1. FP2 Comments

2. Algebra Questions Well done generally Partial fractions tests prime quadratic factors in the numerator necessitate comparing powers to determine constants Summation of series well done on the whole Don’t forget formulae in tables Frequently questions can be completed by mathematical induction as an alternative method

3. General Solution of 3 ½ sin  – cos  = 2 ½ Question was designed to be completed by developing the expression on the left using harmonic form Many candidates squared and solved without realising this introduced spurious solutions If squaring is used - each solution must be checked in the original equation!

4. Complex root questions Candidates should be ready to identify the roots of a polynomial in a complex variable. (Implicit knowledge of the fundamental theorem of algebra is assumed) Real coefficients => complex roots occur in conjugate pairs

5. Proof by Mathematical Induction Prove u n = 2(3 n ) + 1 given u 1 = 7 and u n+1 = 3u n – 2 where n is a positive integer Many candidates have yet to grasp the real thrust of this method of proof

6. Proof by Mathematical Induction A framework of proof might be: Identify the need to prove u n = 2(3 n ) + 1 Quote u 1 = 7 (i) Assume u k = 2(3 k ) + 1 Prove u k+1 = 2(3 k+1 ) +1(ii) State that (i) and (ii) imply that P(n) is true for all n>=1.

7. Maclaurin and de Moivre question A solution which starts sin(3  ) = Im (cos(  ) + i sin(  )) 3 does not include the full development of the answer A small number attempted to derive a series expansion from first principles which involves rather more effort

8. Ellipse and Parabola questions Typical questions involve some bookwork such as –verifying that the parametric point satisfies the Cartesian equation –Deriving the general equation of the tangent or normal Questions build to proving some property of the conic section in question

9. FP3 Comments

10. Integration by substitution question do what the question asks only full credit will be given to an answer involving substitution some candidates failed to return to the original variable

11. Calculus

12. Many candidates failed to spot that the product rule was need for the second term of (i) Look for connections between parts.

13. Hyperbolic Functions

14. Many candidates failed to use the quicker method of using the inverse cosh function The majority solved the quadratic in the exponential function

15. Vectors, lines and planes

16. Remember AC = OC – OA Many found it difficult to determine the line through a point perpendicular to a plane There is a formula for the distance from a point to a plane in the tables

17. Calculus and hyperbolic functions

18. Many candidates failed to determine the nature of the stationary points correctly Be ready to integrate an inverse hyperbolic or trig function

20. Reduction Formulae Many candidates failed to recognise that the ln(x) part is differentiated by function of a function Read The Question – many candidates calculated the VOLUME of revolution and not the area under the curve!!