Binomial Theorem - IB Math AA Question Bank

Subjects

Features

Review

HOT

Tutoring

Binomial Theorem

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

Verified by Karim

Paper

Difficulty

Status

Level

This content can't be printed from the browser. Use Export PDF with Pro for an offline copy.

Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

Consider the expansion of $(2-3x)^5$ in ascending powers of $x$ .

Find the coefficient of $x^2$ .

[2]

Find the first three terms of the expansion in ascending powers of $x$ .

[3]

Question 2

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

In the expansion of $(1+kx)^6$ , where $k$ is a positive constant, the coefficient of $x^2$ is $60$ .

Show that $k=2$ .

[3]

Hence find the coefficient of $x^3$ .

[2]

Question 3

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Consider the expansion of $(2-3x)^8$ .

Write down the number of terms in the expansion.

[1]

Find the coefficient of $x^2$ .

[3]

Question 4

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

A row of Pascal's triangle is numbered $n$ , starting with row $0$ . In one row, the second entry is $11$ .

Write down the value of $n$ .

[1]

Using a GDC or otherwise, find all values of $r$ such that $\binom{11}{r}=330$ .

[2]

Hence find the coefficient of $a^7b^4$ in the expansion of $(a+b)^{11}$ .

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The coefficient of $x^3$ in the expansion of $(1+2x)^n$ is $8$ times the coefficient of $x^2$ , where $n\in\mathbb{Z}^+$ and $n\geq 3$ .

Write down expressions, in terms of $n$ , for the coefficients of $x^2$ and $x^3$ .

[3]

Find $n$ .

[3]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the expansion of

\left(x+\frac{2}{x^2}\right)^9

where $x\neq 0$ .

Find the exponent of $x$ in the general term corresponding to $r$ .

[2]

Find the constant term in the expansion.

[4]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The expansion of $(1+x)^p$ in ascending powers of $x$ begins

1+px+10x^2+\cdots

where $p\in\mathbb{Z}^+$ and $p>1$ .

Find the possible values of $p$ .

[3]

Hence find the coefficient of $x^3$ in the expansion.

[2]

Question 8

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The expansion of $(1+kx)^6$ is considered, where $k$ is a real constant.

Find, in terms of $k$ , the coefficient of $x^2$ .

[2]

Given that the coefficient of $x^2$ is $240$ , find the possible values of $k$ .

[2]

For $k=4$ , find the coefficient of $x^5$ .

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 9

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Consider the expansion of

\left(x^2+\frac{3}{x}\right)^9, \quad x\ne 0

Show that the constant term occurs when $r=6$ .

[2]

Find the constant term.

[3]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the coefficients in the expansion of $(1+x)^{15}$ .

Using a GDC table or otherwise, determine all values of $r$ for which $\binom{15}{r}>3000$ .

[2]

Hence determine how many terms in the expansion have coefficients greater than $3000$ .

[2]

Question 11

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The first three terms in the expansion of $(1+ax)^8$ in ascending powers of $x$ are

1+24x+bx^2+\cdots

where $a$ and $b$ are constants.

Find $a$ .

[2]

Find $b$ .

[2]

Using your value of $a$ , find the coefficient of $x^2$ in $(1+ax)^8(1-x)^2$ .

[3]

Question 12

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

In the expansion of $(3x-2)^7$ , the terms are written in descending powers of $x$ .

Find the fourth term.

[3]

Find the ratio of the coefficient of $x^4$ to the coefficient of $x^5$ .

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

In the expansion of $(1+mx)^7(1-2x)$ , the coefficient of $x$ is $5$ .

Show that $m=1$ .

[3]

Hence find the coefficient of $x^3$ in the expansion of $(1+mx)^7(1-2x)$ .

[3]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the expansion of

\left(2x^2-\frac{1}{x}\right)^8

where $x\neq 0$ .

Find the general term in the expansion corresponding to $r$ .

