Let m, n, p, q be four positive integers. If ₀²π} ^m x ⁿ x \, dx = 4 ₀^{π/2} ^m x ⁿ x \, dx ₀²π} ^p x ⁿ x \, dx = 0 ₀^{π} ^p x ^q x \, dx = 0, a = m+n+p { and } b = m+n+q, { then }

Correct Answer is : Both aaa and bbb are odd numbers

a is even number and b is odd number

a is odd number and b is even number

Correct Answer is : Both aaa and bbb are odd numbers

Test Index

TG EAMCET 2 May 2025 Shift 2 Paper

Question : 77 of 160

Marks: +1, -0

Let

m, n, p, q

\int\limits_{0}^{2\pi} \sin^m x \cos^n x \, dx = 4 \int\limits_{0}^{\frac{\pi}{2}} \sin^m x \cos^n x \, dx \int\limits_{0}^{2\pi} \sin^p x \cos^n x \, dx = 0

\int\limits_{0}^{\pi} \sin^p x \cos^q x \, dx = 0, a = m+n+p \text{ and } b = m+n+q, \text{ then }

Solution:

Given,

m, n, p, q

be four positive inegers

\int\limits_{0}^{2\pi} \sin^m x \cos^n x \, dx = 4 \int\limits_{0}^{\frac{\pi}{2}} \sin^m x \cos^n x \, dx \dots \dots (i)

\int\limits_{0}^{2\pi} \sin^p x \cos^n x \, dx = 0

\text{ And } \int\limits_{0}^{2\pi} \sin^p x \cos^q x \, dx = 0

a = m+n+p, b = m+n+q

From Eq. (i), we know that

\int\limits_{0}^{2\pi} f(x) \, dx = 4 \int\limits_{0}^{\frac{\pi}{2}} f(x)

f

is periodic with

\pi

and symmetric in all four quadrants

\Rightarrow \; m

and

n

are even

\int\limits_{0}^{2\pi} \sin^p x \cos^n x \, dx = 0

\Rightarrow

Integrand is odd over symmetric interval

[0, 2\pi]

Since,

n

is even so,

p

is odd also,

\int\limits_{0}^{\pi} \sin^p x \cos^q x \, dx = 0

\Rightarrow \sin^p x \cos^q x \text{ is odd about } x = \frac{\pi}{2}

\Rightarrow q \text{ is odd }

\therefore a = m+n+p

= \text{ even } + \text{ even } + \text{ odd } = \text{ odd }

\text{ And } b = m+n+p

= \text{ even } + \text{ even } + \text{ odd } = \text{ odd }

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