mudassir khan assignment

TOPIC 10

Algebraic Expressions & Manipulation

1. Factorise completely

(a)
4x² − 25y²
Answer (a):
(2x − 5y)(2x + 5y)
(b)
5ax − 5a² − 2x + 2a
Answer (b):
(5a − 2)(x − a)

2. (a) Factorise completely

(i)
3x² − 12x
Answer (a)(i):
3x(x − 4)
(ii)
x² − xy − 2y²
Answer (a)(ii):
(x − 2y)(x + y)

2. (b) Simplify

$\frac{x^{2} + 4 x}{x^{2} - 16}$
Answer (b):
$\frac{x (x + 4)}{(x - 4) (x + 4)} = \frac{x}{x - 4}$

4. (a) Factorise completely $16a^2-6a$

Answer (a) ..................................... $2a(8a-3)$

(b) Factorise completely $6x+3xy-4y-8$

Answer (b) ..................................... $(3x-4)(y+2)$

5. Factorise

(a) $4t^2-9$ ,

Answer (a) ..................................... $(2t-3)(2t+3)$

(b) $3x^2+5x-2$

Answer (b) ..................................... $(3x-1)(x+2)$

6. Factorise completely

(a) $12ab^2-8a^2b$

Answer ..................................... $4ab(3b-2a)$

(b) $2x^2+3x-20$

Answer ..................................... $(2x-5)(x+4)$

7. (a) Factorise completely $9pq-12q^2$
Answer ..................................... $3q(3p-4q)$
(b) Factorise completely $8px+4py-6x-3y$
Answer ..................................... $(4p-3)(2x+y)$

8. Factorise completely $2xy-3x-10y+15$
Answer ..................................... $(2y-3)(x-5)$

9. Factorise completely

(a) $12x^2-15x^3$
Answer ..................................... $3x^2(4-5x)$
(b) $x^2-x-6$
Answer ..................................... $(x-3)(x+2)$

10. (a) Factorise
(i) $x^2+x-12$
Answer ..................................... $(x+4)(x-3)$
(ii) $25x^2-4y^2$
Answer ..................................... $(5x-2y)(5x+2y)$
(b) Write as a single fraction
$\frac{4}{3p}+\frac{1}{6p}$
Answer ..................................... $\dfrac{3}{2p}$
$-----------------------------------------------------------------------------------------------------------------------------$
11. Factorise completely
(a) 20p + 25p²

Answer …………… 5p(4 + 5p)

(b) 9 − 4t²

Answer …………… (3 − 2t)

(c) 9 + 35x − 4x²

Answer …………… (9 − x)(4x + 1)
12. Expand the brackets and simplify
(a) 6k − 2(l − k) + 3

Answer …………… 8k − 2l + 3

(b) (2x − 3)(x + 4)

Answer …………… 2x² + 5x − 12
13. Factorise completely
(a) 16p + 4p²

Answer …………… 4p(4 + p)

(b) xy + 2ay + 3ax + 6a²

Answer …………… (x + 2a)(y + 3a)
14. (a) Factorise fully
10x²y + 15xy²

Answer …………… 5xy(2x + 3y)

(b) Factorise $25a^2 - b^2$
Answer …………… $(5a - b)(5a + b)$
(c) Simplify $\dfrac{3}{(x+1)^2} - \dfrac{2}{x+1}$
Answer …………… $\dfrac{1 - 2x}{(x+1)^2}$
(d) Simplify $\dfrac{3a^2}{10bc} \times \dfrac{9a}{5b^2c}$
Answer …………… $\dfrac{27a^3}{50b^3c^2}$

