##### Question 1:

Give five examples of expressions containing one variable and five examples of expressions containing two variables.

##### Answer:

##### Question 2:

Show on the number line x, x - 4, 2x + 1, 3x - 2.

##### Answer:

Let us say the variable x has a position X on the number line

The value of x - 4 will be 4 units to the left of X

The position of 2x will be point A, the distance of A from the origin will be twice the distance of X from the origin. The position B of 2x + 1 will be 1 unit to the right of A.

The position of 3x will be point B; the distance of B from origin will be three times the distance of X from the origin. The position P of 3x - 2 will be 2 units to the left of B.

##### Question 3:

Identify the coefficients of each term in the expression:

##### Answer:

Coefficient of x^{2}y^{2} is 1

Coefficient of x^{2}y is -10

Coefficient of xy^{2} is 5

Constant term is -20.

##### Question 4:

Classify the following polynomials as monomials, binomials, trinomials:

-z + 5, x + y + z, y + z + 100, ab - ac, 17.

##### Answer:

Monomials: 17

Binomials: -z + 5, ab - ac

Trinomials: x + y + z, y + z + 100.

##### Question 5:

Construct 3 binomials with only x as a variable.

##### Answer:

Binomials having x as a variable:

(i) 3x^{2} + 5x (ii) 6x^{3} + 2x (iii) 9x + 4x^{3}

##### Question 6:

Construct 3 binomials with x and y as variable.

##### Answer:

Binomials having x and y as variables:

(i) 3x^{2} y + 4xy (ii) 9xy + 6xy^{2} (iii) 5xy + 8x^{2} y

##### Question 7:

Construct 3 monomials with x and y as variables.

##### Answer:

Monomials with x and y as variables:

(i) 4x^{2}y (ii) 5xy^{2} (iii) 3xy

##### Question 8:

Construct 2 polynomials with 4 or more terms.

##### Answer:

Polynomials having 4 or more terms:

(i) a + b + c + d

(ii) 2x^{3}+ 5x^{2} + 3x + 18

(iii) 3x^{2} + 4xy + y^{2} + y + x + 9.

##### Question 9:

Write two terms which are like:

(i) 7xy (ii) 4 mn^{2} (iii) 2l.

##### Answer:

(i) Terms which are like 7xy:

3xy, 9xy etc.

(ii) Terms which are like 4 mn^{2}:

9mn^{2}, mn^{2} etc.

(iii) Terms which are like 2l:

6l, 9l, 5l etc.

##### Question 10:

Identify the terms, their coefficients for each of the following expressions:

##### Answer:

##### Question 11:

Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?

##### Answer:

##### Question 12:

Add the following:

ab - bc, bc - ca, ca - ab

##### Answer:

##### Question 13:

Add the following:

a - b + ab, b - c + bc, c - a + ac.

##### Answer:

##### Question 14:

Add the following:

2p^{2}q^{2} - 3pq + 4, 5 + 7pq - 3p^{2}q^{2}

##### Answer:

##### Question 15:

Add the following:

l^{2} + m^{2}, m^{2} + n^{2}, n^{2} + l^{2}, 2lm + 2mn + 2nl

##### Answer:

##### Question 16:

Subtract 4a - 7ab + 3b +12 from 12a - 9ab + 5b - 3

##### Answer:

##### Question 17:

Subtract 3xy + 5yz - 7zx from 5xy - 2yz - 2zx + 10xyz

##### Answer:

##### Question 18:

Subtract 4p^{2}q - 3pq + 5pq^{2} - 8p + 7q - 10 from

18 - 3p - 11q + 5pq - 2pq^{2} + 5p^{2}q.

##### Answer:

##### Question 19:

Find 4x x 5y x 7z.

Does the order in which you carry out the multiplication matter?

##### Answer:

The result is same in both the cases.

No, the order of multiplication does not alter the result.

##### Question 20:

Find the product of the following pairs of monomials:

(i) 4, 7p

(ii) -4p, 7p

(iii) -4p, 7pq

(iv) 4p3, -3p

(v) 4p, 0

##### Answer:

(i) 4 x 7p = 28p

(ii) -4p x 7p = (-4 x 7) x (p x p) = -28p^{2}

(iii) -4p x 7pq = (-4 x 7) x (p x pq)
= -28p^{2}q

(iv) 4p3 x -3p = [4 x (-3)] x (p^{3} x p)
= -12p^{4}

(v) 4p x 0 = 0.

