Class 8 - Maths - Algebraic Expressions and Identities

Exercise 9.1

Question 1:

Identify the terms, their coefficients for each of the following expressions:

(i) 5xyz2 - 3zy                       (ii) 1+ x + x2                        (iii) 4x2 y2 - 4x2 y2 z2

(iv) 3 - pq + qr - rp              (v) x/2 + y/2 - xy                (vi) 0.3a - 0.6ab + 0.5b

Answer:

(i) Terms: 5xyz2 and -3zy

Coefficient in 5xyz2 is 5 and in -3zy is -3.

(ii) Terms: 1, x and x2.

Coefficient of x and coefficient of x2 is 1.

(iii) Terms: 4x2 y2, -4 x2 y2 z2 and z2.

Coefficient in 4x2 y2 is 4, coefficient of -4 x2 y2 z2 is -4 and coefficient of z2 is 1.

(iv) Terms: 3, -pq, qr and -rp

Coefficient of –pq is -1, coefficient of qr is 1 and coefficient of –rp is -1.

(v) Terms: x/2, y/2 and and -xy

Coefficient of x/2 is 1/2, coefficient of y/2 is 1/2 and coefficient of –xy is -1.

(vi) Terms: 0.3a, 0.6ab and 0.5b

Coefficient of 0.3a is 0.3, coefficient of -0.6ab is -0.6 and coefficient of 0.5b is 0.5.

Question 2:

Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories:

x + y, 1000,    x + x2 + x3 + x4,   7 + y + 5x,   2y – 3y2 ,   2y - 3y + 4y,    5x – 4y + 3xy,

4z – 15z2,   ab + bc + cd +  da,    pqr, p2 q + pq2,  2p + 2q

Answer:

(i) Since x + y contains two terms. Therefore it is binomial.

(ii) Since 1000 contains one term. Therefore it is monomial.

(iii) Since x + x2 + x3 + x4 contains four terms. Therefore it is a polynomial and it does not fit in

above three categories.

(iv) Since 7 + y + 5x contains three terms. Therefore it is trinomial.

(v) Since 2y – 3y2 contains two terms. Therefore it is binomial.

(vi) Since 2y – 3y2 + 4y3 contains three terms. Therefore it is trinomial.

(vii) Since 5x – 4y + 3xy contains three terms. Therefore it is trinomial.

(viii) Since 4x - 15z2 contains two terms. Therefore it is binomial.

(ix) Since ab + bc + cd + da contains four terms. Therefore it is a polynomial and it does not fit

in above three categories.

(x) Since pqr contains one terms. Therefore it is monomial.

(xi) Since p2 q + pq2 contains two terms. Therefore it is binomial.

(xii) Since 2p + 2q contains two terms. Therefore it is binomial.

Question 3:

Add the following:

(i) ab – bc, bc – ca, ca - ab

(ii) a - b + ab, b - c + bc, c - a + ac

(iii) 2p2 q2 - 3pq + 4, 5 + 7pq - 3p2 q2

(iv) l2 + m2, m2 + n2, n2 + l2 + 2lm + 2mn + 2nl

Answer:

(i) ab – bc + bc – ca + ca – ab = ab – ab + bc – bc + ca – ca = 0

(ii) a - b + ab + b - c + bc + c - a + ac

= a – a + b – b + ab + c – c + bc + ac

= ab + bc + ac

(iii) 2p2 q2 - 3pq + 4 + 5 + 7pq - 3p2 q2

= -p2 q2 + 4pq + 9

(iv) l2 + m2 + m2 + n2 + n2 + l2 + 2lm + 2mn + 2nl

= 2 l2 + 2m2 + 2n2 + 2lm + 2mn + 2nl

Question 4:

(a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b - 3

(b) Subtract 3xy + 5yz - 7zx from 5xy – 2yz – 2zx + 10xyz

(c) Subtract 4p2 q - 3pq + 5pq2 - 8p + 7q – 10 from 18 – 3p – 11p + 5pq – 2pq2 + 5p2 q

Answer:

(a) 12a – 9ab + 5b – 3 – (4a – 7ab + 3b + 12)

