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Page No 84:

Question 1:

Add:
8ab, −5ab, 3ab, −ab

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:

    8ab-5ab    3ab  -ab
________

   5ab

Page No 84:

Question 2:

Add:
7x, −3x, 5x, −x, −2x

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:

    7x-3x    5x  -x-2x
_____
  6x

Page No 84:

Question 3:

Add:
3a − 4b + 4c, 2a + 3b − 8c, a − 6b + c

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:

3a - 4b + 4c2a + 3b - 8c  a - 6b +  c
___________
6a -7b-3c

Page No 84:

Question 4:

Add:
5x − 8y + 2z, 3z − 4y − 2x, 6yzx and 3x − 2z − 3y

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:

    5x-8y+2z-2x-4y+3z  -x+6y- z   3x-3y-2z   5x-9y+2z

Page No 84:

Question 5:

Add:
6ax − 2by + 3cz, 6by − 11axcz and 10cz − 2ax − 3by

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:

       6ax-2by+ 3cz-11ax+ 6by-  cz  -2ax-3by+10cz   - 7ax+  by+12cz

Page No 84:

Question 6:

Add:
2x3 − 9x2 + 8, 3x2 − 6x − 5, 7x3 − 10x + 1 and 3 + 2x − 5x2 − 4x3

Answer:

On arranging the terms of the given expressions in the descending powers of x and adding column-wise:

     2x3- 9x2+  0x+8   0x3+ 3x2 - 6x-5   7x3+ 0x2-10x+1-4x3-5x2+ 2x+3     5x3-11x2-14x+7

Page No 84:

Question 7:

Add:
6p + 4q r + 3, 2r − 5p − 6, 11q − 7p + 2r − 1 and 2q − 3r + 4

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise:

    6p+  4q - r+3-5p+  0q+2r-6-7p+11q+2r-1   0p+  2q-3r+4-6p+17q+0r+0=-6p+17q

Page No 84:

Question 8:

Add:
4x2 − 7xy + 4y2 − 3, 5 + 6y2 − 8xy + x2 and 6 − 2xy + 2x2 − 5y2

Answer:

On arranging the terms of the given expressions in the descending powers of x and adding column-wise:

4x2+4y2-7xy-3 x2+ 6y2-8xy+02x2-5y2-2xy+67x2+5y2-17xy+3

Page No 84:

Question 9:

Subtract:
3a2b from −5a2b

Answer:

On arranging the terms of the given expressions in the descending powers of x and subtracting:

-5a2b   3a2b--8a2b

Page No 84:

Question 10:

Subtract:
−8pq from 6pq

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:

                     6pq   -8pq   +                  14pq

Page No 84:

Question 11:

Subtract:
−2abc from −8abc

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:

     -8abc -2abc  +     -6abc

Page No 84:

Question 12:

Subtract:
−16p from −11p

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:

  -11p -16p +     5p

Page No 84:

Question 13:

Subtract:
2a − 5b + 2c − 9 from 3a − 4b c + 6

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:

     3a-4b- c +6    2a-5b+2c-9 -    +     -    +      a + b-3c+15

Page No 84:

Question 14:

Subtract:
−6p + q + 3r + 8 from p − 2q − 5r − 8

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:

      p-2q-5r-8-6p+  q+3r+8+    -     -   -   7p-3q-8r-16   

Page No 84:

Question 15:

Subtract:
x3 + 3x2 − 5x + 4 from 3x3x2 + 2x − 4

Answer:

On arranging the terms of the given expressions in the descending powers of x and subtracting column-wise:

   3x3-x2+2x-4  x3+3x2-5x+4-    -    +    -  2x3-4x2+7x-8

Page No 84:

Question 16:

Subtract:
5y4 − 3y3 + 2y2 + y − 1 from 4y4 − 2y3 − 6y2y + 5

Answer:

Arranging the terms of the given expressions in the descending powers of x and subtracting column-wise:

     4y4-2y3-6y2-y+5   5y4-3y3+2y2+y-1 -     +    -     -   +   -y4+ y3- 8y2-2y+6

Page No 84:

Question 17:

Subtract:
4p2 + 5q2 − 6r2 + 7 from 3p2 − 4q2 − 5r2 − 6

Answer:

Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:

     3p2-4q2-5r2-6 4p2+5q2-6r2+7-    -      +      - -p2-9q2+r2-13

Page No 84:

Question 18:

What must be subtracted from 3a2 − 6ab − 3b2 − 1 to get 4a2 − 7ab − 4b2 + 1?

Answer:

Let the required number be x.
(3a2-6ab-3b2-1)-x=4a2-7ab-4b2+1
(3a2-6ab-3b2-1)-(4a2-7ab-4b2+1)=x

    3a2-6ab-3b2-1 4a2-7ab-4b2+1-     +    +     -  -a2+ ab+   b2-2

∴ Required number = -a2+ab+b2-2

Page No 84:

Question 19:

The two adjacent sides of a rectangle are 5x2 − 3y2 and x2 + 2xy. Find the perimeter.

