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UPSC NDA II 2023 Mathematics Solved Paper

Question Numbers: 31-32

Consider the following for the next items that follow:

Consider the equation

(1 - x)4 + (5 - x)^{4} = 82

Question : 31 of 120

Marks: +1, -0

What is the number of real roots of the equation?

Solution:

Concept:

Use substitution to reduce the equation to a quadratic in

t^2

Explanation:

Let

t = x - 3

. Then

(1-x) = -(t+2)

and

(5-x) = -(t-2)

. Since the fourth power removes the sign, the equation becomes

(t+2)^4 + (t-2)^4 = 82

Expand:

(t+2)^4 = t^4 + 8t^3 + 24t^2 + 32t + 16

(t-2)^4 = t^4 - 8t^3 + 24t^2 - 32t + 16

Add them:

2t^4 + 48t^2 + 32 = 82

Simplify:

2t^4 + 48t^2 - 50 = 0

→ divide by 2:

t^4 + 24t^2 - 25 = 0

Let

u = t^2 \ge 0

u^2 + 24u - 25 = 0

→

(u+25)(u-1)=0

→

u = 1

u = -25

(not possible).

Thus

t^2 = 1

→

t = \pm 1

. Hence

x = 3 \pm 1

, i.e.,

x = 2

and

x = 4

. Both are real.

Answer:

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