CAIE P1 2024 June — Question 11 13 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTangents, normals and gradients
TypeIncreasing/decreasing intervals
DifficultyStandard +0.8 This question requires finding where f'(x) < 0 by solving a cubic inequality (part a), then finding equations of a normal and tangent at specific points, determining their intersection, and calculating a triangle area (part b). The multi-step nature, algebraic manipulation of the cubic inequality, and geometric construction with coordinate geometry make this moderately challenging, though all techniques are standard A-level material.
Spec1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07o Increasing/decreasing: functions using sign of dy/dx
\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.
  1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
  2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]
Question 11:
AnswerMarks
11(a)dy 12 3
 
AnswerMarks
dx x4 x2B1
dy 12 3
  0 leading to 3x4 12x2 0 or -12 + 3x2 = 0
AnswerMarks Guidance
dx x4 x2M1 Set = 0 or uses , ⩽ and simplifies.
dy A B
Must be from   .
dx x4 x2
AnswerMarks Guidance
3x2 x2 4  0 leading to x2 onlyA1 SC B1 for x2 if M0 scored.
2x0 and 0x2 or (-2, 0) and (0, 2) or 2x2 and x 0B1FT Allow and/or.
B1FTAllow 2 x0 and/or 0 x2 but only B1B0 if 0
included in either or both.
Allow [–2, 0) and (0, 2].
Allow B1B0 for 2x2 or (–2, 2).
dy A B
Must be from   .
dx x4 x2
AnswerMarks
5dy A B
B marks only available if   .
dx x4 x2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
11(b)[At x1] y 3 and mtan 9 *M1
Using their .
dx
1 1
m norm 
AnswerMarks
9 9DM1
1  1 26
Equation of normal is y 3 x1 leading to y  x
 
AnswerMarks
9  9 9 A1
At x1, y1,m9M1
Equation of tangent is y19x1 leading to y9x8A1
1 26  49
Meet when x  9x8 leading to x  1.19512,
 
AnswerMarks Guidance
9 9  41 M1 Equates their tangent and their normal.
1 26 
Area = their1 .19512their  8
AnswerMarks Guidance
2  9 M1 If y  y is used integration must be correct and
2 1
substitution shown.
AnswerMarks Guidance
6.51A1 AWRT
Accept fraction wrt 6.51
8
Question 11:
--- 11(a) ---
11(a) | dy 12 3
 
dx x4 x2 | B1
dy 12 3
  0 leading to 3x4 12x2 0 or -12 + 3x2 = 0
dx x4 x2 | M1 | Set = 0 or uses , ⩽ and simplifies.
dy A B
Must be from   .
dx x4 x2
3x2 x2 4  0 leading to x2 only | A1 | SC B1 for x2 if M0 scored.
2x0 and 0x2 or (-2, 0) and (0, 2) or 2x2 and x 0 | B1FT | Allow and/or.
B1FT | Allow 2 x0 and/or 0 x2 but only B1B0 if 0
included in either or both.
Allow [–2, 0) and (0, 2].
Allow B1B0 for 2x2 or (–2, 2).
dy A B
Must be from   .
dx x4 x2
5 | dy A B
B marks only available if   .
dx x4 x2
Question | Answer | Marks | Guidance
--- 11(b) ---
11(b) | [At x1] y 3 and mtan 9 | *M1 | dy
Using their .
dx
1 1
m norm 
9 9 | DM1
1  1 26
Equation of normal is y 3 x1 leading to y  x
 
9  9 9  | A1
At x1, y1,m9 | M1
Equation of tangent is y19x1 leading to y9x8 | A1
1 26  49
Meet when x  9x8 leading to x  1.19512,
 
9 9  41  | M1 | Equates their tangent and their normal.
1 26 
Area = their1 .19512their  8
2  9  | M1 | If y  y is used integration must be correct and
2 1
substitution shown.
6.51 | A1 | AWRT
Accept fraction wrt 6.51
8
\includegraphics{figure_11}

A function is defined by f$(x) = \frac{4}{x^3} - \frac{3}{x} + 2$ for $x \neq 0$. The graph of $y = \text{f}(x)$ is shown in the diagram.

\begin{enumerate}[label=(\alph*)]
\item Find the set of values of $x$ for which f$(x)$ is decreasing. [5]

\item A triangle is bounded by the $y$-axis, the normal to the curve at the point where $x = 1$ and the tangent to the curve at the point where $x = -1$.

Find the area of the triangle. Give your answer correct to 3 significant figures. [8]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q11 [13]}}
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