CAIE Further Paper 3 2021 June — Question 7 5 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2021
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProjectiles
TypeVelocity direction at specific time/point
DifficultyChallenging +1.2 Part (c) is a straightforward application of the standard projectile formulas R = u²sin(2θ)/g and H = u²sin²(θ)/(2g), requiring simple algebraic manipulation. Part (d) is more substantial, requiring differentiation of the trajectory equation and solving an inequality involving tan(45°) = 1, but follows a clear method once the approach is identified. The 4-mark allocation and 'or otherwise' hint suggest this is a standard further maths mechanics question rather than requiring novel insight.
Spec3.02i Projectile motion: constant acceleration model
It is given that \(R = \frac{4H}{\sqrt{3}}\).
  1. Show that \(\theta = 60°\). [1]
It is given also that \(u = \sqrt{40}\) m s\(^{-1}\).
  1. Find, by differentiating the equation of the trajectory or otherwise, the set of values of \(x\) for which the direction of motion makes an angle of less than \(45°\) with the horizontal. [4]
Question 7:
AnswerMarks
7(a)R2
y = 0 in trajectory equation: Rtanθ−g =0
( cosθ)2
AnswerMarks
2u2M1
R=)2u2sinθcosθ
(
only
AnswerMarks Guidance
gA1 Any equivalent single term expression, for example:
u2sin2θ 2u2tanθ
, , at least one intermediate line of
g g sec2θ
working, not just quoting a result.
SC B1 using SUVAT.
2
AnswerMarks
7(b)u2sinθcosθ
x=their
and substitute in trajectory equation.
AnswerMarks Guidance
gM1 Or use SUVAT.
( sinθ)2
u2
H =
AnswerMarks Guidance
2gA1 Single term.
2
AnswerMarks
7(c)4H
Use R= and simplify: tanθ= 3, θ = 60°
AnswerMarks Guidance
3B1 AG
1
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
7(d)dy gx
=tanθ−
( cosθ)2
AnswerMarks Guidance
dx u2M1 Differentiate with respect to x.
x
tanθ− =±1
used
( cosθ)2
AnswerMarks Guidance
4M1 dy
Use =±1 as limiting case.
dx
AnswerMarks Guidance
x = 3 +1, x = 3 −1A1
3 −1< x < 3 +1A1 Strict inequality, exact values.
Alternative method for question 7(d)
1 dy
y= 3x− x2, = 3−x
AnswerMarks Guidance
2 dxM1 Differentiate with respect to x.
dy
=±1 used
AnswerMarks Guidance
dxM1 dy
Use =±1 as limiting case.
dx
AnswerMarks Guidance
x = 3 +1 , x = 3 −1A1
3 −1< x < 3 +1A1 Strict inequality, exact values.
QuestionAnswer Marks
7(d)Alternative method for question 7(d)
v =±v
When moving at 45° to horizontal,
AnswerMarks Guidance
x yM1 Used, both cases considered.
v = 40 cosθ , v = 40sinθ−10t
x y
1 ( ) 1 ( )
t= 30− 10 , t= 30+ 10
AnswerMarks Guidance
10 10M1
x = 3 +1 , x = 3 −1A1
3 −1< x < 3 +1A1 Strict inequality, exact values.
4
Question 7:
--- 7(a) ---
7(a) | R2
y = 0 in trajectory equation: Rtanθ−g =0
( cosθ)2
2u2 | M1
R=)2u2sinθcosθ
(
only
g | A1 | Any equivalent single term expression, for example:
u2sin2θ 2u2tanθ
, , at least one intermediate line of
g g sec2θ
working, not just quoting a result.
SC B1 using SUVAT.
2
--- 7(b) ---
7(b) | u2sinθcosθ
x=their
and substitute in trajectory equation.
g | M1 | Or use SUVAT.
( sinθ)2
u2
H =
2g | A1 | Single term.
2
--- 7(c) ---
7(c) | 4H
Use R= and simplify: tanθ= 3, θ = 60°
3 | B1 | AG
1
Question | Answer | Marks | Guidance
--- 7(d) ---
7(d) | dy gx
=tanθ−
( cosθ)2
dx u2 | M1 | Differentiate with respect to x.
x
tanθ− =±1
used
( cosθ)2
4 | M1 | dy
Use =±1 as limiting case.
dx
x = 3 +1, x = 3 −1 | A1
3 −1< x < 3 +1 | A1 | Strict inequality, exact values.
Alternative method for question 7(d)
1 dy
y= 3x− x2, = 3−x
2 dx | M1 | Differentiate with respect to x.
dy
=±1 used
dx | M1 | dy
Use =±1 as limiting case.
dx
x = 3 +1 , x = 3 −1 | A1
3 −1< x < 3 +1 | A1 | Strict inequality, exact values.
Question | Answer | Marks | Guidance
7(d) | Alternative method for question 7(d)
v =±v
When moving at 45° to horizontal,
x y | M1 | Used, both cases considered.
v = 40 cosθ , v = 40sinθ−10t
x y
1 ( ) 1 ( )
t= 30− 10 , t= 30+ 10
10 10 | M1
x = 3 +1 , x = 3 −1 | A1
3 −1< x < 3 +1 | A1 | Strict inequality, exact values.
4
It is given that $R = \frac{4H}{\sqrt{3}}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that $\theta = 60°$. [1]
\end{enumerate}

It is given also that $u = \sqrt{40}$ m s$^{-1}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find, by differentiating the equation of the trajectory or otherwise, the set of values of $x$ for which the direction of motion makes an angle of less than $45°$ with the horizontal. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2021 Q7 [5]}}
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