[2]

Find the coefficient of $x^4$ .

[4]

Question 15

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A student uses the binomial theorem to approximate $(1.01)^{12}$ .

Write down the first three terms in the expansion of $(1+x)^{12}$ in ascending powers of $x$ .

[3]

Use these three terms to approximate $(1.01)^{12}$ .

[1]

Using your GDC, calculate the percentage error in this approximation.

[2]

Question 16

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let $n\in\mathbb{N}$ . In the expansion of $(3+x^2)^n$ , the coefficient of $x^4$ is $1215$ .

Form an equation in $n$ .

[3]

Hence find $n$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the polynomial

(1+2x)^8(1-x)^5

Write down the coefficient of $x^3$ in the expansion of $(1+2x)^8$ .

[1]

Determine the coefficient of $x^4$ in the polynomial.

[5]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $a>0$ . In the expansion of

\left(ax^2-\frac{3}{x}\right)^9, \quad x\ne 0

the constant term is $489888$ .

Show that the constant term is $61236a^3$ .

[3]

Find the value of $a$ .

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

In the expansion of $(1+ax)^n$ , where $a>0$ and $n\in\mathbb{N}$ , the coefficient of $x$ is $18$ and the coefficient of $x^2$ is $135$ .

Form two equations involving $a$ and $n$ .

[3]

Hence determine the value of $n$ and the value of $a$ .

[3]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $n>4$ , in the expansion of $(2+x)^n$ , the coefficients of $x^3$ and $x^4$ are equal.

Find the value of $n$ .

[3]

Hence find the coefficient of $x^5$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 21

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $p$ and $q$ be positive constants. The polynomial

P(x)=(1+px)^5(1-qx)^3

has coefficient of $x$ equal to $1$ and coefficient of $x^2$ equal to $-23$ .

I.

Find, in terms of $p$ and $q$ , the coefficient of $x$ in $P(x)$ .

[2]

II.

Find, in terms of $p$ and $q$ , the coefficient of $x^2$ in $P(x)$ .

[3]

Determine the values of $p$ and $q$ .

[4]

Hence find the coefficient of $x^3$ in $P(x)$ .

[2]

Question 22

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

In the expansion of $(a+bx)^7$ in ascending powers of $x$ , the first three terms are

128+448x+672x^2+\cdots

where $a>0$ and $b>0$ .

I.

Find the value of $a$ .

[2]

II.

Find the value of $b$ .

[2]

Verify that the coefficient of $x^2$ is $672$ .

[2]

Using your values of $a$ and $b$ , find the coefficient of $x^4$ in

(a+bx)^7(1-x)^2

[4]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

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

E_n=\binom{n}{0}+\binom{n}{2}2^2+\binom{n}{4}2^4+\cdots

and

O_n=\binom{n}{1}2+\binom{n}{3}2^3+\binom{n}{5}2^5+\cdots

where the sums continue over the possible even and odd indices respectively.

I.

Write down an expression for $E_n+O_n$ .

[1]

II.

Find an expression for $E_n-O_n$ .

[2]

Hence prove that

E_n=\frac{3^n+(-1)^n}{2}

[4]

Find the value of $E_6$ .

[2]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $n\in\mathbb{Z}^+$ with $n\geq 3$ . In the expansion of $(1+x)^n$ , the coefficient of $x^3$ is twice the coefficient of $x^2$ .

I.

Write down expressions for the coefficients of $x^2$ and $x^3$ .

[2]

II.

Find the value of $n$ .

[3]

Using your value of $n$ , find the coefficient of $x^4$ in

(1+x)^n(1-x)^2

[4]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The coefficients of three consecutive terms in the expansion of $(1+x)^n$ are $15$ , $20$ and $15$ , in that order, where $n\in\mathbb{Z}^+$ .

Let the coefficient $15$ be $\binom{n}{r}$ . Write two equations involving $n$ and $r$ .

