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Show that the equation $y=16 x-17$ is a tangent to the curve
[5]
1 Show $3 x^{2}+4 x-5$
2 If $x$ is small and measured in addans, show that
\[
y=3 x^{2}+4 x-5
\]
3 The parametric equations of $a$ curve are given by
\[
4 \sin \left(x+\frac{\pi}{6}\right)-8 \cos \left(x-\frac{\pi}{6}\right)=(2 \sqrt{3}-1) x^{2}+(2 \sqrt{3}-4) x+2-4 \sqrt{3} .
\]
Express the equation of the curve in terms of $x$ and $y$ in the form $a y=b x^{2}+c x+d$ where $a, b, c$ and $d$ are constants.
[6]
4 By using the substitution $u=e^{2}$, solve the equation
[6]
$2 e^{2 x}-5 e^{x}+2=0$. leaving solutions in exact form.
5 A polynomial $f(x)=4 x^{2}+a x^{2}+b x-5$ where $a$ and $b$ are constants. When $f(x)$ is divided by $(x-2)$ it leaves a temainder of 7 and when divided by $(x+1)$ it leaves a remainder of -5 . Find $a$ and $b$
6 Express $\frac{3 x^{3}+18 x^{2}+19 x+20}{(x+3)\left(x^{2}+2\right)}$ in partial fractions.
[7]
7 The table below shows the expenmental values of two vanables $x$ and $y$.
$x$ | 1 | 4 | 10 | 25 | 40
$y$ | 1.6 | 2 | 2.6 | 3.8 | 47
Given that $x$ and $y$ are related by the equation $y^{2}=a x-b$ where $a$ and $b$ are constants.
(a) draw a straight line graph of $y^{1}$ against $x$ to represent the data given in
[4] the table above.
(b) use the graph to estimate the value of $a$ and $b$ correct to 1 decimal place
(3)
8 The function $f(x)$ is denoted by
$\omega f(x)=5-\ln (x-4), x \in R, x>_{4}$
(a) Sketch the graph of $f(x)$ showing clearly where it crosses the axis.
(b) Hence or otherwise state the range of $f(x)$.
[3]
195
(1)
(c) Find the inverse of $f(x)$ stating clearly the domain.
[3]
$y=x_{2}$
$28 / 5$