Q192 / 270 · NDA 2022 (Phase 2) · Mathematics

Paragraph: $A B C$ is a triangular plot with $A B=16 \mathrm{~m}, B C=10 \mathrm{~m}$ and $C A=10 \mathrm{~m}$. A lamp post is situated at the middle point of the side $A B$. The lamp post subtends an angle $45^{\circ}$ at the vertex $B$.
Question: What is $\cos A+\cos B+\cos C$ equal to ?

  • A
    $1$
  • B
    $\frac{41}{25}$
  • C
    $\frac{37}{25}$
  • Correct answer
    D
    $\frac{33}{25}$

Solution

$\begin{aligned} & \cos A+\cos B+\cos C \\ & \frac{b^2+c^2-a^2}{2 b c}+\frac{a^2+c^2-b^2}{2 a c}+\frac{a^2+b^2-c^2}{2 a b} \\ & =\frac{100+256-100}{2 \times 160}+\frac{100+256-100}{2 \times 160} \\ & +\frac{100+100-256}{200} \\ & =\frac{256}{160}-\frac{56}{200}=\frac{1280-224}{800}=\frac{1056}{800}=\frac{33}{25}\end{aligned}$

Hence, the correct answer is option: 'D'.

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