Polynomial intersection with algebra

Edexcel P1 2019 January Q11

12 marks Moderate -0.3

11. (a) On Diagram 1 sketch the graphs of

$y = x ( 3 - x )$
$y = x ( x - 2 ) ( 5 - x )$ showing clearly the coordinates of the points where the curves cross the coordinate axes.
(b) Show that the $x$ coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation $x \left( x ^ { 2 } - 8 x + 13 \right) = 0$ The point $P$ lies on both curves. Given that $P$ lies in the first quadrant,
(c) find, using algebra and showing your working, the exact coordinates of $P$.

\includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
\section*{Diagram 1}

Edexcel C12 2017 January Q13

11 marks Standard +0.3

13. (a) On separate axes sketch the graphs of

$y = c ^ { 2 } - x ^ { 2 }$
$y = x ^ { 2 } ( x - 3 c )$ where $c$ is a positive constant.
Show clearly the coordinates of the points where each graph crosses or meets the $x$-axis and the $y$-axis.
(b) Prove that the $x$ coordinate of any point of intersection of $$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$ where $c$ is a positive constant, is given by a solution of the equation $$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$ Given that the graphs meet when $x = 2$ (c) find the exact value of $c$, writing your answer as a fully simplified surd.

Edexcel C1 2007 January Q10

13 marks Moderate -0.3

10. (a) On the same axes sketch the graphs of the curves with equations

$y = x ^ { 2 } ( x - 2 )$,
$y = x ( 6 - x )$,
and indicate on your sketches the coordinates of all the points where the curves cross the $x$-axis.
(b) Use algebra to find the coordinates of the points where the graphs intersect.

Edexcel C1 2010 June Q10

15 marks Moderate -0.3

10. (a) On the axes below sketch the graphs of

$y = x ( 4 - x )$
$y = x ^ { 2 } ( 7 - x )$ showing clearly the coordinates of the points where the curves cross the coordinate axes.
(b) Show that the $x$-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation $x \left( x ^ { 2 } - 8 x + 4 \right) = 0$ The point $A$ lies on both of the curves and the $x$ and $y$ coordinates of $A$ are both positive.
(c) Find the exact coordinates of $A$, leaving your answer in the form ( $p + q \sqrt { } 3 , r + s \sqrt { } 3$ ), where $p , q , r$ and $s$ are integers. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}

OCR MEI C1 Q2

13 marks Moderate -0.3

2 You are given that $\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12$.

Show that $x = - 3$ is a root of $\mathrm { f } ( x ) = 0$ and hence factorise $\mathrm { f } ( x )$ fully.
Sketch the curve $y = \mathrm { f } ( x )$.
Find the $x$-coordinates of the points where the line $y = 4 x + 12$ intersects $y = \mathrm { f } ( x )$.

OCR MEI C1 Q4

13 marks Moderate -0.3

4

Sketch the graph of $y = x ( x - 3 ) ^ { 2 }$.
Show that the equation $x ( x - 3 ) ^ { 2 } = 2$ can be expressed as $x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0$.
Show that $x = 2$ is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).

Edexcel AEA 2009 June Q1

8 marks Challenging +1.2

(a) On the same diagram, sketch

$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the $x$-coordinates of the points of intersection of these two curves.

Edexcel PURE 2024 October Q4

Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-10_812_853_255_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curves $C _ { 1 }$ and $C _ { 2 }$ Given that $C _ { 1 }$

has equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x )$ is a quadratic function
cuts the $x$-axis at the origin and at $x = 4$
has a minimum turning point at ( $2 , - 4.8$ )
1. find $\mathrm { f } ( x )$

Given that $C _ { 2 }$

The curves $C _ { 1 }$ and $C _ { 2 }$ meet in the first quadrant at the point $P$, shown in Figure 1.

Use algebra to find the coordinates of $P$.

Edexcel C1 Q10

13 marks Moderate -0.3

On the same axes sketch the graphs of the curves with equations
1. $y = x^2(x - 2)$, [3]
2. $y = x(6 - x)$, [3]
and indicate on your sketches the coordinates of all the points where the curves cross the $x$-axis.
Use algebra to find the coordinates of the points where the graphs intersect. [7]

OCR MEI C1 Q5

12 marks Moderate -0.3

Solve, by factorising, the equation $2x^2 - x - 3 = 0$. [3]
Sketch the graph of $y = 2x^2 - x - 3$. [3]
Show that the equation $x^2 - 5x + 10 = 0$ has no real roots. [2]
Find the $x$-coordinates of the points of intersection of the graphs of $y = 2x^2 - x - 3$ and $y = x^2 - 5x + 10$. Give your answer in the form $a \pm \sqrt{b}$. [4]

WJEC Unit 1 2022 June Q5

9 marks Moderate -0.8

The curve $C_1$ has equation $y = -x^2 + 2x + 3$ and the curve $C_2$ has equation $y = x^2 - x - 6$. The two curves intersect at the points $A$ and $B$.

Determine the coordinates of $A$ and $B$. [4]
On the same set of axes, sketch the graphs of $C_1$ and $C_2$. Clearly label the points where the two curves intersect. [3]
In the diagram drawn in part (b), shade the region satisfying the following inequalities: [2] $$x > 0,$$ $$y < -x^2 + 2x + 3,$$ $$y > x^2 - x - 6.$$