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Calculate an unbiased estimate of
(a) the mean time taken to drive to school; (b) the variance of the time taken to drive to school. Working: Answers:
(Total 6 marks) 552. The diagram below shows the graph of y1 = f(x). y2 x 367 On the axes below, sketch the graph of y2 = f′ (x). y2 x
(Total 6 marks) 553. The function f is defined by f(x) =
(a) (i) Show that f′ (x) =
, for x > 0.
2x 2 x – x 2 ln 2
2x Obtain an expression for f″(x), simplifying your answer as far as possible.
(5) (i) Find the exact value of x satisfying the equation f′(x) = 0 (ii) (b) Show that this value gives a maximum value for f(x).
(4) (c) Find the x-coordinates of the two points of inflexion on the graph of f.
(Total 12 marks) 368 554. The variables x, y, z satisfy the simultaneous equations
x + 2y + z = k
2x + y + 4z = 6
x – 4y + 5z = 9
where k is a constant.
(a) (i) Show that these equations do not have a unique solution. (ii) Find the value of k for which the equations are consistent (that is, they can be
(6) (b) For this value of k, find the general solution of these equations.
(Total 9 marks) 555. (a) Prove, using mathematical induction, that for a positive integer n,
(cosθ + i sinθ)n = cosnθ + i sinnθ where i2 = –1.
(5) (b) The complex number z is defined by z = cosθ + i sinθ. 1
= cos(–θ) + i sin(–θ).
z (i) Show that (ii) Deduce that zn + z–n = 2cos nθ.
(5) (c) (i)
(ii) Find the binomial expansion of (z + z–l)5. 1
(a cos5θ + b cos3θ + c cosθ),
where a, b, c are positive integers to be found.
Hence show that cos5θ = (5)
(Total 15 marks) 369 556. A business man spends X hours on the telephone during the day. The probability density
function of X is given by 1
3 (8 x – x ), for 0 ≤ x ≤ 2
f(x) = 12
otherwise. (a) (i) Write down an integral whose value is E(X). (ii) Hence evaluate E(X).
(3) (b) (i) Show that the median, m, of X satisfies the equation
m4 – 16m2 + 24 = 0. (ii) Hence evaluate m.
(5) (c) Evaluate the mode of X.
(Total 11 marks) π
557. The function f with domain 0, is defined by f(x) = cos x + 2 3 sin x. This function may also be expressed in the form R cos(x – α) where R > 0 and 0 < α <
2 Find the exact value of R and of α.
(3) (b) (i) Find the range of the function f. (ii) State, giving a reason, whether or not the inverse function of f exists.
(5) (c) Find the exact value of x satisfying the equation f(x) = 2.
(3) 370 (d) Using the result ∫ sec xdx = lnsec x + tan x+ C, where C is a constant,
0 ∫ dx
= ln(3 + 2 3 ).
f ( x) 2
(Total 16 marks) 558. Give all numerical answers to this question correct to two decimal places.
A radar records the speed, v kilometres per hour, of cars on a road. The speed of these cars is
normally distributed. The results for 1000 cars are recorded in the following table.
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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