= 12a – 9ab + 5b – 3 – 4a + 7ab - 3b – 12

= 8a – 2ab + 2b - 15

(b) 5xy – 2yz – 2zx + 10xyz – (3xy + 5yz - 7zx)

= 5xy – 2yz – 2zx + 10xyz – 3xy - 5yz + 7zx

= 2xy – 7yz + 5zx + 10xyz

(c) 18 – 3p – 11p + 5pq – 2pq2 + 5p2 q – (4p2 q - 3pq + 5pq2 - 8p + 7q – 10)

= 18 – 3p – 11p + 5pq – 2pq2 + 5p2 q – 4p2 q + 3pq - 5pq2 + 8p - 7q + 10

= p2 q - 7pq2 + 8pq - 18q + 5p + 28

Exercise 9.2

Question 1:

Find the product of the following pairs of monomials:

(i) 4,7p              (ii) -4p, 7p          (iii) -4p, 7pq             (iv) 4p3, -3p            (iv) 4p, 0

Answer:

(i) 4 *7p = 4 * 7 * p = 28p

(ii) -4p * 7p = (-4 * 7) * (p * p) = -28p2

(iii) -4p * 7pq = (-4 * 7) * (p * pq) = -28p2 q

(iv) 4p3 * -3p (4 * -3) * (p3 * p) = -12p4

(iv) 4p * 0 = (4 * 0) * p = 0

Question 2:

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:

(p, q), (10m, 5n), (20x2, 5y2), (4x, 3x), (3mn, 4np)

Answer:

(i) Area of rectangle = length * breadth

= p * q = pq sq. units

(ii) Area of rectangle = length * breadth

= 10m * 5n = 50mn sq. units

(iii) Area of rectangle = length * breadth

= 20x2, 5y2 = 100 x2 y2 sq. units

(iv) Area of rectangle = length * breadth

= 4x * 3x2 = (4 * 3) * (x * x2) = 12x3 sq. units

(v) Area of rectangle = length * breadth

= 3mn * 4np = (3 * 4) * (mn * np) = 12 mn2 p sq. units

Question 3:

Complete the table of products: Answer: Question 4:

Obtain the volume of rectangular boxes with the following length, breadth and height respectively:

(i) 5a, 3a2, 7a4           (ii) 2p, 4q, 8r          (iii) xy, 2x2 y, 2xy2              (iv) a, 2b, 3c

Answer:

(i) Volume of rectangular box = length * breadth * height

= 5a * 3a2 * 7a4

= 105 a7 cubic units

(ii) Volume of rectangular box = length * breadth * height

= 2p * 4q * 8r

= (2 * 4 * 8) * (p * q * r)

= 64 pqr cubic units

(iii) Volume of rectangular box = length * breadth * height

= xy * 2x2 y * 2xy2

= (1 * 2 * 2) * (xy * x2 y * xy2)

= 4x4 y4 cubic units

(iv) Volume of rectangular box = length * breadth * height

= a * 2b * 3c

= (1 * 2 * 3) * (a * b * c) = 6abc cubic units

Question 5:

Obtain the product of:

(i) xy, yz, zx                (ii) –a, a2, a3                (iii) 2, 4y, 8y2, 16y3                           (iv) a, 2b, 3c, 6abc

(v) m, -mn, mnp

Answer:

(i) xy * yz * zx = x2 y2 z2

(ii) –a * a2 * a3 = -a6

(iii) 2 * 4y * 8y2 * 16y3 = 1024y6

(iv) a * 2b * 3c * 6abc = 36a2 b2 c2

(v) m * -mn * mnp = -m3 n2 p

Exercise 9.3

Question 1:

Carry out the multiplication of the expressions in each of the following pairs:

(i) 4p, q + r                         (ii) ab, a - b                            (iii) a + b, 7a2 b2                 (iv) a2 - 9, 4a

(v) pq + qr + rp, 0

Answer:

