The cubic equation whose roots are thrice to each of the roots of

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.
Advertisement

 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


Advertisement

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


A.

 x3 + 6x2 - 36x + 27 = 0

Given equation isx3 + 2x2 - 4x +1 = 0Let α,  β and γ be the roots of the given equation α + β + γ = -2αβ + βγ + γα = - 4and αβγ = - 1Let the required cubic equation has the roots 3α, 3β and 3γ3α + 3β + 3γ = - 6,3α . 3β + 3β . 3γ + 3γ . 3α = - 36and 3α . 3β . 3γ = - 27Required equation isx3 - - 6x2 + - 36x - 27 = 0 x3 + 6x2 - 36x + 27 = 0


Advertisement
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

Advertisement

 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


Advertisement
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


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


Advertisement