Let a, b and c be positive real numbers. The following system of equations in x, y and z,

$$\begin{gathered} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} - \frac{{\mathrm{z}}^2}{{\mathrm{c}}^2} = 1, \frac{{\displaystyle {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} - \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} + \frac{{\displaystyle {\mathrm{z}}^2}}{{\displaystyle {\mathrm{c}}^2}} = 1 \\ \mathrm{and} \frac{{\displaystyle - {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} + \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} + \frac{{\displaystyle {\mathrm{z}}^2}}{{\displaystyle {\mathrm{c}}^2}} = 1 \mathrm{has} \end{gathered}$$

• finitely many solutions

• no solution

• unique solution

• infinitely many solutions

If $∆\left(\mathrm{r}\right) = \left|\begin{array}{cc}\mathrm{r}& {\mathrm{r}}^3\\ 1& \mathrm{n}\left(\mathrm{n} + 1\right)\end{array}\right|$, then $\sum _{\mathrm{r} = 1}^{\mathrm{n}}∆\left(\mathrm{r}\right)$ is equal to

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}{\mathrm{r}}^2$

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}{\mathrm{r}}^3$

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}\mathrm{r}$

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}{\mathrm{r}}^4$

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

378.

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