Further Mathematics

1. 1.

   Convert the polar point r = 4, theta = pi/3 to Cartesian coordinates. Give its distance from the origin.

   [3 marks] · no calculator

   Answer explanation

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

   1. Use the exact sine and cosine of pi/3 rather than rounded calculator values.
   2. sqrt(x^2 + y^2) = sqrt(4 + 12), confirming the radius.

   Marking points

   • x = r cos theta = 2.
   • y = r sin theta = 2sqrt(3).
   • Distance is 4 units.

   Examiner tip: The angle is in radians, not degrees.

2. 2.

   Convert r = 6cos theta to Cartesian form and identify its centre and radius.

   [3 marks] · no calculator

   Answer explanation

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

   1. Multiplying by r avoids dividing by a radius that can be zero at the origin.
   2. The resulting circle passes through the pole and the point (6,0).

   Marking points

   • Multiply by r: r^2 = 6r cos theta, giving x^2 + y^2 = 6x.
   • Complete the square: (x - 3)^2 + y^2 = 9.
   • Centre (3,0), radius 3.

   Examiner tip: The coefficient 6 is the diameter, not the radius.

3. 3.

   Find the area swept by r = 2theta for 0 <= theta <= pi/2. Give an exact answer.

   [3 marks] · no calculator

   Answer explanation

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

   1. The radius increases continuously over a non-overlapping quarter-turn sector.
   2. The upper-limit cube is pi^3/8; multiply by 2/3 to get pi^3/12.

   Marking points

   • Use area = (1/2) integral r^2 dtheta.
   • Integrate 2theta^2 to 2theta^3/3.
   • Area is pi^3/12 square units.

   Examiner tip: Polar area integrates r squared, not r.

4. 4.

   For r = 1 + cos theta, find dy/dx at theta = pi/2.

   [3 marks] · no calculator

   Answer explanation

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

   1. Differentiate both products before evaluating the angle; changing r contributes to each derivative.
   2. Take the ratio of the two parameter derivatives, not dr/dtheta.

   Marking points

   • x = r cos theta and y = r sin theta, with r' = -sin theta.
   • At pi/2, dx/dtheta = -1 and dy/dtheta = -1.
   • dy/dx = 1.

   Examiner tip: A polar radial derivative is not a Cartesian tangent gradient.

5. 5.

   For r = 3cos(2theta), identify the theta interval for the petal centred on the positive x-axis and calculate its area exactly.

   [3 marks] · no calculator

   Answer explanation

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

   1. At theta = 0 the radius is maximal and positive, locating the desired petal.
   2. The sine boundary terms cancel to zero and the interval width pi/2 leaves integral value pi/4 before the area factor.

   Marking points

   • The petal runs from theta = -pi/4 to pi/4, between adjacent zeros of r.
   • Area = (9/2) integral cos^2(2theta) dtheta over that interval.
   • Using cos^2(2theta) = (1 + cos(4theta))/2 gives 9pi/8 square units.

   Examiner tip: Integrating over a full rotation counts more than the single requested petal.

6. 6.

   Find the area inside r = 1 + cos theta but outside the unit circle. State the intersection angles used.

   [3 marks] · no calculator

   Answer explanation

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

   1. Subtract squared inner radius from squared outer radius along each ray.
   2. The integrand simplifies to 2cos theta + cos^2 theta; their integrals are 4 and pi/2 before halving.

   Marking points

   • r = 1 at theta = +/-pi/2; the cardioid is outside the circle for -pi/2 <= theta <= pi/2.
   • Area = (1/2) integral [(1 + cos theta)^2 - 1] dtheta over that interval.
   • The area is 2 + pi/4 square units.

   Examiner tip: Subtract areas only on angles where the cardioid is the outer curve.

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