Let the numbers 2, b, c be in an A.P. and A = $\left|\begin{array}{ccc}1& 1& 1\\ 2& \mathrm{b}& \mathrm{c}\\ 4& {\mathrm{b}}^2& {\mathrm{c}}^2\end{array}\right|$. If det(A) $\in$ [2, 16], then c lies in the interval :

• [4, 6]

• (2 + 2^3/4, 4)

• [2, 3)

• [3, 2 + 2^3/4]

If the system of equation 2x + 3y - z = 0, x + ky - 2z = 0 and 2x - y + z = 0 has a non – trivial solution (x, y, z) , then $\frac{\mathrm{x}}{\mathrm{y}} +\hspace{0.17em}\frac{\mathrm{y}}{\mathrm{z}} + \frac{\mathrm{z}}{\mathrm{x}} + \mathrm{k}$ is equal to

• $\frac{3}{4}$

• $\frac{1}{2}$

• - 4

• - $\frac{1}{4}$

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation x² + x + 1 = 0 . Then for y $\ne$ 0 in R, $\left|\begin{array}{ccc}\mathrm{y} + 1& \mathrm{\alpha }& \mathrm{\beta }\\ \mathrm{\alpha }& \mathrm{y} +\hspace{0.17em}\mathrm{\beta }& 1\\ \mathrm{\beta }& 1& \mathrm{y} + \mathrm{\alpha }\end{array}\right|$ is equal to :

• y(y² - 1)

• y³

• y(y² - 3)

• y³ - 1

If the line y = mx + 7$\sqrt{3}$ is normal to the hyperbola $\frac{{\mathrm{x}}^2}{24} - \frac{{\mathrm{y}}^2}{18} = 1$, then a value of m is :

• $\frac{\sqrt{5}}{2}$

• $\frac{\sqrt{15}}{2}$

• $\frac{{\displaystyle 2}}{{\displaystyle \sqrt{15}}}$

• $\frac{{\displaystyle 3}}{{\displaystyle \sqrt{15}}}$

85.

If ${△}_1 = \left|\begin{array}{ccc}\mathrm{x}& \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& - \mathrm{x}& 1\\ \cos \left(\mathrm{\theta }\right)& 1& \mathrm{x}\end{array}\right| \mathrm{and} \left|\begin{array}{ccc}\mathrm{x}& \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\\ - \sin \left(2\mathrm{\theta }\right)& - \mathrm{x}& 1\\ \cos \left(2\mathrm{\theta }\right)& 1& \mathrm{x}\end{array}\right|$, $\mathrm{x} \ne 0$; then for all $\mathrm{\theta } \in \left(0, \frac{\mathrm{\pi }}{2}\right)$ :

• ${△}_1 - {△}_2 = \mathrm{x}\left(\cos \left(2\mathrm{\theta }\right) - \cos \left(4\mathrm{\theta }\right)\right)$

• ${△}_1 + {△}_2 = - 2{\mathrm{x}}^3$

• ${△}_1 + {△}_2 = - 2\left({\mathrm{x}}^3 + \mathrm{x} - 1\right)$

• ${△}_1 - {△}_2 = - 2{\mathrm{x}}^3$

The solution of the equation

$\left[\begin{array}{ccc}1& 0& 1\\ - 1& 1& 0\\ 0& - 1& 1\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] = \left[\begin{array}{c}1\\ 1\\ 2\end{array}\right]$ is (x, y, z) = 

• (1, 1, 1)

• (0, - 1, 2)

• (- 1, 2, 2)

• (- 1, 0, 2)

The value of 'a' for which the system of equations

a³x + (a + 1)³y + (a + 2)³z = 0

     ax + (a - 1)y + (a + 2)z = 0

                          x + y + z = 0

• 1

• 0

• - 1

• None of these

If $\frac{7}{2} \mathrm{and} 1$ are the roots of the equation $\left|\begin{array}{ccc}2\mathrm{x}& 3& 7\\ 2& 2\mathrm{x}& 2\\ 7& 6& 2\mathrm{x}\end{array}\right|$ = 0, then third root is

• $- \frac{7}{2}$

• $- \frac{9}{2}$

• $- \frac{3}{2}$

• $- \frac{5}{2}$

The value of determinant $\left|\begin{array}{ccc}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{a}$}\right.& \mathrm{bc}& {\mathrm{a}}^3\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{b}$}\right.& \mathrm{ca}& {\mathrm{b}}^3\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{c}$}\right.& \mathrm{ab}& {\mathrm{c}}^3\end{array}\right|$

• a²b²c²(a - b)(b - c)(c - a)

• 0

• (a - b)(b - c)(c - a)

• None of the above

The number of solutions of the system of equations 2x + y - z = 7, x - 3y + 2z = 1 and x + 4y - 3z = 5 is

• 0

• 1

• 2

• 3