Proofs - IB Math AA Question Bank

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Proofs

Practice exam-style IB Math AA questions for Proofs, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

Let $n\in\mathbb{Z}$ .

Show that $(2n+3)^2-(2n-1)^2\equiv 16n+8$ .

[2]

Hence prove that the difference of the squares of any two odd integers which differ by $4$ is divisible by $8$ .

[3]

Question 2

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

State the values of $x$ for which the expression $\frac{1}{x-2}-\frac{1}{x+2}$ is not defined.

[1]

Show that $\frac{1}{x-2}-\frac{1}{x+2}\equiv \frac{4}{x^2-4}$ , for all allowed values of $x$ .

[3]

Explain why substituting $x=0$ into both sides is not a proof of the identity in part (b).

[1]

Question 3

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Consider the statement

$(2x-1)^2-(x+3)(x-3)\equiv 3x^2-4x+10$

Prove the identity by starting with the left-hand side.

[2]

Check the identity using $x=-2$ .

[1]

Explain why the calculation in part (b) is not a proof of the identity.

[2]

Question 4

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Show exactly that

$\frac{1}{6}+\frac{1}{15}+\frac{1}{10}=\frac{1}{3}$

[2]

Prove the identity

$\frac{1}{k}-\frac{1}{k+3}\equiv \frac{3}{k(k+3)}$

where $k\neq 0,-3$ .

[2]

Explain why the decimal calculation $0.166666\ldots+0.066666\ldots+0.1=0.333333\ldots$ is not the preferred proof for part (a).

[1]

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Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Show that $\frac{1}{6}+\frac{1}{12}+\frac{1}{20}=\frac{3}{10}$ .

[2]

Show that $\frac{1}{m(m+1)}+\frac{1}{(m+1)(m+2)}+\frac{1}{(m+2)(m+3)}\equiv \frac{3}{m(m+3)}$ , where $m\neq 0,-1,-2,-3$ .

[4]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $n\in\mathbb{Z}$ .

Show that $n(n+1)(n+2)(n+3)+1\equiv (n^2+3n+1)^2$ .

[4]

Hence prove that the product of four consecutive integers, increased by $1$ , is a perfect square.

[2]

Question 7

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Show that $(\sqrt{3}+\sqrt{2})^2=5+2\sqrt{6}$ and $(\sqrt{3}-\sqrt{2})^2=5-2\sqrt{6}$ .

[3]

Hence show that $\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}=2\sqrt{2}$ .

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $a$ be a non-zero rational number and let $b$ be an irrational number.

Prove by contradiction that $ab$ is irrational.

[4]

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Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A student makes the following claim.

For every $n\in\mathbb{N}$ , the number $n^2+n+17$ is prime.

Disprove the claim by giving a counterexample. You must explain why your example is a counterexample.

[4]

Question 10

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the statement:

The sum of any two irrational numbers is irrational.

Disprove the statement by giving a counterexample. You must explain why your example is a counterexample.

[4]

Question 11

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The following identity is defined for suitable real values of $x$ :

$\frac{1}{x-2}-\frac{1}{x+2}\equiv \frac{4}{x^2-4}$

State the values of $x$ that must be excluded.

[1]

Show that the identity is true.

[3]

Hence solve exactly

$\frac{1}{x-2}-\frac{1}{x+2}=\frac{1}{2}$

[2]

Question 12

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Consider the identity

$\frac{x^2-9}{x^2+6x+9}\equiv \frac{x-3}{x+3}$

State the restriction on $x$ .

[1]

Prove the identity.

[2]

Hence solve exactly

$\frac{x^2-9}{x^2+6x+9}=\frac{1}{4}$

[2]

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Question 13

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

$f(x)=(x+1)^3-(x-1)^3$

Use your GDC to find $f(2)$ and $f(5)$ .

[1]

Prove that $f(x)\equiv 6x^2+2$ .

[2]

A student says that the two values found in part (a) prove the result in part (b). Explain why the student is incorrect.

[2]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $r$ be a non-zero rational number and let $s$ be an irrational number. Prove by contradiction that $rs$ is irrational.

[5]

Question 15

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the statement:

For every $n\in\mathbb{N}$ , the number $n^2+n+17$ is prime.

