Practice Paper 1 (Edexcel IAL Maths)

Divide $x^{3}-19x-30$ by $(x-5)$.

(i) Complete the square of $x^{2}+4x$.

(ii) Complete the square of $y^{2}-6y$.

(i) Use your answers to part (a) to show that the equation $x^{2}+y^{2}+4x-6y+4=0$ can be written in the form $\left(x + 2\right)^{2} + \left(y - 3\right)^{2} = 9$.

(ii) Hence, write down the centre and the radius of the circle with equation $x^{2}+y^{2}+4x-6y+4=0.$

Find the values of $x$ for which $f(x)=x^{3}-5x^{2}+3x-2$ is a decreasing function.

Solve the equation

$5^{2x}-8\times5^{x}+12=0,$

giving your answers in the form $\mathrm{log}_{a}b$.

The trapezium rule is to be used to estimate the integral

$\int_{- 2}^{0} x^{3} + 8 d x$

By completing the table of values below, use the trapezium rule to estimate the integral given above.

i] By finding the exact value of the integral above, find the percentage error of your estimate from part [a].

ii] Explain how you could increase the accuracy of your trapezium rule estimate.

Prove that the sum of two odd numbers is even.

Write down the next three terms in these arithmetic sequences

(i) 30, 18, 6, ...

(ii) $\frac{1}{4},\frac{5}{12},\frac{7}{12},\frac{3}{4},.....$

Find the first three terms in the expansion of $(5+2x)^{5}$.

Use your answer to part (a) to estimate the value of $(5.04)^{5}$.

A geometric sequence has first term 900 and a common ratio $r$ where $r>0$. The 18th term of the sequence is 18.

Show that $r$ satisfies the equation $17\mathrm{log}r+\mathrm{log}50=0.$

Hence or otherwise find the value of $r$ correct to 3 significant figures.

Mary wishes to open a savings account and is comparing two options.

The first option is an account offering an interest rate of 0.85% per annum but the maximum that can be added to the account each year is £4000.

The second option is an account offering an interest rate of 1.1% per annum but the maximum that can be added to the account each year is £3000.

Mary will open her account with the maximum amount allowed and will add the maximum yearly amount at the start of every year in order to maximise the interest. Mary’s aim is for her account to reach £40 000 in the quickest time possible.

Which account should Mary choose and how long (in whole number of years) will it take for her to achieve her aim?

A convergent geometric series is given by $1-4x+16x^{2}-64x^{3}+...$

Write down the range of possible values of $x$.

Given that $\sum_{n = 1}^{ \infty } ( - 4 x )^{n - 1} = 24$

Calculate the value of $x$.

Solve the equation $6\mathrm{cos}^{2}2θ=\mathrm{sin}2θ+5$ for $-180^{\circ}\leqθ\leq180^{\circ}$, giving your answers to 1 decimal place where appropriate.

Use polynomial division to show that $(x-2)$ is a factor of $2x^{3}+3x^{2}-18x+8$.

The points $A$(-2, 3), $B$(0, 6) and $C$($k$, -1) lie on a circle, where $BC$ is the diameter of the circle.

Find the value of $k$.