Prelim Practice Paper: Preliminary Examination — question paper

Section A: Structured questions[80 marks]

Answer all questions in the spaces provided. Show all necessary working — omission of essential working will result in loss of marks. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless told otherwise.

1. 1

   Answer all parts of this question.

   1. (a)

      Express 6,340,000 in standard form.

      Answer =[1]
   2. (b)

      Giving your answer in standard form, evaluate (2.4 × 10⁻³) ÷ (8 × 10⁶).

      [2]
   3. (c)

      Evaluate √52.4, correct to 2 decimal places.

      [1]
2. 2

   Answer all parts of this question.

   1. (a)

      Express the ratio 2.4 kg : 600 g in its simplest form.

      [2]
   2. (b)

      The price of a helmet rose from $45 to $51.30. Find the percentage increase.

      [2]
3. 3

   A loan club holds $3,600 for a member.

   1. (a)

      Find the simple interest earned in 8 months at 2.5% per year.

      [2]
   2. (b)

      Find the value of the sum after 2 years at 2.5% per year, compounded yearly.

      [2]
4. 4

   Answer all parts of this question.

   1. (a)

      Factorise x² − 11x + 24.

      [2]
   2. (b)

      Simplify (2x + 3)/(x − 1) − (x + 1)/(x − 1).

      [2]
5. 5

   Solve the following.

   1. (a)

      Solve 3x² − 5x − 2 = 0.

      [3]
   2. (b)

      Solve (x + 2)/3 = (2x − 1)/5.

      [1]
6. 6

   A curve has equation y = x² − 6x + 5.

   1. (a)

      Factorise x² − 6x + 5 and hence write down the roots of y = 0.

      [1]
   2. (b)

      By completing the square, find the coordinates of the turning point.

      [2]
   3. (c)

      Write down the equation of the axis of symmetry of the curve.

      [1]
7. 7

   In a survey of 100 shoppers, 63 buy bread, 49 buy milk, and 21 buy neither.

   1. (a)

      Find the number of shoppers who buy both.

      [2]
   2. (b)

      Find the number who buy bread only.

      [1]
   3. (c)

      A shopper is chosen at random. Find the probability that the shopper buys exactly one of the two items.

      [1]
8. 8

   Matrices M = [[3, 1], [−2, 4]] and N = [[0, 2], [5, −1]].

   1. (a)

      Find 3M.

      [1]
   2. (b)

      Find MN.

      [3]
   3. (c)

      Find M + N.

      [1]
9. 9

   Answer all parts of this question on polygon geometry.

   1. (a)

      Find the sum of the interior angles of a polygon with 7 sides.

      [1]
   2. (b)

      A regular octagon is drawn. Find its interior angle.

      [2]
   3. (c)

      In a triangle, two interior angles are 2x and 3x, and the exterior angle at the third vertex is 110°. Find x.

      [1]
10. 10

    Two similar triangles P and Q have corresponding sides in the ratio 2 : 5. Triangle P has a side of 6 cm and an area of 24 cm²; a small prism similar to a larger one has the same 2 : 5 length ratio and a volume of 48 cm³.

    1. (a)

       Find the length of the corresponding side of triangle Q.

       [1]
    2. (b)

       Find the area of triangle Q.

       [2]
    3. (c)

       Find the volume of the larger prism.

       [2]
11. 11

    A, B and C lie on a circle with centre O. Angle AOC = 120°, and B lies on the major arc AC. The radius of the circle is 9 cm.

    1. (a)

       Find angle ABC, stating the circle property used.

       [2]
    2. (b)

       M is the midpoint of the chord AC. State the size of angle OMA and give the circle property that justifies it.

       [1]
    3. (c)

       Find the length of the chord AC, correct to 3 significant figures.

       [2]
12. 12

    P is due west of Q, with PQ = 20 km. R is positioned so that angle PQR = 90°, QR = 15 km, and R is due north of Q.

    1. (a)

       Find PR.

       [2]
    2. (b)

       Find the bearing of R from P, correct to 1 decimal place.

       [3]
    3. (c)

       Find the shortest distance from Q to the road PR.

       [1]
13. 13

    A sector of a circle has radius 9 cm and angle 140°.

    1. (a)

       Find the arc length of the sector, correct to 3 significant figures.

       [2]
    2. (b)

       Find the area of the sector, correct to 3 significant figures.

       [2]
    3. (c)

       The sector is folded into a cone by joining its two straight edges. Write down the circumference of the cone’s base, with a reason.

       [1]
14. 14

    The points A(−2, 1) and B(4, 5) are marked on a grid.

    1. (a)

       Find the gradient of AB.

       [1]
    2. (b)

       Find the midpoint of AB.

       [1]
    3. (c)

       Find the equation of the line AB, in the form y = mx + c.

       [2]
15. 15

    It is given that a = (4, −3) and b = (−2, 6), written as column vectors.

    1. (a)

       Find the magnitude of a.

       [2]
    2. (b)

       Find 2a + b as a column vector.

       [1]
    3. (c)

       The vector c = (m, 8) is parallel to (1, −4). Find the value of m.

       [1]
16. 16

    Ten test scores are: 5, 7, 7, 8, 10, 11, 12, 12, 12, 16.

    1. (a)

       Find the mean score.

       [2]
    2. (b)

       Find the median score.

       [1]
    3. (c)

       Using the statistics mode of a calculator, find the standard deviation, correct to 3 significant figures.

       [2]
17. 17

    A bag contains 5 red and 4 blue counters. Two counters are drawn at random, one after the other, without replacement.

    1. (a)

       Write the probabilities on the branches of a fully labelled tree diagram.

       [1]
    2. (b)

       Find the probability that the two counters have the same colour.

       [2]
    3. (c)

       Write down the probability that the two counters have different colours.

       [1]
18. 18

    A shuttle bus on an airport route accelerates uniformly from rest to 12 m/s in 15 seconds, cruises at 12 m/s for 45 seconds, then decelerates uniformly to rest in 20 seconds.

    1. (a)

       Find the acceleration of the bus during the first stage.

       [1]
    2. (b)

       Find the total distance travelled by the bus.

       [2]
    3. (c)

       Find the average speed for the whole journey, correct to 3 significant figures.

       [2]