CAIE Further Paper 2 2023 November — Question 5 10 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric differentiation
TypeFind second derivative d²y/dx²
DifficultyChallenging +1.2 This is a standard Further Maths parametric question requiring arc length calculation and second derivative analysis. Part (a) involves computing dx/dt and dy/dt, then integrating √(1+(dy/dx)²) which simplifies nicely. Part (b) requires finding d²y/dx² using the chain rule formula. While it requires multiple techniques and careful algebra, these are well-practiced procedures for Further Maths students with no novel insight needed. The 10 marks and straightforward setup place it moderately above average difficulty.
Spec1.07s Parametric and implicit differentiation4.08f Integrate using partial fractions
The curve C has parametric equations $$x = \frac{5}{3}t^{\frac{3}{2}} - 2t^{\frac{1}{2}}, \quad y = 2t + 5, \quad \text{for } 0 < t \leq 3.$$
  1. Find the exact length of C. [5]
  2. Find the set of values of \(t\) for which \(\frac{d^2y}{dx^2} > 0\). [5]
Question 5:
AnswerMarks
5(a)1 1
−
AnswerMarks Guidance
x=t2 −t 2, y=2B1 Differentiates x and y with respect to t.
2 2
 1 1   1 1 
− −
x2 + y2 =t2 −t 2 +4=t+2+t−1 =t2 +t 2
   
AnswerMarks Guidance
   M1 A1 Factorises x2 + y2.
3
1 1  3 1
3 − 2
 t2 +t 2dt = t2 +2t2 =4 3
0  3 
AnswerMarks Guidance
0M1 A1 Applies correct formula for arc length.
3
M1 for their  x2 + y2 dt.
0
Answer must be simplified to 4 3 for A1.
5
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
5(b)dy y 2
= =
dx x 1 1
−
AnswerMarks Guidance
t2 −t 2B1 Finds first derivative.
 1 3
1 − 1 −
−2 t 2 + t 2
   
d  2   2 2 
=
dt 1 1  2
−  1 1 
t2 −t 2  t2 −t − 2
 
AnswerMarks Guidance
 B1 dy
Differentiates with respect to t.
dx
 1 3
1 − 1 −
−2 t 2 + t 2
   
d2y d  2  dt  2 2  t+1
=  = =−
dx2 dt 1 − 1  dx  1 1  3 (t−1)3
t2 −t 2  t2 −t − 2
 
AnswerMarks Guidance
 M1 A1 Applies chain rule. OE. Does not have to be simplified
for A1.
AnswerMarks Guidance
0t1A1 Accept −1t1. CWO.
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
--- 5(a) ---
5(a) | 1 1
−
x=t2 −t 2, y=2 | B1 | Differentiates x and y with respect to t.
2 2
 1 1   1 1 
− −
x2 + y2 =t2 −t 2 +4=t+2+t−1 =t2 +t 2
   
    | M1 A1 | Factorises x2 + y2.
3
1 1  3 1
3 − 2
 t2 +t 2dt = t2 +2t2 =4 3
0  3 
0 | M1 A1 | Applies correct formula for arc length.
3
M1 for their  x2 + y2 dt.
0
Answer must be simplified to 4 3 for A1.
5
Question | Answer | Marks | Guidance
--- 5(b) ---
5(b) | dy y 2
= =
dx x 1 1
−
t2 −t 2 | B1 | Finds first derivative.
 1 3
1 − 1 −
−2 t 2 + t 2
   
d  2   2 2 
=
dt 1 1  2
−  1 1 
t2 −t 2  t2 −t − 2
 
  | B1 | dy
Differentiates with respect to t.
dx
 1 3
1 − 1 −
−2 t 2 + t 2
   
d2y d  2  dt  2 2  t+1
=  = =−
dx2 dt 1 − 1  dx  1 1  3 (t−1)3
t2 −t 2  t2 −t − 2
 
  | M1 A1 | Applies chain rule. OE. Does not have to be simplified
for A1.
0t1 | A1 | Accept −1t1. CWO.
5
Question | Answer | Marks | Guidance
The curve C has parametric equations
$$x = \frac{5}{3}t^{\frac{3}{2}} - 2t^{\frac{1}{2}}, \quad y = 2t + 5, \quad \text{for } 0 < t \leq 3.$$

\begin{enumerate}[label=(\alph*)]
\item Find the exact length of C. [5]
\item Find the set of values of $t$ for which $\frac{d^2y}{dx^2} > 0$. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q5 [10]}}
This paper (8 questions)
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Q1 4 Q2 5 Q3 6 Q4 10 Q5 10 Q6 14 Q7 11 Q8 15