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Algebraic Expressions And Identities : Exercise 9.3 (Mathematics NCERT Class 8th)


Q.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,7{a^2}{b^2} (iv) {a^2} - 9,4a

(v) pq + qr + rp,0

Sol. (i) 4p,q + r

4p \times (q + r) = 4p \times q + 4p \times r = 4pq + 4pr

(ii) ab,a - b

ab \times (a - b) = ab \times a - ab \times b = {a^2}b + a{b^2}

(iii) a + b,7{a^2}{b^2}

(a + b) \times 7{a^2}{b^2} = a \times 7{a^2}{b^2} + b \times 7{a^2}{b^2} = 7{a^3}{b^2} + 7{a^2}{b^3}

(iv) {a^2} - 9,4a

({a^2} - 9) \times 4a = {a^2} \times 4a - 4a \times 9 = 4{a^3} - 36a

(v) pq + qr + rp,0

(pq + qr + rp) \times 0 = pq \times 0 + qr \times 0 + rp \times 0 = 0 + 0 + 0 = 0

Q.2 Complete the table.

  First expression Second expression Product
(i)

(ii)

(iii)

(iv)

(v)

a

x + y - 5

p

4{p^2}{q^2}

a + b + c

b + c + d

5xy

6{p^2} - 7p + 5

{p^2} - {q^2}

abc

...

...

...

...

...

Sol. The complete table is as follows:

  First expression Second expression Product
(i)

(ii)

(iii)

(iv)

(v)

a

x + y - 5

p

4{p^2}{q^2}

a + b + c

b + c + d

5xy

6{p^2} - 7p + 5

{p^2} - {q^2}

abc

ab + ac + ad

5{x^2}y + 5x{y^2} - 25xy

6{p^3} - 7{p^2} + 5p

4{p^4}{q^2} - 4{p^2}{q^4}

{a^2}bc + a{b^2}c + ab{c^2}

Q.3 Find the product.

(i) ({a^2}) \times (2{a^{22}}) \times (4{a^{26}}) (ii)\left( {{2 \over 3}xy} \right) \times \left( {{{ - 9} \over {10}}{x^2}{y^2}} \right)

(iii) \left( { - {{10} \over 3}p{q^3}} \right) \times \left( {{6 \over 5}{p^3}q} \right) (iv)x \times {x^2} \times {x^3} \times {x^4}

Sol. (i) ({a^2}) \times (2{a^{22}}) \times (4{a^{26}})

= 2 \times 4 \times {a^2} \times {a^{22}} \times {a^{26}} = 8{a^{50}}

(ii) \left( {{2 \over 3}xy} \right) \times \left( {{{ - 9} \over {10}}{x^2}{y^2}} \right)

= \left( {{2 \over 3}} \right) \times \left( {{{ - 9} \over {10}}} \right) \times x \times y \times {x^2} \times {y^2} = {{ - 3} \over 5}{x^3}{y^3}

(iii) \left( { - {{10} \over 3}p{q^3}} \right) \times \left( {{6 \over 5}{p^3}q} \right)

= \left( {{{ - 10} \over 3}} \right) \times \left( {{6 \over 5}} \right) \times p{q^3} \times {p^3}q = - 4{p^4}{q^4}

(iv) x \times {x^2} \times {x^3} \times {x^4}

=x \times {x^2} \times {x^3} \times {x^4} = {x^{10}}

Q.4 (a) Simplify 3x (4x – 5) + 3 and find its values for (i) x = 3 (ii) x = {1 \over 2}.

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

Sol. (a) 3x(4x - 5) + 3

3x(4x - 5) + 3 = 12{x^2} - 15x + 3

For (i) x = 3

12{x^2} - 15x + 3= 12{(3)^2} - 15(3) + 3

= 108 – 45 + 3

= 66

(ii) x = {1 \over 2}

12{x^2} - 15x + 3= 12{\left( {{1 \over 2}} \right)^2} - 15\left( {{1 \over 2}} \right) + 3

= 3 - {{15} \over 2} + 3

=6 - {{15} \over 2}

= {{12 - 15} \over 2}={{ - 3} \over 2}

(b) a({a^2} + a + 1) + 5

a({a^2} + a + 1) + 5= {a^3} + {a^2} + a + 5

(i) a = 0

{a^3} + {a^2} + a + 5= {(0)^3} + {(0)^2} + 0 + 5

= 5

(ii) a = 1

{a^3} + {a^2} + a + 5= {(1)^3} + {(1)^2} + 1 + 5

= 1 + 1 + 1 + 5

= 8

(iii) a = – 1

{a^3} + {a^2} + a + 5= {( - 1)^3} + {( - 1)^2} + ( - 1) + 5

= -1 + 1 - 1 + 5

= 4

Q.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 – 4 m + 5 n) from 4l ( 10 n – 3 m + 2 l )

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

Sol. (a) p(p - q) + q(q - r) + r(r - p)

= {p^2} - pq + {q^2} - qr + {r^2} - rp

= {p^2} + {q^2} + {r^2} - pq - qr - rp

(b) 2x(z - x - y) + 2y(z - y - x)

= 2xz - 2{x^2} - 2xy + 2yz - 2{y^2} - 2xy

=  - 2{x^2} - 2{y^2} - 4xy + 2yz + 2xz

(c)4l(10n - 3m + 2l) - 3l(l - 4m + 5n)

= 40ln - 12lm + 8{l^2} - 3{l^2} + 12lm - 15ln

= 8{l^2} - 3{l^2} - 12lm + 12lm + 40ln - 15ln

=5{l^2} + 25ln

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

=  - 4ac + 4bc + 4{c^2} - [3{a^2} + 3ab + 3ac - 2ab + 2{b^2} - 2bc]

=  - 4ac + 4bc + 4{c^2} - [3{a^2} + 2{b^2} + 3ab - 2bc + 3ac - 2ab]

=  - 4ac + 4bc + 4{c^2} - [3{a^2} + 2{b^2} + ab - 2bc + 3ac]

=  - 4ac + 4bc + 4{c^2} - 3{a^2} - 2{b^2} - ab + 2bc - 3ac

=  - 3{a^2} - 2{b^2} + 4{c^2} - ab + 6bc - 7ac



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