CAIE Further Paper 2 2021 November — Question 5 10 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2021
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric differentiation
TypeFind second derivative d²y/dx²
DifficultyStandard +0.3 This is a straightforward parametric differentiation question requiring standard techniques: computing dx/dt and dy/dt, algebraic manipulation to verify an identity, arc length integration, and second derivative calculation. The algebra is slightly involved but follows routine procedures with no novel insights required, making it slightly easier than average for Further Maths.
Spec1.07s Parametric and implicit differentiation
The curve \(C\) has parametric equations $$x = 3t + 2t^{-1} + at^3, \quad y = 4t - \frac{3}{2}t^{-1} + bt^3, \quad \text{for } 1 \leq t \leq 2,$$ where \(a\) and \(b\) are constants.
  1. It is given that \(a = \frac{2}{3}\) and \(b = -\frac{1}{2}\). Show that \(\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \frac{25}{4}(t^2 + t^{-2})^2\) and find the exact length of \(C\). [6]
  2. It is given instead that \(a = b = 0\). Find the value of \(\frac{d^2y}{dx^2}\) when \(t = 1\). [4]
Question 5:
AnswerMarks
5Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
AnswerMarks Guidance
5(a)x=3−2t −2 +2t2 y =4+ 3t −2 −3t2
2 2B1 Differentiates x and y with respect to t.
4 12 23 9t4 12 9
4t4 +12t2 + − +1+ + + + −12t2
AnswerMarks Guidance
t4 t2 2 4 t2 4t4M1 ( )2
Expands x2 + y2. Accept 25+ 25 t −2 −t2 .
4
( )2
25t4 + 25t −4 + 25 = 25 t2 +t −2
AnswerMarks Guidance
4 4 2 4M1 A1 Factorises x2 + y2. AG.
5 2 t2 +t −2 dt = 51t3 −t −1 2 = 85
AnswerMarks Guidance
2 1 23  1 12M1 A1 Applies correct formula for arc length.
6
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
5(b)dy y 4+ 3t −2
= = 2
AnswerMarks Guidance
dx x 3−2t −2B1 dy
Finds .
dx
( )( ) ( )( )
d 4+ 3t −2  3−2t −2 −3t −3 − 4+ 3t −2 4t −3
 2  = 2
AnswerMarks Guidance
dt  3−2t −2   ( 3−2t −2 )2B1 dy
Differentiates with respect to t.
dx
d2y d 4+ 3t −2  dt −25t −3
=  2  × = =−25 when t =1.
AnswerMarks Guidance
dx2 dt  3−2t −2   dx ( 3−2t −2 )3M1 A1 Applies chain rule.
4
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
--- 5(a) ---
5(a) | x=3−2t −2 +2t2 y =4+ 3t −2 −3t2
2 2 | B1 | Differentiates x and y with respect to t.
4 12 23 9t4 12 9
4t4 +12t2 + − +1+ + + + −12t2
t4 t2 2 4 t2 4t4 | M1 | ( )2
Expands x2 + y2. Accept 25+ 25 t −2 −t2 .
4
( )2
25t4 + 25t −4 + 25 = 25 t2 +t −2
4 4 2 4 | M1 A1 | Factorises x2 + y2. AG.
5 2 t2 +t −2 dt = 51t3 −t −1 2 = 85
2 1 23  1 12 | M1 A1 | Applies correct formula for arc length.
6
Question | Answer | Marks | Guidance
--- 5(b) ---
5(b) | dy y 4+ 3t −2
= = 2
dx x 3−2t −2 | B1 | dy
Finds .
dx
( )( ) ( )( )
d 4+ 3t −2  3−2t −2 −3t −3 − 4+ 3t −2 4t −3
 2  = 2
dt  3−2t −2   ( 3−2t −2 )2 | B1 | dy
Differentiates with respect to t.
dx
d2y d 4+ 3t −2  dt −25t −3
=  2  × = =−25 when t =1.
dx2 dt  3−2t −2   dx ( 3−2t −2 )3 | M1 A1 | Applies chain rule.
4
Question | Answer | Marks | Guidance
The curve $C$ has parametric equations

$$x = 3t + 2t^{-1} + at^3, \quad y = 4t - \frac{3}{2}t^{-1} + bt^3, \quad \text{for } 1 \leq t \leq 2,$$

where $a$ and $b$ are constants.

\begin{enumerate}[label=(\alph*)]
\item It is given that $a = \frac{2}{3}$ and $b = -\frac{1}{2}$.

Show that $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \frac{25}{4}(t^2 + t^{-2})^2$ and find the exact length of $C$. [6]

\item It is given instead that $a = b = 0$.

Find the value of $\frac{d^2y}{dx^2}$ when $t = 1$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 2 2021 Q5 [10]}}
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Q1 5 Q2 7 Q3 8 Q4 10 Q5 10 Q6 11 Q7 11 Q8 13