The GATE Grind

GATE 2014 CS (CS1) – Question 16

Engineering Mathematics · Calculus · 1 mark · Multiple choice

Let $f(\theta)=\begin{vmatrix}\sin\theta&\cos\theta&\tan\theta\\\sin(\pi/6)&\cos(\pi/6)&\tan(\pi/6)\\\sin(\pi/3)&\cos(\pi/3)&\tan(\pi/3)\end{vmatrix}$ for $\theta\in[\pi/6,\pi/3]$. Which claims are TRUE? I: Some $\theta\in(\pi/6,\pi/3)$ has $f^\prime(\theta)=0$. II: Some θ in that interval has $f^\prime(\theta)\ne0$.

  1. I only
  2. II only
  3. Both I and II
  4. Neither I nor II

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) Both I and II

Explanation

At θ=π/6 andπ/3, two determinant rows coincide, so both endpoint values are zero. Rolle’s theorem guarantees an interior zero derivative. The determinant is not identically zero between these endpoints, so the mean-value theorem also guarantees an interior nonzero derivative. Thus **both I and II, C**.