Use your GDC, or otherwise, to find a value of $n$ that may be used as a counterexample.

[1]

Show that your value is a counterexample.

[2]

Explain why this disproves the statement.

[1]

Question 16

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the expression $\frac{x^3-1}{x-1}$ .

Show that $\frac{x^3-1}{x-1}\equiv x^2+x+1$ , for $x\neq 1$ .

[2]

Determine the solution of $\frac{x^3-1}{x-1}=3$ , giving a reason why any rejected value is not a solution.

[3]

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Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Use mathematical induction to prove that $\sum_{r=1}^{n} r(r+1)=\frac{n(n+1)(n+2)}{3}$ for all $n\in\mathbb{Z}^+$ .

[6]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Prove by contradiction that $\sqrt{6}$ is irrational. You may use the fact that if a prime divides $a^2$ , then it divides $a$ .

[3]

Hence prove that $\sqrt{2}+\sqrt{3}$ is irrational.

[3]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

For all real values of $x$ ,

$a(x-1)^2+b(x+2)\equiv 3x^2-4x+c$

where $a$ , $b$ and $c$ are constants.

Expand the left-hand side and collect like terms.

[2]

Determine the values of $a$ , $b$ and $c$ .

[3]

Check your result by substituting $x=2$ into both sides of the identity.

[1]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Use mathematical induction to prove that

$\sum_{r=1}^{n} r(r+1)=\frac{n(n+1)(n+2)}{3}$

for all $n\in\mathbb{Z}^+$ .

[6]

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Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Use mathematical induction to prove that $7$ divides

$3^{2n+1}+2^{n+2}$

for all $n\in\mathbb{Z}^+$ .

[6]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Prove by contradiction that $\sqrt{7}$ is irrational.

[6]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

For real values of $x$ , consider

A(x)=\frac{x^2+x-6}{x^2+4x+3}

Algebraic form and domain

I.

State the values of $x$ for which $A(x)$ is not defined.

[2]

II.

Show that

A(x)\equiv \frac{x-2}{x+1}

for all allowed values of $x$ .

[3]

Further algebraic manipulation

I.

Show that

\frac{x-2}{x+1}-1\equiv -\frac{3}{x+1}

[2]

II.

Hence solve $A(x)=\dfrac12$ .

[2]

Calculate both sides when $x=0$ and explain whether this check proves the identity in part (a)(ii).

[3]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $n\in\mathbb{Z}$ and define

F(n)=(n+2)^3-(n-1)^3

I.

Show that

F(n)\equiv 9(n^2+n+1)

[4]

II.

Hence prove that $F(n)$ is divisible by $9$ for every integer $n$ .

[2]

A student calculates $F(0)$ and $F(1)$ to support the result in part (a).

I.

Find $F(0)$ and $F(1)$ .

[2]

II.

Explain why these two calculations are not a proof of the result in part (a)(ii).

[2]

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Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the exact surd expressions

P=\sqrt{7+4\sqrt3},\qquad Q=\sqrt{7-4\sqrt3}

AI.

Show that $P=2+\sqrt3$ and $Q=2-\sqrt3$ .

[3]

AII.

Hence find the exact value of $P-Q$ .

[2]

BI.

Show that $PQ=1$ .

[2]

BII.

Find the exact value of $P+Q$ and hence solve

(P-Q)y+(P+Q)=8

[3]

Explain why a decimal approximation to $P-Q$ is not a proof of the exact value found in part (a)(ii).

[2]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Use mathematical induction to prove that $7$ divides $3^{2n+1}+2^{n+2}$ for all $n\in\mathbb{Z}^+$ .

[6]

Question 27

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

For real values of $x$ , consider the expression

E(x)=\frac{2}{x-1}+\frac{3}{x+2}

I.

State the values of $x$ for which $E(x)$ is not defined.

[2]

II.

Prove that

\frac{2}{x-1}+\frac{3}{x+2}\equiv \frac{5x+1}{x^2+x-2}

for all allowed values of $x$ .

[3]

Hence solve exactly

\frac{2}{x-1}+\frac{3}{x+2}=1

[3]

I.

Use your GDC to check the identity in part (a)(ii) when $x=4$ .

[1]

II.

Explain why the calculation in part (c)(i) is not a proof of the identity.

