MCQ
The least positive integer $n$ for which $\sqrt[3]{n+1}-\sqrt[3]{n} < \frac{1}{12}$ is
- A$6$
- B$7$
- ✓$8$
- D$9$
We have,
$\sqrt[3]{n+1}-\sqrt[3]{n} < \frac{1}{12}$
$\sqrt[3]{n+1} < \sqrt[3]{n}+\frac{1}{12}$
Cubing both sides, we get
$n+1 < n+3(n)^{23} \times \frac{1}{12}+3 \sqrt[3]{n} \times \frac{1}{144}+\frac{1}{1728}$
$\Rightarrow \quad 1 < \frac{3 n^{1 / 3}}{}-\left(n^{1 / 3}+\frac{1}{12}\right)+\frac{1}{1728}$
$\Rightarrow \quad n^{1 / 3}\left(n^{1 / 3}+\frac{1}{12}\right) > 1-\frac{1}{1728}$
$\Rightarrow \quad n^{1 / 3}\left(n^{1 / 3}+\frac{1}{12}\right) > \frac{1727}{432}$
Put $n=8$ only possible least positive integers.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$K _1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _1$;
$E_n$ : ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_{n}^2}=1$ of largest area inscribed in $R_{n-1}, n>1$;
$R _{ n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _{ n }, n >1$.
Then which of the following options is/are correct?
$(1)$ The eccentricities of $E _{18}$ and $E _{19}$ are NOT equal
$(2)$ The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
$(3)$ The length of latus rectum of $E_Q$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N\left(\right.$ area of $\left.R_2\right)<24$, for each positive integer $N$