If n is an integer which leaves remainder one when divided by thr

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 Multiple Choice QuestionsMultiple Choice Questions

231.

If α + β = - 2 and α3 + β3 = - 56, thenthe quadratic equation whose roots are α and β 

  • x2 + 2x - 16 = 0

  • x2 + 2x + 15 = 0

  • x2 + 2x - 12 = 0

  • x2 + 2x - 8 = 0


232.

The cubic equation whose roots are thrice to each of the roots of x3 + 2x2 - 4x + 1 = 0 is

  •  x3 + 6x2 - 36x + 27 = 0

  •  x3 + 6x2 + 36x + 27 = 0

  •  x3 - 6x2 - 36x + 27 = 0

  •  x3 - 6x2 + 36x + 27 = 0


233.

The sum of the fourth powers of the roots of the equation

x3 + x + 1 = 0 is

  • - 2

  • - 1

  • 1

  • 2


 Multiple Choice QuestionsMatch The Following

234.

let α and β be the roots of the quadratic equation ax2 + bx + c = 0. Observe the lists given below
  List-I   List-II
(i) α = β (A) (ac2)1/3 + (a2c)1/3 + b = 0
(ii) α = 2β (B) 2b2 = 9ac
(iii) α = 3β (C) b2 = 6ac
(iv) α = β2 (D) 3b2 = 16ac
    (E) b2 = 4ac
    (F) (ac2)1/3 + (a2c)1/3 = b

The correct match of List-I from List-II is

A. (i) (ii) (iii) (iv) (i) E B D F
B. (i) (ii) (iii) (iv) (ii) E B A D
C. (i) (ii) (iii) (iv) (iii) E D B F
D. (i) (ii) (iii) (iv) (iv) E B D A

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 Multiple Choice QuestionsMultiple Choice Questions

235.

The roots (x - a) (x - a - 1) + (x - a - 1) (x - a - 2) + (x - a) (x - a - 2) = 0, a  R are always

  • equal

  • imaginary

  • real and distinct

  • rational and equal


236.

Let f(x) = x + ax + b, where a, b  R. If f(x) = 0 has all-its roots imaginary, then the roots of f(x) + f'(x) + f"(x) = 0 are

  • real and distinct

  • imaginary

  • equal

  • rational and equal


237.

If α, β, γ are the roots of x3 + 4x + 1 = 0, then the equation whose roots are α3β + γ, β2γ + α, γ2α + β is

  • x3 - 4x - 1 = 0

  • x3 - 4x + 1 = 0

  • x3 + 4x - 1 = 0

  • x3 + 4x + 1 = 0


238.

If α and β are the roots of x2 - 2x + 4 = 0, then the value of α6 + β6 is

  • 32

  • 64

  • 128

  • 256


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239.

If n is an integer which leaves remainder one when divided by three, then 1 + 3in + 1 - 3in equals

  • - 2n + 1

  • 2n + 1

  • - (- 2)n

  • - 2n


C.

- (- 2)n

Now, 1 + 3in + 1 - 3in= 21 + 3i2n + 21 - 3i2n= - 2w2n + - 2wn= - 2nw23r + 1 + w3r + 1       n = 3r + 1, where r is an integer= - 2nw2 + w = - - 2n


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240.

If ∝, ß, y are the roots of the equation x3 - 6x2 + 11x - 6 = 0 and if a = ∝2 + ß2 + γ2, b = ∝ß + ßγ + γ∝ and  c = (∝ + ß)(ß + γ)(γ + ∝), then the correct inequality among the following is

  • a < b < c

  • b < a < c

  • b < c < a

  • c < a < b


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