[1]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

For all real values of $x$ ,

A(x-2)^2+B(x+1)+C\equiv 4x^2-7x+9

where $A$ , $B$ and $C$ are constants.

I.

Expand the left-hand side and collect like terms.

[2]

II.

Determine the values of $A$ , $B$ and $C$ .

[3]

Let $F(x)=4(x-2)^2+9(x+1)-16$ and $G(x)=4x^2-7x+9$ .

I.

Use your GDC to find $F(-3)$ and $G(-3)$ .

[2]

II.

Explain why the result in part (b)(i) is consistent with, but does not prove, the identity.

[2]

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Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=(2x+1)^3-(2x-3)^3

I.

Use your GDC to find $f(0)$ and $f(2)$ .

[2]

II.

Prove that $f(x)\equiv 48x^2-48x+28$ .

[2]

Hence solve $f(x)=200$ , giving exact answers.

[3]

A student claims that the two values found in part (a)(i) prove the identity in part (a)(ii). Explain why this claim is incorrect.

[2]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Use mathematical induction to prove that, for $x>0$ ,

$\frac{d^n}{dx^n}(\ln x)=\frac{(-1)^{n-1}(n-1)!}{x^n}$

for all $n\in\mathbb{Z}^+$ .

[6]

Question 31

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A sequence is defined by $u_1=1$ and $u_{n+1}=5u_n+4$ for $n\in\mathbb{Z}^+$ . This question investigates a closed form and divisibility properties of the sequence.

$n$	$u_n$	$u_n+1$	$u_n^2-1$
1
2
3
4

I.

Calculate $u_2$ , $u_3$ and $u_4$ .

[2]

II.

Use your values to conjecture a formula for $u_n+1$ .

[1]

I.

Use mathematical induction to prove that

u_n=2\cdot 5^{n-1}-1

for all $n\in\mathbb{Z}^+$ .

[5]

I.

Hence prove that $8$ divides $u_n^2-1$ for every integer $n\geq 2$ .

[2]

II.

A student claims that $32$ divides $u_n^2-1$ for every integer $n\geq 2$ . Disprove the claim by giving a counterexample.

[1]

Question 32

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

For each $n\in\mathbb{Z}^+$ , define

a_n=\sqrt{n^2+1}-n

This question investigates exact forms and irrationality.

n	$a_n$ exact	$a_n$ decimal approx.
1	$\sqrt{2}-1$	0.414214
2	$\sqrt{5}-2$	0.236068
3		0.162278
4	$\sqrt{17}-4$	0.123106
5	$\sqrt{26}-5$	0.099020
6	$\sqrt{37}-6$	0.082763
7	$\sqrt{50}-7$	0.071068
8	$\sqrt{65}-8$	0.062258

I.

Show that

a_n\equiv \frac{1}{\sqrt{n^2+1}+n}

[2]

II.

Use the identity in part (a)(i) to find an exact expression for $a_3$ .

[2]

Prove by contradiction that $\sqrt{n^2+1}$ is irrational for every $n\in\mathbb{Z}^+$ .

[4]

I.

Hence prove that $a_n$ is irrational for every $n\in\mathbb{Z}^+$ .

[1]

II.

Disprove the statement: $\sqrt{n^2+7}$ is irrational for every $n\in\mathbb{Z}^+$ .

[1]

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Question 33

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A student investigates the quadratic expression

T_n=n^2+n+11

where $n\in\mathbb{N}$ . The student claims that $T_n$ is prime for every $n\in\mathbb{N}$ .

n	$T_n$	Prime?
0	11	Yes
1	13	Yes
2	17	Yes
3	23	Yes
4	31	Yes
5	41	Yes
6	53	Yes
7	67	Yes
8	83	Yes
9	101	Yes

I.

Calculate $T_0$ , $T_1$ and $T_2$ .

[2]

II.

Explain why the calculations in part (a)(i) do not prove the student's claim.

[1]

Find and justify a counterexample to the student's claim.

[2]

I.

Show that, for every $k\in\mathbb{N}$ ,

T_{11k+10}=11(11k^2+21k+11)

[3]

II.

Hence explain why there are infinitely many values of $n$ for which $T_n$ is not prime.

