bx+y-a-b          a2-b2=a+ba-b=x+y+a-bx+y-a+b

Answer:

25x2-10x+1-36y2=5x2-2×5x×1+12-6y2=5x-12-6y2                                        a2-2ab+b2=a-b2=5x-1+6y5x-1-6y                   a2-b2=a+ba-b=5x+6y-15x-6y-1

Page No 105:

Question 25:

25x2-10x+1-36y2=5x2-2×5x×1+12-6y2=5x-12-6y2                                        a2-2ab+b2=a-b2=5x-1+6y5x-1-6y                   a2-b2=a+ba-b=5x+6y-15x-6y-1

Answer:

a-b-a2+b2=a-b-a2-b2                       =a-b-a-ba+b                       =a-b1-a+b                       =a-b1-a-b

Page No 105:

Question 26:

a-b-a2+b2=a-b-a2-b2                       =a-b-a-ba+b                       =a-b1-a+b                       =a-b1-a-b

Answer:

a2-b2-4ac+4c2=a2-4ac+4c2-b2                              =a2-2×2a×c +2c2-b2                              =a-2c2-b2                              =a-2c+ba-2c-b

Page No 105:

Question 27:

a2-b2-4ac+4c2=a2-4ac+4c2-b2                              =a2-2×2a×c +2c2-b2                              =a-2c2-b2                              =a-2c+ba-2c-b

Answer:

9-a2+2ab-b2=9-a2-2ab+b2                            =32-a-b2                            =3-a-b3+a-b                            =3-a+b3+a-b

Page No 105:

Question 28:

9-a2+2ab-b2=9-a2-2ab+b2                            =32-a-b2                            =3-a-b3+a-b                            =3-a+b3+a-b

Answer:

x3-5x2-x+5=x2x-5-1x-5                         =x-5x2-1                         =x-5x2-12                         =x-5x-1x+1

Page No 105:

Question 29:

x3-5x2-x+5=x2x-5-1x-5                         =x-5x2-1                         =x-5x2-12                         =x-5x-1x+1

Answer:

1+2ab-a2+b2=1+2ab-a2-b2=1-a2+2ab-b2=12-a2-2ab+b2

=12-a-b2                                        a2-2ab+b2=a-b2=1+a-b1-a-b                    a2-b2=a+ba-b=1+a-b1-a+b

Page No 105:

Question 30:

1+2ab-a2+b2=1+2ab-a2-b2=1-a2+2ab-b2=12-a2-2ab+b2

=12-a-b2                                        a2-2ab+b2=a-b2=1+a-b1-a-b                    a2-b2=a+ba-b=1+a-b1-a+b

Answer:

9a2+6a+1-36b2=3a2+2×3a×1+12-6b2=3a+12-6b2                              a2+2ab+b2=a+b2=3a+1-6b3a+1+6b              a2-b2=a-ba+b=3a-6b+13a+6b+1

Page No 105:

Question 31:

9a2+6a+1-36b2=3a2+2×3a×1+12-6b2=3a+12-6b2                              a2+2ab+b2=a+b2=3a+1-6b3a+1+6b              a2-b2=a-ba+b=3a-6b+13a+6b+1

Answer:

 x2-y2+6y-9=x2-y2-6y+9                          =x2-y2-2×y×3 +32                          =x2-y-32                                                    =x+y-3x-y-3                          =x+y-3x-y+3

Page No 105:

Question 32:

 x2-y2+6y-9=x2-y2-6y+9                          =x2-y2-2×y×3 +32                          =x2-y-32                                                    =x+y-3x-y-3                          =x+y-3x-y+3

Answer:

4x2-9y2-2x-3y=4x2-9y2-2x+3y                               =2x2-3y2-2x+3y                               =2x-3y2x+3y-12x+3y                               =2x+3y2x-3y-1                                

Page No 105:

Question 33:

4x2-9y2-2x-3y=4x2-9y2-2x+3y                               =2x2-3y2-2x+3y                               =2x-3y2x+3y-12x+3y                               =2x+3y2x-3y-1                                

Answer:

9a2+3a-8b-64b2=9a2-64b2+3a-8b                                   =3a2-8b2+3a-8b                                   =3a-8b3a+8b+13a-8b                                   =3a-8b3a+8b+1

Page No 105:

Question 34:

9a2+3a-8b-64b2=9a2-64b2+3a-8b                                   =3a2-8b2+3a-8b                                   =3a-8b3a+8b+13a-8b                                   =3a-8b3a+8b+1

Answer:

x2+1x2-3=x2+1x2-2-1=x2+1x2-2×x×1x-1=x-1x2-12                                   a2-2ab+b2=a-b2=x-1x+1x-1x-1                   a2-b2=a-ba+b

Page No 105:

Question 35:

x2+1x2-3=x2+1x2-2-1=x2+1x2-2×x×1x-1=x-1x2-12                                   a2-2ab+b2=a-b2=x-1x+1x-1x-1                   a2-b2=a-ba+b

Answer:

x2-2+1x2-y2=x2-2×x×1x+1x2-y2=x-1x2-y2                                   a2-2ab+b2=a-b2=x-1x+yx-1x-y                   a2-b2=a-ba+b

Disclaimer: The expression of the question should be x2-2+1x2-y2. The same has been done before solving the question.

Page No 105:

Question 36:

x2-2+1x2-y2=x2-2×x×1x+1x2-y2=x-1x2-y2                                   a2-2ab+b2=a-b2=x-1x+yx-1x-y                   a2-b2=a-ba+b

Disclaimer: The expression of the question should be x2-2+1x2-y2. The same has been done before solving the question.

Answer:

x4+4x4=x4+4x4+4-4=x22+2x22+2×x2×2x2-22=x2+2x22-22                                      a2+2ab+b2=a+b2=x2+2x2+2x2+2x2-2                   a2-b2=a+ba-b

Page No 105:

Question 37:

x4+4x4=x4+4x4+4-4=x22+2x22+2×x2×2x2-22=x2+2x22-22                                      a2+2ab+b2=a+b2=x2+2x2+2x2+2x2-2                   a2-b2=a+ba-b

Answer:

x8-1=x42-12=x4+1x4-1                               a2-b2=a+ba-b=x4+1x22-12

=x4+1x2+1x2-1                   a2-b2=a+ba-b=x4+1x2+1x2-12=x4+1x2+1x+1x-1            a2-b2=a+ba-b

Page No 105:

Question 38:

x8-1=x42-12=x4+1x4-1                               a2-b2=a+ba-b=x4+1x22-12

=x4+1x2+1x2-1                   a2-b2=a+ba-b=x4+1x2+1x2-12=x4+1x2+1x+1x-1            a2-b2=a+ba-b

Answer:

16x4-1=4x22-12=4x2+14x2-1                               a2-b2=a+ba-b=4x2+12x2-12=4x2+12x+12x-1                     a2-b2=a+ba-b

Page No 105:

Question 39:

16x4-1=4x22-12=4x2+14x2-1                               a2-b2=a+ba-b=4x2+12x2-12=4x2+12x+12x-1                     a2-b2=a+ba-b

Answer:

81x4-y4=9x22-y22=9x2+y29x2-y2                               a2-b2=a+ba-b=9x2+y23x2-y2=9x2+y23x+y3x-y                      a2-b2=a+ba-b

Page No 105:

Question 40:

81x4-y4=9x22-y22=9x2+y29x2-y2                               a2-b2=a+ba-b=9x2+y23x2-y2=9x2+y23x+y3x-y                      a2-b2=a+ba-b

Answer:

x4-625=x22-252=x2+25x2-25                               a2-b2=a+ba-b=x2+25x2-52=x2+25x+5x-5                         a2-b2=a+ba-b



Page No 114:

Question 1:

x4-625=x22-252=x2+25x2-25                               a2-b2=a+ba-b=x2+25x2-52=x2+25x+5x-5                         a2-b2=a+ba-b

Answer:

We have:
x2+11x+30
We have to split 11 into two numbers such that their sum of is 11 and their product is 30.
Clearly, 5+6=11 and 5×6=30.

 x2+11x+30 = x2+5x+6x+30                             = x(x+5)+6(x+5)                             =(x+5)(x+6)

Page No 114:

Question 2:

We have:
x2+11x+30
We have to split 11 into two numbers such that their sum of is 11 and their product is 30.
Clearly, 5+6=11 and 5×6=30.

 x2+11x+30 = x2+5x+6x+30                             = x(x+5)+6(x+5)                             =(x+5)(x+6)

Answer:

We have:
x2+18x+32
We have to split 18 into two numbers such that their sum is 18 and their product is 32.
Clearly, 16+2=18 and 16×2=32.

x2+18x+32=x2+16x+2x+32                           =x(x+16)+2(x+16)                           =(x+16)(x+2)

Page No 114:

Question 3:

We have:
x2+18x+32
We have to split 18 into two numbers such that their sum is 18 and their product is 32.
Clearly, 16+2=18 and 16×2=32.

x2+18x+32=x2+16x+2x+32                           =x(x+16)+2(x+16)                           =(x+16)(x+2)

Answer:


x2+20x-69=x2+23x-3x-69=xx+23-3x+23=x+23x-3

Page No 114:

Question 4:


x2+20x-69=x2+23x-3x-69=xx+23-3x+23=x+23x-3

Answer:


x2+19x-150=x2+25x-6x-150=xx+25-6x+25=x+25x-6

Page No 114:

Question 5:


x2+19x-150=x2+25x-6x-150=xx+25-6x+25=x+25x-6

Answer:


x2+7x-98=x2+14x-7x-98=xx+14-7x+14=x+14x-7

Page No 114:

Question 6:


x2+7x-98=x2+14x-7x-98=xx+14-7x+14=x+14x-7

Answer:


x2+23x24=x2+43x-23x-24=xx+43-23x+43=x+43x-23

Page No 114:

Question 7:


x2+23x24=x2+43x-23x-24=xx+43-23x+43=x+43x-23

Answer:


x2-21x+90=x2-15x-6x+90=xx-15-6x-15=x-6x-15

Page No 114:

Question 8:


x2-21x+90=x2-15x-6x+90=xx-15-6x-15=x-6x-15

Answer:


x2-22x+120=x2-12x-10x+120=xx-12-10x-12=x-10x-12

Page No 114:

Question 9:


x2-22x+120=x2-12x-10x+120=xx-12-10x-12=x-10x-12

Answer:


x2-4x+3=x2-3x-x+3=xx-3-1x-3=x-1x-3

Page No 114:

Question 10:


x2-4x+3=x2-3x-x+3=xx-3-1x-3=x-1x-3

Answer:


x2+76x+60=x2+56x+26x+60=xx+56+26x+56=x+56x+26

Page No 114:

Question 11:


x2+76x+60=x2+56x+26x+60=xx+56+26x+56=x+56x+26

Answer:


x2+33x+6=x2+23x+3x+6=xx+23+3x+23=x+23x+3

Page No 114:

Question 12:


x2+33x+6=x2+23x+3x+6=xx+23+3x+23=x+23x+3

Answer:


x2+66x+48=x2+46x+26x+48=xx+46+26x+46=x+46x+26

Page No 114:

Question 13:


x2+66x+48=x2+46x+26x+48=xx+46+26x+46=x+46x+26

Answer:


x2+55x+30=x2+35x+25x+30=xx+35+25x+35=x+35x+25

Page No 114:

