A Level Further Mathematics 9231 Paper 12, Oct/Nov 2024

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9231/12 · Oct/Nov 2024

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The sequence $u_{1}, u_{2}, u_{3}, \dots$ is such that $u_{1} = 4$ and $u_{n + 1} = 3 u_{n} - 2$ for $n ⩾ 1$ . Prove by induction that $u_{n} = 3^{n} + 1$ for all positive integers $n$ .

[5]

The line $l_{1}$ has equation $r = i + 3 j - k + λ (i - j - 4 k)$ . The plane $Π$ contains $l_{1}$ and is parallel to the vector $2 i + 5 j - 4 k$ .

Find the equation of $Π$ , giving your answer in the form $a x + b y + c z = d$ .

[4]

The line $l_{2}$ is parallel to the vector $5 i - 5 j - 2 k$ . Find the acute angle between $l_{2}$ and $Π$ .

[3]

It is given that $\begin{aligned} α + β + γ + δ & = 2, \\ α^{2} + β^{2} + γ^{2} + δ^{2} & = 3, \\ α^{3} + β^{3} + γ^{3} + δ^{3} & = 4. \end{aligned}$

Find the value of $α β + α γ + α δ + β γ + β δ + γ δ$ .

[2]

Find the value of $α^{2} β + α^{2} γ + α^{2} δ + β^{2} α + β^{2} γ + β^{2} δ + γ^{2} α + γ^{2} β + γ^{2} δ + δ^{2} α + δ^{2} β + δ^{2} γ$ .

[3]

c(i)

It is given that $α, β, γ, δ$ are the roots of the equation $6 x^{4} - 12 x^{3} + 3 x^{2} + 2 x + 6 = 0.$ Find the value of $α^{4} + β^{4} + γ^{4} + δ^{4}$ .

[3]

c(ii)

Find the value of $α^{5} + β^{5} + γ^{5} + δ^{5}$ .

[2]

The matrices $A$ , $B$ and $C$ are given by $A = (\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array}), B = (\begin{array}{cc} 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array}) and C = (\begin{array}{ccc} - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array}) .$

Show that $C A B = (\begin{array}{cc} 3 & - 7 \\ - 9 & 3 \end{array})$ .

[3]

Find the equations of the invariant lines, through the origin, of the transformation in the $x$ - $y$ plane represented by $C A B$ .

[5]

Let $M = (\begin{array}{cc} 3 & 0 \\ 0 & 1 \end{array})$ . Give full details of the transformation represented by $M$ .

[2]

Find the matrix $N$ such that $N M = C A B$ .

[3]

It is given that $S_{n} = \sum_{r = 1}^{n} u_{r}$ , where $u_{r} = x^{f (r)} - x^{f (r + 1)}$ and $x > 0$ .

Find $S_{n}$ in terms of $n$ , $x$ and the function $f$ .

[2]

Given that $f (r) = \ln r$ , find the set of values of $x$ for which the infinite series $u_{1} + u_{2} + u_{3} + \dots$ is convergent and give the sum to infinity when this exists.

[3]

Given instead that $f (r) = 2 \log_{x} r$ where $x \neq 1$ , use standard results from the List of formulae (MF19) to find $\sum_{n = 1}^{N} S_{n}$ in terms of $N$ . Fully factorise your answer.

[4]

The curve $C$ has equation $y = \frac{x^{2} + 3}{x^{2} + 1}$ .

Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote.

[2]

Show that $1 < y ⩽ 3$ for all real values of $x$ .

[4]

Find the coordinates of any stationary points on $C$ .

[2]

Sketch $C$ , stating the coordinates of any intersections with the axes and labelling the asymptote.

[3]

Sketch the curve with equation $y = \frac{x^{2} + 1}{x^{2} + 3}$ and find the set of values of $x$ for which $\frac{x^{2} + 1}{x^{2} + 3} < \frac{1}{2}$ .

[4]

The curve $C_{1}$ has polar equation $r = a (\cos θ + \sin θ)$ for $- \frac{1}{4} π ⩽ θ ⩽ \frac{3}{4} π$ , where $a$ is a positive constant.

Find a Cartesian equation for $C_{1}$ and show that it represents a circle, stating its radius and the Cartesian coordinates of its centre.

[4]

Sketch $C_{1}$ and state the greatest distance of a point on $C_{1}$ from the pole.

[3]

The curve $C_{2}$ with polar equation $r = a θ$ intersects $C_{1}$ at the pole and the point with polar coordinates $(a ϕ, ϕ)$ . Verify that $1.25 < ϕ < 1.26$ .

[2]

Show that the area of the smaller region enclosed by $C_{1}$ and $C_{2}$ is equal to $\frac{1}{2} a^{2} (\frac{3}{4} π + \frac{1}{3} ϕ^{3} - ϕ + \frac{1}{2} \cos 2 ϕ)$ and deduce, in terms of $a$ and $ϕ$ , the area of the larger region enclosed by $C_{1}$ and $C_{2}$ .

[7]

Mark scheme

9231/12 · Oct/Nov 2024

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1 · 5 marks

B1
Shows base case
B1
States inductive hypothesis.
M1
Substitutes into recursion formula.
A1
$u_{k + 1} = 3^{k + 1} + 1$
A1
States conclusion.

