Theory of Equations Important Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    10 x 1 = 10
  1. A zero of x3 + 64 is

    (a)

    0

    (b)

    4

    (c)

    4i

    (d)

    -4

  2. The number of real numbers in [0,2π] satisfying sin4x-2sin2x+1 is

    (a)

    2

    (b)

    4

    (c)

    1

    (d)

  3. If x3+12x2+10ax+1999 definitely has a positive zero, if and only if

    (a)

    a≥0

    (b)

    a>0

    (c)

    a<0

    (d)

    a≤0

  4. The polynomial x3+2x+3 has

    (a)

    one negative and two real roots

    (b)

    one positive and two imaginary roots

    (c)

    three real roots

    (d)

    no solution

  5. Ifj(x) = 0 has n roots, thenf'(x) = 0 has __________ roots

    (a)

    n

    (b)

    n -1

    (c)

    n+1

    (d)

    (n-r)

  6. If x is real and \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then

    (a)

    \(\frac{1}{3}\) ≤k≤

    (b)

    k≥5

    (c)

    k≤0

    (d)

    none

  7. If ∝, β, ૪ are the roots of the equation x3-3x+11=0, then ∝+β+૪ is __________.

    (a)

    0

    (b)

    3

    (c)

    -11

    (d)

    -3

  8. If ∝, β,૪ are the roots of 9x3-7x+6=0, then ∝ β ૪ is __________

    (a)

    \(\frac{-7}{9}\)

    (b)

    \(\frac{7}{9}\)

    (c)

    0

    (d)

    \(\frac{-2}{3}\)

  9. If ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

    (a)

    c>0

    (b)

    c<0

    (c)

    c=0

    (d)

    c≥0

  10. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

    (a)

    no

    (b)

    1

    (c)

    2

    (d)

    infinite

  11. 5 x 1 = 5
  12. 6x6 - 35x5 + 56x4 -56x2 + 35x - 6 = 0

  13. (1)

    4 Change of sign

  14. p(x)=xn.p\(\left( \frac { 1 }{ x } \right) \)

  15. (2)

    4 Change of sign

  16. 2x7-3x6-4x5+5x4+6x3-7x+8 = 0

  17. (3)

    Reciprocal equation of type I

  18. 9x9+2x5-x4-7x2+2=0

  19. (4)

    even degree reciprocal equation of type II

  20. x9-5x8 -14x7=0

  21. (5)

    at least 6 imaginary roots

    5 x 2 = 10
  22. Find the sum of squares of roots of the equation 2x4-8x+6x2-3=0.

  23. 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\(\delta\).

  24. Find all zeros of the polynomial x6-3x5-5x4+22x3-39x2-39x+135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.

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

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

  27. 5 x 3 = 15
  28. If α and β are the roots of the quadratic equation 17x2+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

  29. Find the condition that the roots of ax3+bx2+cx+d=0 are in geometric progression. Assume a,b,c,d ≠0.

  30. If the roots of x3+px2+qx+r=0 are in H.P. prove that 9pqr = 27r3+2p.

  31. Solve: 2x+2x-1+2x-2=7x+7x-1+7x-2

  32. Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

  33. 2 x 5 = 10
  34. Show that, if p,q,r are rational, the roots of the equation x2−2px+p2−q2+2qr−r2=0 are rational.

  35. If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

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