[2]

Determine $n$ and $r$ .

[5]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The first three terms in the expansion of $(a+bx)^6$ in ascending powers of $x$ are

64+192x+240x^2+\cdots

where $a>0$ and $b>0$ .

Find $a$ .

[2]

Find $b$ .

[3]

Verify that the coefficient of $x^2$ is $240$ .

[2]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

S=\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots

be the sum of the even-indexed binomial coefficients in the expansion of $(1+x)^n$ , where $n\in\mathbb{Z}^+$ .

Write down the value of $(1+1)^n$ as a sum of binomial coefficients.

[2]

Using $(1-1)^n$ , prove that $S=2^{n-1}$ .

[4]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A student uses the binomial theorem to approximate powers of numbers close to $1$ .

Consider the expansion of $(1+x)^{10}$ in ascending powers of $x$ .

I.

Write down the first four non-zero terms of the expansion of $(1+x)^{10}$ .

[3]

II.

Use your answer to part (a)(i) to approximate $(1.025)^{10}$ .

[2]

Using your GDC, calculate the percentage error in the approximation from part (a)(ii).

[3]

Let $A(x)=1+10x+45x^2+120x^3$ . Determine the largest value of $x$ , where $0\leq x\leq 0.05$ , for which the percentage error in using $A(x)$ to approximate $(1+x)^{10}$ is less than $0.010\%$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let

P(x)=(2+px)^7(1-qx)^3

where $p>0$ and $q>0$ . In the expansion of $P(x)$ , the coefficient of $x$ is $64$ and the coefficient of $x^2$ is $-288$ .

Use the given information to find $p$ and $q$ .

I.

Show that $7p-6q=1$ .

[3]

II.

Hence find $p$ and $q$ .

[3]

For these values of $p$ and $q$ , find the coefficient of $x^3$ in $P(x)$ .

[3]

For these values of $p$ and $q$ , find the coefficient of $x^4$ in $P(x)$ .

[2]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let

F(x)=\left(x^2+\frac{a}{x}\right)^8

where $x\ne 0$ and $a>0$ .

Consider the general term in the expansion of $F(x)$ corresponding to the index $r$ , where $0\leq r\leq 8$ .

I.

Write down the general term.

[2]

II.

Hence state all possible exponents of $x$ in the expansion.

[2]

Given that the coefficient of $x^4$ is $5670$ , find $a$ .

[3]

Using your value of $a$ , find the coefficient of $x^{-2}$ in $F(x)$ .

[2]

Question 31

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Rows of Pascal's triangle are numbered starting from row $0$ . The entry in position $r$ of row $n$ is $\binom{n}{r}$ , where $r=0,1,\ldots,n$ .

row	entries
$0$	$1$
$1$	$1,\,1$
$2$	$1,\,2,\,1$
$3$	$1,\,3,\,3,\,1$
$4$	$1,\,4,\,6,\,4,\,1$
$5$	$1,\,5,\,10,\,10,\,5,\,1$
$6$	$1,\,6,\,15,\,20,\,15,\,6,\,1$
$n$	$\binom{n}{0},\,\binom{n}{1},\,\binom{n}{2},\,\binom{n}{3},\,\binom{n}{4},\,\binom{n}{5},\,\binom{n}{6}$

In a particular row, the entry in position $4$ is five times the entry in position $3$ .

I.

In the row labelled $n$ in the table, the entry in position $4$ is five times the entry in position $3$ . Show that the row number is $23$ .

[3]

II.

Using a GDC table or otherwise, find the largest entry in row $23$ .

[1]

Find the coefficient of $x^4$ in the expansion of $(2+x)^{23}$ .

[3]

Use a counting argument to justify Pascal's identity

\binom{n+1}{r}=\binom{n}{r-1}+\binom{n}{r}

where $1\leq r\leq n$ .

[3]

Question 32

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A grid route consists of $12$ moves. Each move is either east or north. A route with exactly $r$ north moves has weight $3^{12-r}2^r$ .