[2]

Question 34

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Let $f(x)=xe^x$ . For $n\in\mathbb{Z}^+$ , let $f^{(n)}(x)$ denote the $n$ th derivative of $f$ . This question investigates a formula for $f^{(n)}(x)$ .

Derivative order	Simplified expression
0	$xe^x$
1	$(x+1)e^x$
2	$(x+2)e^x$
3	$(x+3)e^x$

I.

Find $f'(x)$ and $f''(x)$ .

[2]

II.

Conjecture a formula for $f^{(n)}(x)$ .

[1]

Use mathematical induction to prove your conjecture from part (a)(ii). If you did not obtain a conjecture, prove that $f^{(n)}(x)=(x+n)e^x$ .

[5]

I.

Find the value of $n$ for which $f^{(n)}(0)=10$ .

[1]

II.

Explain why finding $f'(x)$ , $f''(x)$ and $f'''(x)$ would not prove the formula for every $n\in\mathbb{Z}^+$ .

[1]

Question 35

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $x$ be a real number for which $\sec x$ and $\tan x$ are defined.

Part (a)

I.

Show that

\sec x-\tan x\equiv \frac{1-\sin x}{\cos x}

[2]

II.

Show that

(\sec x-\tan x)(\sec x+\tan x)\equiv 1

[3]

Part (b)

I.

Hence write $\dfrac{1}{\sec x+\tan x}$ in the form $\sec x-\tan x$ .

[1]

II.

Solve exactly

\sec x+\tan x=\sqrt3

for $0\leq x<2\pi$ .

[3]

Explain why verifying the identity in part (a)(ii) at $x=0$ is not a proof of the identity.

[2]

Question 36

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

The function $f$ is defined by

f(x)=\frac{(x+1)^4-(x-1)^4}{8},\qquad x\in\mathbb{R}

I.

Show that

f(x)\equiv x^3+x

[4]

II.

Hence prove that $f(n)$ is even for every integer $n$ .

[3]

I.

Solve exactly $f(x)=6x$ .

[3]

II.

Explain why the identity in part (a)(i) justifies using $x^3+x$ in part (b)(i).

[1]

Check one of the non-zero solutions from part (b)(i) by substituting it into $x^3+x=6x$ .

[2]

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Question 37

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

For real values of $x$ , let

B(x)=\frac{x}{x-1}-\frac{2}{x+1}

I.

State the values of $x$ for which $B(x)$ is not defined.

[1]

II.

Show that

B(x)\equiv \frac{x^2-x+2}{x^2-1}

for all allowed values of $x$ .

[3]

I.

Show that

x^2-x+2\equiv \left(x-\frac12\right)^2+\frac74

[2]

II.

Hence prove that $B(x)>0$ for $x>1$ .

[2]

A student attempts to solve $B(x)=0$ by using only the numerator of the expression in part (a)(ii). Explain why $B(x)=0$ has no real solutions.

[2]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $n\in\mathbb{Z}^+$ , define

S_n=\sum_{r=1}^{n} r(2r+1)

I.

Find $S_3$ .

[2]

II.

Verify that $S_3=\dfrac{3(3+1)(4\cdot3+5)}{6}$ .

[1]

Use mathematical induction to prove that

S_n=\frac{n(n+1)(4n+5)}{6}

for all $n\in\mathbb{Z}^+$ .

[7]

Hence find

\sum_{r=4}^{n} r(2r+1),\qquad n\geq4

[2]

Question 39

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the statement $P(n)$ : $n^3+5n$ is divisible by $6$ , where $n\in\mathbb{Z}^+$ .

I.

Verify $P(1)$ and $P(2)$ .

[2]

II.

Use mathematical induction to prove $P(n)$ for all $n\in\mathbb{Z}^+$ .

[6]

A student claims that $n^3+5n$ is divisible by $12$ for every $n\in\mathbb{Z}^+$ . Disprove this claim by giving a counterexample.

[3]

Question 40

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A sequence $(u_n)$ is defined by

u_1=4,\qquad u_{n+1}=2u_n+3

for $n\in\mathbb{Z}^+$ .

I.

Find $u_2$ and $u_3$ .

[2]

II.

Verify that $u_n=7\cdot2^{n-1}-3$ for $n=1,2,3$ .

