Practice Paper 1 (Edexcel IAL Maths)

Show that the graph of $y=p(x)$ where $p(x)=2x+1$ maps onto the graph of its inverse under the transformations described by $\frac{1}{2}p(\frac{1}{2}x)-1$.

It is given

$f(x)=4x^{3}+4x^{2}-7x+2$

Write down the domain and range of the function f(x).

Sketch the graph of y = f(x), stating the coordinates of any intersections with the coordinate axes. (You do not need to give the coordinates of any turning points.)

One of the three algebraic fractions below is improper (‘top-heavy’).

Identify which fraction is improper and write it in the form $A+\frac{B}{x+2}$, where $A$ and $B$ are integers to be found.

A curve has the equation $y=e^{-3x}+\mathrm{ln}x,x>0.$

Show that the equation of the tangent to the curve at the point with x-coordinate 1 is

$y = ( \frac{e^{3} - 3}{e^{3}} ) x + \frac{4 - e^{3}}{e^{3}}$

The following equations can be used for exponential models.

State whether each one would represent exponential growth or exponential decay.

(i) $y=3^{-2x}$

(ii) $y = 20 ( 2 )^{x}$

(iii) $y=30a^{-x}$ where $a>0$

Rewrite the equation $y=3.6x^{-0.4}$ in the form $\mathrm{log}y=\mathrm{log}A-b\mathrm{log}x$

Sketch the graph of $\mathrm{log}y$ against $\mathrm{log}x$.

Given that

$\mathrm{cos}2θ\equiv2\mathrm{cos}^{2}θ-1$

use calculus to find the exact value of

integral _ pi / 4 end _ ^ pi / 2 end ^ cos ^2 theta d theta

A solution to the equation $\mathrm{f} ( x ) = 0$ is $x=3.1$, correct to two significant figures.

(i) Write down the lower bound, l, and the upper bound, u, of 3.1.

(ii) Assuming $f(x)$ is continuous in the interval $l<x<u$, what can you say about the values of $f(u)$ and $\mathrm{f} ( l )$?

Show that $2$ cosec $2A\equiv$cosec $A$ sec $A$.

Solve the equation

$3$ sec⁴ $θ+16=16+16$tan² $θ,-π\leqθ\leqπ$

giving your answers to three significant figures where appropriate.

Sketch, in the interval $-2π\leqθ\leqπ$, the graph of y = 2 + 3 sec ( theta + pi / 2 ) , include asymptotes and label the coordinates of all maximum and minimum points.

Deduce the maximum and minimum values of (1)/(2 + 3 sec ( theta + pi / 2 )).

Show that

1 / ( (sqrt( 3)/(2) cos theta - 1 half sin theta ) ^2 + 1 / ( (sqrt( 3)/(2) sin theta + 1 half cos theta ) ^2 === 4 cosec ^2 ( 2 theta + pi / 3 )

The diagram below shows the graph of $y=g(x)$ where

$g(x)=\frac{1}{x},x\neq0$

(i) Write down the equations of the two asymptotes.

(ii) Determine the equations of the two asymptotes on the graph of $y=g(x-1)+5$.