Number Skills

[Maximum mark: 6]

After solving a problem, John has an exact answer of $z = 0.1475$.

1. Write down the exact value of $z$ in the form $a\times10^k$, where $1 \leq a < 10, k\in \mathbb{Z}$.[2]
   

2. State the value of $z$ given correct to $2$ significant figures. [1]
   

3. Calculate the percentage error if $z$ is given correct to $2$ significant figures. [3]
   

[Maximum mark: 6]

Given that $z = \dfrac{10\sin \alpha}{3x+y}$, where $\alpha = \ang{30}$, $x = 6$ and $y = 46$.

1. Find the exact value of $z$. [2]
   

2. Write your answer to part (a)

   1. correct to $2$ decimal places;

   2. correct to $3$ significant figures;

   3. in the form $a\times10^k$, where $1 \leq a < 10$ and $k\in \mathbb{Z}$.[4]
      

[Maximum mark: 6]

Let $A = \sqrt{\dfrac{\sin \alpha - \sin \beta}{x^2 + 2y}}$, where $\alpha = \ang{54}$, $\beta = \ang{18}$, $x = 24$ and $y = 18.25$.

1. Find the value of $A$. Write down your full calculator display. [2]
   

2. Give your answer to part (a) correct to

   1. three significant figures;

   2. three decimal places. [2]
      

3. Give the answer to part (b) (i) in the form $a\times10^k$, where $1 \leq a < 10$, $k \in \mathbb{Z}$.[2]
   

[Maximum mark: 6]

Let $Q = \dfrac{(\sin 2x + b)(2\sin x - 1)}{a^2 - 4\tan x}$, where $x = \ang{45}$, $a = 18$ and $b = \sqrt{2}$.

1. Find the exact value of $Q$. [2]
   

2. Give your answer to part (a) correct to

   1. three decimal places;

   2. three significant figures. [2]
      

3. Calculate the percentage error if $Q$ is given to three decimal places. [2]
   

[Maximum mark: 6]

The volume of a hemisphere, $V$, is given by the formula

where $S$ is the total surface area.

The total surface area of a given hemisphere is $529$ cm$^2$.

1. Calculate the volume of this hemisphere in cm$^3$. Give your answer correct to one decimal place. [3]
   

2. Write down your answer to part (a) correct to the nearest integer. [1]
   

3. Write down your answer to part (b) in the form $a\times10^k$, where $1 \leq a < 10$ and $k \in \mathbb{Z}$.[2]
   

[Maximum mark: 6]

Four cement bags labelled, "5 kg", were delivered to a customer. The customer measured each bag to check their weights and recorded the following:

1. 1. Find the mean of the customer's measurements.

   2. Calculate the percentage error between the mean and the stated,
      approximate weight of $5$ kg. [3]
      

2. Calculate $\sqrt{2.15^8-5.12^{-0.8}}$, giving your answer

   1. correct to the nearest integer;

   2. in the form $a\times10^k$, where $1 \leq a < 10$ and $k\in \mathbb{Z}$. [3]
      

[Maximum mark: 6]

The distance between two points with coordinates $(x_1,y_1)$ and $(x_2,y_2)$ is equal to $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

1. Calculate the distance between points A$(40,-100)$ and B$(1,-2)$. Give your answer correct to three significant figures. [3]
   

2. Give your answer from part (a) correct to one decimal place. [1]
   

3. Write the answer to part (b) in the form $a\times10^k$, where $1 \leq a < 10$, $k \in \mathbb{Z}$. [2]
   

[Maximum mark: 6]

The following diagram shows a rectangle with sides of length $7.6\times10^2$ cm and $1.5\times10^3$ cm.

1. Write down the area of the rectangle in the form $a\times10^k$, where
   $1 \leq a < 10$ and $k \in \mathbb{Z}$. [3]
   

Natalie estimates the area of the rectangle to be $1\hspace{0.1em}200\hspace{0.15em}000$ cm$^2$.

2. Find the percentage error in Natalie's estimate. [3]
   

[Maximum mark: 6]

Let $F = \dfrac{(4\sin 2z-1)(2\tan 3z+1)}{x^2-y^2}$, where $x = 12$, $y = 8$ and $z = \ang{15}$.

1. Calculate the exact value of $F$. [2]
   

2. Give your answer to $F$ correct to

   1. two significant figures;

   2. two decimal places. [2]
      

Sasha estimates the value of $F$ to be $0.03$.

3. Calculate the percentage error in Sasha's estimate. [2]
   

[Maximum mark: 6]

Given $r = 2a - \dfrac{\sqrt{b}}{c}$, $a = 0.975$, $b = 4.41$ and $c = 35$,

1. calculate the value of $r$. [2]
   

Albert first writes $a$, $b$ and $c$ correct to one significant figure and then uses these values to estimate the value of $r$.

2. 1. Write down $a$, $b$ and $c$ each correct to one significant figure.

   2. Find Albert's estimate of the value of $r$. [2]
      

3. Calculate the percentage error in Albert's estimate of the value of $r$. [2]
   

[Maximum mark: 8]

A cuboid has the following dimensions: $\text{length} = 9.6\hspace{0.25em}$cm, $\text{width} = 7.4\hspace{0.25em}$cm, and $\text{height} = 5.2\hspace{0.25em}$cm, measured correct to the nearest millimetre.

1. Using these measurements, calculate the volume of the cuboid, in cm$^3$. Give your answer to two decimal places. [2]
   

The lower and upper bounds for the length of the cuboid can be expressed as $9.55 \leq l < 9.65$.

2. Write similar expressions for

   1. the width;

   2. the height. [2]
      

3. Hence, calculate the minimum volume of the cuboid. Give your answer to three significant figures. [2]
   

4. Write your answer to part (c) in the form $a\times10^k$, where $1 \leq a < 10$ and $k \in \mathbb{Z}$. [2]