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Formulae List

The roots of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 are 𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎

Sine rule: 𝑎

sin𝐴=

𝑏

sin𝐵=

𝑐

sin𝐶

Cosine Rule: 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐cos𝐴 or cos𝐴 =𝑏2+𝑐2−𝑎2

2𝑏𝑐

Area of a triangle: 𝐴 =1

2𝑎𝑏sin𝐶

Volume of a sphere: 𝑉 =4

3𝜋𝑟3

Volume of a cone: 𝑉 =1

3𝜋𝑟2ℎ

Volume of a pyramid: 𝑉 =1

3𝐴ℎ

Standard deviation: 𝑠 = √Σ(𝑥−�̅�)2

𝑛−1= √

Σ𝑥2−(Σ𝑥)2/𝑛

𝑛−1, where n is the sample size

St Andrew’s High School National 5 Mathematics S4 Practice Prelim Papers

Paper 1 (Non-calc) Duration – 1 hour Total marks – 40 Paper 2 (Calc) Duration – 1 hour 30 minutes Total marks – 50

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Practice Prelim A – Paper 1 1. Calculate

12 + 8.4 ÷ 7 1

2. Calculate

12

7×

5

9 2

3. (a) Multiply out the brackets and collect like terms

)25)(32( 2 xxx 3

(b) Factorise: (i) 81𝑦2 − 16𝑥2

(ii) 972 2 xx 4 4. Two functions are given below.

35)(

12)( 2

xxg

xxxf

Find the values of x for which )()( xgxf . 3

5. Andrew and Doreen each book in at the Sleepwell Youth Hostel. (a) Andrew stays for 3 nights and has breakfast on 2 mornings. His bill is £34 Write down an algebraic equation to illustrate this. 1 (b) Doreen stays for 5 nights and has breakfast on 4 mornings. Her bill is £58. Write down an algebraic equation to illustrate this. 1 (c) Find the cost of one breakfast. 3

6. Express 2

2

5

2

1

a

aa in its simplest form. 2

7. Find |u|, the magnitude of vector u =

3

2

6

. 2

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8. Shown below is the straight line which goes through A(-2,-1) and B(3,9).

(a) Find the equation of the straight line AB 3 (b) Find the point where the line cuts the y-axis 1 (c) Find the point where the line cuts the x-axis 1

9. Triangle PQR is shown.

If sin P = , calculate the area of triangle PQR. 3

1

4

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10. The diagram below shows part of the graph 344 2 xxy

The graph cuts the y-axis at A and the x-axis at B and C.

(a) Write down the coordinates of A. 1 (b) Find the coordinates of B and C . 3

(c) Calculate the minimum value of 344 2 xx . 2 11. A mirror is shaped like part of a circle.

The radius of the circle, centre C, is 10 centimetres. The height of the mirror is 18 centimetres. Calculate the length of the base of the mirror, represented in the diagram by AB. 4

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Practice Prelim A – Paper 2 1. One hundred milligrams of an antibiotic are given to a patient. At the end of each hour the number of milligrams of antibiotic in the body is 20% less than at the

beginning of that hour. How many milligrams of antibiotic remain in the body at the end of three hours? 3 2. The annual profit (£) of a company was 3·2 × 109 for the year 1997. What profit did the company make per second? Give your answer to three significant figures. 3

3. The diagram below shows the graph of 2axy .

Find the value of a. 2

4. Triangle DEF is shown below

It has sides of length 10.4 metres, 13.2 metres and 19.6 metres. Calculate the size of angle EDF. Do not use a scale drawing. 3 5. Solve the equation 𝑥2 − 6𝑥 − 2 = 0, giving the roots correct to one decimal place. 4

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6. A child’s toy is in the shape of a hemisphere with a cone on top, as shown in the diagram. The toy is 10 centimetres wide and 16 centimetres high. Calculate the volume of the toy. Give your answer correct to 2 significant figures. 5

7. Marmalade is on special offer. Each jar on special offer contains 12·5% more than a standard jar. A jar on special offer contains 450g of marmalade. How much does the standard jar contain? 3 8. Two fridge magnets are mathematically similar. One magnet is 4 centimetres long and the other is 10 centimetres long.

The area of the smaller magnet is 18 square centimetres. Calculate the area of the larger magnet. 3 9. The top of a garden table is in the shape of a regular hexagon.

The three diagonals of the hexagon which are shown as dotted lines in the diagram below each have length 50 centimetres.

