IB Maths AA SL Topic 1: Number and algebra questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

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Question 1

Paper 1, no calculatorMedium6 marks

A geometric sequence has nth term $u_n$ where $u_3=18$ and $u_6=486$

(a)

Determine the first term and the common ratio of the sequence.

[3]

(b)

Hence, write $S_6$ the sum of the first six terms, in sigma notation and determine its value.

[3]

Question 2

Paper 2, calculatorMedium6 marks

A small venue records its ticket sales over five weeks.

Week	1	2	3	4	5
Tickets sold	420	458	491	530	563

The sales do not form an exact arithmetic sequence. An arithmetic model is used, with its first and fifth terms matching the recorded sales for weeks 1 and 5.

(a)

Determine the common difference of this model.

[1]

(b)

Hence estimate the number of tickets sold in week 9.

[2]

(c)

State the first term and common difference of your model. Write the model’s total predicted ticket sales for weeks 1 to 9 in sigma notation, and determine this total. You may use your GDC.

[2]

(d)

Give one reason why the model’s estimate for week 9 may differ from the actual number of tickets sold.

[1]

Question 3

Paper 1, no calculatorMedium6 marks

Let $p=\log_2 3$ and $q=\log_2 5$ Consider the equation

4^{x+\frac12}=\frac{5\times81^{\frac34}}{3},\qquad x\in\mathbb R.

(a)

Find the value of $81^{\frac34}$

[1]

(b)

Show that

\log_2\left(\frac{5\times81^{\frac34}}{3}\right)=2p+q.

[2]

(c)

Hence solve the equation, giving your answer in terms of $p$ and $q$

[3]

Question 4

Paper 2, calculatorMedium6 marks

Two account balances, in dollars, are modelled for $t\geq 0$ years after opening by

A(t)=240(1.04)^t,\qquad B(t)=300(1.025)^t.

(a)

Determine the time, in years, at which the two modelled balances are equal. Give your answer correct to two decimal places.

[4]

(b)

Hence determine the first whole number of years after opening at which $A(t)>B(t)$

[2]

Question 6

Paper 1, no calculatorHard6 marks

For $n\in\mathbb Z$ define

S_n=(n+1)^5+(n-1)^5.

A student conjectures that $S_n$ is divisible by $32$ for every odd integer $n$

(a)

Calculate $S_1$ directly.

[1]

(b)

Starting from $S_n$ prove the identity

S_n\equiv 2n^5+20n^3+10n

for all $n\in\mathbb Z$ Use $=$ between successive expressions and $\equiv$ for the final identity.

[2]

(c)

Hence prove the student’s conjecture for every odd integer $n$

[3]

Question 7

Paper 1, no calculatorHard6 marks

Let $m\in\mathbb Z^+$ Figure 1 shows a rectangle with side lengths $m+1$ and $m+2$ divided into four smaller rectangles.

It is claimed that integers $A$ and $B$ can be chosen so that, for every $m\in\mathbb Z^+$

\frac{1}{m+1}+\frac{1}{m+2}+\frac{Am+B}{m^2+3m+2} =\frac{3}{m+1}.

(a)

Using the areas shown in Figure 1, write an identity relating the area of the whole rectangle to the areas of the four smaller rectangles. Use $\equiv$ to denote an identity.

[1]

(b)

If the claim is true, the equality for $m=1$ requires a particular relation between $A$ and $B$ Determine this relation.

[1]

(c)

Determine $A$ and $B$ so that the claim is an identity for every $m\in\mathbb Z^+$ Then, starting with the left-hand side, prove the resulting identity. Use your answer to part (b).

[4]

Question 8

Paper 2, calculatorHard6 marks

In the expansion of $(1+x)^n$ where $n$ is an integer with $1\leq n\leq 12$ the coefficients of $x^r$ and $x^{r+1}$ are 120 and 210, respectively. Here $r$ is an integer with $0\leq r<n$

(a)

Determine $n$ and $r$

[4]

(b)

Hence, for this value of $n$ use your GDC to generate a table of $\binom{n}{k}$ for integer $k$ from 0 to $n$ Use the table to determine all powers of $x$ whose coefficients are 120. Verify your answers using the formula for $\binom{n}{k}$

[2]

Question 9

Paper 1, no calculatorHard6 marks

For positive integers $n$ and $a$ consider the expansion of $(1+ax)^n$

The coefficient of $x^2$ is five times the coefficient of $x$ and the coefficient of $x^3$ is $\frac{8}{3}$ times the coefficient of $x^2$

(a)

Determine $n$ and $a$

[4]

(b)

Hence determine the sum of the coefficients of all even powers of $x$ including the constant term, in the expansion.

[2]

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