Rational curve intersections

Questions involving sketching a rational function (reciprocal or reciprocal squared) with another curve (polynomial or linear) to determine number of intersections or solve equations.

10 questions · Standard +0.0

1.02n Sketch curves: simple equations including polynomials 1.02o Sketch reciprocal curves: y=a/x and y=a/x^2 1.02q Use intersection points: of graphs to solve equations

Sort by: Default | Easiest first | Hardest first

Edexcel P1 2022 January Q10

9 marks Challenging +1.2

10. The curve $C$ has equation $$y = \frac { 1 } { x ^ { 2 } } - 9$$

Sketch the graph of $C$. On your sketch
The curve $D$ has equation $y = k x ^ { 2 }$ where $k$ is a constant. Given that $C$ meets $D$ at 4 distinct points,
find the range of possible values for $k$.

Edexcel P1 2020 October Q7

11 marks Standard +0.3

7. The curve $C$ has equation $$y = \frac { 1 } { 2 - x }$$

Sketch the graph of $C$. On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line $l$ has equation $y = 4 x + k$, where $k$ is a constant. Given that $l$ meets $C$ at two distinct points,
show that $$k ^ { 2 } + 16 k + 48 > 0$$
Hence find the range of possible values for $k$.

Edexcel C1 2009 January Q8

7 marks Moderate -0.3

8. The point $P ( 1 , a )$ lies on the curve with equation $y = ( x + 1 ) ^ { 2 } ( 2 - x )$.

Find the value of $a$.
On the axes below sketch the curves with the following equations:
1. $y = ( x + 1 ) ^ { 2 } ( 2 - x )$,
2. $y = \frac { 2 } { x }$. On your diagram show clearly the coordinates of any points at which the curves meet the axes.
With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x } .$$
\includegraphics[max width=\textwidth, alt={}]{871f5957-180d-4379-88ce-186432f57bad-10_1347_1344_1245_297}

Edexcel C1 2011 January Q10

8 marks Moderate -0.3

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

$y = x ( x + 2 ) ( 3 - x )$
$y = - \frac { 2 } { x }$ showing clearly the coordinates of all the points where the curves cross the coordinate axes.
(b) Using your sketch state, giving a reason, the number of real solutions to the equation $$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}

OCR C1 Q4

6 marks Standard +0.3

4. (i) Sketch on the same diagram the curves $y = x ^ { 2 } - 4 x$ and $y = - \frac { 1 } { x }$.
(ii) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

OCR H240/01 2018 March Q4

9 marks Moderate -0.3

Sketch the curves $y = \frac { 3 } { x ^ { 2 } }$ and $y = x ^ { 2 } - 2$ on the axes provided in the Printed Answer Booklet.
In this question you must show detailed reasoning. Find the exact coordinates of the points of intersection of the curves $y = \frac { 3 } { x ^ { 2 } }$ and $y = x ^ { 2 } - 2$.

Edexcel C1 Q4

6 marks Moderate -0.3

Sketch on the same diagram the curves $y = x^2 - 4x$ and $y = -\frac{1}{x}$. [4]
State, with a reason, the number of real solutions to the equation $$x^2 - 4x + \frac{1}{x} = 0.$$ [2]

OCR MEI C1 Q2

12 marks Standard +0.3

\includegraphics{figure_2} Fig. 12 shows the graph of $y = \frac{1}{x-2}$.

Draw accurately the graph of $y = 2x + 3$ 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 $y = 2x + 3$. [3]
Show algebraically that the $x$-coordinates of the points of intersection of $y = \frac{1}{x-2}$ and $y = 2x + 3$ satisfy the equation $2x^2 - x - 7 = 0$. Hence find the exact values of the $x$-coordinates of the points of intersection. [5]
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 C1 Q3

13 marks Moderate -0.3

\includegraphics{figure_3} Fig. 12 shows the graph of $y = \frac{1}{x-3}$.

Draw accurately, on the copy of Fig. 12, the graph of $y = x^2 - 4x + 1$ for $-1 < x < 5$. Use your graph to estimate the coordinates of the intersections of $y = \frac{1}{x-3}$ and $y = x^2 - 4x + 1$. [5]
Show algebraically that, where the curves intersect, $x^3 - 7x^2 + 13x - 4 = 0$. [3]
Use the fact that $x = 4$ is a root of $x^3 - 7x^2 + 13x - 4 = 0$ to find a quadratic factor of $x^3 - 7x^2 + 13x - 4$. Hence find the exact values of the other two roots of this equation. [5]

SPS SPS SM 2020 October Q7

11 marks Moderate -0.3

Sketch the curves $y = \frac{3}{x^2}$ and $y = x^2 - 2$ on the axes provided below. \includegraphics{figure_1} [3]
In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves $y = \frac{3}{x^2}$ and $y = x^2 - 2$. [6]
Hence, solve the inequality $\frac{3}{x^2} \leq x^2 - 2$, giving your answer in interval notation. [2]