| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring ellipse sketching, discriminant analysis for intersection conditions, completing the square in two variables, and applying tangency conditions. While each individual technique is standard FP1 material, the question requires sustained reasoning across multiple parts with the final part building on earlier work. The discriminant inequality and the translation/tangency synthesis elevate this above routine exercises, but it remains within expected FP1 scope without requiring exceptional insight. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
TOTAL
Question 9:
9
TOTAL
Answer all questions.
Answer each question in the space provided for that question.
1 A curve passes through the point ð9, 6Þ and satisfies the differential equation
dy 1
¼ p
ffiffiffi
dx 2þ x
Use a step-by-step method with a step length of 0:25 to estimate the value of y at
x ¼ 9:5. Give your answer to four decimal places.
[5 marks]
2 The quadratic equation
2x2 þ8xþ1 ¼ 0
has roots a and b.
(a) Write down the value of a þb and the value of ab.
[2 marks]
(b)(i) Find the value of a2 þb2.
[2 marks]
449
(ii) Hence, or otherwise, show that a4 þb4 ¼ .
2
[2 marks]
(c) Find a quadratic equation, with integer coefficients, which has roots
1 1
2a4 þ and 2b4 þ
b2 a2
[5 marks]
n n
X X
r3 r2
3 Use the formulae for and to find the value of
r¼1 r¼1
60
X
r2ðr(cid:2)6Þ
r¼3
[4 marks]
4 Find the complex number z such that
5izþ3z(cid:3) þ16 ¼ 8i
Give your answer in the form aþbi, where a and b are real.
[6 marks]
5 A curve C has equation y ¼ xðxþ3Þ.
(a) Find the gradient of the line passing through the point ð(cid:2)5, 10Þ and the point on C
with x-coordinate (cid:2)5þh. Give your answer in its simplest form.
[3 marks]
(b) Show how the answer to part (a) can be used to find the gradient of the curve C at
the point ð(cid:2)5, 10Þ. State the value of this gradient.
[2 marks]
1
6 A curve C has equation y ¼ .
xðxþ2Þ
(a) Write down the equations of all the asymptotes of C.
[2 marks]
(b) The curve C has exactly one stationary point. The x-coordinate of the stationary point
is (cid:2)1.
(i) Find the y-coordinate of the stationary point.
[1 mark]
(ii) Sketch the curve C.
[2 marks]
(c) Solve the inequality
1 1
4
xðxþ2Þ 8
[5 marks]
7 (a) Write down the 2(cid:4)2 matrix corresponding to each of the following transformations:
(i) a reflection in the line y ¼ (cid:2)x;
[1 mark]
(ii) a stretch parallel to the y-axis of scale factor 7.
[1 mark]
(b) Hence find the matrix corresponding to the combined transformation of a reflection in
the line y ¼ (cid:2)x followed by a stretch parallel to the y-axis of scale factor 7.
[2 marks]
p
(cid:3) ffiffiffi (cid:4)
(cid:2)3 (cid:2) 3
(c) The matrix A is defined by A ¼ p .
ffiffiffi
(cid:2) 3 3
(i) Show that A2 ¼ kI, where k is a constant and I is the 2(cid:4)2 identity matrix.
[1 mark]
(ii) Show that the matrix A corresponds to a combination of an enlargement and a
reflection. State the scale factor of the enlargement and state the equation of the line
of reflection in the form y ¼ ðtanyÞx.
[5 marks]
8 (a) Find the general solution of the equation
p
(cid:5) (cid:6) ffiffiffi
5 p 2
cos x(cid:2) ¼
4 3 2
giving your answer for x in terms of p.
[5 marks]
(b) Use your general solution to find the sum of all the solutions of the equation
p
(cid:5) (cid:6) ffiffiffi
5 p 2
cos x(cid:2) ¼ that lie in the interval 04x420p. Give your answer in the
4 3 2
form kp, stating the exact value of k.
[4 marks]
9 An ellipse E has equation
x2 y2
þ ¼ 1
16 9
(a) Sketch the ellipse E, showing the values of the intercepts on the coordinate axes.
[2 marks]
(b) Given that the line with equation y ¼ xþk intersects the ellipse E at two distinct
points, show that (cid:2)5 < k < 5.
[5 marks]
(cid:3) (cid:4)
a
(c) The ellipse E is translated by the vector to form another ellipse whose equation
b
is 9x2 þ16y2 þ18x(cid:2)64y ¼ c. Find the values of the constants a, b and c.
[5 marks]
(d) Hence find an equation for each of the two tangents to the ellipse
9x2 þ16y2 þ18x(cid:2)64y ¼ c that are parallel to the line y ¼ x.
[3 marks]An ellipse $E$ has equation
$$\frac{x^2}{16} + \frac{y^2}{9} = 1$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the ellipse $E$, showing the values of the intercepts on the coordinate axes. [2 marks]
\item Given that the line with equation $y = x + k$ intersects the ellipse $E$ at two distinct points, show that $-5 < k < 5$. [5 marks]
\item The ellipse $E$ is translated by the vector $\begin{bmatrix} a \\ b \end{bmatrix}$ to form another ellipse whose equation is $9x^2 + 16y^2 + 18x - 64y = c$. Find the values of the constants $a$, $b$ and $c$. [5 marks]
\item Hence find an equation for each of the two tangents to the ellipse $9x^2 + 16y^2 + 18x - 64y = c$ that are parallel to the line $y = x$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2014 Q9 [15]}}