Number Toolkit (DP IB Analysis & Approaches (AA): SL): Exam Questions

Let $P=\frac{\left(4\sin 2q-2\right)\left(6\tan q+2\right)}{10{\left(r+s\right)}^2}$, where $q=\frac{\mathrm{\pi }}{6}$, $r=6$ 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 $W=\frac{\left(2\cos 2x+y\right)\left(\tan {\displaystyle \frac{x}{2}}-z\right)}{10\left(5\sin x+z^2\right)}$, where $x=\frac{\mathrm{\pi }}{2},$ $y=-1$ 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.

A prism has a cross sectional area of $5.50\times {10}^3{\mathrm{cm}}^2$ and volume of $4.40\times {10}^4{\mathrm{cm}}^3$.

Calculate the length of the prism.

The cross-sectional area of the prism is in the shape of a trapezium and its parallel sides measure 2 m and 2.4 m.

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

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

Write down the exact answer of as a decimal.

Give your answer from part (a) correct to

(i) three decimal places

(ii) one significant figure.

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 $\sqrt{\frac{\sin a}{\sin b}}\mathrm{cm}.$

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

A medium rare steak should have an internal temperature of $55°\mathrm{C}$ to $56°\mathrm{C}$. Max decides to go to 10 different steak houses, he measures the internal temperature of a medium rare steak at each establishment and records the following:

51.0,   52.1,  62.9,  49.0,  59.8,  50.2,  54.3,  47.7,  48.6,   65.4

Find the mean internal temperature of Max’s recordings.

Max goes to 5 more steak houses and calculates the mean of all 15 restaurants to be $55.2°\mathrm{C}.$

Calculate the mean internal temperature from the 5 additional steak houses Max went to.

Max records one last steak that has an internal temperature of $T°\mathrm{C}.$

Calculate the interval of $T$ such that the mean internal temperature for all 16 steaks is within the temperature range for a medium rare steak.

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.

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

Find the exact value of YT.

Point A has coordinates ($-$1, 7) and point B has coordinates (11, 12).

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.

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

Calculate the distance between points A and B.

The formula for the coordinates of the midpoint of a line segment with endpoints $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given in your formula booklet.

$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculate the midpoint of the line segment with endpoints A and B.

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.

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$\left(q-p,\frac{p}{q}\right)$.

The mean height of the four tallest students in a classroom is 176 cm and the mean height of the six tallest students is 165 cm. The fifth tallest student is 4 cm taller than the sixth tallest student.

Find the heights of the fifth and sixth tallest students.