15. (a) Expand and simplify $(t - 5)(t + 3)$

Answer …………… $t^2 - 2t - 15$
(b) Factorise $64x^2 - 9y^2$
Answer …………… $(8 x - 3 y)$
(c) Factorise $6ab - 2a - 3a^2 + 4b$
Answer …………… $(3 a - 2) (2 b - a)$
(d) (i) Write $x^2 - 6x + 3$ in the form $(x - a)^2 + b$
Answer …………… $(x - 3)^2 - 6$
(ii) Hence solve $x^2 - 6x + 3 = 0$ leaving your answer in the form $p \pm \sqrt{q}$
Answer $x =$ …………… $3 \pm \sqrt{6}$

16. (a) Factorise $25r^2 - 4$

Answer …………… $(5 r - 2)$
(b) Factorise completely $6r^2h - 2r^2h$
Answer …………… $2 r^{2} h (3 r - 1)$
(c) Factorise completely $8xy + 4x - 6y - 3$
Answer …………… $(4x - 3)(2y + 1)$

17. (a) Solve $2(5^p) = 250$

Answer $p =$ …………… 3
(b) Simplify
(i) $1 + x^{-5}$
Answer …………… $\dfrac{x^5 + 1}{x^5}$
(ii) $\dfrac{3a}{4} \times \dfrac{9a^2}{8}$
Answer …………… $\dfrac{27a^3}{32}$

18. (a) Expand and simplify

(i) $4(2t + 3) + 5$
Answer …………… $8t + 17$
(ii) $6p + 3q - 2(2p - 5q)$
Answer …………… $2p + 13q$
-----------------------------------------------------------------------------------------------------------------------------
(b) Factorise completely
25x²y² − 15x²y

Answer 5x²y(5y − 3)

19. (a) Expand and simplify (2x + 1)(x + 4).

Answer 2x² + 9x + 4

(b) Write $\dfrac{3}{x} + \dfrac{4}{x+2}$ as a single fraction in its simplest form.

Answer $\dfrac{7x + 6}{x(x + 2)}$

(c) Solve $\dfrac{10}{x} = x + 3$

Answer x = 2 or x = −5

20. Factorise 2ac − 3bc − 6bd + 4ad.

Answer (2a − 3b)(c + 2d)

21. (a) Factorise completely 4a − 16a².

Answer 4a(1 − 4a)

(b) Factorise 9b² − c².

Answer (3b − c)(3b + c)

-----------------------------------------------------------------------------------------------------------------------------
(c) Factorise x² − 5y − xy + 5x.
Answer (x − 5)(x − y)
22. Express $\dfrac{1}{x+2} + \dfrac{3}{x+1}$ as a single fraction in its simplest form.

Answer $\dfrac{4x + 7}{(x+1)(x+2)}$
23. (a) Factorise completely 3 − 12a².
Answer 3(1 − 4a²)
(b) Factorise x² − 6y + 2xy − 3x.
Answer (x − 3)(x + 2y)

24. (a) Factorise completely p²q − pq.
Answer pq(p − 1)
(b) (i) Factorise 5x² + x − 4.
Answer (5x − 4)(x + 1)
(ii) Hence solve 5x² + x − 4 = 0.

Answer x = $\dfrac{4}{5}$ or x = −1

25. (a) Expand and simplify 10 − 3(3x − 2).
Answer 16 − 9x
(b) Simplify fully
$\dfrac{3x^2 + 16x + 5}{9x^2 - 1}$
Answer $\dfrac{3x + 1}{3x - 1}$
26. (a) Factorise
(i) 4p² − 9q².
Answer (2p − 3q)(2p + 3q)
(ii) 2n² + 5n − 3.
Answer (2n − 1)(n + 3)
(b) Express $\dfrac{3}{4x} + \dfrac{2}{3y}$ as a single fraction.

