# Algebraic Expressions

## Exercise 12.3

1). If *m *= 2, find the value of:

(i) *m *– 2 (ii) 3*m *– 5

(iii) 9 – 5*m *(iv) 3*m*^{2} – 2*m *– 7

(v) ^{5m}/_{2} – 4

Solution:

(i) *m *– 2

Put *m* = 2 in the given expression

*m *– 2 = 2 – 2 = 0

(ii) 3*m *– 5

Put *m* = 2 in the given expression

3*m *– 5 = 3(2) – 5

= 6 – 5

= 1

(iii) 9 – 5*m *

Put *m* = 2 in the given expression

9 – 5*m * = 9– 5(2)

= 9 – 10

= –1

(iv) 3*m*^{2} – 2*m *– 7

Put *m* = 2 in the given expression

3*m*^{2} – 2*m *– 7 = 3(2)^{2} – 2(2) – 7

= 3(4) – 4 – 7

= 12 – 4 – 7

= 1

(v) ^{5m}/_{2} – 4

Put *m* = 2 in the given expression

^{5m}/_{2} – 4 =( ^{5 x 2}/_{2} ) – 4

= (^{10}/_{2} ) – 4

= 5 – 4

= 1

2). If *p *= – 2, find the value of:

(i) 4*p *+ 7 (ii) – 3*p*^{2} + 4*p *+ 7

(iii) – 2*p*^{3} – 3*p*^{2} + 4*p *+ 7

Solution:

(i) 4*p *+ 7

Put *p *= – 2 in the given expression

4*p *+ 7 = 4(–2) + 7

= – 8 + 7

= – 1

(ii) – 3*p*^{2} + 4*p *+ 7

Put *p *= – 2 in the given expression

– 3*p*^{2} + 4*p *+ 7 = – 3(– 2)^{2} + 4(– 2) + 7

= – 3(4) – 8 + 7

= – 12 – 8 + 7

= – 13

(iii) – 2*p*^{3} – 3*p*^{2} + 4*p *+ 7

Put *p *= – 2 in the given expression

– 2*p*^{3} – 3*p*^{2} + 4*p *+ 7 = – 2(– 2)^{3} – 3(– 2)^{2} + 4(– 2) + 7

= – 2(– 8) – 3(4) – 8 + 7

= 16 – 12 – 8 + 7

= 3

3). Find the value of the following expressions, when *x *= –1:

(i) 2*x *– 7 (ii) – *x *+ 2

(iii) *x*^{2} + 2*x *+ 1 (iv) 2*x*^{2} – *x *– 2

Solution:

(i) 2*x *– 7

Put *x *= –1 in the given expression

2*x *– 7 = 2(– 1) – 7

= – 2 – 7

= – 9

(ii) – *x *+ 2

Put *x *= –1 in the given expression

– *x *+ 2 = – (–1)+ 2

= 1 + 2

= 3

(iii) *x*^{2} + 2*x *+ 1

Put *x *= –1 in the given expression

*x*^{2} + 2*x *+ 1 =(–1)^{2} + 2(–1) + 1

= 1– 2 + 1

= 0

(iv) 2*x*^{2} – *x *– 2

Put *x *= –1 in the given expression

2*x*^{2} – *x *– 2 = 2(–1)^{2} – (–1) – 2

= 2(1) + 1– 2

= 2 + 1 –2

= 1

4). If *a *= 2, *b *= – 2, find the value of:

(i) *a*^{2} + *b*^{2} (ii) *a*^{2} + *ab *+ *b*^{2}

(iii) *a*^{2} – *b*^{2}

Solution:

(i) *a*^{2} + *b*^{2}

Put *a *= 2, *b *= – 2 in the given expression

*a*^{2} + *b*^{2} = (2)^{2} + (– 2)^{2}

= 4 + (4)

= 4 + 4

= 8

(ii) *a*^{2} + *ab *+ *b*^{2}

Put *a *= 2, *b *= – 2 in the given expression

*a*^{2} + *ab *+ *b*^{2} = (2)^{2} + 2 X (– 2) + (– 2)^{2}

= 4 + (– 4) + (4)

= 4 – 4 + 4

= 4

(iii) *a*^{2} – *b*^{2}

Put *a *= 2, *b *= – 2 in the given expression

*a*^{2} – *b*^{2} = (2)^{2} – (– 2)^{2}

= 4 – (4)

= 4 – 4

= 0

5). When *a *= 0, *b *= – 1, find the value of the given expressions:

(i) 2*a *+ 2*b *(ii) 2*a*^{2} + *b*^{2} + 1

(iii) 2*a*^{2}*b *+ 2*ab*^{2} + *ab *(iv) *a*^{2} + *ab *+ 2

Solution:

(i) 2*a *+ 2*b*

Put *a *= 0, *b *= – 1 in the given expression

2*a *+ 2*b* = 2(0) + 2(– 1)

