CAIE P3 2024 June — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2024
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTangents, normals and gradients
TypeFind stationary points
DifficultyStandard +0.3 This is a straightforward application of the product rule to find dy/dx, setting it equal to zero, and solving a basic trigonometric equation. While it requires careful algebraic manipulation of exponentials and trig functions, it follows a standard procedure with no conceptual surprises, making it slightly easier than average.
Spec1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation
Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]
Question 2:
AnswerMarks Guidance
2Use correct product rule
cos2x may be 1 – 2sin2x or …M1 ae2xsin2x + e2xbcos2x. Need a or b = 2.
Allow M1 if only error is ex instead of e2x in one of terms, then
maximum 1/5.
AnswerMarks Guidance
Obtain correct derivative 2e2xsin2x2e2xcos2xA1 OE, e.g. 4e2xsinxcosx2e2x cos2xsin2 x  .
Equate derivative of the form ae2xsin2x + e2xbcos2x to 0 and solve
for 2x or x using a correct method
AnswerMarks Guidance
Note may have substituted for sin2x and/or cos2xM1 Obtain 2x = tan−1(− their b/their a) OE.
Allow one slip in rearranging.
Allow degrees.
Variety of other methods available, such as solving quadratic
equation in sinx or tanx e.g. tan²x – 2tanx – 1 = 0 leading to x
= tan-1(1+√2).
Obtain x = 3π only or exact equivalent
AnswerMarks Guidance
8A1 CWO
67.5° gets A0.
π
Ignore any answers outside interval 0 ⩽ x ⩽ .
2
1 3 π
Obtain y = 2e4 only or exact simplified equivalent
AnswerMarks Guidance
2A1 CWO, ISW.
3 3 
π
Not sin πe4 .
4 
 
π
Ignore any answers using x outside interval 0 ⩽ x ⩽ .
2
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 2:
2 | Use correct product rule
cos2x may be 1 – 2sin2x or … | M1 | ae2xsin2x + e2xbcos2x. Need a or b = 2.
Allow M1 if only error is ex instead of e2x in one of terms, then
maximum 1/5.
Obtain correct derivative 2e2xsin2x2e2xcos2x | A1 | OE, e.g. 4e2xsinxcosx2e2x cos2xsin2 x  .
Equate derivative of the form ae2xsin2x + e2xbcos2x to 0 and solve
for 2x or x using a correct method
Note may have substituted for sin2x and/or cos2x | M1 | Obtain 2x = tan−1(− their b/their a) OE.
Allow one slip in rearranging.
Allow degrees.
Variety of other methods available, such as solving quadratic
equation in sinx or tanx e.g. tan²x – 2tanx – 1 = 0 leading to x
= tan-1(1+√2).
Obtain x = 3π only or exact equivalent
8 | A1 | CWO
67.5° gets A0.
π
Ignore any answers outside interval 0 ⩽ x ⩽ .
2
1 3 π
Obtain y = 2e4 only or exact simplified equivalent
2 | A1 | CWO, ISW.
3 3 
π
Not sin πe4 .
4 
 
π
Ignore any answers using x outside interval 0 ⩽ x ⩽ .
2
5
Question | Answer | Marks | Guidance
Find the exact coordinates of the stationary point of the curve $y = e^{2x} \sin 2x$ for $0 \leqslant x < \frac{1}{2}\pi$. [5]

\hfill \mbox{\textit{CAIE P3 2024 Q2 [5]}}
This paper (12 questions)
View full paper
Q1 4 Q2 5 Q3 5 Q4 4 Q5 5 Q6 7 Q7 3 Q7 3 Q8 8 Q9 11 Q10 11 Q11 9