Expressions and Formulae

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1 Topic by Topic Revision Part 1 Expressions and Formulae Attempt Section A Start with Surds and Indices Q1 (a) (c). If you can attempt these three questions successfully (without checking your notes or asking for help) go on to Q2. If you check your notes or ask for help, then attempt the rest of Q1 before moving on to Q2. Repeat the process above for Q2 and Q3. You must attempt Q4. Now attempt the questions for Multiplying Out Brackets and Factorisation; Algebraic Fractions and Gradient, Sectors and Volume. If you feel you are repeating a skill you have mastered, then it is reasonable to move on to the next question or topic. Attempt all questions in Section B You will have a week to complete this revision before you are assessed under exam conditions.

2 Section A Surds and Indices 1. Simplify (a) 27 (b) 12 (c) 32 (d) 75 (e) 48 (f) 8 (g) 50 (h) 125 (i) Simplify (a) x 4 x 3 x 5 (b) y 3 y 6 y 2 (c) a 4 a 3 a 6 (d) (g) t 5 t t 3 (e) b 3 b 3 b f 5 f 3 f 3 (h) s 5 s 1 (f) s 3 (i) r 3 r 2 r 1 d 2 d 7 d 3 3. Simplify (a) 4x 2 2x 3 (b) 3x 3 5x 5 (c) 2x 3 6x 1 (d) 5x 2 3x 1 2 (e) 3x 2 7x 1 3 (f) 8x 3 2x 1 2 (g) 4x 2 3x 1 2 (h) 3x 3 10x 1 3 (i) 9x 2 3x A satellite travels miles in a day. A higher orbit satellite travel 12 times this distance each day. Calculate the distance the higher orbit satellite travels each day. Give your answer in scientific notation.

3 Multiplying Out Brackets and Factorisation 1. Multiply out the brackets (a) t(3t u) (b) (a + 3)(a + 4) (c) y(2y z) (d) 2b(3b c) (e) (d + 2)(d + 5) (f) x(4x + 5) (g) (e + 7)(e + 3) (h) (g + 4)(g + 6) (i) 5a(3a 2) 2. Factorise Fully (a) p 2 3p (b) x 2 + 5x + 6 (c) x 2 + 5x (d) x 2 25 (e) x x + 24 (f) x 2 1 (g) x 2 + 6x + 8 (h) a 2 + 4a (i) r 2 4 (j) x x + 21 (k) y 2 16 (l) 5p 2 15pq 3. Express each of the following in the form (x + p) 2 + q. (a) x x + 3 (b) x 2 6x + 7 (c) x 2 + 8x + 4 (d) x 2 10x + 6 (e) x 2 + 4x + 9 (f) x 2 + 2x 2 (g) x 2 + 6x 9 (h) x 2 14x + 2 (i) x x 3

4 Algebraic Fractions 1. Write each algebraic fraction in its simplest form. (a) (2x 3)(x+1) (x+1) 2, x 1 (b) (3x 1)(x+2) (x+2) 2, x 2 (c) (x+3)(2x+1) (x+3) 2, x 3 (d) (x 1)(x+1) (x 1) 2, x 1 (e) (5x 2)(x+1) (x 3)(x+1), x 1 or 3 (f) (x 1) 2 (x 1)(x+2), x 2 or 1 (g) (x+3) 2 (x 2)(x+3), x 3 or 2 (h) (x 1)(x+8) (x 1)(x+1), x 1 or 1 2. Write each of the following as a single fraction (a) 2 3 x, y 0 x y (b) x t t, x, y 0 6 y (c) 3 x + 4 y x, y 0 (d) x t t, x, y 0 8 y (e) x, y 0 x y (f) 1 x 1 y x, y 0 (g) x, y 0 x y (h) x t t, x, y 0 10 y (i) 9 x 2 y x, y 0 (j) x t t, x, y 0 5 y (k) x t t, x, y y (l) 5 x 3 y x, y 0