Answer $\dfrac{9y + 8x}{12xy}$

27. Factorise completely 3xy − 20 + 5x − 12y.
Answer (3x − 12)(y + 1)

28. Simplify fully
$\dfrac{4x^2 - 1}{2x^2 - 9x - 5}$
Answer $\dfrac{2x + 1}{2x - 5}$
29. Factorise completely
(a) $5 - 20r^2$
Answer $5(1 - 2r)(1 + 2r)$
(b) $3y^2 - 2xy - 6x + 9y$
Answer $(y + 3)(3y - 2x)$
30. (a) Given that $a = 3$ and $b = -7$ , evaluate
(i) $2 a - b$
Answer $13$
(ii) $a^2 + b^2$
Answer $58$
(b) $A = 2r^2 + 5$
Rearrange the formula to make $r$ the subject.
Answer $r = \sqrt{\dfrac{A - 5}{2}}$
31. Simplify fully
$\frac{4 x^{2} - 9}{2 x^{2} - 13 x + 15}$
Answer $\dfrac{2x + 3}{x - 5}$
32. (a) Simplify $8 - 3(2t + 1)$

Answer $5 - 6t$
(b) Simplify $\dfrac{(2x^2y)^3}{6x^4y^4}$
Answer $\dfrac{4x^2}{3y}$

33. (a) Factorise $25a^2 - 5a$

Answer $5a(5a - 1)$
(b) Factorise $9b^2 - 16$
Answer $(3b - 4)(3b + 4)$
(c) Factorise $4xy + 3 + 6y + 2x$
Answer $(2x + 3)(2y + 1)$

34. (a) Factorise $9a^2 - 6a$

Answer $3a(3a - 2)$
(b) Factorise $4 - 25r^2$
Answer $(2 - 5r)(2 + 5r)$
(c) Factorise $6cd - xy + 2cx - 3dy$
Answer $(2c - y)(3d + x)$

35. Factorise completely

(a) $2ax - 3by + 6bx - ay$
Answer $(a + 3 b) (2 x - y)$
(b) $27x^2 - 3y^2$
Answer $3(3x - y)(3x + y)$

36. (a) Factorise $25r^2 - 4$

Answer $(5r - 2)(5r + 2)$

(b) Factorise
$x^2 - 6x - 3xy + 18y$
Answer $(x - 6)(x - 3y)$

37. Express each of the following as a single fraction in its simplest form

(a) $\dfrac{2}{3a} + \dfrac{5}{2a}$
Answer $\dfrac{19}{6a}$
(b) $\dfrac{5}{2b^2} + \dfrac{15}{4b^3}$
Answer $\dfrac{10b + 15}{4b^3}$

38. (a) Factorise completely $15a + 3ab$

Answer $3a(5 + b)$
(b) Factorise $6 k - x y + 2 k r - 3 y$
Answer $(2k - y)(3 + r)$

39. (a) Simplify $4c - 3(2c - 5)$

Answer $15 - 2c$
(b) Factorise $8 - 10 y + 12 x - 15 x y$
Answer $(4 - 5y)(2 + 3x)$

40. Factorise

(a) $25x - 5$
Answer $5(5x - 1)$
(b) $2x^2 - 18y^2$
Answer $2(x - 3y)(x + 3y)$

41. (a) Expand and simplify $(x - 3)^2$

Answer $x^2 - 6x + 9$
(b) Factorise $18 - 6y + 5xy - 15x$
Answer $(3 - y)(6 - 5x)$

42. (a) Write $x^2 - 7x + 5$ in the form $(x - a)^2 - b$

Answer $(x - \tfrac{7}{2})^2 - \tfrac{29}{4}$
(b) Hence write down the minimum value of $x^2 - 7x + 5$
Answer $-\dfrac{29}{4}$

43. (a) Factorise $1 - 36p^2$

Answer $(1 - 6 p) (1 + 6 p)$
(b) Factorise $4x + 3y + xy + 12$
Answer $(x + 3)(y + 4)$

44. (a) Simplify $7 - 3 (5 k - 2)$

Answer $13 - 15k$
(b) Solve the equation $5x^2 - 3x = 0$
Answer $x = 0$ or $x = \dfrac{3}{5}$

45. Factorise

(a) $49 - 9r^2$
Answer $(7 - 3r)(7 + 3r)$
(b) $15xy + 5x - 6y - 2$
Answer $(3y + 1)(5x - 2)$

mudassir khan assignment

Algebraic Expressions & Manipulation

1. Factorise completely

(a)4x² − 25y²Answer (a):(2x − 5y)(2x + 5y)(b)5ax − 5a² − 2x + 2aAnswer (b):(5a − 2)(x − a)

2. (a) Factorise completely

(i)3x² − 12xAnswer (a)(i):3x(x − 4)(ii)x² − xy − 2y²Answer (a)(ii):(x − 2y)(x + y)

2. (b) Simplify

4.

