AS June 2022 Paper 1 Q13

AQACurrent spec10 marksGraphs & Inequalities

13 A curve \(C_1\) has equation

\[y = \frac{2x + 7}{3x + 5}\]

(a) Write down the equations of the asymptotes of curve \(C_1\) [2 marks]

(b) On the axes below, sketch the graph of curve \(C_1\)

Indicate the values of the intercepts of the curve with the axes. [3 marks]

Blank axes: x-axis and y-axis crossing at O

(d) Curve \(C_2\) is a reflection of curve \(C_1\) in the line \(y = -x\)

Find an equation for curve \(C_2\) in the form \(y = \mathrm{f}(x)\) [3 marks]

Mark scheme (a)

Scheme	Marks	AO
Obtains at least one correct asymptote.	M1	1.1a
Obtains two correct asymptotes and no incorrect asymptotes.	A1	1.1b
	(2)

Typical solution

\[y = \frac{2}{3} \quad \text{and} \quad x = -\frac{5}{3}\]

Sketch of y = (2x + 7)/(3x + 5) with dashed asymptotes x = −5/3 and y = 2/3; the right branch crosses the y-axis at 7/5 and the left branch crosses the x-axis at −7/2 — Mark scheme (b)

Scheme	Marks	AO
Sketches a curve asymptotic to either a horizontal asymptote or a vertical asymptote. Only one branch required for this mark.	B1	1.1b
Sketches a curve with two branches asymptotic to their horizontal asymptote and vertical asymptote.	M1	1.1a
Deduces the shape of the curve and sketches it correctly with the correct axis intercepts. Must draw the asymptotes – condone solid lines.	A1	2.2a
	(3)

Mark scheme (c)

Scheme	Marks	AO
Identifies both critical values and no others. Or identifies one correct set of \(x\)-values. FT their asymptote or \(x\)-intercept.	M1	3.1a
Obtains the correct set of \(x\)-values.	A1	3.2a
	(2)

Typical solution

\[x \leqslant -\frac{7}{2}, \quad x \gt -\frac{5}{3}\]

Mark scheme (d)

Scheme	Marks	AO
Selects a method to find the equation of the reflected curve. Swaps \(-x\) for \(y\) and \(-y\) for \(x\) or any two of: \(x = -\dfrac{2}{3}\) \(\quad y = \dfrac{5}{3}\) \(x\)-intercept \(= -1.4\) \(\quad y\)-intercept \(= 3.5\) Follow through their asymptotes and/or intercepts from parts (a) and (b)	M1	3.1a
Multiplies an equation of the form \(\pm x = \dfrac{\pm 2y + 7}{\pm 3y + 5}\) throughout by the denominator and attempts to isolate \(y\) or any three of: \(x = -\dfrac{2}{3}\) \(\quad y = \dfrac{5}{3}\) \(x\)-intercept \(= -1.4\) \(\quad y\)-intercept \(= 3.5\) Follow through their asymptotes and/or intercepts from parts (a) and (b)	M1	1.1a
Obtains a correct equation in the correct format.	A1	3.2a
	(3)
	(10 marks)

Typical solution

\[-x = \frac{2(-y) + 7}{3(-y) + 5}\]\[-x(-3y + 5) = -2y + 7\]\[3xy - 5x = -2y + 7\]\[3xy + 2y = 5x + 7\]\[y(3x + 2) = 5x + 7\]\[y = \frac{5x + 7}{3x + 2}\]