Prediction Exams and November 2023 Past Paper Solutions available now!    🚀 Math AA HL Bootcamps are in beta! 🚀

Topic 1 All - Number & Algebra

All Questions for Topic 1 (Number & Algebra). Sequences & Series, Exponents & Logs, Binomial Theorem, Proofs

Question Type

Paper

Paper 1
Paper 2

Difficulty

Easy
Medium
Hard

View

Question 1

no calculator

easy

[Maximum mark: 4]

Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your answer.

easy

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 2

calculator

easy

[Maximum mark: 6]

An arithmetic sequence has $u_1= 40$, $u_2 = 32$, $u_3 = 24$.

1. Find the common difference, $d$. [2]

2. Find $u_8$. [2]

3. Find $S_8$. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 3

no calculator

easy

[Maximum mark: 7]

Find the value of each of the following, giving your answer as an integer.

1. $\log_{10} 100$. [2]

2. $\log_{10} 50 + \log_{10} 2$. [2]

3. $\log_{10} 4 - \log_{10} 40$. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 4

no calculator

easy

[Maximum mark: 6]

Find the value of each of the following, giving your answer as an integer.

1. $\log_6 6$. [2]

2. $\log_6 9 + \log_6 4$. [2]

3. $\log_6 72 - \log_6 2$. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 5

no calculator

easy

[Maximum mark: 6]

Consider an arithmetic sequence $2,6,10,14,\dots$

1. Find the common difference, $d$. [2]

2. Find the $10$th term in the sequence. [2]

3. Find the sum of the first $10$ terms in the sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 6

calculator

easy

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

$\begin{array}{rcccccl} a_n &=& \dfrac{1}{3},\,\dfrac{1}{4},\,\dfrac{1}{5},\,\dfrac{1}{6},\,\dots &\,\hspace{4em}\,& c_n &=& 3,\,1,\,\dfrac{1}{3},\,\dfrac{1}{9},\,\dots \\[12pt] b_n &=& 2.5,\,5,\,7.5,\,10,\,\dots &\,\hspace{4em}\,& d_n &=& 1,\,3,\,6,\,10,\,\dots \end{array}$
1. State which sequence is arithmetic and find the common difference of the sequence. [2]

2. State which sequence is geometric and find the common ratio of the sequence.[2]

3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 7

calculator

easy

[Maximum mark: 6]

Consider the infinite geometric sequence $4480$, $-3360$, $2520$, $-1890,\dots$

1. Find the common ratio, $r$. [2]

2. Find the $20$th term. [2]

3. Find the exact sum of the infinite sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 8

calculator

easy

[Maximum mark: 4]

Expand $(2x - 3)^4$ in descending powers of $x$ and simplify your answer.

easy

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 9

no calculator

easy

[Maximum mark: 4]

Prove that the sum of three consecutive positive integers is divisible by $3$.

easy

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 10

calculator

easy

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

$\begin{array}{rcccccl} a_n &=& 1,\,5,\,10,\,15,\,\dots &\,\hspace{4em}\,& c_n &=& 1.5,\,3,\,4.5,\,6,\,\dots \\[12pt] b_n &=& \dfrac{1}{2},\,\dfrac{2}{3},\,\dfrac{3}{4},\,\dfrac{4}{5},\,\dots &\,\hspace{4em}\,& d_n &=& 2,\,1,\,\dfrac{1}{2},\,\dfrac{1}{4},\,\dots \end{array}$
1. State which sequence is arithmetic and find the common difference of the sequence. [2]

2. State which sequence is geometric and find the common ratio of the sequence.[2]

3. For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 11

no calculator

easy

[Maximum mark: 4]

Consider two consecutive positive integers, $k$ and $k+1$.

Show that the difference of their squares is equal to the sum of the two integers.

easy

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 12

calculator

easy

[Maximum mark: 6]

The table shows the first four terms of three sequences: $u_n$, $v_n$, and $w_n$.

1. State which sequence is

1. arithmetic;

2. geometric. [2]

2. Find the sum of the first $50$ terms of the arithmetic sequence. [2]

3. Find the exact value of the $13$th term of the geometric sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 13

no calculator

easy

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.

easy

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 14

calculator

easy

[Maximum mark: 6]

Consider the expansion of $(2x-1)^9$.

1. Write down the number of terms in this expansion. [1]

2. Find the coefficient of the term in $x^2$. [5]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 15

calculator

easy

[Maximum mark: 6]

Jeremy invests $\8000$ into a savings account that pays an annual interest rate of $5.5$ %, compounded annually.

1. Write down a formula which calculates that total value of the investment after $n$ years. [2]

2. Calculate the amount of money in the savings account after:

1. $1$ year;

2. $3$ years. [2]

3. Jeremy wants to use the money to put down a $\10\hspace{0.15em}000$ deposit on an apartment. Determine if Jeremy will be able to do this within a $5$-year timeframe.[2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 16

calculator

easy

[Maximum mark: 5]

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

1. Write down the number of terms in this expansion. [1]

2. Find the coefficient of the term in $x^6$. [4]

easy

Formula Booklet

Mark Scheme

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Revisit

Question 17

calculator

easy

[Maximum mark: 6]

Consider the infinite geometric sequence $9000$, $-7200$, $5760$, $-4608$, ...