(a) Factorise completely 16a2−6a16a^2-6a Answer (a) ..................................... 2a(8a−3)2a(8a-3) (b) Factorise completely 6x+3xy−4y−86x+3xy-4y-8 Answer (b) ..................................... (3x−4)(y+2)(3x-4)(y+2)

5. Factorise

(a) 4t2−94t^2-94t2−9, Answer (a) ..................................... (2t−3)(2t+3)(2t-3)(2t+3) (b) 3x2+5x−23x^2+5x-2 Answer (b) ..................................... (3x−1)(x+2)(3x-1)(x+2)

6. Factorise completely

7.

(a) Factorise completely 9pq−12q29pq-12q^2Answer ..................................... 3q(3p−4q)3q(3p-4q)(b) Factorise completely 8px+4py−6x−3y8px+4py-6x-3yAnswer ..................................... (4p−3)(2x+y)(4p-3)(2x+y)

8.

Factorise completely 2xy−3x−10y+152xy-3x-10y+15Answer ..................................... (2y−3)(x−5)(2y-3)(x-5)

9. Factorise completely

(a) 12x2−15x312x^2-15x^3Answer ..................................... 3x2(4−5x)3x^2(4-5x)(b) x2−x−6x^2-x-6Answer ..................................... (x−3)(x+2)(x-3)(x+2)

10.

15. (a) Expand and simplify (t−5)(t+3)(t - 5)(t + 3)

16. (a) Factorise 25r2−425r^2 - 4

Answer …………… (5r−2)(b) Factorise completely 6r2h−2r2h6r^2h - 2r^2hAnswer …………… 2r2h(3r−1)(c) Factorise completely 8xy+4x−6y−38xy + 4x - 6y - 3Answer …………… (4x−3)(2y+1)(4x - 3)(2y + 1)

17. (a) Solve 2(5p)=2502(5^p) = 250

Answer p=p =p= …………… 3(b) Simplify(i) 1+x−51 + x^{-5}Answer …………… x5+1x5\dfrac{x^5 + 1}{x^5}​(ii) 3a4×9a28\dfrac{3a}{4} \times \dfrac{9a^2}{8}​Answer …………… 27a332\dfrac{27a^3}{32}​

18. (a) Expand and simplify

33. (a) Factorise 25a2−5a25a^2 - 5a25a2−5a

Answer 5a(5a−1)5a(5a - 1)(b) Factorise 9b2−169b^2 - 16Answer (3b−4)(3b+4)(3b - 4)(3b + 4)(c) Factorise 4xy+3+6y+2x4xy + 3 + 6y + 2xAnswer (2x+3)(2y+1)(2x + 3)(2y + 1)

34. (a) Factorise 9a2−6a9a^2 - 6a

Answer 3a(3a−2)3a(3a - 2)(b) Factorise 4−25r24 - 25r^2Answer (2−5r)(2+5r)(2 - 5r)(2 + 5r)(c) Factorise 6cd−xy+2cx−3dy6cd - xy + 2cx - 3dyAnswer (2c−y)(3d+x)(2c - y)(3d + x)

35. Factorise completely

(a) 2ax−3by+6bx−ay2ax - 3by + 6bx - ayAnswer (a+3b)(2x−y)(b) 27x2−3y227x^2 - 3y^2Answer 3(3x−y)(3x+y)3(3x - y)(3x + y)

