A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

Find the monic polynomial equation of minimum degree with real coefficients having 2 -\(\sqrt{3}\)i as a root.

Find a polynomial equation of minimum degree with rational coefficients, having 2 +√3 i as a root.

Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

Show that the polynomial 9x⁹+ 2x⁵- x⁴- 7x²+ 2 has at least six imaginary roots.

Solve: \(2\sqrt { \frac { x }{ a } } +3\sqrt { \frac { a }{ x } } =\frac { b }{ a } +\frac { 6a }{ b } \)

Show that the equation x⁹- 5x⁵+ 4x⁴+ 2x²+ 1 = 0 has atleast 6 imaginary solutions.

Determine the number of positive and negative roots of the equation x⁹- 5x⁸-14x⁷= 0.

Examine for the rational roots of x⁸- 3x + 1 = 0

Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+ 2x³+ 5x²+11

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.

Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

Find the number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2