In the right-angled triangle $\Delta ABC$, angle $C$ is a right angle. $AB=13$ cm, $BC=5$ cm

Question

In the right-angled triangle $\Delta ABC$, angle $C$ is a right angle. $AB=13$ cm, $BC=5$ cm and $X$ is a point on $BC$ such that $\angle BAX$ is $\alpha$ radians. Find the exact value of $\cos \angle BAC$.[1] By considering triangle $\Delta AXC$, or otherwise, show that $AX = \frac{156}{12 \cos \alpha + 5 \sin \alpha}$[3] Given that $\alpha$ is a sufficiently small angle, show that $AX \approx 13 + p\alpha + q\alpha^2$, where $p$ and $q$ are constants to be determined exactly. [4]

Accepted Answer

(a) $\dfrac{12}{13}$ (b) $AX=\dfrac{156}{12\cos\alpha+5\sin\alpha}$ (c) $p=-\dfrac{65}{12},\ q=\dfrac{1261}{144}$

2025 EJC P1 Q3

2025 EJC P1 Q3

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