Calculate the area of the top of the table. 4

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10. The heights, in centimetres, of seven netball players in Team A are given below. 173 176 168 166 170 180 171 For this sample, calculate: (a) the mean; 1 (b) the standard deviation. Show clearly all your working. 3 (c) Team B also has seven players. Their mean height is 170 and their standard deviation is 6·3. Make two valid comparisons of the heights of the teams 2

11. Part of the graph of axby cos is shown in the diagram.

State the values of a and b 2 12. A cone is formed from a paper circle with a sector removed as shown. The radius of the paper circle is 30 cm. Angle AOB is 100o.

(a) Calculate the area of paper used to make the cone. 3 (b) Calculate the circumference of the base of the cone. 3

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13. (a) There are three mooring points A, B and C on Lake Sorling.

From A, the bearing of B is 074o. From C, the bearing of B is 310o. Calculate the size of the angle ABC. 2 (b) B is 230 metres from A and 110 metres from C. Calculate the direct distance from A to C. Give your answer to 3 significant figures. 4 To view answers visit department website: sahsmathematics.wikispaces.com

Practice Prelim B – Paper 1

1. Calculate 31

4÷

5

8 2

2. 52 bRP Change the subject of the formula to R. 3

3. Multiply out the brackets and collect like terms 8𝑥 + (3𝑥 − 2)(2𝑥 + 5). 3

4. Solve the inequality 5 2( 1)x x 3

5. 2( ) 2 5f x x x

Find ( 2)f 2

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6. A rectangular picture frame is to be made. It is 80 centimetres high and 60 centimetres wide, as shown. To check that the frame is rectangular, the diagonal, d, is measured. It is 1 metre long. Is the frame rectangular? 4 Justify your answer.

7. A parabola has equation 𝑦 = 𝑥2 + 4𝑥 − 6. (a) Write the equation in the form 𝑦 = (𝑥 + 𝑝)2 + 𝑞. 2 (b) State the co-ordinates of the turning point of the parabola. 1

8. Solve algebraically the system of equations

5x + 3y = 11

3x 2y = 18 3

9. Simplify

3

24 Express your answer as a fraction with a rational denominator 3

10. A straight line is represented by the equation 2x + y − 4 = 0.

(a) The gradient of this line is

A −4 B −2 C 1 D 2 Write down the letter that corresponds to the correct gradient. 1

(b) Write down the coordinates of the point where this line crosses the y-axis. 1

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11. QR and TR are tangents to the circle. The angle TQR = 65°. Calculate the size of angle QPT. 3

12. Express

2 3 2( )y y y

in its simplest form. 2

13. In triangle ABC:

angle ACB = 90° AB = 8 centimetres AC = 4 centimetres. Calculate the length of BC. Give your answer as a surd in its simplest form. 3

14. The equilateral triangle and rectangle shown below have the same perimeter. Find an expression for the breadth of the rectangle. 4

O R

Q

T P

65°

A

4 cm 8 cm

C B

breadth 2x + 2

2x 1

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Practice Prelim B – Paper 2

1. The value of a boat decreased from £35 000 to £32 200 in one year.

(a) What was the percentage decrease? 1

(b) If the value of the boat continued to fall at this rate, what would its value be after a further 3 years? Give your answer to the nearest hundred pounds. 3

2. Factorise

a) 100 − 81𝑦2 b) 862 xx 3 3.

3 4. The sketch shows a parallelogram PQRS.

(a) Calculate the size of angle PQR

Do not use a scale drawing. 3 (b) Calculate the area of the parallelogram. 2

P

Q R

S

12.6 cm 8.4 cm

11.2 cm

Find the equation of the straight line above in terms of K and L.

𝑳

𝑲 0

6

(10, -4)

3

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5. The temperature, in degrees Celsius, at mid-day in a seaside town and the sales, in pounds, of umbrellas are shown in the scattergraph below. A line of best fit has been drawn.

(a) Find the equation of the line of best fit. 3

(b) Use your answer to part (a) to predict the sales for a day when the temperature is 32 degrees Celsius. 1

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6.

The equation of the parabola in the above diagram is

2

3 16y x .

(a) State the coordinates of the minimum turning point of the parabola. 2 (b) Find the coordinates of C. 2 (c) A is the point (-1, 0). State the coordinates of B. 1

7. The diagram shows a cube placed on top of a cuboid, relative to the coordinate axes.

A is the point (11, 5, 8). (a) Write down the coordinates of B and C. 2 (b) Find the components of the vector B. 1

8. Bottles of juice should contain 50 millilitres.

The contents of 7 bottles are checked in a random sample. The actual volume in millilitres are as shown below.

