CAIE P1 2023 November — Question 3 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2023
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIndefinite & Definite Integrals
TypeCurve properties and tangent/normal
DifficultyModerate -0.8 This is a straightforward integration question requiring basic power rule application and using a point to find the constant of integration, plus finding a normal line using the negative reciprocal of the derivative. Both parts are routine A-level techniques with no conceptual challenges—simpler than average but not trivial.
Spec1.07m Tangents and normals: gradient and equations1.08a Fundamental theorem of calculus: integration as reverse of differentiation
The equation of a curve is such that \(\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}\). The curve passes through the point \(P(2, 8)\).
  1. Find the equation of the normal to the curve at \(P\). [2]
  2. Find the equation of the curve. [4]
Question 3:
AnswerMarks
3(a) 
−1 −1 2
[Gradient of normal =]  = − 
11 11 11
Their  
AnswerMarks Guidance
2  2 M1 Tangent gradient must come from x = 2 substituted into the
given expression.
y−8 2 2x 92
=− or 1 1y + 2x = 92 or y= − +
AnswerMarks Guidance
x−2 11 11 11A1 OE
2
AnswerMarks
3(b)1  72  x2 24 
y=  x2 2+ −3 +c  − +c
AnswerMarks Guidance
2  x3   4 x3 B1, B1 One mark for each correct unsimplified { }.
1 24
8= 4− +c
AnswerMarks Guidance
4 8M1 Substitution of x = 2, y = 8 into their integrated expression,
defined by at least one correct power. Two terms and + c
needed.
1  24
y= or 0.25 x2 − +10
AnswerMarks Guidance
4  x3A1 Both coefficients must be simplified but allow x−3. Condone
c = 10 as line as long as either y or f(x) = is seen elsewhere.
4
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
--- 3(a) ---
3(a) |  
−1 −1 2
[Gradient of normal =]  = − 
11 11 11
Their  
2  2  | M1 | Tangent gradient must come from x = 2 substituted into the
given expression.
y−8 2 2x 92
=− or 1 1y + 2x = 92 or y= − +
x−2 11 11 11 | A1 | OE
2
--- 3(b) ---
3(b) | 1  72  x2 24 
y=  x2 2+ −3 +c  − +c
2  x3   4 x3  | B1, B1 | One mark for each correct unsimplified { }.
1 24
8= 4− +c
4 8 | M1 | Substitution of x = 2, y = 8 into their integrated expression,
defined by at least one correct power. Two terms and + c
needed.
1  24
y= or 0.25 x2 − +10
4  x3 | A1 | Both coefficients must be simplified but allow x−3. Condone
c = 10 as line as long as either y or f(x) = is seen elsewhere.
4
Question | Answer | Marks | Guidance
The equation of a curve is such that $\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}$. The curve passes through the point $P(2, 8)$.

\begin{enumerate}[label=(\alph*)]
\item Find the equation of the normal to the curve at $P$. [2]

\item Find the equation of the curve. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2023 Q3 [6]}}
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