Binomial Theorem - IB Math AA Question Bank

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 $x$ .

A

Find the coefficient of $x^2$ .

[2]

B

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 $60$ .

A

Show that $k=2$ .

[3]

B

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 $(2-3x)^8$ .

A

Write down the number of terms in the expansion.

[1]

B

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 $11$ .

A

Write down the value of $n$ .

[1]

B

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

[2]

C

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

[1]

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 $n\geq 3$ .

A

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

[3]

B

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 $x\neq 0$ .

A

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

[2]

B

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 $p>1$ .

A

Find the possible values of $p$ .

[3]

B

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.

A

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

[2]

B

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

[2]

C

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

[1]

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

A

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

[2]

B

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 $(1+x)^{15}$ .

A

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

[2]

B

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.

A

Find $a$ .

[2]

B

Find $b$ .

[2]

C

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 $x$ .

A

Find the fourth term.

[3]

B

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

[3]

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 $5$ .

A

Show that $m=1$ .

[3]

B

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 $x\neq 0$ .

A

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

[2]

B

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 $(1.01)^{12}$ .

A

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

[3]

B

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

[1]

C

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 $1215$ .

A

Form an equation in $n$ .

[3]

B

Hence find $n$ .

[2]

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the polynomial

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

A

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

[1]

B

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 $489888$ .

A

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

[3]

B

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 $135$ .

A

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

[3]

B

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.

A

Find the value of $n$ .

[3]

B

Hence find the coefficient of $x^5$ .

[2]

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 $-23$ .

A

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]

B

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

[4]

C

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 $b>0$ .

A

I.

Find the value of $a$ .

[2]

II.

Find the value of $b$ .

[2]

B

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

[2]

C

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.

A

I.

Write down an expression for $E_n+O_n$ .

[1]

II.

Find an expression for $E_n-O_n$ .

[2]

B

Hence prove that

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

[4]

C

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 $x^2$ .

A

I.

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

[2]

II.

Find the value of $n$ .

[3]

B

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

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

[4]

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

A

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

[2]

B

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 $b>0$ .

A

Find $a$ .

[2]

B

Find $b$ .

[3]

C

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

A

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

[2]

B

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 $1$ .

A

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]

B

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

[3]

C

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]

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 $-288$ .

A

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

I.

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

[3]

II.

Hence find $p$ and $q$ .

[3]

B

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

[3]

C

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 $a>0$ .

A

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]

B

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

[3]

C

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

A

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]

B

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

[3]

C

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 $3^{12-r}2^r$ .

A simple grid-route diagram showing a start point, an end point after 12 total moves, and examples of east and north moves labelled with weights 3 and 2 respectively.

A

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]

B

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

[2]

C

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

[3]

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

A

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]

B

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 $10752$ .

A

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]

B

Find the value of $k$ .

[3]

C

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 $104$ .

A

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]

B

Find the value of $ab$ .

[2]

C

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

[2]

D

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 $90$ .

A

I.

Write an equation using the coefficient of $x$ .

[1]

II.

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

[2]

B

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

[4]

C

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

[2]

D

Hence find the coefficient of $x^3$ in

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

[2]

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 $x^2$ .

A

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]

B

Let

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

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

[4]

C

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 $405$ .

A

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]

B

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

[2]

C

Let

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

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

[3]

D

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 $720$ .

A

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]

B

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 $220$ .

A

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]

B

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

[4]

C

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

[2]

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

A

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]

B

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]

C

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.

A portion of Pascal's triangle with one shallow diagonal corresponding to entries of the form binomial r choose 2 highlighted, and the adjacent entry at the end of the diagonal indicated but not numerically evaluated.

A

I.

Calculate $D_6$ .

[1]

II.

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

[2]

B

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]

C

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 $1-0.48t+0.1008t^2+\cdots$ .

Exact and cubic approximation of the remaining proportion.

A

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]

B

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

[3]

C

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

A

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]

B

Determine $n$ and $q$ .

[4]

C

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]

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.

Rows of Pascal's triangle near an even-numbered row, with the central entry of the even row highlighted and the two adjacent entries indicated.

A

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]

B

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

[3]

C

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 $(R+U)^{p+q}$ .

A rectangular grid from $(0,0)$ to $(7,5)$ with a marked intermediate point $(3,2)$. Right and up directions are labelled, and no path counts are displayed.

A

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]

B

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]

C

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]

D

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 $\binom{n}{r}$ .

A segment of one row of Pascal's triangle showing two adjacent boxed entries labelled 495 and 792, with row number $n$ and position $r$ indicated symbolically.

A

I.

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

[2]

II.

Determine $n$ and $r$ .

[3]

B

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

[2]

C

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 $r>n$ .

A

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]

B

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]

C

Hence find

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

[2]

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

A

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]

B

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

[4]

C

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

[3]

D

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 $(1+x)^n$ .

A

I.

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

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

[3]

B

Deduce that

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

[3]

C

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]

D

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}

A

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]

B

Hence prove that

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

[3]

C

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 $(1+x)^n$ .

A

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]

B

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

[2]

C

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]

D

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

[3]

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.

A

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]

B

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

[2]

C

Let

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

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

[4]

D

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

Exact and 4th-degree curves for (1-3x)^15 on 0≤x≤0.03.

A

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]

B

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

[2]

C

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

[3]

D

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 $E(x)$ .

Graph of the constant term function C(a) for a>0.

A

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]

B

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

[4]

C

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

A

I.

Calculate $E_4$ directly from the definition.

[2]

II.

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

[1]

B

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

[4]

C

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

[3]

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.

A

I.

Find $a$ in terms of $n$ .

[2]

II.

Find $b$ in terms of $n$ .

[3]

B

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

[3]

C

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

A

I.

Calculate $S_4$ directly from the definition.

[2]

II.

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

[1]

B

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

[4]

C

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

A

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]

B

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

[3]

C

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 $m$ .

A

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]

B

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]

C

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

[2]

D

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