MCQ
$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{{n^3}}}\left[ {{1^2}\sin \frac{1}{n} + {2^2}\sin \frac{2}{n} + {3^2}\sin \frac{3}{n} + ....+{n^2}\sin \frac{n}{n}} \right]$ =
- A$cos1 + 2sin1$
- B$2sin1 -2$
- C$cos1 -2sin1 -2$
- D$cos1 + 2sin1 -2$
$\left[x^{2}(-\cos x)-2 x(-\sin x)+2 \cos x\right]_{0}^{1}$
$=(-\cos 1+2 \sin 1+2 \cos 1)-2$
$=\cos 1+2 \sin 1-2$
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$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
પ્રમાણે વ્યાખ્યાયિત છે, જ્યાં $a, b, c \in R$ અને $[t]$ એ $t$ અથવા તેથી નાનો મહત્તમ પૂર્ણક દર્શાવે છે. તો નીચેના પૈકી કયું વિધાન સાયું છે $?$