If α and β are the roots of the equation x2 – x +1 =0, then

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 Multiple Choice QuestionsMultiple Choice Questions

11.

If z is a complex number of unit modulus and argument θ, then arg open parentheses fraction numerator 1 plus straight z over denominator 1 plus straight z with bar on top end fraction close parentheses is equal to

  • π/2-θ

  • θ

  • θ

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12.

The equation esinx-e-sinx -4 = 0 has

  • infinite number of real roots

  • No real root

  • exactly one real root

  • exactly one real root

312 Views

13.

Let α,  β be real and z be a complex number. If z2 + αz + β = 0 has two distinct roots on the line Re z = 1, then it is necessary that

  • β ∈(0, 1)

  • β ∈(-1, 0)

  • |β| = 1

  • |β| = 1

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14.

If ω(≠1) is a cube root of unity, and (1 + ω)7 = A + Bω.Then (A, B) equals

  • (0,1)

  • (1,1)

  • (1,0)

  • (1,0)

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15.

The Number of complex numbers z such that |z– 1| = |z + 1| = |z – i| equals 

  • 0

  • 1

  • 2

  • 2

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16.

If α and β are the roots of the equation x2 – x +1 =0, then α2009 + β2009 =

  • -2

  • -1

  • 1

  • 1


C.

1

The quadratic equation ax2 + bx +c = 0 has roots α and β,
Then α + β = - b/a, α β = c/a
Also, if ax2+ bx +c = 0
Then, straight x space equals space fraction numerator negative straight b plus-or-minus space square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction
We know that 1,ω, ω2 are cube roots of unity.
1+ω + ω2 = 0  (ω2 = 1)
and straight omega space equals space fraction numerator negative 1 space plus space straight i square root of 3 over denominator 2 end fraction comma space straight omega squared space equals space fraction numerator negative 1 minus straight i square root of 3 over denominator 2 end fraction
Since α and β are roots of the equations
x2-x+1 = 0
⇒ α + β = 1 , α β =1
rightwards double arrow space straight x space equals space fraction numerator 1 space plus-or-minus space square root of 3 straight i over denominator 2 end fraction
rightwards double arrow space straight x space equals space fraction numerator 1 plus space square root of 3 straight i over denominator 2 end fraction
rightwards double arrow space equals space fraction numerator 1 minus square root of 3 straight i over denominator 2 end fraction
x = ω2, or -ω
α = -ω2, then β =-ω
or α = -ω, then β =-ω2, (where ω3 = 1)
Hence,  α2009 + β2009 =(-ω)2009 + (-ω2)2009
 = - [(ω3)669. ω2 + (ω3)1337.ω]
 = - [ω2 + ω]
 = -(-1) = 1

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17.

If, for a positive integer n, the quadratic equation,
x(x + 1) + (x + 1) (x + 2) + .....
+ (x + n -1 ) (x + n) = 10n
has two consecutive integral solutions, then n is equal to :

  • 11

  • 12

  • 9

  • 9

2171 Views

18.

If the roots of the equation bx2+ cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is

  • greater than 4ab

  • less than 4ab

  • greater than -4ab

  • greater than -4ab

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19.

The conjugate of a complex number is fraction numerator 1 over denominator straight i minus 1 end fraction. Then the complex number is

  • fraction numerator negative 1 over denominator straight i minus 1 end fraction
  • fraction numerator 1 over denominator straight i plus 1 end fraction
  • fraction numerator negative 1 over denominator 1 plus straight i end fraction
  • fraction numerator negative 1 over denominator 1 plus straight i end fraction
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20.

The quadratic equations x2 – 6x + a = 0 and x2 – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

  • 1

  • 4

  • 3

  • 3

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