Use binomial coefficients to count routes.

I.

Write down the number of routes with exactly $r$ north moves.

[1]

II.

Find the total weight of all routes with exactly $5$ north moves.

[2]

Show that the total weight of all possible routes is $5^{12}$ .

[2]

A route is selected with probability proportional to its weight. Find the probability that the selected route has exactly $5$ north moves.

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 33

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the product

\left(x+\frac{2}{x^2}\right)^7(1-x)^8, \quad x\ne 0

Find all pairs $(r,s)$ that can contribute to the constant term, where $r$ is the index in the expansion of $\left(x+\frac{2}{x^2}\right)^7$ and $s$ is the index in the expansion of $(1-x)^8$ .

[2]

Determine the constant term in the product.

[4]

Question 34

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $k$ be a positive constant and let

E(x)=\left(x^2+\frac{k}{x}\right)^8(1-x)^4,\quad x\neq 0

The constant term in the expansion of $E(x)$ is $10752$ .

I.

Write down the general term of $\left(x^2+\frac{k}{x}\right)^8$ corresponding to the index $r$ , and simplify the power of $x$ .

[3]

II.

Show that the only contribution to the constant term in $E(x)$ occurs when $r=6$ and the term $x^2$ is chosen from $(1-x)^4$ .

[2]

Find the value of $k$ .

[3]

Hence find the coefficient of $x$ in $E(x)$ .

[2]

Question 35

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $a$ and $b$ be positive constants. In the expansion of

(1+ax)^4(1+bx)^4

the coefficient of $x$ is $16$ and the coefficient of $x^2$ is $104$ .

I.

Show how the coefficient of $x$ gives $a+b=4$ .

[2]

II.

Express the coefficient of $x^2$ in terms of $a+b$ and $ab$ .

[3]

Find the value of $ab$ .

[2]

Determine the possible values of $a$ and $b$ .

[2]

Find the coefficient of $x^3$ in $(1+ax)^4(1+bx)^4$ .

[2]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $n\in\mathbb{Z}^+$ and $k>0$ . In the expansion of $(1+kx)^n$ , the coefficient of $x$ is $15$ and the coefficient of $x^2$ is $90$ .

I.

Write an equation using the coefficient of $x$ .

[1]

II.

Write an equation using the coefficient of $x^2$ .

[2]

Find the values of $n$ and $k$ .

[4]

Find the coefficient of $x^3$ in $(1+kx)^n$ .

[2]

Hence find the coefficient of $x^3$ in

(1+kx)^n(1-x)^4

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 37

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Let $n\in\mathbb{Z}^+$ , $n\geq 3$ . In the expansion of $(1+x)^n$ , the coefficient of $x^3$ is twice the coefficient of $x^2$ .

Work with the coefficients of $x^2$ and $x^3$ in $(1+x)^n$ .

I.

Write down expressions for these two coefficients.

[2]

II.

Hence determine $n$ .

[4]

Let

Q(x)=(1+x)^8(1-2x)^4

Find the coefficient of $x^5$ in $Q(x)$ .

[4]

Using a GDC, determine the greatest magnitude of any coefficient in the expansion of $(1+x)^8(1-2x)^4$ .

[2]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $n\in\mathbb{Z}^+$ and $k>0$ . In the expansion of $(1+kx)^n$ , the coefficient of $x$ is $30$ and the coefficient of $x^2$ is $405$ .

Use the given coefficients to find $n$ and $k$ .

I.

Form two equations involving $n$ and $k$ .

[2]

II.

Hence determine $n$ and $k$ .

[4]

For these values of $n$ and $k$ , find the coefficient of $x^3$ in $(1+kx)^n$ .

[2]

Let

R(x)=(1+3x)^{10}(1-x)^6

Find the coefficient of $x^7$ in $R(x)$ .

