CAIE P3 2018 June — Question 9 8 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2018
SessionJune
Marks8
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TopicConnected Rates of Change
TypeCurve motion: find x-coordinate
DifficultyStandard +0.3 This is a straightforward connected rates of change question requiring standard integration with substitution (part i), basic application of the chain rule dy/dt = (dy/dx)(dx/dt) (part ii), and routine differentiation to verify a constant product (part iii). All techniques are standard P3 material with no novel problem-solving required, making it slightly easier than average.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08b Integrate x^n: where n != -1 and sums
A curve is such that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \sqrt{(4x + 1)}\) and \((2, 5)\) is a point on the curve.
  1. Find the equation of the curve. [4]
  2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \((2, 5)\). [2]
  3. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \frac{\mathrm{d}y}{\mathrm{d}x}\) is constant. [2]
Question 9:
AnswerMarks Guidance
9(i)Use a correct method to find a constant M1
Obtain one of the values A = – 3, B = 1, C = 2A1
Obtain a second valueA1
Obtain the third valueA1
4
AnswerMarks
9(ii)Use a correct method to find the first two terms of the expansion
of ( 3−x )−1 ,  1− 1 x   −1 , ( 2+x2 )−1 or  1+ 1 x2   −1
AnswerMarks Guidance
 3   2 M1 Symbolic binomial coefficients are not sufficient for the M1.
Obtain correct unsimplified expansions up to the term in x3 of
AnswerMarks Guidance
each partial fractionA1Ft + A1Ft The ft is on A, B and C.
−1  1+ x + x2 + x3 ...   + x+2 1− x2 ...  
 3 9 27  2  2 
−1− x − x2 − x3 +1− x2 + x − x3
3 9 27 2 2 4
Multiply out their expansion, up to the terms in x3, by
AnswerMarks
Bx + C, where BC ≠ 0M1
1 11 31
Obtain final answer x− x2 − x3, or equivalent
AnswerMarks
6 18 108A1
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 9:
--- 9(i) ---
9(i) | Use a correct method to find a constant | M1
Obtain one of the values A = – 3, B = 1, C = 2 | A1
Obtain a second value | A1
Obtain the third value | A1
4
--- 9(ii) ---
9(ii) | Use a correct method to find the first two terms of the expansion
of ( 3−x )−1 ,  1− 1 x   −1 , ( 2+x2 )−1 or  1+ 1 x2   −1
 3   2  | M1 | Symbolic binomial coefficients are not sufficient for the M1.
Obtain correct unsimplified expansions up to the term in x3 of
each partial fraction | A1Ft + A1Ft | The ft is on A, B and C.
−1  1+ x + x2 + x3 ...   + x+2 1− x2 ...  
 3 9 27  2  2 
−1− x − x2 − x3 +1− x2 + x − x3
3 9 27 2 2 4
Multiply out their expansion, up to the terms in x3, by
Bx + C, where BC ≠ 0 | M1
1 11 31
Obtain final answer x− x2 − x3, or equivalent
6 18 108 | A1
5
Question | Answer | Marks | Guidance
A curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x} = \sqrt{(4x + 1)}$ and $(2, 5)$ is a point on the curve.

\begin{enumerate}[label=(\roman*)]
\item Find the equation of the curve. [4]

\item A point $P$ moves along the curve in such a way that the $y$-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the $x$-coordinate when $P$ passes through $(2, 5)$. [2]

\item Show that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \frac{\mathrm{d}y}{\mathrm{d}x}$ is constant. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2018 Q9 [8]}}
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