If A is matrix and B is a matrix such that A'B and BA' are both defined, then B is of the type

• $3 \times 4$

• $4 \times 3$

• $3 \times 3$

The symmetric part of the matrix A = $\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& - 2& 7\end{array}\right]$ is

• $\left[\begin{array}{ccc}0& - 2& - 1\\ - 2& 0& - 2\\ - 1& - 2& 0\end{array}\right]$

• $\left[\begin{array}{ccc}1& 4& 3\\ 2& 8& 0\\ 3& 0& 7\end{array}\right]$

• $\left[\begin{array}{ccc}0& - 2& - 1\\ 2& 0& 2\\ - 1& 2& 0\end{array}\right]$

• $\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$

If A is a matrix of order 3 such that A (adj A) = 10I, then $\left|\mathrm{adj} \mathrm{A}\right|$ is equal to

• 1

• 10

• 100

• 10I

Consider the following statements

(i) If any two rows or columns of a determinant are identical, then the value of the determinant is zero.
(ii) Ifthe corresponding rows and columns of a determinant are interchanged, then the value of determmant does not change.
(iii) If any two rows (or columns) of a determinant are mterchanged, then the value of the determinant changes in sign.

Which of these are correct?

• (i) and (iii)

• (i) and (ii)

• (i), (ii) and (iii)

• (ii) and (iii)

The inverse ofthe matrix A = $\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$ is

• $\frac{1}{24}\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

• $\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

• $\frac{1}{24}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

• $\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& \frac{1}{3}& 0\\ 0& 0& \frac{1}{4}\end{array}\right]$

If a, b and c are in AP, then the value of $\left|\begin{array}{ccc}\mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + \mathrm{a}\\ \mathrm{x} + 4& \mathrm{x} + 5& \mathrm{x} +\hspace{0.17em}\mathrm{b}\\ \mathrm{x} +\hspace{0.17em}6& \mathrm{x} + 7& \mathrm{x} + \mathrm{c}\end{array}\right|$ is

• 0

• x - (a + b + c)

• a + b + c

• 9x² + a + b + c

17.

The local minimum value of the function f' given by f(x) = 3 + $\left|\mathrm{x}\right|, \mathrm{x} \in \mathrm{R}$ is

• - 1

• 3

• 1

• 0

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At that instant, when the radius of circular wave is 8 cm, how fast is the enclosed area

• $6\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $8\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{8}{3} {\mathrm{cm}}^2/\mathrm{s}$

• $80\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

A gardeneris digging a plot of land. As he gets tired, he works more slowly. After 't' minutes he is digging at a rate of $\frac{2}{\sqrt{\mathrm{t}}}$ m²/min . How long will it take him to dig an area of 40 sq m ?

• 100 min

• 10 min

• 30 min

• 40 min

Area of the region bounded by two parabolas y = x² and x = y² is

• $\frac{1}{4}$

• $\frac{{\displaystyle 1}}{{\displaystyle 3}}$

• 4

• 3