KVPY SA Exam 2016 Question Paper

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Question : 61 of 80

Marks: +1, -0

Part-2 Mathematics

Suppose a is a positive real number such that

a ^5 - a ^3 + a = 2

. Then

$a^6 < 2$
$2 < a^6 < 3$
$3 < a^6 < 4$
$4 ≤ a^6$

Solution:

a^5- a^3 + a = 2

;

a ∈ R^{+}

Let

f(a) = a^5- a^3 + a - 2

; {Note f'(a) > 0 ∀ a ∈ R}
for

a^6 = 3 ⇒ a = 3^{1/6} = 1.2

{use calculator}
we get f(1.2) < 0 and at

a = 4^{1/6}

⇒ f(4^{1/6}) > 0

so one root in a ∈ (3, 4)

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