Exercise 12.2 Page: 239

**1. Simplify combining like terms:**

**(i) 21b – 32 + 7b – 20b**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then,

= (21b + 7b – 20b) – 32

= b (21 + 7 – 20) – 32

= b (28 – 20) – 32

= b (8) – 32

= 8b – 32

**(ii) – z ^{2} + 13z^{2} – 5z + 7z^{3} – 15z**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then,

= 7z^{3} + (-z^{2} + 13z^{2}) + (-5z – 15z)

= 7z^{3 }+ z^{2 }(-1 + 13) + z (-5 – 15)

= 7z^{3 }+ z^{2} (12) + z (-20)

= 7z^{3 }+ 12z^{2} – 20z

**(iii) p – (p – q) – q – (q – p)**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then,

= p – p + q – q – q + p

= p – q

**(iv) 3a – 2b – ab – (a – b + ab) + 3ab + b – a**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then,

= 3a – 2b – ab – a + b – ab + 3ab + b – a

= 3a – a – a – 2b + b + b – ab – ab + 3ab

= a (1 – 1- 1) + b (-2 + 1 + 1) + ab (-1 -1 + 3)

= a (1 – 2) + b (-2 + 2) + ab (-2 + 3)

= a (1) + b (0) + ab (1)

= a + ab

**(v) 5x ^{2}y – 5x^{2} + 3yx^{2} – 3y^{2} + x^{2} – y^{2} + 8xy^{2} – 3y^{2}**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then,

= 5x^{2}y + 3yx^{2} – 5x^{2} + x^{2} – 3y^{2} – y^{2} – 3y^{2}

= x^{2}y (5 + 3) + x^{2} (- 5 + 1) + y^{2} (-3 – 1 -3) + 8xy^{2}

= x^{2}y (8) + x^{2} (-4) + y^{2} (-7) + 8xy^{2}

= 8x^{2}y – 4x^{2} – 7y^{2} + 8xy^{2}

**(vi) (3y ^{2} + 5y – 4) – (8y – y^{2} – 4)**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then,

= 3y^{2} + 5y – 4 – 8y + y^{2} + 4

= 3y^{2} + y^{2} + 5y – 8y – 4 + 4

= y^{2} (3 + 1) + y (5 – 8) + (-4 + 4)

= y^{2} (4) + y (-3) + (0)

= 4y^{2} – 3y

**2. Add:**

**(i) 3mn, – 5mn, 8mn, – 4mn**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= 3mn + (-5mn) + 8mn + (- 4mn)

= 3mn – 5mn + 8mn – 4mn

= mn (3 – 5 + 8 – 4)

= mn (11 – 9)

= mn (2)

= 2mn

**(ii) t – 8tz, 3tz – z, z – t**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= t – 8tz + (3tz – z) + (z – t)

= t – 8tz + 3tz – z + z – t

= t – t – 8tz + 3tz – z + z

= t (1 – 1) + tz (- 8 + 3) + z (-1 + 1)

= t (0) + tz (- 5) + z (0)

= – 5tz

**(iii) – 7mn + 5, 12mn + 2, 9mn – 8, – 2mn – 3**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= – 7mn + 5 + 12mn + 2 + (9mn – 8) + (- 2mn – 3)

= – 7mn + 5 + 12mn + 2 + 9mn – 8 – 2mn – 3

= – 7mn + 12mn + 9mn – 2mn + 5 + 2 – 8 – 3

= mn (-7 + 12 + 9 – 2) + (5 + 2 – 8 – 3)

= mn (- 9 + 21) + (7 – 11)

= mn (12) – 4

= 12mn – 4

**(iv) a + b – 3, b – a + 3, a – b + 3**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= a + b – 3 + (b – a + 3) + (a – b + 3)

= a + b – 3 + b – a + 3 + a – b + 3

= a – a + a + b + b – b – 3 + 3 + 3

= a (1 – 1 + 1) + b (1 + 1 – 1) + (-3 + 3 + 3)

= a (2 -1) + b (2 -1) + (-3 + 6)

= a (1) + b (1) + (3)

= a + b + 3

**(v) 14x + 10y – 12xy – 13, 18 – 7x – 10y + 8xy, 4xy**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= 14x + 10y – 12xy – 13 + (18 – 7x – 10y + 8xy) + 4xy

= 14x + 10y – 12xy – 13 + 18 – 7x – 10y + 8xy + 4xy

= 14x – 7x + 10y– 10y – 12xy + 8xy + 4xy – 13 + 18

= x (14 – 7) + y (10 – 10) + xy(-12 + 8 + 4) + (-13 + 18)

