The MSc Business Analytics & AI at ImperiaBusiness 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 and determine whether you are suitable to apply for the programme, please complete the Maths test below - you should get at least five out of seven questions correct.

Please note, this Maths test only applies to candidates considering the MSc Business Analytics & AI.

All questions are required.

1. Optimisation

The profit of a firm depends on two positive input factors as follows. P of x one and x two equals thirty times x one minus x one squared plus sixty times the natural logarithm of x two minus four times x two.

Determine the input levels that maximise the profit function and find the maximum profit. For simplicity, round the profit to two decimal places. (required)

2. Matrix algebra

The firm AC/DC Associates manufactures ballet tutus and tights, in competition with Bob Dylan Co. and Queen Plc. Last season the Gaga Ballet Company purchased items from all three companies. Purchases — tutus: 20 from AC and DC Associates, 10 from Bob and Dylan Co., 20 from Queen Plc.; tights in pairs: 20, 30 and 10 respectively. Cost of items — tutus: 30, 40 and 50; tights in pairs: 20, 20 and 15 respectively. Let Q be the 2 by 3 matrix of purchases and C the 2 by 3 matrix of costs per item.

Compute the product C transpose Q. What do the diagonal entries of the product represent? (required)

3. Simultaneous Equations and Linear Algebra

Find the solution vector of the following linear system. (required)

3 x-sub-1 plus 2 x-sub-2 minus 2 x-sub-3 equals 1; 2 x-sub-1 minus x-sub-2 plus x-sub-3 equals 3; x-sub-1 plus 3 x-sub-2 minus 2 x-sub-3 equals 1.

4. Multivariable Calculus

Let f of x one and x two equal x one squared plus two times x one times x two plus three times x two squared.

What is the gradient of f at the point one, two? (required)

5. Expectation and Variance

Let X and Y be independent random variables with expected value of X equal to one, variance of X equal to four, expected value of Y equal to two, and variance of Y equal to nine.

What are the expected value and variance of two times X minus Y? (required)

6. Conditional Probability

The prevalence of a particular heart disease is zero point one percent, or one in a thousand people. A diagnostic test has one hundred percent sensitivity, meaning it always tests positive when the disease is present, and ninety-five percent specificity, meaning it tests negative for ninety-five percent of people who do not have the disease.

If a randomly selected person tests positive, what is the probability that the person actually has the disease? (required)

7. Central Limit Theorem

Let X one, X two, through X n be independent and identically distributed random variables with expected value of X i equal to a and variance of X i equal to b.

What is the approximate distribution of the sample average, X one plus X two through X n, all divided by n, when n is large? (required)