[1]

Use mathematical induction to prove that

u_n=7\cdot2^{n-1}-3

for all $n\in\mathbb{Z}^+$ .

[6]

I.

Find the value of $n$ for which $u_n=109$ .

[2]

II.

A student claims that $u_n$ is divisible by $5$ for every odd positive integer $n$ . Disprove this claim by giving a counterexample.

[2]

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Question 41

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

For $x\geq 1$ , define

R(x)=\sqrt{x+1}-\sqrt{x-1}

I.

Show that

R(x)\equiv \frac{2}{\sqrt{x+1}+\sqrt{x-1}}

[3]

II.

Use your GDC to check the identity when $x=10$ . Give both values to three significant figures.

[2]

Solve exactly $R(x)=0.4$ .

[4]

Question 42

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , let

S_n=\sum_{r=1}^{n}\frac{1}{r(r+2)}

I.

Show that

\frac{1}{r(r+2)}\equiv \frac{1}{2}\left(\frac{1}{r}-\frac{1}{r+2}\right)

[3]

II.

Use your GDC to find $S_{20}$ , giving your answer to three significant figures.

[1]

Hence show that

S_n\equiv \frac{3}{4}-\frac{1}{2(n+1)}-\frac{1}{2(n+2)}

[4]

Prove that $S_n<\frac{3}{4}$ for all $n\in\mathbb{Z}^+$ .

[2]

Question 43

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

For real $\theta$ , define

T(\theta)=\frac{1-\cos \theta}{\sin \theta}+\frac{\sin \theta}{1-\cos \theta}

I.

State the values of $\theta$ in the interval $0\leq \theta\leq 2\pi$ for which $T(\theta)$ is not defined.

[2]

II.

Prove that

T(\theta)\equiv \frac{2}{\sin \theta}

for all allowed values of $\theta$ .

[3]

Hence solve $T(\theta)=3$ for $0\leq \theta\leq 2\pi$ . Give your answers in radians to three significant figures.

[2]

Use your GDC to check the identity in part (a)(ii) at $\theta=1.2$ , and explain why this check is not a proof.

[2]

Question 44

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A student investigates the numbers

P(n)=n^2-n+11,\qquad n\in\mathbb{N}

The student claims that $P(n)$ is prime for every $n\in\mathbb{N}$ .

$n$	$P(n)$	Prime?
1	11	yes
2	13	yes
3	17	yes
4	23	yes
5	31	yes
6	41	yes
7	53	yes
8	67	yes
9	83	yes
10	not shown	not shown
11	not shown	not shown

I.

Use your GDC to find $P(10)$ and $P(11)$ .

[2]

II.

State which input value of $n$ gives a counterexample.

[1]

Show that $n=11$ is a counterexample.

[3]

For an integer $a>1$ , define $Q_a(n)=n^2-n+a$ . Show that the statement “ $Q_a(n)$ is prime for every $n\in\mathbb{N}$ ” is false whenever $a$ is prime.

[3]

Explain why the values of $P(n)$ for $1\leq n\leq 10$ cannot prove the student’s claim.

[1]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

This question investigates a matrix representation of the Fibonacci sequence. The Fibonacci numbers are defined by $F_0=0$ , $F_1=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 1$ . Let

A=\begin{pmatrix}1&1\\1&0\end{pmatrix}

n	Fibonacci terms	$A^{n}$
1	1, 1, 0	$\begin{pmatrix}1&1\\1&0\end{pmatrix}$
2	1, 1, 1	$\begin{pmatrix}2&1\\1&1\end{pmatrix}$
3	2, 1, 1	$\begin{pmatrix}3&2\\2&1\end{pmatrix}$
4	3, 2, 1	$\begin{pmatrix}5&3\\3&2\end{pmatrix}$
5	5, 3, 2	$\begin{pmatrix}8&5\\5&3\end{pmatrix}$
6	8, 5, 3	$\begin{pmatrix}13&8\\8&5\end{pmatrix}$

I.

Calculate $A^2$ and $A^3$ .

[2]

II.

Verify, using your answers to part (a)(i), that

A^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}

for $n=2$ and $n=3$ .

[2]

Use mathematical induction to prove that

A^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}

for all $n\in\mathbb{Z}^+$ .