(i) 4p * (q + r) = 4p * q + 4p * r = 4pq + 4pr

(ii) ab * (a – b) = ab * a – ab * b = a2 b – ab2

(iii) (a + b) * 7a2 b2 = a * 7a2 b2 + b * 7a2 b2 = 7a3 b2 + 7a2 b3

(iv) (a2 – 9) * 4a = a2 * 4a – 9 * 4a = 4a3 – 36a

(v) (pq + qr + rp) * 0 = pq * 0 + qr * 0 + rp * 0 = 0

Question 2:

Complete the table: Answer: Question 3:

Find the product:

(i)  a2 * 2a22 * 4a26                     (ii) (2xy/3) * (-9x2 y2/10)                        (iii) (-10pq3/3) * (6p3 q/5) (iv) x * x2 * x3 * x4

Answer:

(i)  a2 * 2a22 * 4a26 = (1 * 2 * 4) * (a2 * a22 * a26) = 8 * a2+22+26 = 8a50

(ii) (2xy/3) * (-9x2 y2/10) = (2/3 * -9/10)  *(xy * x2 y2) = -3/5 x3 y3

(iii) (-10pq3/3) * (6p3 q/5) = (-10/3 * 6/5) * (pq3 * p3 q) = -4 p4 q4

(iv) x * x2 * x3 * x4 = x1+2+3+4 = x10

Question 4:

(a) Simplify: 3x(4x - 5) + 3 and find values for (i) x = 3 (ii) x = 1/2

(b) Simplify: a(a2 + a + 1) + 5 and find its value for (i) a = 0 (ii) a =1 (iii) a = -1.

Answer:

(a)  3x(4x - 5) + 3 = 3x * 4x – 3x * 5 + 3 = 12x2 – 15x + 3

(i) For x = 3

12(3)2 – 15 * 3 + 3 = 12 * 9 – 45 + 3 = 108 – 45 + 3 = 111 – 45 = 66

(ii) For x = 1/2

12(1/2)2 – 15 * 1/2 + 3 = 12 * 1/4 - 15/2 + 3

= 3 – 15/2 + 3

= 6 – 15/2

= (12 - 15)/2

= -3/2

(b)  a(a + a + 1) + 5 = a * a2 + a * a + a * 1 + 5 = a3 + a2 + a + 5

For a = 0,

03 + 02 + 0 + 5 = 5

For a = 1,

13 + 12 + 1 + 5 = 1 + 1 + 1 + 5 = 8

For a = -1,

(-1)3 + (-1)2 + (-1) + 5 = -1 + 1 – 1 + 5 = -2 + 6 = 4

Question 5:

(a) Add: p(p - q), q(q - r)and r(r – p)

(b) Add: 2x(z - x – y) and 2y(z -y – zx)

(c) Subtract: 3l(l - 4m + 5n) from 4l(10n – 3m + 2l)

(d) Subtract: 3a(a + b + c) – 2b(a – b + c) from 4c(-a + b + c)

Answer:

(a) p(p - q) + q(q - r)+  r(r – p) = p2 – pq + q2 – qr + r2 - rp

= p2 + q2 + r2 – pq – qr - rp

(b) 2x(z - x – y) + 2y(z - y – x) = 2xz – 2x2 – 2xy + 2yz – 2y2 – 2xy

= 2xz – 2x2 – 4xy + 2yz – 2y2

= -2x2 – 2y2 – 4xy + 2yz + 2zx

(c) 4l(10n – 3m + 2l) - 3l(l - 4m + 5n) = 40ln – 12lm + 8l2 – 3l2 + 12lm – 15ln

= 8l2 – 3l2 + 52 – 12lm + 12 lm + 40ln – 15ln

= 5l2 + 25ln

(d) 4c(-a + b + c) – [3a(a + b + c) – 2b(a – b + c)]

= -4ac + 4ab + 4c2 – (3a2 + 3ab + 3ac – 2ab + 2b2 – 2ac)

= -4ac + 4ab + 4c2 – 3a2 - 3ab - 3ac + 2ab - 2b2 + 2ac

= -3a2 – 2b2 + 4c2 – ab + 4bc + 2bc – 4ac – 3ac

= -3a2 – 2b2 + 4c2 – ab + 6bc + 2bc – 7ac

Exercise 9.4

Question 1:

