Module 5HigherPaper 1 Non-CalculatorSpecimen Paper (2 Tier) 2008

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

0n this grid draw a plan from A

0n this grid draw the front elevation from B

1 Mark

1 Mark

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

(b) Show that x + y + z = 180

Angle BAC = z Angles of a triangle add up to 180ºHence x + y + z = 180º

2 Marks

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

4. Bag A contains x countersBag 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

Bag C has 2x counters

(b) Show that the total number of counters in bags A, B and C is 4(x + 2)

2 Marks

x + x + 8 + 2x

4x + 8

4 ( x + 2 )2 Marks

5. On the grid draw the graph y = 2x + 3 for values of x from 0 to 4

x

x

x

x

x

3 Marks

Solve 2x + 3 = 7.5

x = 2.25

OR

2x + 3 = 7.52x = 4.5x = 2.25

2 Marks

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

Answer is 245º 1 Mark

(b) The point C is South –East of A on a bearing of 100º from B. Draw C on the diagram

N

100º This line is South East

from A

CC is where the 2 lines intersect2 Marks

7. The diagram shows two triangles A and B

Describe fully the single transformation that maps triangle A onto triangle B

EnlargementScale factor ½Centre ( 1, 3 ) 3 Marks

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

9) Here are three fractions

8

3

16

5

5

2 Which fraction is closest to ¼You must show all your working

The fractions are equivalent to

80

30

80

25

80

32 = 4

1

80

20

Closest is 16

5

3 Marks

10. Solve the equation below. You must show all your working.

3

1x+ 5

2x= 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

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

÷ thro’ by 2 t = ( w – v ) ÷ 2

2 marks

Factorise h2 - 25

Answer ( h + 5 ) ( h – 5 )1 Mark

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

13. Triangles ABC and PQR are similar. AC = 3.2cm, AB = 4cm and PR = 4.8cm

(a) Explain why sinx = 0.8

Sin x = hyp

opp

= 3.2 ÷ 4= 0.8 0r 4.8 ÷ 6

= 0.8

(b) Calculate the length of PQ

4.8 ÷ 3.2 = 1.5

PQ = 1.5 x 4 = 6

1 Mark 3 marks

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

15. The diagram shows the points A (-2,2) and B(8,7)

Find the equation of the line perpendicular to AB and passing through (0,7)( -2, 2)

(8, 7)

Gradient of AB is28

27

= 2

1

gradient of perpendicular line is 1

2

required line : y = -2x + c

When x = 0 y = 77 = 0 + c c = 7

Answer y = -2x + 7 3 marks

16. This is the graph of y= sin x for 0º ≤ x ≤ 360º

Draw the graphs indicated for 0º ≤ x ≤ 360ºIn each case the graph of y = sin x is shown to help you

(a) y = 2 sin x

X

X

X

XX

Stretch scale factor 2 in the y-direction

Reflection in the x-axis

Stretch scale factor ½ in the x-direction

17. 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)

(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 nn2

1 Mark

3 Marks

18. A shape is made from two trapezia

2

h

2

b

Multiply both sides by 2: 2A = ah + bh + ab + bh 2A – 2bh = ah + ab 2A – 2bh = a(h + b)

bh

bhA

22

a =

19. 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 + PQOQ = 2p + 2rOM = ½ OQOM = p + r 1 Mark

(b) Use vectors to show that M is the midpoint of PR

PM = PO + OMPM = -2p + p + rPM = - p + r

PR = RO + OPPR = -2p + 2r= 2 ( -p + r )

PR is twice PMHence M is midpoint of PR

3 Marks

Total: out of 70 - a rough guide

grade D C B A A*

score 16 26 36 46 56