Question 14:


x2+55x+30=x2+35x+25x+30=xx+35+25x+35=x+35x+25

Answer:


x2-24x-180=x2-30x+6x-180=xx-30+6x-30=x-30x+6

Page No 114:

Question 15:


x2-24x-180=x2-30x+6x-180=xx-30+6x-30=x-30x+6

Answer:


x2-32x-105=x2-35x+3x-105=xx-35+3x-35=x-35x+3

Page No 114:

Question 16:


x2-32x-105=x2-35x+3x-105=xx-35+3x-35=x-35x+3

Answer:


x2-11x-80=x2-16x+5x-80=xx-16+5x-16=x-16x+5

Page No 114:

Question 17:


x2-11x-80=x2-16x+5x-80=xx-16+5x-16=x-16x+5

Answer:


-x2-x+6=-x2-3x+2x+6=-xx+3+2x+3=x+3-x+2=x+32-x

Page No 114:

Question 18:


-x2-x+6=-x2-3x+2x+6=-xx+3+2x+3=x+3-x+2=x+32-x

Answer:


x2-3x-6=x2-23x+3x-6=xx-23+3x-23=x-23x+3

Page No 114:

Question 19:


x2-3x-6=x2-23x+3x-6=xx-23+3x-23=x-23x+3

Answer:


-x2+3x+40=-x2+8x-5x+40=-xx-8-5x-8=x-8-x-5=8-xx+5

Page No 114:

Question 20:


-x2+3x+40=-x2+8x-5x+40=-xx-8-5x-8=x-8-x-5=8-xx+5

Answer:


x2-26x+133=x2-19x-7x+133=xx-19-7x-19=x-19x-7

Page No 114:

Question 21:


x2-26x+133=x2-19x-7x+133=xx-19-7x-19=x-19x-7

Answer:


x2-23x-24=x2-43x+23x-24=xx-43+23x-43=x-43x+23

Page No 114:

Question 22:


x2-23x-24=x2-43x+23x-24=xx-43+23x-43=x-43x+23

Answer:


x2-35x-20=x2-45x+5x-20=xx-45+5x-45=x-45x+5

Page No 114:

Question 23:


x2-35x-20=x2-45x+5x-20=xx-45+5x-45=x-45x+5

Answer:


x2+2x-24=x2+42x-32x-24=xx+42-32x+42=x+42x-32

Page No 114:

Question 24:


x2+2x-24=x2+42x-32x-24=xx+42-32x+42=x+42x-32

Answer:


x2-22x-30=x2-52x+32x-30=xx-52+32x-52=x-52x+32

Page No 114:

Question 25:


x2-22x-30=x2-52x+32x-30=xx-52+32x-52=x-52x+32

Answer:

We have:
x2-x-156
We have to split (-1) into two numbers such that their sum is (-1) and their product is (-156).
Clearly, -13+12=-1 and -13×12=-156.

x2-x-156=x2-13x+12x-156                        =x(x-13)+12(x-13)                        =(x-13)(x+12)

Page No 114:

Question 26:

We have:
x2-x-156
We have to split (-1) into two numbers such that their sum is (-1) and their product is (-156).
Clearly, -13+12=-1 and -13×12=-156.

x2-x-156=x2-13x+12x-156                        =x(x-13)+12(x-13)                        =(x-13)(x+12)

Answer:


x2-32x-105=x2-35x+3x-105=xx-35+3x-35=x-35x+3

Page No 114:

Question 27:


x2-32x-105=x2-35x+3x-105=xx-35+3x-35=x-35x+3

Answer:


9x2+18x+8=9x2+12x+6x+8=3x3x+4+23x+4=3x+43x+2

Page No 114:

Question 28:


9x2+18x+8=9x2+12x+6x+8=3x3x+4+23x+4=3x+43x+2

Answer:


6x2+17x+12=6x2+9x+8x+12=3x2x+3+42x+3=2x+33x+4

Page No 114:

Question 29:


6x2+17x+12=6x2+9x+8x+12=3x2x+3+42x+3=2x+33x+4

Answer:

We have:
18x2+3x-10
We have to split 3 into two numbers such that their sum is 3 and their product is (-180), i.e., 18×-10.
Clearly, 15+-12=3 and 15×-12=-180.

​18x2+3x-10=18x2+15x-12x-10                             =3x6x+5-26x+5                             =6x+53x-2

Page No 114:

Question 30:

We have:
18x2+3x-10
We have to split 3 into two numbers such that their sum is 3 and their product is (-180), i.e., 18×-10.
Clearly, 15+-12=3 and 15×-12=-180.

​18x2+3x-10=18x2+15x-12x-10                             =3x6x+5-26x+5                             =6x+53x-2

Answer:

We have:
2x2+11x-21
We have to split 11 into two numbers such that their sum is 11 and their product is (-42), i.e., 2×-21.
Clearly, 14+-3=11 and 14×-3=-42.

2x2+11x-21=2x2+14x-3x-21                             =2xx+7-3x+7                             =x+72x-3

Page No 114:

Question 31:

We have:
2x2+11x-21
We have to split 11 into two numbers such that their sum is 11 and their product is (-42), i.e., 2×-21.
Clearly, 14+-3=11 and 14×-3=-42.

2x2+11x-21=2x2+14x-3x-21                             =2xx+7-3x+7                             =x+72x-3

Answer:

We have:
15x2+2x-8
We have to split 2 into two numbers such that their sum is 2 and their product is (-120), i.e., 15×-8.
Clearly, 12+-10=2 and 12×-10=-120.

15x2+2x-8=15x2+12x-10x-8                          =3x5x+4-25x+4                          =5x+43x-2

Page No 114:

Question 32:

We have:
15x2+2x-8
We have to split 2 into two numbers such that their sum is 2 and their product is (-120), i.e., 15×-8.
Clearly, 12+-10=2 and 12×-10=-120.

15x2+2x-8=15x2+12x-10x-8                          =3x5x+4-25x+4                          =5x+43x-2

Answer:


21x2+5x-6=21x2+14x-9x-6=7x3x+2-33x+2=3x+27x-3

Page No 114:

Question 33:


21x2+5x-6=21x2+14x-9x-6=7x3x+2-33x+2=3x+27x-3

Answer:

We have:
24x2-41x+12
We have to split (-41) into two numbers such that their sum is (-41) and their product is 288, i.e., 24×12.
Clearly, -32+-9=-41 and -32×-9=288.

24x2-41x+12=24x2-32x-9x+12                               =8x3x-4-33x-4                               =3x-48x-3

Page No 114:

Question 34:

We have:
24x2-41x+12
We have to split (-41) into two numbers such that their sum is (-41) and their product is 288, i.e., 24×12.
Clearly, -32+-9=-41 and -32×-9=288.

24x2-41x+12=24x2-32x-9x+12                               =8x3x-4-33x-4                               =3x-48x-3

Answer:

3x2-14x+8=3x2-12x-2x+8                      =3xx-4-2x-4                      =x-43x-2

Hence, factorisation of 3x2 – 14x + 8 is x-43x-2.

Page No 114:

Question 35:

3x2-14x+8=3x2-12x-2x+8                      =3xx-4-2x-4                      =x-43x-2

Hence, factorisation of 3x2 – 14x + 8 is x-43x-2.

Answer:

We have:
2x2+3x-90
We have to split 3 into two numbers such that their sum is 3 and their product is (-180), i.e., 2×-90.
Clearly, -12 + 15 = 3 and -12×15 = -180.

2x2+3x-90=2x2-12x+15x-90                          =2xx-6+15x-6                          =x-62x+15

Page No 114:

Question 36:

We have:
2x2+3x-90
We have to split 3 into two numbers such that their sum is 3 and their product is (-180), i.e., 2×-90.
Clearly, -12 + 15 = 3 and -12×15 = -180.

2x2+3x-90=2x2-12x+15x-90                          =2xx-6+15x-6                          =x-62x+15

Answer:

We have:
5x2+2x-35
We have to split 2 into two numbers such that their sum is 2 and product is (-15), i.e.,5×-35.
Clearly, 5+-3=2 and 5×-3=-15.

5x2+2x-35=5x2+5x-3x-35                                  =5xx+5-3x+5                                  =x+55x-3

Page No 114:

Question 37:

We have:
5x2+2x-35
We have to split 2 into two numbers such that their sum is 2 and product is (-15), i.e.,5×-35.
Clearly, 5+-3=2 and 5×-3=-15.

5x2+2x-35=5x2+5x-3x-35                                  =5xx+5-3x+5                                  =x+55x-3

Answer:

We have:
23x2+x-53
We have to split 1 into two numbers such that their sum is 1 and product is 30, i.e.,23×-53.
Clearly, 6+-5=1 and 6×-5=-30.

23x2+x-53=23x2+6x-5x-53                                  =23xx+3-5x+3                                  =x+323x-5

Page No 114:

Question 38:

We have:
23x2+x-53
We have to split 1 into two numbers such that their sum is 1 and product is 30, i.e.,23×-53.
Clearly, 6+-5=1 and 6×-5=-30.

23x2+x-53=23x2+6x-5x-53                                  =23xx+3-5x+3                                  =x+323x-5

Answer:

We have:
7x2+214x+2
We have to split 214 into two numbers such that their sum is 214 and product is 14.
Clearly, 14+14=214 and 14×14=14.
7x2+214x+2=7x2+14x+14x+2                                 =7x7x+2+27x+2                                 =7x+27x+2                                 =7x+22

Page No 114:

Question 39:

We have:
7x2+214x+2
We have to split 214 into two numbers such that their sum is 214 and product is 14.
Clearly, 14+14=214 and 14×14=14.
7x2+214x+2=7x2+14x+14x+2                                 =7x7x+2+27x+2                                 =7x+27x+2                                 =7x+22

Answer:

We have:
63x2-47x+53
Now, we have to split (-47) into two numbers such that their sum is (-47) and their product is 90.
Clearly, -45+-2=-47 and -45×-2=90.

63x2-47x+53 =63x2-2x-45x+53                                        =2x33x-1-5333x-1                                        =33x-12x-53

Page No 114:

Question 40:

We have:
63x2-47x+53
Now, we have to split (-47) into two numbers such that their sum is (-47) and their product is 90.
Clearly, -45+-2=-47 and -45×-2=90.

63x2-47x+53 =63x2-2x-45x+53                                        =2x33x-1-5333x-1                                        =33x-12x-53

Answer:

We have:
55x2+20x+35
We have to split 20 into two numbers such that their sum is 20 and their product is 75.
Clearly, 
15+5=20 and 15×5=75

55x2+20x+35=55x2+15x+5x+35                                       =5x5x+3+5(5x+3)                                       =5x+35x+5

Page No 114:

Question 41:

We have:
55x2+20x+35
We have to split 20 into two numbers such that their sum is 20 and their product is 75.
Clearly, 
15+5=20 and 15×5=75

55x2+20x+35=55x2+15x+5x+35                                       =5x5x+3+5(5x+3)                                       =5x+35x+5

Answer:

3x2+10x+83=3x2+6x+4x+83                                 =3xx+23+4x+23                                 =x+233x+4

Hence, factorisation of 3x2+10x+83 is x+233x+4.

Page No 114:

Question 42:

3x2+10x+83=3x2+6x+4x+83                                 =3xx+23+4x+23                                 =x+233x+4

Hence, factorisation of 3x2+10x+83 is x+233x+4.

