Mathematics

1. 1.

   A circle has centre (3, -2) and radius 5. Write down its equation and show that the point (6, 2) lies on the circle.

   [3 marks] · no calculator
2. 2.

   The vectors a = (2, -1, 3) and b = (1, 4, -2). Find the scalar product a.b and the angle between a and b, to 1 decimal place.

   [3 marks]
3. 3.

   Points A and B have position vectors (1, 2, 3) and (4, 0, 5). Find a vector equation of the line AB and show that the point C(7, -2, 7) lies on it.

   [4 marks] · no calculator
4. 4.

   The lines l1: r = (1, 0, 2) + lambda(1, 1, 1) and l2: r = (2, 3, 2) + mu(1, -1, 2) intersect. Find the coordinates of the point of intersection.

   [4 marks] · no calculator
5. 5.

   The circle C has equation x^2 + y^2 - 6x + 4y - 12 = 0. The line y = x + k is a tangent to C. Find the two possible values of k, giving exact answers.

   [5 marks] · no calculator
6. 6.

   A curve has parametric equations x = t^2, y = 2t. Show that the tangent at P(p^2, 2p) has equation py = x + p^2. Hence find where the tangent at P meets the x-axis, and the equation of the tangent when p = 3.

   [5 marks] · no calculator

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