Formulate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.

Construct a cubic equation with roots 2, −2, and 4.

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

Find a polynomial equation of the lowest degree with rational co-efficients having \(\sqrt 3\) and 1 - 2i as two of its roots.

If the roots of the equation \(x^{3}+\mathrm{p} x^{2}+\mathrm{q} x+\mathrm{r}=0\) are in arithmetic progression, show that 2p³- 9 pq + 27r = 0

Form an equation whose roots are three times those of the equation \(x^{3}-x^{2}+x+1=0\)

Form an equation whose roots are the reciprocals of the roots of \(x^{4}-5 x^{3}+7 x^{2}-4 x+5=0\)