**1. Carry out the multiplication of the expressions in each of the following pairs.**

**(i) 4p, q + r**

**Solution:**

= 4p(q + r)

= 4pq + 4pr

**(ii) ab, a – b**

**Solution:**

= ab(a – b)

= a^{2} b – a b^{2}

**(iii) a + b, 7a²b²**

**Solution:**

= (a + b) (7a^{2}b^{2})

= 7a^{3}b^{2} + 7a^{2}b^{3}

**(iv) a² – 9, 4a**

**Solution:**

= (a^{2} – 9)(4a)

= 4a^{3} – 36a

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

**Solution:**

= (pq + qr + rp) × 0

= 0

**2. Complete the table.**

**Solution:**

**3. Find the product.**

**i) a ^{2} x (2a^{22}) x (4a^{26})**

**Solution:**

= a^{2} x (2a^{22}) x (4a^{26})

= (2 × 4) (a^{2} × a^{22} × a^{26})

= 8 × a^{2} ^{+ 22 + 26}

= 8a^{50}

**ii) (2/3 xy) ×(-9/10 x ^{2}y^{2})**

**Solution:**

= (2xy/3) ×(-9x^{2}y^{2}/10)

= (2/3 × -9/10) (x × x^{2} × y × y^{2})

= (-3/5 x^{3}y^{3})

(iii) ((-10/3)pq^{3}) × ((6/5)p^{3}q)

**Solution:**

= (-10pq^{3}/3) ×(6p^{3}q/5)

= ( -10/3 × 6/5 ) (p × p^{3}× q^{3 }× q)

= (-4p^{4}q^{4})

**(iv) (x) × (x ^{2}) × (x^{3}) × (x^{4})**

**Solution:**

= (x) x (x^{2}) x (x^{3}) x (x^{4})

= x ^{1 + 2 + 3 + 4}

= x^{10}

**4. (a) Simplify 3x (4x – 5) + 3 and find its values for (i) x = 3 (ii) x =1/2**

**Solution:**

= 3x (4x – 5) + 3

= (3x (4x) – 3x (5)) +3

= 12x^{2} – 15x + 3

Now,

(i)Putting x=3 in the equation:

= 12x^{2} – 15x + 3

= 12(3^{2}) – 15 (3) +3

= 108 – 45 + 3

= 66

(ii) Putting x=1/2 in the equation:

= 12x^{2} – 15x + 3

= 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) Simplify a (a ^{2}+ a + 1) + 5 and find its value for (i) a = 0, (ii) a = 1 (iii) a = – 1.**

**Solution:**

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

= a x a^{2} + a x a + a x 1 + 5

=a^{3}+a^{2}+a+ 5

(i)putting a=0 in the equation:

= 0^{3}+0^{2}+0+5

=5

(ii) putting a=1 in the equation:

1^{3} + 1^{2 }+ 1+5

= 1 + 1 + 1+5

= 8

(iii) Putting a = -1 in the equation:

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

= -1 + 1 – 1+5

= 4

**5. (a) Add: p ( p – q), q ( q – r) and r ( r – p)**

**Solution:**

= p ( p – q) + q ( q – r) + r ( r – p)

= (p^{2} – pq) + (q^{2} – qr) + (r^{2} – pr)

= p^{2 }+ q^{2} + r^{2} – pq – qr – pr

**(b) Add: 2x (z – x – y) and 2y (z – y – x)**

**Solution:**

= 2x (z – x – y) + 2y (z – y – x)

= (2xz – 2x^{2} – 2xy) + (2yz – 2y^{2} – 2xy)

= 2xz – 4xy + 2yz – 2x^{2} – 2y^{2}

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

**Solution:**

= 4l (10 n – 3 m + 2 l) – 3l (l – 4 m + 5 n)

= (40ln – 12lm + 8l^{2}) – (3l^{2} – 12lm + 15ln)

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

= 25 ln + 5l^{2}

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

**Solution:**

= 4c (– a + b + c) – (3a (a + b + c) – 2 b (a – b + c))

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

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

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

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

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