Download Pure mathematics - University of London International Programmes

This paper is not to be removed from the Examination Halls
UNIVERSITY OF LONDON
FP0007 ZA
International Foundation Programme
Foundation Course: Pure Mathematics
Wednesday, 22 April 2015 : 14:30 to 16:30
This paper contains 5 questions. Candidates should answer ALL questions. All
questions carry equal numbers of marks.
Candidates are strongly advised to divide their time accordingly.
A list of formulae is provided at the end of this paper.
A calculator may be used when answering questions on this paper and it must comply
in all respects with the specification given with your Admission Notice. The make and
type of machine must be clearly stated on the front cover of the answer book.
PLEASE TURN OVER
© University of London 2015
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1.
(a) Suppose that p and q are propositions. Use a truth table to show that the propositions
¬(p ⇒ q) and p ∧ (¬q) are logically equivalent.
(5 marks)
(b) Prove that, if n is an odd number, then n2 + 1 is an even number.
(5 marks)
(c) Suppose that A and B are sets. Prove that A ∩ (B \ A) is the empty set. (5 marks)
(d) A function, f , is given by
f (x) = x3 + 2x2 + x + k,
where k is a constant and, when f (x) is divided by x − 1, the remainder is 4.
Find k and hence find the solutions of the equation f (x) = 0.
2.
(5 marks)
(a) The roots of the cubic equation
x3 + 6x2 + 7x − 2 = 0,
are α, β and γ. Find the equation whose roots are α + 2, β + 2 and γ + 2.
Use this new equation to solve the original equation.
(10 marks)
2+x
in partial fractions.
(b) Express
(x + 1)(x − 2)
Hence find the first three non-zero terms of its binomial expansion. For what values
of x is this expansion valid?
(10 marks)
3.
(a) Prove the trigonometric identity
tan(θ) + tan(φ) =
sin(θ + φ)
.
cos(θ) cos(φ)
(4 marks)
(b) Hence, for 0 ≤ x ≤ π, solve the trigonometric equation
tan(x) + tan(3x) = tan(4x).
(8 marks)
(c) Evaluate the definite integral
Z
π/6
0
sin(3x)
dx,
cos(x) cos(2x)
leaving your answer as a single logarithm.
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(8 marks)
D1
4.
(a) Find the following indefinite integrals.
(i)
Z
tan2 (x) dx.
(3 marks)
(ii)
Z
√
(4 marks)
x ln(3x) dx.
(b) A curve is given by the parametric equation
x = 2 cos2 (θ) and y = 3 sin(2θ).
Find the value of y ′ (x) at the point on this curve where θ = π/8.
Express y 2 as a function of x and hence verify your answer.
5.
(13 marks)
 
1
3

A line in R , L1 , goes through the point (1, 1, 1) in the direction of the vector 0.
1
Another line in R3 , L2 , goes through the points (1, 0, 2) and (3, 2, 2).
(a) Find the vector equations of these two lines.
(5 marks)
(b) Find the point of intersection of these two lines.
(5 marks)
(c) Find the angle between these two lines.
(4 marks)
 
2
3

(d) A plane in R goes through the point (0, 1, 0) with normal vector 1.
3
Find the Cartesian equation of this plane. At what point does the line L1 intersect
this plane?
(6 marks)
END OF PAPER
END OF PAPER
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Formula sheet
n
(n)(n − 1)(n − 2) · · · (n − r + 1)
Binomial coefficients:
=
.
r
r!
n
n 2
n
n
Binomial theorem for n ∈ N: (1 + x) = 1 +
x+
x + ··· +
xn−1 + xn .
1
2
n−1
n
n 2
n 3
n
Binomial theorem for n 6∈ N: If |x| < 1, then (1 + x) = 1 +
x+
x +
x + · · ·.
1
2
3
Compound angle formulae:
sin(A ± B) = sin(A) cos(B) ± cos(A) sin(B).
cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B).
tan(A ± B) =
Derivatives:
function
tan(x)
cosec(x)
sec(x)
cot(x)
sin−1 (x)
cos−1 (x)
tan−1 (x)
tan(A) ± tan(B)
.
1 ∓ tan(A) tan(B)
derivative
sec2 (x)
− cosec(x) cot(x)
sec(x) tan(x)
− cosec2 (x)
(1 − x2 )−1/2
−(1 − x2 )−1/2
(1 + x2 )−1
The chain rule: If f (x) = f (g) for some function g(x), then
df
df dg
=
.
dx
dg dx
df
dg
d
f (x)g(x) =
g(x) + f (x) .
The product rule:
dx
dx
dx
d f (x)
1
df
dg
The quotient rule:
=
g(x) − f (x)
.
dx g(x)
[g(x)]2 dx
dx
Z
Z
′
Integration by parts:
f (x)g (x) dx = f (x)g(x) − f ′ (x)g(x) dx.
Linear differential equations: The solution of y ′ (x) + p(x)y(x) = q(x) is given by
Z
1
µ(x)q(x) dx where µ(x) = eP (x)
y(x) =
µ(x)
and P (x) is an antiderivative of p(x).
Dot products: If a, b ∈ R2 , then a · b = a1 b1 + a2 b2 = |a||b| cos(θ).
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