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

Easy Medium Hard

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 5

Paper 2, calculatorMedium7 marks

Let $A(x)=(x+1)^2-1$ and $B(x)=x(x+2)$ where $x\in\mathbb{R}$

(a)

Find $A(3)$ and $B(3)$

[2]

(b)

Starting with $A(x)$ show by a chain of algebraic equalities that $A(x)$ has the same value as $B(x)$ for every $x\in\mathbb{R}$

[3]

(c)

Complete each statement using either $=$ or $\equiv$

A(3)\ \square\ B(3)

A(x)\ \square\ B(x),\quad \text{for all }x\in\mathbb{R}

[2]

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