Answer:

Sides of the rectangle are l and b.
l=5x2-3y2b=x2+2xy
Perimeter of the rectangle is (2l+2b).

Perimeter = 2(5x2-3y2) + 2(x2+2xy)                =10x2-6y2+2x2+4xy     10x2-6y2   2x2         + 4xy    12x2-6y2+4xyHence, the perimeter of the rectangle is 12x2-6y2+4xy.

Page No 84:

Question 20:

The perimeter of a triangle is 6p2 − 4p + 9 and two of its sides are p2 − 2p + 1 and 3p2 − 5p + 3. Find the third side of the triangle.

Answer:

Let a, b and c be the three sides of the triangle.

∴ Perimeter of the triangle =(a+b+c)

Given perimeter of the triangle = 6p2-4p+9
One side (a)  = p2-2p+1
Other side (b) = 3p2-5p+3
Perimeter = (a+b+c)
(6p2-4p+9)=(p2-2p+1)+(3p2-5p+3)+c6p2-4p+9-p2+2p-1-3p2+5p-3=c(6p2-p2-3p2)+(-4p+2p+5p)+(9-1-3)=c2p2+3p+5 = c

Thus, the third side is 2p2+3p+5.



Page No 87:

Question 1:

Find the product:
(5x + 7) × (3x + 4)

Answer:

By horizontal method:
(5x+7)×(3x+4)=5x(3x+4)+7(3x+4)=15x2+20x+21x+28=15x2+41x+28

Page No 87:

Question 2:

Find the product:
(4x + 9) × (x − 6)

Answer:

By horizontal method:

(4x+9)×(x-6)=4x(x-6)+9(x-6)=4x2-24x+9x-54=4x2-15x-54

Page No 87:

Question 3:

Find the product:
(2x + 5) × (4x − 3)

Answer:

By horizontal method:

(2x+5)×(4x-3)=2x(4x-3)+5(4x-3)=8x2-6x+20x-15=8x2+14x-15

Page No 87:

Question 4:

Find the product:
(3y − 8) × (5y − 1)

Answer:

By horizontal method:

(3y-8)×(5y-1)=3y(5y-1)-8(5y-1)=15y2-3y-40y+8=15y2-43y+8

Page No 87:

Question 5:

Find the product:
(7x + 2y) × (x + 4y)

Answer:

By horizontal method:

(7x+2y)×(x+4y)=7x(x+4y)+2y(x+4y)=7x2+28xy+2xy+8y2=7x2+30xy+8y2

Page No 87:

Question 6:

Find the product:
(9x + 5y) × (4x + 3y)

Answer:

By horizontal method:

(9x+5y)×(4x+3y)9x(4x+3y)+5y(4x+3y)=36x2+27xy+20xy+15y2=36x2+47xy+15y2

Page No 87:

Question 7:

Find the product:
(3m − 4n) × (2m − 3n)

Answer:

By horizontal method:

(3m-4n)×(2m-3n)=3m(2m-3n)-4n(2m-3n)=6m2-9mn-8mn+12n2=6m2-17mn+12n2

Page No 87:

Question 8:

Find the product:
(x2a2) × (xa)

Answer:

By horizontal method:

(x2-a2)×(x-a)=x2(x-a)-a2(x-a)=x3-ax2-a2x+a3
i.e (x3+a3)-ax(x-a)

Page No 87:

Question 9:

Find the product:
(x2y2) × (x + 2y)

Answer:

By horizontal method:

(x2-y2)×(x+2y)=x2(x+2y)-y2(x+2y)=x3+2x2y-xy2-2y3i.e(x3-2y3)+xy(2x-y)

Page No 87:

Question 10:

Find the product:
(3p2 + q2) × (2p2 − 3q2)

Answer:

By horizontal method:

(3p2+q2)×(2p2-3q2)=3p2(2p2-3q2)+q2(2p2-3q2)=6p4-9p2q2+2p2q2-3q4i.e6p4-7p2q2-3q4

Page No 87:

Question 11:

Find the product:
(2x2 − 5y2) × (x2 + 3y2)

Answer:

By horizontal method:

(2x2-5y2)×(x2+3y2)=2x2(x2+3y2)-5y2(x2+3y2)=2x4+6x2y2-5x2y2-15y4=2x4+x2y2-15y4

Page No 87:

Question 12:

Find the product:
(x3y3) × (x2 + y2)

Answer:

By horizontal method:

(x3-y3)×(x2+y2)=x3(x2+y2)-y3(x2+y2)=x5+x3y2-x2y3-y5=x5-y5+x2y2x-y

Page No 87:

Question 13:

Find the product:
(x4 + y4) × (x2y2)

Answer:

By horizontal method:
(x4+y4)×(x2-y2)=x4(x2-y2)+y4(x2-y2)=x6-x4y2+y4x2-y6=x6-y6-x2y2x2-y2

Page No 87:

