CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2

33 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)8 marksSolve, for x > 0, the equation 3\log_8 x = 2\log_x 8 - 5.
2. 1(b)(i)3 marksCopy and complete the table for values of 2^x and e^{-x} using a calculator, approximating values to 2 decimal places.
3. 1(b)(ii)5 marksOn the same pair of axes using a scale of 4 cm for 1 unit on the x-axis and 4 cm for 1 unit on the y-axis, draw the graphs of y = 2^x and y = e^{-x} for -1 \le x \le 3, x \in \mathbb{R}.
4. 1(b)(iii)a)2 marksUse your graphs to find the value of x satisfying 2^x - e^{-x} = 0.
5. 1(b)(iii)b)2 marksUse your graphs to find the range of values of x for which 2^x - e^{-x} < 0.
6. 2(a)3 marksShow that for n \ge 2, \tan^n x = \tan^{n-2} x \sec^2 x - \tan^{n-2} x.
7. 2(b)3 marksFind \frac{dy}{dx} when y = \tan^n x.
8. 2(c)(i)7 marksBy using the result in (a) above, show that I_n + I_{n-2} = \frac{1}{n-1}.
9. 2(c)(ii)7 marksHence evaluate I_4.
10. 3(a)9 marksProve by Mathematical Induction that u_n = n! for all n \in \mathbb{N}.
11. 3(b)(i)5 marksFind the n-th term of S.
12. 3(b)(ii)2 marksShow that S is a geometric progression.
13. 3(b)(iii)2 marksFind the first term and common ratio of S.
14. 3(b)(iv)2 marksDeduce the sum to infinity of S.
15. 4(a)(i)4 marksShow that f(x) = 0 has a root \alpha in the interval (0, 1).
16. 4(a)(ii)5 marksIf x_1 is a first approximation to \alpha of f(x) = 0 in (0, 1), show that the Newton-Raphson method gives a second approximation x_2 in (0, 1) satisfying x_2 = \frac{3x_1^4 - 1}{4(x_1^3 - 1)}.
17. 4(b)(i)5 marksShow that P = 570.
18. 4(b)(ii)6 marksFind, in terms of n, 1 \le n \le 12, an expression for the remaining debt on the loan after John has paid the n-th instalment.
19. 5(a)(i)2 marksDetermine the number of ways of selecting the six marbles if there are no restrictions.
20. 5(a)(ii)4 marksFind the probability that the marbles chosen contain more black marbles than white marbles.
21. 5(b)(i)2 marksDetermine the probability that a person selected at random is a female.
22. 5(b)(ii)4 marksDetermine the probability that a person selected at random is a male or likes watching the News.
23. 5(b)(iii)2 marksDetermine the probability that a person selected at random is a female that likes watching Friends.
24. 5(b)(iv)2 marksDetermine the probability that a person selected at random does not like watching Matlock.
25. 5(c)(i)2 marksCalculate the value of p.
26. 5(c)(ii)2 marksDetermine the probability that there are more than 3 accidents in a week.
27. 6(a)(i)1 markWrite the system in matrix form.
28. 6(a)(ii)1 markWrite down the augmented matrix.
29. 6(a)(iii)3 marksReduce the augmented matrix to echelon form.
30. 6(a)(iv)1 markDeduce the value of \alpha for which the system is consistent.
31. 6(a)(v)4 marksFind ALL solutions corresponding to this value of \alpha.
32. 6(b)(i)3 marksFind kI - A, where I is the 3 \times 3 identity matrix and k is a real number.
33. 6(b)(ii)7 marksFind the values of k for which |kI - A| = 0.