# Category Archives: STEP and MAT

# STEP I 2001 Q.1 (triangle in square)

STEP I/2001/1 – The points A, B, and C lie on the sides of a square of side 1 cm and no two points lie on the same side. Show that the length of at least one side of the triangle ABC must be less than or equal to cm.

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# Worked answers or part-answers for first group of mechanics STEP problems and for first “spiral” set of problems

# Worked answers, STEP I 2011

# Answer to STEP I 2013 Q.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 III 2007 Q.11

Mechanics of a wheel falling off the edge of a table. 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

where *k* and α are constants satisfying *k* > 0 and

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