Answer:

We have:
2x2+3x+2
We have to split 3 into two numbers such that their sum is 3 and their product is 2, i.e., 2×2.
Clearly, 2+1=3 and 2×1=2.

2x2+3x+2=2x2+2x+x+2                                =2xx+2+1x+2                                =x+22x+1

Page No 114:

Question 43:

We have:
2x2+3x+2
We have to split 3 into two numbers such that their sum is 3 and their product is 2, i.e., 2×2.
Clearly, 2+1=3 and 2×1=2.

2x2+3x+2=2x2+2x+x+2                                =2xx+2+1x+2                                =x+22x+1

Answer:

We have:
2x2+33x+3
We have to split 33 into two numbers such that their sum is 33 and their product is 6, i.e.,2×3.
Clearly, 23+3=33 and 23×3=6.

2x2+33x+3=2x2+23x+3x+3                              =2xx+3+3x+3                              =x+32x+3

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Question 44:

We have:
2x2+33x+3
We have to split 33 into two numbers such that their sum is 33 and their product is 6, i.e.,2×3.
Clearly, 23+3=33 and 23×3=6.

2x2+33x+3=2x2+23x+3x+3                              =2xx+3+3x+3                              =x+32x+3

Answer:

We have:
15x2-x-28
We have to split (-1) into two numbers such that their sum is (-1) and their product is (-420), i.e., 15×-28.
Clearly, -21+20=-1 and -21×20=-420.

15x2-x-28=15x2-21x+20x-28                          =3x(5x-7)+4(5x-7)                          =(5x-7)(3x+4)

Page No 114:

Question 45:

We have:
15x2-x-28
We have to split (-1) into two numbers such that their sum is (-1) and their product is (-420), i.e., 15×-28.
Clearly, -21+20=-1 and -21×20=-420.

15x2-x-28=15x2-21x+20x-28                          =3x(5x-7)+4(5x-7)                          =(5x-7)(3x+4)

Answer:

We have:
6x2-5x-21
We have to split (-5) into two numbers such that their sum is (-5) and their product is (-126), i.e., 6×-21.
Clearly, 9+-14=-5 and 9×-14=-126.

6x2-5x-21=6x2+9x-14x-21                          =3x2x+3-72x+3                          =2x+33x-7

Page No 114:

Question 46:

We have:
6x2-5x-21
We have to split (-5) into two numbers such that their sum is (-5) and their product is (-126), i.e., 6×-21.
Clearly, 9+-14=-5 and 9×-14=-126.

6x2-5x-21=6x2+9x-14x-21                          =3x2x+3-72x+3                          =2x+33x-7

Answer:

We have:
2x2-7x-15
We have to split (-7) into two numbers such that their sum is (-7) and their product is (-30), i.e., 2×-15.
Clearly, -10+3=-7 and -10×3=-30.

2x2-7x-15=2x2-10x+3x-15                          =2xx-5+3x-5                          =x-52x+3

Page No 114:

Question 47:

We have:
2x2-7x-15
We have to split (-7) into two numbers such that their sum is (-7) and their product is (-30), i.e., 2×-15.
Clearly, -10+3=-7 and -10×3=-30.

2x2-7x-15=2x2-10x+3x-15                          =2xx-5+3x-5                          =x-52x+3

Answer:

We have:
5x2-16x-21
We have to split (-16) into two numbers such that their sum is (-16) and their product is (-105), i.e., 5×-21.
Clearly, -21+5=-16 and -21×5=-105.

5x2-16x-21=5x2+5x-21x-21                             =5xx+1-21x+1                             =x+15x-21

Page No 114:

Question 48:

We have:
5x2-16x-21
We have to split (-16) into two numbers such that their sum is (-16) and their product is (-105), i.e., 5×-21.
Clearly, -21+5=-16 and -21×5=-105.

5x2-16x-21=5x2+5x-21x-21                             =5xx+1-21x+1                             =x+15x-21

Answer:

6x2-11x-35=6x2-21x+10x-35                         =3x2x-7+52x-7                         =2x-73x+5

Hence, factorisation of 6x2 – 11x – 35 is 2x-73x+5.
 

Page No 114:

Question 49:

6x2-11x-35=6x2-21x+10x-35                         =3x2x-7+52x-7                         =2x-73x+5

Hence, factorisation of 6x2 – 11x – 35 is 2x-73x+5.
 

Answer:

9x2-3x-20=9x2-15x+12x-20                       =3x3x-5+43x-5                       =3x-53x+4

Hence, factorisation of 9x2 – 3x – 20 is 3x-53x+4.

Page No 114:

Question 50:

9x2-3x-20=9x2-15x+12x-20                       =3x3x-5+43x-5                       =3x-53x+4

Hence, factorisation of 9x2 – 3x – 20 is 3x-53x+4.

Answer:

We have:
10x2-9x-7

We have to split (-9) into two numbers such that their sum is (-9) and their product is (-70), i.e., 10×-7.
Clearly, -14+5=-9 and -14×5=-70.

10x2-9x-7=10x2+5x-14x-7                          =5x2x+1-72x+1                          =2x+15x-7

Page No 114:

Question 51:

We have:
10x2-9x-7

We have to split (-9) into two numbers such that their sum is (-9) and their product is (-70), i.e., 10×-7.
Clearly, -14+5=-9 and -14×5=-70.

10x2-9x-7=10x2+5x-14x-7                          =5x2x+1-72x+1                          =2x+15x-7

Answer:

We have:x2-2x+716=16x2-32x+716=11616x2-32x+7

Now, we have to split (-32) into two numbers such that their sum is (-32) and their product is 112, i.e., 16×7.
Clearly, -4+-28=-32 and -4×-28=112.

x2 - 2x + 716 =116(16x2-32x+7)                                =116(16x2-4x-28x+7)                                =1164x(4x-1)-7(4x-1)                                =116(4x-1)(4x-7)

Page No 114:

Question 52:

We have:x2-2x+716=16x2-32x+716=11616x2-32x+7

Now, we have to split (-32) into two numbers such that their sum is (-32) and their product is 112, i.e., 16×7.
Clearly, -4+-28=-32 and -4×-28=112.

x2 - 2x + 716 =116(16x2-32x+7)                                =116(16x2-4x-28x+7)                                =1164x(4x-1)-7(4x-1)                                =116(4x-1)(4x-7)

Answer:

13x2-2x-9=x2-6x-273                       =x2-9x+3x-273                       =xx-9+3x-93                       =x-9x+33                       =x-93×x+31                       =13x-3x+3

Hence, factorisation of 13x2-2x-9 is 13x-3x+3.

Page No 114:

Question 53:

13x2-2x-9=x2-6x-273                       =x2-9x+3x-273                       =xx-9+3x-93                       =x-9x+33                       =x-93×x+31                       =13x-3x+3

Hence, factorisation of 13x2-2x-9 is 13x-3x+3.

Answer:

x2+1235x+135=35x2+12x+135                           =35x2+7x+5x+135                           =7x5x+1+15x+135                           =5x+17x+135                           =5x+17x+15×7                           =5x+15×7x+17                           =x+15x+17

Hence, factorisation of x2+1235x+135 is x+15x+17.

Page No 114:

Question 54:

x2+1235x+135=35x2+12x+135                           =35x2+7x+5x+135                           =7x5x+1+15x+135                           =5x+17x+135                           =5x+17x+15×7                           =5x+15×7x+17                           =x+15x+17

Hence, factorisation of x2+1235x+135 is x+15x+17.

Answer:

21x2-2x+121=21x2-x-x+121                           =21xx-121-1x-121                           =x-12121x-1

Hence, factorisation of 21x2-2x+121 is x-12121x-1.

Page No 114:

Question 55:

21x2-2x+121=21x2-x-x+121                           =21xx-121-1x-121                           =x-12121x-1

Hence, factorisation of 21x2-2x+121 is x-12121x-1.

Answer:

32x2+16x+10=32x2+15x+x+10                           =3x12x+5+1x+10                           =32xx+10+1x+10                           =x+1032x+1

Hence, factorisation of 32x2+16x+10 is x+1032x+1.

Page No 114:

Question 56:

32x2+16x+10=32x2+15x+x+10                           =3x12x+5+1x+10                           =32xx+10+1x+10                           =x+1032x+1

Hence, factorisation of 32x2+16x+10 is x+1032x+1.

Answer:

23x2-173x-28=23x2-8x+73x-28                             =2x13x-4+713x-4                             =13x-42x+7

Hence, factorisation of 23x2-173x-28 is 13x-42x+7.

Page No 114:

Question 57:

23x2-173x-28=23x2-8x+73x-28                             =2x13x-4+713x-4                             =13x-42x+7

Hence, factorisation of 23x2-173x-28 is 13x-42x+7.

Answer:

35x2-195x+4=35x2-3x-45x+4                           =3x15x-1-415x-1                           =15x-13x-4

Hence, factorisation of 35x2-195x+4 is 15x-13x-4.

Page No 114:

Question 58:

35x2-195x+4=35x2-3x-45x+4                           =3x15x-1-415x-1                           =15x-13x-4

Hence, factorisation of 35x2-195x+4 is 15x-13x-4.

Answer:

2x2-x+18=2x2-12x-12x+18                    =2xx-14-12x-14                    =x-142x-12

Hence, factorisation of 2x2-x+18 is x-142x-12.

Page No 114:

Question 59:

2x2-x+18=2x2-12x-12x+18                    =2xx-14-12x-14                    =x-142x-12

Hence, factorisation of 2x2-x+18 is x-142x-12.

Answer:

We have:
2x+y2-9x+y-5Let:(x+y)=u
Thus, the given expression becomes
2u2-9u-5
Now, we have to split (-9) into two numbers such that their sum is (-9) and their product is (-10).
Clearly, -10+1=-9 and -10×1=-10.

2u2-9u-5=2u2-10u+u-5                         =2u(u-5)+1(u-5)                         =(u-5)(2u+1)
Putting u=(x+y), we get:
2x+y2 - 9x+y - 5 = x+y-52x+y+1                                          = x+y-52x+2y+1

Page No 114:

Question 60:

We have:
2x+y2-9x+y-5Let:(x+y)=u
Thus, the given expression becomes
2u2-9u-5
Now, we have to split (-9) into two numbers such that their sum is (-9) and their product is (-10).
Clearly, -10+1=-9 and -10×1=-10.

2u2-9u-5=2u2-10u+u-5                         =2u(u-5)+1(u-5)                         =(u-5)(2u+1)
Putting u=(x+y), we get:
2x+y2 - 9x+y - 5 = x+y-52x+y+1                                          = x+y-52x+2y+1

Answer:

We have:
9(2a-b)2-4(2a-b)-13Let:(2a-b)=p
Thus, the given expression becomes
9p2-4p-13
Now, we must split (-4) into two numbers such that their sum is (-4) and their product is (-117).
Clearly, -13+9=-4 and -13×9=-117.
9p2-4p-13=9p2+9p-13p-13                           =9p(p+1)-13(p+1)                           =(p+1)(9p-13)
Putting p=(2a-b), we get:
92a-b2-42a-b-13=2a-b+192a-b-13                                           =2a-b+118a-9b-13



Page No 115:

Question 61:

We have:
9(2a-b)2-4(2a-b)-13Let:(2a-b)=p
Thus, the given expression becomes
9p2-4p-13
Now, we must split (-4) into two numbers such that their sum is (-4) and their product is (-117).
Clearly, -13+9=-4 and -13×9=-117.
9p2-4p-13=9p2+9p-13p-13                           =9p(p+1)-13(p+1)                           =(p+1)(9p-13)
Putting p=(2a-b), we get:
92a-b2-42a-b-13=2a-b+192a-b-13                                           =2a-b+118a-9b-13

Answer:

7x-2y2-25x-2y+12=7x-2y2-21x-2y-4x-2y+12                                              =7x-2yx-2y-3-4x-2y-3                                              =7x-2y-4x-2y-3                                              =7x-14y-4x-2y-3

Hence, factorisation of 7x-2y2-25x-2y+12 is 7x-14y-4x-2y-3.

