Question

If the function \(f(x)=\left\{\begin{array}{cl}\frac{\cos a x-\cos b x}{\cos c x-\cos b x} & , \text { if } x \neq 0 \\ -1 & , \text { if } x=0\end{array}\right.\) is continuous at \(x=0\), then \(\mathrm{a}^2, \mathrm{~b}^2, \mathrm{c}^2\) are in

• A. Geometric progression
• B. Arithmetic progression
• C. Harmonic progression
• D. Arithmetico-Geometric progression

Hint:

For continuity, the limit as x tends to 0 must equal -1; use the cosine expansion for small x.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Continuity at 0 needs \(\lim_{x\to0}f(x)=f(0)=-1\).

Step 2: Expand
For small x, \(\cos\theta x\approx1-\dfrac{\theta^2x^2}{2}\). So \(\cos ax-\cos bx\approx\dfrac{(b^2-a^2)x^2}{2}\) and \(\cos cx-\cos bx\approx\dfrac{(b^2-c^2)x^2}{2}\).

Step 3: Limit
\(\lim f(x)=\dfrac{b^2-a^2}{b^2-c^2}=-1\), so \(b^2-a^2=c^2-b^2\).

Step 4: Compare the options
This gives \(2b^2=a^2+c^2\), so \(a^2,b^2,c^2\) are in arithmetic progression. Option (B) is correct.

Final Answer:
The squares a^2, b^2, c^2 are in arithmetic progression. \[ \boxed{\text{(B) }\text{Arithmetic progression}} \]