Question 14:

Find the product:
x4+1x4×x+1x

Answer:

By horizontal method:

x4+1x4×x+1x=x4x+1x+1x4x+1x=x5+x3+1x3+1x5i.e x3(x2+1)+1x31+1x2

Page No 87:

Question 15:

Find the product:
(x2 − 3x + 7) × (2x + 3)

Answer:

By horizontal method:

(x2-3x+7)×(2x+3)=2x(x2-3x+7)+3(x2-3x+7)=2x3-6x2+14x+3x2-9x+21=2x3-3x2+5x+21

Page No 87:

Question 16:

Find the product:
(3x2 + 5x − 9) × (3x − 5)

Answer:

By horizontal method:
(3x2+5x-9)×(3x-5)=3x(3x2+5x-9)-5(3x2+5x-9)=9x3+15x2-27x-15x2-25x+45=9x3-52x+45

Page No 87:

Question 17:

Find the product:
(x2xy + y2) × (x + y)

Answer:

By horizontal method:
(x2-xy+y2)×(x+y)=x(x2-xy+y2)+y(x2-xy+y2)=x3-x2y+y2x+x2y-xy2+y3=x3+y3

Page No 87:

Question 18:

Find the product:
(x2 + xy + y2) × (xy)

Answer:

By horizontal method:

(x2+xy+y2)×(x-y)x(x2+xy+y2)-y(x2+xy+y2)=x3+x2y+xy2-x2y-xy2-y3=x3-y3

Page No 87:

Question 19:

Find the product:
(x3 − 2x2 + 5) × (4x − 1)

Answer:

By horizontal method:

(x3-2x2+5)×(4x-1)=4x(x3-2x2+5)-1(x3-2x2+5)=4x4-8x3+20x-x3+2x2-5=4x4-9x3+2x2+20x-5

Page No 87:

Question 20:

Find the product:
(9x2x + 15) × (x2 − 3)

Answer:

By horizontal method:

(9x2-x+15)×(x2-3)=x2(9x2-x+15)-3(9x2-x+15)=9x4-x3+15x2-27x2+3x-45=9x4-x3-12x2+3x-45

Page No 87:

Question 21:

Find the product:
(x2 − 5x + 8) × (x2 + 2)

Answer:

By horizontal method:

(x2-5x+8)×(x2+2)=x2(x2-5x+8)+2(x2-5x+8)=x4-5x3+8x2+2x2-10x+16=x4-5x3+10x2-10x+16

Page No 87:

Question 22:

Find the product:
(x3 − 5x2 + 3x + 1) × (x3 − 3)

Answer:

By horizontal method:

(x3-5x2+3x+1)×(x2-3)=x2(x3-5x2+3x+1)-3(x3-5x2+3x+1)=x5-5x4+3x3+x2-3x3+15x2-9x-3=x5-5x4+16x2-9x-3

Page No 87:

Question 23:

Find the product:
(3x + 2y − 4) × (xy + 2)

Answer:

By horizontal method:

(3x+2y-4)×(x-y+2)x(3x+2y-4)-y(3x+2y-4)+2(3x+2y-4)=3x2+2xy-4x-3xy-2y2+4y+6x+4y-8=3x2-2y2-xy+2x+8y-8

Page No 87:

Question 24:

Find the product:
(x2 − 5x + 8) × (x2 + 2x − 3)

Answer:

By horizontal method:

(x2-5x+8)×(x2+2x-3)=x2(x2-5x+8)+2x(x2-5x+8)-3(x2-5x+8)=x4-5x3+8x2+2x3-10x2+16x-3x2+15x-24=x4-3x3-5x2+31x-24

Page No 87:

Question 25:

Find the product:
(2x2 + 3x − 7) × (3x2 − 5x + 4)

Answer:

By horizontal method:

(2x2+3x-7)×(3x2-5x+4)=2x2(3x2-5x+4)+3x(3x2-5x+4)-7(3x2-5x+4)=6x4-10x3+8x2+9x3-15x2+12x-21x2+35x-28=6x4-x3-28x2+47x-28

Page No 87:

Question 26:

Find the product:
(9x2x + 15) × (x2x − 1)

Answer:

By horizontal method:

(9x2-x+15)×(x2-x-1)=x2(9x2-x+15)-x(9x2-x+15)-1(9x2-x+15)=9x4-x3+15x2-9x3+x2-15x-9x2+x-15=9x4-10x3+7x2-14x-15



Page No 90:

Question 1:

Divide:
(i) 24x2y3 by 3xy
(ii) 36xyz2 by −9xz
(iii) −72x2y2z by −12xyz
(iv) −56mnp2 by 7mnp

Answer:

(i) 24x2y3 by 3xy

24x2y3 3xy243x2-1y3-18xy2.

Therefore, the quotient is 8xy2.

(ii) 36xyz2 by −9xz

36xyz2 -9xz36-9x1-1y1-0z2-1-4yz


Therefore, the quotient is 4yz.

