The line joining two points A(2, 0), B(3, 1) is rotated about A in anti-clockwise direction through an angle of 15°. The equation of the line in the now position, is

• √3x - y - 2√3 = 0

• x - 3√y - 2 = 0

• √3x + y - 2√3 = 0

• x - √3y - 2 = 0

The line 2x + $\sqrt{6}$y = 2 is a tanent to the curve x² - 2y² = 4. The point of contact is

• $\left(4, - \sqrt{6}\right)$

• $\left(7, - 2\sqrt{6}\right)$

• (2, 3)

• $\left(\sqrt{6}, 1\right)$

The number of integral points (integral points means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is

• 133

• 190

• 233

• 105

If a tangent having slope of - $\frac{4}{3}$ to the ellipse $\frac{{\mathrm{x}}^2}{18} + \frac{{\mathrm{y}}^2}{32} = 1$ intersects the major and minor axes in points A and B respectively, then the area of $∆\mathrm{OAB}$ is equal to (O is centre of the ellipse)

• 12 sq units

• 48 sq units

• 64 sq units

• 24 sq units

The locus of mid-points of tangents intercepted between the axes of ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ will be

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 1$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 2$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 3$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 4$

If PQ is a double ordinate of hyperbola (x²/a²) - (y²/b²) = 1 such that OPQ is a equilateral triangle, O being the centre of the hyperbola, then the eccentricity 'e' of the hyperbola satisfies

• 1 < e < 2/√3

• e = 2/√3

• e = √3/2

• e > 2/√3

The sides AB, BC and CA of a $∆\mathrm{ABC}$ have respectively 3, 4 and 5 points lying on them. number of triangles that can be constructed using these points as vertices is

• 205

• 220

• 210

• None of these

18.

In the expansion of $\frac{\mathrm{a} + \mathrm{bx}}{{\mathrm{e}}^{\mathrm{x}}}$, the coefficient of x^r is

• $\frac{\mathrm{a} - \mathrm{b}}{\mathrm{r}!}$

• $\frac{\mathrm{a} -\mathrm{br}}{\mathrm{r}!}$

• ${\left(- 1\right)}^{\mathrm{r}}\frac{\mathrm{a} - \mathrm{br}}{\mathrm{r}!}$

• None of the above

If n = (1999)!, Then $\sum _{\mathrm{x} = 1}^{1999}{\log }_{\mathrm{n}}\left(\mathrm{x}\right)$ is equal to

• 1

• 0

• $\sqrt[1999]{1999}$

• - 1

P is a fixed point (a, a, a) on a line through the origin is equally inclined to the axes, then any plane through P perpendicular to Op, makes intercepts on the axes, the sum of whose reciprocal is equal to

• a

• $\frac{3}{2\mathrm{a}}$

• $\frac{3\mathrm{a}}{2}$

• None of these