Question - 1 : -
Carry out themultiplication of the expressions in each of the following pairs:
(i) 4p, q + r
(ii) ab, a – b
(iii) a + b, 7a²b²
(iv) a² – 9, 4a
(v) pq + qr + rp, 0

Answer - 1 : -

(i) 4p × (q + r) = (4p × q) + (4p × r) = 4pq + 4pr
(ii) ab, a – b = ab × (a – b) = (ab × a) – (ab × b) = a²b – ab²
(iii) (a + b) × 7a²b² =(a × 7a²b²) + (b× 7a²b²) = 7a³b² +7a²b³
(iv) (a² –9) × 4a = (a² ×4a) – (9 × 4a) = 4a³ –36a
(v) (pq + qr + rp) × 0 = 0
[∵ Anynumber multiplied by 0 is = 0]

Question - 2 : -
Complete the table.

S.No.

First Expression

Second
Expression

Product

(i)

a

b + c + d

–

(ii)

x + y – 5

5xy

–

(iii)

p

6p² – 7p + 5

–

(iv)

4p²q²

p² – q²

–

(v)

a + b + c

abc

–

Answer - 2 : -

(i) a × (b + c + d) = (a × b) + (a × c) + (a × d) = ab + ac + ad
(ii) (x + y – 5) (5xy) = (x × 5xy) + (y × 5xy) – (5 × 5xy) = 5x²y + 5xy² –25xy
(iii) p × (6p² –7p + 5) = (p × 6p²) – (p× 7p) + (p × 5) = 6p³ –7p² +5p
(iv) 4p²q² ×(p² –q²) = 4p²q² ×p² –4p²q² ×q² =4p⁴q² –4p²q⁴
(v) (a + b + c) × (abc) = (a × abc) + (b × abc) + (c × abc) = a²bc + ab²c + abc²

Completed Table:

S.No.	First Expression	Second Expression	Product
(i)	a	b + c + d	ab + ac + ad
(ii)	x + y – 5	5xy	5x²y + 5xy² – 25xy
(iii)	p	6p² – 7p + 5	6p³ – 7p² + 5p
(iv)	4p²q²	p²– q²	4p4q2 – 4p²q⁴
(v)	a + b + c	abc	a²bc + ab²c + abc²

Question - 3 : - Find the products.

Answer - 3 : -

Question - 4 : -
(a) Simplify: 3x(4x – 5) + 3 and find its 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 - 4 : -

(a) We have 3x(4x – 5) + 3 = 4x × 3x – 5 × 3x + 3 = 12x² –15x + 3
(i) For x = 3, we have
12 × (3)² –15 × 3 + 3 = 12 × 9 – 45 + 3 = 108 – 42 = 66

(b) We have a(a² +a + 1) + 5
= (a² ×a) + (a × a) + (1 × a) + 5
= a³ +a² +a + 5
(i) For a = 0, we have
= (0)³ +(0)² +(0) + 5 = 5
(ii) For a = 1, we have
= (1)³ +(1)² +(1) + 5 = 1 + 1 + 1 + 5 = 8
(iii) For a = -1, we have
= (-1)³ +(-1)² +(-1) + 5 = -1 + 1 – 1 + 5 = 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 – x)
(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 - 5 : -

(a) p(p – q) + q(q – r) + r(r – p)
= (p × p) – (p × q) + (q × q) – (q × r) + (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)
= (2x × z) – (2x × x) – (2x × y) + (2y × z) – (2y × y) – (2y × x)
= 2xz – 2x² –2xy + 2yz – 2y² –2xy
= -2x² –2y² +2xz + 2yz – 4xy
= -2x² –2y² –4xy + 2yz + 2xz
(c) 4l(10n – 3m + 2l) – 3l(l – 4m + 5n)
= (4l × 10n) – (4l × 3m) + (4l × 2l) – (3l × l) – (3l × -4m) – (3l × 5n)
= 40ln – 12lm + 8l² –3l² +12lm – 15ln
= (40ln – 15ln) + (-12lm + 12lm) + (8l² –3l²)
= 25ln + 0 + 5l²
= 25ln + 5l²
= 5l² +25ln
(d) [4c(-a + b + c)] – [3a(a + b + c) – 2b(a – b + c)]
= (-4ac + 4bc + 4c²) – (3a² +3ab + 3ac – 2ab + 2b² –2bc)
= -4ac + 4bc + 4c² –3a² –3ab – 3ac + 2ab – 2b² +2bc
= -3a² –2b² +4c² –ab + 6bc – 7ac

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