Verulam School Mathematics Year 9 Revision Material (with answers) Page 1
Q1. (a) Simplify a 2 a 4 Answer... (b) Simplify b 9 b 3 Answer... (c) Simplify c 5 c c 5 Answer... (Total 3 marks) Q2. (a) Expand and simplify (x + 4) 2 Answer... (b) Hence or otherwise, show that (x + 4) 2 4(x + 4) x (x + 4) (Total 4 marks) Page 2
Q3. (a) Simplify 5c + 3c c Answer... (b) Simplify 7x + 2y 3x 5y Answer... (Total 3 marks) Q4. (a) Here is a flow chart. (i) The final answer is 36 What are the two possible starting numbers?... Page 3
... Answer... or... (ii) Explain why starting with an odd number always gives an even number as the answer............. (b) The middle column below shows the steps in a Think of a Number problem. (i) Starting with a value of 5, work through the left-hand column, writing in all the missing numbers. (ii) Starting with a value of n, work through the right-hand column, writing in all the missing expressions. (3) (Total 8 marks) Page 4
Q5. (a) Solve the equation Answer x =... (3) (b) Solve the inequality 3x + 8 < 29 Answer... (Total 5 marks) Q6. The radius of the semicircle is 10 cm. Not drawn accurately Work out the area of the semicircle. State the units of your answer............. Answer... (Total 3 marks) Page 5
Q7. (a) Expand and simplify 4(2x 1) + 3(x + 6) Answer... (b) Expand x 2 (4 2x) Answer... (Total 4 marks) Q8. (a) Complete the table of values for y = x 2 2 x 2 1 0 1 2 y 2 2 1 2 (b) On the grid below, draw the graph of y = x 2 2 for values of x from 2 to +2 (c) Explain how the graph can be used to estimate the solutions of x 2 2 = 0 (Total 4 marks) Page 6
Q9. (a) Show clearly that (p + q) 2 p 2 + 2pq + q 2 (b) Hence, or otherwise, write the expression below in the form ax 2 + bx + c (2x + 3) 2 + 2(2x + 3)(x 1) + (x 1) 2 Answer... (3) (Total 4 marks) Q10. (a) Solve the inequality 3x + 7 > x + 8 Answer... (b) Make a the subject of the formula Answer... (Total 4 marks) Page 7
Q11. The diagram shows two triangles, P and Q. drawn accurately Not In which of these triangles does a 2 + b 2 = c 2? Explain your answer. Triangle... Explanation............... (Total 2 marks) Q12. You are given that c = 8.7 10 8 correct to two significant figures and d = 1.3 10 5 correct to two significant figures. Find the maximum value of. Give your answer as an ordinary number.......... Answer... (Total 3 marks) Page 8
Q13. (a) The diagram shows a circle with centre O. Work out the size of the angle marked x. Answer... degrees (b) The diagram shows a different circle with centre O. Work out the size of the angle marked y. Answer... degrees Page 9
(c) A, B and C are points on the circumference of a circle with centre O. BOC is a straight line. Angle ABC = 20 Work out the size of the angle marked z. Explain your answer. Answer... degrees (Total 4 marks) Page 10
Q14. O is the centre of the circle. Angle PRS = 134 Not drawn accurately Work out the size of the reflex angle POQ. You must show your working....... Answer... degrees (Total 3 marks) Q15. Solve the equation 3x 2 5x 7 = 0 Give your answers to 2 decimal places................ Answer... (Total 3 marks) Page 11
M 1. (a) a 6 (b) b 6 (c) c 11 [3] M2. (a) x 2 + 4x + 4x + 16 At least three terms correct x 2 + 8x + 16 (b) 4x 16 or (x + 4) (x + 4 4) x 2 + 4x = x(x + 4) or (x + 4) x [4] M3. (a) 7c (b) 4x 3y For 4x or 3y B2 [3] Page 12
M4. (a) (i) 11 8 (ii) Odd 3 = Odd Odd Odd = Odd Odd + 3 = Even Odd + Odd = Even (b) (i) 10, 16, 8 (ii) 2n 2n + 6 n + 3 [8] M5. (a) 23 2x = 15 4.6 0.4x = 3 gets allow one error 23 15 = 2x 1.6 = 0.4x 4 f.t. if awarded. ft (b) 3x < 21 3x = 21 gets iff recovered Page 13
(b) 4x 2 2x 3 each term fw such as = 2x 5 only give x < 7 Must have inequality in answer. Accept.. [5] M6. (0.5 ) π 10 10 Allow π = 3 or 3.1 or 3.14( ) 157 or 50π 155-158 cm 2 [3] M7. (a) 8x 4 + 3x + 18 Allow one error 11x + 14 fw that does not contradict is not penalised but fw such as = 25x do not award 4x 2 2x 2 = 2x 2 is, 4x 2 2x 2 = 6x 4 is BO B2 [4] Page 14
M8. (a) 1 (b) Correct curve from x = 2 to 2 ± 2 mm from integer points Four points plotted correctly from ( 2, 2) (their) ( 1, 1), (0, 2), (1, 1) and (2, 2) B2 (c) (x values at points where) curve crosses x-axis oe [3] M9. (a) Convincing algebra Must see or box method and (b) Allow one sign or coefficient error For middle term accept or ft if awarded and no further errors ft [4] Page 15
0. (a) 3x x > 8 7 x > oe (b) a + 3 = b 2 a = b 2 3 [4] 1. P and valid explanation. eg, a clear indication that P has a right angle and Q does not (all angle calculations shown must be correct) and a 2 + b 2 = c 2 (or Pythagoras rule) only works in a right angled triangle or or An incomplete/missing/incorrect explanation and either: P and 90 (and 100 ) correct and shown P (and 90 ) and 100 correct and shown (P and) 90 and 100 correct and shown B2 [2] 2. Any correct bound from 8.65 10 8, 8.75 10 8 1.25 10 5, 1.35 10 5 oe using ordinary numbers Page 16
oe maximum using accepted recurring notation Their maximum c must be > 8.7 10 8, Their minimum d must be < 1.3 10 5 if correct 7 000 SC2 Only digit 7 (eg, 70 000, 70 etc) This credits candidates whose only errors involve powers of 10, and failing to score either or because they failed to show clearly how they worked out the answer [3] 3. (a) 100 (b) 130 (c) 70 seen Full explanation. E.g 180 (90 + 20) Minimum requirement 90 20 [4] Page 17
4. Angle PRQ = 180 134 or 46 Angle POQ = 2 their 46 or 92 Reflex angle POQ = 268 [3] 5. Allow one error or [5 ± 59] 6 (5 ± 109) 6 2.57 and 0.91 SC1 2.3 and 0.45 [3] Page 18