Exercise 2.4 Solutions | Class 9 Maths

1. Use suitable identities to find the following products:

(i) $(x + 4) (x + 10)$
Answer: $x^{2} + 14 x + 40$

(ii) $(x + 8) (x - 10)$
Answer: $x^{2} - 2 x - 80$

(iii) $(3 x + 4) (3 x - 5)$
Answer: $9 x^{2} - 3 x - 20$

(iv) $(y^{2} + \frac{3}{2}) (y^{2} - \frac{3}{2})$
Answer: $y^{4} - \frac{9}{4}$

(v) $(3 - 2 x) (3 + 2 x)$
Answer: $9 - 4 x^{2}$

2. Evaluate the following products without multiplying directly:

(i) $103 \times 107$
Write as $(100 + 3) (100 + 7) = 100^{2} + (3 + 7) 100 + 3 \cdot 7$
Answer: $11021$

(ii) $95 \times 96$
Write as $(100 - 5) (100 - 4) = 100^{2} - (5 + 4) 100 + 5 \cdot 4$
Answer: $9120$

(iii) $104 \times 96$
Write as $(100 + 4) (100 - 4) = 100^{2} - 4^{2}$
Answer: $9984$

3. Factorize using appropriate identities:

(i) $9 x^{2} + 6 x y + y^{2}$
This is $(3 x + y)^{2}$

(ii) $4 y^{2} - 4 y + 1$
This is $(2 y - 1)^{2}$

(iii) $x^{2} - \frac{y^{2}}{100}$
Write as $x^{2} - (\frac{y}{10})^{2} = (x - \frac{y}{10}) (x + \frac{y}{10})$

4. Expand each using suitable identities:

Use $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$

(i) $(x + 2 y + 4 z)^{2} = x^{2} + 4 y^{2} + 16 z^{2} + 4 x y + 16 y z + 8 x z$

(ii) $(2 x - y + z)^{2} = 4 x^{2} + y^{2} + z^{2} - 4 x y - 2 y z + 4 x z$

(iii) $(- 2 x + 3 y + 2 z)^{2} = 4 x^{2} + 9 y^{2} + 4 z^{2} - 12 x y + 12 y z - 8 x z$

(iv) $(3 a - 7 b - c)^{2} = 9 a^{2} + 49 b^{2} + c^{2} - 42 a b + 14 b c - 6 c a$

(v) $(- 2 x + 5 y - 3 z)^{2} = 4 x^{2} + 25 y^{2} + 9 z^{2} - 20 x y - 30 y z + 12 x z$

(vi) $(\frac{1}{4} a - \frac{1}{2} b + 1)^{2} = \frac{1}{16} a^{2} + \frac{1}{4} b^{2} + 1 - \frac{1}{4} a b - b + \frac{1}{2} a$

5. Factorise:

(i) $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z$
Recognize as $(2 x + 3 y - 4 z)^{2}$

(ii) $2 x^{2} + y^{2} + 8 z^{2} - 2 \sqrt{2} x y + 4 \sqrt{2} y z - 8 x z$
Write as $(- \sqrt{2} x + y - 2 \sqrt{2} z)^{2}$

6. Write the following cubes in expanded form:

Use $(x + y)^{3} = x^{3} + y^{3} + 3 x y (x + y)$ and $(x - y)^{3} = x^{3} - y^{3} - 3 x y (x - y)$

(i) $(2 x + 1)^{3} = 8 x^{3} + 12 x^{2} + 6 x + 1$

(ii) $(2 a - 3 b)^{3} = 8 a^{3} - 27 b^{3} - 36 a^{2} b + 54 a b^{2}$

(iii) $(\frac{3}{2} x + 1)^{3} = \frac{27}{8} x^{3} + \frac{27}{4} x^{2} + \frac{9}{2} x + 1$

(iv) $(x - \frac{2}{3} y)^{3} = x^{3} - \frac{8}{27} y^{3} - 2 x^{2} y + \frac{4}{3} x y^{2}$

