Practice Paper 1 (Edexcel IGCSE Further Maths)

$\mathrm{f} ( x ) = 3 - 4 x - 9 x^{2}$

Given that $\mathrm{f} ( x )$can be expressed in the form $A - B ( x + C )^{2}$ where $A$, $B$ and $C$ are positive constants

find the value of $A$, the value of $B$and the value of $C$

$\mathrm{f} ( x ) = 3 - 4 x - 9 x^{2}$

Given that $\mathrm{f} ( x )$can be expressed in the form $A - B ( x + C )^{2}$ where $A$, $B$ and $C$ are positive constants

Hence write down the maximum value of $\mathrm{f} ( x )$

The equation $\mathrm{f} ( x ) = 0$ has roots $α$ and $β$

Without solving the equation $\mathrm{f} ( x ) = 0$, form a quadratic equation, with integer coefficients, that has roots $\frac{3α}{β}$ and $\frac{3β}{α}$

Show that $(x+y)^{3}=x^{3}+y^{3}+3xy(x+y)$

$g ( x ) = 3 x^{2} + q x + r$ where $q$and $r$ are constants

The equation $g(x)=0$ has roots $α^{2}-β$ and $β^{2}-α$ where $α$ and $β$are the roots of the equation $\mathrm{f} ( x ) = 0$

Using your answer to part (d), find in simplified exact form, the value of $q$ and the value of $r$

The curve $S$ with equation $y=\frac{x^{2}}{4}+2$ where x greater-than or slanted equal to 0 and the line $l$ with equation $2y-x-4=0$ where x greater-than or slanted equal to 0 intersect at the points $A$and $B$, as shown in Figure 2.

(i) Show that the coordinates of point $A$ are (0, 2)

(ii) Find the coordinates of the point $B$

The finite region bounded by $S$ and $l$, shown shaded in Figure 2, is rotated through $2π$ radians about the $y$-axis.

Use algebraic integration to find the volume of the solid generated. Give your answer in terms of $π$

A particle $P$ is moving along a straight line.

At time $t$ seconds $\left( t \geq 0 \right)$, the velocity, $v$ m/s, of $P$ is given by

$v=3t^{2}-16t+5$

Find the values of $t$ when $P$ is instantaneously at rest.

At time $t$ seconds the acceleration of $P$ is $a$ m/s²

Find the range of values of $t$ for which $a>0$

Find the distance, in m, that $P$ travels in the interval 1 space less-than or slanted equal to space t space less-than or slanted equal to space 4

Figure 3 shows part of the curve $C$ with equation $y=4-e^{2x}$ The curve $C$ crosses the $y$-axis at the point $A$and the $x$-axis at the point $B$.

(i) Write down the $y$ coordinate of point $A$.

(ii) Show that the $x$ coordinate of $B$ is $x=\mathrm{ln}2$

The line $l$ is the normal to $C$ at the point $B$.

Find an equation for $l$ , giving your answer in the form $y=mx+c$

The finite region $R$ is bounded by $C$, $l$ and the $y$-axis.

Using calculus, find the area of $R$. Give your answer to one decimal place.

Figure 3 shows a sketch of the curve with equation $y=f(x)$, which passes through the points with coordinates $(-1,0),(b,0)$ and $(a,0)$ where $0<b<a.$

Given that $f'(x)=6x^{2}--26x+12$ find,

(i) the value of $a$,

(ii) the value of $b$.

On the grid below, draw the line with equation

(i) $3x+4y=24$

(ii) $2x-5y+10=0$

Show, by shading on the grid, the region $R$ defined by the inequalities

3 x space plus space 4 y space less-than or slanted equal to space 24 space space space space space 2 x space minus space 5 y space plus 10 space greater-than or slanted equal to space 0 space space space space y space less-than or slanted equal to space 5 space space space space space x space greater-than or slanted equal to space minus 1

Label the region $R$