Further Pure 1: Coordinate systemsEdexcel A-Level Further Maths: Topic test

20 questions, 54 marks

Edexcel A-Level Further Maths

Further Pure 1: Coordinate systems topic test

Total 54 marks

Name

Class

Date

1
The parabola $C$ has equation $y^2=12x$ and focus $S$ .
(a)
What are the coordinates of $S$ ?
[1 mark]
A $(3,0)$
B $(12,0)$
C $(0,3)$
D $(6,0)$
(b)
What is the equation of the directrix of $C$ ?
[1 mark]
A $x=3$
B $x=-6$
C $y=-3$
D $x=-3$
(c)
The point $P$ on $C$ is a distance of $15$ from the directrix. Find the coordinates of the possible positions of $P$ .
[2 marks]
Total for question 1: 4 marks
2
The ellipse $E$ has parametric equations $x=6\cos t$ , $y=2\sqrt5\sin t$ , so its Cartesian equation is $\frac{x^2}{36}+\frac{y^2}{20}=1$ .
(a)
What is the eccentricity of $E$ ?
[1 mark]
A $\frac49$
B $\frac{\sqrt5}{3}$
C $\frac59$
D $\frac23$
(b)
What are the equations of the directrices of $E$ ?
[1 mark]
A $x=\pm4$
B $x=\pm9$
C $x=\pm6$
D $x=\pm13.5$
(c)
The point $P$ on $E$ has parameter $t=\frac{\pi}{3}$ . Find the exact coordinates of $P$ .
[2 marks]
Total for question 2: 4 marks
3
The rectangular hyperbola $H$ has equation $xy=9$ . The point $P\left(3t,\frac3t\right)$ , where $t>0$ , lies on $H$ .
(a)
Show that the tangent to $H$ at $P$ has equation $x+t^2y=6t$ .
[3 marks]
(b)
The tangent at $P$ meets the $x$ -axis at $A$ and the $y$ -axis at $B$ , and $O$ is the origin. Show that the area of triangle $OAB$ does not depend on $t$ , and find its value.
[4 marks]
Total for question 3: 7 marks
4
The parabola $C$ has equation $y^2=16x$ and focus $S$ . The ellipse $E$ has its centre at the origin, its foci on the $x$ -axis, eccentricity $\frac47$ , and has $S$ as one of its foci.
(a)
Find the equation of $E$ in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hence find the coordinates of the points where $C$ and $E$ intersect, giving $y$ -coordinates to 3 significant figures.
[6 marks]
(b)
The point $P\left(4t^2,8t\right)$ , where $t\neq0$ , lies on $C$ . The tangent to $C$ at $P$ meets the directrix of $C$ at $Q$ . Show that angle $PSQ$ is $90^\circ$ .
[6 marks]
Total for question 4: 12 marks
5
The hyperbola $H$ has equation $\frac{x^2}{25}-\frac{y^2}{144}=1$ .
(a)
What is the eccentricity of $H$ ?
[1 mark]
A $\frac{12}5$
B $\frac{13}5$
C $\frac{169}{25}$
D $\frac{13}{12}$
(b)
What are the coordinates of the foci of $H$ ?
[1 mark]
A $(\pm12,0)$
B $\left(\pm\frac{13}5,0\right)$
C $\left(\pm\sqrt{119},0\right)$
D $(\pm13,0)$
(c)
Find the distance between the two directrices of $H$ .
[2 marks]
Total for question 5: 4 marks
6
The parabola $C$ has equation $y^2=24x$ . The point $P$ on $C$ has parameter $t=2$ , where a general point of $C$ is $(6t^2,12t)$ .
(a)
What are the coordinates of $P$ ?
[1 mark]
A $(24,24)$
B $(12,24)$
C $(24,12)$
D $(6,12)$
(b)
What is the gradient of the normal to $C$ at $P$ ?
[1 mark]
A $\frac12$
B $2$
C $-2$
D $-\frac12$
(c)
Find the equation of the normal to $C$ at $P$ , giving your answer in the form $ax+by+c=0$ where $a$ , $b$ and $c$ are integers.
[2 marks]
Total for question 6: 4 marks
7
The ellipse $E$ has equation $\frac{x^2}{16}+\frac{y^2}{9}=1$ . The line $y=mx+c$ is a tangent to $E$ .
(a)
Show that $c^2=16m^2+9$ .
[3 marks]
(b)
Hence find the equations of the two tangents to $E$ that pass through the point $(0,5)$ .
[4 marks]
Total for question 7: 7 marks
8
A point $P(x,y)$ moves so that its distance from the point $S(4,0)$ is always twice its perpendicular distance from the line $l$ with equation $x=1$ .
(a)
Show that the locus of $P$ is a hyperbola and find its equation in the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .
[6 marks]
(b)
The parabola $C$ has its vertex at the origin and has $S$ as its focus. Find the coordinates of the points where $C$ meets the locus of $P$ .
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).