IFYMB004 Maths (Business) Exam 1
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THE NCUK INTERNATIONAL FOUNDATION YEAR
IFYMB004 Mathematics (Business)
Examination
Question 1
A bag holds 6 red beads and 4 blue beads.
Two beads are drawn, one after the other with no replacement.
Find the probability that both beads are of the same colour.
Question 2
An investment of £1800 gives £226 compound interest after 4 years at a rate of T%.
Find the value of T.
Question 3
Points A, B and C lie on line l1 . Point A lies at (2, −4), point B lies at (8,
11) and point C lies at (n + 5, 3n).
(a) Find the value of n. [ 3 ]
Line l2 passes through point A and is perpendicular to line l1 .
(b) Find the equation of line l2 .
Give your answer in the form ax + by + c = 0 where a, b and c are integers. [ 3 ]
Question 4
(a) A geometric series is defined as 3 + 3√2 + 6 + …
Find the sum of the first 20 terms.
Give your answer to 3 significant figures. [ 3 ]
In this question, 1 mark will be given for the correct use of significant figures.
(b) Find the value of t if
√5243t15 × 4t 1
20t2 ÷ 4t−5 = 19 5
Question 5
(a) y
=
(4, 5)
(1, 2)
y = x + 1
O x
Figure 1
Figure 1 shows the curve y = x 2 − 4x + 5 and the line y = x + 1.
The line and the curve intersect at (1, 2) and (4, 5) .
Find the area, which is shaded on the diagram, that is bounded by the curve [ 4 ]
y = x 2 − 4x + 5 and the line y = x + 1.
(b) Find [ 3 ]
∫ dt.
Question 6
Eight pairs of readings (x, y) are recorded and the results are summarised below.
∑ x = 288; ∑ y = 264; ∑ x 2 = 12814; ∑ y 2 = 9110 ; ∑ xy = 10408
(a) Find sx , sy and sxy .
Hence find the Product Moment Correlation Coefficient.
(b) Describe the correlation between x and y.
(c) The equipment used to take the readings is found to be faulty and all the y − readings should have been lower by 0.5.
State the new Product Moment Correlation Coefficient.
Question 7
(a) A circle has centre at (−1, 3).
The point with coordinates (2, 7) lies on the circle.
Find the equation of the circle.
(b) Divide X 3 + X 2 − 22X + 30 by (X − 3).
(c)
Question 8
(a)
NOT TO SCALE
20 cm
25 cm
B
C
Figure 2
Figure 2 shows the acute-angled triangle ABC with AB = 20 cm, AC = 25 cm
and angle A = e° where cos e = .
Show that sin e = and find the area of triangle ABC.
Find also the length of BC and an expression for sin B .
Give each of your answers in surd form.
(b) Solve 125 cos 3e = 26 (0° ≤ e ≤ 270°) [ 5 ]
Question 9
ℎ cm
T cm
Figure 3
Figure 3 shows a solid cylinder with radius T cm and height ℎ cm. The volume of the cylinder is 2662 cm3 .
(a) Express ℎ in terms of T and hence show that the surface area of the
cylinder, A, is given by
A = 2T 2 +
(b) Use calculus to find the value of T which gives the minimum surface area
and confirm that your value gives a minimum.
Question 10
(a) The variables t and y are connected by the formula
y = 4e2t
Find the value of t when y = 3 − e t . Give your answer in exact form.
(b) Solve the equation log9 (x2 − 1) − log9 (x2 − 3x + 2) = (x > 2)
(c) Show that ln (2 × 4 × 8 × 16 × … ) can be written as the arithmetic series ln 2 + 2 ln 2 + 3 ln 2 + 4 ln 2 + ⋯
and hence find the sum of the first n terms of the series.
Give your answer in terms of n and ln 2, and in fully factorised form.
Question 11
(a) At a checkout in a shop, the times taken by 20 customers are recorded and
the results are shown in the table below.
Time, t, in minutes |
Frequency |
1 ≤ t ≤ 2 |
4 |
2 < t ≤ 3 |
6 |
3 < t ≤ 5 |
5 |
5 < t ≤ 7 |
3 |
7 < t ≤ 10 |
2 |
(You may wish to copy and extend this table to help you answer some of the questions below.)
Estimate the mean and standard deviation. Working must be shown.
Explain why your answers are estimates.
(b) Write down the largest possible value of the range.
(c) In which interval does the lower quartile lie?
Question 12
y |
The Venn diagram shows two events A and B with their various probabilities.
(a) Find P(A), P(A ∩ B), P(B′) and P(B|A).
Give each answer in terms of x and/or y where appropriate.
(b) You are given y = . Find the value of x.
(c) Are events A and B are independent? Give a reason and show your working.
Question 13
(a) A discrete random variable, X, has probability distribution as given in the
table below.
x |
0 |
1 |
4 |
6 |
10 |
12 |
P(X = x) |
k |
0.2 |
0.1 |
0.12 |
0.24 |
0.15 |
Find E(X) and Var(X).
(b) The probability that a student is right handed is p.
A sample of 10 students is chosen. Find the probability that 2 or less
students are left handed.
Give your answer in terms of p and in its simplest form. [You may assume that nobody is both left and right handed.]
(c) The masses of oranges can be assumed to follow a Normal distribution with standard deviation 25 grams.
A sample of 20 oranges is taken and the mean mass of this sample is found to be 163 grams.
Find a 98% confidence interval of the mean masses of all the oranges.
Question 14
(a) A curve has equation 2x3 + 4x2y − 3y2 = 6.
dy
dx
(b) Another curve has equation y = .
Use the Quotient Rule to find and hence find the coordinates of the
stationary values.
Question 15
(a)
Express
2x2 − 9x + 16 (x + 1)(x − 2)2
in partial fractions.
(b) Find
∫ (4x3 + 6x2 ) lnx dx.
(c) Evaluate
2
∫ dx.
Give your answer in the form |
m n |
where m and n are integers. |
All working must be shown. Just giving the answer, even the correct one, will score no marks if this working is not seen.
2023-06-02