Partial Fractions, Indices & Standard Form (DP IB Analysis & Approaches (AA): 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.

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 $k$.

Write $\frac{3}{x^2+5x+4}$ as a sum of partial fractions.

Write $\frac{9-x}{x^2+3x-10}$ as a sum of partial fractions.

Write $\frac{3x-23}{2x^2-5x-12}$ as a sum of partial fractions.

Write $\frac{33-12x}{\left(x+1\right){\left(x-2\right)}^2}$ as the sum of partial fractions in the form $\frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{{\left(x-2\right)}^2}$.

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 $W=\frac{\left(2\cos 2x+y\right)\left(\tan \frac{x}{2}-z\right)}{10\left(5\sin x+z^2\right)}$, 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 positive integers and $a$ is prime. 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.

Write  $\frac{2x+17}{(x+4)(x-5)}$  as a sum of partial fractions.

Write $\frac{8x-7}{2x^2-x-3}$ as a sum of partial fractions.

Write $\frac{37-7x}{2x^2+3x-20}$ as a sum of partial fractions.

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.

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

The expression $\frac{5x^2+7x}{\left(2x+1\right)\left(x^2-1\right)}$ can be written as the sum of partial fractions in the form

$\frac{A}{2x+1}+\frac{B}{x+a}+\frac{C}{x-a}$

where $a,A,B,C\in \mathrm{\mathbb{Z}}$ and $a>0$.

Write down the value of $a$.

Find the values of $A$, $B$ and $C$.

Write $\frac{4x+5}{2x+1}$in the form $A+\frac{B}{2x+1}$

where $A$ and $B$ are integers to be found.

The expression $\frac{6ax+b}{a+bx}$ can be written in the form $4-\frac{5}{a+bx}$.

Find the value of $a$ and the value of $b$.

The expression $\frac{-3x^2-x+34}{x^3-5x^2+3x+9}$ can be written as the sum of partial fractions in the form

$\frac{A}{x+1}+\frac{B}{x-a}+\frac{C}{{\left(x-a\right)}^2}$

where $a,A,B,C\in \mathrm{\mathbb{Z}}$.

Find the value of $a$.

Find the values of $A$, $B$ and $C$.