Page No 115:

Question 62:

7x-2y2-25x-2y+12=7x-2y2-21x-2y-4x-2y+12                                              =7x-2yx-2y-3-4x-2y-3                                              =7x-2y-4x-2y-3                                              =7x-14y-4x-2y-3

Hence, factorisation of 7x-2y2-25x-2y+12 is 7x-14y-4x-2y-3.

Answer:

103x+1x2-3x+1x-3=103x+1x2-63x+1x+53x+1x-3                                                =23x+1x53x+1x-3+153x+1x-3                                                =53x+1x-323x+1x+1                                                =15x+5x-36x+2x+1

Hence, factorisation of 103x+1x2-3x+1x-3 is 15x+5x-36x+2x+1.

Page No 115:

Question 63:

103x+1x2-3x+1x-3=103x+1x2-63x+1x+53x+1x-3                                                =23x+1x53x+1x-3+153x+1x-3                                                =53x+1x-323x+1x+1                                                =15x+5x-36x+2x+1

Hence, factorisation of 103x+1x2-3x+1x-3 is 15x+5x-36x+2x+1.

Answer:

62x-3x2+72x-3x-20=62x-3x2+152x-3x-82x-3x-20                                                  =32x-3x22x-3x+5-422x-3x+5                                                  =22x-3x+532x-3x-4                                                  =4x-6x+56x-9x-4

Hence, factorisation of 62x-3x2+72x-3x-20 is 4x-6x+56x-9x-4.

Page No 115:

Question 64:

62x-3x2+72x-3x-20=62x-3x2+152x-3x-82x-3x-20                                                  =32x-3x22x-3x+5-422x-3x+5                                                  =22x-3x+532x-3x-4                                                  =4x-6x+56x-9x-4

Hence, factorisation of 62x-3x2+72x-3x-20 is 4x-6x+56x-9x-4.

Answer:

a+2b2+101a+2b+100=a+2b2+100a+2b+1a+2b+100                                                 =a+2ba+2b+100+1a+2b+100                                                 =a+2b+1a+2b+100                                                 =a+2b+1a+2b+100

Hence, factorisation of a+2b2+101a+2b+100 is a+2b+1a+2b+100.

Page No 115:

Question 65:

a+2b2+101a+2b+100=a+2b2+100a+2b+1a+2b+100                                                 =a+2ba+2b+100+1a+2b+100                                                 =a+2b+1a+2b+100                                                 =a+2b+1a+2b+100

Hence, factorisation of a+2b2+101a+2b+100 is a+2b+1a+2b+100.

Answer:

4x4+7x2-2=4x4+8x2-x2-2                      =4x2x2+2-1x2+2                      =4x2-1x2+2

Hence, factorisation of 4x4 + 7x2 – 2 is 4x2-1x2+2.

Page No 115:

Question 66:

4x4+7x2-2=4x4+8x2-x2-2                      =4x2x2+2-1x2+2                      =4x2-1x2+2

Hence, factorisation of 4x4 + 7x2 – 2 is 4x2-1x2+2.

Answer:

9992-1=9992-12                   =999-1999+1                   =9981000                   =998000

Hence, {(999)2 – 1} = 998000.



Page No 119:

Question 1:

9992-1=9992-12                   =999-1999+1                   =9981000                   =998000

Hence, {(999)2 – 1} = 998000.

Answer:

i a+2b+5c2=a2 + 2b2 +5c2+2a2b+22b5c+25ca                           =a2+4b2+25c2+4ab+20bc+10ac

ii 2a-b+c2=2a+-b+c2                          =2a2+-b2+c2+22a-b+2-bc+4ac                          =4a2+b2+c2-4ab-2bc+4ac

iii a-2b-3c2=a+-2b+-3c2                             =a2+-2b2+-3c2+2a-2b+2-2b-3c+2a-3c                             =a2+4b2+9c2-4ab+12bc-6ac

Page No 119:

Question 2:

i a+2b+5c2=a2 + 2b2 +5c2+2a2b+22b5c+25ca                           =a2+4b2+25c2+4ab+20bc+10ac

ii 2a-b+c2=2a+-b+c2                          =2a2+-b2+c2+22a-b+2-bc+4ac                          =4a2+b2+c2-4ab-2bc+4ac

iii a-2b-3c2=a+-2b+-3c2                             =a2+-2b2+-3c2+2a-2b+2-2b-3c+2a-3c                             =a2+4b2+9c2-4ab+12bc-6ac

Answer:

i 2a-5b-7c2=2a+-5b+-7c2                             =2a2+-5b2+-7c2+22a-5b+2-5b-7c+22a-7c                             =4a2+25b2+49c2-20ab+70bc-28ac

ii -3a+4b-5c2=-3a+4b+-5c2                                 =-3a2+4b2+-5c2+2-3a4b+24b-5c+2-3a-5c                                 =9a2+16b2+25c2-24ab-40bc+30ac

iii 12a-14b+22=a2+-b4+22                                  =a22+-b42+22+2a2-b4+2-b42+2a22                                  =a24+b216+4-ab4-b+2a

Page No 119:

Question 3:

i 2a-5b-7c2=2a+-5b+-7c2                             =2a2+-5b2+-7c2+22a-5b+2-5b-7c+22a-7c                             =4a2+25b2+49c2-20ab+70bc-28ac

ii -3a+4b-5c2=-3a+4b+-5c2                                 =-3a2+4b2+-5c2+2-3a4b+24b-5c+2-3a-5c                                 =9a2+16b2+25c2-24ab-40bc+30ac

iii 12a-14b+22=a2+-b4+22                                  =a22+-b42+22+2a2-b4+2-b42+2a22                                  =a24+b216+4-ab4-b+2a

Answer:


We have: 4x2+9y2+16z2+12xy-24yz-16xz=2x2+3y2+-4z2+2(2x)(3y)+2(3y)(-4z)+2(-4z)(2x)=2x+3y+-4z2=2x+3y-4z2

Page No 119:

Question 4:


We have: 4x2+9y2+16z2+12xy-24yz-16xz=2x2+3y2+-4z2+2(2x)(3y)+2(3y)(-4z)+2(-4z)(2x)=2x+3y+-4z2=2x+3y-4z2

Answer:

We have:9x2+16y2+4z2-24xy+16yz-12xz=-3x2+4y2+2z2+2(-3x)(4y)+2(4y)(2z)+2(2z)(-3x)=-3x+4y+2z2=-3x+4y+2z2

Page No 119:

Question 5:

We have:9x2+16y2+4z2-24xy+16yz-12xz=-3x2+4y2+2z2+2(-3x)(4y)+2(4y)(2z)+2(2z)(-3x)=-3x+4y+2z2=-3x+4y+2z2

Answer:

We have:25x2+4y2+9z2-20xy-12yz+30xz=5x2+-2y2+3z2+2(5x)(-2y)+2(-2y)(3z)+2(3z)(5x)=5x+-2y+3z2=5x-2y+3z2

Page No 119:

Question 6:

We have:25x2+4y2+9z2-20xy-12yz+30xz=5x2+-2y2+3z2+2(5x)(-2y)+2(-2y)(3z)+2(3z)(5x)=5x+-2y+3z2=5x-2y+3z2

Answer:

16x2+4y2+9z2-16xy-12yz+24xz=4x2+-2y2+3z2+24x-2y+2-2y3z+23z4x=4x-2y+3z2               using a2+b2+c2+2ab+2bc+2ca=a+b+c2

Hence, 16x2 + 4y2 + 9z2 – 16xy – 12yz + 24xz = 4x-2y+3z2.

Page No 119:

Question 7:

16x2+4y2+9z2-16xy-12yz+24xz=4x2+-2y2+3z2+24x-2y+2-2y3z+23z4x=4x-2y+3z2               using a2+b2+c2+2ab+2bc+2ca=a+b+c2

Hence, 16x2 + 4y2 + 9z2 – 16xy – 12yz + 24xz = 4x-2y+3z2.

Answer:

i 992=100-12             =100+-12             = 1002+2×100×-1+-12             =10000-200+1             =9801         

ii 9952=1000-52                =1000+-52                =10002+2×1000×-5+-52                =1000000-10000+25                =990025

iii 1072=100+72                 =1002+2×100×7+72                 =10000+1400+49                 =11449



Page No 123:

Question 1:

i 992=100-12             =100+-12             = 1002+2×100×-1+-12             =10000-200+1             =9801         

ii 9952=1000-52                =1000+-52                =10002+2×1000×-5+-52                =1000000-10000+25                =990025

iii 1072=100+72                 =1002+2×100×7+72                 =10000+1400+49                 =11449

Answer:

i 3x+23=3x3+3×3x2x2+3×3x×22+23                   =27x3+54x2+36x+8
ii 3a+14b3=3a3+14b3+33a214b+33a14b2                          =27a3+164b3+27a24b+9a16b2
iii 1+23a3=23a3+3×23a2×1+3a23a×12+13                         =827a3+43a2+2a+1

Page No 123:

Question 2:

i 3x+23=3x3+3×3x2x2+3×3x×22+23                   =27x3+54x2+36x+8
ii 3a+14b3=3a3+14b3+33a214b+33a14b2                          =27a3+164b3+27a24b+9a16b2
iii 1+23a3=23a3+3×23a2×1+3a23a×12+13                         =827a3+43a2+2a+1

Answer:

i 5a-3b3=5a3-3b3-35a23b+35a3b2                        =125a3-27b3-225a2b+135ab2

ii 3x-5x3=3x3-5x3-33x25x+33x5x2                        =27x3-125x3-135x+225x


iii 45a-23=45a3-23-345a22+345a22                        =64125a3-8-9625a2+485a

Page No 123:

Question 3:

i 5a-3b3=5a3-3b3-35a23b+35a3b2                        =125a3-27b3-225a2b+135ab2

ii 3x-5x3=3x3-5x3-33x25x+33x5x2                        =27x3-125x3-135x+225x


iii 45a-23=45a3-23-345a22+345a22                        =64125a3-8-9625a2+485a

Answer:

8a3+27b3+36a2b+54ab2=2a3+3b3+32a23b+32a3b2                                                 =2a+3b3

Hence, factorisation of 8a3+27b3+36a2b+54ab2 is 2a+3b3.

Page No 123:

Question 4:

8a3+27b3+36a2b+54ab2=2a3+3b3+32a23b+32a3b2                                                 =2a+3b3

Hence, factorisation of 8a3+27b3+36a2b+54ab2 is 2a+3b3.

Answer:

64a3-27b3-144a2b+108ab2=4a3-3b3-34a23b+34a3b2                                                       =4a-3b3

Hence, factorisation of 64a3-27b3-144a2b+108ab2 is 4a-3b3.

