Question 3 - A-Level Maths

CAIE M1 2021 March — Question 3 5 marks

Exam Board	CAIE
Module	M1 (Mechanics 1)
Year	2021
Session	March
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Forces, equilibrium and resultants
Type	Particle suspended by strings
Difficulty	Moderate -0.8 This is a straightforward two-string equilibrium problem requiring resolution of forces in two directions (horizontal and vertical) with standard angles (30° and 60°). The setup is clear, the method is routine (resolve horizontally and vertically, solve simultaneous equations), and the trigonometry is simple. This is easier than average for A-level mechanics questions.
Spec	3.03m Equilibrium: sum of resolved forces = 0 3.03n Equilibrium in 2D: particle under forces

3 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-05_518_616_255_767} A particle \(Q\) of mass 0.2 kg is held in equilibrium by two light inextensible strings \(P Q\) and \(Q R . P\) is a fixed point on a vertical wall and \(R\) is a fixed point on a horizontal floor. The angles which strings \(P Q\) and \(Q R\) make with the horizontal are \(60 ^ { \circ }\) and \(30 ^ { \circ }\) respectively (see diagram). Find the tensions in the two strings.

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Question 3:

Answer	Marks	Guidance
Answer	Marks	Guidance
For attempting to resolve forces in either direction	M1	Correct number of relevant terms
\(T_P \cos 60 = T_R \cos 30\)	A1
\(T_P \sin 60 = T_R \sin 30 + 0.2g\)	A1
Attempt to solve simultaneously for either tension	M1	From 2 equations, with correct number of relevant terms
\(T_P = 3.46\) N and \(T_R = 2\) N	A1	Both correct. Allow \(T_P = 2\sqrt{3}\) N
Alternative method:
\(\frac{T_P}{\sin 60} = \frac{T_R}{\sin 150} = \frac{0.2g}{\sin 150}\)	M1	Attempt one pair of Lami's equations. Correct angles
One pair correct	A1
Equations all correct	A1
Solve for \(T_P\) or \(T_R\)	M1	From equations of the correct form
\(T_P = 3.46\) N and \(T_R = 2\) N	A1	Both correct. Allow \(T_P = 2\sqrt{3}\) N
5

## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| For attempting to resolve forces in either direction | M1 | Correct number of relevant terms |
| $T_P \cos 60 = T_R \cos 30$ | A1 | |
| $T_P \sin 60 = T_R \sin 30 + 0.2g$ | A1 | |
| Attempt to solve simultaneously for either tension | M1 | From 2 equations, with correct number of relevant terms |
| $T_P = 3.46$ N and $T_R = 2$ N | A1 | Both correct. Allow $T_P = 2\sqrt{3}$ N |
| **Alternative method:** | | |
| $\frac{T_P}{\sin 60} = \frac{T_R}{\sin 150} = \frac{0.2g}{\sin 150}$ | M1 | Attempt one pair of Lami's equations. Correct angles |
| One pair correct | A1 | |
| Equations all correct | A1 | |
| Solve for $T_P$ or $T_R$ | M1 | From equations of the correct form |
| $T_P = 3.46$ N and $T_R = 2$ N | A1 | Both correct. Allow $T_P = 2\sqrt{3}$ N |
| | **5** | |

Show LaTeX source

3\\
\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-05_518_616_255_767}

A particle $Q$ of mass 0.2 kg is held in equilibrium by two light inextensible strings $P Q$ and $Q R . P$ is a fixed point on a vertical wall and $R$ is a fixed point on a horizontal floor. The angles which strings $P Q$ and $Q R$ make with the horizontal are $60 ^ { \circ }$ and $30 ^ { \circ }$ respectively (see diagram).

Find the tensions in the two strings.\\

\hfill \mbox{\textit{CAIE M1 2021 Q3 [5]}}

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Q1 3 Q2 6 Q3 5 Q4 6 Q5 9 Q6 11 Q7 10