(iii)

-72x2y2z by -12xyz-72x2y2z -12xyz-72-12x2-1y2-1z1-16xy

Therefore, the quotient is 6xy.

(iv) −56mnp2 by 7mnp


-56mnp2 7mnp-567m1-1n1-1p2-1-8p

Therefore, the quotient is −8p.

Page No 90:

Question 2:

Divide:
(i) 5m3 − 30m2 + 45m by 5m
(ii) 8x2y2 − 6xy2 + 10x2y3 by 2xy
(iii) 9x2y − 6xy + 12xy2 by − 3xy
(iv) 12x4 + 8x3 − 6x2 by − 2x2

Answer:

(i) 5m3 − 30m2 + 45m by 5m

(5m3-30m2 +45m) ÷ 5m5m35m-30m25m+ 45m 5mm2 -6m + 9

Therefore, the quotient is m2 6m + 9.

(ii) 8x2y2 − 6xy2 + 10x2y3 by 2xy

(8x2y2 - 6xy2 + 10x2y3 )÷ 2xy8x2y22xy- 6xy22xy+ 10x2y3 2xy4xy - 3y + 5xy2

Therefore, the quotient is 4xy 3y + 5xy2.

(iii) 9x2y − 6xy + 12xy2 by − 3xy

(9x2y - 6xy + 12xy2 )÷ -3xy9x2y-3xy-6xy-3xy+12xy2 -3xy-3x + 2 -4y

Therefore, the quotient is −3x + 2 4y.

(iv) 12x4 + 8x3 − 6x2 by − 2x2

(12x4 + 8x3 - 6x2 )÷ -2x212x4 -2x2+8x3-2x2-6x2-2x2-6x2-4x+32 

Therefore the quotient is −6x2 4x + 3.

Page No 90:

Question 3:

Write the quotient and remainder when we divide:
(x2 − 4x + 4) by (x − 2)

Answer:



Therefore, the quotient is x-2 and the remainder is 0.

Page No 90:

Question 4:

Write the quotient and remainder when we divide:
(x2 − 4) by (x + 2)

Answer:



Therefore, the quotient is x−2 and the remainder is 0.

Page No 90:

Question 5:

Write the quotient and remainder when we divide:
(x2 + 12x + 35) by (x + 7)

Answer:

(x2 + 12x + 35) by (x + 7)



Therefore, the quotient is x+5 and the remainder is 0.

Page No 90:

Question 6:

Write the quotient and remainder when we divide:
(15x2 + x − 6) by (3x + 2)

Answer:



Therefore, the quotient is 5x-3 and the remainder is 0.

Page No 90:

Question 7:

Write the quotient and remainder when we divide:
(14x2 − 53x + 45) by (7x − 9)

Answer:



Therefore, the quotient is 2x - 5 and the remainder is 0.

Page No 90:

Question 8:

Write the quotient and remainder when we divide:
(6x2 − 31x + 47) by (2x − 5)

Answer:



Therefore, the quotient is 3x - 8 and the remainder is 7.

Page No 90:

Question 9:

Write the quotient and remainder when we divide:
(2x3 + x2 − 5x − 2) by (2x + 3)

Answer:



Therefore, the quotient is x2-x-1 and the remainder is 1.

Page No 90:

Question 10:

Write the quotient and remainder when we divide:
(x3 + 1) by (x + 1)

Answer:



Therefore, the quotient is x2-x+1 and the remainder is 0.

Page No 90:

Question 11:

Write the quotient and remainder when we divide:
(x4 − 2x3 + 2x2 + x + 4) by (x2 + x + 1)

Answer:



Therefore, the quotient is ( x2 - 3x + 4) and remainder is 0.

Page No 90:

Question 12:

Write the quotient and remainder when we divide:
(x3 − 6x2 + 11x − 6) by (x2 − 5x + 6)

Answer:



Therefore, the quotient is (x-1) and the remainder is 0.

Page No 90:

Question 13:

Write the quotient and remainder when we divide:
(5x3 − 12x2 + 12x + 13) by (x2 − 3x + 4)

Answer:



Therefore, the quotient is ( 5x+ 3) and the remainder is (x + 1).

Page No 90:

Question 14:

Write the quotient and remainder when we divide:
(2x3 − 5x2 + 8x − 5) by (2x2 − 3x + 5)

Answer:



Therefore, the quotient is (x-1) and the remainder is 0.

Page No 90:

Question 15:

Write the quotient and remainder when we divide:
(8x4 + 10x3 − 5x2 − 4x + 1) by (2x2 + x − 1)

Answer:



Therefore, the quotient is ( 4x2+ 3x -2) and the remainder is ( x-1).



