If a 1- a 2- a 3- a 4- a 5 are the roots of the equation 6 x 5-41 x 4-97 x 3-97 x 2-41 x -6-0- such that - a 1- - a 2- - a 3- - a 4- - a 5- thenwhich of the following is-are correct-A- the equation has three real roots and two imaginary roots- B- a 3- a 4- a 5 are in A-P-C- a 1- a 2- a 3 are in G-P-D- a 1- a 2- a 3 are in H-P-

The correct options are B a_3,a_4,a_5 are in A.P. nbsp; D a_1,a_2,a_3 are in H.P. nbsp; nbsp;This is a reciprocal equation of odd degree with opposite sig ...

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Byju's Answer

Standard XII

Mathematics

Harmonic Mean

If a 1, a 2, ...

Question

If $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ are the roots of the equation $6 x^{5} - 41 x^{4} + 97 x^{3} - 97 x^{2} + 41 x - 6 = 0,$ such that $| a_{1} | \leq | a_{2} | \leq | a_{3} | \leq | a_{4} | \leq | a_{5} |,$ then which of the following is/are correct?

the equation has three real roots and two imaginary roots.

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a_{3}, a_{4}, a_{5}

are in A.P.

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a_{1}, a_{2}, a_{3}

are in G.P.

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a_{1}, a_{2}, a_{3}

are in H.P.

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Solution

The correct options are
B $a_{3}, a_{4}, a_{5}$ are in A.P.
D $a_{1}, a_{2}, a_{3}$ are in H.P.
This is a reciprocal equation of odd degree with opposite signs.
$∴ x = 1$ is a root.
Dividing L.H.S. by $x - 1$ , the given equation reduces to
$6 x^{4} - 35 x^{3} + 62 x^{2} - 35 x + 6 = 0$
Divide above equation by $x^{2}$ ,
$6 (x^{2} + \frac{1}{x^{2}}) - 35 (x + \frac{1}{x}) + 62 = 0$
Put $x + \frac{1}{x} = y$
$\Rightarrow (x^{2} + \frac{1}{x^{2}}) = y^{2} - 2$
$\Rightarrow 6 (y^{2} - 2) - 35 y + 62 = 0 \Rightarrow (3 y - 10) (2 y - 5) = 0 \Rightarrow y = x + \frac{1}{x} = \frac{10}{3} or \frac{5}{2}$
$\Rightarrow 3 x^{2} - 10 x + 3 = 0 or 2 x^{2} - 5 x + 2 = 0 \Rightarrow (3 x - 1) (x - 3) = 0 or (2 x - 1) (x - 2) = 0 \Rightarrow x = \frac{1}{3}, 3 or x = \frac{1}{2}, 2$
$∴ (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}) = (\frac{1}{3}, \frac{1}{2}, 1, 2, 3)$
$\Rightarrow a_{3}, a_{4}, a_{5}$ are in A.P.
$\Rightarrow a_{1}, a_{2}, a_{3}$ are in H.P.

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If a1,a2,a3,a4,a5 are the roots of the equation 6x5−41x4+97x3−97x2+41x−6=0, such that |a1|≤|a2|≤|a3|≤|a4|≤|a5|, then which of the following is/are correct?

If $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ are the roots of the equation $6 x^{5} - 41 x^{4} + 97 x^{3} - 97 x^{2} + 41 x - 6 = 0,$ such that $| a_{1} | \leq | a_{2} | \leq | a_{3} | \leq | a_{4} | \leq | a_{5} |,$ then which of the following is/are correct?