7. Evaluate the following using identities:

(i) $99^{3} = (100 - 1)^{3} = 1000000 - 1 - 300 (99) = 970299$

(ii) $102^{3} = (100 + 2)^{3} = 1000000 + 8 + 600 (102) = 1061208$

(iii) $998^{3} = (1000 - 2)^{3} = 1000000000 - 8 - 6000 (998) = 994011992$

8. Factorise each of the following (cubic-type factorizations):

(i) $8 a^{3} + b^{3} + 12 a^{2} b + 6 a b^{2} = (2 a + b)^{3}$

(ii) $8 a^{3} - b^{3} - 12 a^{2} b + 6 a b^{2} = (2 a - b)^{3}$

(iii) $27 - 125 a^{3} - 135 a + 225 a^{2} = (3 - 5 a)^{3}$

(iv) $64 a^{3} - 27 b^{3} - 144 a^{2} b + 108 a b^{2} = (4 a - 3 b)^{3}$

(v) $27 p^{3} - \frac{1}{216} - \frac{9}{2} p^{2} + \frac{1}{4} p = (3 p - \frac{1}{6})^{3}$

9. Verify identities

(i) $x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})$ . Verified by expanding RHS.

(ii) $x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})$

Verified by expanding RHS.

(standard verifications using cube identities.)

10. Factorise sums/differences of cubes:

(i) $27 y^{3} + 125 z^{3} = (3 y + 5 z) (9 y^{2} - 15 y z + 25 z^{2})$

(ii) $64 m^{3} - 343 n^{3} = (4 m - 7 n) (16 m^{2} + 28 m n + 49 n^{2})$

11. Factorise

$27 x^{3} + y^{3} + z^{3} - 9 x y z$

Write as $(3 x)^{3} + y^{3} + z^{3} - 3 (3 x) y z$ . Using the identity $x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$ we get

$(3 x + y + z) (9 x^{2} + y^{2} + z^{2} - 3 x y - y z - 3 x z)$

12. Verify

$x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]$

Verified by expanding RHS (standard identity; equivalently derived from $(x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$

13. If $x + y + z = 0$ , show $x^{3} + y^{3} + z^{3} = 3 x y z$

From identity $x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$

If $x + y + z = 0$ RHS is $0$ , hence $x^{3} + y^{3} + z^{3} = 3 x y z$

14. Without actually calculating the cubes, find:

(i) $(- 12)^{3} + 7^{3} + 5^{3}$ . Since $- 12 + 7 + 5 = 0$ , use result of Q13: value $= 3 (- 12) (7) (5) = - 1260$

(ii) $28^{3} + (- 15)^{3} + (- 13)^{3}$ . Since $28 - 15 - 13 = 0$ , value $= 3 \cdot 28 \cdot (- 15) \cdot (- 13) = 16380$

15. Give possible expressions for length and breadth of rectangles whose areas are:

(i) Area $= 25 a^{2} - 35 a + 12$ . Factorize:

$25 a^{2} - 35 a + 12 = (5 a - 3) (5 a - 4)$

Possible dimensions: $5 a - 4$ and $5 a - 3.$

(ii) Area $= 35 y^{2} + 13 y - 12$ . Factorize:

$35 y^{2} + 13 y - 12 = (7 y - 3) (5 y + 4)$

Possible dimensions: $7 y - 3$ and $5 y + 4.$

16. Possible expressions for dimensions of cuboids with given volumes:

(i) Volume $= 3 x^{2} - 12 x = 3 x (x - 4)$ . Possible dimensions: $3, x, x - 4$

(ii) Volume $= 12 k y^{2} + 8 k y - 20 k = 4 k (3 y^{2} + 2 y - 5) = 4 k (3 y + 5) (y - 1)$ . Possible dimensions: $4 k, 3 y + 5, y - 1$