If for non-zero x, af(x) + bf ((1)/(x) ) = (1)/(x) - 25 where a ≠ b then 12 f(x)dx is A) (1)/(a2 - b2) [ a(ln 2 - 25) + (47b)/(2) ] B) (1)/(a2 - b2) [ a(2ln 2 - 25) - (47b)/(2) ] C) (1)/(a2 - b2) [ a(2ln 2 - 25) + (47b)/(2) ] D) (1)/(a2 - b2) [ a(ln 2 - 25) - (47b)/(2) ]

B) (1)/(a2 - b2) [ a(2ln 2 - 25) - (47b)/(2) ]

C) (1)/(a2 - b2) [ a(2ln 2 - 25) + (47b)/(2) ]

D) (1)/(a2 - b2) [ a(ln 2 - 25) - (47b)/(2) ]

GATE CSE 2015 Set 3 · Q45

MCQ2 marksMedium

Time00:00

If for non-zero $x$ , $af(x) + bf\left(\frac{1}{x}\right) = \frac{1}{x} - 25$ where $a \neq b$ then $\int_1^2 f(x)dx$ is

Select one answerSingle correct

Pick an option to check

Solution

Official answer: A) (1)/(a2 - b2) [ a(ln 2 - 25) + (47b)/(2) ]

Calculus

We are given the functional equation:

af(x) + bf\left(\frac{1}{x}\right) = \frac{1}{x} - 25 \quad \text{--- (1)}

Replacing $x$ with $\frac{1}{x}$ :

af\left(\frac{1}{x}\right) + bf(x) = x - 25 \quad \text{--- (2)}

Multiply equation (1) by $a$ and equation (2) by $b$ :

a^2 f(x) + ab f\left(\frac{1}{x}\right) = \frac{a}{x} - 25a

b^2 f(x) + ab f\left(\frac{1}{x}\right) = bx - 25b

Subtracting the second equation from the first:

(a^2 - b^2) f(x) = \frac{a}{x} - bx - 25(a - b)

f(x) = \frac{1}{a^2 - b^2} \left[ \frac{a}{x} - bx - 25(a - b) \right]

Now, integrate $f(x)$ from $1$ to $2$:

\int_1^2 f(x) \, dx = \frac{1}{a^2 - b^2} \int_1^2 \left( \frac{a}{x} - bx - 25(a - b) \right) dx

= \frac{1}{a^2 - b^2} \left[ a \ln x - \frac{b x^2}{2} - 25(a - b)x \right]_1^2

= \frac{1}{a^2 - b^2} \left[ a(\ln 2 - \ln 1) - \frac{b}{2}(4 - 1) - 25(a - b)(2 - 1) \right]

= \frac{1}{a^2 - b^2} \left[ a \ln 2 - \frac{3b}{2} - 25a + 25b \right]

= \frac{1}{a^2 - b^2} \left[ a(\ln 2 - 25) + b\left(25 - \frac{3}{2}\right) \right]

= \frac{1}{a^2 - b^2} \left[ a(\ln 2 - 25) + \frac{47b}{2} \right]

This matches option (A).

Try the question first, then check your work.

Think the official key or this write-up is wrong?

Examiner trap

This Calculus previous year question is Q45 of the GATE CSE 2015 Set 3 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

Free diagnostic

Take the 20-minute diagnostic: the 3 topics that cost you the most marks, and a rank range. Or sit 5 related Calculus PYQs now with no account.

An official question by email at 7 AM IST, with a link to its verified solution. Unsubscribe in one click.

Calculus has 40 GATE CSE PYQs across 25 papers from 2011 to 2026. 8 of them are 2-mark MCQs like this one, and GATE CSE 2015 Set 3 has 3.

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.