The equation a 8 X 8- a 7 X 7- a 6 X 6- a 0-0 has all its roots positive and real where a -8-1- a 7-4- a 0-1-28- then which of thefollowing is-are true- A- a 1-1- ggB- a 1-1-244C- a 2-D- a 3-1-22

The correct options are B a_1=1/2^4 C a_2=7/2^4Let the roots be α_1, α_2,⋯α_8. then, α_1, α_2,⋯α_8=4 α_1 α_2 ⋯α_8 = 1/2^8 ⇒ (α_1 α_2 ⋯α_8)^1/8=1/2 =α_1+ ...

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Standard XII

Mathematics

Common Roots

The equation ...

Question

The equation $a_{8} X^{8} + a_{7} X^{7} + a_{6} X^{6} + \dots + a_{0} = 0$ has all its roots positive and real $(w h e r e a_{8} = 1, a_{7} = - 4, a_{0} = \frac{1}{2^{8}})$ , then which of the following is/are true?

a_{1} = \frac{1}{2^{8}}

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a_{1} = \frac{1}{2^{4}}

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a_{2} = \frac{7}{2^{4}}

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a_{3} = \frac{1}{2^{2}}

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Solution

The correct options are
B $a_{1} = \frac{1}{2^{4}}$
C $a_{2} = \frac{7}{2^{4}}$
Let the roots be $α_{1}, α_{2}, \dots α_{8} .$ then,
$α_{1}, α_{2}, \dots α_{8} = 4 α_{1} α_{2} \dots α_{8} = \frac{1}{2^{8}} \Rightarrow (α_{1} α_{2} \dots α_{8})^{\frac{1}{8}} = \frac{1}{2}$
$= \frac{α_{1} + α_{2} + \dots + α_{8}}{8} = (α_{1} α_{2} \dots α_{8})^{\frac{1}{8}}$
$\Rightarrow A M = G M \Rightarrow$ all the roots are equal $\frac{1}{2}$
$\Rightarrow a_{1} = -^{8} C_{7} {(\frac{1}{2})}^{7} = - \frac{1}{2^{4}} a_{2} = -^{8} C_{6} {(\frac{1}{2})}^{6} = \frac{7}{2^{4}} a n d a_{3} = -^{8} C_{5} {(\frac{1}{2})}^{5}$

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The equation $a_{8} x^{8} + a_{7} x^{7} + a_{6} x^{6} + . . . + a_{0} = 0$ has all its roots positive and real $(w h e r e a_{8} = 1, a_{7} = - 4, a_{0} = {\frac{1}{2}}^{8})$ , then

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The equation a8X8+a7X7+a6X6+⋯+a0=0 has all its roots positive and real (where a8=1,a7=−4,a0=128), then which of the following is/are true?

The correct options are B a1=124 C a2=724Let the roots be α1, α2,⋯α8. then, α1, α2,⋯α8=4 α1α2⋯α8=128 ⇒(α1α2⋯α8)18=12 =α1+α2+⋯+α88=(α1α2⋯α8)18 ⇒AM=GM⇒ all the roots are equal 12 ⇒a1=−8C7(12)7=−124 a2=−8C6(12)6=724 and a3=−8C5(12)5

The equation $a_{8} X^{8} + a_{7} X^{7} + a_{6} X^{6} + \dots + a_{0} = 0$ has all its roots positive and real $(w h e r e a_{8} = 1, a_{7} = - 4, a_{0} = \frac{1}{2^{8}})$ , then which of the following is/are true?