MCQ
The numbers $(\sqrt 2 + 1),\;1,\;(\sqrt 2 - 1)$ will be in
- A$A.P.$
- ✓$G.P.$
- C$H.P.$
- DNone of these
$\therefore $ ${(1)^2} = (\sqrt 2 + 1)(\sqrt 2 - 1) = {(\sqrt 2 )^2} - {(1)^2} = 2 - 1 = 1$.
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$f(0)=1 \text { and } \int_0^{\frac{\pi}{3}} f( t ) dt =0$
Then which of the following statements is (are) $TRUE$?
$(A)$ The equation $f( x )-3 \cos 3 x =0$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(B)$ The equation $f( x )-3 \sin 3 x =-\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(C)$ $\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) d t}{1- e ^{x^2}}=-1$
$(D)$ $\lim _{ x \rightarrow 0} \frac{\sin x \int_0^{ x } f( t ) dt }{ x ^2}=-1$