Module 5 Higher Paper 1 Non-Calculator Specimen Paper (2 Tier) 2008

1. Module 5 Higher Paper 1 Non-Calculator Specimen Paper (2 Tier) 2008

2. 1. This 3-D shape is made from 7 cubes. It is drawn on an isometric grid. (a) Tim looks down on the shape from A. One of the faces of a cube that he sees is shaded. Shade all the other faces that he sees. 1 Mark

3. 0n this grid draw a plan from A 0n this grid draw the front elevation from B 1 Mark 1 Mark

4. 2. Work out the size of angles a and b a = 40º ( alternate angle between parallel lines are equal) b = 180 – (40 + 20 ) = 120º 3 Marks

5. (b) Show that x + y + z = 180 Angle BAC = z Angles of a triangle add up to 180º Hence x + y + z = 180º 2 Marks

6. 3. Jasmin has a pond in her garden . The surface of the pond is a circle of diameter 6 metres Calculate the area of a circle of diameter 6 metres. Give your answer in terms of π. Answer Area = π X 32 = 9π 2 marks

7. Bag A contains x counters • Bag B contains 8 more counters than bag A. • Bag C contains twice as many counters as bag A Write down the number of counters in bags B and C Bag B has x + 8 counters 2 Marks Bag C has 2x counters (b) Show that the total number of counters in bags A, B and C is 4(x + 2) x + x + 8 + 2x 4x + 8 2 Marks 4 ( x + 2 )

8. 5. On the grid draw the graph y = 2x + 3 for values of x from 0 to 4 Solve 2x + 3 = 7.5 x x x = 2.25 x OR x 2x + 3 = 7.5 2x = 4.5 x = 2.25 x 2 Marks 3 Marks

9. 6. The diagram shows a scale drawing of two points, A and B (a) Measure and write down the bearing of B from A Need to measure this angle and add to 180º N Answer is 245º 1 Mark This line is South East from A 100º (b) The point C is South –East of A on a bearing of 100º from B. Draw C on the diagram C C is where the 2 lines intersect 2 Marks

10. 7. The diagram shows two triangles A and B Describe fully the single transformation that maps triangle A onto triangle B Enlargement Scale factor ½ Centre ( 1, 3 ) 3 Marks

11. 8. A cuboid is made from centimetre cubes. The area of the base of the cuboid is 5 cm2. the volume of the cuboid is 10cm3. Work out the surface area of the cuboid. State the units of your answer. Dimensions of base must be 1cm by 5 cm Height of cuboid must be 2cm Surface Area: 2 x ( 1 x 5 + 1 x 2 + 2 x 5) = 34cm2 4 Marks

12. Here are three fractions Which fraction is closest to ¼ You must show all your working The fractions are equivalent to = Closest is 3 Marks

13. 10. Solve the equation below. You must show all your working. + = 1 Multiply thro’ by 15 5 ( x + 1 ) + 3 ( x + 2 ) = 15 Multiply out brackets 5x + 5 + 3x + 6 = 15 Collect terms 8x + 11 = 15 Subtract 11 from both sides 8x = 4 Divide both sides by 8 x = 0.5 4 Marks

14. 11 (a) Expand and simplify (x + 5 ) ( x + 4 ) ( x + 5 ) ( x + 4 ) = x2 + 4x + 5x + 20 = x2 + 9x + 20 2 Marks (b) Make t the subject of the formula w = 2t + v w = 2t + v Subtract v w – v = 2t 2 marks ÷ thro’ by 2 t = ( w – v ) ÷ 2 Factorise h2 - 25 1 Mark Answer ( h + 5 ) ( h – 5 )

15. 12. Solve the equation z2 - 8z + 15 = 0 Factorise ( z – 5 ) ( z – 3 ) = 0 Either ( z – 5 ) = 0 or ( z – 3 ) = 0 z= 5 and z = 3 3 Marks

16. 13. Triangles ABC and PQR are similar. AC = 3.2cm, AB = 4cm and PR = 4.8cm (a) Explain why sinx = 0.8 (b) Calculate the length of PQ Sin x = 4.8 ÷ 3.2 = 1.5 PQ = 1.5 x 4 = 6 = 3.2 ÷ 4 = 0.8 4.8 ÷ 6 = 0.8 0r 1 Mark 3 marks

17. 14. O is the centre of the circle. A, B and C are points on the circumference. Write down the value of angle x. Answer x = 52º 1 Mark (b) P, Q and R are points on the circumference of the circle. NPT is the tangent to the circle. Calculate the value of z. Give reasons for each step of your working. Angle at Q is 52º. ( alternate segment theorem) z = 180 – ( 70 + 52) = 58º 3 marks

18. gradient of perpendicular line is 15. The diagram shows the points A (-2,2) and B(8,7) (8, 7) Find the equation of the line perpendicular to AB and passing through (0,7) ( -2, 2) required line : y = -2x + c Gradient of AB is = When x = 0 y = 7 7 = 0 + c  c = 7 3 marks Answer y = -2x + 7

19. 16. This is the graph of y= sin x for 0º ≤ x ≤ 360º Draw the graphs indicated for 0º ≤ x ≤ 360º In each case the graph of y = sin x is shown to help you X (a) y = 2 sin x X X X X Stretch scale factor 2 in the y-direction

20. Reflection in the x-axis Stretch scale factor ½ in the x-direction

21. The triangle number sequence is • 1, 3, 6, 10, 15, 21,………………. • The nth term of this sequence is given by • ½n ( n + 1) • (a) Write down an algebraic expression for the (n-1) th term. Answer : ½ (n-1)( n-1 +1) ½ n (n-1) 1 Mark (b) Prove algebraically that the sum of any two consecutive triangle numbers is a square number. ½ n ( n – 1) + ½ n ( n + 1 ) ½ n2 - ½n + ½ n2 + ½ n n2 3 Marks

22. 18. A shape is made from two trapezia The area of the shape is given by A = ( a + b ) + ( a + h) Rearrange the formula to make a the subject Multiply both sides by 2: 2A = ah + bh + ab + bh 2A – 2bh = ah + ab 2A – 2bh = a(h + b) a =

23. OPQR is a parallelogram. M is the midpoint of the diagonal OQ. • OP = 2p and OR = 2r (a) Express OM in terms of p and r OQ = OP + PQ OQ = 2p + 2r OM = ½ OQ OM = p + r 1 Mark (b) Use vectors to show that M is the midpoint of PR PM = PO + OM PM = -2p + p + r PM = - p + r PR is twice PM Hence M is midpoint of PR PR = RO + OP PR = -2p + 2r= 2 ( -p + r ) 3 Marks

24. Total: out of 70 - a rough guide