If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

If ax² + bx + c is divided by x + 3, x − 5, and x − 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)

Solve the system: x + y − 2z = 0, 2x − 3y + z = 0, 3x − 7y + 10z = 0, 6x − 9y + 10z = 0.

Solve the following systems of linear equations by Cramer’s rule:
 3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

Solve the equation z³+ 27 = 0

Solve the following equation: x⁴-10x³+ 26x²-10x + 1 = 0

Find the principal value of cosec⁻¹(−1)

Find the equation of the ellipse whose eccentricity is \(\frac { 1 }{ 2 } \), one of the foci is(2, 3) and a directrix is x = 7. Also find the length of the major and minor axes of the ellipse.

At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)