Solved Examples

Example 1:

        If f(θ) = Then find the value of f(∏/6)

Solution:

= 3/4((1/4)0 + √3/4(-√3/4)-1/2(-(3/8)-(1/8))) (Expanding along first row)

       => f(∏/6) = 9/16 + 3/16 + 4/16 = 1                                   (Ans.)

Example 2:

Show that the determinant

        is divisible by x².

Solution:

At x = 0, Δ = 0 with three columns identical. Therefore (x - 0) is a repeated root of the equation Δ = 0 => x² is a factor of determinant.

=> Δ is divisible by x².                                                   (Proved)

Example 3:

Determine the value of 'a' for which the following system of equation has a non trivial solution.

a³ x + (a+1)³y = (a+2)³z = 0

ax + (a+1)y + (a+2)z = 0

x + y + z = 0

Solution:

For this system of equations to have a non trivial solution.

                => (3a² + 3a + 1)(2) + (6a² + 12a + 8)(-1) = 0

                => 6a² + 6a + 2 - 6a² - 12a - 8 = 0

                => -6a - 6 = 0

                => a = - 1                                                          (Proved)