CAPE Pure Mathematics Unit 1 · 2022 · Paper 2

30 questions and parts from this paper. Open one to see it in full, then practise it on Quelpr and get it marked against the mark scheme.

1. 1(a)(i)3 marksLet p and q be any two propositions. Complete the truth table below.
2. 1(a)(ii)2 marksHence, state whether the statements (\sim q \wedge p) and p \vee (\sim q \wedge p) are logically equivalent. Justify your response.
3. 1(b)9 marksDetermine the values of a, b, and c, such that 2x^2 - 7x + 12 = a(x - 2)(x - 1) - b(x - 3) + c.
4. 1(c)5 marksSolve the inequality |3x + 2| \ge 4.
5. 1(d)6 marksSolve the logarithmic equation \log_5(x + 2) + \log_5(x + 6) = 1.
6. 2(a)(i)3 marksLet f(x) = \frac{3x + 1}{x} and g(x) = e^{-2x} + 1. Show that f^{-1}(x) = \frac{1}{x - 3}.
7. 2(a)(ii)2 marksHence, or otherwise, write an expression for (f^{-1} \circ g)(x).
8. 2(b)8 marksSolve the equation 6 - \frac{7}{2^{2x}} - \frac{3}{4^{2x}} = 0.
9. 2(c)6 marksThe function f(x) = 2x^3 - px^2 + qx - 10 is divisible by x - 1 and has a remainder of -6 when divided by x + 1. Calculate the values of p and q.
10. 2(d)6 marksThe roots of the equation 2x^3 - x^2 + 3x - 1 = 0 are \alpha, \beta, and \gamma. Given that \alpha\beta + \alpha\gamma + \beta\gamma = \frac{3}{2} and…
11. 3(a)6 marksProve that \frac{\tan\theta \sin\theta}{1 - \cos\theta} = 1 + \frac{1}{\cos\theta}.
12. 3(b)6 marksSolve the equation \tan^2\theta - 2\tan\theta = 3 for 0 \le \theta \le 2\pi.
13. 3(c)(i)5 marksShow that 4\cos\theta + 3\sin\theta = 5\sin(\theta + 0.927^c).
14. 3(c)(ii)4 marksHence, or otherwise, solve the equation 4\cos\theta + 3\sin\theta = 0.
15. 3(d)4 marksGiven that \sin A = \frac{1}{3} and A is obtuse, calculate the value of \cos A without using a calculator.
16. 4(a)(i)4 marksA circle has equation x^2 + y^2 - 10x + 4y - 5 = 0. Determine the centre and radius of the circle.
17. 4(a)(ii)4 marksDetermine the equation of the tangent to the circle at the point (2, 3).
18. 4(b)(i)3 marksDetermine the equation of the plane that passes through the point (0, 2, -2) and which is perpendicular to the vector \mathbf{v} = 3\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}.
19. 4(b)(ii)3 marksDetermine the angle between the vector 3\mathbf{i} - 2\mathbf{j} and the x-axis.
20. 4(c)5 marksThe parametric equations of a line are given as x = 2 + 5\lambda, y = -3 + 4\lambda, z = 4 - 2\lambda. Determine the coordinates of the point where the line crosses the xy plane.
21. 4(d)6 marksA point P(x, y) moves in the xy plane such that it is the same distance from the point A(1, 2) as it is from the line x = 3. Determine the equation of the locus of P.
22. 5(a)5 marksDetermine \lim_{x \to 2} \frac{x^2 + 3x - 10}{x^2 + x - 6}.
23. 5(b)7 marksA ladder which is 10\text{ metres} long is leaning against a wall. The bottom of the ladder is sliding away from the base of the wall at a rate of 4\text{ m/s}. Determine the rate at which the top of the ladder is…
24. 5(c)(i)6 marksA function f is given as f(x) = 4x^3 - 3x^2 + 1, for -1 \le x \le 1. Determine the coordinates of the stationary points of the function f.
25. 5(c)(ii)4 marksDetermine the nature of these stationary points.
26. 5(c)(iii)3 marksDetermine the absolute maximum and minimum values of the function f.
27. 6(a)5 marksUsing the substitution u = x^2 + 1, determine \int 4x(x^2 + 1)^5\, dx.
28. 6(b)6 marksThe diagram below shows two curves, y = x + x^2 and y = x(3 - x). Calculate the area of the shaded region.
29. 6(c)6 marksCalculate the volume of the solid generated by revolving the region bounded by the line y = 6x and the parabola y = 6x^2 about the x-axis.
30. 6(d)8 marksSolve the differential equation \frac{dy}{dx} = \frac{x + 4x^2}{y^2}, given that y = 6 when x = 0.