##### Question 21:

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:

(p, q); (10 m, 5n); (20x^{2}, 5y^{2}); (4x, 3x^{2}); (3mn, 4np).

##### Answer:

(i) Length of rectangle = p

Breadth of rectangle = q

Area of rectangle = length x breadth
= p x q = pq.

(ii) Length of rectangle = 10 m

Breadth of rectangle = 5n

Area of rectangle = length x breadth
= 10m x 5n

= 50mn.

(iii) Length of rectangle = 20x^{2}

Breadth of rectangle = 5y^{2}

Area of rectangle = length x breadth

= 20x^{2} x 5y^{2}

= 100x^{2}y^{2}.

(iv) Length of rectangle = 4x

Breadth of rectangle = 3x^{2}

Area of rectangle = length x breadth

= 4x x 3x^{2}

= 12x^{3}.

(v) Length of rectangle = 3mn

Breadth of rectangle = 4np

Area of rectangle = length x breadth
= 3mn x 4np

= 12mn^{2}p.

##### Question 22:

Complete the table of products:

##### Answer:

Complete table of products is as under:

##### Question 23:

Obtain the volume of rectangular boxes with the following length, breadth and height respectively.

##### Answer:

##### Question 24:

##### Answer:

##### Question 25:

Find the product:

(i) 2x(3x + 5xy) (ii) a^{2}(2ab – 5c)

##### Answer:

(i) (2x) x (3x + 5xy)

= (2x) x (3x) + (2x) x (5xy)

= 6x^{2} + 10x^{2}y.

(ii) a^{2} (2ab – 5c)

= a^{2} x 2ab – a^{2} x 5c

= 2a^{3}b – 5a^{2}c.

##### Question 26:

Find the product: (4p^{2} + 5p + 7) x 3p.

##### Answer:

4p^{2} x 3p + 5p x 3p + 7 x 3p

= 12p^{3} + 15p^{2} + 21p.

##### Question 27:

Carry out the multiplication of the expressions in each of the following pairs:

(i) 4p, q + r (ii) ab, a – b

(iii) a + b, 7a^{2}b^{2} (iv) a^{2} – 9, 4a

(v) pq + qr + rp, 0

##### Answer:

##### Question 28:

Complete the table:

##### Answer:

##### Question 29:

##### Answer:

##### Question 30:

##### Answer:

##### Question 31:

##### Answer:

##### Question 32:

Multiply the binomials: (2x + 5) and (4x – 3)

##### Answer:

(2x + 5) x (4x – 3)

= 2x x (4x – 3) + 5 (4x – 3)

= 2x x 4x – 2x x 3 + 5 x 4x – 5 x 3

= 8x^{2} – 6x + 20x – 15

= 8x^{2} + 14x – 15

##### Question 33:

Multiply the binomials: (y – 8) and (3y – 4)

##### Answer:

(y – 8) x (3y – 4)

= y (3y – 4) – 8 x (3y – 4)

= 3y^{2} – 4y – 24y + 32

= 3y^{2} – 28y + 32

##### Question 34:

Multiply the binomials: (2.5l – 0.5m) and (2.5l + 0.5m)

##### Answer:

(2.5l – 0.5m) x (2.5l + 0.5m)

= 2.5l x (2.5l + 0.5m) – 0.5m (2.5l + 0.5m)

= 6.25l^{2} + 1.25lm – 1.25lm – 0.25m^{2}

= 6.25l^{2} – 0.25m^{2}

##### Question 35:

Multiply the binomials: (a + 3b) and (x + 5)

##### Answer:

(a + 3b) x (x + 5)

= a x (x + 5) + 3b x (x + 5)

= ax + 5a + 3bx + 15b

= 5a + (a + 3b) x + 15b

##### Question 36:

Multiply the binomials: (2pq + 3q^{2}) and (3pq – 2q^{2})

##### Answer:

(2pq + 3q^{2}) x (3pq – 2q^{2})

= 2pq x (3pq – 2q^{2}) + 3q^{2} x (3pq – 2q^{2})

= 6p^{2}q^{2} – 4pq^{3} + 9pq^{3} – 6q^{4}

= 6p^{2}q^{2} + 5pq^{3} – 6q^{4}.

