Determinants

If the system of equations x + ky - z = 0, 3x - ky - z = 0 and x - 3y + z =0, has non-zero solution, then k is equal to

• - 1

• 0

• 1

• 2

The value of the determinant

$\left|\begin{array}{ccc}1& \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right)& \cos \left(\mathrm{\alpha }\right)\\ \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right)& 1& \cos \left(\mathrm{\beta }\right)\\ \cos \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\beta }\right)& 1\end{array}\right|$ is

• ${\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2$

• ${\mathrm{\alpha }}^2 - {\mathrm{\beta }}^2$

• 1

• 0

If a, b and c are in AP, then determinnant

$\left|\begin{array}{ccc}\mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + 2\mathrm{a}\\ \mathrm{x} + 3& \mathrm{x} +\hspace{0.17em}4& \mathrm{x} +\hspace{0.17em}2\mathrm{b}\\ \mathrm{x} + 4& \mathrm{x} +\hspace{0.17em}5& \mathrm{x} + 2\mathrm{c}\end{array}\right|$ is

• 0

• 1

• x

• 2x

The value of the determinant $\left|\begin{array}{ccc}\cos \left(\mathrm{\alpha }\right)& - \sin \left(\mathrm{\alpha }\right)& 1\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)& 1\\ \cos \left(\mathrm{\alpha } + \mathrm{\beta }\right)& - \sin \left(\mathrm{\alpha } + \mathrm{\beta }\right)& 1\end{array}\right|$ is

• independent of $\mathrm{\alpha }$

• independent of $\mathrm{\beta }$

• independent of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$

• None of the above

Which of the following is correct?

• Determinant is a square matrix

• Determinant is a number associated to a matrix

• Determinant is a number associated to a square matrix

• All of the above

If $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ are the roots of x³ + ax² + b = 0, then the value of $\left|\begin{array}{ccc}\mathrm{\alpha }& \mathrm{\beta }& \mathrm{\gamma }\\ \mathrm{\beta }& \mathrm{\gamma }& \mathrm{\alpha }\\ \mathrm{\gamma }& \mathrm{\alpha }& \mathrm{\beta }\end{array}\right|$ is

• - a³

• a³ - 3b

• a³

• a² - 3b

The system of equations 2x + y - 5 = 0, x - 2y + 1 = 0, 2x - 14y - a= 0, is consistent. Then, a is equal to

• 1

• 2

• 5

• None of these

If x, y, z are all positive and are the pth , qth and rth terms of a geometric progression respectively, then the value of determinant $\left|\begin{array}{ccc}\log \left(\mathrm{x}\right)& \mathrm{p}& 1\\ \log \left(\mathrm{y}\right)& \mathrm{q}& 1\\ \log \left(\mathrm{z}\right)& \mathrm{r}& 1\end{array}\right|$ equals

• log(xyz)

• (p - 1)(q - 1)(r - 1)

• pqr

• 0

The system of equations

x + y + z = 0

2x + 3y + z = 0

and x + 2y = 0

has

• a unique solution; x = 0, y = 0, z = 0

• infinite solutions

• no solution

• finite number of non-zero solutions

If D = diag (d₁, d₂, ..., d_n), where d_i $\ne$ 0, for i = 1, 2, ... , n then D^-1 is equal to

• D^T

• D

• adj(D)

• $\mathrm{diag}\left({{\mathrm{d}}_1}^{-1}, {{\mathrm{d}}_2}^{-1}, ..., {{\mathrm{d}}_{\mathrm{n}}}^{-1}\right)$