A Level Paper 5: Mixed Applications

Question 1 [2 marks]

Exponentials and Logarithms

Without using a calculator, find the exact value of log_2(32).

Question 2 [2 marks]

Moments

A uniform beam of weight 60 N and length 4 m is suspended horizontally in equilibrium by two vertical strings, one at each end.

Find the tension in each string.

Question 3 [3 marks]

Sampling and Data Presentation

The table shows the age, in years, of 40 people attending a clinic: 0-20 (6 people), 20-40 (14 people), 40-60 (12 people), 60-80 (8 people).

Using the midpoint of each class, estimate the mean age.

Question 4 [3 marks]

Sequences, Series and the Binomial Expansion

The first three terms of a geometric sequence are 8, 12, 18.

Find the common ratio and the 6th term.

Question 5 [4 marks]

Vectors

Find the magnitude of the vector v = 3i - 4j, and find the vector of magnitude 15 in the same direction as v.

Question 6 [4 marks]

Sampling and Data Presentation

The box plot summarising the times, in minutes, taken by 200 runners to complete a race gives: minimum 28, lower quartile 35, median 41, upper quartile 50, maximum 68.

Find the interquartile range, and state, with a reason, whether a time of 66 minutes would be classed as an outlier using the rule 'more than 1.5 x IQR above the upper quartile'.

Question 7 [5 marks]

Kinematics

A train accelerates uniformly from rest at 0.5 m/s^2 for 20 seconds, then travels at constant speed for a further 90 seconds.

Find the total distance travelled by the train.

Question 8 [5 marks]

The Normal Distribution and Hypothesis Testing

A machine fills bags of flour with a mean mass of 1000 g. A sample of 60 bags is taken and found to have a mean mass of 1015 g, with population standard deviation known to be 40 g.

Test, at the 5% significance level, whether there is evidence that the mean mass has increased. State your hypotheses, test statistic and conclusion.

Question 9 [5 marks]

Vectors

Vectors a and b are such that |a| = 5, |b| = 3 and the angle between a and b is 60 degrees.

Find a.b, and find |a + b|.

Question 10 [5 marks]

Moments

A non-uniform plank AB has length 5 m and weight 120 N, and rests horizontally on two supports at A and B. When a block of weight 60 N is placed at a point 2 m from A, the reaction at A is 100 N.

Find the reaction at B in this situation, and find the distance of the centre of mass of the plank from A.

Question 11 [4 marks]

Sampling and Data Presentation

A sample of 12 pairs of data gives a product moment correlation coefficient of r = 0.62. The critical value for a sample of size 12 at the 5% significance level (one-tail) is 0.497.

Test, at the 5% significance level, whether this provides evidence of positive correlation between the variables in the population.

Question 12 [6 marks]

Moments

A non-uniform ladder AB of length 5 m and weight 240 N rests with end A on rough horizontal ground and end B against a smooth vertical wall. The centre of mass of the ladder is 2 m from A. The coefficient of friction between the ladder and the ground is 0.4.

Find the least angle the ladder can make with the ground without slipping.

Question 13 [6 marks]

Kinematics

A particle P moves in a straight line so that its displacement s metres from a fixed point O at time t seconds (t >= 0) is given by s = t^3 - 9t^2 + 24t.

Find the velocity and acceleration of P as functions of t, find the times at which P is instantaneously at rest, and find the total distance travelled by P in the first 5 seconds.

Question 14 [6 marks]

Moments

A uniform ladder AB of length 6 m and weight 180 N rests with end A on rough horizontal ground and end B against a smooth vertical wall, making an angle of 60 degrees with the ground. A person of weight 700 N stands on the ladder at a point 4 m from A.

Given that the ladder is on the point of slipping, find the coefficient of friction between the ladder and the ground.