Multiply the binomial

(i) (2x + 5) and (4x - 3)                    (ii) (y - 8) and (3y - 4)             (iii) (2.5l – 0.5m) and (2.5l + 0.5m)

(iv) (a + 3b) and (x + 5)                   (v) (2pq + 3q2) and (3pq – 2q2)

(vi) (3a2/4 + 3b2) and 4(a2 – 2b2/3)

Answer:

(i) (2x + 5) * (4x - 3) = 2x(4x - 3) + 5(4x - 3)

= 8x2 – 6x + 20x – 15

= 8x2 + 14x – 15

(ii) (y - 8) * (3y - 4) = y(3y - 4) - 8(3y - 4)

= 3y2 – 4y – 24y + 32

=  3y2 – 28y + 32

(iii) (2.5l – 0.5m) * (2.5l + 0.5m) = 2.5l(2.5l + 0.5m) – 0.5m(2.5l + 0.5m)

= 6.25l2 + 1.25lm – 1.25l – 0.25m2

= 6.25l2 – 0.25m2

(iv) (a + 3b) * (x + 5) = a(x + 5) + 3b(x + 5)

= ax + 5a + 3bx + 15

(v) (2pq + 3q2) * (3pq – 2q2) = 2pq(3pq – 2q2) + 3q2(3pq – 2q2)

= 6p2 q2 – 4pq3 + 9pq3 – 6q4

= 6p2 q2 + 5pq3 – 6q4

(vi) (3a2/4 + 3b2) * 4(a2 – 2b2/3) = (3a2/4 + 3b2) * (4a2 – 8b2/3)

=3a2/4(4a2 – 8b2/3) + 3b2(4a2 – 8b2/3)

= 3a2/4 * 4a2 – 3a2/4 * 8b2/3 + 3b2 * 4a2 – 3b2 * 8b2/3

= 3a4 – 2a2 b2 + 12a2 b2 – 8b4

= 3a4 + 10a2 b2 – 8b4

Question 2:

Find the product:

(i) (5 – 2x) * (3 + x)                                   (ii) (x + 7y) * (7x - y)

(iii) (a2 + b) *(a + b2)                                (iv) (p2 – q2) *(2p + q)

Answer:

(i) (5 – 2x) * (3 + x) = 5(3 + x) – 2x(3 + x)

= 15 + 5x – 6x – 2x2

= 15 – x – 2x2

(ii) (x + 7y) * (7x - y) = x(7x - y) + 7y(7x - y)

= 7x2 – xy + 49xy – 7y2

= 7x2 + 48xy – 7y2

(iii) (a2 + b) *(a + b2) = a2(a + b2) + b(a + b2)

= a3 + a2 b2 + ab + b3

(iv) (p2 – q2) * (2p + q) = p2(2p + q) – q2(2p + q)

= 2p3 + p2 q – 2pq2 – q3

Question 3:

Simplify:

(i) (x2 - 5) * (x + 5) + 25

(ii) (a2 + 5) * (b2 + 3) + 5

(iii) (t + s2) * (t2 - s)

(iv) (a + b) *(c - d) + (a - b) *(c + d) + 2(ac + bd)

(v) (x + y) * (2x + y) + (x + 2y) * (x - y)

(vi) (x + y) * (x2 – xy + y2)

(vii) (1.5x – 4y) * (1.5x + 4y + 3) – 4.5x + 12y

(viii) (a + b + c) * (a + b - c)

Answer:

(i) (x2 - 5) * (x + 5) + 25 = x2(x + 5) - 5(x + 5) + 25

= x3 + 5x2 – 5x – 25 + 25

= x3 + 5x2 – 5x

(ii) (a2 + 5) * (b2 + 3) + 5 = a2(b2 + 3) + 5(b2 + 3) + 5

= a2b2 + 3a2 + 5b2 + 15 + 5

= a2b2 + 3a2 + 5b2 + 20
(iii) (t + s2) * (t2 - s) = t(t2 - s) + s2(t2 - s)