5 Gradient, Sectors and Volume 1. Find the gradient of the straight line between the two given points (a) T(3, 2) and R(4, 4) (b) A( 1, 3) and Q(4, 8) (c) C( 3, 2) and S(7, 3) (d) V(0, 3) and L( 3, 9) (e) B(1, 4) and H( 1, 2) (f) G( 3, 4) and W( 1, 8) (g) K(9, 2) and N(5, 12) (h) X( 7, 4) and E( 3, 2) 2. Find the volume of the following shapes and round your answer to the given number of significant figures (a) Find the volume of a sphere of radius 1 8 metres. Give your answer to 2 significant figures. 1 8 m (b) Find the volume of a cone of radius 12 centimetres and height 15 centimetres. 15 cm Give your answer to 2 significant figures. 12 cm

6 (c) Find the volume of a pyramid with a base of 230 square centimetres and a height of 25 centimetres. Give your answer to 2 significant figures. 25 cm 230 cm 2 50 cm 0 12 m (d) Find the volume of a cylinder of radius 0 12 metres and height 50 centimetres. Give your answer to 2 significant figures. (e) Find the volume of a sphere of radius 6 12 centimetres. Give your answer to 3 significant figures cm (f) Find the volume of a cone of radius 3 4 metres and height 7 metres. 7 m Give your answer to 3 significant figures. 3 4 m

7 3. Calculate the length of the minor arc giving your answer to one decimal place. (a) (b) O 120 o 15 cm 12 cm O 95 o A B C D 4. Calculate the area of the minor sector giving your answer to one decimal place. (a) (b) O 11 cm 80 o A O 72 o 9 cm B C D

8 Section B Mixed Questions 1. (a) Simplify 2 18 (b) Simplify (c) Hence show that 2 18 = Remove the brackets and simplify a 1 2 (a 1 2 2) 3. Simplify m 3 m 4. Express p 3 (p 2 p 3 ) in its simplest form 5. Factorise x 2 5x Multiply out the brackets and collect like terms (x + 5)(2x 2 3x 1) 7. A right angles triangle has dimensions shown. Calculate the length of AC, leaving your answer as a surd in its simplest form.

9 8. (a) Factorise x 2 + x 6 (b) Multiply out the brackets and collect like terms (3x + 2)(x 2 + 5x 1) 9. (a) Simplify m 5 m 3 (b) Express as a surd in its simplest form. 10. Remove the brackets and simplify (2x + 3) 2 3(x 2 6) 11. Factorise fully 5x Given that x 2 10x + 18 = (x a) 2 + b find the values of a and b. 13. Expand and fully simplify x(x 1) Express 5p2 8 p 2 as a fraction in its simplest form. 15. One atom of gold weighs grams. How many atoms will there ne in one kilogram of gold? Give your answer in scientific notation correct to 2 significant figures. 16. There are platelets per millilitre of blood. On average, a person has 5 5 litres of blood. On average, how many platelets does a person have in their blood? Give your answer in scientific notation.

17. Simplify ab 6 a 3 b 2. 18. A glass ornament in the shape of a cone is part filled with coloured water. What is the volume of water.

10 17. Simplify ab 6 a 3 b A glass ornament in the shape of a cone is part filled with coloured water. What is the volume of water. Give your answer correct to 2 significant figures. 19. A square of side x centimetres has a diagonal 6 centimetres long. Calculate the value of x, giving your answer as a surd in its simplest form.

$20. Simplify 3x 15 (x 5) 2 21. Express 3 4, x 0, x 1 as a single fraction in its simplest form. x x+1 22. A cod liver oil capsule is shown in the diagram below.$

11 20. Simplify 3x 15 (x 5) Express 3 4, x 0, x 1 as a single fraction in its simplest form. x x A cod liver oil capsule is shown in the diagram below. Each capsule is in the shape of a cylinder with hemispherical ends. Calculate the volume of the cod liver oil capsule. 23. The battle of Largs in 1263 is commemorated by a monument known as The Pencil. This monument is in the shape of a cylinder with a cone on top. The cylinder part has a 3-metre diameter and a height of 15 metres. (a) Calculate the volume of the cylindrical part of the pencil. The volume of the cone part of The Pencil is 5 7 cubic metres. (b) Calculate the total height of the pencil.

