Line intersecting reciprocal curve - Simultaneous equations

CAIE P1 2002 June Q1

4 marks Moderate -0.8

1 The line $x + 2 y = 9$ intersects the curve $x y + 18 = 0$ at the points $A$ and $B$. Find the coordinates of $A$ and $B$.

CAIE P1 2004 June Q6

8 marks Moderate -0.3

6 The curve $y = 9 - \frac { 6 } { x }$ and the line $y + x = 8$ intersect at two points. Find

the coordinates of the two points,
the equation of the perpendicular bisector of the line joining the two points.

CAIE P1 2003 November Q1

4 marks Moderate -0.8

1 Find the coordinates of the points of intersection of the line $y + 2 x = 11$ and the curve $x y = 12$.

Edexcel C1 2008 June Q6

9 marks Moderate -0.8

6. The curve $C$ has equation $y = \frac { 3 } { x }$ and the line $l$ has equation $y = 2 x + 5$.

On the axes below, sketch the graphs of $C$ and $l$, indicating clearly the coordinates of any intersections with the axes.
Find the coordinates of the points of intersection of $C$ and $l$. \includegraphics[max width=\textwidth, alt={}, center]{9451ec48-d955-44a8-9988-68f7c0fb9821-07_1137_1141_1046_397}

OCR MEI C1 Q3

4 marks Standard +0.3

3 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e93e3c51-ae2b-420b-abb8-bf0c483caff8-3_1270_1219_326_463} \captionsetup{labelformat=empty} \caption{Fig. 12}

\end{figure} Fig. 12 shows the graph of $y = \frac { 1 } { x - 2 }$.

Draw accurately the graph of $y = 2 x + 3$ on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of $y = \frac { 1 } { x - 2 }$ and $y = 2 x + 3$.
Show algebraically that the $x$-coordinates of the points of intersection of $y = \frac { 1 } { x - 2 }$ and $y = 2 x + 3$ satisfy the equation $2 x ^ { 2 } - x - 7 = 0$. Hence find the exact values of the $x$-coordinates of the points of intersection.
Find the quadratic equation satisfied by the $x$-coordinates of the points of intersection of $y = \frac { 1 } { x - 2 }$ and $y = - x + k$. Hence find the exact values of $k$ for which $y = - x + k$ is a tangent to $y = \frac { 1 } { x - 2 }$. [4]

OCR MEI FP1 2005 January Q3

7 marks Standard +0.3

3

Solve the equation $\frac { 1 } { x + 2 } = 3 x + 4$.
Solve the inequality $\frac { 1 } { x + 2 } \leqslant 3 x + 4$.

OCR MEI AS Paper 1 2021 November Q9

9 marks Moderate -0.8

9

Sketch both of the following on the axes provided in the Printed Answer Booklet.
1. The curve $\mathrm { y } = \frac { 12 } { \mathrm { x } }$, stating the coordinates of at least one point on the curve.
2. The line $y = 2 x + 8$, stating the coordinates of the points at which the line crosses the axes.
In this question you must show detailed reasoning. Determine the exact coordinates of the points of intersection of the curve and the line.

OCR MEI AS Paper 2 2022 June Q7

7 marks Moderate -0.3

7

On the pair of axes in the Printed Answer Booklet, sketch the graphs of

OCR MEI C4 2006 January Q1

5 marks Moderate -0.3

1 Solve the equation $\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3$.

OCR MEI C1 Q4

12 marks Moderate -0.8

Answer the whole of this question on the insert provided. The insert shows the graph of $y = \frac{1}{x}$, $x \neq 0$.

Use the graph to find approximate roots of the equation $\frac{1}{x} = 2x + 3$, showing your method clearly. [3]
Rearrange the equation $\frac{1}{x} = 2x + 3$ to form a quadratic equation. Solve the resulting equation, leaving your answers in the form $\frac{p \pm \sqrt{q}}{r}$. [5]
Draw the graph of $y = \frac{1}{x} + 2$, $x \neq 0$, on the grid used for part (i). [2]
Write down the values of $x$ which satisfy the equation $\frac{1}{x} + 2 = 2x + 3$. [2]

Pre-U Pre-U 9794/2 Specimen Q2

4 marks Standard +0.3

Solve the simultaneous equations $$x - 2y = 5,$$ $$\frac{4}{x} - \frac{2}{y} = 5.$$ [4]