= t3 – ts + s2 t2 – s3

(iv) (a + b) *(c - d) + (a - b) *(c + d) + 2(ac + bd)

= a(c - d) + b(c - d) + a(c + d) - b(c + d) + 2ac + 2bd

= ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd

=ac + ac + 2ac – bd – bd + bc – bc – ad + ad + 2bd

= 4ac

(v) (x + y) * (2x + y) + (x + 2y) * (x - y)

= x(2x + y) + y(2x + y) + x(x - y) + 2y(x - y)

= 2x2 + xy + 2xy + y2 + x2 – xy + 2xy – 2y2

= 3x2 + 4xy – y2

(vi) (x + y) * (x2 – xy + y2) = x(x2 – xy + y2) + y(x2 – xy + y2)

= x3 – x2 y + xy2 + x2 y – xy2 + y3

= x3 + y3

(vii) (1.5x – 4y) * (1.5x + 4y + 3) – 4.5x + 12y

= 1.5x(1.5x + 4y + 3) – 4y(1.5x + 4y + 3) – 4.5x + 12y

= 2.25x2 + 6.0xy + 4.5x – 6.0xy – 16y2 – 12y – 4.5x + 12y

= 2.25x2 – 16y2

(viii) (a + b + c) * (a + b - c) = a(a + b - c) + b(a + b - c) + c(a + b - c)

= a2 + ab – ac + ab + b2 – bc + ac + bc – c2

= a2 + b2 – c2 + 2ab

Exercise 9.5

Question 1:

Use a suitable identity to get each of the following products:

(i) (x + 3)*(x + 3)       (ii) (2y + 5)*(2y + 5)       (iii) (2a - 7)*(2a - 7)        (iv)(3a – 1/2)*(3a – 1/2)

(v) (1.1m – 0.4)*(1.1m + 0.4)                    (vi) (a2 + b2)*(-a2 + b2)          (vii) (6x - 7)*(6x + 7)

(viii) (-a + c)*(-a + c)                           (ix) (x/2 + 3y/4)*(x/2 + 3y/4)        (x) (7a – 9b)*(7a – 9b)

Answer:

(i)  (x + 3)*(x + 3) = (x + 3)2

= x2 + 2 * x * 3 + 32        [Using (a + b)2 = a2 + 2ab + b2 ]

= x2 + 6x + 9

(ii)  (2y + 5)*(2y + 5) = (2y + 5)

= (2y)2 + 2 * 2y * 5 + 52        [Using (a + b)2 = a2 + 2ab + b2 ]

= 4y2 + 20y + 25

(iii)  (2a - 7)*(2a - 7) = (2a - 7)2

= (2y)2 - 2 * 2y * 7 + 72        [Using (a - b)2 = a2 - 2ab + b2 ]

= 4y2 - 28y + 49

(iv) (3a – 1/2)*(3a – 1/2) = (3a – 1/2)2

= (3a)2 - 2 * 3a * 1/2 + (1/2)2        [Using (a - b)2 = a2 - 2ab + b2 ]

= 9a2 – 3a + 1/4

(v)  (1.1m – 0.4)*(1.1m + 0.4) = (1.1m)2 – (0.4)2   [Using (a - b)*(a + b) = a2 – b2]

= 1.21 m2 – 0.16

(vi)  (a2 + b2)*(-a2 + b2) = (b2 + a2)*(b2 – a2)

= (b2)2 – (a2)2   [Using (a - b)*(a + b) = a2 – b2]

= b4 – a4

(vii)  (6x - 7)*(6x + 7) = (6x)2 – (7)2   [Using (a - b)*(a + b) = a2 – b2]

= 36 x2 – 49

(viii)  (-a + c)*(-a + c) = (c - a)*(c - a)

= (c – a)2

= (c)2 - 2 * c * a + a2        [Using (a - b)2 = a2 - 2ab + b2 ]

= c2 – 2ac + a2

(ix)  (x/2 + 3y/4)*(x/2 + 3y/4) = (x/2 + 3y/4)2

= (x/2)2 + 2 * x/2 * 3y/4 + (3y/4)2      [Using (a + b)2 = a2 + 2ab + b2 ]