= x (7) + y (0) + xy(0) + (5)

= 7x + 5

**(vi) 5m – 7n, 3n – 4m + 2, 2m – 3mn – 5**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= 5m – 7n + (3n – 4m + 2) + (2m – 3mn – 5)

= 5m – 7n + 3n – 4m + 2 + 2m – 3mn – 5

= 5m – 4m + 2m – 7n + 3n – 3mn + 2 – 5

= m (5 – 4 + 2) + n (-7 + 3) – 3mn + (2 – 5)

= m (3) + n (-4) – 3mn + (-3)

= 3m – 4n – 3mn – 3

**(vii) 4x ^{2}y, – 3xy^{2}, –5xy^{2}, 5x^{2}y**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= 4x^{2}y + (-3xy^{2}) + (-5xy^{2}) + 5x^{2}y

= 4x^{2}y + 5x^{2}y – 3xy^{2} – 5xy^{2}

= x^{2}y (4 + 5) + xy^{2 }(-3 – 5)

= x^{2}y (9) + xy^{2} (- 8)

= 9x^{2}y – 8xy^{2}

**(viii) 3p ^{2}q^{2} – 4pq + 5, – 10 p^{2}q^{2}, 15 + 9pq + 7p^{2}q^{2}**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= 3p^{2}q^{2} – 4pq + 5 + (- 10p^{2}q^{2}) + 15 + 9pq + 7p^{2}q^{2}

= 3p^{2}q^{2} – 10p^{2}q^{2} + 7p^{2}q^{2} – 4pq + 9pq + 5 + 15

= p^{2}q^{2} (3 -10 + 7) + pq (-4 + 9) + (5 + 15)

= p^{2}q^{2 }(0) + pq (5) + 20

= 5pq + 20

**(ix) ab – 4a, 4b – ab, 4a – 4b**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

= ab – 4a + (4b – ab) + (4a – 4b)

= ab – 4a + 4b – ab + 4a – 4b

= ab – ab – 4a + 4a + 4b – 4b

= ab (1 -1) + a (4 – 4) + b (4 – 4)

= ab (0) + a (0) + b (0)

= 0

**(x) x ^{2} – y^{2} – 1, y^{2} – 1 – x^{2}, 1 – x^{2} – y^{2}**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to add the like terms

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

= x^{2} – y^{2} – 1 + y^{2} – 1 – x^{2} + 1 – x^{2} – y^{2}

= x^{2} – x^{2} – x^{2} – y^{2} + y^{2} – y^{2 }– 1 – 1 + 1

= x^{2} (1 – 1- 1) + y^{2} (-1 + 1 – 1) + (-1 -1 + 1)

= x^{2} (1 – 2) + y^{2} (-2 +1) + (-2 + 1)

= x^{2} (-1) + y^{2} (-1) + (-1)

= -x^{2} – y^{2} -1

**3. Subtract:**

**(i) –5y ^{2} from y^{2}**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= y^{2} – (-5y^{2})

= y^{2} + 5y^{2}

= 6y^{2}

**(ii) 6xy from –12xy**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= -12xy – 6xy

= – 18xy

**(iii) (a – b) from (a + b)**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= (a + b) – (a – b)

= a + b – a + b

= a – a + b + b

= a (1 – 1) + b (1 + 1)

= a (0) + b (2)

= 2b

**(iv) a (b – 5) from b (5 – a)**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= b (5 -a) – a (b – 5)

= 5b – ab – ab + 5a

= 5b + ab (-1 -1) + 5a

= 5a + 5b – 2ab

**(v) –m ^{2} + 5mn from 4m^{2} – 3mn + 8**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= 4m^{2} – 3mn + 8 – (- m^{2} + 5mn)

= 4m^{2} – 3mn + 8 + m^{2} – 5mn

= 4m^{2} + m^{2} – 3mn – 5mn + 8

= 5m^{2 }– 8mn + 8

**(vi) – x ^{2} + 10x – 5 from 5x – 10**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= 5x – 10 – (-x^{2} + 10x – 5)

= 5x – 10 + x^{2} – 10x + 5

= x^{2} + 5x – 10x – 10 + 5

= x^{2} – 5x – 5

**(vii) 5a ^{2} – 7ab + 5b^{2} from 3ab – 2a^{2} – 2b^{2}**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= 3ab – 2a^{2} – 2b^{2} – (5a^{2} – 7ab + 5b^{2})

= 3ab – 2a^{2} – 2b^{2} – 5a^{2 }+ 7ab – 5b^{2}

= 3ab + 7ab – 2a^{2} – 5a^{2} – 2b^{2} – 5b^{2}

= 10ab – 7a^{2} – 7b^{2}

**(viii) 4pq – 5q ^{2} – 3p^{2} from 5p^{2} + 3q^{2} – pq**

**Solution:-**

When term have the same algebraic factors, they are like terms.