36. (a) Factorise 25r2−425r^2 - 425r2−4

Answer (5r−2)(5r+2)(5r - 2)(5r + 2) (b) Factorise x2−6x−3xy+18yx^2 - 6x - 3xy + 18yAnswer (x−6)(x−3y)(x - 6)(x - 3y)

37. Express each of the following as a single fraction in its simplest form

(a) 23a+52a\dfrac{2}{3a} + \dfrac{5}{2a}​Answer 196a\dfrac{19}{6a}​(b) 52b2+154b3\dfrac{5}{2b^2} + \dfrac{15}{4b^3}​Answer 10b+154b3\dfrac{10b + 15}{4b^3}​

38. (a) Factorise completely 15a+3ab15a + 3ab

Answer 3a(5+b)3a(5 + b)(b) Factorise 6k−xy+2kr−3yAnswer (2k−y)(3+r)(2k - y)(3 + r)

39. (a) Simplify 4c−3(2c−5)4c - 3(2c - 5)

Answer 15−2c15 - 2c(b) Factorise 8−10y+12x−15xyAnswer (4−5y)(2+3x)(4 - 5y)(2 + 3x)

40. Factorise

(a) 25x−525x - 52Answer 5(5x−1)5(5x - 1)(b) 2x2−18y22x^2 - 18y^2Answer 2(x−3y)(x+3y)2(x - 3y)(x + 3y)

41. (a) Expand and simplify (x−3)2(x - 3)^2

Answer x2−6x+9x^2 - 6x + 9x(b) Factorise 18−6y+5xy−15x18 - 6y + 5xy - 15xAnswer (3−y)(6−5x)(3 - y)(6 - 5x)

42. (a) Write x2−7x+5x^2 - 7x + 5x2−7x+5 in the form (x−a)2−b(x - a)^2 - b(x−a)2−b

Answer (x−72)2−294(x - \tfrac{7}{2})^2 - \tfrac{29}{4}​(b) Hence write down the minimum value of x2−7x+5x^2 - 7x + 5Answer −294-\dfrac{29}{4}​

43. (a) Factorise 1−36p21 - 36p^2

Answer (1−6p)(1+6p)(b) Factorise 4x+3y+xy+124x + 3y + xy + 12Answer (x+3)(y+4)(x + 3)(y + 4)

44. (a) Simplify 7−3(5k−2)

Answer 13−15k13 - 15k(b) Solve the equation 5x2−3x=05x^2 - 3x = 0Answer x=0x = 0x=0 or x=35x = \dfrac{3}{5}​

45. Factorise

(a) 49−9r249 - 9r^2Answer (7−3r)(7+3r)(7 - 3r)(7 + 3r)(b) 15xy+5x−6y−215xy + 5x - 6y - 2Answer (3y+1)(5x−2)(3y + 1)(5x - 2)

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(a)
4x² − 25y²
Answer (a):
(2x − 5y)(2x + 5y)
(b)
5ax − 5a² − 2x + 2a
Answer (b):
(5a − 2)(x − a)

(i)
3x² − 12x
Answer (a)(i):
3x(x − 4)
(ii)
x² − xy − 2y²
Answer (a)(ii):
(x − 2y)(x + y)

(a) Factorise completely $16a^2-6a$

Answer (a) ..................................... $2a(8a-3)$

(b) Factorise completely $6x+3xy-4y-8$

Answer (b) ..................................... $(3x-4)(y+2)$

(a) $4t^2-9$ ,

Answer (a) ..................................... $(2t-3)(2t+3)$

(b) $3x^2+5x-2$

Answer (b) ..................................... $(3x-1)(x+2)$

(a) Factorise completely $9pq-12q^2$
Answer ..................................... $3q(3p-4q)$
(b) Factorise completely $8px+4py-6x-3y$
Answer ..................................... $(4p-3)(2x+y)$

Factorise completely $2xy-3x-10y+15$
Answer ..................................... $(2y-3)(x-5)$

