Value of the determinant $\left|\begin{array}{ccc}1 +\hspace{0.17em}\mathrm{a}& 1\mathrm{a}& 1\\ 1& 1 + \mathrm{b}& 1\\ 1& 1& 1 +\hspace{0.17em}\mathrm{c}\end{array}\right|$ is :

• 1 + abc + ab + bc + ca

• abc

• 4abc

• $\mathrm{abc}\left(1 + \frac{1}{\mathrm{a}} + \frac{1}{\mathrm{b}} + \frac{1}{\mathrm{c}}\right)$

If the value of the determinant $\left|\begin{array}{ccc}\mathrm{x} + 1& 1& 1\\ 2& \mathrm{x} + 2& 2\\ 3& 3& \mathrm{x} + 3\end{array}\right|$ is equal to zero, then x is :

• 0 and - 6

• 0 and 6

• 6

• - 6

The value of a for which the matrix A = $\left[\begin{array}{cc}\mathrm{a}& 2\\ 2& 4\end{array}\right]$ is singular :

• $\mathrm{a} \ne 1$

• a = 1

• a = 0

• a = - 1

If A = $\left[\begin{array}{cc}2& - 1\\ - 1& 2\end{array}\right]$ and I is the unit matrix of order two, then A² is equal to :

• 4A - 3I

• 3A - 4I

• A - I

• A + I

If A and B are two square matrices of the same order, then (A - B)² :

• A² - AB - BA + B²

• A² - 2AB + B²

• A² - 2BA + B²

• A² - B²

If $\mathrm{P} = \left[\begin{array}{ccc}\mathrm{i}& 0& - \mathrm{i}\\ 0& - \mathrm{i}& \mathrm{i}\\ - \mathrm{i}& \mathrm{i}& 0\end{array}\right] \mathrm{and} \mathrm{Q} = \left[\begin{array}{cc}- \mathrm{i}& \mathrm{i}\\ 0& 0\\ \mathrm{i}& - \mathrm{i}\end{array}\right]$, then PQ is equal to

• $\left[\begin{array}{cc}- 2& 2\\ 1& - 1\\ 1& - 1\end{array}\right]$

• $\left[\begin{array}{cc}2& - 2\\ - 1& 1\\ - 1& 1\end{array}\right]$

• $\left[\begin{array}{cc}2& - 2\\ - 1& 1\end{array}\right]$

• $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

If I is the unit matrix of order 10, then the determinant of I is equal to :

• 10

• 1

• $\frac{1}{10}$

• 9

8.

The equation of the tangent to the curve (1 + x²)y = 2 - x where it crosses the x-axis, is :

• x + 5y = 2

• x - 5y = 2

• 5x - y = 2

• 5x + y - 2 = 0

The sides of an equilateral triangle are increasing at the rate of 2 cm/s. The rate at which the area increases, when the side is 10 cm, is:

• $\sqrt{3} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• 10 sq cm/s

• $10\sqrt{3} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{10}{\sqrt{3}} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

If f(x) = $$\begin{gathered} \left\{\frac{{\mathrm{x}}^2 - 9}{\mathrm{x} - 3}, \mathrm{if} \mathrm{x} \ne 3 \\ 2\mathrm{x} + \mathrm{k}, \mathrm{otherwise}\right\} \end{gathered}$$ is continuous at x = 3, then k is equal to :

• 3

• 0

• - 6

• $\frac{1}{6}$