Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Are the square roots of all positive integers irrational? If not, give an example of the square of a number which is a rational number.

Which of the following hold good? If $n$ is a positive integer, then

A $n^{n}>1.3 \times 5 \times \ldots \times (2n-1)$

B $2.4 \times 6 \times \ldots \times 2n < (n+1)^{n}$

C $(n!)^{3} < n^{n} \left(\frac{n+1}{2}\right)^{2n}$

D $\left[1^{r}+2^{r}+3^{r}+\ldots+n^{r}\right]^{n} > n^{n} \times (n!)^{r}$

Radha decorated the door of her house with garlands on the occasion of Diwali. Each garland forms the shape of a parabola.

Evaluate the number of zeroes of a quadratic polynomial: A. more than 2 B. at most 2 C. less than 2 D. equal to 1

The ellipse $E_{1}: \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0,4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E_{2}$ is:

A) $\frac{\sqrt{2}}{2}$

B) $\frac{\sqrt{3}}{2}$

C) $\frac{1}{2}$

D) $\frac{3}{4}$

State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.

(iii) Every real number is an irrational number.