Proof by deduction

Given that n is a positive integer, express

as a single fraction not involving surds.

Hence, deduce that

is a rational number for all positive integer values of n

Assume that a and b are integers such that

Prove that a is even.

Hence, prove that 2b 1 is even and explain why this is a contradiction.

Explain what can be deduced about the solutions of the equation

Given that

use the factor theorem to prove that (5x 1) is a factor of P(x).

Factorise P (x) completely.

Hence, prove that 250n³ 300n² 110n 12 is a multiple of 12 when n is a positive whole number.

a and b are two positive irrational numbers.

The sum of a and b is rational.

The product of a and b is rational.

Caroline is trying to prove is rational.

Here is her proof:

Step 2 2 is rational and a b is non-zero and rational.

Identify Caroline’s mistake.

Write down a correct version of the proof.

Prove by contradiction that the difference of any rational number and any irrational number is irrational.

x² bx c and x² dx e have a common factor (x 2)

Fully justify your answer.

A curve has equation

2 2 4

Prove that the curve does not intersect the coordinate axes.

Prove that the curve has no stationary points.

In the case when x 0, find the equation of the tangent to the curve when y 1

Section B

all questions in the spaces provided.

Which of these statements is correct?

Tick one box.