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PERTH MODERN SCHOOL

YR11 MATHEMATICS SPECIALIST – 2019

TEST 1 – Reasoning & Permutations

NAME: _____________________________________ DATE:_______________

Teacher’s Name______________________________

To achieve full marks working and reasoning should be shown.

This is a Calculator Assumed Assessment – 45 minutes / 38 marks

You may have notes on one side of an A4 sheet of paper.

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[ 3 marks]

Suppose a mathematical statement of the form PQ is true. Then (Circle your answer)

Its converse will always be true.

Its converse will always be false.

Its negation will be always be true.

Its contrapositive will always be true.

Its contrapositive will always be false.

Suppose n is an integer. Consider the statement: If n2 is even then n is even.

The converse of this statement is: (Circle your answer)

If n2 is odd, then n is even.

If n2 is even, then n is odd.

If n2 is odd, then n is odd.

(d) If n is odd, then n2 is odd.

If n is even, then n2 is even.

Consider the statement: The number 5n2-4n+1 is a composite number for every positive integer n.

The negation of this statement is: (Circle your answer)

The number 5n2-4n+1 is a prime number for some prime number n.

The number 5n2-4n+1 is a prime number for every prime number n.

The number 5n2-4n+1 is a prime number for every positive integer n.

The number 5n2-4n+1 is a prime number for some positive integer n.

(e) The number 5n2-4n+1 is a composite number for some positive integer n.

2. [6 marks]

Use mathematical induction to prove that 72n-1+5 is divisible by 12 for all integers n > 1.

3. [8 marks]

For n=1,2,3,…. , let Sn=12+22+32…+n2

Use mathematical induction to prove that, for all integers n with n=1,2,3,…,

Sn=16n(n+1)(2n+1)

4. [6 marks]

Use the fact that if n2 is divisible by 3, then n is divisible by 3 to prove that 3 is irrational.

5. [5 marks]

Suppose that a, b ∈ and consider the statement: If ab is irrational then either a or b is irrational.

Write down the contrapositive of this statement.

Prove the contrapositive of this statement.

6. [5 marks]

From the letters of the word FACTORISE, words of 5 letters are arranged without repeating letters.

How many of these arrangements of 5 letters:-

(a) are possible altogether

(b) begin with the letters AR in that particular order

(c) end with the letter T

start with AR in any order (ie AR or RA) and end with T

7. [5 marks]

Using the digits from this list: 0, 3, 4, 5, 6, 8 determine:

a) How many 4 digit numbers can be made that are greater than 4000?

(No repetition allowed. You cannot start the number with zero)

b) How many 4 digit numbers are even and greater than 4000?

(No repetition allowed. You cannot start the number with zero)

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