End of Chapter Exercises

Q1 (i)

Use suitable identities to find the following products: \((−3x + 4)^3\)

Q1 (ii)

Use suitable identities to find the following products: \((2s + 7)(2s − 7)\)

Q1 (iii)

Use suitable identities to find the following products: \((p^2 + \frac{1}{2})(p^2 − \frac{1}{2})\)

Q1 (iv)

Use suitable identities to find the following products: \((2n + 7)(2n − 7)\)

Q1 (ix)

Use suitable identities to find the following products: \((\frac{7}{2}k − \frac{2}{3}m)^3\)

Q1 (v)

Use suitable identities to find the following products: \((s − 2t)(s^2 + 2st + 4t^2)\)

Q1 (vi)

Use suitable identities to find the following products: \((\frac{1}{2r} − 4r)^2\)

Q1 (vii)

Use suitable identities to find the following products: \((−3m + 4k − l)^2\)

Q1 (viii)

Use suitable identities to find the following products: \((x − \frac{1}{3}y)^3\)

Q2 (i)

Find the values using suitable identities: 17 × 21

Q2 (ii)

Find the values using suitable identities: 104 × 96

Q2 (iii)

Find the values using suitable identities: 24 × 16

Q2 (iv)

Find the values using suitable identities: \(147^3\)

Q2 (v)

Find the values using suitable identities: \(199^3\)

Q2 (vi)

Find the values using suitable identities: \(127^3\)

Q2 (vii)

Find the values using suitable identities: \((−107^3)\)

Q2 (viii)

Find the values using suitable identities: \((−299^3)\)

Q3 (i)

Factor the following algebraic expressions: \(4y^2 + 1 + \frac{1}{16y^2}\)

Q3 (ii)

Factor the following algebraic expressions: \(9m^2 − \frac{1}{25n^2}\)

Q3 (iii)

Factor the following algebraic expressions: \(27b^3 − \frac{1}{64b^3}\)

Q3 (iv)

Factor the following algebraic expressions: \(x^2 + \frac{5x}{6} + \frac{1}{6}\)

Q3 (ix)

Factor the following algebraic expressions: \(9x^3 − \frac{8}{3}y^3 + \frac{z^3}{3} + 6xyz\)

Q3 (v)

Factor the following algebraic expressions: \(27u^3 − \frac{1}{125} − \frac{27u^2}{5} + \frac{9u}{25}\)

Q3 (vi)

Factor the following algebraic expressions: \(63y^3 + \frac{1}{125}z^3\)

Q3 (vii)

Factor the following algebraic expressions: \(p^3 + 27q^3 + r^3 − 9pqr\)

Q3 (viii)

Factor the following algebraic expressions: \(9m^2 − 12m + 4\)

Q3 (x)

Factor the following algebraic expressions: \(4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy\)

Q3 (xi)

Factor the following algebraic expressions: \(27u^3 − \frac{1}{216} − \frac{9u^2}{2} + \frac{u}{4}\)

Q4 (i)

Simplify the following: \(\frac{4x^2 + 4x + 1}{4x^2 − 1}\)

Q4 (ii)

Simplify the following: \(\frac{9(3a^3 − 24b^3}{9a^2 − 36b^2}\)

Q4 (iii)

Simplify the following: \(\frac{s^3 + 125t^3}{s^2 − 2st − 35t^2}\)

Q5 (i)

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units. \(25a^2 − 30ab + 9b^2\)

Q5 (ii)

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units. \(36s^2 − 49t^2\)

Q6 (i)

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units. \(6a^2 − 24b^2\)

Q6 (ii)

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units. \(3ps^2 − 15ps + 12p\)

Q7

The village playground is shaped as a square of side 40 metres. A path of width s metres is created around the playground for people to walk. Find an expression for the area of the path in terms of s.

Q8

If a number plus its reciprocal equals \(\frac{10}{3}\), find the number.

Q9

A rectangular pool has area 2x2 + 7x + 3 square hastas. If its width is 2x + 1 hastas, find its length. Hasta was a unit used to measure length.

Q10

If both \(x − 2\) and \(x − \frac{1}{2}\) are factors of \(px^2 + 5x + r\), show that p = r.

Q11

If a+ b+ c = 5 and ab + bc + ca = 10, then prove that \(a^3 + b^3 + c^3 − 3abc = − 25\).

Q12

By factoring the expression, check that \(n^3 − n\) is always divisible by 6 for all natural numbers n. Give reasons.

Q13 (i)

Find the value of \(x^3 + y^3 − 12xy + 64\), when \(x + y = −4\).

Q13 (ii)

Find the value of \(x^3 − 8y^3 − 36xy − 216\), when \(x = 2y + 6\).