JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let [t] denote the greatest integer less than or equal to t. If the function
\(f(x)=\left\{\begin{array}{cl}b^2 \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x) \cos x\right]\right), & x<0 \\ \frac{\sin x-\frac{1}{2} \sin 2 x}{x^3} & , x>0 \\ a & , x=0\end{array}\right.\)
is continuous at \(x =0\), then \(a ^2+ b ^2\) is equal to
- A \(\frac{5}{8} \)
- B \(\frac{9}{16} \)
- C \(\frac{3}{4} \)
- D \(\frac{1}{2} \)
Answer & Solution
Correct Answer
(C) \(\frac{3}{4} \)
Step-by-step Solution
Detailed explanation
\(f(0)=a\) \(R H L=\lim _{x \rightarrow 0^{+}} \frac{\sin x(1-\cos x)}{x^3}=\frac{1}{2}\) \(L H L=\lim _{x \rightarrow 0^{-}}\left(b^2 \sin \frac{\pi}{2}\left[\frac{\pi}{2}(\sin x+\cos x) \cos x\right]\right)=b^2\) \(\therefore a=\frac{1}{2}~\&~b^2=\frac{1}{2}\)…
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