Topic 1 All - Number & Algebra

[Maximum mark: 4]

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

[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]
   

[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]
   

[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]
   

[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]
   

[Maximum mark: 6]

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

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]
   

[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]
   

[Maximum mark: 4]

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

[Maximum mark: 4]

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

[Maximum mark: 6]

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

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]
   

[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.

[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]
   

[Maximum mark: 4]

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

Prove that the result is a perfect cube.

[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]
   

[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]
   

[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]
   

[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]
   

[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]
   

[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]
   

[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]
   

[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]
   

[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 $\$6200$.

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

[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]
   

[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]
   

[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.

2. 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]
   

[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]
   

[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]
   

[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.

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

[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]
   

[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]
   

[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]
   

[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.

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

[Maximum mark: 6]

1. Show that $(2n-1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]
   

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
   

[Maximum mark: 6]

A tennis ball bounces on the ground $n$ times. The heights of the bounces, $h_1, h_2, h_3, \dots,h_n,$ form a geometric sequence. The height that the ball bounces the first time, $h_1$, is $80$ cm, and the second time, $h_2$, is $60$ cm.

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

2. Find the height that the ball bounces the tenth time, $h_{10}$. [2]
   

3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to $2$ decimal places. [2]
   

[Maximum mark: 5]

In an arithmetic sequence, the sum of the 2nd and 6th term is $32$.
Given that the sum of the first six terms is $120$, determine the first term and common difference of the sequence.

[Maximum mark: 6]

Ali bought a car for $\$18\hspace{0.15em}000$. The value of the car depreciates by $10.5$ % each year.

1. Find the value of the car at the end of the first year. [2]
   

2. Find the value of the car after $4$ years. [2]
   

3. Calculate the number of years it will take for the car to be worth exactly half its original value. [2]