If α and β are the roots of the qu

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 Multiple Choice QuestionsMultiple Choice Questions

71.

Let α, β be the roots of x2 - x - 1 = 0 and Sn = αn + βn, for all integers n  1. Then, for every integer n  2

  • Sn + Sn - 1 = Sn +1

  • Sn - Sn - 1 = Sn +1

  • Sn - 1 = Sn +1

  • Sn + Sn - 1 = 2Sn +1


72.

If α, β are the roots of the quadratic equation x2 + px + q = 0, then the values of α3 + β3 and α4 + α2β2 + β4 are respectively

  • 3pq - p3 and p4 - 3p2q + 3q2

  • - p(3q - p2) and (p2 - q)(p2 + 3q)

  • pq - 4 and p4 - q4

  • 3pq - p3 and (p2 - q)(p2 - 3q)


73.

Let α, β denote the cube roots of unity other than 1 and α  β. Let S = n = 0302- 1nαβn. Then, the value of S is

  • either - 2w or - 2w2

  • either - 2w or 2w2

  • either 2w or - 2w2

  • either 2w or 2w2


74.

Let p (x) be a quadratic polynomial with constant term 1. Suppose p(x), when divided by x - 1 leaves remainder 2 and when divided by x + 1 leaves remainder 4. Then, the sum of the roots of p(x) = 0 is

  • - 1

  • 1

  • - 12

  • 12


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75.

If z1 = 2 + 3i and z2 = 3 + 4i  be two points on the complex plane. Then, the set of complex number z satisfying z - z12 + z - z22 = z1 - z22 represnts

  • a straight line

  • a point

  • a circle

  • a pair of straight line


76.

If α and β are roots of x2 - x + 1 = 0, then the value of α2013 + β2013 is

  • 2

  • - 2

  • - 1

  • 1


77.

If α and β  B are the roots of the quadratic equation is, x2 + ax + b = 0, (b 0), then the quadratic equation whose roots are α - 1β, β - 1α, is

  • ax2 + a(b - 1)x + (a - 1)2 = 0

  • bx2 + a(b - 1)x + (b - 1)2 = 0

  • x2 + ax + b = 0

  • abx2 + bx + a = 0


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78.

If α and β are the roots of the quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac, then

  • α = 4β or β = 4α

  • α = - 4β or β = - 4α

  • α = 3β or β = 3α

  • α = - 3β or β = - 3α


C.

α = 3β or β = 3α

Given that, (α, β) are roots of the quadratic equation

ax2 + bx + c = 0

and 3b2 = 16ac         ...(i)

 α + β = - ba and αβ = ca    ...(ii)

From Eq. (i), we get

                   3b . b = 16a . c

                   3ba2 = 16ca               divide by a2              3α + β2 = 16αβ          from Eq. (ii) 3α2 + 3β2 + 6αβ = 16αβ             3α2 + 3β2 = 10αβ       3αβ + 3βα = 10

Let                              αβ = xThen,                 3x + 3x = 10           3x2 - 10x + 3 = 0 3xx - 3 - 1x - 3 = 0          x - 33x - 1 = 0                                 x = 3, 13                               αβ = 13, 3                β = 3α or α = 3β


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79.

The number of solutions of the equation

12log3x + 1x +5 + logex + 52 = 1

  • 0

  • 1

  • 2

  • infinite


80.

If sinα, cosα  be the roots of the equation x2 - bx + c = 0, Then, which of the following statements is/are correct?

  • c 12

  • b  2

  • c > 12

  • > 2


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