[5]

I.

Hence write down $F_8$ .

[1]

II.

By considering determinants, prove Cassini's identity

F_{n+1}F_{n-1}-F_n^2=(-1)^n

for all $n\in\mathbb{Z}^+$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $z=\cos\theta+i\sin\theta$ , where $\theta\in\mathbb{R}$ . This question investigates powers of complex numbers on the unit circle.

I.

Show that $z^2=\cos 2\theta+i\sin 2\theta$ .

[2]

II.

Write down $z^3$ in the form $\cos a+i\sin a$ .

[1]

Use mathematical induction to prove De Moivre's theorem,

(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta

for all $n\in\mathbb{Z}^+$ .

[5]

I.

Let $\omega=\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5}$ . Prove that

1+\omega+\omega^2+\omega^3+\omega^4=0

[2]

II.

Hence prove that

\cos\frac{2\pi}{5}+\cos\frac{4\pi}{5}=-\frac12

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , define

S_n=\sum_{r=1}^{n}\frac{1}{r(r+2)}

This question investigates a telescoping sum and two different proof methods.

n	Exact S_n	Decimal approx.
1	1/3	0.333333
2	11/24	0.458333
4	17/30	0.566667
8	29/45	0.644444
16	106/153	0.692810
32	404/561	0.720143
64	1576/2145	0.734732

I.

Show that

\frac{1}{r(r+2)}\equiv \frac12\left(\frac1r-\frac{1}{r+2}\right)

for all allowed values of $r$ .

[2]

II.

Calculate $S_3$ exactly.

[2]

Use part (a)(i) to show that

S_n=\frac34-\frac{1}{2(n+1)}-\frac{1}{2(n+2)}

[3]

Prove the result in part (b) by mathematical induction. If you did not obtain the formula in part (b), use

S_n=\frac34-\frac{1}{2(n+1)}-\frac{1}{2(n+2)}

[5]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

This question investigates identities involving binomial coefficients. The notation $\binom{n}{r}$ has its usual meaning, and $n\in\mathbb{Z}^+$ .

$n$	$r=0$	$r=1$	$r=2$	$r=3$	$r=4$	$r=5$	$r=6$
1	1	1	—	—	—	—	—
2	1	2	1	—	—	—	—
3	1	3	3	1	—	—	—
4	1	4	6	4	1	—	—
5	1	5	10	10	5	1	—
6	1	6	15	20	15	6	1

I.

Calculate

\sum_{r=0}^{3}\binom{3}{r}^2

[2]

II.

Write down $\binom{6}{3}$ .

[1]

By considering the coefficient of $x^n$ in $(1+x)^n(1+x)^n$ , prove that

\sum_{r=0}^{n}\binom{n}{r}^2=\binom{2n}{n}

[4]

I.

Use mathematical induction to prove that

\sum_{r=0}^{n}\binom{n}{r}=2^n

for all $n\in\mathbb{Z}^+$ .

[4]

II.

A student claims that $\binom{2n}{n}=2^n$ for all $n\in\mathbb{Z}^+$ . Disprove the claim.

[1]

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Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

This question investigates Bernoulli's inequality:

(1+x)^n\geq 1+nx

Assume throughout parts (a), (b) and (d) that $x\geq -1$ and $n\in\mathbb{Z}^+$ .

I.

Verify Bernoulli's inequality for $n=2$ by simplifying the difference $(1+x)^2-(1+2x)$ .

[2]

II.

State the value of $x$ for which equality holds in part (a)(i).

[1]

Use mathematical induction to prove Bernoulli's inequality for all $n\in\mathbb{Z}^+$ .

[6]

Disprove the statement that Bernoulli's inequality is true for all real $x$ and all $n\in\mathbb{Z}^+$ .

[1]

I.

Use Bernoulli's inequality to show that

\left(1+\frac1n\right)^n\geq 2

for all $n\in\mathbb{Z}^+$ .

[1]

II.

Explain why checking the inequality for $n=1,2,3$ would not prove the result in part (d)(i).

[1]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

This question investigates a divisibility result involving powers. Define

D_n=5^{2n}-3^n

for $n\in\mathbb{Z}^+$ .

