MHTCET 16 Apr 2026 Shift 2 Paper

Concept:Use the cosine rule to compute $\cos A$ and $\cos C$, then compare $\cos C$ with $\cos 2A$ to identify the relation between $C$ and $A$.Explanation:In $\triangle ABC$, the sides $a=4$, $b=5$, and $c=6$ are opposite to angles $A$, $B$, and $C$ respectively.Apply the cosine rule for angle $A$:$$\cos A = \frac{b^{2}+c^{2}-a^{2}}{2bc}$$Substitute the given side lengths:$$\cos A = \frac{5^{2}+6^{2}-4^{2}}{2 \cdot 5 \cdot 6} = \frac{25+36-16}{60} = \frac{45}{60} = \frac{3}{4}$$Now apply the cosine rule for angle $C$:$$\cos C = \frac{a^{2}+b^{2}-c^{2}}{2ab}$$Substitute the side lengths:$$\cos C = \frac{4^{2}+5^{2}-6^{2}}{2 \cdot 4 \cdot 5} = \frac{16+25-36}{40} = \frac{5}{40} = \frac{1}{8}$$Next, compute $\cos 2A$ using the double-angle formula:$$\cos 2A = 2\cos^{2}A - 1$$Since $\cos A = \frac{3}{4}$:$$\cos 2A = 2\left(\frac{3}{4}\right)^{2} - 1 = 2 \cdot \frac{9}{16} - 1 = \frac{18}{16} - 1 = \frac{2}{16} = \frac{1}{8}$$Thus, $\cos C = \cos 2A = \frac{1}{8}$.As both $C$ and $2A$ lie in $(0, \pi)$ where cosine is one-to-one, we get $C = 2A$.Answer:The angle $C$ equals $2A$, so the correct option is Option B ($2A$).