Page No 123:

Question 5:

64a3-27b3-144a2b+108ab2=4a3-3b3-34a23b+34a3b2                                                       =4a-3b3

Hence, factorisation of 64a3-27b3-144a2b+108ab2 is 4a-3b3.

Answer:

1+27125a3+9a5+27a225=13+35a3+31235a+3135a2                                            =1+35a3

Hence, factorisation of 1+27125a3+9a5+27a225 is 1+35a3.

Page No 123:

Question 6:

1+27125a3+9a5+27a225=13+35a3+31235a+3135a2                                            =1+35a3

Hence, factorisation of 1+27125a3+9a5+27a225 is 1+35a3.

Answer:

125x3-27y3-225x2y+135xy2=5x3-3y3-35x23y+35x3y2                                                        =5x-3y3

Hence, factorisation of 125x3-27y3-225x2y+135xy2 is 5x-3y3.

Page No 123:

Question 7:

125x3-27y3-225x2y+135xy2=5x3-3y3-35x23y+35x3y2                                                        =5x-3y3

Hence, factorisation of 125x3-27y3-225x2y+135xy2 is 5x-3y3.

Answer:

a3x3-3a2bx2+3ab2x-b3=ax3-b3-3ax2b+3axb2                                               =ax-b3

Hence, factorisation of a3x3-3a2bx2+3ab2x-b3 is ax-b3.

Page No 123:

Question 8:

a3x3-3a2bx2+3ab2x-b3=ax3-b3-3ax2b+3axb2                                               =ax-b3

Hence, factorisation of a3x3-3a2bx2+3ab2x-b3 is ax-b3.

Answer:

64125a3-9625a2+485a-8=45a3-23-345a22+345a22                                               =45a-23

Hence, factorisation of 64125a3-9625a2+485a-8 is 45a-23.

Page No 123:

Question 9:

64125a3-9625a2+485a-8=45a3-23-345a22+345a22                                               =45a-23

Hence, factorisation of 64125a3-9625a2+485a-8 is 45a-23.

Answer:

a3-12aa-4-64=a3-12a2+48a-64                                  =a3-43-3a24+3a42                                  =a-43

Hence, factorisation of a3 – 12a(a – 4) – 64 is a-43.

Page No 123:

Question 10:

a3-12aa-4-64=a3-12a2+48a-64                                  =a3-43-3a24+3a42                                  =a-43

Hence, factorisation of a3 – 12a(a – 4) – 64 is a-43.

Answer:

i 1033=100+33                =1003+33+310023+310032                =1000000+27+90000+2700                =1092727ii 993=100-13              =1003-13-310021+310012              =1000000-1-30000+300              =1000300-30001              =970299



Page No 129:

Question 1:

i 1033=100+33                =1003+33+310023+310032                =1000000+27+90000+2700                =1092727ii 993=100-13              =1003-13-310021+310012              =1000000-1-30000+300              =1000300-30001              =970299

Answer:

x3+27=x3+33            =x+3x2-3x+32            =x+3x2-3x+9

Page No 129:

Question 2:

x3+27=x3+33            =x+3x2-3x+32            =x+3x2-3x+9

Answer:

We know that 
x3+y3=x+yx2+y2-xy
Given: 27a3 + 64b3
x = 3a, y = 4b
27a3 + 64b3=3a+4b9a2+16b2-12ab
 

Page No 129:

Question 3:

We know that 
x3+y3=x+yx2+y2-xy
Given: 27a3 + 64b3
x = 3a, y = 4b
27a3 + 64b3=3a+4b9a2+16b2-12ab
 

Answer:

125a3+18=5a3+123                   =5a+125a2-5a×12+122                   =5a+1225a2-5a2+14

Page No 129:

Question 4:

125a3+18=5a3+123                   =5a+125a2-5a×12+122                   =5a+1225a2-5a2+14

Answer:

216x3+1125=6x3+153                        =6x+156x2-6x×15+152                        =6x+1536x2-6x5+125

Page No 129:

Question 5:

216x3+1125=6x3+153                        =6x+156x2-6x×15+152                        =6x+1536x2-6x5+125

Answer:


16x4+54x=2x8x3+27                    =2x2x3+33                    =2x2x+32x2-2x×3+32                    =2x2x+34x2-6x+9

Page No 129:

Question 6:


16x4+54x=2x8x3+27                    =2x2x3+33                    =2x2x+32x2-2x×3+32                    =2x2x+34x2-6x+9

Answer:

7a3+56b3=7a3+8b3                   =7a3+2b3                   =7a+2ba2-a×2b+2b2                   =7a+2ba2-2ab+4b2

Page No 129:

Question 7:

7a3+56b3=7a3+8b3                   =7a3+2b3                   =7a+2ba2-a×2b+2b2                   =7a+2ba2-2ab+4b2

Answer:

x5+x2=x2x3+1            =x2x3+13            =x2x+1x2-x×1+12            =x2x+1x2-x+1

Page No 129:

Question 8:

x5+x2=x2x3+1            =x2x3+13            =x2x+1x2-x×1+12            =x2x+1x2-x+1

Answer:

a3+0.008=a3+0.23                  =a+0.2a2-a×0.2+0.22                  =a+0.2a2-0.2a+0.04

Page No 129:

Question 9:

a3+0.008=a3+0.23                  =a+0.2a2-a×0.2+0.22                  =a+0.2a2-0.2a+0.04

Answer:

1-27a3=13-3a3              =1-3a12+1×3x+3a2              =1-3a1+3a+9a2

Page No 129:

Question 10:

1-27a3=13-3a3              =1-3a12+1×3x+3a2              =1-3a1+3a+9a2

Answer:

64a3-343=4a3-73                   =4a-716a2+4a×7+72                   =4a-716a2+28a+49

Page No 129:

Question 11:

64a3-343=4a3-73                   =4a-716a2+4a×7+72                   =4a-716a2+28a+49

Answer:

x3-512 =x3-83               =x-8x2+8x+82               =x-8x2+8x+64

Page No 129:

Question 12:

x3-512 =x3-83               =x-8x2+8x+82               =x-8x2+8x+64

Answer:

a3-0.064=a3-0.43                  =a-0.4a2+a×0.4+0.42                  =a-0.4a2+0.4a+0.16

Page No 129:

Question 13:

a3-0.064=a3-0.43                  =a-0.4a2+a×0.4+0.42                  =a-0.4a2+0.4a+0.16

Answer:

8x3-127y3=2x3-13y3                    =2x-13y2x2+2x×13y+13y2                    =2x-13y4x2+2x3y+19y2

Page No 129:

Question 14:

8x3-127y3=2x3-13y3                    =2x-13y2x2+2x×13y+13y2                    =2x-13y4x2+2x3y+19y2

Answer:

We know
a3-b3=a-ba2+b2+ab
We have,
 x3216-8y3=x63-2y3
So, a=x6,b=2y
x3216-8y3=x6-2yx62+x6×2y+2y2=x6-2yx236+xy3+4y2

Page No 129:

Question 15:

We know
a3-b3=a-ba2+b2+ab
We have,
 x3216-8y3=x63-2y3
So, a=x6,b=2y
x3216-8y3=x6-2yx62+x6×2y+2y2=x6-2yx236+xy3+4y2

Answer:

x-8xy3=x1-8y3              =x13-2y3              =x1-2y12+1×2y+2y2              =x1-2y1+2y+4y2

Page No 129:

Question 16:

x-8xy3=x1-8y3              =x13-2y3              =x1-2y12+1×2y+2y2              =x1-2y1+2y+4y2

Answer:

 32x4  500x=4x8x3-125=4x2x3-53we knowa3-b3=a-ba2+b2+aba=2x,b=532x4  500x=4x2x3-53=4x2x-54x2+25+10x

Page No 129:

Question 17:

 32x4  500x=4x8x3-125=4x2x3-53we knowa3-b3=a-ba2+b2+aba=2x,b=532x4  500x=4x2x3-53=4x2x-54x2+25+10x

Answer:

3a7b-81a4b4=3a4ba3-27b3                         =3a4ba3-3b3                         =3a4ba-3ba2+a×3b+3b2                         =3a4ba-3ba2+3ab+9b2

Page No 129:

Question 18:

3a7b-81a4b4=3a4ba3-27b3                         =3a4ba3-3b3                         =3a4ba-3ba2+a×3b+3b2                         =3a4ba-3ba2+3ab+9b2

Answer:

Using the identity 
a3-b3=a-ba2+b2+ab
x4 y4xy=xyx3y3-1=xyxy-1x2y2+1+xy

Page No 129:

Question 19:

Using the identity 
a3-b3=a-ba2+b2+ab
x4 y4xy=xyx3y3-1=xyxy-1x2y2+1+xy

Answer:

8x2y3x5=x28y3-x3=x22y-x4y2+x2+2xy

Page No 129:

Question 20:

8x2y3x5=x28y3-x3=x22y-x4y2+x2+2xy

Answer:

10293x3=3343-x3=373-x3=37-x49+x2+7x

Page No 129:

Question 21:

10293x3=3343-x3=373-x3=37-x49+x2+7x

Answer:

x6-729=x23-93             =x2-9x22+x2×9+92             =x2-32x4+9x2+81             =x+3x-3x4+18x2+81-9x2             =x+3x-3x22+2×x2×9+92-9x2             =x+3x-3x2+92-3x2             =x+3x-3x2+9+3xx2+9-3x             =x+3x-3x2+3x+9x2-3x+9

Page No 129:

Question 22:

x6-729=x23-93             =x2-9x22+x2×9+92             =x2-32x4+9x2+81             =x+3x-3x4+18x2+81-9x2             =x+3x-3x22+2×x2×9+92-9x2             =x+3x-3x2+92-3x2             =x+3x-3x2+9+3xx2+9-3x             =x+3x-3x2+3x+9x2-3x+9

Answer:

x9y9=x33-y33we knowa3-b3=a-ba2+b2+aba=x3,b=y3So,x9y9=x33-y33=x3-y3x6+y6+x3y3=x-yx2+y2+xyx6+y6+x3y3

Page No 129:

Question 23:

x9y9=x33-y33we knowa3-b3=a-ba2+b2+aba=x3,b=y3So,x9y9=x33-y33=x3-y3x6+y6+x3y3=x-yx2+y2+xyx6+y6+x3y3

Answer:

a + b3-a-b3=a+b-a-ba+b2+a+ba-b+a-b2                            =a+b-a+ba2+2ab+b2+a2-b2+a2-2ab+b2                            =2b3a2+b2

Page No 129:

Question 24:

a + b3-a-b3=a+b-a-ba+b2+a+ba-b+a-b2                            =a+b-a+ba2+2ab+b2+a2-b2+a2-2ab+b2                            =2b3a2+b2

Answer:

8a3-b3-4ax+2bx=2a3-b3-2x2a-b                                    =2a-b2a2+2ab+b2-2x2a-b                                    =2a-b4a2+2ab+b2-2x2a-b                                    =2a-b4a2+2ab+b2-2x

Page No 129:

Question 25:

8a3-b3-4ax+2bx=2a3-b3-2x2a-b                                    =2a-b2a2+2ab+b2-2x2a-b                                    =2a-b4a2+2ab+b2-2x2a-b                                    =2a-b4a2+2ab+b2-2x

Answer:

a3+3a2b+3ab2+b3-8=a3+b3+3a2b+3ab2-8                                            =a3+b3+3aba+b-8                                            =a+b3-23                                            =a+b-2a+b2+2a+b+22                                            =a+b-2a+b2+2a+b+4                                            