Page No 93:

Question 1:

Find each of the following products:
(i) (x + 6)(x + 6)
(ii) (4x + 5y)(4x + 5y)
(iii) (7a + 9b)(7a + 9b)
(iv) 23x+45y23x+45y
(v) (x2 + 7)(x2 + 7)
(vi) 56a2+256a2+2

Answer:

(i) We have:

(x+6)(x+6)=(x+6)2=x2+62+2×x×6                [using (a+b)2=a2+b2+2ab]=x2+36+12x

(ii) We have:

(4x+5y)(4x+5y)=(4x+5y)2=4x2+5y2+2×4x×5y          [using (a+b)2=a2+b2+2ab]=16x2+25y2+40xy

(iii) We have:
(7a+9b)(7a+9b)=(7a+9b)2=7a2+9b2+2×7a×9b            [using (a+b)2=a2+b2+2ab]=49a2+81b2+126ab

(iv) We have:
23x+45y23x+45y=23x+45y2=23x2+45y2+2×23x×45y              [using (a+b)2=a2+b2+2ab]=49x2+1625y2+1615xy


(v) We have:
(x2+7)(x2+7)=(x2+7)2=x22+72+2×x2×7             [using (a+b)2=a2+b2+2ab]=x4+49+14x2

(vi) We have:
56a2+256a2+2=56a2+22=56a22+22+2×56a2×2               [using (a+b)2=a2+b2+2ab]=2536a4+4+103a2

Page No 93:

Question 2:

Find each of the following products:
(i) (x − 4)(x − 4)
(ii) (2x − 3y)(2x − 3y)
(iii) 34x-56y34x-56y
(iv) x-3xx-3x
(v) 13x2-913x2-9
(vi) 12y2-13y12y2-13y

Answer:

(i) We have:
(x-4)(x-4)=(x-4)2=x2-2×x×4+42                   [using (a-b)2=a2-2ab+b2]=x2-8x+16

(ii) We have:
(2x-3y)(2x-3y)=(2x-3y)2=2x2-2×2x×3y+3y2                [using (a-b)2=a2-2ab+b2]=4x2-12xy+9y2

(iii) We have:
34x-56y34x-56y=34x-56y2=34x2-2×34x×56y+56y2           [using (a-b)2=a2-2ab+b2]=916x2-1512xy+2536y2

(iv) We have:
x-3xx-3x=x-3x2=x2-2×x×3x+3x2              [using (a-b)2=a2-2ab+b2]=x2-6+9x2

(v) We have:
13x2-913x2-9=13x2-92=13x22-2×13x2×9+92             [using (a-b)2=a2-2ab+b2]=19x4-6x2+81

(vi) We have:
12y2-13y12y2-13y=12y2-13y2=12y22-2×12y2×13y+13y2            [using (a-b)2=a2-2ab+b2]=14y4-13y3+19y2

Page No 93:

Question 3:

Expand:
(i) (8a + 3b)2
(ii) (7x + 2y)2
(iii) (5x + 11)2
(iv) a2+2a2
(v) 3x4+2y92
(vi) (9x − 10)2
(vii) (x2yyz2)2
(viii) xy-yx2
(ix) 3m-45n2

Answer:

We shall use the identities (a+b)2 =a2 +b2 +2ab and (a-b)2 =a2 +b2 -2ab.

(i) We have:
(8a+3b)2=8a2+2×8a×3b+3b2=64a2+48ab+9b2

(ii)We have:
(7x+2y)2=7x2+2×7x×2y+2y2=49x2+28xy+4y2

(iii) We have :
(5x+11)2=5x2+2×5x×11+112=25x2+110x+121

(iv) We have:
a2+2a2=a22+2×a2×2a+2a2=a42+2+4a2

(v) We have:
3x4+2y92=3x42+2×3x4×2y9+2y92=9x162+13xy+4y281

(vi) We have:
(9x-10)29x2-2×9x×10+102=81x2-180x+100

(vii) We have:
(x2y-yz2)2x2y2-2×x2y×yz2+yz22=x4y2-2x2y2z2+y2z4

(viii) We have:
xy-yx2=xy2-2×xy×yx+yx2=x2y2-2+y2x2

(ix) We have:
3m-45n2=3m2-2×3m×45n+45n2=9m2-24mn5+1625n2



Page No 94:

Question 4:

Find each of the following products:
(i) (x + 3)(x − 3)
(ii) (2x + 5)(2x − 5)
(iii) (8 + x)(8 − x)
(iv) (7x + 11y)(7x − 11y)
(v) 5x2+34y25x2-34y2
(vi) 4x5-5y34x5+5y3
(vii) x+1xx-1x
(viii) 1x+1y1x-1y

Answer:

(i) We have:

(x+3)(x-3)=x2-9                                [using (a+b)(a-b)=a2-b2]

(ii) We have:

(2x+5)(2x-5)=4x2-25                              [using (a+b)(a-b)=a2-b2]

(iii) We have:

(8+x)(8-x)=64-x2                                 [using (a+b)(a-b)=a2-b2]

(iv) We have:

(7x+11y)(7x-11y)=49x2-121y2                        [using (a+b)(a-b)=a2-b2]

(v) We have:

5x2+34y25x2-34y2=25x4-916y4                       [using (a+b)(a-b)=a2-b2]

