31.

The straight line x + y - 1 = 0 meets the circle x² + y² - 6x - 8y = 0 at A and B. Then the equation of the circle of which AB is a diameter is

• x² + y² - 2y - 6 = 0

• x² + y² + 2y - 6 = 0

• 2(x² + y²) + 2y - 6

• 3(x² + y²) + 2y - 6 = 0

If t₁ and t₂ be the parameters of the end points of a focal chord for the parabola y² = 4ax, then which one is true?

• t₁t₂ = 1

• $\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2} = 1$

• t₁t₂ = - 1

• t₁ + t₂ = - 1

S and T are the foci of an ellipse and B is end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

For different values of a, the locus of the point of intersection of the two straight lines $\sqrt{3}\mathrm{x} - \mathrm{y} - 4\sqrt{3}\mathrm{\alpha } = 0$ and $\sqrt{3}\mathrm{\alpha x} + \mathrm{\alpha y} - 4\sqrt{3} = 0$

• a hyperbola with eccentricity 2

• an ellipse with eccentricity $\sqrt{\frac{2}{3}}$

• a hyperbola with eccentricity $\sqrt{\frac{19}{16}}$

• an ellipse with eccentricity $\frac{3}{4}$

The value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) - 1}{{\mathrm{x}}^2}$

• 1

• $\frac{1}{2}$

• $\frac{- 1}{2}$

• 0

The value of $\underset{\mathrm{x}\to 0}{\lim }{\left(\frac{1 + 5{\mathrm{x}}^2}{1 + 3{\mathrm{x}}^2}\right)}^{\frac{1}{{\mathrm{x}}^2}}$

• e²

• e

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

If f(5)=7 and f'(5)=7 then $\underset{\mathrm{x}\to 5}{\lim }\frac{\mathrm{xf}\left(5\right) - 5\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x} - 5}$ is given by

• 35

• - 35

• 28

• - 28

The value of f(0) so that the function $\frac{1 - \cos \left(1 - \cos \left(\mathrm{x}\right)\right)}{{\mathrm{x}}^4}$ is continuous everywhere is

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

• $\frac{1}{8}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left|\mathrm{x}\right|}{\mathrm{x}}$ is equal to 

• 1

• 0

• positive infinity

• does not exist

The coordinates of the point on the curve y = x² - 3x + 2 where the tangent is perpendicular to the straight line y = x are 

• (0, 2)

• (1, 0)

• (- 1, 6)

• (2, - 2)