MCQ
For the ellipse $\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{28}} = 1$, the eccentricity is
- ✓$\frac{3}{4}$
- B$\frac{4}{3}$
- C$\frac{2}{{\sqrt 7 }}$
- D$1\over3$
${e^2} = \frac{{36}}{{64}}$
==> $e = \frac{3}{4}$.
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| Column $I$ | Column $II$ |
| $(A)$ $\int_{-1}^1 \frac{\mathrm{dx}}{1+\mathrm{x}^2}$ | $(p)$ $\frac{1}{2} \log \left(\frac{2}{3}\right)$ |
| $(B)$ $\int_0^1 \frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}^2}}$ | $(q)$ $2 \log \left(\frac{2}{3}\right)$ |
| $(C)$ $\int_2^3 \frac{\mathrm{dx}}{1-\mathrm{x}^2}$ | $(r)$ $\frac{\pi}{3}$ |
| $(D)$ $\int_1^2 \frac{d x}{x \sqrt{x^2-1}}$ | $(s)$ $\frac{\pi}{2}$ |
$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$