The point of contact of 3x + 4y + 7 = 0 and x² + y² - 4x - 6y -12 = 0 is

• (1, 1)

• (- 1, 1)

• (1, - 1)

• ( -1, - 1)

The equation x² - 7xy + 12y² = 0 represents a

• circle

• pair of parallel straight lines

• pair of perpendicular straight lines

• pair of non-perpendicular straight lines

If x is a complex root of the equation $\left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x}\\ \mathrm{x}& 1& \mathrm{x}\\ \mathrm{x}& \mathrm{x}& 1\end{array}\right| + \left|\begin{array}{ccc}1 - \mathrm{x}& 1& 1\\ 1& 1 - \mathrm{x}& 1\\ 1& 1& 1 - \mathrm{x}\end{array}\right| = 0$, then ${\mathrm{x}}^{2005} + \frac{1}{{\mathrm{x}}^{2005}}$ is equal to

• 1

• - 1

• i

• w

For f(x) = (x - 1)^2/3, the mean value theorem is applicable to f(x) in the interval

• [2, 4]

• [0, 2]

• [- 2, 2]

• any finite interval

The value of ${\sin }^{-1}\left(\frac{3}{5}\right) + {\sin }^{-1}\left(\frac{8}{17}\right)$ is

• ${\cos }^{-1}\left(\frac{36}{85}\right)$

• ${\sin }^{-1}\left(\frac{15}{85}\right)$

• ${\sin }^{-1}\left(\frac{36}{85}\right)$

• ${\cos }^{-1}\left(\frac{15}{85}\right)$

Let f(x) = a - (x - 3)^8/9, then maxima of f(x) is

• 3

• a - 3

• a

• None of these

The differential coefficient of f(log(x)) with respect to x, where f(x) = log(x) is

• $\frac{\mathrm{x}}{\log \left(\mathrm{x}\right)}$

• $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}$

• ${\left(\mathrm{xlog}\left(\mathrm{x}\right)\right)}^{- 1}$

• None of these

28.

If f(x) = $$\begin{gathered} \left[\left[\mathrm{x}\right] + \left[- \mathrm{x}\right], \mathrm{x} \ne 2 \\ \mathrm{\lambda }, \mathrm{x} = 2\right] \end{gathered}$$, then f is continuous at x = 2, provided $\mathrm{\lambda }$ is equal to

• 1

• 0

• - 1

• 2

If A² - A + I = 0, then matrix A^-1 will be equal to

• A + I

• I - A

• A

• A - I

The derivative of ${\sec }^{-1}\left(\frac{1}{2{\mathrm{x}}^2 - 1}\right)$ with respect to $\sqrt{1 - {\mathrm{x}}^2}$ at x = 1/2 will be

• 1/4

• ${\sec }^{-1}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)$

• 4

• 0