Determinants

If x, y, z are different from zero and $∆ = \left|\begin{array}{ccc}\mathrm{a}& \mathrm{b} - \mathrm{y}& \mathrm{c} - \mathrm{z}\\ \mathrm{a} - \mathrm{x}& \mathrm{b}& \mathrm{c} - \mathrm{z}\\ \mathrm{a} - \mathrm{x}& \mathrm{b} - \mathrm{y}& \mathrm{c}\end{array}\right|$ = 0, then the value of the expression $\frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}}$ is

• 0

• - 1

• 1

• 2

If x = - 5 is a root of $\left|\begin{array}{ccc}2\mathrm{x} + 1& 4& 8\\ 2& 2\mathrm{x}& 2\\ 7& 6& 2\mathrm{x}\end{array}\right|$ = 0, then the other roots are :

• 3, 3.5

• 1, 3.5

• 1, 7

• 2, 7

The simultaneous equations Kx + 2y - z = 1, (K - I)y - 2z = 2 and (K + 2)z = 3 have only one solution when :

• K = - 2

• K = - 1

• K = 0

• K = 1

If the matrix M_r is given by Mr = $\left[\begin{array}{cc}\mathrm{r}& \mathrm{r} - 1\\ \mathrm{r} - 1& \mathrm{r}\end{array}\right]$, r = 1, 2, 3, ..., then the value of det(M₁) + det(M₂) + ... + det(M₂₀₀₈) is

• 2007

• 2008

• (2008)²

• (2007)²

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the cube roots of unity, then the value of the determinant $\left|\begin{array}{ccc}{\mathrm{e}}^{\mathrm{\alpha }}& {\mathrm{e}}^{2\mathrm{\alpha }}& \left({\mathrm{e}}^{3\mathrm{\alpha }} - 1\right)\\ {\mathrm{e}}^{\mathrm{\beta }}& {\mathrm{e}}^{2\mathrm{\beta }}& \left({\mathrm{e}}^{3\mathrm{\beta }} - 1\right)\\ {\mathrm{e}}^{\mathrm{\gamma }}& {\mathrm{e}}^{2\mathrm{\gamma }}& \left({\mathrm{e}}^{3\mathrm{\gamma }} - 1\right)\end{array}\right|$ is equal to

• - 2

• - 1

• 0

• 1

If the three linear equations

x + 4ay + az = 0

x + 3by + bz = 0

x + 2cy + cz = 0

have a non-trivial solution, where a$\mathrm{a} \ne 0, \mathrm{b} \ne 0, \mathrm{c} \ne 0$, then ab + bc is equal to

• 2ac

• - ac

• ac

• - 2ac

If A = $\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$, then the value of the determinant $\left|{\mathrm{A}}^{2009} - 5{\mathrm{A}}^{2008}\right|$ is

• - 6

• - 5

• - 4

• 4

The value of the determinant $\left|\begin{array}{ccc}15!& 16!& 17!\\ 16!& 17!& 18!\\ 17!& 18!& 19!\end{array}\right|$ is equal to

• 15! + 16!

• 2(15!)(16!)(17!)

• 15! + 16! + 17!

• 16! + 17!

If A = $\left[\begin{array}{cc}\log \left(\mathrm{x}\right)& - 1\\ - \log \left(\mathrm{x}\right)& 2\end{array}\right]$ and if det(A) = 2, then the value of x is equal to

• 2

• e²

• - 2

• e

If $\left|\begin{array}{cc}{\sin }^2\left(\mathrm{\alpha }\right)& {\cos }^2\left(\mathrm{\alpha }\right)\\ {\cos }^2\left(\mathrm{\alpha }\right)& {\sin }^2\left(\mathrm{\alpha }\right)\end{array}\right|$ = 0, $\mathrm{\alpha } \in \left(0, \mathrm{\pi }\right)$, then the value of $\mathrm{\alpha }$ are

• $\frac{\mathrm{\pi }}{2} \mathrm{and} \frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{2} \mathrm{and} \frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4} \mathrm{and} \frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6} \mathrm{and} \frac{\mathrm{\pi }}{3}$