Then, we have to subtract the like terms

= 5p^{2} + 3q^{2} – pq – (4pq – 5q^{2} – 3p^{2})

= 5p^{2} + 3q^{2} – pq – 4pq + 5q^{2} + 3p^{2}

= 5p^{2} + 3p^{2} + 3q^{2} + 5q^{2} – pq – 4pq

= 8p^{2} + 8q^{2} – 5pq

**4. (a) What should be added to x ^{2} + xy + y^{2} to obtain 2x^{2} + 3xy?**

**Solution:-**

Let us assume p be the required term

Then,

p + (x^{2} + xy + y^{2}) = 2x^{2} + 3xy

p = (2x^{2} + 3xy) – (x^{2} + xy + y^{2})

p = 2x^{2} + 3xy – x^{2} – xy – y^{2}

p = 2x^{2} – x^{2} + 3xy – xy – y^{2}

p = x^{2} + 2xy – y^{2}

**(b) What should be subtracted from 2a + 8b + 10 to get – 3a + 7b + 16?**

**Solution:-**

Let us assume x be the required term

Then,

2a + 8b + 10 – x = -3a + 7b + 16

x = (2a + 8b + 10) – (-3a + 7b + 16)

x = 2a + 8b + 10 + 3a – 7b – 16

x = 2a + 3a + 8b – 7b + 10 – 16

x = 5a + b – 6

**5. What should be taken away from 3x ^{2} – 4y^{2} + 5xy + 20 to obtain – x^{2} – y^{2} + 6xy + 20?**

**Solution:-**

Let us assume a be the required term

Then,

3x^{2} – 4y^{2} + 5xy + 20 – a = -x^{2} – y^{2} + 6xy + 20

a = 3x^{2} – 4y^{2} + 5xy + 20 – (-x^{2} – y^{2} + 6xy + 20)

a = 3x^{2} – 4y^{2} + 5xy + 20 + x^{2} + y^{2} – 6xy – 20

a = 3x^{2} + x^{2} – 4y^{2} + y^{2} + 5xy – 6xy + 20 – 20

a = 4x^{2} – 3y^{2} – xy

**6. (a) From the sum of 3x – y + 11 and – y – 11, subtract 3x – y – 11.**

**Solution:-**

First we have to find out the sum of 3x – y + 11 and – y – 11

= 3x – y + 11 + (-y – 11)

= 3x – y + 11 – y – 11

= 3x – y – y + 11 – 11

= 3x – 2y

Now, subtract 3x – y – 11 from 3x – 2y

= 3x – 2y – (3x – y – 11)

= 3x – 2y – 3x + y + 11

= 3x – 3x – 2y + y + 11

= -y + 11

**(b) From the sum of 4 + 3x and 5 – 4x + 2x ^{2}, subtract the sum of 3x^{2} – 5x and**

**–x ^{2} + 2x + 5.**

**Solution:-**

First we have to find out the sum of 4 + 3x and 5 – 4x + 2x^{2}

= 4 + 3x + (5 – 4x + 2x^{2})

= 4 + 3x + 5 – 4x + 2x^{2}

= 4 + 5 + 3x – 4x + 2x^{2}

= 9 – x + 2x^{2}

= 2x^{2} – x + 9 … [equation 1]

Then, we have to find out the sum of 3x^{2} – 5x and – x^{2} + 2x + 5

= 3x^{2} – 5x + (-x^{2} + 2x + 5)

= 3x^{2} – 5x – x^{2} + 2x + 5

= 3x^{2} – x^{2} – 5x + 2x + 5

= 2x^{2} – 3x + 5 … [equation 2]

Now, we have to subtract equation (2) from equation (1)

= 2x^{2 }– x + 9 – (2x^{2} – 3x + 5)

= 2x^{2} – x + 9 – 2x^{2} + 3x – 5

= 2x^{2} – 2x^{2} – x + 3x + 9 – 5

= 2x + 4