Number Toolkit (DP IB Applications & Interpretation (AI): HL): Exam Questions

Let $Q=\frac{30\sin 2a}{8b}$, where $a=45$° and $b=2$.

Calculate the exact value of $Q$.

Give your answer from part (a) correct to

(i) two decimal places

(ii) two significant figures.

Nina estimates the value of $Q$ to be 2.

Calculate the percentage error in Nina’s estimate.

Let $R=\frac{4x}{6\cos 5y}$, where $x=1.25$ and $y=36$°.

Find the value of $R$. Give your answer as a fraction.

Give your answer from part (a) to

(i) one decimal place

(ii) three significant figures.

Kieran estimates the value of $R$ to be $-1$.

Calculate the percentage error in Kieran’s estimate.

Consider the numbers $a=4.14\times {10}^6$ and $b=2.54\times {10}^{-7}$.

Calculate $C=\sqrt[10]{{\left(\frac{a}{b}\right)}^3}$. Give your answer correct to the nearest integer.

Give your answer to part (a) in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

Calculate the percentage error if $C$ was approximated to be 9000.

A cylindrical grain silo has radius 12.5 m and height 28.0 m.

Find the volume of the silo, expressing your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

The silo is emptied at a constant rate of 45 m³ per hour.

Find the time taken to empty the silo.

A rectangular field has length, $L$, of 25.2 m and width, $W$, of 21.4 m, each correct to one decimal place.

Write down the lower bound and the upper bound of

(i) $L$

(ii) $W$.

Find the lower bound and the upper bound of

(i) the perimeter, $P$, of the field

(ii) the area, $A$, of the field.

Consider the following four numbers.

Write down

(i) the number that is in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$

(ii) the largest of these numbers.

(i) Find the value of $p+q-r+s$.

Your answer to part (b)(i) can be written in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

(ii) Find the value of $a$ and the value of $k$.

Five Olympic barbells, each labelled “2.2 m in length”, were delivered to a weightlifting team. The coach measured the length of each barbell, in metres, and recorded the following results.

(i) Find the mean length of the five barbells.

(ii) Taking the labelled length of 2.2 m as the exact value, calculate the percentage error in the mean length.

Each barbell is labelled as having a weight of 20 kg. The coach also weighed each barbell, in kilograms, and recorded the following results.

(i) Find the mean weight of the five barbells.

(ii) Taking the labelled weight of 20 kg as the exact value, calculate the percentage error in the mean weight.

In a game show, there is a transparent box filled with identical cubes. Contestants must estimate the number of cubes in the box. The box is 60 cm wide, 80 cm long and 20 cm tall.

Find the volume of the box.

Monica estimates that the volume of one cube is $300{\text{ cm}}^3$. She uses this value to estimate the number of cubes in the box.

Find Monica’s estimate of the number of cubes in the box.

The actual number of cubes in the box is 280.

Find the percentage error in Monica’s estimated number of cubes in the box.

A prism has a cross-sectional area of $5.83\times {10}^7{\text{ cm}}^2$ and a volume of $4.22\times {10}^8{\text{ cm}}^3$.

Find the height of the prism.

Lily estimates the height of the prism to be 7 cm.

Calculate the percentage error in Lily’s estimate.

Let $P=\frac{\left(4\sin 2q-2\right)\left(6\tan q+2\right)}{10{\left(r+s\right)}^2}$, where $q=30$°, $r=6$ and $s=2$.

Calculate the exact value of $P$.

Give your answer from part (a) correct to

(i) two decimal places.

(ii) two significant figures.

Michael estimates the value of $P$ to be $0.015$

Calculate the percentage error of Michael’s estimate.

Let $W=\frac{\left(2\cos 2x+y\right)\left(\tan \frac{x}{2}-z\right)}{10\left(5\sin x+z^2\right)}$, where $x=90$°, $y=-1$ and $z=2$.

Find the value of $W$. Give your answer as a fraction.

Give your answer from part (a) to

(i) three decimal places.

(ii) three significant figures.

Louis estimates the value of $W$ to be 0.03.

Calculate the percentage error of Louis’ estimate.

Let $S=\left(a{\sin }^{2}4b\right){\left(c^2{\tan }^{2}12d\right)}^{-1}\left(\sqrt{a}+c-\cos 48b\right)$, where $a=16$, $b=7.5$°, $c=3$ and $d=5$°.