##### Question 37:

Multiply the binomials:

##### Answer:

##### Question 38:

Find the product: (5 – 2x) (3 + x)

##### Answer:

(5 – 2x) (3 + x)

= 5 x (3 + x) – 2x x (3 + x)

= 15 + 5x – 6x – 2x^{2}

= 15 – x – 2x^{2}

##### Question 39:

Find the product: (x + 7y) (7x – y)

##### Answer:

(x + 7y) (7x – y)

= x x (7x – y) + 7y x (7x – y)

= 7x^{2} – xy + 49xy – 7y^{2}

= 7x^{2} + 48xy – 7y^{2}

##### Question 40:

Find the product: (a^{2} + b) (a + b^{2})

##### Answer:

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

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

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

##### Question 41:

Find the product: (p^{2} – q^{2}) (2p + q)

##### Answer:

(p^{2} – q^{2}) (2p + q) = p^{2} (2p + q) – q^{2} (2p + q)

= 2p^{3} + p^{2}q – 2pq^{2} – q^{3}

##### Question 42:

Simplify:

(x^{2} – 5) (x + 5) + 25

##### Answer:

(x^{2} – 5) (x + 5) + 25

= x^{2} (x + 5) – 5 (x + 5) + 25

= x^{3} + 5x^{2} – 5x – 25 + 25

= x^{3} + 5x^{2} – 5x

##### Question 43:

Simplify:

(a^{2} + 5) (b^{3} + 3) + 5

##### Answer:

(a^{2} + 5) (b^{3} + 3) + 5

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

= a^{2}b^{3} + 3a^{2} + 5b^{3} + 15 + 5

= a^{2}b^{3} + 3a^{2} + 5b^{3} + 20

##### Question 44:

Simplify:

(t + s^{2}) (t^{2} – s)

##### Answer:

(t + s^{2}) (t^{2} – s)

= t (t^{2} – s) + s^{2} (t^{2} – s)

= t^{3} – ts + s^{2}t^{2} – s^{3}

##### Question 45:

Simplify:

(a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)

##### Answer:

(a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)

= a(c – d) + b(c – d) + a(c + d) – b(c + d) + 2ac + 2bd

= ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd

= 4ac

##### Question 46:

Simplify:

(x + y) (2x + y) + (x + 2y) (x – y)

##### Answer:

(x + y) (2x + y) + (x + 2y) (x – y)

= x(2x + y) + y(2x + y) + x(x – y) + 2y(x – y)

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

= 3x^{2} + 4xy – y^{2}

##### Question 47:

Simplify:

(x + y) (x^{2} – xy + y^{2})

##### Answer:

(x + y) (x^{2} – xy + y^{2})

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

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

= x^{3} + y^{3}

##### Question 48:

Simplify:

(1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y

##### Answer:

(1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y

= 1.5x(1.5x + 4y + 3) – 4y (1.5x + 4y + 3) – 4.5x + 12y

= 2.25x^{2} + 6xy + 4.5x – 6xy – 16y^{2} – 12y – 4.5x + 12y

= 2.25x^{2} – 16y^{2}

##### Question 49:

Simplify:

(a + b + c) (a + b – c)

##### Answer:

(a + b + c) (a + b – c)

= a (a + b – c) + b (a + b – c) + c (a + b – c)

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

= a^{2} + b^{2} – c^{2} + 2ab

##### Question 50:

Put –b in place of b in Identity (i): (a + b)^{2} = a^{2} + 2ab + b^{2}. Do you get Identity (ii): (a – b)^{2} = a^{2} – 2ab + b^{2}?

##### Answer:

##### Question 51:

Verify identity (iv): (x + a)(x + b) = x^{2} + x(a + b) + ab for a = 2, b = 3, x = 5.

##### Answer:

##### Question 52:

Consider, the special case of identity (iv): (x + a)(x + b)= x^{2} + x(a + b) + ab with a = b, what do you get? Is it related to Identity (i): (x + a)^{2} = x^{2} + 2ax + a^{2}?

##### Answer:

The Identity (iv) is

(x + a) (x + b) = x^{2} + (a + b)x + ab

For a = b the above identity becomes

(x + a)(x + a) = x^{2} + (a + a) x + a.a

$\Rightarrow $ (x + a)^{2} = x^{2} + 2ax + a^{2}

Yes, it is related to identity (x + a)^{2} = x^{2} + 2ax + a^{2}.

