Number Toolkit (DP IB Analysis & Approaches (AA): 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.

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

Write the angle of $y$ in radians.

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

Give your answer from part (b) to

(i) one decimal place

(ii) three significant figures.

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.

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 $k$.

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

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 $x=\frac{\pi }{2}$, $y=-1$ 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.

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 $r$ equal to $\sqrt{\frac{\sin a}{\sin b}}$ cm.

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

Give your answer from part (b) correct to three significant figures.

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

Find the exact value of $YT$.

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.

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 $\left(x_1,y_1\right)$ 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 points A$\left(p+q,p-q\right)$ and B$\left(q-p,\frac{p}{q}\right)$.

Simplify the following:

$\frac{\left(4x{y}^{-2}\right)\left(-12x^{-4}{y}^{12}\right)}{6x^{2}y}$

${\left(2x^{-1}{y}^{-2}\right)}^{-3}{\left(4x^2{y}^3\right)}^4.$

${\sqrt[2]{\left(9x^6{y}^{-2}{z}^4\right)}}^3{\left(3xyz\right)}^{-2}$

Solve the equation $2-x\sqrt{3}=\frac{7x}{\sqrt{3}},$ giving your answer in the form $\frac{\sqrt{a}}{b}$ where $a$ and $b$ are integers. State the values of $a$ and $b$.

$\sqrt{425}$ can be written in the form $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is as small as possible. Find the values of $a$ and $b$. Show all of your working.

A solid metal prism has a cross-section of area $5.50\times {10}^3{\text{ cm}}^2$ and a volume of $4.40\times {10}^4{\text{ cm}}^3$.

Calculate the length of the prism.

The cross-section of the prism is a trapezium whose parallel sides measure 2 m and 2.4 m.

Calculate the height of the trapezium. Give your answer in cm.