= x2/4 + 3xy/4 + 9y2 /16

(x)  (7a – 9b)*(7a – 9b) = (7a – 9b)2

= (7a)2 - 2 * 7a * 9b + (9b)2        [Using (a - b)2 = a2 - 2ab + b2 ]

= 49a2 – 126ab + 91b2

Question 2:

Use the identity (x + a)(x + b) = x2 + (a + b)x + ab to find the following products:

(i) (x + 3)(x + 7)       (ii) (4x + 5)(4x + 1)         (iii) (4x - 5)(4x - 1)          (iv) (4x + 5)(4x - 1)

(v) (2x + 5y)(2x + 3y)        (vi) (2a2 + 9)(2a2 + 5)            (vii) (xyz - 4)(xyz - 2)

Answer:

(i) (x + 3)(x + 7) = x2 + (3 + 7)x + 3*7 = x2 + 10x + 21

(ii) (4x + 5)(4x + 1) = (4x)2 + (5 + 1)4x + 5*1 = 16x2 + 24x + 5

(iii) (4x - 5)(4x - 1) = (4x)2 + (-5 - 1)4x + (-5)*(-1) = 16x2 - 24x + 5

(iv) (4x + 5)(4x - 1) = (4x)2 + (5 - 1)4x + 5 *(-1) = 16x2 - 16x - 5

(v) (2x + 5y)(2x + 3y) = (2x)2 + (5y + 3y)2x + 5y * 3y = 4x2 + 16xy + 15y2

(vi) (2a2 + 9)(2a2 + 5) = (2a2)2 + (9 + 5) 2a2 + 9 * 5

= 4a4 + 14 * 2a2 + 45

= 4a4 + 28a2 + 45

(vii) (xyz - 4)(xyz - 2) = (xyz)2 + (-4 - 2)xyz + (-4)*(-2)

= x2 y2 z2 – 6xyz + 8

Question 3:

Find the following squares by using identities:

(i) (b - 7)2                         (ii) (xy + 3z)2                (iii) (6x2 – 5y)2                           (iv) (2m/3 + 3n/2)2

(v) (0.4p – 0.5q)2           (vi) (2xy + 5y)2

Answer:

(i) (b - 7)2 = b2 – 2*b*7 + 72           [Using (a - b)2 = a2 - 2ab + b2 ]

= b2 – 14b + 49

(ii) (xy + 3z)2 = (xy)2 + 2*xy*3z + (3z)2           [Using (a + b)2 = a2 + 2ab + b2 ]

= x2 y2 – 6xyz + 9z2

(iii) (6x2 – 5y)2 = (6x)2 - 2*6x*5y + (5y)2           [Using (a - b)2 = a2 - 2ab + b2 ]

= 36x2 – 60xy + 25y2

(iv) (2m/3 + 3n/2)2 = (2m/3)2 + 2 * 2m/3 * 3n/2 + (3n/2)2           [Using (a + b)2 = a2 + 2ab + b2 ]

= 4m2/9 + 2mn + 9y2/4

(v) (0.4p – 0.5q)2 = (0.4p)2 – 2 * 0.4p * 0.5q + (0.5q)2           [Using (a - b)2 = a2 - 2ab + b2 ]

= 0.16p2 – 0.40pq + 0.25q2

(vi) (2xy + 5y)2 = (2xy)2 - 2*2xy*5y + (5y)2           [Using (a - b)2 = a2 - 2ab + b2 ]

= 4x2y2 – 20xy2 + 25y2

Question 4:

Simplify:

(i) (a2 – b2)2             (ii) (2x + 5)2 – (2x - 5)2             (iii) (7m – 8n)2 + (7m + 8n)2

(iv) (4m + 5n)2 + (5m + 4n)2       (v) (2.5p – 1.5q)2 – (1.5p – 2.5q)2     (vi) (ab + bc)2 – 2ab2 c

(vii) (m2 – n2 m2) + 2m3 n2

Answer:

(i) (a2 – b2)2 = (a2)2 – 2 * a2 * b2 + (b2)2           [Using (a - b)2 = a2 - 2ab + b2 ]

