IBMathsDPAnalysis & Approaches (AA)HLExam Questions1. Number & AlgebraSimple Proof & Reasoning

Simple Proof & Reasoning (DP IB Analysis & Approaches (AA): HL): Exam Questions

3 hours34 questions
Medium
Hard
Very Hard
1
3 marks

Prove that left parenthesis 4 x minus 1 right parenthesis left parenthesis 2 x plus 3 right parenthesis minus left parenthesis 2 x plus 1 right parenthesis squared equals 2 left parenthesis 2 x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis.

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2
3 marks

Prove that left parenthesis a minus b right parenthesis squared minus left parenthesis a plus b right parenthesis squared equals negative 4 a b.

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3
3 marks

Prove that the sum of any three consecutive integers is a multiple of 3.

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4
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2 marks

Prove that x squared plus 2 greater or equal than 2 for all values of x.

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5
3 marks

 Prove that the square of an even number is a multiple of 4.

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6a
1 mark

Factorise n squared plus 3 n plus 2.

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6b
1 mark

Hence show that n cubed plus 3 n squared plus 2 n equals n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis.

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6c
2 marks

Given that n is even, write down whether open parentheses n plus 1 close parentheses and open parentheses n plus 2 close parentheses are odd or even.

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6d
2 marks

Hence deduce whether n cubed plus 3 n squared plus 2 n is odd or even. Justify your answer.

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7a
2 marks

Show that left parenthesis 3 n plus 2 right parenthesis squared minus left parenthesis n plus 2 right parenthesis squared = 8 n squared plus 8 n, where n element of straight integer numbers.

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7b
2 marks

Hence, or otherwise, prove that left parenthesis 3 n plus 2 right parenthesis squared minus left parenthesis n plus 2 right parenthesis squared is a multiple of 8.

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8
3 marks

Prove that x squared minus 3 x plus 3 is positive for all real values of x. 

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9
4 marks

Given z equals x plus y i

(i) prove that z z to the power of asterisk times equals open vertical bar z close vertical bar open vertical bar z to the power of asterisk times close vertical bar,

(ii) prove that, for x greater or equal than 0, arg open parentheses z close parentheses plus arg open parentheses z to the power of asterisk times close parentheses equals 0.

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10
8 marks

Determine, with appropriate reasoning, whether the following statements are true or false: 

(i)  Given n element of straight integer numbers and n squared is divisible by 4, then n is divisible by 4.

(ii)  Given n element of straight integer numbers then n squared minus 1 is a prime number.

(iii)  Given n element of straight integer numbers and n squared is divisible by 3, then n is divisible by 3.

(iv)  Given an integer is a multiple 8 and 6, then it is a multiple of 48.

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1
4 marks

Show that fraction numerator 1 over denominator n plus 1 end fraction plus fraction numerator 1 over denominator n squared plus n end fraction equals 1 over n.

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2
4 marks

Forspace f left parenthesis x right parenthesis equals x squared minus 10 x plus 17, prove thatspace f left parenthesis x right parenthesis greater or equal than negative 8 for all values of x.

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3
5 marks

Prove that the exterior angle in any triangle is equal to the sum of the two opposite interior angles. You may use the diagram below to help.

q3-1-4-ib-aa-sl-proof-and-reasoning

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4
4 marks

Consider the functionspace f left parenthesis x right parenthesis equals 5 x squared plus 4 x plus 1. Show thatspace f open parentheses x close parentheses is positive for all values of x.

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5
4 marks

Consider two consecutive positive integers, n and n plus 1.

Show that the difference of their squares is equal to the sum of the two integers.

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6
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4 marks

Prove that left parenthesis 2 q minus 1 right parenthesis left parenthesis q minus 3 right parenthesis minus 3 left parenthesis q minus 4 right parenthesis squared equals negative q squared plus 17 q minus 45.

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7
4 marks

Prove that the square of an odd number is always odd.

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8
4 marks

Prove that the sum of the squares of any two consecutive odd integers is even.

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9
4 marks

Prove that the sum of any three consecutive even numbers is a multiple of 6.

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10
4 marks

The product of three consecutive integers is added to the middle integer. 

