MCQ
The greatest integer less than or equal to $\int_1^2 \log _2\left(x^3+1\right) d x+\int_1^{\log 2}\left(2^x-1\right)^{\frac{1}{3}} d x$ is. . . . .
- A$2$
- B$3$
- C$4$
- ✓$5$
$x^3+1=2^y \Rightarrow x=\left(2^y-1\right)^{1 / 3}=f^{-1}(y)$
$f^{-1}(x)=\left(2^x-1\right)^{1 / 3}$
$=\int_1^2 \log _2\left(x^3+1\right) d x+\int_1^{\log _2 9}\left(2^x-1\right)^{1 / 3} d x$
$=\int_1^2 f(x) d x+\int_1^{\log _2 9} f^{-1}(x) d x=2 \log _2 9-1$
$=8<9<2^{7 / 2} \Rightarrow 3<\log _2 9<\frac{7}{2}$
$=5<2 \log _2 9-1<6$
${\left[2 \log _2 9-1\right]=5}$
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$-x+2 y+5 z=b_1$
$2 x-4 y+3 z=b_2$
$x-2 y+2 z=b_3$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each$\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$ $\in$ $S$ ?
$(A)$ $x+2 y+3 z=b_1, 4 y+5 z=b_2$ and $x+2 y+6 z=b_3$
$(B)$ $x+y+3 z=b_1, 5 x+2 y+6 z=b_2$ and $-2 x-y-3 z=b_3$
$(C)$ $-x+2 y-5 z=b_1, 2 x-4 y+10 z=b_2$ and $x-2 y+5 z=b_3$
$(D)$ $x+2 y+5 z=b_1, 2 x+3 z=b_2$ and $x+4 y-5 z=b_3$