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Unformatted text preview: 9x 2 . Working: Answers:
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(Total 4 marks) k 96. Find the real number k > 1 for which ∫ 1 + x1 dx = 2 1 3.
2 Working: Answers:
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(Total 4 marks) 67 97. (a) a − 4 − 6 Find the values of a and b given that the matrix A = − 8
5
7 is the inverse of
− 5
3
4 1 the matrix B = 3 −1 (b) 2
b
1 −2 1 .
− 3 For the values of a and b found in part (a), solve the system of linear equations
x + 2y – 2z = 5
3x + by + z = 0
–x + y – 3z = a – 1. Working: Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks) 68 98. The local Football Association consists of ten teams. Team A has a 40 % chance of winning any
game against a higherranked team, and a 75 % chance of winning any game against a
lowerranked team. If A is currently in fourth position, find the probability that A wins its next
game.
Working: Answers:
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(Total 4 marks) 99. The polynomial p(x) – (ax + b)3 leaves a remainder of –1 when divided by (x + 1), and a
remainder of 27 when divided by (x – 2). Find the values of the real numbers a and b.
Working: Answers:
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(Total 4 marks) 69 100. The acceleration, a(t) m s–2, of a fast train during the first 80 seconds of motion is given by
a( t ) = – 1 t + 2
20
where t is the time in seconds. If the train starts from rest at t = 0, find the distance travelled by
the train in the first minute.
Working: Answers:
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(Total 4 marks) 101. For what values of k is the straight line y = kx + 1 a tangent to the circle with centre (5, 1) and
radius 3?
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(Total 4 marks) 70 102. Calculate the shortest distance from the point A(0, 2, 2) to the line
r
r
r
r
r
r
r
r = 5 i + 9 j + 6 k + t( i + 2 j + 2 k ) where t is a scalar.
Working: Answers:
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(Total 4 marks) 103. Solve the differential equation
dy
= y tan x + 1,
dx 0≤x< π ,
2 if y = 1 when x = 0.
Working: Answers:
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(Total 4 marks) 71 104. In the diagram, PTQ is an arc of the parabola y = a2 – x2, where a is a positive constant, and
PQRS is a rectangle. The area of the rectangle PQRS is equal to the area between the arc PTQ
of the parabola and the xaxis.
y
T S R P Q
O x
y=a2–x2 Find, in terms of a, the dimensions of the rectangle.
Working: Answers: ....……………………………………..........
(Total 4 marks) 72 105. (a) Evaluate (1 + i)2, where i = −1 .
(2) (b) Prove, by mathematical induction, that (1 + i)4n = (–4)n, where n ∈ *.
(6) (c) Hence or otherwise, find (1 + i)32.
(2)
(Total 10 marks) 106. Let z1 = (a) 6 −i 2...
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 Fall '13
 Apple
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