$n$	$D_n = 5^{2n} - 3^n$	$D_n / 11$
5	9765382	887762
6	244139896	22194536
3	15598	1418
4	390544	35504

I.

Calculate $D_1$ and $D_2$ .

[2]

II.

Verify that $11$ divides both values found in part (a)(i).

[1]

Use mathematical induction to prove that $11$ divides $5^{2n}-3^n$ for all $n\in\mathbb{Z}^+$ .

[5]

I.

A student claims that $121$ divides $5^{2n}-3^n$ for all $n\in\mathbb{Z}^+$ . Disprove this claim.

[1]

II.

Explain why part (b) does not imply the student's claim in part (c)(i).

[1]

Question 51

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $p_1,p_2,\ldots,p_k$ denote the first $k$ prime numbers, and define

N_k=p_1p_2\cdots p_k+1

This question investigates Euclid's argument for the infinitude of primes.

$k$	$p_1\cdots p_k$	$N_k$	Nature of $N_k$
1	2	3	Prime
2	6	7	Prime
3	30	31	Prime
4	210	211	Prime
5	2310	2311	Prime
6	30030	30031	Composite

I.

Calculate $N_3$ .

[1]

II.

State whether $N_3$ is prime.

[1]

Use proof by contradiction to prove that there are infinitely many prime numbers.

[5]

I.

A student claims that $N_k$ is prime for every $k\in\mathbb{Z}^+$ . Use the case $k=6$ to disprove the claim.

[2]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

z=\cos\theta+{\mathrm{i}}\sin\theta

where $\theta\in\mathbb{R}$ .

I.

Show that $z\overline z=1$ , where $\overline z$ is the complex conjugate of $z$ .

[1]

II.

Find $z^2$ in the form $a+{\mathrm{i}}b$ .

[2]

Use mathematical induction to prove that

z^n=\cos(n\theta)+{\mathrm{i}}\sin(n\theta)

for all $n\in\mathbb{Z}^+$ .

[6]

I.

By expanding $z^3$ , show that the real part of $z^3$ is $\cos^3\theta-3\cos\theta\sin^2\theta$ .

[2]

II.

Hence show that

\cos(3\theta)=4\cos^3\theta-3\cos\theta

[2]

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Question 53

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

This question concerns irrational numbers.

I.

Prove by contradiction that $\sqrt[3]{4}$ is irrational. You may use the fact that if a prime $p$ divides $a^3$ , then $p$ divides $a$ .

[5]

II.

Hence prove that $5-\sqrt[3]{4}$ is irrational.

[3]

Disprove the statement: the product of any two irrational numbers is irrational.

[2]

Question 54

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

This question considers Euclid's argument about prime numbers. Assume, for contradiction, that $p_1,p_2,\ldots,p_k$ is a complete finite list of all primes, with the primes distinct, and define

N=p_1p_2\cdots p_k+1

Assume, for contradiction, that there are only finitely many primes, $p_1,p_2,\ldots,p_k$ . Define

N=p_1p_2\cdots p_k+1

I.

Show that none of $p_1,p_2,\ldots,p_k$ divides $N$ .

[3]

II.

Explain why $N$ has a prime divisor.

[2]

III.

Hence complete the proof by contradiction that there are infinitely many primes.

[2]

A student claims that for every finite set of primes, the product of the primes plus $1$ is itself prime.

[3]

Explain why the counterexample in part (b) does not invalidate the proof in part (a).

[2]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , let

S_n=\sum_{r=1}^{n} r2^r

The formula for $S_n$ is assumed to have the form

S_n=(an+b)2^{n+1}+c

I.

Use your GDC to find $S_1$ , $S_2$ and $S_3$ .

[1]

II.

Determine $a$ , $b$ and $c$ .

[2]

Use mathematical induction to prove that

\sum_{r=1}^{n} r2^r=(n-1)2^{n+1}+2

for all $n\in\mathbb{Z}^+$ .

[6]

Use the formula to determine the least value of $n$ for which $S_n>10^6$ .

[2]

Question 56

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , define

A_n=5^{2n}-(-1)^n

I.

Use your GDC to find $A_1$ , $A_2$ and $A_3$ .

[1]

II.

State a conjecture about the divisibility of $A_n$ .

[2]

Use mathematical induction to prove your conjecture.