1. Find the common ratio. [2]

2. Find the $25$th term. [2]

3. Find the exact sum of the infinite sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 18

no calculator

easy

[Maximum mark: 6]

Let $\log_2 a = p$, $\log_2 b = q$, $\log_2 c = r$. Write down the following expressions in terms of $p$, $q$ and $r$.

1. $\log_2\Big(\dfrac{ab}{c}\Big)$ [2]

2. $\log_2\Big(\dfrac{a^2c}{b^3}\Big)$ [2]

3. $\log_a b$ [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 19

no calculator

easy

[Maximum mark: 7]

An arithmetic sequence is given by $3$, $5$, $7,\dots$

1. Write down the value of the common difference, $d$. [1]

2. Find

1. $u_{10}$;

2. $S_{10}$. [4]

3. Given that $u_n = 253$, find the value of $n$. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 20

no calculator

easy

[Maximum mark: 7]

Let $p=\ln 2$ and $q = \ln 6$. Write down the following expressions in terms of $p$ and $q$.

1. $\ln 12$ [2]

2. $\ln 3$ [2]

3. $\ln 48$ [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 21

no calculator

easy

[Maximum mark: 6]

Consider the following sequence of figures.

Figure 1 contains $6$ line segments.

1. Given that Figure $n$ contains $101$ line segments, show that $n = 20$.[3]

2. Find the total number of line segments in the first $20$ figures. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 22

calculator

easy

[Maximum mark: 6]

Hannah buys a car for $\24\hspace{0.15em}900$. The value of the car depreciates by $16$ % each year.

1. Find the value of the car after $10$ years. [3]

Patrick buys a car for $\12\hspace{0.15em}000$. The car depreciates by a fixed percentage each year, and after $6$ years it is worth $\6200$.

1. Find the annual rate of depreciation of the car. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 23

calculator

easy

[Maximum mark: 6]

A $3$D printer builds a set of $49$ Ei$\text{f}$fel Tower Replicas in different sizes. The height of the largest tower in this set is $64$ cm. The heights of successive smaller towers are $95$ % of the preceding larger tower, as shown in the diagram below.

1. Find the height of the smallest tower in this set. [3]

2. Find the total height if all $49$ towers were placed one on top of another. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 24

no calculator

easy

[Maximum mark: 6]

Let $a = \log_5b$, where $b > 0$. Write down each of the following expressions
in terms of $a$.

1. $\log_5b^4$ [2]

2. $\log_5 (25b)$ [2]

3. $\log_{25}b$ [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 25

calculator

easy

[Maximum mark: 6]

Julia wants to buy a house that requires a deposit of $74\hspace{0.15em}000$ Australian dollars (AUD).

Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of $5.5$ %, compounded monthly.

1. Find the amount of AUD Julia needs to invest to reach $74\hspace{0.15em}000$ AUD after $8$ years. Give your answer correct to the nearest dollar. [3]

Julia's parents offer to add $5000$ AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of $3.5$ %, compounded quarterly.

1. Find the number of years it would take Julia to save the $74\hspace{0.15em}000$ AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 26

calculator

easy

[Maximum mark: 6]

The fifth term, $u_5$, of a geometric sequence is $125$. The sixth term, $u_6$, is $156.25$.

1. Find the common ratio of the sequence. [2]

2. Find $u_1$, the first term of the sequence. [2]

3. Calculate the sum of the first $12$ terms of the sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 27

calculator

easy

[Maximum mark: 6]

In an arithmetic sequence, $u_5 = 24$, $u_{13} = 80$.

1. Find the common difference. [2]

2. Find the first term. [2]

3. Find the sum of the first $20$ terms in the sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 28

calculator

easy

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits $4000$ Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of $6$ %, compounded semi-annually.

1. Find the amount of interest that Mia will earn over the next $2.5$ years. [3]

Ella also deposits AUD into a bank account. Her bank pays a nominal annual $\text{interest}$ rate of $4$ %, compounded monthly. In $2.5$ years, the total amount in Ella's account will be $4000$ AUD.

1. Find the amount that Ella deposits in the bank account. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 29

calculator

easy

[Maximum mark: 5]

Maria invests $\25\hspace{0.15em}000$ into a savings account that pays a nominal annual interest rate of $4.25$%, compounded monthly.

1. Calculate the amount of money in the savings account after $3$ years. [2]

2. Calculate the number of years it takes for the account to reach $\40\hspace{0.15em}000$. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 30

no calculator

easy

[Maximum mark: 6]

The first three terms of a geometric sequence are $u_1 = 32$, $u_2 = -16$, $u_3 = 8$.

1. Find the value of the common ratio, $r$. [2]

2. Find $u_6$. [2]

3. Find $S_{\infty}$. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 31

no calculator

easy

[Maximum mark: 6]

In an arithmetic sequence, $u_4 = 12$, $u_{11} = -9$.

1. Find the common difference. [2]

2. Find the first term. [2]

3. Find the sum of the first $11$ terms in the sequence. [2]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Question 32

calculator

easy

[Maximum mark: 6]

Emily deposits $2000$ Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of $4$ %, compounded monthly.

1. Find the amount of money that Emily will have in her bank account after $5$ years. Give your answer correct to two decimal places. [3]

Emily will withdraw the money back from her bank account when the amount reaches $3000$ AUD.

1. Find the time, in months, until Emily withdraws the money from her bank account. [3]

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

1. Show that $\left(2n-1{\right)}^{3}+\left($