##### Question 53:

Consider, the special case of identity (iv): (x + a)(x + b)= x^{2} + x(a + b) + ab with a = –c and b = –c. What do you get? Is it related to Identity (ii): (x – c)^{2} = x^{2} – 2cx + c^{2}?

##### Answer:

The Identity (iv) is

(x + a) (x + b) = x^{2} + (a + b) x + ab

For a = – c and b = – c the identity (iv) becomes

[x + (–c)][x + (– c)] = x^{2} + [( – c) + (–c)]x + (– c)(– c)

$\Rightarrow $ (x – c) (x – c) = x^{2} + ( –2c) x + c^{2}

$\Rightarrow $ (x – c) (x – c) = x^{2} – 2 cx + c^{2}

$\Rightarrow $ (x – c)^{2} = x^{2} – 2cx + c^{2}

Yes, it is related to identity (ii).

##### Question 54:

Consider the special case of identity (iv): (x + a)(x + b)= x^{2} + x(a + b) + ab with b = –a. What do you get? Is it related to identity (iii): (x – c)^{2} = x^{2} – 2cx + c^{2}.

##### Answer:

The identity (iv) is

(x + a) (x + b) = x^{2} + (a + b) x + ab

For b = – a, identity (iv) becomes

(x + a) [x + (– a)] = x^{2} + [(a) + (–b)]x + (a)(– a)

$\Rightarrow $ (x + a) (x – a) = x^{2} + (a – a)x – a^{2}

$\Rightarrow $ (x + a) (x – a) = x^{2} + (0)x – a^{2}

$\Rightarrow $ (x + a) (x – a) = x^{2} – a^{2}.

Yes, it is related to identity (iii): (x – c)^{2} = x^{2} – 2cx + c^{2}.

##### Question 55:

Use a suitable identity to get product of (x + 3) (x + 3)

##### Answer:

##### Question 56:

Use a suitable identity to get product of (2y + 5) (2y + 5)

##### Answer:

##### Question 57:

Use a suitable identity to get product of (2a – 7) (2a – 7)

##### Answer:

##### Question 58:

Use a suitable identity to get product of

##### Answer:

##### Question 59:

Use a suitable identity to get product of (1.1m – 0.4) (1.1m + 0.4)

##### Answer:

(1.1m – 0.4) (1.1m + 0.4)

(a + b) (a – b) = a^{2} – b^{2}

a = 1.1m, b = 0.4

$\therefore $ (1.1m – 0.4) (1.1m + 0.4) = (1.1m)^{2} – (0.4)^{2}

= 1.21m^{2} – 0.16

##### Question 60:

Use a suitable identity to get product of

(a^{2} + b)^{2} (– a^{2} + b^{2})

##### Answer:

##### Question 61:

Use a suitable identity to get product of (6x – 7) (6x + 7)

##### Answer:

##### Question 62:

Use a suitable identity to get product of (– a + c) (– a + c)

##### Answer:

##### Question 63:

Use a suitable identity to get each of the following products:

##### Answer:

##### Question 64:

Use a suitable identity to get product of (7a – 9b) (7a – 9b)

##### Answer:

##### Question 65:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (x + 3) (x + 7)

##### Answer:

##### Question 66:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (4x + 5) (4x + 1)

##### Answer:

##### Question 67:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (4x – 5) (4x – 1)

##### Answer:

##### Question 68:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (4x + 5) (4x – 1)

##### Answer:

##### Question 69:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (2x + 5y) (2x + 3y)

##### Answer:

##### Question 70:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (2a^{2} + 9) (2a^{2} + 5)

##### Answer:

##### Question 71:

Use the identity (x + a) (x + b) = x^{2} + (a + b) x + ab to find the following product: (xyz – 4) (xyz – 2)

##### Answer:

##### Question 72:

##### Answer:

##### Question 73:

##### Answer:

##### Question 74:

##### Answer:

##### Question 75:

##### Answer:

a = 1, b = 0.05

$\Rightarrow $ (1 + 0.05) (1 – 0.05)

= (1)^{2} – (0.05)^{2}

= 1 – 0.0025 = 0.9975

$\therefore $ 1.05 × 9.5 = 10 × 0.9975 = 9.975