Theory of EquationseasyFree

Theory of Equations: Roots Equation Value Equal

JEE Maths question with a full step-by-step solution.

Question

\alpha, \beta, \gamma

are roots of the equation

x^3 - 2x^2 - 1 = 0

and

T_n = \alpha^n + \beta^n + \gamma^n

, then the value of

\dfrac{T_{11} - T_8}{T_{10}}

is equal to

1

2

correct

-1

3

Solution

Step 1: From

x^3 - 2x^2 - 1 = 0

: each root satisfies

\alpha^3 = 2\alpha^2 + 1

, so

\alpha^3 - 1 = 2\alpha^2

. Step 2:

T_{11} - T_8 = \sum \alpha^{11} - \sum \alpha^8 = \sum \alpha^8(\alpha^3 - 1) = \sum \alpha^8 \cdot 2\alpha^2 = 2\sum \alpha^{10} = 2T_{10}

Step 3:

\frac{T_{11} - T_8}{T_{10}} = \frac{2T_{10}}{T_{10}} = 2

Correct answer: (2) Note: This question can also solved by newton's formula

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