Question 1 - A-Level Maths

CAIE M1 2016 June — Question 1 5 marks

Exam Board	CAIE
Module	M1 (Mechanics 1)
Year	2016
Session	June
Marks	5
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 finding the resultant. While it involves multiple forces and trigonometry (with sin α = 3/5 given, so cos α = 4/5), it's a standard M1 technique with no conceptual challenges—just systematic component resolution and Pythagoras/tan for the final answer. Slightly easier than average due to the nice ratio given.
Spec	3.03p Resultant forces: using vectors

1 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_373_591_260_776} Coplanar forces of magnitudes \(7 \mathrm {~N} , 6 \mathrm {~N}\) and 8 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the magnitude and direction of the resultant of the three forces.

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Question 1:

Answer	Marks	Guidance
Answer	Mark	Guidance
\(X = 7 - 8\cos\alpha - 6\sin\alpha = -3\)	M1	For resolving forces horizontally
\(X = 7 - 8 \times (4/5) - 6 \times (3/5) = -3\)	A1	Allow \(\alpha = 36.9\) used
\(Y = 8\sin\alpha - 6\cos\alpha = 0\)	M1	For resolving forces vertically
\(Y = 8 \times (3/5) - 6 \times (4/5) = 0\)	A1	Allow \(\alpha = 36.9\) used
Resultant force is 3N to the left	B1	Total: 5

## Question 1:

| Answer | Mark | Guidance |
|--------|------|----------|
| $X = 7 - 8\cos\alpha - 6\sin\alpha = -3$ | M1 | For resolving forces horizontally |
| $X = 7 - 8 \times (4/5) - 6 \times (3/5) = -3$ | A1 | Allow $\alpha = 36.9$ used |
| $Y = 8\sin\alpha - 6\cos\alpha = 0$ | M1 | For resolving forces vertically |
| $Y = 8 \times (3/5) - 6 \times (4/5) = 0$ | A1 | Allow $\alpha = 36.9$ used |
| Resultant force is 3N to the left | B1 | **Total: 5** |

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1\\
\includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_373_591_260_776}

Coplanar forces of magnitudes $7 \mathrm {~N} , 6 \mathrm {~N}$ and 8 N act at a point in the directions shown in the diagram. Given that $\sin \alpha = \frac { 3 } { 5 }$, find the magnitude and direction of the resultant of the three forces.

\hfill \mbox{\textit{CAIE M1 2016 Q1 [5]}}

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Q1 5 Q2 5 Q3 6 Q4 7 Q5 7 Q6 8 Q7 12