Page No 129:

Question 26:

a3+3a2b+3ab2+b3-8=a3+b3+3a2b+3ab2-8                                            =a3+b3+3aba+b-8                                            =a+b3-23                                            =a+b-2a+b2+2a+b+22                                            =a+b-2a+b2+2a+b+4                                            

Answer:

a3-1a3-2a+2a=a3-1a3-2a-1a                                 =a3-1a3-2a-1a                                 =a-1aa2+a×1a+1a2-2a-1a                                 =a-1aa2+1+1a2-2a-1a                                 =a-1aa2+1+1a2-2                                 =a-1aa2-1+1a2

Page No 129:

Question 27:

a3-1a3-2a+2a=a3-1a3-2a-1a                                 =a3-1a3-2a-1a                                 =a-1aa2+a×1a+1a2-2a-1a                                 =a-1aa2+1+1a2-2a-1a                                 =a-1aa2+1+1a2-2                                 =a-1aa2-1+1a2

Answer:

2a3+16b3-5a-10b=2a3+8b3-5a+2b                                        =2a3+2b3-5a+2b                                        =2a+2ba2-a×2b+2b2-5a+2b                                        =2a+2ba2-2ab+4b2-5a+2b                                        =a+2b2a2-2ab+4b2-5

Page No 129:

Question 28:

2a3+16b3-5a-10b=2a3+8b3-5a+2b                                        =2a3+2b3-5a+2b                                        =2a+2ba2-a×2b+2b2-5a+2b                                        =2a+2ba2-2ab+4b2-5a+2b                                        =a+2b2a2-2ab+4b2-5

Answer:

a6+b6=a23+b23            =a2+b2a22-a2b2+b22            =a2+b2a4-a2b2+b4

Page No 129:

Question 29:

a6+b6=a23+b23            =a2+b2a22-a2b2+b22            =a2+b2a4-a2b2+b4

Answer:

a12 – b12
=a6+b6a6-b6=a23+b23a32-b32=a2+b2a4+b4-a2b2a3-b3a3+b3=a2+b2a4+b4-a2b2a-ba2+b2+aba+ba2+b2-ab=a-ba2+b2+aba+ba2+b2-aba2+b2a4+b4-a2b2

 

Page No 129:

Question 30:

a12 – b12
=a6+b6a6-b6=a23+b23a32-b32=a2+b2a4+b4-a2b2a3-b3a3+b3=a2+b2a4+b4-a2b2a-ba2+b2+aba+ba2+b2-ab=a-ba2+b2+aba+ba2+b2-aba2+b2a4+b4-a2b2

 

Answer:

Let x3=y
So, the equation becomes 
y2-7y-8=y2-8y+y-8=yy-8+y-8=y-8y+1=x3-8x3+1=x-2x2+4+2xx+1x2+1-x

Page No 129:

Question 31:

Let x3=y
So, the equation becomes 
y2-7y-8=y2-8y+y-8=yy-8+y-8=y-8y+1=x3-8x3+1=x-2x2+4+2xx+1x2+1-x

Answer:

x3 – 3x+ 3x + 7
=x33x2+3x+7=x33x2+3x+8-1=x33x2+3x-1+8=x33x2+3x-1+8=x-13+23=x-1+2x-12+4-2x-1=x+1x2+1-2x+4-2x+2=x+1x2-4x+7

Page No 129:

Question 32:

x3 – 3x+ 3x + 7
=x33x2+3x+7=x33x2+3x+8-1=x33x2+3x-1+8=x33x2+3x-1+8=x-13+23=x-1+2x-12+4-2x-1=x+1x2+1-2x+4-2x+2=x+1x2-4x+7

Answer:

(x +1)3 + (x – 1)3
=x+1+x-1x+12+x-12-x-1x+1=2xx+12+x-12-x2-1=2xx2+1+2x+x2+1-2x-x2+1=2xx2+3

 

Page No 129:

Question 33:

(x +1)3 + (x – 1)3
=x+1+x-1x+12+x-12-x-1x+1=2xx+12+x-12-x2-1=2xx2+1+2x+x2+1-2x-x2+1=2xx2+3

 

Answer:

(2a +1)3 + (a – 1)3  
=2a+1+a-12a+12+a-12-2a+1a-1=3a4a2+1+4a+a2+1-2a-2a2+2a-a+1=3a3a2+3a+3=9aa2+a+1

Page No 129:

Question 34:

(2a +1)3 + (a – 1)3  
=2a+1+a-12a+12+a-12-2a+1a-1=3a4a2+1+4a+a2+1-2a-2a2+2a-a+1=3a3a2+3a+3=9aa2+a+1

Answer:

8(x +y)3 – 27(x – y)3
=2x+y3-3x-y3=2x+2y-3x+3y4x+y2+9x-y2+6x2-y2=-x+5y4x2+y2+2xy+9x2+y2-2xy+6x2-y2=-x+5y4x2+4y2+8xy+9x2+9y2-18xy+6x2-6y2=-x+5y19x2+7y2-10xy

Page No 129:

Question 35:

8(x +y)3 – 27(x – y)3
=2x+y3-3x-y3=2x+2y-3x+3y4x+y2+9x-y2+6x2-y2=-x+5y4x2+y2+2xy+9x2+y2-2xy+6x2-y2=-x+5y4x2+4y2+8xy+9x2+9y2-18xy+6x2-6y2=-x+5y19x2+7y2-10xy

Answer:

(x +2)3 + (x – 2)3
=x+2+x-2x+22+x-22-x2-4=2xx2+4+4x+x2+4-4x-x2+4=2xx2+12

Page No 129:

Question 36:

(x +2)3 + (x – 2)3
=x+2+x-2x+22+x-22-x2-4=2xx2+4+4x+x2+4-4x-x2+4=2xx2+12

Answer:

(x + 2)3 – (x – 2)3
=x+2-x+2x+22+x-22+x2-4=4x2+4+4x+x2+4-4x+x2-4=43x2+4

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Question 37:

(x + 2)3 – (x – 2)3
=x+2-x+2x+22+x-22+x2-4=4x2+4+4x+x2+4-4x+x2-4=43x2+4

Answer:

LHS:0.85×0.85×0.85+0.15×0.15×0.150.85×0.85-0.85×0.15+0.15×0.15=0.853+0.1530.852-0.85×0.15+0.152We knowa3+b3=a+ba2+b2-abHere a=0.85,b=0.15
0.853+0.1530.852-0.85×0.15+0.152=0.85+0.150.852-0.85×0.15+0.1520.852-0.85×0.15+0.152=0.85+0.15=1:RHS
Thus, LHS = RHS
 

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Question 38:

LHS:0.85×0.85×0.85+0.15×0.15×0.150.85×0.85-0.85×0.15+0.15×0.15=0.853+0.1530.852-0.85×0.15+0.152We knowa3+b3=a+ba2+b2-abHere a=0.85,b=0.15
0.853+0.1530.852-0.85×0.15+0.152=0.85+0.150.852-0.85×0.15+0.1520.852-0.85×0.15+0.152=0.85+0.15=1:RHS
Thus, LHS = RHS
 

Answer:

59×59×59-9×9×959×59+59×9+9×9=593-93592+59×9+92
We knowa3+b3=a+ba2+b2-abHere a=59,b=9So,59-9592+92+59×9592+92+59×9=59-9=50:RHS
Thus, LHS=RHS



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Question 1:

59×59×59-9×9×959×59+59×9+9×9=593-93592+59×9+92
We knowa3+b3=a+ba2+b2-abHere a=59,b=9So,59-9592+92+59×9592+92+59×9=59-9=50:RHS
Thus, LHS=RHS

Answer:

   x+y-zx2+y2+z2-xy+yz+zx=x+y+-zx2+y2+-z2-xy-y×-z--z×x=x3+y3+-z3-3x×y×-z=x3+y3-z3+3xyz

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Question 2:

   x+y-zx2+y2+z2-xy+yz+zx=x+y+-zx2+y2+-z2-xy-y×-z--z×x=x3+y3+-z3-3x×y×-z=x3+y3-z3+3xyz

Answer:

(x – y − z) (x2 + y2 + z2 + xy – yz + xz)
=(x+-y+z) (x2+y2+z2+xyyz+xz)We knowa+b+ca2+b2+c2-ab-bc-ca=a3+b3+c3-3abcHere, a=x,b=-y,c=-z(x+-y+z) (x2+y2+z2+xyyz+xz)=x3-y3-z3-3xyz

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Question 3:

(x – y − z) (x2 + y2 + z2 + xy – yz + xz)
=(x+-y+z) (x2+y2+z2+xyyz+xz)We knowa+b+ca2+b2+c2-ab-bc-ca=a3+b3+c3-3abcHere, a=x,b=-y,c=-z(x+-y+z) (x2+y2+z2+xyyz+xz)=x3-y3-z3-3xyz

Answer:

   x − 2y + 3x2+4y2+2xy-3x+6y+9=x − 2y + 3x2+4y2+9+2xy+6y-3x=x+-2y+3x2+-2y2+32-x×-2y--2y×3-3×x=x3+-2y3+33-3x-2y3=x3-8y3+27+18xy

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Question 4:

   x − 2y + 3x2+4y2+2xy-3x+6y+9=x − 2y + 3x2+4y2+9+2xy+6y-3x=x+-2y+3x2+-2y2+32-x×-2y--2y×3-3×x=x3+-2y3+33-3x-2y3=x3-8y3+27+18xy

Answer:

3x-5y+49x2+25y2+15xy-20y+12x+16=3x+-5y+49x2+25y2+16+15xy-20y+12x
a+b+ca2+b2+c2-ab-bc-ca=a3+b3+c3-3abcHere, a=3x,b=-5y,c=4
3x+-5y+49x2+25y2+16+15xy-20y+12x=3x3+-5y3+43-3×3x-5y4=27x3-125y3+64+180xy

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Question 5:

3x-5y+49x2+25y2+15xy-20y+12x+16=3x+-5y+49x2+25y2+16+15xy-20y+12x
a+b+ca2+b2+c2-ab-bc-ca=a3+b3+c3-3abcHere, a=3x,b=-5y,c=4
3x+-5y+49x2+25y2+16+15xy-20y+12x=3x3+-5y3+43-3×3x-5y4=27x3-125y3+64+180xy

Answer:

125a3+b3+64c3-60abc=5a3+b3+4c3-3×5a×b×4c                                              =5a+b+4c5a2+b2+4c2-5a×b-b×4c-5a×4c                                              =5a+b+4c25a2+b2+16c2-5ab-4bc-20ac                          

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Question 6:

125a3+b3+64c3-60abc=5a3+b3+4c3-3×5a×b×4c                                              =5a+b+4c5a2+b2+4c2-5a×b-b×4c-5a×4c                                              =5a+b+4c25a2+b2+16c2-5ab-4bc-20ac                          

Answer:

a3+8b3+64c3-24abc=a3+2b3+4c3-3×a×2b×4c                                          =a+2b+4ca2+2b2+4c2-a×2b-2b×4c-4c×a                                          =a+2b+4ca2+4b2+16c2-2ab-8bc-4ca

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Question 7:

a3+8b3+64c3-24abc=a3+2b3+4c3-3×a×2b×4c                                          =a+2b+4ca2+2b2+4c2-a×2b-2b×4c-4c×a                                          =a+2b+4ca2+4b2+16c2-2ab-8bc-4ca