[3]

Determine the largest coefficient in the expansion of $(1+3x)^{10}$ .

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

G(x)=\left(x^2+\frac{a}{x}\right)^{10}(1+bx)^5

where $x\ne 0$ , $a>0$ and $b>0$ . In the expansion of $\left(x^2+\frac{a}{x}\right)^{10}$ , the coefficient of $x^{14}$ is $720$ .

Find $a$ .

I.

Write down the general term in the expansion of $\left(x^2+\frac{a}{x}\right)^{10}$ .

[2]

II.

Hence find $a$ .

[3]

Let $b=2$ . Find the constant term in the expansion of $G(x)$ . If you did not obtain $a=4$ , use $a=4$ for this part.

[7]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

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

H_n(x)=(1+x)^n(1+x^2)^n

In the expansion of $H_n(x)$ , the coefficient of $x^3$ is $220$ .

Find $n$ .

I.

Show that the coefficient of $x^3$ in $H_n(x)$ is $\binom{n}{3}+n^2$ .

[3]

II.

Hence find $n$ .

[2]

Find the coefficient of $x^6$ in $H_{10}(x)$ .

[4]

Using a GDC, determine the largest coefficient in the expansion of $H_{10}(x)$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A family of calibration polynomials is defined by $A(x)=(1+kx)^n$ , where $n\in\mathbb{Z}^+$ and $k>0$ . The coefficient of $x^r$ in the expansion of $A(x)$ is denoted by $c_r$ . A partial output from a GDC for one such polynomial is shown.

$r$	$c_r$
0	1
1	24
2	252
3	1512

I.

Write down an expression for $c_r$ in terms of $n$ , $k$ and $r$ .

[2]

II.

Show that $\dfrac{c_{r+1}}{c_r}=\dfrac{k(n-r)}{r+1}$ .

[3]

For the polynomial shown in the table, $c_1=24$ and $c_2=252$ . Determine the value of $n$ and the value of $k$ .

[4]

Hence find the coefficient of $x^4$ in $A(x)(1-x)^5$ .

[3]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A diagonal sum in Pascal's triangle is defined by $D_m=\sum_{r=0}^{m}\binom{r}{2}$ , where entries with lower index greater than upper index are taken as zero.

I.

Calculate $D_6$ .

[1]

II.

Write down the value of $\binom{7}{3}$ and comment on your answer to part (a)(i).

[2]

I.

Show that $\binom{r}{2}+\binom{r}{3}=\binom{r+1}{3}$ for $r\geq 2$ .

[2]

II.

Prove that $D_m=\binom{m+1}{3}$ for all integers $m\geq 2$ .

[2]

The coefficient of $x^2$ in $\sum_{r=0}^{m}(1+x)^r$ is $560$ . Hence determine $m$ .

[3]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A laboratory model for the remaining proportion of a substance after time $t$ is $R(t)=(1-ct)^m$ , where $m\in\mathbb{Z}^+$ and $0<c<1$ . The first three terms in ascending powers of $t$ are $1-0.48t+0.1008t^2+\cdots$ .

I.

Write down expressions, in terms of $m$ and $c$ , for the coefficients of $t$ and $t^2$ in the expansion of $(1-ct)^m$ .

[2]

II.

Find $m$ and $c$ .

[4]

Using terms up to and including $t^3$ , approximate $R(3)$ .

[3]

Calculate the percentage error in the approximation from part (b), using the exact value of $R(3)$ .

[2]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A weighted row is generated from the expansion of $(1+qx)^n$ , where $n\in\mathbb{Z}^+$ and $q>0$ . The coefficient of $x^r$ is denoted by $w_r$ . In one row, $w_2=405$ and $\dfrac{w_3}{w_2}=8$ .

r	w_r
0	1
1	30
2	405
3	3240
4	17010
5	61236
6	153090
7	262440
8	295245
9	196830
10	59049

I.

