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CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 2

31 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.

Practise this paper
  1. 1(a)8 marksSolve the simultaneous equations x² + xy = 6 and x - 3y + 1 = 0.
  2. 1(b)(i)2 marksstate the values of α + β and αβ
  3. 1(b)(ii)3 marksfind the value of α² + β²
  4. 1(b)(iii)7 marksfind the equation whose roots are 1 + 1/α and 1 + 1/β
  5. 2(a)10 marksProve, by Mathematical Induction, that Σ (from r=1 to n) r = (1/2)n(n + 1).
  6. 2(b)1 markExpress, in terms of n and in the SIMPLEST form,
  7. 2(b)(i)2 marksΣ (from r=1 to 2n) r
  8. 2(b)(ii)4 marksΣ (from r=n+1 to 2n) r.
  9. 2(c)4 marksFind n if Σ (from r=n+1 to 2n) r = 100.
  10. 3(a)(i)4 marksFind the coordinates of the centre and radius of the circle x² + 2x + y² – 4y = 4.
  11. 3(a)(ii)5 marksBy writing x + 1 = 3 sin θ, show that the parametric equations of this circle are x = -1 + 3 sin θ, y = 2 + 3 cos θ.
  12. 3(a)(iii)4 marksShow that the x-coordinates of the points of intersection of this circle with the line x + y = 1 are x = -1 + (3√2)/2.
  13. 3(b)7 marksFind the general solutions of the equation cos θ = 2 sin²θ – 1.
  14. 4(a)(i)7 marksfind the values of R and α correct to one decimal place
  15. 4(a)(ii)2 markshence, find ONE value of x between 0° and 360° for which the curve y = 4 sin x - cos x has a stationary point.
  16. 4(b)(i)1 markFind in the form a + bi, a, b ∈ R,
  17. 4(b)(i)a)1 markz₁ + z₂
  18. 4(b)(i)b)3 marksz₁z₂
  19. 4(b)(i)c)5 marksz₁/z₂
  20. 4(b)(ii)2 marksFind the quadratic equation whose roots are z₁ and z₂.
  21. 5(a)(i)1 markState the value of lim (as δx → 0) (sin 8x / δx).
  22. 5(a)(ii)2 marksGiven that sin 2(x + δx) – sin 2x = 2 cos A sin B, find A and B in terms of x and/or δx.
  23. 5(a)(iii)7 marksHence, or otherwise, differentiate with respect to x, from first principles, the function y = sin 2x.
  24. 5(b)1 markFind
  25. 5(b)(i)5 marksthe values of the constants h and k
  26. 5(b)(ii)5 marksthe equation of the tangent to the curve at the point where x = 1/2.
  27. 6(a)(i)6 marksShow that the area S is approximately 1/n² + 2/n² + 3/n² + .... + (n-1)/n².
  28. 6(a)(ii)2 marksGiven that Σ (from r=1 to n-1) r = (1/2)n(n - 1), show that S = (1/2)(1 - 1/n).
  29. 6(b)(i)4 marksShow that for f(x) = 2x/(x²+4), f'(x) = (8-2x²)/(x²+4)².
  30. 6(b)(ii)3 marksHence, evaluate ∫ (from 0 to 1) (24-6x²)/(x²+4)² dx.
  31. 6(c)5 marksFind the value of u > 0 if ∫ (from 1 to u) (2/x⁴) dx = 7/192.

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