Answer:

1+b3+8c3-6bc=13+b3+2c3-3×1×b×2c                               =1+b+2c12+b2+2c2-1×b-b×2c-1×2c                               =1+b+2c1+b2+4c2-b-2bc-2c

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Question 8:

1+b3+8c3-6bc=13+b3+2c3-3×1×b×2c                               =1+b+2c12+b2+2c2-1×b-b×2c-1×2c                               =1+b+2c1+b2+4c2-b-2bc-2c

Answer:

216+27b3+8c3-108abc=63+3b3+2c3-3×6×3b×2c                                               =6+3b+2c62+3b2+2c2-6×3b-3b×2c-2c×6                                               =6+3b+2c36+9b2+4c2-18b-6bc-12c

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Question 9:

216+27b3+8c3-108abc=63+3b3+2c3-3×6×3b×2c                                               =6+3b+2c62+3b2+2c2-6×3b-3b×2c-2c×6                                               =6+3b+2c36+9b2+4c2-18b-6bc-12c

Answer:

27a3-b3+8c3+18abc=3a3+-b3+2c3-3×3a×-b×2c                                         =3a+-b+2c3a2+-b2+2c2-3a-b--b2c-3a×2c                                         =3a-b+2c9a2+b2+4c2+3ab+2bc-6ac

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Question 10:

27a3-b3+8c3+18abc=3a3+-b3+2c3-3×3a×-b×2c                                         =3a+-b+2c3a2+-b2+2c2-3a-b--b2c-3a×2c                                         =3a-b+2c9a2+b2+4c2+3ab+2bc-6ac

Answer:

8a3+125b3-64c3+120abc=2a3+5b3+-4c3-3×2a×5b×-4c                                                   =2a+5b-4c2a2+5b2+-4c2-2a5b-5b-4c-2a×-4c                                                   =2a+5b-4c4a2+25b2+16c2-10ab+20bc+8ac

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Question 11:

8a3+125b3-64c3+120abc=2a3+5b3+-4c3-3×2a×5b×-4c                                                   =2a+5b-4c2a2+5b2+-4c2-2a5b-5b-4c-2a×-4c                                                   =2a+5b-4c4a2+25b2+16c2-10ab+20bc+8ac

Answer:

8-27b3-343c3-126bc=23+-3b3+-7c3-3×2×-3b×-7c                                             =2+-3b+-7c22+-3b2+-7c2-2-3b--3b-7c-2-7c                                             =2-3b-7c4+9b2+49c2+6b-21bc+14c

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Question 12:

8-27b3-343c3-126bc=23+-3b3+-7c3-3×2×-3b×-7c                                             =2+-3b+-7c22+-3b2+-7c2-2-3b--3b-7c-2-7c                                             =2-3b-7c4+9b2+49c2+6b-21bc+14c

Answer:

125-8x3-27y3-90xy=53+-2x3+-3y3-3×5×-2x×-3y                                          =5+-2x +-3y52+-2x2+-3y2-5×-2x--2x-3y-5×-3y                                          =5-2x-3y25+4x2+9y2+10x-6xy+15y



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Question 13:

125-8x3-27y3-90xy=53+-2x3+-3y3-3×5×-2x×-3y                                          =5+-2x +-3y52+-2x2+-3y2-5×-2x--2x-3y-5×-3y                                          =5-2x-3y25+4x2+9y2+10x-6xy+15y

Answer:

22a3+162b3+c3-12abc=2a3+22b3+c3-3×2a×22b×c                                                    =2a+22b+c2a2+22b2+c2-2a×22b-22b×c-2a×c                                                    =2a+22b+c2a2+8b2+c2-4ab-22bc-2ac

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Question 14:

22a3+162b3+c3-12abc=2a3+22b3+c3-3×2a×22b×c                                                    =2a+22b+c2a2+22b2+c2-2a×22b-22b×c-2a×c                                                    =2a+22b+c2a2+8b2+c2-4ab-22bc-2ac

Answer:

27x3-y3z39xyz=3x3-y3-z3-3×3x×-y×-zWe know, a3+b3+c3-3abc=a+b+ca2+b2+c2-ab-bc-caa=3x,b=-y,c=-z3x3-y3-z3-3×3x×-y×-z=3x-y-z9x2+y2+z2+3xy-yz+3xz

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Question 15:

27x3-y3z39xyz=3x3-y3-z3-3×3x×-y×-zWe know, a3+b3+c3-3abc=a+b+ca2+b2+c2-ab-bc-caa=3x,b=-y,c=-z3x3-y3-z3-3×3x×-y×-z=3x-y-z9x2+y2+z2+3xy-yz+3xz

Answer:

22a3+33b3+c3-36abc=2a3+3b3+c3-32a3bcWe knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx=2a,y=3b,z=c2a3+3b3+c3-32a3bc=2a+3b+c2a2+3b2+c2-6ab-3bc-2ac

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Question 16:

22a3+33b3+c3-36abc=2a3+3b3+c3-32a3bcWe knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx=2a,y=3b,z=c2a3+3b3+c3-32a3bc=2a+3b+c2a2+3b2+c2-6ab-3bc-2ac

Answer:

33a3-b3-55c3-315abc=3a3+-b3+-5c3-33a-b-5cWe knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxHere, x=3a,y=(-b),z=-5c33a3-b3-55c3-315abc=3a3+-b3+-5c3-33a-b-5c=3a-b-5c3a2+b2+5c2+3ab-5bc+15c

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Question 17:

33a3-b3-55c3-315abc=3a3+-b3+-5c3-33a-b-5cWe knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxHere, x=3a,y=(-b),z=-5c33a3-b3-55c3-315abc=3a3+-b3+-5c3-33a-b-5c=3a-b-5c3a2+b2+5c2+3ab-5bc+15c

Answer:

a-b3+b-c3+c-a3
Putting a-b=x, b-c=y and c-a=z, we get:a-b3+b-c3+c-a3=x3+y3+z3        [Where x+y+z=a-b+b-c+c-a=0]=3xyz                 x+y+z=0 x3+y3+z3=3xyz=3a-bb-cc-a

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Question 18:

a-b3+b-c3+c-a3
Putting a-b=x, b-c=y and c-a=z, we get:a-b3+b-c3+c-a3=x3+y3+z3        [Where x+y+z=a-b+b-c+c-a=0]=3xyz                 x+y+z=0 x3+y3+z3=3xyz=3a-bb-cc-a

Answer:

We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=a-3b,y=(3b-c),z=c-a
a-3b3+3b-c3+c-a3=a-3b+3b-c+c-aa-3b2+3b-c2+c-a2-a-3b3b-c-3b-cc-a-c-aa-3b+3a-3b3b-cc-a=0+3a-3b3b-cc-a=3a-3b3b-cc-a

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Question 19:

We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=a-3b,y=(3b-c),z=c-a
a-3b3+3b-c3+c-a3=a-3b+3b-c+c-aa-3b2+3b-c2+c-a2-a-3b3b-c-3b-cc-a-c-aa-3b+3a-3b3b-cc-a=0+3a-3b3b-cc-a=3a-3b3b-cc-a

Answer:

Put 3a-2b=x, 2b-5c=y and 5c-3a=z.We have:x+y+z = 3a-2b+2b-5c+5c-3a=0Now,3a-2b3+2b-5c3+5c-3a3=x3+y3+z3                                                   =3xyz    Here, x+y+z=0. So, x3 + y3 +z3 = 3xyz                                                   =33a-2b2b-5c5c-3a

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Question 20:

Put 3a-2b=x, 2b-5c=y and 5c-3a=z.We have:x+y+z = 3a-2b+2b-5c+5c-3a=0Now,3a-2b3+2b-5c3+5c-3a3=x3+y3+z3                                                   =3xyz    Here, x+y+z=0. So, x3 + y3 +z3 = 3xyz                                                   =33a-2b2b-5c5c-3a

Answer:

Put 5a-7b=x, 9c-5a=z and 7b-9c=y.Here, x+y+z = 5a - 7b + 9c-5a+7b-9c=0Thus, we have:5a-7b3+9c-5a3+7b-9c3=x3+z3+y3                                                   =3xzy   When x+y+z=0, x3+y3+z3 = 3xyz.                                                   =3 5a-7b9c-5a7b-9c

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Question 21:

Put 5a-7b=x, 9c-5a=z and 7b-9c=y.Here, x+y+z = 5a - 7b + 9c-5a+7b-9c=0Thus, we have:5a-7b3+9c-5a3+7b-9c3=x3+z3+y3                                                   =3xzy   When x+y+z=0, x3+y3+z3 = 3xyz.                                                   =3 5a-7b9c-5a7b-9c

Answer:

We have:a3b-c3+b3c-a3+c3a-b3 = ab-c3+bc-a3+ca-b3Put ab-c = x      bc-a = y      ca-b = z Here, x+y+z = ab-c+bc-a+ca-b              =ab - ac + bc - ab + ac - bc              =0Thus, we have:a3b-c3+b3c-a3+c3a-b3 =x3 + y3+ z3                                                   =3xyz      When x+y+z =0, x3 + y3+ z3 =3xyz.                                                   =3 ab-cbc-aca-b                                                   =3abca-bb-cc-a

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Question 22:

We have:a3b-c3+b3c-a3+c3a-b3 = ab-c3+bc-a3+ca-b3Put ab-c = x      bc-a = y      ca-b = z Here, x+y+z = ab-c+bc-a+ca-b              =ab - ac + bc - ab + ac - bc              =0Thus, we have:a3b-c3+b3c-a3+c3a-b3 =x3 + y3+ z3                                                   =3xyz      When x+y+z =0, x3 + y3+ z3 =3xyz.                                                   =3 ab-cbc-aca-b                                                   =3abca-bb-cc-a

Answer:

(i) (–12)+ 7+ 53
We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=-12,y=7,z=5
-123+73+53=-12+7+5-122+72+52-7-12-35+60+3-12×35=0-1260=-1260

(ii) (28)3 + (–15)3 + (–13)3

We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=-28,y=-15,z=-13
283+-153+-133=28-15-13282+-152+-132-28-15--15-13-28-13+3×28-15-13=0+16380=16380
 

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Question 23:

(i) (–12)+ 7+ 53
We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=-12,y=7,z=5
-123+73+53=-12+7+5-122+72+52-7-12-35+60+3-12×35=0-1260=-1260

(ii) (28)3 + (–15)3 + (–13)3

We knowx3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zxx3+y3+z3=x+y+zx2+y2+z2-xy-yz-zx+3xyzHere, x=-28,y=-15,z=-13
283+-153+-133=28-15-13282+-152+-132-28-15--15-13-28-13+3×28-15-13=0+16380=16380
 

Answer:

a+b+c3=a+b+c3=a+b3+c3+3a+bca+b+ca+b+c3=a3+b3+3aba+b+c3+3a+bca+b+ca+b+c3-a3+b3-c3=3aba+b+3a+bca+b+ca+b+c3-a3+b3-c3=3a+bab+ca+cb+c2a+b+c3-a3+b3-c3=3a+bab+c+cb+ca+b+c3-a3+b3-c3=3a+bb+ca+c
 

