CAIE Further Paper 1 2024 November — Question 6 13 marks

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2024
SessionNovember
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCurve Sketching
TypeMultiple transformations including squared
DifficultyChallenging +1.2 This is a comprehensive curve sketching question requiring multiple techniques (asymptotes, stationary points, modulus transformation) but all are standard Further Maths procedures. The rational function is straightforward to analyze, and while the question has many parts (13 marks total), each step follows established methods without requiring novel insight or particularly complex algebra.
Spec1.02l Modulus function: notation, relations, equations and inequalities1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives
The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).
  1. Find the equations of the asymptotes of \(C\). [2]
  2. Find the coordinates of any stationary points on \(C\). [4]
  3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
  4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]
Question 6:
AnswerMarks
6(a)1
x= , x=3
AnswerMarks Guidance
2B1 Vertical asymptotes.
y=2B1 Horizontal asymptote.
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
6(b)dy (2x2 −7x+3)(8x+1)−(4x2 +x+1)(4x−7)
=
AnswerMarks Guidance
dx ( 2x2 −7x+3 )2M1 dy
Finds . Allow top line only for M1.
dx
AnswerMarks Guidance
−3x2 +2x+1=0M1 Sets equal to 0 and forms equation.
( −1,1) (1,−3)
,
AnswerMarks
3 5A1 A1
4
AnswerMarks Guidance
6(c)xy B1
or clear intersection with axes at correct place).
AnswerMarks
B1x3 correctly approaching asymptotes, not too
truncated.
AnswerMarks
B11 x3 correct.
2
AnswerMarks
B1x 1 correct.
2
( 0,1)
AnswerMarks Guidance
3B1 States coordinates of intersection with axis. May
be seen on their graph.
5
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
6(d)xy B1FT
k3B1
2
AnswerMarks Guidance
QuestionAnswer Marks 
Question 6:
--- 6(a) ---
6(a) | 1
x= , x=3
2 | B1 | Vertical asymptotes.
y=2 | B1 | Horizontal asymptote.
2
Question | Answer | Marks | Guidance
--- 6(b) ---
6(b) | dy (2x2 −7x+3)(8x+1)−(4x2 +x+1)(4x−7)
=
dx ( 2x2 −7x+3 )2 | M1 | dy
Finds . Allow top line only for M1.
dx
−3x2 +2x+1=0 | M1 | Sets equal to 0 and forms equation.
( −1,1) (1,−3)
,
3 5 | A1 A1
4
--- 6(c) ---
6(c) | xy | B1 | Axes and asymptotes. Clear identification (label
or clear intersection with axes at correct place).
B1 | x3 correctly approaching asymptotes, not too
truncated.
B1 | 1 x3 correct.
2
B1 | x 1 correct.
2
( 0,1)
3 | B1 | States coordinates of intersection with axis. May
be seen on their graph.
5
Question | Answer | Marks | Guidance
--- 6(d) ---
6(d) | xy | B1FT | FT from sketch in (c). At least two branches.
k3 | B1
2
Question | Answer | Marks | Guidance
The curve $C$ has equation $y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}$.

\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$. [2]

\item Find the coordinates of any stationary points on $C$. [4]

\item Sketch $C$, stating the coordinates of any intersections with the axes. [5]

\item Sketch the curve with equation $y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|$ and state the set of values of $k$ for which $\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k$ has 4 distinct real solutions. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q6 [13]}}
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Q1 10 Q2 6 Q3 10 Q4 8 Q5 13 Q6 13 Q7 15