If x1 - x2 - x3 - x4 - x5 - x6 are non-zero distinct real roots of the equation x6 - ax4 - bx - - - 0 for some a- b- - - R- then the value of -i-16 1-xi - -i-16 1-xi is equal to

(x x_1)(x x_2)(x x_3)(x x_4)(x x_5) (x x_6) = x^6 +ax^4 + bx + λ ⇒ (1 x_1)(1 x_2).......(1 x_6) = 1 +a+ b + λ and (1+x_1)(1+x_2).....(1+x_6) = 1 + a b + λ ⇒∏_i= ...

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Byju's Answer

Standard XII

Mathematics

Slope of a Line Connecting Two Points

If x1 , x2 , ...

Question

If $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}$ are non-zero distinct real roots of the equation $x^{6} + a x^{4} + b x + λ = 0$ for some $a, b, λ \in R,$ then the value of $\prod_{i = 1}^{6} (1 - x_{i}) - \prod_{i = 1}^{6} (1 + x_{i})$ is equal to

2 λ

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Zero

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- 2 b

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2 b

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Solution

The correct option is D $2 b$
$(x - x_{1}) (x - x_{2}) (x - x_{3}) (x - x_{4}) (x - x_{5}) (x - x_{6}) = x^{6} + a x^{4} + b x + λ$
$\Rightarrow (1 - x_{1}) (1 - x_{2}) . . . . . . . (1 - x_{6}) = 1 + a + b + λ$ and $(1 + x_{1}) (1 + x_{2}) . . . . . (1 + x_{6}) = 1 + a - b + λ$
$\Rightarrow \prod_{i = 1}^{6} (1 - x_{i}) - \prod_{i = 1}^{6} (1 + x_{i}) = 2 b$

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If x1,x2,x3,x4,x5,x6 are non-zero distinct real roots of the equation x6+ax4+bx+λ=0 for some a,b,λ ∈ R, then the value of ∏6i=1(1−xi)−∏6i=1(1+xi) is equal to

The correct option is D 2b(x−x1)(x−x2)(x−x3)(x−x4)(x−x5)(x−x6)=x6+ax4+bx+λ ⇒(1−x1)(1−x2).......(1−x6)=1+a+b+λ and (1+x1)(1+x2).....(1+x6)=1+a−b+λ ⇒∏6i=1(1−xi)−∏6i=1(1+xi)=2b

If $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}$ are non-zero distinct real roots of the equation $x^{6} + a x^{4} + b x + λ = 0$ for some $a, b, λ \in R,$ then the value of $\prod_{i = 1}^{6} (1 - x_{i}) - \prod_{i = 1}^{6} (1 + x_{i})$ is equal to