Write down $w_r$ in terms of $n$ , $q$ and $r$ .

[1]

II.

Show that $\dfrac{w_3}{w_2}=\dfrac{q(n-2)}{3}$ .

[3]

Determine $n$ and $q$ .

[4]

Hence find the sum of the coefficients of the odd powers of $x$ in $(1+qx)^n$ , and interpret this as a coefficient sum.

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For an even row $2m$ of Pascal's triangle, let $M_m=\binom{2m}{m}$ be the central coefficient.

I.

Show that $\dfrac{\binom{2m}{m}}{\binom{2m}{m-1}}=\dfrac{m+1}{m}$ .

[2]

II.

Explain why $M_m$ is greater than each adjacent coefficient in row $2m$ .

[2]

Using a GDC table or otherwise, determine the smallest even row number $2m$ for which $M_m>100000$ .

[3]

Prove that $\binom{2m}{m}=2\binom{2m-1}{m-1}$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

On a rectangular grid, a shortest path from $(0,0)$ to $(p,q)$ consists of $p$ moves right and $q$ moves up. Such paths can be represented by terms in the expansion of $(R+U)^{p+q}$ .

I.

Find the number of shortest paths from $(0,0)$ to $(7,5)$ .

[2]

II.

Find the number of shortest paths from $(0,0)$ to $(7,5)$ that pass through $(3,2)$ .

[2]

Show that the number of shortest paths from $(0,0)$ to $(p,q)$ is the coefficient of $R^pU^q$ in $(R+U)^{p+q}$ .

[3]

Prove Vandermonde's identity $\sum_{r=0}^{m}\binom{k}{r}\binom{n-k}{m-r}=\binom{n}{m}$ , where terms with invalid lower indices are zero.

[3]

Hence calculate $\sum_{r=0}^{4}\binom{6}{r}\binom{8}{4-r}$ .

[2]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two adjacent entries in a row of Pascal's triangle are recorded as $495$ and $792$ , in that order from left to right. The row is numbered $n$ , starting with row $0$ , and the first of the two entries is $\binom{n}{r}$ .

I.

Show that $\dfrac{\binom{n}{r+1}}{\binom{n}{r}}=\dfrac{n-r}{r+1}$ .

[2]

II.

Determine $n$ and $r$ .

[3]

Hence find the coefficient of $x^8$ in the expansion of $(2+x)^n$ .

[2]

Using Pascal's identity, show that the entry directly below $495$ and $792$ in the next row is $1287$ .

[3]

Question 48

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $n\in\mathbb{Z}^+$ . Consider the product

(1+x)^n(1-x)^n

In this question, take $\binom{n}{r}=0$ when $r<0$ or $r>n$ .

I.

Write down an expression, as a summation, for the coefficient of $x^k$ in $(1+x)^n(1-x)^n$ .

[3]

II.

Simplify $(1+x)^n(1-x)^n$ .

[2]

I.

Prove that, for odd $k$ ,

\sum_{r=0}^{k}(-1)^r\binom{n}{r}\binom{n}{k-r}=0

[2]

II.

Prove that, for $0\leq m\leq n$ ,

\sum_{r=0}^{2m}(-1)^r\binom{n}{r}\binom{n}{2m-r}=(-1)^m\binom{n}{m}

[3]

Hence find

\sum_{r=0}^{4}(-1)^r\binom{6}{r}\binom{6}{4-r}

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 49

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider

F(x)=\left(x^2+\frac{1}{x}\right)^8(1+x)^6,\quad x\neq 0

I.

Write the general term of $\left(x^2+\frac{1}{x}\right)^8$ corresponding to index $r$ .

[2]

II.

Write the condition on $r$ and $s$ for a term from this factor and a term $\binom{6}{s}x^s$ from $(1+x)^6$ to contribute to the coefficient of $x^j$ .

[2]

Find the constant term in the expansion of $F(x)$ .

