# Self-assessment Maths test

## Self-assessment Maths test

Our MSc Business Analytics at Imperial College Business School is a highly rigorous and technically demanding programme. Our students are required to have a high level of quantitative skills. To test your quantitative ability to determine if you are suitable to apply for our Business Analytics programme, please complete the Maths test below. Please note that in order to be a suitable candidate for this programme you should get at least 5 out of 7 questions correct.

Please note, this Maths test only applies to candidates considering the MSc Business Analytics and MSc Business Analytics (online delivery, part-time).

## 1. Optimisation

The profit of a firm depends on three positive input factors and as follows:

### P(x1,x2) = 30x1 - x12 + 60lnx2 - 4x2

Determine input factors which maximise the profit function and find the maximum profit. For simplicity please round to 2 decimal places.

## 2. Matrix algebra

The firm AC/DC Associates manufactures ballet tutus and tights, and is in intense competition with Bob Dylan Co. and Queen Plc. All three companies are competing fiercely over the quality of their tutus and tights, and this has been a challenge recently. Ballet companies have been switching between the three companies in pursuit of finding the best tutus and tights for their dancers. Last season, the Gaga Ballet Company purchased items from all three companies as follows:

Table 1. Gaga Ballet Company's Purchases

Purchases
AC & DC Associates
Bob & Dylan Co.
Queen Plc.
Tutus
20
10
20
Tights (pairs)
20
30
10

The cost of these items are given in the following chart:

Table 2: Cost of items

Purchases
AC & DC Associates
Bob & Dylan Co.
Queen Plc.
Tutus
30
40
50
Tights (pairs)
20
20
15

Let Q be the 2x3 matrix corresponding to the purchases, and let C be the 2x3 matrix corresponding to the costs per item.

Compute the product CTQ. What do the diagonal entries of the product represent?

## 3. Simultaneous Equations and Linear Algebra

Find the solution vector of the following linear system

3x1 + 2x2 - 2x3 = 1

2x1 - x2 + x3 = 3

x1 + 3x2 - 2x3 = 1

## 4. Ordinary Differential Equations

Find the solution for the differential equation dy/dx + 7x = 0 given y(0)=3. Evaluate this answer for x=1.

## 5.

One in a thousand people has a prevalence for a particular heart disease. There is a test to detect this disease and the test is 100% accurate for people who have the disease and 95% accurate for those who don't have it. If a randomly selected person tests positive what is the probability that the person actually has the disease?

## 6.

We have n IID random variables X_i for i = 1, ... , n with E[X_i] = a and Var(X_i) = b. That is, the mean and variance of each X_i are a and b, respectively. What is the approximate distribution of (x_1 + X_2 + ... + X_n)/n, i.e. the average of the X_i's, when n is large?

## 7. Probability

(This one is harder, so don’t worry if you find this one too difficult)

Let p denote the probability that a firm fiddles its books. Suppose that p = 0.1 . Also suppose that a financial professional services firm keeps auditing firms until they have found 3 ‘bad’ ones.

Compute the probability that they will need to audit exactly 6 firms.