= a4 – 2a2 b2 + b4

(ii) (2x + 5)2 – (2x - 5)2 = (2x)2 + 2 * 2x * 5 + (5)2  - {(2x)2 + 2 * 2x * 5 + (5)2 }

= 4x2 + 20x + 25 – (4x2 - 20x + 25)

= 4x2 + 20x + 25 – 4x2 + 20x - 25

= 40x

(iii) (7m – 8n)2 + (7m + 8n)2  = (7m)2 - 2 * 7m * 8n + (8n)2  + {(7m)2 + 2 * 7m * 8n + (8n)2 }

= 49m2 – 112mn + 64n2 + (49m2 + 112mn + 64n2)

= 49m2 – 112mn + 64n2 + 49m2 + 112mn + 64n2

= 98m2 + 128n2

(iv) (4m + 5n)2 + (5m + 4n)2 = (4m)2 + 2 * 4m * 5n + (5n)2  + {(5m)2 + 2 * 5m * 4n + (4n)2 }

= 16m2 + 40mn + 25n2 + 25m2 + 40mn + 16n2

= 41m2 + 80mn + 41n2

(v) (2.5p – 1.5q)2 – (1.5p – 2.5q)2

= (2.5p)2 - 2 * 2.5p * 1.5q + (1.5q)2  - {(1.5p)2 - 2 * 1.5p * 2.5q + (2.5q)2 }

= 6.25p2 – 7.50pq + 2.25q2 – (2.2.5p2 – 7.50pq + 6.25q2)

= 6.25p2 – 7.50pq + 2.25q2 – 2.2.5p2 + 7.50pq - 6.25q2

= 4p2 – 4q2

(vi) (ab + bc)2 – 2ab2 c = (ab)2 + 2 * ab * bc + (bc)2 – 2ab2 c

= a2 b2 + 2ab2 c + b2 c2 - 2ab2 c

= a2 b2 + b2 c2

(vii) (m2 – n2 m) + 2m3 n2 = (m2)2 – 2 * m2 * n2 m + (n2 m)2 - 2m3 n2

= m4 – 2m3 n2 + n4 m2 - 2m3 n2

= m4 + n4 m2

Question 5:

Show that:

(i) (3x + 7)2 – 84x = (3x - 7)2

(ii) (9p – 5q)2 + 180pq = (9p + 5q)2

(iii) (4m/3 – 3n/4)2 + 2mn = 16m2/9 + 9n2/16

(iv) (4pq + 3q)2 - (4pq - 3q)2 = 48pq2

(v) (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = 0

Answer:

(i)  LHS:

(3x + 7)2 – 84x = (3x)2 + 2 * 3x * 7 + (7)2 – 84x

= 9x2 + 42x + 49 – 84x

= 9x2 - 42x + 49

= (3x)2 - 2 * 3x * 7 + (7)2

= (3x - 7)2

= RHS

(ii) LHS:

(9p – 5q)2 + 180pq = (9p)2 - 2 * 9p * 5q + (5q)2 + 180pq

= 81p2 – 90pq + 25q2 + 180pq

= 81p2 + 90pq + 25q2

= (9p)2 + 2 * 9p * 5q + (5q)2

= (9p + 5q)2

= RHS

(iii) LHS:

(4m/3 – 3n/4)2 + 2mn = (4m/3)2 – 2 * 4m/3 * 3n/4 + (3n/4)2 + 2mn

= 16m2/9 – 2mn + 9n2/16 + 2mn

= 16m2/9 + 9n2/16

= RHS

(iv) LHS:

(4pq + 3q)2 - (4pq - 3q)2 = (4pq)2 + 2 * 4pq * 3q + (3q)2 – {(4pq)2 – 2 * 4pq * 3q + (3q)2}

= (4pq)2 + 24pq2 + (3q)2 – {(4pq)2 – 24pq2 + (3q)2}

= (4pq)2 + 24pq2 + (3q)2 – (4pq)2 + 24pq2 - (3q)2

= 48 pq2

= RHS

(v) LHS:

(a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = a2 – b2 + b2 – c2 + c2 – a2

