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CAPE Pure Mathematics Unit 2 · May/June 2016 · 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.

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  1. 1(a)(i)3 marksCalculate (alpha + beta) and (alpha * beta).
  2. 1(a)(ii)6 marksHence, show that an equation with roots 1/(alpha - 2) and 1/(beta - 2) is given by 10x^2 + 2x + 1 = 0.
  3. 1(b)(i)1 markComplete the Argand diagram to illustrate u.
  4. 1(b)(ii)2 marksOn the same Argand plane, sketch the circle with equation |z - u| = 3.
  5. 1(b)(iii)6 marksCalculate the modulus and principal argument of z = (u / v)^5.
  6. 1(c)7 marksDetermine the x-coordinates of the two stationary values of f.
  7. 2(a)4 marksDetermine partial derivative of w with respect to x.
  8. 2(b)6 marksDetermine the integral of e^(2x) sin(e^x) dx.
  9. 2(c)(i)5 marksUse the trapezium rule with three equal intervals to estimate the area bounded by f and the lines y = 0, x = 2 and x = 5.
  10. 2(c)(ii)6 marksUsing partial fractions, show that f(x) = 3/(x - 1) - (2x)/(x^2 + 1).
  11. 2(c)(iii)4 marksHence, determine the value of the integral from 2 to 5 of f(x) dx.
  12. 3(a)4 marksGiven that u_8 = 13x + 1 and that u_10 = 34x + 1, find (u_9)'.
  13. 3(b)(i)7 marksShow that S_n = (n(n^2 - 1)) / 3.
  14. 3(b)(ii)5 marksHence, or otherwise, evaluate sum_{r=10}^20 r(r - 1).
  15. 3(c)(i)4 marksGiven that ^n P_r = n! / (n - r)!, show that (^2r P_r * ^n P_r) / ((2r)!) is equal to the binomial coefficient ^n C_r.
  16. 3(c)(ii)5 marksDetermine the coefficient of the term in x^3 in the binomial expansion of (3x + 2)^5.
  17. 4(a)(i)3 marksShow that f(x) = (1 + 2x)^(1/3).
  18. 4(a)(ii)5 marksDetermine the series expansion of f up to and including the term in x^4.
  19. 4(a)(iii)3 marksHence, approximate f(0.4) correct to 2 decimal places.
  20. 4(b)(i)3 marksShow that h(x) = 0 has a root on the interval [0, 1].
  21. 4(b)(ii)6 marksUse the iteration x_(n+1) = 1 / (x_n^2 + 1) with initial estimate x_1 = 0.7 to estimate the root of h correct to 2 decimal places.
  22. 4(c)5 marksUse two iterations of the Newton-Raphson method with initial estimate x_1 = 1 to approximate the root of the equation g(x) = e^(4x - 3) - 4 in the interval [1, 2]. Give your answer correct to 3 decimal places.
  23. 5(a)(i)2 marksDetermine the number of possible seating arrangements of the passengers who boarded the bus at the terminal.
  24. 5(a)(ii)4 marksAt the first stop, no passengers will get off the bus but there are eight other persons waiting to board the same bus. Among those waiting are three friends who must sit together. Determine the number of possible groups…
  25. 5(b)5 marksWhat is the probability that Gavin and Alexander are the opening pair for a given match?
  26. 5(c)(i)4 marksFind |A|, the determinant of A.
  27. 5(c)(ii)10 marksHence, or otherwise, find A^(-1), the inverse of A.
  28. 6(a)(i)3 marksCalculate the number of outcomes in the sample space.
  29. 6(a)(ii)2 marksFind the probability of obtaining exactly one head.
  30. 6(a)(iii)4 marksCalculate the probability of obtaining at least one head on the coins and an even number on the die on a particular attempt.
  31. 6(b)6 marksDetermine whether y = C_1 x + C_2 x^2 is a solution to the differential equation (x^2 / 2) y'' - x y' + y = 0, where C_1 and C_2 are constants.
  32. 6(c)(i)7 marksShow that the general solution to the differential equation 3(x^2 + x) dy/dx = 2y(1 + 2x) is y = C * ((x^2 + x)^2)^(1/3), where C in R.
  33. 6(c)(ii)3 marksHence, given that y(1) = 1, solve 3(x^2 + x) dy/dx = 2y(1 + 2x).

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