52 50 51 49 52 53 50

Calculate the mean and standard deviation of the sample. 4

y

xA

C

BO

(11, 5, 8)

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9. Prove that

𝑠𝑖𝑛2𝐴

1−𝑠𝑖𝑛2𝐴= 𝑡𝑎𝑛2𝐴

2

10. (a) Simplify

3227 2

(b) Express in its simplest form

238 )( yy 2

11. A cylindrical paperweight of radius 3 centimetres and height 4 centimetres is filled with sand.

Calculate the volume of sand in the paperweight. 2 (b) Another paperweight, in the shape of a hemisphere, is filled with sand.

It contains the same volume of sand as the first paperweight. Calculate the radius of the hemisphere. 3

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12. A railway goes through an underground tunnel.

The diagram below shows the cross-section of the tunnel. It consists of part of a circle

with a horizontal base.

The centre of the circle is O. XY is a chord of the circle. XY is 2∙2 metres. The radius of the circle is 3∙7 metres.

Find the height of the tunnel. 4

13. A helicopter, at point H, hovers between two aircraft carriers at points A and B which are 1250 metres apart.

From carrier A, the angle of elevation of the helicopter is 60.

From carrier B, the angle of elevation of the helicopter is 50. Calculate the distance from the helicopter to the nearer carrier. 4

To view answers visit department website: sahsmathematics.wikispaces.com

B A

H

1250 m

50 60

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Practice Prelim A – Paper 1 1. 13.2

2. 5

7

3. a. 2𝑥3 − 7𝑥2 − 11𝑥 + 6 b.

i. (9y-4x)(9y+4x) ii. (2x-9)(x+1)

4. x = -1 or x = 4 5.

a. 3n + 2b = 34 b. 5n + 4b = 58 c. b = 2

6. a 7. 7 8.

a. 𝑦 + 1 = 2(𝑥 + 2) or 𝑦 − 9 = 2(𝑥 − 3) b. (0,3)

c. (−3

2, 0)

9. A = 40cm2. 10.

a. A(0, -3)

b. (−3

2, 0) (

1

2, 0)

c. y = -4 11. 12cm

Practice Prelim B – Paper 1

1. 26

5= 5

1

5

2. 𝑅 = √𝑃+5

𝑏

3. 6𝑥2 + 19𝑥 − 10 4. 𝑥 < 1 5. -24 6. Yes, from Pythagoras (a2 = b2 + c2) LHS = 10000cm and RHS =

10000cm 7.

a. 𝑦 = (𝑥 + 2)2 − 10 b. TP(-2, -10)

8. x=4, y=-3

9. 3√2

2

10. a. B b. (0, 4)

11. 65o

12. 𝑦5 − 1

13. 4√3 14. 𝑏 = 𝑥 + 4

Practice Prelim A – Paper 2 1. 51.2mg 2. £101 per second 3. a = -3 4. D = 111.75o 5. x = -0.3 or x = 6.3 6. 550cm3 7. 400g 8. 112.5cm2 9. 1623.7976cm2 10.

a. 172 b. 4.7958 c. On average, Team A’s players are taller as their

mean is higher Results from Team A are less varied as their standard deviation is lower.

11. a = 4 and b = 5 12.

a. 2042.035cm2 b. 136.136cm

13. a. 124o b. 305.44m

Practice Prelim B – Paper 2 1.

a. 8% decrease b. £25 073.75

2. a. (10 − 9𝑦)(10 + 9𝑦) b. (𝑥 − 4)(𝑥 − 2)

3. L = -K + 6 4.

a. Q = 78.58 b. Area = 46.11cm2

5. a. S = -4T + 130 b. £2

6. a. TP(3, -16) b. C(0, -7) c. (7, 0)

7. a. B(11, 5, 13), C(6, 0, 13)

b. 𝐵 (115

13)

8. a. Mean = 51, SD = 1.4142

9. 𝑠𝑖𝑛2𝐴

1−𝑠𝑖𝑛2𝐴=

𝑠𝑖𝑛2𝐴

𝑐𝑜𝑠2𝐴= 𝑡𝑎𝑛2𝐴

10.

a. 5√3 b. y2

11. a. 113.097cm3 b. 3.7798cm

12. 7.23m 13. 1019.009m