Let denote the greatest integer less than or equal to . If the function is continuous at , then is equal to [2026]

Question

Let $[t]$ denote the greatest integer less than or equal to $t$ . If the function

$f (x) =$ ${\begin{matrix} b^{2} \sin (\frac{π}{2} [\frac{π}{2} (\cos x + \sin x) \cos x]), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2 x}{x^{3}}, & x > 0 \\ a, & x = 0 \end{matrix}$

is continuous at $x = 0$ , then $a^{2} + b^{2}$ is equal to [2026]

Smart Achievers · Accepted Answer

(4)

$f (0) = a$

$RHL = \lim_{x \to 0^{+}} \frac{\sin x (1 - \cos x)}{x^{3}} = \frac{1}{2}$

$LHL = \lim_{x \to 0^{-}} (b^{2} \sin \frac{π}{2} [\frac{π}{2} (\sin x + \cos x) \cos x]) = b^{2}$

$∴$ $a = \frac{1}{2}, b^{2} = \frac{1}{2}$

$so (a^{2} + b^{2}) = \frac{1}{4} + \frac{1}{2} = \frac{3}{4}$

Solution

1 $\frac{9}{16}$

2 $\frac{1}{2}$

3 $\frac{5}{8}$

4 $\frac{3}{4}$

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Solution

Q. Let [t] denote the greatest integer less than or equal to t. If the function f(x)={b2sin(π2[π2(cosx+sinx)cosx]),x<0sinx-12sin2xx3,x>0a,x=0 is continuous at x=0, then a2+b2 is equal to [2026]

1 916

2 12

3 58

4 34

Ans. (4) f(0)=a RHL=limx→0+sinx(1-cosx)x3=12 LHL=limx→0-(b2sinπ2[π2(sinx+cosx)cosx])=b2 ∴ a=12, b2=12 so (a2+b2)=14+12=34

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1 $\frac{9}{16}$

2 $\frac{1}{2}$

3 $\frac{5}{8}$

4 $\frac{3}{4}$