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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

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