(vi) We have:

4x5-5y34x5+5y3=16x225-25y29                     [using (a+b)(a-b)=a2-b2)]

(vii) We have:
x+1xx-1x=x2-1x2                            [using (a+b)(a-b)=a2-b2]

(viii) We have:
1x+1y1x-1y=1x2-1y2                      [using (a+b)(a-b)=a2-b2]

(ix) We have:
2a+3b2a-3b=4a2-9b2                     [using (a+b)(a-b)=a2-b2]

Page No 94:

Question 5:

Using the formula for squaring a binomial, evaluate the following:
(i) (54)2
(ii) (82)2
(iii) (103)2
(iv) (704)2

Answer:

We shall use the identity (a+b)2 =a2 +b2 +2ab.

(i)
542=(50+4)2=502+2×50×4+42=2500+400+16=2916

(ii)
822=(80+2)2=802+2×80×2+22=6400+320+4=6724

(iii)
1032=(100+3)2=1002+2×100×3+32=10000+600+9=10609

(iv)
7042=(700+4)2=7002+2×700×4+42=490000+5600+16=495616

Page No 94:

Question 6:

Using the formula for squaring a binomial, evaluate the following:
(i) (69)2
(ii) (78)2
(iii) (197)2
(iv) (999)2

Answer:

We shall use the identity (a-b)2 = a2 +b2 -2ab.

(i)
692=(70-1)2=702-2×70×1+1=4900-140+1=4761

(ii)
782=(80-2)2=802-2×80×2+4=6400-320+4=6084

(iii)
1972=(200-3)2=2002-2×200×3+9=40000-1200+9=38809

(iv)
9992=(1000-1)2=10002-2×1000×1+1=1000000-2000+1=998001

Page No 94:

Question 7:

Find the value of:
(i) (82)2 − (18)2
(ii) (128)2 − (72)2
(iii) 197 × 203
(iv) 198×198-102×10296
(v) (14.7 × 15.3)
(vi) (8.63)2 − (1.37)2

Answer:

We shall use the identity (a-b) (a+b)=a2 - b2.

(i)
(82)2-(18)2=(82-18)(82+18)=(64)(100)=6400

(ii)
(128)2-(72)2=(128-72)(128+72)=(56)(200)=11200

(iii)
197×203=(200-3)(200+3)=2002-32=40000-9=39991

(iv)
198×198-102×10296=1982-102296=(198-102)(198+102)96=(96)(300)96=300

(v)
(14.7×15.3)=(15-0.3)×(15+0.3)=(15)2-(0.3)2=225-0.09=224.91

(vi)
(8.63)2-(1.37)2=(8.63-1.37)(8.63+1.37)=(7.26)(10)=72.6

Page No 94:

Question 8:

Find the value of the expression (9x2 + 24x + 16), when x = 12.

Answer:

9x2 + 24x + 16Given, x = 123x2 + 2 3x4 + 42  3x + 42312+4236 + 42402 = 1600

Therefore, the value of the expression (9x2 + 24x + 16), when x = 12, is 1600.

Page No 94:

Question 9:

Find the value of the expression (64x2 + 81y2 + 144xy), when x = 11 and y=43.

Answer:

64x2+81y2+144xyGiven: x=11  y =438x2 + 9y2 + 28x9y8x +9y 28(11) +9(43)288 +122100210000
                                                                                                                                               
Therefore, the value of the expression (64x2 + 81y2 + 144xy), when x = 11 and y = 43, is 10000.y=43

Page No 94:

Question 10:

Find the value of the expression (36x2 + 25y2 − 60xy), when x=23 and y=15.

Answer:

36x2+25y2-60xyx=23, y=15=6x2 + 5y2 - 26x5y=6x - 5y2=6(23) -5(15)2=4 - 12=329

Page No 94:

Question 11:

If x+1x=4, find the values of
(i) x2+1x2 and
(ii) x4+1x4.

Answer:

(i)  x+1x= 4Squaring both the sides:x+1x2= 42x2+1x2+2x1x= 16x2+1x2+2 =16x2+1x2=16-2x2+1x2= 14

Therefore, the value of  x2+1x2 is 14.

x2 + 1x2 = 14Squaring both the sides:x4 + 1x4+2x21x2= 142x4 + 1x4+2 = 196x4 + 1x4 = 196-2x4 + 1x4=194

Therefore, the value of x4 + 1x4 is 194.

Page No 94:

Question 12:

If x-1x=5, find the values of
(i) x2+1x2
(ii) x4+1x4.

Answer:

(i)  x-1x= 5Squaring both the sides:x-1x2= 52x2+1x2-2x1x= 25x2+1x2-2 =25x2+1x2=25+2x2+1x2= 27Therefore, the value of x2+1x2 is  27.

x2 + 1x2 = 27Squaring both the sides:x4 + 1x4-2x21x2= 272x4 + 1x4-2 = 729x4 + 1x4 = 729+2x4 + 1x4=731Therefore, the value of x4 + 1x4 is 731.