= 0

= RHS

Question 6:

Using identities, evaluate:

(i) 712                      (ii) 992                     (iii) 1022                 (iv) 9982          (v) 5.22         (vi) 297 * 303

(vii) 78 * 82           (viii) 8.92                  (ix) 1.05 * 9.5

Answer:

(i) 712 = (70 + 1)2

= 702 + 2 * 70 * 1 + 12            [Using (a + b)2 = a2 + 2ab + b2 ]

= 4900 + 140 + 1

= 5041

(ii) 992 = (100 - 1)2

= 1002 - 2 * 100 * 1 + 12            [Using (a - b)2 = a2 - 2ab + b2 ]

= 10000 - 200 + 1

= 9801

(iii) 1022 = (100 + 2)2

= 1002 + 2 * 100 * 1 + 22            [Using (a + b)2 = a2 + 2ab + b2 ]

= 10000 + 200 + 4

= 10404

(iv) 9982 = (1000 - 2)2

= 10002 - 2 * 1000 * 1 + 22            [Using (a - b)2 = a2 - 2ab + b2 ]

= 1000000 - 2000 + 4

= 996004

(v) 5.22 = (5 + 0.2)2

= 52 + 2 * 5 * 0.2 + 0.22            [Using (a + b)2 = a2 + 2ab + b2 ]

= 25 + 2.0 + 0.04

= 27.04

(vi) 297 * 303 = (300 - 3)*(300 + 3)

= 3002 – 32                            [(a - b)*(a + b) = a2 – b2

= 90000 – 9

= 89991

(vii) 78 * 82 = (80 - 2)*(80 + 2)

= 802 – 22                            [(a - b)*(a + b) = a2 – b2

= 6400 – 4

= 6396

(viii) 8.92 = (8 + 0.9)2

= 82 + 2 * 8 * 0.9 + 0.92            [Using (a + b)2 = a2 + 2ab + b2 ]

= 64 + 14.4 + 0.81

= 79.21

(ix) 1.05 * 9.5 = (10 + 0.5)*(10 – 0.5)

= 102 – 0.52                            [(a - b)*(a + b) = a2 – b2

= 100 – 0.25

= 99.75

Question 7:

Using a2 – b2 = (a + b)(a - b) , find:

(i) 512 – 492           (ii) (1.02)2 – (0.98)2            (iii) 1532 – 1472            (iv) 12.12 – 7.92

Answer:

(i) 512 – 492 = (51 + 49)(51 - 49)

= 100 * 2

= 200

(ii) (1.02)2 – (0.98)2 = (1.02 + 0.98)(1.02 – 0.98)

= 2.00 * 0.04

= 0.08

(iii) 1532 – 1472 = (153 + 147)(153 - 147)

= 300 * 6

= 1800

(iv) 12.12 – 7.92 = (12.1 + 7.9)(12.1 – 7.9)

= 20.0 * 4.2

= 84

Question 8:

Using (x + a)(x + b) = x2 + (a + b)x + ab, find

(i) 103 * 104                     (ii) 5.1 * 5.2              (iii) 103 * 98              (iv) 9.7 * 9.8

Answer:

(i) 103 * 104 = (100 + 3)(100 + 4)

= 1002 + (3 + 4)100 + 3*4

= 10000 + 7 * 100 + 12

= 10000 + 700 + 12

= 10712

(ii) 5.1 * 5.2 = (5 + 0.1)(5 + 0.2)

= 52 + (0.1 + 0.2)5 + 01. * 0.2

= 25 + 0.3 * 5 + 0.02

= 25 + 1.5 + 0.02

= 26.52

(iii) 103 * 98 = (100 + 3)(100 - 2)

= 1002 + (3 - 2)100 + 3 * (-2)

= 10000 + 100 – 6

= 10094

(iv) 9.7 * 9.8 = (10 – 0.3)(10 – 0.2)

= 102 + {(-0.3) + (-0.2)}10 + (-0.3) * (-0.2)

= 100 – 0.5 * 10 + 0.06

= 100 – 5 + 0.06

= 95.06

.