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Question 24:

a+b+c3=a+b+c3=a+b3+c3+3a+bca+b+ca+b+c3=a3+b3+3aba+b+c3+3a+bca+b+ca+b+c3-a3+b3-c3=3aba+b+3a+bca+b+ca+b+c3-a3+b3-c3=3a+bab+ca+cb+c2a+b+c3-a3+b3-c3=3a+bab+c+cb+ca+b+c3-a3+b3-c3=3a+bb+ca+c
 

Answer:

a+b+c=0a3+b3+c3=3abc

Thus, we have:
a2bc+b2ca+c2ab=a3+b3+c3abc
                     =3abcabc=3

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Question 25:

a+b+c=0a3+b3+c3=3abc

Thus, we have:
a2bc+b2ca+c2ab=a3+b3+c3abc
                     =3abcabc=3

Answer:

a + b + c = 9
a+b+c2=92=81a2+b2+c2+2ab+bc+ca=8135+2ab+bc+ca=81ab+bc+ca=23
We know,
(a+ b3 + c3 – 3abc) = a+b+ca2+b2+c2-ab-bc-ca
=935-23=108
 



Page No 138:

Question 1:

a + b + c = 9
a+b+c2=92=81a2+b2+c2+2ab+bc+ca=8135+2ab+bc+ca=81ab+bc+ca=23
We know,
(a+ b3 + c3 – 3abc) = a+b+ca2+b2+c2-ab-bc-ca
=935-23=108
 

Answer:

(c) 2

x+1 is a factor of 2x2+kx.So, -1 is a zero of 2x2+kx.Thus, we have:2×-12+k×-1=02-k=0k=2

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Question 2:

(c) 2

x+1 is a factor of 2x2+kx.So, -1 is a zero of 2x2+kx.Thus, we have:2×-12+k×-1=02-k=0k=2

Answer:

(249)2 – (248)2
We know
a2-b2=a+ba-bSo, 2492-2482=249-248249+248=497
Hence, the correct answer is option (d).

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Question 3:

(249)2 – (248)2
We know
a2-b2=a+ba-bSo, 2492-2482=249-248249+248=497
Hence, the correct answer is option (d).

Answer:

 (c) 0

   xy+yx=-1x2+y2xy=-1
x2 + y2 = -xy
x2+ y2 + xy = 0

Thus, we have:
x3-y3=x-yx2+y2+xy
         =x-y×0=0

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Question 4:

 (c) 0

   xy+yx=-1x2+y2xy=-1
x2 + y2 = -xy
x2+ y2 + xy = 0

Thus, we have:
x3-y3=x-yx2+y2+xy
         =x-y×0=0

Answer:

(d) 3abc

  a+b+c=0a+b=-c

a+b3=-c3a3+b3+3aba+b=-c3a3+b3+3ab-c=-c3a3+b3+c3=3abc



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Question 5:

(d) 3abc

  a+b+c=0a+b=-c

a+b3=-c3a3+b3+3aba+b=-c3a3+b3+3ab-c=-c3a3+b3+c3=3abc

Answer:

3x+12 3x-12=9x2-p
9x2-14=9x2-p                    a2-b2=a+ba-bp=14
Hence, the correct answer is option (c).

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Question 6:

3x+12 3x-12=9x2-p
9x2-14=9x2-p                    a2-b2=a+ba-bp=14
Hence, the correct answer is option (c).

Answer:

(x + 3)3
=x3+33+9xx+3=x3+27+9x2+27x
So, the coefficient of x in (x + 3)is 27.
Hence, the correct answer is option (d). 

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Question 7:

(x + 3)3
=x3+33+9xx+3=x3+27+9x2+27x
So, the coefficient of x in (x + 3)is 27.
Hence, the correct answer is option (d). 

Answer:

(x + y)3 – (xy3)
=x3+y3+3xyx+y-x3+y3=3xyx+y
Thus, the factors of (x + y)3 – (xy3) are 3xy and (x + y).
Hence, the correct answer is option (d). 
 

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Question 8:

(x + y)3 – (xy3)
=x3+y3+3xyx+y-x3+y3=3xyx+y
Thus, the factors of (x + y)3 – (xy3) are 3xy and (x + y).
Hence, the correct answer is option (d). 
 

Answer:

25x2-1+1+5x2=5x-15x+1+1+5x2=5x+15x-1+1+5x=5x+110x
So, the factors of 25x2-1+1+5x2 are (5x + 1) and 10x
Hence, the correct answer is option (d). 

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Question 9:

25x2-1+1+5x2=5x-15x+1+1+5x2=5x+15x-1+1+5x=5x+110x
So, the factors of 25x2-1+1+5x2 are (5x + 1) and 10x
Hence, the correct answer is option (d). 

Answer:

(b) 5

x+5 is a factor of px=x3-20x+5k. p-5=0-53-20×-5+5k=0-125+100+5k=05k=25k=5

Page No 139:

Question 10:

(b) 5

x+5 is a factor of px=x3-20x+5k. p-5=0-53-20×-5+5k=0-125+100+5k=05k=25k=5

Answer:

(b) m = 7, n = −18

Let:
px=x3+10x2+mx+n
Now,
x+2=0x=-2
(x + 2) is a factor of p(x).
So, we have p(-2)=0
-23+10×-22+m×-2+n=0-8+40-2m+n=032-2m+n=02m-n=32                            .....i
Now,
x-1=0x=1
Also, 
(x - 1) is a factor of p(x).
We have:
p(1) = 0
13+10×12+m×1+n=01+10+m+n=011+m+n=0m+n=-11                              .....iiFrom i and ii, we get:3m=21m=7
By substituting the value of m in (i), we get n = −18.
∴ m = 7 and n = −18

Page No 139:

Question 11:

(b) m = 7, n = −18

Let:
px=x3+10x2+mx+n
Now,
x+2=0x=-2
(x + 2) is a factor of p(x).
So, we have p(-2)=0
-23+10×-22+m×-2+n=0-8+40-2m+n=032-2m+n=02m-n=32                            .....i
Now,
x-1=0x=1
Also, 
(x - 1) is a factor of p(x).
We have:
p(1) = 0
13+10×12+m×1+n=01+10+m+n=011+m+n=0m+n=-11                              .....iiFrom i and ii, we get:3m=21m=7
By substituting the value of m in (i), we get n = −18.
∴ m = 7 and n = −18

Answer:

(b) 9984

104×96=100+4100-4
         =1002-42=10000-16=9984

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Question 12:

(b) 9984

104×96=100+4100-4
         =1002-42=10000-16=9984

Answer:

(c) 93940

305×308=300+5×300+8             =3002+300×5+8+5×8             =90000+3900+40             =93940

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Question 13:

(c) 93940

305×308=300+5×300+8             =3002+300×5+8+5×8             =90000+3900+40             =93940

Answer:

(b) 39951

  207×193=200+7200-7=2002-72=40000-49=39951

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Question 14:

(b) 39951

  207×193=200+7200-7=2002-72=40000-49=39951

Answer:

(a) (2a + b + 2)2

    4a2+b2+4ab+8a+4b+4=4a2+b2+4+4ab+4b+8a=2a2+b2+22+2×2a×b+2×b×2+2×2a×2=2a+b+22

Page No 139:

Question 15:

(a) (2a + b + 2)2

    4a2+b2+4ab+8a+4b+4=4a2+b2+4+4ab+4b+8a=2a2+b2+22+2×2a×b+2×b×2+2×2a×2=2a+b+22

Answer:

(c) (x − 7)(x + 3)

x2-4x-21=x2-7x+3x-21
             =xx-7+3x-7=x-7x+3

Page No 139:

Question 16:

(c) (x − 7)(x + 3)

x2-4x-21=x2-7x+3x-21
             =xx-7+3x-7=x-7x+3

Answer:

(c) (2x + 3) (2x − 1)

4x2+4x-3=4x2+6x-2x-3
             =2x2x+3-12x+3=2x+32x-1

Page No 139:

Question 17:

(c) (2x + 3) (2x − 1)

4x2+4x-3=4x2+6x-2x-3
             =2x2x+3-12x+3=2x+32x-1

Answer:

(b) (2x + 5)(3x + 1)

6x2+17x+5=6x2+15x+2x+5
              =3x2x+5+12x+5=2x+53x+1

Page No 139:

Question 18:

(b) (2x + 5)(3x + 1)

6x2+17x+5=6x2+15x+2x+5
              =3x2x+5+12x+5=2x+53x+1

Answer:

(c) x3 − 2x2 − x − 2

Let:
fx=x3-2x2+x+2
By the factor theorem, (x + 1) will be a factor of f (x) if f (−1) = 0.
We have:
f-1=-13-2×-12+-1+2        =-1-2-1+2        =-20
Hence, (x + 1) is not a factor of fx=x3-2x2+x+2.

Now,
Let:
fx=x3+2x2+x-2

By the factor theorem, (x + 1) will be a factor of f (x) if f (-1) = 0.
We have:
f-1=-13+2×-12+-1-2        =-1+2-1-2        =-20
Hence, (x + 1) is not a factor of fx=x3+2x2+x-2.

Now,
Let:
fx=x3+2x2-x-2

By the factor theorem, (x + 1) will be a factor of f (x) if f (-1) = 0.
We have:
f-1=-13+2×-12--1-2        =-1+2+1-2        =0
Hence, (x + 1) is a factor of fx=x3+2x2-x-2.



Page No 140:

Question 19:

(c) x3 − 2x2 − x − 2

Let:
fx=x3-2x2+x+2
By the factor theorem, (x + 1) will be a factor of f (x) if f (−1) = 0.
We have:
f-1=-13-2×-12+-1+2        =-1-2-1+2        =-20
Hence, (x + 1) is not a factor of fx=x3-2x2+x+2.

Now,
Let:
fx=x3+2x2+x-2

By the factor theorem, (x + 1) will be a factor of f (x) if f (-1) = 0.
We have:
f-1=-13+2×-12+-1-2        =-1+2-1-2        =-20
Hence, (x + 1) is not a factor of fx=x3+2x2+x-2.

Now,
Let:
fx=x3+2x2-x-2

By the factor theorem, (x + 1) will be a factor of f (x) if f (-1) = 0.
We have:
f-1=-13+2×-12--1-2        =-1+2+1-2        =0
Hence, (x + 1) is a factor of fx=x3+2x2-x-2.

Answer:

(d) (3x + 2)(x2 + 1)

3x3+2x2+3x+2=x23x+2+13x+2
                   =3x+2x2+1

Page No 140:

Question 20:

(d) (3x + 2)(x2 + 1)

3x3+2x2+3x+2=x23x+2+13x+2
                   =3x+2x2+1

Answer:

(d) 3

a+b+c=0a3+b3+c3=3abc

Thus, we have:
a2bc+b2ca+c2ab=a3+b3+c3abc
                     =3abcabc=3

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Question 21:

(d) 3

a+b+c=0a3+b3+c3=3abc

Thus, we have:
a2bc+b2ca+c2ab=a3+b3+c3abc
                     =3abcabc=3

Answer:

(a) 108

x3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zx
                   =x+y+zx+y+z2-3xy+yz+zx=9×81-3×23=9×12=108

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Question 22:

(a) 108

x3+y3+z3-3xyz=x+y+zx2+y2+z2-xy-yz-zx
                   =x+y+zx+y+z2-3xy+yz+zx=9×81-3×23=9×12=108

Answer:

    ab+ba=-1a2+b2ab=-1
a2 + b2 = -ab
a2 + b2 + ab = 0

Thus, we have:
a3-b3=a-ba2+b2+ab
         =a-b×0=0



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