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. You may use lg 2 = 0.30 and lg 3 = 0.48.

1. 1

   It is given that 2 sin θ − 5 cos θ can be written as R sin(θ − α), where R > 0 and 0° < α < 90°.

   1. (a)

      State the value of R, giving your answer correct to 3 significant figures, and the value of α correct to 1 decimal place.

      [4]
   2. (b)

      Write down the maximum value of 2 sin θ − 5 cos θ.

      maximum =[1]
   3. (c)

      Hence find the values of θ in 0° ≤ θ ≤ 360° satisfying 2 sin θ − 5 cos θ = 4, correct to 1 decimal place.

      [3]
2. 2

   It is given that cos A = 5/13, where A is an acute angle. Do not use a calculator in this question.

   1. (a)

      Write down the value of sin A.

      sin A =[1]
   2. (b)

      Find the exact values of sin 2A and cos 2A.

      [4]
   3. (c)

      Hence explain, without further calculation, whether A is greater or less than 45°.

      [2]
3. 3

   The points A, B and C lie on a circle with centre O, and angle AOB = 140°, standing on the chord AB. C lies on the major arc AB. The tangent to the circle at A is drawn.

   1. (a)

      Find angle ACB.

      [2]
   2. (b)

      Find angle OAB, explaining your method.

      [3]
   3. (c)

      Find the angle between the tangent at A and the chord AB, and name the theorem used.

      [4]
4. 4

   In triangle PQR, the point M lies on PQ with PM = 6 cm and MQ = 6 cm, and the point N lies on PR with PN = 5 cm and NR = 5 cm.

   1. (a)

      Show that triangles PMN and PQR are similar.

      [3]
   2. (b)

      Given QR = 14 cm, find MN. State the name of the theorem also implied by this result.

      [3]
   3. (c)

      Given that the area of triangle PMN is 8 cm², find the area of triangle PQR.

      [2]
5. 5

   The variables x and y satisfy y = kbˣ, where k and b are constants. The graph of lg y against x is a straight line with gradient 0.30 passing through the point (0, 0.48). You may use lg 2 = 0.30 and lg 3 = 0.48.

   1. (a)

      Find the values of k and b.

      [4]
   2. (b)

      Estimate the value of y when x = 5.

      y =[2]
   3. (c)

      Find the value of x for which y = 12.

      x =[2]
6. 6

   Show that the expression (1 − cos 2A)/sin 2A simplifies to tan A.

   1. Prove the identity with a one-sided layout, quoting each identity you use.

      [8]
7. 7

   The curve y = x³ − 3x² − 9x + 7 has two stationary points.

   1. (a)

      Find dy/dx and show that the x-coordinates of the stationary points are 3 and −1.

      [4]
   2. (b)

      Find the coordinates of each stationary point and determine the nature of each.

      [6]
8. 8

   Calculate the area of the finite region between the parabola y = 4 − x² and the line y = 3x.

   1. Show the intersection working, the integration, and the evaluation at both limits.

      [10]
9. 9

   A particle moves along a straight line with velocity v = t² − 6t + 8 m/s for t ≥ 0, and its displacement is 3 m when t = 0.

   1. (a)

      Find the times at which the particle is at rest.

      [2]
   2. (b)

      Find an expression for the acceleration at time t.

      [2]
   3. (c)

      Find an expression for the displacement s in terms of t.

      [3]
   4. (d)

      Find the displacement of the particle from its starting point when t = 5.

      [2]
   5. (e)

      Find the total distance travelled in the first 5 seconds.

      [3]