Prove that the result is a perfect cube.

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11
4 marks

Prove that there are no non-zero real values of a  and b such that  open parentheses a plus b straight i close parentheses squared equals a plus b straight i.

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12
4 marks

The three statements below are false.

In each case verify the statement is false by use of a counterexample and state an alternative domain that would make the statement true. 

(i) n squared greater than 2 n comma space space n element of straight integer numbers to the power of plus  

(ii) 2 to the power of n minus 1 is a prime number for n element of straight natural numbers comma space 1 less than n less or equal than 4. 

(iii) 5 to the power of n greater than 3 to the power of n plus 4 to the power of n comma space n element of straight integer numbers to the power of plus

How did you do?

1a
3 marks

(i) Prove that

fraction numerator a over denominator open parentheses b over c close parentheses end fraction equals fraction numerator a c over denominator b end fraction

(ii) Specify any cases for which the relation in part (a)(i) is not valid.

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1b
2 marks

Prove that left parenthesis p minus q right parenthesis squared equals left parenthesis q minus p right parenthesis squared for all numbersspace p and q.

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2
4 marks

Prove that the product of two odd numbers is odd.

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3
5 marks

The sum of squares of two consecutive integers is 313.  Find the possible values of the integers.

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4
5 marks

Prove that the sum of the cubes of any two consecutive odd integers is divisible by four.

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5a
4 marks

Prove that fraction numerator a squared minus a minus 6 over denominator a plus 4 end fraction cross times fraction numerator a squared minus 16 over denominator a squared plus 2 a end fraction equals a minus 7 plus 12 over a.

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5b
1 mark

State any values of a for which this mathematical statement does not hold true.

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6
4 marks

Prove that there are no integersspace p and q that satisfy the equation

4 p squared minus q squared equals 49

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7
8 marks

Prove the binomial coefficient identity 

open parentheses table row n row k end table close parentheses equals open parentheses table row cell n minus 1 end cell row k end table close parentheses plus open parentheses table row cell n minus 1 end cell row cell k minus 1 end cell end table close parentheses.

How did you do?

8
8 marks

Prove that the sum of all integers between 600 and 1400 (inclusive) that are not divisible by 7 is equal to 685   885.

How did you do?

9a
2 marks

Write down a comma space b comma space cand d from smallest to largest, given a comma space b comma space c comma space d element of straight real numbers and c greater than d comma space a less than d and a greater than b.

How did you do?

9b
3 marks

Write down p comma space q comma space r and s from smallest to largest, given p comma space q comma space r comma space s element of straight real numbers and

p greater than q

r minus s less than q minus p

p plus q equals r plus s.

How did you do?

9c
3 marks

Prove fraction numerator x over denominator 1 plus x end fraction less than fraction numerator x over denominator 1 plus y end fraction comma space x comma space y element of straight real numbers,  given 0 less or equal than x less than y.

How did you do?

10
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6 marks

Given that the graph of  y equals x to the power of 4 minus 10 x cubed plus 37 x squared minus 60 x plus 36 touches the x-axis at the point with coordinates open parentheses 2 comma 0 close parentheses , prove that y greater or equal than 0 for all real values of x .

How did you do?

11
3 marks

Three of the four statements below are false.

Eliminate the false statements by providing a counterexample and thus deduce the true statement.

(i) open parentheses x minus 1 close parentheses squared not equal to open parentheses x plus 1 close parentheses squared comma space space x element of straight real numbers.

(ii) Every open parentheses 4 n close parenthesesth triangular number is even, n element of straight natural numbers.

(iii) 2 space ln space x greater than ln space 2 x comma space x element of straight real numbers comma space x greater than 0.

(iv) The product of any two distinct positive integers is greater than their sum.

How did you do?

12a
2 marks

The function f open parentheses n close parentheses is given as f open parentheses n close parentheses equals n cubed plus n squared plus 17  where n is an integer.

Find f open parentheses 1 close parentheses comma space f open parentheses 2 close parentheses and f open parentheses 3 close parentheses.

How did you do?

12b
2 marks

Prove that f open parentheses n close parentheses is not prime for all values of n.

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