Page No 94:

Question 13:

Find the continued product:
(i) (x + 1)(x − 1)(x2 + 1)
(ii) (x − 3)(x + 3)(x2 + 9)
(iii) (3x − 2y)(3x + 2y)(9x2 + 4y2)
(iv) (2p + 3)(2p − 3)(4p2 + 9)

Answer:

i x+1x-1x2+1x2-x+x-1x2+1x2-1x2+1x22-122              according to the formula a2-b2 = a+ba-bx4-1.Therefore, the product of x+1x-1x2+1 is x4-1.

ii x-3x+3x2+9x2-32x2+9       according to the formula a2-b2 = a+ba-bx2-9x2+9x22-92                 according to the formula a2-b2 = a+ba-bx4-81Therefore, the product of x-3x+3x2+9 is x4-81.

iii 3x-2y3x+2y9x2+4y23x2-2y29x2+4y2        according to the formula a2-b2 = a+ba-b9x2-4y29x2+4y29x22-4y22                 according to the formula a2-b2 = a+ba-b81x4-16y4.Therefore, the product of 3x-2y3x+2y9x2+4y2 is 81x4-16y4.

iv 2p+32p-34p2+92p2-324p2+9       according to the formula a2-b2 = a+ba-b4p2-94p2+94p22-92                according to the formula a2-b2 = a+ba-b16p4-81.Therefore, the product of 2p+32p-34p2+9 is 16p4-81.

Page No 94:

Question 14:

If x + y = 12 and xy = 14, find the value of (x2 + y2).

Answer:

x+y = 12On squaring both the sides:x+y2 = 122x2+y2+2xy = 144x2+y2 = 144 - 2xyGiven:  xy = 14x2+y2 = 144 - 214x2+y2 = 144 - 28x2+y2 = 116Therefore, the value of x2+y2 is 116.

Page No 94:

Question 15:

If xy = 7 and xy = 9, find the value of (x2 + y2).

Answer:

x-y = 7On squaring both the sides:x-y2 = 72x2+y2-2xy = 49x2+y2 = 49 + 2xyGiven: xy = 9x2+y2 = 49 + 29x2+y2 = 49 + 18x2+y2 = 67.Therefore, the value of x2+y2 is 67.

Page No 94:

Question 1:

Tick (✓) the correct answer:
The sum of (6a + 4bc + 3), (2b − 3c + 4), (11b − 7a + 2c − 1) and (2c − 5a − 6) is
(a) (4a − 6b + 2)
(b) (−3a + 14b − 3c + 2)
(d) (−6a + 17b)
(d) (−6a + 6b + c −4)

Answer:

(c) (−6a + 17b)

   6a  +4b  -c   +3          +2b  -3c +4-7a +11b +2c -1-5a            +2c -6-6a +17b+ 0c +0¯



Page No 95:

Question 2:

Tick (✓) the correct answer:
(3q + 7p2 − 2r3 + 4) − (4p2 − 2q + 7r3 − 3) = ?
(a) (p2 + 2q + 5r3 + 1)
(b) (11p2 + q + 5r3 + 1)
(c) (−3p2 − 5q + 9r3 − 7)
(d) (3p2 + 5q − 9r3 +7)

Answer:

(d) (3p2 + 5q − 9r3 +7)

    7p2 +3q  -2r3 +4   4p2 -2q  +7r3 -3-       +       -      +   3p2+ 5q  -9r3 + 7    ¯

Page No 95:

Question 3:

Tick (✓) the correct answer:
(x + 5)(x − 3) = ?
(a) x2 + 5x − 15
(b) x2 − 3x − 15
(c) x2 + 2x + 15
(d) x2 + 2x − 15

Answer:

(d) x2 + 2x − 15

x+5x-3xx-3+5x-3x2-3x +5x -15x2+2x-15

Page No 95:

Question 4:

Tick (✓) the correct answer:
(2x + 3)(3x − 1) = ?
(a) (6x2 + 8x − 3)
(b) (6x2 + 7x − 3)
(c) 6x2 − 7x − 3
(d) (6x2 − 7x + 3)

Answer:

(b) (6x2 + 7x − 3)

2x+33x-12x3x-1+33x-16x2-2x +9x-36x2+7x-3

Page No 95:

Question 5:

Tick (✓) the correct answer:
(x + 4)(x + 4) = ?
(a) (x2 + 16)
(b) (x2 + 4x + 16)
(c) (x2 + 8x + 16)
(d) (x2 + 16x)

Answer:

(c) (x2 + 8x + 16)

x+4x+4x+42            (according to the formula a+b2 =a2+2ab+b2)x2+2x4+42x2+8x+16

Page No 95:

Question 6:

Tick (✓) the correct answer:
(x − 6)(x − 6) = ?
(a) (x2 − 36)
(b) (x2 + 36)
(c) (x2 − 6x + 36)
(d) (x2 − 12x + 36)

