Question 3 - A-Level Maths

CAIE M1 2019 November — Question 3 6 marks

Exam Board	CAIE
Module	M1 (Mechanics 1)
Year	2019
Session	November
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Vectors Introduction & 2D
Type	Resultant of three coplanar forces
Difficulty	Moderate -0.3 This is a straightforward mechanics problem requiring resolution of forces into components and basic trigonometry. Students need to resolve three forces (with clearly given angles) into horizontal and vertical components, sum them, then find magnitude and direction of the resultant using Pythagoras and tan⁻¹. It's a standard M1 exercise with no conceptual difficulty, though the arithmetic with multiple angles requires care. Slightly easier than average A-level due to being a direct application of a single technique.
Spec	3.03p Resultant forces: using vectors

3 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-05_479_647_264_749} Three coplanar forces of magnitudes \(50 \mathrm {~N} , 60 \mathrm {~N}\) and 100 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of these forces are shown in the diagram. Find the values of \(R\) and \(\alpha\).

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Question 3:

Answer	Marks	Guidance
Answer	Marks	Guidance
Resolving horizontally or vertically	M1
\(50\cos20 + 60 - 100\sin30\) \((= 56.984\ldots)\)	A1
\(100\cos30 - 50\sin20\) \((= 69.501\ldots)\)	A1
\(R = \sqrt{(56.984\ldots)^2 + (69.501\ldots)^2}\) or \(\alpha = \tan^{-1}\left(\dfrac{56.984\ldots}{69.501\ldots}\right)\)	M1	Method to find either \(R\) or \(\alpha\)
\(R = 89.9\ (89.876\ldots)\)	A1
\(\alpha = 39.3\ (39.348\ldots)\)	A1
Total: 6

**Question 3:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Resolving horizontally or vertically | M1 | |
| $50\cos20 + 60 - 100\sin30$ $(= 56.984\ldots)$ | A1 | |
| $100\cos30 - 50\sin20$ $(= 69.501\ldots)$ | A1 | |
| $R = \sqrt{(56.984\ldots)^2 + (69.501\ldots)^2}$ or $\alpha = \tan^{-1}\left(\dfrac{56.984\ldots}{69.501\ldots}\right)$ | M1 | Method to find either $R$ or $\alpha$ |
| $R = 89.9\ (89.876\ldots)$ | A1 | |
| $\alpha = 39.3\ (39.348\ldots)$ | A1 | |
| **Total: 6** | | |

Show LaTeX source

3\\
\includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-05_479_647_264_749}

Three coplanar forces of magnitudes $50 \mathrm {~N} , 60 \mathrm {~N}$ and 100 N act at a point. The resultant of the forces has magnitude $R \mathrm {~N}$. The directions of these forces are shown in the diagram. Find the values of $R$ and $\alpha$.\\

\hfill \mbox{\textit{CAIE M1 2019 Q3 [6]}}

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