Practice Paper 2 (Edexcel IGCSE Further Maths)

Two numbers $x$ and $y$ are such that $2x+y=13$

The sum of the squares of $2x$ and $y$ is $S$.

Show that $S=8x^{2}--52x+169$

Using calculus, find the value of $x$ for which $S$ is a minimum, justifying that this value of $x$ gives a minimum value for $S$.

Using calculus, find the minimum value of $S$.

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.

A particle $P$ is moving along the $x$-axis.

At time $t$ seconds $\left(t \geq 0\right)$ the velocity, $v$ m/s, of $P$is given by $v=4t^{2}--19t+12$

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

When $t=0$, the displacement of $P$ from the origin is −4 m.

Find the displacement of $P$ from the origin when $t=6$

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

Find the value of $t$ when $a=0$

$y = \mathrm{e}^{x ( x^{2} } - 3 x )$

Show that $y-2\frac{dy}{dx}+\frac{d^{2}y}{dx^{2}}=2e^{x}$

Using formulae , show that

(i) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

(ii) $\mathrm{cos}2A=2\mathrm{cos}^{2}A-1$

$f ( \\theta ) = \frac{2 \tan \\theta }{1 + \tan ^{2} \\theta }$

Show that $f(θ)=\mathrm{sin}2θ$

Solve, in radians to 3 significant figures, for negative pi over 2 less-than or slanted equal to x less-than or slanted equal to pi over 2, the equation

$5 \tan ( x + \frac{\pi }{6} ) = [ 1 + \tan ^{2} ( x + \frac{\pi }{6} ) ] [ 1 - 2 \cos ^{2} ( x + \frac{\pi }{6} ) ]$

Using calculus, find the exact value of

$\int_{0}^{ \frac{\pi }{2} } ( \frac{4 \tan \\theta }{1 + \tan ^{2} \\theta } - \cos 5 \\theta + 2 )   \mathrm{d} \\theta$

The curve $C$ has equation $y=\frac{ax-5}{b-x}$ where $a$ and $b$ are integers and $x\neq b$

One intersection of $C$ with the coordinate axes is at the point with coordinates $( \frac{5}{4} , 0 )$

The asymptote parallel to the $y$-axis has equation $x=3$

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

Sketch $C$, showing clearly the asymptotes with their equations and the coordinates of the points of intersection with the coordinate axes.

The straight line $l$ with equation $4y-7x=k$ has no points of intersection with $C$

Show, using algebra, that the range of possible values of $k$ can be written as

$m<k<n$

where $m$ and $n$ are integers to be found.

$\mathrm{f} ( x ) = 6 x^{3} - 13 x^{2} + a x - 10$ where $a$ is a constant

Given that $( 3 x - 2 )$ is a factor of $f(x)$

show that $a=21$

Hence show algebraically that the curve $y=f(x)$ has only one intersection with the $x$-axis.

Find the set of values of $x$ for which $3+x<2x--1$

Find the set of values of $x$ for which $x(x--1)>6$

Find the set of values of $x$ for which $\mathrm{both}3+x<2x--1\mathrm{and}x(x--1)>6$