# Category Archives: STEP and MAT

# STEP 2/2011/5

# STEP 1/2016/6

# STEP 2/2017/8

# STEP 2/2009/8

The non-collinear points A, B, C have position vectors **a**, **b**, **c** respectively. The points P and Q have position vectors **p** and **q**, respectively, given by

**p** = λ **a** + (1 − λ)**b**

**q** = μ **a** + (1 − μ)**c**

where 0 < λ < 1. Draw a diagram showing A, B, C, P and Q

Given that CQ.BP = AB.AC, find μ in terms of λ, and show that, for all values of λ, the line PQ passes through the fixed point D, with position vector **d** given by

**d = − a + b + c**

What can be said about the quadrilateral ABDC?

# STEP 2/2008/8

The points A and B have position vectors **a** and **b**, respectively, relative to the origin O. The points A, B, and O are not collinear. The point P lies on AB between A and B such that

Write down the position vector of P in terms of **a**, **b**, and λ

Given that OP bisects the angle AOB, determine λ in terms of *a* and *b*, where

*a* = |**a**| and *b* = |**b**|

The point Q also lies on AB between A and B, and is such that AP=BQ. Prove that

OQ^{2} − OP^{2} = (b-a)^{2}

# STEP II/1996/3

The Fibonacci numbers F_{n} are defined by the conditions: F_{0}=0, F_{1}=1, and F_{n+1} = F_{n} + F_{n-1} for all n ≥ 1. Show that F_{2}=1, F_{3}=2, F_{4}=3, and compute F_{5}, F_{6} and F_{7}.

Compute F_{n+1}F_{n-1} − F_{0}^{2} for a few values of n; guess a general formula and prove it by induction, or otherwise.

By induction on k, or otherwise, show that F_{n+k}=F_{k}F_{n+1}+F_{k-1}F_{n} for all positive integers n and k.