(a) $12x^2-15x^3$
Answer ..................................... $3x^2(4-5x)$
(b) $x^2-x-6$
Answer ..................................... $(x-3)(x+2)$

15. (a) Expand and simplify $(t - 5)(t + 3)$

16. (a) Factorise $25r^2 - 4$

Answer …………… $(5 r - 2)$
(b) Factorise completely $6r^2h - 2r^2h$
Answer …………… $2 r^{2} h (3 r - 1)$
(c) Factorise completely $8xy + 4x - 6y - 3$
Answer …………… $(4x - 3)(2y + 1)$

17. (a) Solve $2(5^p) = 250$

Answer $p =$ …………… 3
(b) Simplify
(i) $1 + x^{-5}$
Answer …………… $\dfrac{x^5 + 1}{x^5}$
(ii) $\dfrac{3a}{4} \times \dfrac{9a^2}{8}$
Answer …………… $\dfrac{27a^3}{32}$

33. (a) Factorise $25a^2 - 5a$

Answer $5a(5a - 1)$
(b) Factorise $9b^2 - 16$
Answer $(3b - 4)(3b + 4)$
(c) Factorise $4xy + 3 + 6y + 2x$
Answer $(2x + 3)(2y + 1)$

34. (a) Factorise $9a^2 - 6a$

Answer $3a(3a - 2)$
(b) Factorise $4 - 25r^2$
Answer $(2 - 5r)(2 + 5r)$
(c) Factorise $6cd - xy + 2cx - 3dy$
Answer $(2c - y)(3d + x)$

(a) $2ax - 3by + 6bx - ay$
Answer $(a + 3 b) (2 x - y)$
(b) $27x^2 - 3y^2$
Answer $3(3x - y)(3x + y)$

36. (a) Factorise $25r^2 - 4$

Answer $(5r - 2)(5r + 2)$

(b) Factorise
$x^2 - 6x - 3xy + 18y$
Answer $(x - 6)(x - 3y)$

(a) $\dfrac{2}{3a} + \dfrac{5}{2a}$
Answer $\dfrac{19}{6a}$
(b) $\dfrac{5}{2b^2} + \dfrac{15}{4b^3}$
Answer $\dfrac{10b + 15}{4b^3}$

38. (a) Factorise completely $15a + 3ab$

Answer $3a(5 + b)$
(b) Factorise $6 k - x y + 2 k r - 3 y$
Answer $(2k - y)(3 + r)$

39. (a) Simplify $4c - 3(2c - 5)$

Answer $15 - 2c$
(b) Factorise $8 - 10 y + 12 x - 15 x y$
Answer $(4 - 5y)(2 + 3x)$

(a) $25x - 5$
Answer $5(5x - 1)$
(b) $2x^2 - 18y^2$
Answer $2(x - 3y)(x + 3y)$

41. (a) Expand and simplify $(x - 3)^2$

Answer $x^2 - 6x + 9$
(b) Factorise $18 - 6y + 5xy - 15x$
Answer $(3 - y)(6 - 5x)$

42. (a) Write $x^2 - 7x + 5$ in the form $(x - a)^2 - b$

Answer $(x - \tfrac{7}{2})^2 - \tfrac{29}{4}$
(b) Hence write down the minimum value of $x^2 - 7x + 5$
Answer $-\dfrac{29}{4}$

43. (a) Factorise $1 - 36p^2$

Answer $(1 - 6 p) (1 + 6 p)$
(b) Factorise $4x + 3y + xy + 12$
Answer $(x + 3)(y + 4)$

44. (a) Simplify $7 - 3 (5 k - 2)$

Answer $13 - 15k$
(b) Solve the equation $5x^2 - 3x = 0$
Answer $x = 0$ or $x = \dfrac{3}{5}$

(a) $49 - 9r^2$
Answer $(7 - 3r)(7 + 3r)$
(b) $15xy + 5x - 6y - 2$
Answer $(3y + 1)(5x - 2)$