FPM Resit Questions
Answer all 22 questions. There are 308 marks available.
Show all necessary working.
Answers should be hand written. Typed answers will not be marked.
1. Prove each of the following identities:
(i) (ii)
(16 marks)
2. Solve the equation for x in the range
.
(18 marks)
3. The 6th term of an arithmetic progression is 10 and the 14th term is 34.
(i) Find the first term and the common difference.
(ii) Find the sum of the first n terms in terms of n.
(iii) Find the sum of the terms from the 11th to the 30th inclusive.
(16 marks)
4. Expand in ascending powers of z up to and including the term in
.
Hence expand in ascending powers of x up to and including the term in
.
(10 marks)
5. If find
and
.
Hence find the values of the constants a and b so that is identically true.
(12 marks)
6. If where
, find and simplify
.
Find the values of x for which .
Find the greatest and least values of y.
(12 marks)
7. Evaluate: (i) (ii)
.
(16 marks)
8. Using integration by parts, determine:
(i) (ii)
(14 marks)
9. The vectors a, b and c are defined by ,
and
.
(a) Express the vector in terms of i, j and k.
(b) Evaluate the magnitudes of the vectors b, c and .
(c) Evaluate the product .
(12 marks)
10. The position vectors of the points A, B and C are given by ,
and respectively with respect to a fixed origin O.
(a) Show that and
.
Show that the angle ACB is a right angle.
(b) Find the area of the triangle ABC.
Hence, or otherwise, show that the shortest distance from C to AB is .
(24 marks)
11. (a) The complex number z satisfies the equation
.
Solve the equation for z and express z in the form .
(b) Calculate the two square roots of .
(20 marks)
12. The complex numbers are given by
and
.
(a) (i) Evaluate and
.
(ii) Mark on an Argand diagram the points which represent the numbers
respectively.
(b) (i) Evaluate the modulus and argument of .
Write each number in polar form.
(ii) Hence, or otherwise, evaluate the modulus and argument of and
.
(24 marks)
13. Solve = 0
- marks)
14. Find b such that the following set of equations has a non trivial solution:
3x + 3y – z = 0
5x – y – 4z = 0
3x – 4y + bz = 0
- marks)
15. If A = write down and evaluate the expression for detA using the elements and minors of row 1 of A.
- marks)
16. If A = show that A is singular.
(6 marks)
17. Find the solution of the differential equation
that satisfies the condition , expressing your answer in the form
.
(10 marks)
18. Use Maclaurin’s Theorem to express as series of increasing powers of x: (i) tan x, up to the term in (ii) ln(1 + x), up to the term in x4.
(14 marks)
19. If
(a) express in partial fractions,
(b) find the first four terms in the series expansion of in ascending powers of x,
(c) find the coefficient of in the series expansion of
in ascending powers of x,
(d) find the range of values of x for which the series converges.
(18 marks)
20. Show on a diagram the area enclosed between the curve the line
and the y-axis.
By expressing this area as an integral, evaluate the area.
(12 marks)
21. (a) If , evaluate
.
(b) Evaluate (i) (ii)
(iii)
(16 marks)
22. By using mathematical induction, prove that .
By using the series, evaluate the sum of the cubes of the even integers from 2 to 20 inclusive.
(15 marks)
STANDARD DERIVATIVES
| y | y | ||
STANDARD INTEGRALS
| y | y | ||
