21.

The sum of all the two-digit odd numbers is

• 2475

• 2530

• 4905

• 5049

The sum of the first n terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + ... = ?$

• $2^n - n - 1$

• $1 - 2^{ - \mathrm{n}}$

• $2^{ - \mathrm{n}} + \mathrm{n} - 1$

• $2^{\mathrm{n}} - 1$

$$\begin{gathered} \mathrm{The} \mathrm{expansion} \mathrm{of} (\mathrm{x} - \mathrm{y}{)}^{\mathrm{n}}, \mathrm{n} > 5 \mathrm{is} \mathrm{done} \mathrm{in} \mathrm{the} \mathrm{descending} \mathrm{powers} \mathrm{of} \mathrm{r}. \\ \mathrm{If} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{fifth} \mathrm{and} \mathrm{sixth} \mathrm{term} \mathrm{is} \mathrm{zero}, \mathrm{then} \frac{\mathrm{x}}{\mathrm{y}} = ? \end{gathered}$$

• $\frac{\mathrm{n} - 5}{6}$

• $\frac{\mathrm{n} - 4}{5}$

• $\frac{5}{\mathrm{n} - 4}$

• $\frac{6}{\mathrm{n} - 5}$

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\mathrm{\alpha }& 2\\ 2& \mathrm{\alpha }\end{array}\right] \mathrm{and} \det \left({\mathrm{A}}^3\right) = 125, \mathrm{Then} \mathrm{\alpha } = ?$

• $\pm 1$

• $\pm 2$

• $\pm 3$

• $\pm 5$

If B is a non-singular matrix and A is a square matrix, then the value of det (B ^{- 1} AB) is equal to

• det(B)

• det(A)

• $\det \left({\mathrm{B}}^{ - 1}\right)$

• $\det \left({\mathrm{A}}^{ - 1}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{a} \ne \mathrm{b} \ne \mathrm{c}, \mathrm{then} \mathrm{one} \mathrm{value} \mathrm{of} \mathrm{x} \mathrm{which} \mathrm{satisfies} \mathrm{the} \mathrm{equation} \\ \left|\begin{array}{ccc}0& \mathrm{x} - \mathrm{a}& \mathrm{x} - \mathrm{b}\\ \mathrm{x} + \mathrm{a}& 0& \mathrm{x} - \mathrm{c}\\ \mathrm{x} +\hspace{0.17em}\mathrm{b}& \mathrm{x} + \mathrm{c}& 0\end{array}\right| = 0 \mathrm{is} \mathrm{given} \mathrm{by} \end{gathered}$$

• a

• b

• c

• 0

$$\begin{gathered} \mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right] \mathrm{then} \mathrm{what} \mathrm{is} {\mathrm{AA}}^{\mathrm{T}} = ? \\ \left(\mathrm{Where} {\mathrm{A}}^{\mathrm{T}} \mathrm{is} \mathrm{the} \mathrm{transpose} \mathrm{of} \mathrm{A}\right) ? \end{gathered}$$

• Null matrix 

• Identity Matrix

• A

• - A

The equations

x + 2y + 3z = 12

x + y + 3z  = 25

x + 5y + 9z = 4

• have the unique solution

• Have infinitely many solutions

• are inconsistent

• None of the above

What is the value of the determinant

$\left|\begin{array}{ccc}1& 1& 1\\ 1& 1 + \mathrm{xyz}& 1\\ 1& 1& 1 +\hspace{0.17em}\mathrm{xyz}\end{array}\right|$

• 1 + x + y + z

• 2xyz

• x²y²z²

• 2x²y²z²

$\mathrm{If} \left|\begin{array}{ccc}\mathrm{x}& \mathrm{y}& 0\\ 0& \mathrm{x}& \mathrm{y}\\ \mathrm{y}& 0& \mathrm{x}\end{array}\right| = 0, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\frac{\mathrm{x}}{\mathrm{y}} \mathrm{is} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{cube} \mathrm{roots} \mathrm{of} \mathrm{unity}$

• x is one of the cube roots of unity

• y is one of the cube roots of unity

• $\frac{\mathrm{x}}{\mathrm{y}}$ is one of the cube roots - 1