AS June 2019 Paper 1 Q5

AQACurrent spec8 marksGraphs & Inequalities Integration

5 A hyperbola \(H\) has the equation

\[\frac{x^2}{a^2} - \frac{y^2}{4a^2} = 1\]

where \(a\) is a positive constant.

(a) Write down the equations of the asymptotes of \(H\). [1 mark]

(b) Sketch the hyperbola \(H\) on the axes below, indicating the coordinates of any points of intersection with the coordinate axes.
The asymptotes have already been drawn. [2 marks]

Axes crossing at O with two dashed asymptotes drawn through O

(c) The finite region bounded by \(H\), the positive \(x\)-axis, the positive \(y\)-axis and the line \(y = a\) is rotated through \(360^\circ\) about the \(y\)-axis.
Show that the volume of the solid generated is \(ma^3\), where \(m = 3.40\) correct to three significant figures. [5 marks]

Mark scheme (a)

Scheme	Marks	AO
Writes the correct equations with \(a\) removed. Accept any equivalent equations, e.g. \(x = \pm\frac{1}{2}y\)	B1	1.1b

Typical solution

\[\frac{x}{a} = \pm\frac{y}{2a}\]\[y = \pm 2x\]

Sketch of the hyperbola H: two branches approaching the dashed asymptotes, crossing the x-axis at −a and a — Mark scheme (b)

Scheme	Marks	AO
Draws the correct graph, correctly approaching the asymptotes – mark the intention.	B1	1.1b
Writes \((a, 0)\) and \((-a, 0)\) Accept \(a\) and \(-a\) written close to the intercepts.	B1	1.1b

Mark scheme (c)

Scheme	Marks	AO
Formulates an expression for a volume generated by rotating the hyperbola about an axis – must be a clear intent to integrate. A volume expression must be of the form \(\int \mathrm{f}(x)\) or \(\int \mathrm{f}(y)\) where f is a polynomial function of degree 2. Condone missing limits and/or \(\pi\) and/or \(\mathrm{d}y\) (or \(\mathrm{d}x\)).	M1	3.1a
Expresses the volume as \(\pi\displaystyle\int\left(\frac{y^2}{4} + a^2\right)\mathrm{d}y\) or equivalent. Must include \(\pi\) – may be seen later. Condone missing limits and/or \(\mathrm{d}y\).	A1	1.1b
Correctly integrates their expression. Their expression must be of the form \(cy^2 + d\) or \(cx^2 + d\) where \(c\) and \(d\) are constants.	M1	1.1a
Substitutes correct limits into \(py^3 + qy\), where \(p\) and \(q\) are positive constants. May be unsimplified. Accept substitution of 0 not seen.	A1	1.1b
Completes a rigorous mathematical argument, including either \(ka^3\) where \(k \in [3.4005, 3.4045]\) or \(\frac{13}{12}\pi a^3\) (or equivalent), that the volume can be expressed as \(3.40a^3\) to 3 significant figures. Must use \(\mathrm{d}y\) correctly throughout. Must include an appropriate reference to 3 significant figures, e.g. \(\frac{13\pi}{12} = 3.40\) (3sf) Accept substitution of 0 not seen. This mark can only be awarded if M2A2 scored. NMS scores 0/5	R1	2.1
	(8 marks)

Typical solution

\[\begin{aligned}\text{Volume} &= \pi\int x^2\,\mathrm{d}y \\ &= \pi\int_0^a \left(\frac{y^2}{4} + a^2\right)\mathrm{d}y \\ &= \pi\left[\frac{y^3}{12} + a^2y\right]_0^a \\ &= \pi\left(\frac{a^3}{12} + a^3 - 0\right) \\ &= \frac{13\pi}{12}a^3 \\ &= 3.403a^3 \\ &= 3.40a^3 \text{ (3 sig figs)}\end{aligned}\]