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This is the A-Level Pure Mathematics study guide from 2020. It is a free study guide for revision.
Upper $6{ }^{\text {th }}$
LOCKDOWN PURE MATHS TEST 19
Tuesday 11August 2020
ANSWERS
35 MARKS
NB: If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to $\mathbf{2}$ significant figures.
1.
The diagram above shows the curve with equation $y=x^{3}+a x^{2}+b x+c$, where $a$, $b$ and $c$ are constants. The curve crosses the $x$-axis at the point $(-1,0)$ and touches the $x$-axis at the point $(3,0)$. Show that $a=-5$ and find the values of $b$ and $c$.
[6]
The curve crosses the $x$-axis at the point $(-1,0)$ :
\[
\begin{array}{l}
\Rightarrow(-1)^{3}+a(-1)^{2}+b(-1)+c=0
\Rightarrow-1+a-b+c=0
\Rightarrow a-b+c=1
\end{array}
\]
At $(3,0)$ :
\[
\begin{array}{l}
\Rightarrow(3)^{3}+a(3)^{2}+b(3)+c=0
\Rightarrow 27+9 a+3 b+c=0
\Rightarrow 9 a+3 b+c=-27
\end{array}
\]
Equation (ii) - Equation (i):
\[
\begin{array}{l}
\Rightarrow 8 a+4 b=-28
\Rightarrow 4 a+2 b=-14
\end{array}
\]
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