Mathematics

1. 1.

   Solve 2 sin x = 1 for 0 <= x <= 2 pi, giving your answers in terms of pi.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. sin x = 1/2 is positive in the first and second quadrants.
   2. Find the first-quadrant angle, then use pi minus that angle for the second-quadrant solution.

   Marking points

   • Rearranges to sin x = 1/2.
   • The principal solution is x = pi/6.
   • The second solution is pi - pi/6 = 5 pi/6.

   Examiner tip: Sine gives two solutions in 0 to 2 pi: x and pi - x. Check the interval for any extras.

2. 2.

   Show that tan x + cot x = 2/sin 2x.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Express everything in sine and cosine first.
   2. Use sin^2 + cos^2 = 1 on the combined numerator, then the double-angle formula.

   Marking points

   • Writes tan x + cot x = sin x/cos x + cos x/sin x.
   • Combines: (sin^2 x + cos^2 x)/(sin x cos x) = 1/(sin x cos x).
   • Uses sin 2x = 2 sin x cos x to obtain 2/sin 2x.

   Examiner tip: Start from the more complicated side and work towards the other. Do not treat the identity as an equation.

3. 3.

   Express 3 sin x + 4 cos x in the form R sin(x + alpha), where R > 0 and 0 < alpha < pi/2. Hence state the maximum value and the smallest positive x at which it occurs.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Match coefficients of sin x and cos x on both sides to get R cos(alpha) and R sin(alpha).
   2. Square and add for R; divide for tan(alpha). The sine of an angle reaches 1 at pi/2.

   Marking points

   • Expands R sin(x + alpha) = R cos(alpha) sin x + R sin(alpha) cos x and matches R cos(alpha) = 3, R sin(alpha) = 4.
   • R = sqrt(3^2 + 4^2) = 5.
   • alpha = arctan(4/3) = 0.927 radians.
   • The maximum is 5, when x + alpha = pi/2, so x = pi/2 - 0.927 = 0.644.

   Examiner tip: Check alpha is in the correct quadrant: both coefficients are positive, so alpha is acute. Use radians if asked.

4. 4.

   Solve 2cos^2 x + sin x - 1 = 0 for 0 <= x < 2 pi, giving exact answers.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Use the identity to make the equation involve only sin x, then treat it as a quadratic in sin x.
   2. Negative sine lies in the third and fourth quadrants.

   Marking points

   • Replaces cos^2 x with 1 - sin^2 x.
   • Obtains 2 sin^2 x - sin x - 1 = 0, i.e. (2 sin x + 1)(sin x - 1) = 0.
   • sin x = 1 gives x = pi/2.
   • sin x = -1/2 gives x = 7 pi/6 and x = 11 pi/6.

   Examiner tip: After factorising, do not forget the negative-sine solutions, and make sure all answers are in the given interval.

5. 5.

   A sector of a circle of radius r cm has perimeter 24 cm. Show that its area is A = 12r - r^2 cm^2, and find the maximum area and the angle of the sector at that point.

   [5 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The perimeter is two radii plus the arc length r(theta); use it to eliminate theta.
   2. The area formula (1/2) r^2 (theta) needs theta in radians.

   Marking points

   • Perimeter 2r + r(theta) = 24, so r(theta) = 24 - 2r.
   • Area = (1/2) r^2 (theta) = (1/2) r (24 - 2r) = 12r - r^2.
   • dA/dr = 12 - 2r = 0 gives r = 6.
   • Maximum area = 72 - 36 = 36 cm^2 (second derivative is -2 < 0).
   • theta = (24 - 12)/6 = 2 radians.

   Examiner tip: Include the arc length in the perimeter. Angles in sector formulae must be in radians.

6. 6.

   Solve tan 2x = 3 tan x for 0 degrees <= x <= 180 degrees.

   [6 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Write everything in terms of t = tan x using the double-angle formula, then bring all terms to one side.
   2. Factorise out t rather than cancelling it, otherwise the solutions with tan x = 0 are lost. Check by substituting back.

   Marking points

   • Uses tan 2x = 2 tan x/(1 - tan^2 x).
   • Forms 2t = 3t(1 - t^2) with t = tan x.
   • Factorises t(3t^2 - 1) = 0 and does not divide by t.
   • t = 0 gives x = 0 and x = 180.
   • t = 1/sqrt(3) gives x = 30.
   • t = -1/sqrt(3) gives x = 150.

   Examiner tip: Never divide both sides by tan x without checking tan x = 0. Also check the solutions do not make tan undefined.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.