#### Theory of Equations Two Marks Questions

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
15 x 2 = 30
1. 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 cut away.

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

3. Find a polynomial equation of minimum degree with rational coefficients, having 2+$\sqrt{3}$i as a root.

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

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

6. Solve: $2\sqrt { \frac { x }{ a } } +3\sqrt { \frac { a }{ x } } =\frac { b }{ a } +\frac { 6a }{ b }$

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

8. Determine the number of positive and negative roots of the equation x9-5x4-14x2=0.

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

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

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

12. 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.

13. Find th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

14. Find x If $x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } }$

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