Unformatted text preview: rify that your answer to part (b)(i) satisfies the equation
un+1 = 3un – 2un – 1.
(Total 6 marks) 2 – 1
492. The matrix M is defined as M = 1 0 . (a) Find M2, M3 and M4.
(3) (i) State a conjecture for Mn, i.e. express Mn in terms of n, where n ∈ (ii) (b) Prove this conjecture using mathematical induction. + . (7)
(Total 10 marks) 2 3 π
π cos – i sin cos + i sin 4
3 493. Consider the complex number z =
π – i sin cos
24 (a) (i) Find the modulus of z. (ii) Find the argument of z, giving your answer in radians.
(4) (b) Using De Moivre’s theorem, show that z is a cube root of one, i.e. z = 3 1 .
(2) 331 (c) Simplify (l + 2z)(2 + z2), expressing your answer in the form a + bi, where a and b are
exact real numbers.
(Total 11 marks) 494. (a) On the same axes sketch the graphs of the functions, f(x) and g(x), where
f(x) = 4 – (1 – x)2, for – 2 ≤ x ≤ 4,
g(x) = ln (x + 3) – 2, for – 3 ≤ x ≤ 5.
(2) (b) (i) Write down the equation of any vertical asymptotes. (ii) State the x-intercept and y-intercept of g(x).
(3) (c) Find the values of x for which f(x) = g(x).
(2) (d) Let A be the region where f(x) ≥ g(x) and x ≥ 0.
(i) On your graph shade the region A. (ii) Write down an integral that represents the area of A. (iii) Evaluate this integral.
(4) (e) In the region A find the maximum vertical distance between f(x) and g(x).
(Total 14 marks) 495. Consider the points A (1, 3, –17) and B (6, – 7, 8) which lie on the line l.
(a) Find an equation of line l, giving the answer in parametric form.
(4) 332 (b) The point P is on l such that OP is perpendicular to l.
Find the coordinates of P.
(Total 7 marks) 496. Consider the differential equation (a) dy 3 y 2 + x 2
, for x > 0.
2 xy Use the substitution y = vx to show that v + x dv 3v 2 + 1
(3) (b) Hence find the solution of the differential equation, given that y = 2 when x = 1.
(Total 7 marks) 497. (a) At a building site the probability, P(A), that all materials arrive on time is 0.85. The
probability, P(B), that the building will be completed on time is 0.60. The probability that
the materials arrive on time and that the building is completed on time is 0.55.
(i) Show that events A and B are not independent. (ii) All the materials arrive on time. Find the probability that the building will not be
completed on time.
(5) (b) There was a team of ten people working on the building, including three electricians and
two plumbers. The architect called a meeting with five of the team, and randomly
selected people to attend. Calculate the probability that exactly two electricians and one
plumber were called to the meeting.
(2) (c) The number of hours a week the people in the team work is normally distributed with a
mean of 42 hours. 10% of the team work 48 hours or more a week. Find the probability
that both plumbers work more than 40 hours in a given week.
(Total 15 marks) 333 498. Give your answers to four significant figur...
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This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.
- Fall '13
- The Land