Using determinants; find the quadratic defined by f(x) = ax² + bx + c, if f(1) = 0, f(2) = -2 and f(3) = -6.

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2

The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.

Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.

Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.

Use matrices to find the solution set of 
4x+ 8y+ z = - 6
2x - 3y+ 2z = 0
x + 7y - 32 = - 8

If A \(=\left[\begin{array}{cc} \frac{-1+i \sqrt{3}}{2 i} & \frac{-1-i \sqrt{3}}{2 i} \\ \frac{1+i \sqrt{3}}{2 i} & \frac{1-i \sqrt{3}}{2 i} \end{array}\right],\ \mathrm{i}=\sqrt{-1}\) and f(x) = x² + 2 then find f(A).

In a \(\triangle A B C, if \left|\begin{array}{lll}1 & a & b \\ 1 & c & a \\ 1 & b & c\end{array}\right|=0\) , then find the value of \(64\left(\sin ^{2} A+\sin ^{2} B+\sin ^{2} C\right)\)

If D \(=\left|\begin{array}{ccc} 1 & 3 \cos \theta & 1 \\ \sin \theta & 1 & 3 \cos \theta \\ 1 & \sin \theta & 1 \end{array}\right|\) , then find the maximum value of D.

If A \(=\left[\begin{array}{cc} 1 & 2 \\ -2 & -1 \end{array}\right]\)and \(\phi(x)=(1+x)(1-x)^{-1}\) find \(\phi(A) \).

Solve the system of linear equations given by
3x - 2y + 8z = 9
-2x + 2y + z = 3 
x + 2y - 3z = 8

Solve the system of linear equations given by 2y + 3z = 7, 3x + 6y - 12z = -3, 5x - 2y + 2z = -7

Solve by matrix inversion method 2x - y + 3z = 9, x + y + z = 6, x - y + z = 2.

Use Cramer's Rule to solve  3x - y + z = 5, x + 2y -2z = -3, 2x + 3y + z = 4