# STEP I/2013/13

From the integers 1, 2, … 52, I choose seven (distinct) integers at random, all choices being equally likely. From these seven, I discard any pair that sum to 53. Let X be the random variable the value of which is the number of discarded pairs. Find the probability distribution of X and show that E(X) = 7 / 17 Continue reading

# Answer to STEP I 2008 Q.13

Three married couples sit down at a round table at which there are six chairs. All of the possible seating arrangements of the six people are equally likely.

(i) Show that the probability that each husband sits next to his wife is 2/15

(ii) Find the probability that exactly two husbands sit next to their wives.

(iii) Find the probability that no husband sits next to his wife. Continue reading

# Answer to STEP III 2007 Q.9

III/2007/9 – Two small beads, A and B, each of mass m, are threaded on a smooth horizontal circular hoop of radius a and centre O. The angle θ is the acute angle determined by 2 θ = AOB.

The beads are connected by a light straight spring. The energy stored in the spring is
$mk^2a^2(\theta - \alpha)^2$
where k and α are constants satisfying k > 0 and $\frac{\pi}{4} < \alpha < \frac{\pi}{2}$
The spring is held in compression with θ = β and then released. Find the period of oscillations in the two cases that arise according to the value of θ and state the value of β for which oscillations do not occur. Continue reading

# Answers for modulus-argument exercise, Year 12 Further Maths homework for 3/10/17

You can write these answers in different ways, which are all correct. For example, 2 √2 (cos π/4 + i sin π/4) can also be written as 2 √2 cis π/4 or as 2 √2 ∠ π/4. All three ways are correct.

Questions are below

# Answer to STEP I 1999 Q.2

I/1999/2 – A point moves in the x,y plane so that the sum of the squares of its distances from the three fixed points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is always $a^{2}$.

Find the equation of the locus of the point and interpret it geometrically.

Explain why $a^{2}$ cannot be less than the sum of the squares of the distances of the three points from their centroid. Continue reading