IFYMB004 Maths (Business) Exam 2
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THE NCUK INTERNATIONAL FOUNDATION YEAR
IFYMB004 Mathematics (Business)
Examination
Question 1
Figure 1 shows the acute-angled triangle ABC with AB = x cm, AC = cm and
angle A = 60o .
The area of triangle ABC is 40 cm2 . [ 3 ]
Find the value of x .
Question 2
(a) Point J lies at (−2, 3) and point K lies at (1, 12) .
Find the equation of the line through points J and K.
Give your answer in the form y = mx + c.
(b) In this question all working must be seen. Marks will not be given if answers
(even the correct ones) are quoted without showing the working.
Solve the equations 6p + 5q = 8
2p − 3q = −9
Question 3
A student invested P dollars at the beginning of 2020 in an account which pays compound interest at a rate of 4% per year.
At the beginning of 2021 the investment was worth 3328 dollars.
(b) If the interest rate stays at 4%, find the total amount of interest earned over
the 3 years from the beginning of 2020 until the beginning of 2023.
Question 4
(a) In a large box of eggs it is reckoned that 15% of them are bad.
A sample of 20 eggs is taken.
Find the probability that either 3 or 4 eggs in the sample are bad.
(b) Honey is sold in jars and the mass of honey in each jar can be assumed to
follow a Normal distribution with standard deviation 15 grams.
Each jar bears a label which says “Mean mass is 220 grams” .
A sample of 16 jars is selected and the mean mass of honey in each jar is found to be 215 grams. the claim on the label reasonable? Justify your answer.
Question 5
Correlation description |
Correlation Coefficient |
|
A Strong negative |
1 −0.03 |
|
B Weak negative |
2 −0.93 |
|
C Virtually no correlation |
3 |
0.31 |
D Weak positive |
4 |
0.89 |
E Strong positive |
5 −0.28 |
In the table above, the left-hand column (A – E) shows 5 correlation descriptions. The right-hand column (1 - 5) shows 5 correlation coefficients.
Match up each correlation description with a suitable correlation coefficient.
(b) A company sells caravans and its sales during 2020 are shown in the table
below with the 4-point moving average.
Year and Quarter |
Number of sales |
4-point moving average |
2020 First Quarter |
a2 − 1 |
|
|
|
|
Second Quarter |
46 |
|
|
|
6a |
Third Quarter |
33 |
|
|
|
|
Fourth Quarter |
3a + 2 |
|
Find the value of a, given that a < 10.
Question 6
(a) A circle has centre at F(5, −3) and radius 10 units.
Confirm that point G(−1, 5) lies on the circle and find the equation of the tangent to the circle at point G.
Give your answer in the form ax + by + c = 0 where a, b and c are integers.
(b) Function f(x) is defined as f(x) = 4x3 − 8x2 − 9x + 18.
Use the factor theorem to show that (x − 2) is a factor of f(x) and hence factorise f(x) completely.
Question 7
(a) The 6th term of a geometric series is e 13 and the 9th term is e 19 .
Find the common ratio and the first term
Give your answers in terms of e.
(b) The first term of an arithmetic series is 6 and the 5th term is −22.
The sum of the first n terms is −468.
Find the value of n.
Question 8
A curve has equation y = x 3 − x 2 − 8x + 9 .
Find the coordinates of the stationary values.
(b) Determine whether each stationary value is a maximum or a minimum.
(c) Find
∫ y dx .
Question 9
(a)
Figure 2 shows the acute-angled triangle LMN where LM = 21 cm,
LN = 24 cm and MN = 19 cm.
Find the size of angle L. [ 4 ]
Give your answer in radians to 3 significant figures.
In this question, 1 mark will be given for the correct use of significant figures.
(b) Solve 64 cos2 e = 49 (0° ≤ e ≤ 360°) [ 4 ]
Question 10
(a) A survey is carried out on the number of fish in a lake.
The number of fish, N, after t years from when the survey began is given by the formula
N = 480ekt + 120 (t ≤ 8)
where k is a constant.
There are 768 fish in the lake after 2 years.
Show that k ≈ 0. 15 and hence find the value of when t = 3. [ 6 ]
(b) Solve log 3 (x2 + 2x) − log 3 (x2 − 4) = 2 (x > 2) [ 4 ]
Question 11
Figure 3 shows the curve y = x 2 − 6x + 9 and line l which intersects with the curve at point C(2, 1) and point D(6, 9).
(a) Show that line l is not a normal to the curve at point C.
(b) Find the area, which is shaded on the diagram, that is bounded by the curve
y = x 2 − 6x + 9 and line l.
Give your answer in the form where m and n are integers.
Question 12
Events A and B are such that P(A) = 12x, P(B) = 4x and P(A ∩ B) = 3x where x ≠ 0.
(a) Find P(A ∪ B) in terms of x.
(b) If events A and B were independent, what would be the value of x?
(c) You are given instead that x = .
Draw a Venn diagram and hence write down P(A ∩ B′), P(A′ ∪ B) and [ 5 ]
P(B|A).
Question 13
A firm makes computer monitors, and it is claimed that they have an average lifetime of 1450 hours.
The lifetimes of 600 monitors were measured and the results are shown in the table below.
Lifetime (t) in hours |
Frequency |
600 < t ≤ 900 |
36 |
900 < t ≤ 1100 |
64 |
1100 < t ≤ 1300 |
96 |
1300 < t ≤ 1400 |
62 |
1400 < t ≤ 1500 |
78 |
1500 < t ≤ 1800 |
156 |
1800 < t ≤ 2200 |
108 |
(You may wish to copy and extend this table to help you answer some of the questions below)
(a) Estimate the mean. You must show your working.
Does the claim about an average lifetime of 1450 hours seem reasonable? Justify your answer.
(b) On graph paper draw a histogram to show the data.
The firm offers a refund if a monitor fails after less than 1000 hours.
(c) How many monitors will be expected to fail after less than 1000 hours?
Question 14
(a) A die with scores 1, 2, 3, 4, 5 and 6 is biased (unfair). The probability
distribution of X which is the score shown on the die after each throw is shown in the table below.
x |
1 |
2 |
3 |
4 |
5 |
6 |
P(X = x) |
0.05 |
0.3 |
0.12 |
0.18 |
0.25 |
0.1 |
The die is thrown twice.
Find the probability of an odd score on both occasions.
(b) Find E(X) and Var(X).
(c) Another biased die has scores 3, 5, 7, 9, 11 and 13. The probability distribution of Y which is the score shown on this die is shown in the table below.
y |
3 |
5 |
7 |
9 |
11 |
13 |
P(Y = y) |
0.05 |
0.3 |
0.12 |
0.18 |
0.25 |
0.1 |
Write down E(Y) and Var(Y).
Question 15
A curve has equation −2x2 + 2xy + 3y = 8.
(a) Find in terms of x and y.
(b) Show that stationary values occur when y = 2x.
(c) Find the coordinates of the stationary values.
Question 16
(a) Use the Quotient Rule to differentiate
sin x − 1
1 + cos x
You do not need to simplify your answer.
(b) Find [ 3 ]
∫ (3x + 6)e3x dx.
(c) The function f(x) is defined as
f(x) =
where k is a positive constant .
Express f(x) in partial fractions and hence evaluate
3k
∫ f(x) dx.
k
Give your answer in the form ln ( ) where a and b are integers.
2023-06-02