The value of

$1000\left[\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ... + \frac{1}{999 \times 1000}\right]$

• 1000

• 999

• 1001

• $\frac{1}{999}$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$  B are the roots of the quadratic equation is, x² + ax + b = 0, (b $\ne$ 0), then the quadratic equation whose roots are $\mathrm{\alpha } - \frac{1}{\mathrm{\beta }}, \mathrm{\beta } - \frac{1}{\mathrm{\alpha }}$, is

• ax² + a(b - 1)x + (a - 1)² = 0

• bx² + a(b - 1)x + (b - 1)² = 0

• x² + ax + b = 0

• abx² + bx + a = 0

If the distance between the foci of an ellipse is equal to the length of the latusrectum, then its eccentricity is

• $\frac{1}{4}\left(\sqrt{5} - 1\right)$

• $\frac{1}{2}\left(\sqrt{5} + 1\right)$

• $\frac{1}{2}\left(\sqrt{5} - 1\right)$

• $\frac{1}{4}\left(\sqrt{5} + 1\right)$

The equation of the circle passing through the point (1, 1) and the points of intersection of x² + y² -  6x - 8 = 0 and x² + y² - 6 = 0 is 

• x² + y² + 3x - 5 = 0

• x² + y² - 4x + 2 = 0

• x² + y² + 6x - 4 = 0

• x² + y² - 4y - 2 = 0

The number oflines which pass through the point (2, - 3) and are at a distance 8 from the point (- 1, 2) is

• infinite

• 4

• 2

• 0

Six positive numbers are in GP, such that their product is 1000. If the fourth term is 1, then the last term is

• 1000

• 100

• $\frac{1}{100}$

• $\frac{1}{1000}$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the quadratic equation ax² + bx + c = 0 and 3b² = 16ac, then

• $\mathrm{\alpha } = 4\mathrm{\beta } \mathrm{or} \mathrm{\beta } = 4\mathrm{\alpha }$

• $\mathrm{\alpha } = - 4\mathrm{\beta } \mathrm{or} \mathrm{\beta } = - 4\mathrm{\alpha }$

• $\mathrm{\alpha } = 3\mathrm{\beta } \mathrm{or} \mathrm{\beta } = 3\mathrm{\alpha }$

• $\mathrm{\alpha } = - 3\mathrm{\beta } \mathrm{or} \mathrm{\beta } = - 3\mathrm{\alpha }$

The limits of $\sum _{\mathrm{n} = 1}^{1000}{\left(- 1\right)}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}} \mathrm{as} \mathrm{x} \to \infty$

• does not exist

• exists and equals to 0

• exists and approaches to $+ \infty$

• exists and approaches to $- \infty$

39.

Let $\mathrm{f}\left(\mathrm{\theta }\right) = \left(1 + {\sin }^2\left(\mathrm{\theta }\right)\right)\left(2 - {\sin }^2\left(\mathrm{\theta }\right)\right)$. Then, for all values of $\mathrm{\theta }$

• $\mathrm{f}\left(\mathrm{\theta }\right) > \frac{9}{4}$

• $\mathrm{f}\left(\mathrm{\theta }\right) < 2$

• $\mathrm{f}\left(\mathrm{\theta }\right) > \frac{11}{4}$

• $2 \le \mathrm{f}\left(\mathrm{\theta }\right) \le \frac{9}{4}$

If f(x) = e^x(x - 2)², then

• f is increasing in (- $\infty$, 0) and (2, $\infty$) and decreasing in (0, 2).

• f is increasing in (- $\infty , 0$) and decreasing in ($0, \infty$)

• f is increasing in (2, $\infty$) and decreasing in (- $\infty$, 0).

• f is increasing in (0, 2) and decreasing in ($- \infty , 0$) and (2, $\infty$)