LANG00036 Introductory Mathematics

Q1.

(a) Simplify the following algebraic expressions.

(i) (1 + a^1/2 )^2 − 2a^1/2 (2 marks)

(ii) x /(x^2+y^2) − y(x−y)^2/(x^4−y^4) (2 marks)

(b) Solve the following equation.

1/(x + 2) + 1/x = 1 (2 marks)

(c) A computer programmer can complete a job in one hour less than his assistant. Together they do the job in 2 hours 30 minutes. How long would it take each of them to complete job working alone? (4 marks)

Total is 10 marks.

Q2.

(a) Solve the following exponential equations.

(i) 2^x = 3^x (1 mark)

(ii) 3 x − 3 x+3 = −78 (2 marks)

(b) Solve the following trigonometric equations.

(i) Solve 2cos(x)^2 = 0 for 0 ≤ x ≤ 2π (3 marks)

(ii) Find all solutions for the equation cos(2x) − sin(x) = √ 2/2 (4 marks)

Total is 10 marks.

Q3.

(a) You are given the exponential function f(x) = −3 − e^2x .

(i) Write down its domain and range. (2 marks)

(ii) Find f^−1 (x) (3 marks)

(b) Work out the equation of the tangent to the curve y = xe^x for x = −1. (5 marks)

Total is 10 marks.

Q4.

(a) For the following quadratic function find the equation in vertex form y = a(x − h)^2 + k and sketch the graph.

y = 4x^2 − 6x (4 marks)

(b) For the following trigonometric function sketch the graph and write down period, amplitude, phase shift and y-intercept. Sketch the graph.

f(x) = 3sin(2x − π 2 ) + 3 (6 marks)

Total is 10 marks.

Q5.

(a) Given the line l : 3x − 2y = 5 and the point P = (−1, 2), find the equation of another line passing through the point P and perpendicular to l. (3 marks)

(b) Points (a, −7) and (4, b) lie on a circle with the equation x^2+y^2 = 50. Find possible values for a and b. (5 marks)

(c) Find the equation of the line through the point (−2, 1) when the slope is undefined. (2 marks)

Total is 10 marks.

Q6.

(a) Find dy/dx for y = 2e^x/x (2 marks)

(b) Find the points on the graph of f(x) = 4x^3 + 10 at which the tangent is horizontal. (2 marks)

(c) Function N(t) = 100t^2 + t + 5000 describes the number of sales for a particular newspaper after t years.

(i) Find the rate of change of the number of sales with respect to time. (2 marks)

(ii) Work out the rate of change after 6 months and interpret it. (4 marks)

Total is 10 marks.

Q7.

In order to perform a particular research, cells are harvested from the tissue and grown in the laboratory. 20000 cells grow to 5000000 cells in two weeks.The growth of bacteria is modelled by the formula: A = A0e^kt where t is given in weeks.

(a) What is A0? (1 mark)

(b) Work out the parameter k and write down the function for modelling bacteria growth. (4 marks)

(c) After how many weeks the number of bacteria reaches 10000000? (3 marks) (d) Sketch the function for modelling bacteria growth. You must clearly show the y-intercept and the general behaviour (growth or decay). (2 marks)

Total is 10 marks.

Q8.

(a) Suppose £3000 is invested in an account at 1.5% annual interest compounded monthly.

(i) How much will be in this account after 6 months? (2 marks)

(ii) How much will be in this account after 10 years? (2 marks)

(iii) How long will it take for the invested sum to reach £6000 if the annual interest is compounded daily? (3 marks)

(b) Suppose £1000 is invested in an account at annual interest compounded continuously. After 2 years the account contains £1100. What is the annual interest rate? (3 marks)

Total is 10 marks.

Q9.

(a) (i) Find ( 8^√ x + 0.5)dx (2 marks)

(ii) Find 5e^−3xdx (2 marks)

(b) The rate at which students memorise English words is given by the function dM/dt = 0.2t − 0.003t^2 where t is the number of minutes. After 3 minutes the number of memorised words is equal to 5.

(i) Work out the function M(t). (4 marks)

(ii) How many words will be memorised in one hour? (2 marks)

Total is 10 marks.

Q10.

(a) The population (in millions) of a particular city is modelled by the function where t is the number of years since 2010. P(t) = 4e^−0.02t

(i) Find the average population between 2012 and 2018. (4 marks)

(ii) What was the maximum value for the population between 2012 and 2018? (2 marks)

(b) Find the area between two curves f(x) = x and g(x) = x^2 on the interval [0, 2]. (4 marks) Total is 10 marks