Advanced level 2026 BAEBOC pure mathematics with statistics 3

Section 1 (1 hour)

Answer all the questions

1. (a) Define semiconductor. (2 marks)

(b) State two things that can be done to increase the conductivity of a semiconductor. (2 marks)

2. Figure 1 shows a simple machine that can be used in lifting loads.

[Diagram of a pulley system with a 50 N effort and a 200 N load]

Figure 1

(a) State the name of the machine shown in figure 1. (1 mark)

(b) Calculate the mechanical advantage of the machine. (3 marks)

(c) State one reason why the efficiency of the machine is not 100%. (1 mark)

3. (a) A transformer has 1000 turns in the primary coil and 3000 turns in the secondary coil.

(i) Calculate the secondary voltage if the primary voltage is 250 V. (3 marks)

(ii) State two sources of energy lost in a transformer. (2 marks)

(b) Figure 2 shows a bar magnet, held near a coil to which a galvanometer, G, has been connected.

[Diagram of a bar magnet moving toward a coil connected to a galvanometer]

Figure 2

(i) State and explain what is observed when the magnet is moved towards the coil. (3 marks)

(ii) State two things that can be done to increase the observation. (2 marks)

4. The nuclide $^{14}_{6}C$ has a half-life of 5600 years and decays to the nuclide $^{14}_{7}N$.

(a) Define half-life. (2 marks)

(b) Determine the time it will take for the number of undecayed nuclides to reduce to 1/4 the original value. (2 marks)

(c) State the kind of decay undergone by the C-14. Justify your answer. (2 marks)

(d) State and explain how the rate of decay will be affected if the sample is heated to a higher temperature. (2 marks)

5. An object is placed between the principal focus and optical centre of a concave lens.

(a) State the nature of the image formed. (1 mark)

(b) Draw a ray diagram to show how the image is formed. (2 marks)

(c) State one similarity between the image formed and that formed in a plane mirror. (1 mark)

2. In an agricultural experiment the gain in mass, in kilograms of 100 pigs during a certain period were recorded as follows:

a) Calculate the mode, mean and standard deviation of the masses. (3,3,3marks)

b) Draw a cumulative frequency curve and from it estimate the interquartile range. (4marks)

3. A discrete random variable X can take only the values 0, 1, 2, or 3 and the various probabilities are $p(x = 0) = b$, $p(x = 1) = 3b$, $p(x = 2) = 4b$, $p(x = 3) = 5b$ where b is a constant. Find

a) the value of b (3marks)

b) the mean and variance of X (3,2marks)

c) the mean and variance of Y given that $Y = 3X – 2$ (4marks)

4. i) During a weekday, heavy Lorries pass through a check point independently and at random. The Lorries pass at an average rate of 2 per 30 minutes. Find the probability that

a) there will be no lorry passing through the check point in a given 30 minutes period (2marks)

b) at least three lorries will pass through the check point in a given hour (4marks)

ii) In a basketball game the probability of scoring in a single trial is $\frac{1}{10}$

a) find the probability that more than 6 attempts are needed for a player to score the first basket (2marks)

b) What is the smallest value of n if there is to be at least a 90% chance of scoring the first basket on or before the $n^{th}$ attempt? (4marks)

5. The continuous random variable X has probability density function $f(x)$, where

Find

a) the value of the constant k (3marks)

b) the mean and variance of x (3,4marks)

c) If m is the median of X show that $2m^2 – 6m + 3 = 0$ (3marks)

6. A coin is biased so that the probability of a Heads is 3 times the probability of a tail. The coin is tossed 100 times. Find the mean and variance of this binomial distribution if X is the number of tails. Using the normal distribution as an approximation to the binomial distribution. Find the probability that there will be

a) 68 tails or less (3marks)

b) between 67 and 71 tails inclusive (4marks)

c) between 72 and 85 heads exclusive (3marks)