Number Toolkit (DP IB Applications & Interpretation (AI): SL): 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 $1\le a\le 10$ 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 1 decimal place.

Calculate the lower and upper bound for

(i) $L$

(ii) $W$

Calculate the lower and upper bound for the

(i) perimeter, $P$

(ii) area, $A,$ of the field.

Calculate the following, giving your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$

(i) $4\times \left(6.2\times {10}^{-5}\right)$

(ii) $\left(4\times {10}^5\right)-\left(5\times {10}^4\right)$

(iii) $\left({4321}^{-1}\right)\left(1.2\times {10}^{-1}\right)$

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 labelled, “2.2 m in length”, were delivered to an Olympic weightlifting team. The coach measured each barbell to check its length, in metres, and recorded the following:

(i) Find the mean of the coach’s recorded measurements.

(ii) Calculate the percentage error between the mean and the stated length of 2.2 m.

The weights of the barbells are labelled 20 kg. The coach also weighed each barbell, in kg, and recorded the following:

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

(ii) Calculate the percentage error between the mean and the stated weight of 20 kg.

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 the volume of one cube is 300 cm³. She uses this value to estimate the number of cubes in the box.

Find Monica’s estimated 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.

Solve the following systems of linear equations using technology.

(i) $5x+3y-2z=-12$ $3x-4y-z=17$ $10x-10y+z=65$

(ii) $4x-5y+z=50$ $3x+y+3z=-16$ $6x-2z=61+y$

Solve the following systems of linear equations using technology.

(i) $2x-5y-7z=-21$ $3z+x-4y=44$ $x+z-y=12$

(ii) $z-x-y=-11$ $5x+11z-2y=-28$ $3y-4z+x=30$

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 $W=\frac{\left(2\cos 2x+y\right)\left(\tan {\displaystyle \frac{x}{2}}-z\right)}{10\left(5\sin x+z^2\right)}$, 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.

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

Calculate the height of the prism.

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

Calculate the percentage error in Lily’s estimate.

Mary has found the exact answer for $R$ is  $\frac{45}{16}$.

Write down the exact answer of 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 $\sqrt{\frac{\sin a}{\sin b}}\mathrm{cm}.$

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

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 $1\le a<10,k\in \mathrm{\mathbb{Z}}$.

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.

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.

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

Using your answers from part (a), calculate the distance, $d$, between points P and Q. 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.

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

Giving your answer 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 $YT.$

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 ${\mathrm{cm}}^3$.

(ii) in ${\mathrm{m}}^3$.

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 being $Q{\mathrm{cm}}^3$ and the percentage error in his estimate is 5%.

Calculate the exact possible values of $Q$.

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: sin² $\theta$ = (sin $\theta$)²

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

Let $X$ = $\frac{\sqrt{a}+c^2-2sin54d}{\sqrt{\left(a^3\right)}-a-c}$

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.

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

Giving your answers in the form $a\times {10}^k,$ where $1\le a<10,k\in \mathrm{\mathbb{Z}},$ 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 $(x_1,y_1)$ and $\left(x_2,y_2\right)$ is given in your formula booklet.

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

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

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

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%.