12 24. Express s2 t 3t 2s as a fraction in its simplest form. 25. Express 1 2, p 0, p 5 as a single fraction in its simplest form. p (p+5) Answers Section A Surds and Indices Q1 (a) 3 3 (b) 2 3 (c) 4 2 (d) 5 3 (e) 4 3 (f) 2 2 (g) 5 2 (h) 5 5 (i) 2 5 Q2 (a) x 2 (b) y 7 (c) a (d) t 3 (e) b 5 (f) r 6 (g) f 11 (h) s (i) d 2 Q3 (a) 8x 5 (b) 15x 8 (c) 12x 2 (d) 15x 5 2 (e) 21x 7 3 (f) 16x 7 2 (g) 12x 3 2 (h) 30x 8 3 (i) 27x 3 2 Q Multiplying Out Brackets and Factorisation Q1 (a) 3t 2 tu (b) a 2 + 7a + 12 (c) 2y 2 yz (d) 6b 2 2bc (e) d 2 + 7d + 10 (f) 4x 2 + 5x (g) e e + 21 (h) g g + 24 (i) 15a 2 10a Q2 (a) p(p 3) (b) (x + 3)(x + 2) (c) x(x + 5) (d) (x 5)(x + 5) (e) (x + 12)(x + 2) (f) (x 1)(x + 1) (g) (x + 4)(x + 2) (h) a(a + 4) (i) (r 2)(r + 2) (j) (x + 7)(x + 3) (k) (y 4)(y + 4) (l) 5p(p 3q)

13 Q3 (a) (x + 7) 2 46 (b) (x 3) 2 2 (c) (x + 4) 2 12 (d) (x 5) 2 19 (e) (x + 2) (f) (x 1) 2 3 (g) (x + 3) 2 18 (h) (x + 5) 2 28 Algebraic Fractions Q1 (a) 2x 3 x+1 (b) 3x 1 x+2 (c) 2x+1 x+3 (d) x+1 x 1 (e) 5x 2 x 3 (f) x 1 x+2 (g) x+3 x 2 (h) x+8 x+1 Q2 (a) 2y 3x xy (b) tx 6y (c) 3y+4x xy (d) xy 8t (e) 3y+5x xy (f) y x xy (g) 7y+3x xy (h) xt 10y (i) 9y 2x xy (j) xy 5t (k) tx 100y (l) 5y 3x xy Gradient, Sectors and Volume 1. (a) 2 (b) 1 (c) (d) -2 (e) 3 (f) 2 (g) 5 2 (h) 2. (a) 24m³ (b) 2300cm³ (c) 1900cm³ (d) 0 023m³/23000cm³ (e) 960cm³ (f) 84 7m³ 3. (a) 31 4cm (b) 19 9cm 4. (a) 84 5cm² (b) 50 9cm² Section B 1. (a) 6 (b) 4 2 (c) 2. a 2a m p (x 8)(x + 3) 6. 2x 3 + 7x x (a) (x 2)(x + 3) (b) 3x x 2 + 7x

14 9. (a) m 2 (b) x x (x 3)(x + 3) 12. a = 5, b = x 3 2x 2 + x 14. 5p b 4 a cm x = x x x 2 +x mm 2 23 (a) 106 m 3 (b) 17 4 m 24 3s p+5 p 2 +5p

CFE National 5 Resource

Pegasys Educational Publishing CFE National 5 Resource Unit Expressions and Formulae Homework Exercises Homework exercises covering all the Unit topics + Answers + Marking Schemes Pegasys 0 National 5