= 0 – 2

= – 2

(ii) 2*a*^{2} + *b*^{2} + 1

Put *a *= 0, *b *= – 1 in the given expression

2*a*^{2} + *b*^{2} + 1 = 2(0)^{2} + (– 1)^{2} + 1

= 2(0) +(1) + 1

= 0 + 1 + 1

= 2

(iii) 2*a*^{2}*b *+ 2*ab*^{2} + *ab*

Put *a *= 0, *b *= – 1 in the given expression

2*a*^{2}*b *+ 2*ab*^{2} + *ab* = 2 X 0^{2} X (– 1) + 2 X 0 X (– 1)^{2} + 0 X (– 1)

= 0 + 0 + 0

= 0

(iv) *a*^{2} + *ab *+ 2

Put *a *= 0, *b *= – 1 in the given expression

*a*^{2} + *ab *+ 2= 0^{2} + 0 X (– 1) * *+ 2

= 0 + 0 + 2

= 2

6). Simplify the expressions and find the value if *x *is equal to 2

(i) *x *+ 7 + 4 (*x *– 5) (ii) 3 (*x *+ 2) + 5*x *– 7

(iii) 6*x *+ 5 (*x *– 2) (iv) 4(2*x *– 1) + 3*x *+ 11

Solution:

(i) *x *+ 7 + 4 (*x *– 5)

= *x *+ 7 + 4*x *– 20

= *x *+ 4*x* + 7 * *– 20

= 5*x* – 13

= 5(2) – 13 (putting *x* = 2 in the expression)

= 10 – 13

= – 3

(ii) 3 (*x *+ 2) + 5*x *– 7

= 3*x *+ 6 + 5*x *– 7

= 3*x *+ 5*x* + 6 – 7

= 8*x *– 1

= 8 (2) – 1 (putting *x* = 2 in the expression)

= 16 – 1

= 15

(iii) 6*x *+ 5 (*x *– 2)

= 6*x *+ 5*x *– 10

= 11*x *– 10

= 11(2) – 10 (putting *x* = 2 in the expression)

= 22 – 10

= 12

(iv) 4(2*x *– 1) + 3*x *+ 11

= 8*x *– 4 + 3*x *+ 11

= 8*x *+ 3*x *– 4 + 11

= 11*x *+ 7

= 11(2) + 7 (putting *x* = 2 in the expression)

= 22 + 7

= 29

7). Simplify these expressions and find their values if *x *= 3, *a *= – 1,

*b *= – 2.

(i) 3*x *– 5 – *x *+ 9 (ii) 2 – 8*x *+ 4*x *+ 4

(iii) 3*a *+ 5 – 8*a *+ 1 (iv) 10 – 3*b *– 4 – 5*b*

(v) 2*a *– 2*b *– 4 – 5 + *a*

Solution:

(i) 3*x *– 5 – *x *+ 9

= 3*x *– *x *– 5 + 9

= 2*x* + 4

= 2 (3) + 4 (putting *x* = 3 in the expression)

= 6 + 4

= 10

(ii) 2 – 8*x *+ 4*x *+ 4

= 2 + 4 – 8*x *+ 4*x *

= 6 – 4*x*

= 6 – 4(3)

= 6 – 12

= – 6

(iii) 3*a *+ 5 – 8*a *+ 1

= 3*a *– 8*a* + 5 + 1

= – 5*a *+ 6 (putting *a* = –1 in the expression)

= – 5(–1) + 6

= 5 + 6

= 11

(iv) 10 – 3*b *– 4 – 5*b*

= 10 – 4 – 3*b *– 5*b*

= 6 – 8*b *

= 6 – 8(– 2) (putting *b* = –2 in the expression)

= 6 + 16

= 22

(v) 2*a *– 2*b *– 4 – 5 + *a*

= 2*a *+ *a *– 2*b *– 4 – 5

= 3*a *– 2*b *– 9

= 3(–1) – 2(–2) – 9 (putting *a* = –1, *b* = –2 in the expression)

= – 3 + 4 – 9

= – 8

8). (i) If *z *= 10, find the value of *z*^{3} – 3(*z *– 10).

*z*^{3} – 3(*z *– 10) = *z*^{3} – 3*z *+ 30

= (10)^{3} – 3(10) + 30

= 1000 – 30 + 30

= 1000

(ii) If *p *= – 10, find the value of *p*^{2} – 2*p *– 100

*p*^{2} – 2*p *– 100 = (–10)^{2} – 2(–10) – 100

= 100 + 20 – 100

= 20

9). What should be the value of *a *if the value of 2*x*^{2} + *x *– *a *equals to 5, when *x *= 0?

2*x*^{2} + *x *– *a *= 5 when *x *= 0

Put *x* = 0 in the above expression

2(0)^{2} +(0) – *a *= 5

2 X 0 + 0 – *a* = 5

0 + 0 – *a* = 5

– *a* = 5

*a* = – 5

10). Simplify the expression and find its value when *a *= 5 and

*b *= – 3.

2(*a*^{2} + *ab*) + 3 – *ab *

= 2*a*^{2} + 2*ab* + 3 – *ab*

= 2*a*^{2} + 2*ab* – *ab* + 3

= 2*a*^{2} + *ab* + 3

= 2(5)^{2} + 5 X ( – 3) + 3

= 2 (25) – 15 + 3

= 50 – 15 + 3

= 38

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