1 \includegraphics[max width=\textwidth, alt={}, center]{27d3ee31-7c6e-4451-9c3d-aa4cfc0fdb22-2_744_504_255_824} A uniform ladder \(A B\), of length \(3 a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(A C = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(D C\) is in the same vertical plane as the ladder \(A B\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \theta = 2 \tan \alpha\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\).

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $R_B = T\cos\alpha$ | M1 A1 | Resolve horizontally |
| $R_A = W + T\sin\alpha$ | M1 A1 | Resolve vertically |
| $R_B 3a\sin\theta = W(3a/2)\cos\theta + Ta(\sin\alpha\cos\theta + \cos\alpha\sin\theta)$ *or* $+Ta\sin(\alpha+\theta)$ *or* $+T3a\cos\theta\sin\alpha$ | M1 A1 | Moments about $A$ ($a$ may be omitted) |
| $R_A 3a\cos\theta = W(3a/2)\cos\theta + T2a(\sin\alpha\cos\theta + \cos\alpha\sin\theta)$ *or* $+T2a\sin(\alpha+\theta)$ *or* $+T3a\sin\theta\cos\alpha$ | (M1 A1) | Moments about $B$ |
| $R_A a\cos\theta + W(a/2)\cos\theta = R_B 2a\sin\theta$ | (M1 A1) | Moments about $C$ |
| $R_A 3a\cos\theta - W(3a/2)\cos\theta = R_B 3a\sin\theta$ | (M1 A1) | Moments about $D$ |
| $T = W/2\sin\alpha$ *or* $\frac{1}{2}W\operatorname{cosec}\alpha$ | B1 | Solve for $T$, $R_A$, $R_B$ (AEF in $W$ and $\alpha$) |
| $R_A = 3W/2$ | B1 | |
| $R_B = W/2\tan\alpha$ *or* $\frac{1}{2}W\cot\alpha$ | B1 | Total: 9 marks |

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\includegraphics[max width=\textwidth, alt={}, center]{27d3ee31-7c6e-4451-9c3d-aa4cfc0fdb22-2_744_504_255_824}

A uniform ladder $A B$, of length $3 a$ and weight $W$, rests with the end $A$ in contact with smooth horizontal ground and the end $B$ against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point $C$, where $A C = a$. The other end of the rope is fixed to the point $D$ at the base of the wall and the rope $D C$ is in the same vertical plane as the ladder $A B$. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle $\theta$ with the horizontal and the rope making an angle $\alpha$ with the horizontal (see diagram). It is given that $\tan \theta = 2 \tan \alpha$. Find, in terms of $W$ and $\alpha$, the tension in the rope and the magnitudes of the forces acting on the ladder at $A$ and at $B$.

\hfill \mbox{\textit{CAIE FP2 2015 Q1 [9]}}