Answer all questions in the spaces provided.
In questions where more than 1 mark is available, appropriate working must be shown.
A decimal approximation will not be accepted if an exact answer is required to a question.
Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Question 1
A manufacturer makes metal rods whose lengths are normally distributed with mean 140.0 cm and standard
deviation 1.2 cm.
a. Find the probability, correct to three decimal places, that a randomly selected metal rod is longer than
141.5 cm.
2 marks
b.
A rod has a size fault if it is not within d cm either side of the mean. The probability of a rod having a size
fault is 0.15. Find the value of d, correct to one decimal place.
2 marks
c.
A random sample of 12 rods is taken from a crate containing a very large number of rods. Find the
probability, correct to three decimal places, that the sample contains exactly 2 rods with a size fault.
2 marks
d.
A particular box of 25 rods has 4 rods in it which have size faults. A sample of 12 rods is withdrawn without
replacement. Find the probability, correct to three decimal places, that the sample contains at least 2 rods
with a size fault.
2 marks
Question 1 – continued
3
e.
MATH METH (CAS) EXAM 2
The sales manager is considering at what price, x dollars, to sell each rod. The materials cost $5. The rods
are sorted into three bins. 15% of all the rods manufactured have a size fault and another 17% of all the
rods have other faults. The profit, Y dollars, is a random variable whose probability distribution is shown
in the table below.
Bin
Description
Profit($y)
Pr(Y = y)
A
Good rods –
these are sold for $x each
x−5
k
B
Rods with a size fault –
these are not sold but are recycled
0
0.15
C
Rods with other faults –
these are sold at a discount of $3 each
x−8
0.17
i.
Find the value of k.
ii.
Find the mean of Y in terms of x.
iii.
Hence or otherwise, find, correct to the nearest cent, the selling price of good rods so that the mean
profit is zero.
iv.
The rods are stored in bins until there is a large number ready to be sold.
What proportion of the rods ready to be sold are good rods?
1 + 1 + 1 + 1 = 4 marks
Total 12 marks
TURN OVER
MATH METH (CAS) EXAM 2
4
Question 2
Andrew is making a skateboard ramp. He draws a cross-section diagram with coordinate axes as shown
below.
y
3
x
y = 2 − 2 cos
2
A
O
ground
The curve has the equation y = 2 – 2 cos
of the structure is 8 metres.
a.
B
4
x
, −4 ≤ x ≤ 4. All measurements are in metres; the horizontal length
How many metres above the ground is the highest point of the ramp? Give your answer to two decimal
places.
1 mark
b.
Show that the gradient of the ramp is always less than or equal to 1.
2 marks
Question 2 – continued
5
c.
i.
Write a definite integral which gives the area of the shaded region.
ii.
Find the area of the shaded region, correct to two decimal places.
MATH METH (CAS) EXAM 2
2 + 1 = 3 marks
There is a supporting beam AB on the structure as shown. A is a point on the curve one metre vertically above
the x-axis. B is a point on the x-axis such that AB is normal to the curve at A.
d.
i. Find the exact x-coordinate of A.
ii.
Find the exact value of the gradient of the normal to the curve at A.
iii.
Find the exact length of AB.
2 + 2 + 3 = 7 marks
Total 13 marks
TURN OVER
MATH METH (CAS) EXAM 2
Question 3
Consider the function f: R
a. Find f ′(x).
6
R, f (x) = x3e−2x
1 mark
b.
The graph of y = f (x) is as shown.
y
0.1
–1
O
1
2
3
4
x
–0.1
–0.2
–0.3
Find the exact coordinates of the two stationary points and state their nature.
2 marks
Question 3 – continued
7
c.
MATH METH (CAS) EXAM 2
i.
Find an equation of the tangent to the graph of y = f (x) at the point where x = 1.
ii.
Find an equation of the tangent to the curve at the point (0, 0).
iii.
Show that the tangents of parts i. and ii. are the only two tangents to the curve which pass through
the origin.
3 + 1 + 3 = 7 marks
Question 3 – continued
TURN OVER
MATH METH (CAS) EXAM 2
d.
8
Consider the continuous probability density function with rule g(x) = kx3e− 2x for x ≥ 0 and 0 elsewhere,
where k is a positive real number.
i. Find the value of k.
ii.
Find, correct to two decimal places, the median value of the distribution of this probability density
function.
2 + 2 = 4 marks
Total 14 marks
9
MATH METH (CAS) EXAM 2
Question 4
A tranquilliser is injected into a muscle from which it enters the bloodstream. The concentration, x mg/L, of
3t
the tranquilliser in the bloodstream, may be modelled by the equation x =
, t ≥ 0, where t is the number
5 + t2
of hours after the injection is given. The graph of this equation is shown.
x
0.7
O
a.
8
16
t
Find the exact number of hours after the injection is given when the tranquilliser concentration is greatest.
Also find the exact value of this maximum concentration.
2 marks
Question 4 – continued
TURN OVER
MATH METH (CAS) EXAM 2
b.
10
The derivative of x with respect to t gives a measure of the rate of absorption of the tranquilliser into the
bloodstream.
What is the exact rate of absorption one hour after the injection is given?
1 mark
c.
The tranquilliser is effective when the concentration is at least 0.4 mg/L.
Find the exact value of the length of time in hours for which the tranquilliser is effective.
3 marks
d.
i.
ii.
3t
What is the least value of a such that the function g: [a, ∞) → R, g(t) =
, has an inverse
5 + t2
function?
For this value of a, sketch the graph of g−1 on the axes below. Label any end-point with its coordinates.
Label any asymptote with its equation.
O
t
Question 4 – continued
11
iii.
MATH METH (CAS) EXAM 2
Find the rule for g−1.
1 + 3 + 3 = 7 marks
It is discovered that the drug will produce undesirable side-effects if its concentration exceeds 1 mg/L at any
time. A modification to the drug is proposed so that the concentration in the blood, y mg/L, at time t hours after
the injection is given is modelled by the equation
y=
e.
3t
p + t2
, 0 ≤ t ≤ 8, where p is a parameter.
Find the least value which p may take if the concentration is to be always less than 1 mg/L.