Practice Paper 2 (Edexcel IAL Maths)

$f(x)=4x^{3}+4x^{2}-23x-30$

Use the factor theorem to show that $(x+2)$ is a factor of $f(x)$.

Factorise $f(x)$ completely.

Write down all the real roots of the equation $f(x)=0$.

The line $y+2x=11$meets the circle with equation $x^{2}+y^{2}+6x-14y=-38$ .

(i) Show that the line and circle meet at one point only.

(ii) Find the coordinates of the point of intersection.

Show that the point $(2,1)$ is a [local] maximum point on the curve with equation

$y=2x^{2}-\frac{2}{3}x^{3}-\frac{5}{3}$

Solve the equation

$5^{2x}-25=0$

Solve the equation

$3^{2x-1}=4^{3}+4^{2}+1$

Evaluate

$\int_{1}^{5} ( 4 x + 6 x^{2} ) d x$

Prove that $f(x)\geq4$ for all values of $x$, where $f(x)=(3-x)^{2}+4$.

The 4^th and 8^th terms of an arithmetic sequence are 20 and 64 respectively. Find the first term and the common difference.

The 12^th and 16^th terms of an arithmetic sequence differ by 20. Find the possible values of the common difference.

Fully expand $(2-\frac{1}{3}x)^{4}$.

A geometric series has first term $a$ and common ratio $\sqrt{5}$.

Show that the sum of the first ten terms of the series is equal to $ka(\sqrt{5}+1)$, where $k$ is a positive integer to be determined.

Two sequences are being used to model the value of a car, $n$ years after it was new.

At new, the car’s value is £30 000.

Model 1 is an arithmetic sequence where the value of the car, $u_{n}$, at $n$ years old, is given by the formula $u_{n}=30000-5000n$.

Model 2 is a geometric sequence where the value of the car, $u_{n}$, at $n$ years old, is given by the formula $u_{n}=30000(0.6)^{n}$.

[i] Find the age of the car when Model 1 predicts its value has halved.

[ii] Find the age of the car when Model 2 predicts its value has halved.

Which model predicts the greater value for the car when it is 5 years old?

[i] State a problem with Model 1 for higher values of $n$.

[ii] Justify why Model 2 never predicts the value of the car to be £0.

Given that $\sum_{r = 1}^{k} 3 \times ( - 2 )^{r} = - 262146$,

(i) show that $\frac{k-1}{2}=\frac{\mathrm{log}65536}{\mathrm{log}4}$

(ii) hence find the value of $k$.

For this value of $k$, calculate $\sum_{r = 5}^{k + 2} 3 \times ( - 2 )^{r}$.

Solve the equation $x^{2}+x-2=0.$

Hence, or otherwise, solve the equation $\mathrm{cos}^{2}x+\mathrm{cos}x-2=0$ for $0^{\circ}\leq x\leq720^{\circ}.$

Given $x=-2$ is a root of the function $f(x)=x^{3}-3x^{2}-8x+4$, fully factorise $f(x)$.

A circle $C$ has centre (-2, 3) and passes through the point $P$(6, -3).

Find an equation for the circle $C$.

Find an equation for the tangent to the circle at P.

Show that the function $\mathrm{f} ( x ) = \sqrt{x} - \frac{7}{ \sqrt{x} }$ , $x>0$, is increasing for all $x$ in its domain.

Solve $2(2^{2x})+4=9(2^{x}).$

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.