Answer:

(d) (x2 − 12x + 36)

x-6x-6x-62                             (according to the formula a-b2 =a2-2ab+b2)x2-2x6+62x2-12x+36

Page No 95:

Question 7:

Tick (✓) the correct answer:
(2x + 5)(2x − 5) = ?
(a) (4x2 + 25)
(b) (4x2 − 25)
(c) (4x2 − 10x + 25)
(d) (4x2 + 10x − 25)

Answer:

(b) (4x2 − 25)

2x+52x-52x2-52           according to the formula a+ba-b = a2-b24x2 - 25

Page No 95:

Question 8:

Tick (✓) the correct answer:
8a2b3 ÷ (−2ab) = ?
(a) 4ab2
(b) 4a2b
(c) −4ab2
(d) −4a2b

Answer:

(c) −4ab2

8a2b3 ÷ -2ab8-2a2-1b3-1-4ab2

Page No 95:

Question 9:

Tick (✓) the correct answer:
(2x2 + 3x + 1) ÷ (x + 1) = ?
(a) (x + 1)
(b) (2x + 1)
(c) (x + 3)
(d) (2x + 3)

Answer:

(b) (2x + 1)

Page No 95:

Question 10:

Tick (✓) the correct answer:
(x2 − 4x + 4) ÷ (x − 2) = ?
(a) (x − 2)
(b) (x + 2)
(c) (2 − x)
(d) (2 + x + x2)

Answer:

(a) (x − 2)

Page No 95:

Question 11:

Tick (✓) the correct answer:
(a + 1)(a − 1)(a2 + 1) = ?
(a) (a4 − 2a2 − 1)
(b) (a4a2 − 1)
(c) (a4 − 1)
(d) (a4 + 1)

Answer:

(c) (a4 − 1)

i a+1a-1a2+1a2 -12a2+1       according to the formula a2-b2 = a+ba-ba2-1a2+1a22-122                 according to the formula a2-b2 = a+ba-ba4-1

Page No 95:

Question 12:

Tick (✓) the correct answer:
1x+1y1x-1y=?
(a) 1x2-1y2
(b) 1x2+1y2
(c) 1x2+1y2-1xy
(d) 1x2-1y2+1xy

Answer:

a) 1x2-1y2

1x+1y1x-1yAccording to the formula a+ba-b=a2-b2:1x2-1y2(1x21y2)

Page No 95:

Question 13:

Tick (✓) the correct answer:
If x+1x=5, then x2+1x2=?
(a) 25
(b) 27
(c) 23
(d) 25125

Answer:

(c) 23

  x+1x= 5Squaring both the sides:x+1x2= 52x2+1x2+2x1x= 25x2+1x2+2 =25x2+1x2=25-2x2+1x2= 23

Page No 95:

Question 14:

Tick (✓) the correct answer:
If x-1x=6, then x2+1x2=?
(a) 36
(b) 38
(c) 32
(d) 36136

Answer:

(b) 38

x-1x= 6Squaring both the sides:x-1x2= 62x2+1x2-2x1x= 36x2+1x2-2 =36x2+1x2=36+2x2+1x2= 38

Page No 95:

Question 15:

Tick (✓) the correct answer:
(82)2 − (18)2 = ?
(a) 8218
(b) 6418
(c) 6400
(d) 7204

Answer:

(c) 6400

822-182=82 + 1882 - 18=10064=6400                           [using the identity (a-b)(a+b)=a2 -b2]

Page No 95:

Question 16:

Tick (✓) the correct answer:
(197 × 203) = ?
(a) 39991
(b) 39999
(c) 40009
(d) 40001

Answer:

(a) 39991

197×203200-3200+32002-3240000-939991                   [using the identity (a+b) (a-b) = a2  -b2]

Page No 95:

Question 17:

Tick (✓) the correct answer:
If (a + b) = 12 and ab = 14, then (a2 + b2) = ?
(a) 172
(b) 116
(c) 165
(d) 126

Answer:

(b) 116

a+b =12Squaring both the sides:a+b2 = 122a2+ b2 +2ab = 144a2+ b2 = 144-2aba2+ b2 = 144 -214a2+ b2 = 144 -28a2+ b2 = 116

Page No 95:

Question 18:

Tick (✓) the correct answer:
If (ab) = 7 and ab = 9, then (a2 + b2) = ?
(a) 67
(b) 31
(c) 40
(d) 58

Answer:

(a) 67

a-b =7Squaring both the sides:a-b2 = 72a2+ b2 -2ab = 49a2+ b2 = 49+2aba2+ b2 = 49 +29a2+ b2 =  49+18a2+ b2 = 67

Page No 95:

Question 19:

Tick (✓) the correct answer:
If x = 10, then the value of (4x2 + 20x + 25) = ?
(a) 256
(b) 425
(c) 625
(d) 575

Answer:

(c) 625

4x2+20x+252x2+22x5+522x + 52210+5220+52252625



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