[4]

Find the coefficient of $x^2$ in the expansion of $F(x)$ .

[3]

Explain why there are no other contributions to the coefficient of $x^2$ .

[1]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $n\in\mathbb{Z}^+$ . This question concerns sums involving the binomial coefficients in the expansion of $(1+x)^n$ .

I.

Prove that, for $1\leq r\leq n$ ,

r\binom{n}{r}=n\binom{n-1}{r-1}

[3]

Deduce that

\sum_{r=0}^{n}r\binom{n}{r}=n2^{n-1}

[3]

By considering the second derivative of $(1+x)^n$ , show that

\sum_{r=0}^{n}r(r-1)\binom{n}{r}=n(n-1)2^{n-2}

[3]

Hence find, in terms of $n$ ,

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

and evaluate this sum when $n=7$ .

[3]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $p,q\in\mathbb{Z}^+$ and consider the identity

(1+x)^p(1+x)^q=(1+x)^{p+q}

I.

By comparing coefficients of $x^k$ , prove that

\sum_{r=0}^{k}\binom{p}{r}\binom{q}{k-r}=\binom{p+q}{k}

[5]

Hence prove that

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

[3]

Use this result to find

\sum_{r=0}^{6}\binom{6}{r}^2

[4]

Question 52

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , consider the coefficients in the expansion of $(1+x)^n$ .

Use coefficients to prove an identity.

I.

Show that the coefficient of $x^n$ in $(1+x)^n(1+x)^n$ is $\displaystyle \sum_{r=0}^{n}\binom{n}{r}^2$ .

[3]

II.

Hence prove that

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

[3]

Evaluate $\displaystyle \sum_{r=0}^{12}\binom{12}{r}^2$ .

[2]

Two subsets are chosen independently at random from a set with $12$ elements. Find the probability that the two subsets have the same size.

[4]

Using a GDC table or otherwise, determine the least positive integer $n$ for which $\binom{2n}{n}>1000000$ .

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $a>0$ . In the expansion of $(a+x)^{12}$ , the coefficients of $x^4$ and $x^5$ are equal.

Find $a$ .

I.

Write down the coefficients of $x^4$ and $x^5$ in terms of $a$ .

[2]

II.

Hence find $a$ in exact form.

[2]

Using $a=\frac{8}{5}$ , find the coefficient of $x^6$ in $(a+x)^{12}$ .

[2]

Let

J(x)=\left(\frac{8}{5}+x\right)^{12}(1-x)^4

Find the coefficient of $x^6$ in $J(x)$ .

[4]

For $n\in\mathbb{Z}^+$ and $a>0$ , with $0\le r\le n-1$ , justify that adjacent coefficients of $x^r$ and $x^{r+1}$ in $(a+x)^n$ are equal if and only if

a=\frac{n-r}{r+1}

[3]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A fourth-degree binomial approximation is used for $(1-3x)^{15}$ :

B(x)=1-45x+945x^2-12285x^3+110565x^4

Verify the expression for $B(x)$ using the binomial theorem.

I.

Write down the general term in the expansion of $(1-3x)^{15}$ .

[2]

II.

Show that the coefficient of $x^4$ in $B(x)$ is $110565$ .

[3]

Use $B(x)$ to approximate $(0.94)^{15}$ .

[2]

Using your GDC, calculate the percentage error in the approximation in part (b).

[3]

Determine the largest value of $x$ in $0\leq x\leq 0.03$ for which the percentage error in using $B(x)$ to approximate $(1-3x)^{15}$ is less than $0.100\%$ .

[2]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a>0$ and $x\neq 0$ , consider $E(x)=\left(x^2+\dfrac{a}{x}\right)^6(1-2x)^4$ . Let $C(a)$ be the constant term in the expansion of $E(x)$ .

I.

Find the power of $x$ in the general term formed by taking index $r$ from $\left(x^2+\dfrac{a}{x}\right)^6$ and index $s$ from $(1-2x)^4$ .

