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#### Theory of Equations - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Construct a cubic equation with roots 1,2, and 3

2. If α, β, γ  and $\delta$ are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + $\delta$ and αβ૪$\delta$.

3. Show that the equation 2x2−6x+7=0 cannot be satisfied by any real values of x.

4. Show that, if p,q,r are rational, the roots of the equation x2−2px+p2−q2+2qr−r2=0 are rational.

5. Obtain the condition that the roots of x3+px2+qx+r=0 are in A.P.

6. Discuss the nature of the roots of the following polynomials:
x2018+1947x1950+15x8+26x6+2019

7. Solve the equations:
6x4-35x3+62x2-35x+6=0

8. Find the exact number of real roots and imaginary of the equation x9+9x7+7x5+5x3+3x.

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

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

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