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JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Area Under Curves

The odd natural number a, such that the area of the region bounded by

y = 1, y = 3, x = 0, x = y^{a}

is

\frac{364}{3}

, equal to :

(A)3
(B)5✓ Correct
(C)7
(D)9

Answer: (B)

Solution

The area is given by:

A = \int_{1}^{3} y^{a} d y = [\frac{y ^{a + 1}}{a + 1}]_{1}^{3} = \frac{3 ^{a + 1} - 1}{a + 1} = \frac{364}{3}

Thus

3^{a + 1} - 1 = \frac{364 ( a + 1 )}{3}

.

Testing odd values: for

a = 5

,

3^{6} - 1 = 729 - 1 = 728 = \frac{364 \cdot 6}{3} = \frac{2184}{3} = 728

. ✓

Therefore

a = 5

.

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Single correct option

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