[2]

II.

Find all pairs $(r,s)$ which contribute to the constant term.

[1]

Show that $C(a)=15a^4-192a^5$ .

[4]

I.

Determine the interval of values of $a$ for which $C(a)>0$ .

[2]

II.

Determine the value of $a$ for which $C(a)$ is a maximum, and find this maximum value.

[4]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , define $E_n$ to be the sum of the even-indexed terms in the expansion of $(1+3x)^n$ when $x=1$ ; that is, $E_n=\binom{n}{0}+\binom{n}{2}3^2+\binom{n}{4}3^4+\cdots$ .

n	E_n
1	1
2	10
3	28
4	136
5	496
6	2080

I.

Calculate $E_4$ directly from the definition.

[2]

II.

Write down the corresponding sum of the odd-indexed terms for $n=4$ .

[1]

Prove that $E_n=\dfrac{4^n+(-2)^n}{2}$ .

[4]

Hence solve $E_n=524800$ for $n\in\mathbb{Z}^+$ .

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , define $P_n(x)=(1+x)^n(1+ax+bx^2)$ . The constants $a$ and $b$ are chosen so that the coefficients of $x$ and $x^2$ in $P_n(x)$ are both zero.

I.

Find $a$ in terms of $n$ .

[2]

II.

Find $b$ in terms of $n$ .

[3]

Show that the coefficient of $x^3$ in $P_n(x)$ is $\binom{n+2}{3}$ .

[3]

For one value of $n$ , the coefficient of $x^3$ is $84$ . Hence determine $n$ and find the coefficient of $x^4$ in $P_n(x)$ .

[3]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ , define $S_n=\sum_{r=0}^{n}r\binom{n}{r}2^r$ . This is a weighted sum of the coefficients in the expansion of $(1+2x)^n$ when $x=1$ .

n	S_n
1	2
2	12
3	54
6	2916
5	810

I.

Calculate $S_4$ directly from the definition.

[2]

II.

Calculate $\dfrac{S_4}{3^4}$ .

[1]

Prove that $S_n=2n3^{n-1}$ .

[4]

Hence find $n$ if $\dfrac{S_n}{3^n}=18$ .

[3]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A checksum uses the last two digits of powers of $9$ . The binomial expansion of $9^n=(10-1)^n$ can be used to investigate these last two digits for $n\in\mathbb{Z}^+$ .

n	last two digits of 9^n
1	09
2	81
3	29
4	61
5	49
6	41
7	69
8	21
9	89
10	01
11	09
12	81

I.

Expand $(10-1)^n$ using the binomial theorem, writing the first three terms.

[2]

II.

Explain why all terms from the third term onwards do not affect the last two digits.

[2]

Show that, modulo $100$ , $9^n\equiv 1-10n$ if $n$ is even and $9^n\equiv 10n-1$ if $n$ is odd.

[3]

Determine all values of $n$ with $1\leq n\leq 50$ for which the last two digits of $9^n$ are $21$ .

[3]

Question 60

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

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

5^n=(1+4)^n

An integer is said to be divisible by $16$ if it can be written in the form $16m$ for some integer $m$ .

I.

Write the first three terms in the binomial expansion of $(1+4)^n$ .

[3]

II.

Show that $5^n$ can be written in the form

5^n=1+4n+16M

where $M$ is an integer.

[2]

I.

Prove that $5^n-1$ is divisible by $16$ if $n$ is divisible by $4$ .

[2]

II.

Prove the converse: if $5^n-1$ is divisible by $16$ , then $n$ is divisible by $4$ .

[2]

Hence determine whether $5^{2028}-1$ is divisible by $16$ .

[2]

Find the remainder when $5^{2027}$ is divided by $16$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Complex Numbers

The Ultimate All-in-One
IB Platform

All content on this website has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate and IB are registered trademarks owned by the International Baccalaureate Organization.