4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude \(X \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of \(X\) is 23.1 , correct to 3 significant figures, and find the greatest possible value of \(X\).

## Question 4:

| Answer | Mark | Guidance |
|--------|------|----------|
| $R = 15g\cos20°$ | B1 | 140.95 |
| $F = \mu R = 0.2 \times 15g\cos20°$ | B1 | 28.19 |
| | M1 | For resolving parallel to the plane ($F$ acting up plane) |
| $X + 0.2 \times 15g\cos20° = 15g\sin20°$ | A1 | |
| Least value of $X$ is 23.1 | A1 | AG |
| $[X = 15g\sin20° + 0.2 \times 15g\cos20°]$ | M1 | For resolving parallel to the plane ($F$ acting down plane) |
| Greatest value of $X$ is 79.5 | A1 | **[7 marks]** |

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4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of $20 ^ { \circ }$ to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude $X \mathrm {~N}$ acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of $X$ is 23.1 , correct to 3 significant figures, and find the greatest possible value of $X$.

\hfill \mbox{\textit{CAIE M1 2016 Q4 [7]}}