[5]

Hence find the remainder when $5^{2026}$ is divided by $13$ .

[2]

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Question 57

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $y=x^2e^x$ .

I.

Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ .

[2]

II.

Use your answers to suggest a formula for $\frac{d^ny}{dx^n}$ .

[2]

Use mathematical induction to prove that

\frac{d^n}{dx^n}(x^2e^x)=(x^2+2nx+n(n-1))e^x

for all $n\in\mathbb{Z}^+$ .

[6]

Use the formula and your GDC to find the minimum value of $\frac{d^5y}{dx^5}$ for $-8\leq x\leq 1$ , giving your answer to three significant figures.

[2]

Question 58

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

You may use the fact that if a prime number divides $a^3$ , where $a\in\mathbb{Z}$ , then it divides $a$ .

I.

Use your GDC to find a decimal approximation to $\sqrt[3]{4}$ .

[1]

II.

Prove by contradiction that $\sqrt[3]{4}$ is irrational.

[5]

Let $p$ and $q$ be rational numbers with $q\ne 0$ . Prove by contradiction that $p+q\sqrt[3]{4}$ is irrational.

[4]

Question 59

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

S_n=\sum_{r=0}^{n} i^r

where $i^2=-1$ and $n$ is a non-negative integer.

I.

Use your GDC, or otherwise, to find $S_0$ , $S_1$ , $S_2$ and $S_3$ .

[2]

II.

State a formula for $S_n$ in terms of $i^{n+1}$ .

[1]

Use mathematical induction to prove that

S_n=\frac{1-i^{n+1}}{1-i}

for all integers $n\geq 0$ .

[5]

Hence determine all integers $n$ such that $0\leq n\leq 50$ and $S_n=0$ .

[2]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A sequence is defined by

x_1=\sqrt2,\qquad x_{n+1}=\sqrt{2+x_n}

for $n\in\mathbb{Z}^+$ . This question investigates bounds and the limiting behaviour suggested by the sequence.

I.

Calculate $x_2$ and $x_3$ exactly in nested radical form.

[2]

II.

Use your GDC to give $x_5$ to three significant figures.

[1]

Use mathematical induction to prove that $1<x_n<2$ for all $n\in\mathbb{Z}^+$ .

[5]

Show that the sequence is increasing.

[3]

I.

Assuming the sequence converges to a limit $L$ , find $L$ .

[1]

II.

Explain why the induction proof in part (b) alone does not prove that the sequence converges.

[1]

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n	$T_n$	Prime?
0	11	Yes
1	13	Yes
2	17	Yes
3	23	Yes
4	31	Yes
5	41	Yes
6	53	Yes
7	67	Yes
8	83	Yes
9	101	Yes

$n$	$r=0$	$r=1$	$r=2$	$r=3$	$r=4$	$r=5$	$r=6$
1	1	1	—	—	—	—	—
2	1	2	1	—	—	—	—
3	1	3	3	1	—	—	—
4	1	4	6	4	1	—	—
5	1	5	10	10	5	1	—
6	1	6	15	20	15	6	1

n	$T_n$	Prime?
0	11	Yes
1	13	Yes
2	17	Yes
3	23	Yes
4	31	Yes
5	41	Yes
6	53	Yes
7	67	Yes
8	83	Yes
9	101	Yes

$n$	$r=0$	$r=1$	$r=2$	$r=3$	$r=4$	$r=5$	$r=6$
1	1	1	—	—	—	—	—
2	1	2	1	—	—	—	—
3	1	3	3	1	—	—	—
4	1	4	6	4	1	—	—
5	1	5	10	10	5	1	—
6	1	6	15	20	15	6	1

n	$T_n$	Prime?
0	11	Yes
1	13	Yes
2	17	Yes
3	23	Yes
4	31	Yes
5	41	Yes
6	53	Yes
7	67	Yes
8	83	Yes
9	101	Yes

$n$	$r=0$	$r=1$	$r=2$	$r=3$	$r=4$	$r=5$	$r=6$
1	1	1	—	—	—	—	—
2	1	2	1	—	—	—	—
3	1	3	3	1	—	—	—
4	1	4	6	4	1	—	—
5	1	5	10	10	5	1	—
6	1	6	15	20	15	6	1