2a · 4 marks

M1
Finds vector perpendicular to the plane.
A1
Correct vector perpendicular to the plane.
M1
Uses point in the plane.
A1
$24 x - 4 y + 7 z = 5$

2b · 3 marks

M1
Uses dot product of $5 i - 5 j - 2 k$ and their normal.
A1
Correct $\cos α = \frac{126}{\sqrt{641} \sqrt{54}}$ FT.
A1
Acute angle between $l_{2}$ and $Π$ is $90 - α = {42.6}^{\circ}$

3a · 2 marks

M1
Substitutes into formula for sum of squares.
A1
$α β + α γ + α δ + β γ + β δ + γ δ = \frac{1}{2}$

3b · 3 marks

M1
Factorises and substitutes.
A1
Correct expression in terms of sums of powers.
A1
$2 (3) - 4 = 2$

3c(i) · 3 marks

M1
Sums and substitutes.
A1
Correct substituted equation.
A1
$S_{4} = \frac{11}{6}$

3c(ii) · 2 marks

M1
Multiplies equation through by $x$ , sums and substitutes.
A1
$S_{5} = - \frac{4}{3}$

4a · 3 marks

M1
Multiplying two matrices correctly, correct dimensions.
A1
Correct product of first pair of matrices.
B1
Convincingly completing matrix multiplication, AG.

4b · 5 marks

B1
Transforms $(\begin{array}{c} x \\ y \end{array})$ to $(\begin{array}{c} X \\ Y \end{array})$ .
M1
Uses $y = m x$ and $Y = m X$ .
A1
$- 9 x + 3 m x = m (3 x - 7 m x)$
A1
$- 9 + 3 m = 3 m - 7 m^{2} ⟹ 7 m^{2} = 9$
A1
$y = \frac{3}{\sqrt{7}} x$ and $y = - \frac{3}{\sqrt{7}} x$

4c · 2 marks

B1
Stretch
B1
parallel to the $x$ -axis, scale factor 3.

4d · 3 marks

B1
$M^{- 1} = \frac{1}{3} (\begin{array}{cc} 1 & 0 \\ 0 & 3 \end{array})$
M1
Correct order
A1
$= (\begin{array}{cc} 1 & - 7 \\ - 3 & 3 \end{array})$

5a · 2 marks

M1
Writes at least three terms, including last.
A1
$= x^{f (1)} - x^{f (n + 1)}$

5b · 3 marks

B1
$[0 <] x \leq 1$
B1
$\sum_{r = 1}^{\infty} u_{r} = 1$ [for $x < 1$ ] Without wrong working.
B1
$\sum_{r = 1}^{\infty} u_{r} = 0$ for $x = 1$

5c · 4 marks

M1
Uses $x^{2 \log_{x} r} = r^{2}$
A1
$S_{n} = 1 - (n + 1)^{2} = - n^{2} - 2 n$
M1
Substitutes formulae from MF19.
A1
$- \frac{1}{6} N (N + 1) (2 N + 7)$

6a · 2 marks

B1
$x^{2} + 1 = 0$ has no real roots.
B1
$y = 1$ Horizontal asymptote.

6b · 4 marks

M1
Forms quadratic in $x$ or uses $y = 1 + \frac{2}{x^{2} + 1}$ .
A1
$(1 - y) x^{2} - y + 3 = 0$
M1
Uses that discriminant is $\geq 0$ or $0 < \frac{2}{x^{2} + 1} \leq 2$ .
A1
$1 < y \leq 3$ Explanation of why $y \neq 1$ AG.

6c · 2 marks

M1
Differentiates (or using inequality from (b))
A1
$(0, 3)$

6d · 3 marks

B1
Axes and correct asymptote labelled.
B1
Correct shape and position.
B1
States $(0, 3)$ coordinates of intersection with axes, may been seen on diagram.

6e · 4 marks

B1
FT from sketch in (d)
M1
Finds critical points.
A1
$x^{2} = 1 ⟹ x = \pm 1$
A1
$- 1 < x < 1$

7a · 4 marks

M1
Uses $x = r \cos θ$ and $y = r \sin θ$ to eliminate $θ$
M1
Obtains Cartesian equation (using $r^{2} = x^{2} + y^{2}$ ).
A1
$x^{2} - a x + y^{2} - a y = 0$ OE
B1
Centre $(\frac{a}{2}, \frac{a}{2})$ and radius $\frac{a}{\sqrt{2}}$

7b · 3 marks

B1
Circle
B1
Correct position, passing through $O$ .
B1
States maximum distance or labels sketch.

7c · 2 marks

M1
$\cos ϕ + \sin ϕ - ϕ = 0$
A1
Shows sign change.

7d · 7 marks

M1
Uses area formula on both curves.
A1
Forms area of smaller region enclosed by $C_{1}$ and $C_{2}$ with correct limits.
M1
Applies relevant identities for the circle integral to produce integrable form.
A1
For correct integration of the circle part.
A1
Integrates spiral and substitutes correct limits. AG.
M1
Forms area of larger region enclosed by $C_{1}$ and $C_{2}$ .
A1
$\frac{a^{2}}{2} (- \frac{ϕ^{3}}{3} + \frac{1}{4} π + ϕ - \frac{1}{2} \cos 2 ϕ)$

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