Note: ${\sin }^2\theta ={\left(\sin \theta \right)}^2$

Find the value of $S$, giving your answer as a fraction.

Let $X=\frac{\sqrt{a}+c^2-2\sin 54d}{\sqrt{a^3}-a-c}$.

Find the value of $X$, giving your answer as a fraction.

Calculate the value of $SX,$ giving your answer as a fraction.

John estimates the value of $SX$ to be 0.3.

Calculate the percentage error in John’s estimate.

Mary has found that the exact value of $R$ is $\frac{45}{16}$.

Write down the exact value of $R$ as a decimal.

Mary rounds her exact answer so that the percentage error is 6.67%.

State the number of significant figures Mary rounded the exact answer to and write down the approximate value of $R$ used in calculating the percentage error.

It is given that $\sin a=\frac{\sqrt{3}}{2}$ and $\sin b=\frac{1}{2}$, where $0\le a\le 90$° and $0\le b\le 90$°.

Find the size of the angles $a$ and $b$.

A circle has radius equal to $\sqrt{\frac{\sin a}{\sin b}}$ cm.

Find the area of the circle, giving your answer in terms of $\pi$.

James rounds the answer from part (b) to the nearest integer.

Calculate the percentage error of James’ estimate.

The diameter of Earth is $1.274\times {10}^7$ m, correct to four significant figures.  The circumference of Mars is $2.134\times {10}^7$ m, correct to four significant figures. 

Modelling Earth and Mars as perfect spheres, find the difference between the volume of Earth and the volume of Mars, giving your answer in the form $a\times {10}^k$, where $1\le a<10,k\in \mathrm{\mathbb{Z}}$.

A cuboid has length $l=0.102$ m, width $w=9.4$ cm and height $h=0.25$ m.

Calculate the exact volume of the cuboid

(i) in cm³

(ii) in m³.

Write your answers to part (a) (i) and (ii) in the form $a\times {10}^k$, where $1\le a<10,$ $k\in \mathrm{\mathbb{Z}}$.

William estimates the volume of the cuboid as $Q$ cm³, and the percentage error in his estimate is 5%.

Calculate the exact possible values of $Q$.

Let $Y={\left(pq\right)}^{-1}r$ and $T=pq{r}^{-1}$, where $p=\sin 60$°, $q=\sqrt{3}$ and $r=2$.

Giving your answers to 1 decimal place, calculate the value of

(i) $Y$

(ii) $T$.

Using your answers to part (a), estimate the value of $YT$. Give your answer as a fraction.

Calculate the percentage error between your estimated value of $YT$ found in part (b) and the exact value of $YT.$

Consider the numbers $p=2.41\times {10}^4$ and $q=4.12\times {10}^5$.

Giving your answers correct to three significant figures in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$, calculate

(i) $p+q$

(ii) $p-q$

(iii) $q-p$

(iv) $\frac{p}{q}$.

The formula for the distance, $d$, between two points with coordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given in your formula booklet.

$d=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$

Using your answers to part (a), estimate the distance between the points A $\left(p+q,p-q\right)$ and B $\left(q-p,\frac{p}{q}\right)$.

Calculate the percentage error between the estimate of the distance AB found in part (b) and the exact distance.

Consider the numbers  $a=11\sqrt{2},b=(5+6\pi ),c=\sqrt{2},d=6(\pi -1)$.

Giving your answer to 1 decimal place, calculate the value of

(i) $a$

(ii) $b$

(iii) $c$

(iv) $d$

Points P and Q have coordinates $\left(a,b\right)$ and $\left(c,d\right)$ respectively.

Using your answers from part (a), calculate the distance PQ. Give your answer correct to 1 decimal place.

Find the percentage error between the distance, correct to 1 decimal place, found in part (b) and the exact distance between points P and Q.

A shop sells bags of potatoes labelled as “5 kg”. The shop owner weighs five bags, in kilograms, at random and recorded the following:

4.96,   4.89,   5.07,   5.11,   5.02

(i) Find the mean of the shop owner’s recorded weights.

(ii) Calculate the percentage error between the mean and the stated weight of 5 kg.

The shop owner shares his findings with his potato supplier, who weighs another five bags and recorded the following:

5.05,   5.01,   4.97,   5.09,   X

The supplier allows a maximum percentage error of 1%.

Find the interval for the values of X such that the percentage error from the five bags that the supplier weighed is less than 1%.

Find the interval for the values of X such that the percentage error from all the ten bags weighed is less than 1%.