A straight line through the point A (3, 4) is such that its intercept between the axes is bisected at A. Its equation is

• 3x - 4y + 7 = 0

• 4x + 3y = 24

• 3x + 4y = 25

• x + y = 7

The tangent at (1, 7) to the curve x² = y - 6 touches the circle x² + y2 + 16x + 12y + c = 0 at

• (6, 7)

• (- 6, 7)

• (6, - 7)

• (- 6, - 7)

The equation of straight line through the intersection of the lines x - 2y = 1 and x + 3y = 2 and parallel to 3x + 4y = 0 is

• 3x + 4y + 5 = 0

• 3x + 4y - 10 = 0

• 3x + 4y - 5 = 0

• 3x + 4y + 6 = 0

The eccentricity of the ellipse, which meets the straight line $\frac{\mathrm{x}}{7} + \frac{\mathrm{y}}{2} = 1$ on the axis of x and the staraight line $\frac{\mathrm{x}}{3} - \frac{\mathrm{y}}{5} = 1$ on the axis of y and whose axes lie along the axes of coordinate, is

• $\frac{3\sqrt{2}}{7}$

• $\frac{2\sqrt{6}}{7}$

• $\frac{\sqrt{3}}{7}$

• None of the above

15.

If $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \left(\mathrm{a} > \mathrm{b}\right)$ and x² - y² = c² cut at right angles, then

• a² + b² = 2c²

• b² - a² = 2c²

• a² - b² = 2c²

• a²b² = 2c²

The equation of the conic with focus at (1, - 1) directrix along x - y + 1 = 0 and with eccentricity $\sqrt{2}$, is

• x² - y² = 1

• xy = 1

• 2xy - 4x + 4y + 1 = 0

• 2xy + 4x - 4y - 1 = 0

There are 5 letters and 5 different envelopes. The number of ways in which all the letters can be put in wrong envelope, is

• 119

• 44

• 59

• 40

The sum of the series

$1 + \frac{1^2 + 2^2}{2!} + \frac{1^2 + 2^2 + 3^2}{3!} + \frac{1^2 + 2^2 + 3^2 + 4^2}{4!} + ...$ is

• 3e

• $\frac{17}{6}$e

• $\frac{13}{6}$e

• $\frac{19}{6}$e

The coefficient of xⁿ in the expansion of ${\log }_{\mathrm{a}}\left(1 + \mathrm{x}\right)$ is

• $\frac{{\left(- 1\right)}^{\mathrm{n} - 1}}{\mathrm{n}}$

• $\frac{{\left(- 1\right)}^{\mathrm{n} - 1}}{\mathrm{n}}{\log }_{\mathrm{a}}\left(\mathrm{e}\right)$

• $\frac{{\left(- 1\right)}^{\mathrm{n} - 1}}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\mathrm{a}\right)$

• $\frac{{\left(- 1\right)}^{\mathrm{n}}}{\mathrm{n}}{\log }_{\mathrm{a}}\left(\mathrm{e}\right)$

The value of $\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{\pi }}{2} - {\tan }^{-1}\left(\mathrm{x}\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$ is

• 0

• 1

• - 1

• e