NDA 2 2025 Maths, question 56

Topic: Analytical geometry

Passage · Q55 to Q56

Let $A(1,-1,0)$ , $B(-2,1,8)$ and $C(-1,2,7)$ are three consecutive vertices of a parallelogram $ABCD$ .

Q56

If angle $BCD$ is $\theta$ , then what is $\cos ^{2}\theta$ equal to?

(a) $26/77$
(b) $27/77$ Correct answer
(c) $82/237$
(d) $83/237$

Solution.

Answer: (b) $27/77$

At C use CB=B−C=(−1,−1,1) and CD=D−C=(3,−2,−8).
Their dot product is −9 and their squared lengths are 3 and 77.
Therefore cos²θ=81/(3×77)=27/77.
Squaring removes the negative sign of the obtuse interior angle.