Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f\left(x\right)=\left{\begin{matrix}{b}^{2}\text{sin}\left(\frac{\pi }{2}\left[\frac{\pi }{2}\left(\text{cos}x+\text{sin}x\right)\text{cos}x\right]\right),\text{ }x 0 \ \text{ }a\text{ },\text{ }x=0\end{matrix}\right.$ is continuous at $x = 0$, then $a^2+b^2$ is equal to [JEE Main 2026, 24 Jan (Shift 2)]

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YESPYQ · Accepted Answer

Correct answer: d) $\frac{3}{4}$. Given $f(x)$ is continuous at $x=0$ $f(0)=a$ $\mathrm{RHL}=\lim _{x\to {0}^{+}}\frac{\sin x(1−\cos x)}{{x}^{3}}\ =\lim _{x\to {0}^{+}}\left(\frac{\sin x}{x}\right)\left(\frac{1-\cos x}{{x}^{2}}\right)\ =\frac{1}{2}$ $\mathrm{LHL}=\lim _{x\to {0}^{−}}\left({b}^{2}\sin \left(\frac{\pi }{2}\left[\frac{\pi }{2}(\sin x+\cos x)\cos x\right]\right)\right)\ ={b}^{2}$ $∴a=\frac{1}{2}$ ${b}^{2}=\frac{1}{2}$ So, $\left({a}^{2}+{b}^{2}\right)=\frac{1}{4}+\frac{1}{2}=\frac{3}{4}$

Let \([t]\) denote the greatest integer less than or equal to \(t\). If the function \(f\left(x\right)=\left{\begin{matr…

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