Match List I with List – II
| List – I | List – II | ||
| A. | An even function | I. | \(x^2 + \cos x\) |
| B. | For an even function, \(\int_{-a}^{a}\displaystyle f(x)\,dx =\) | II. | 0 |
| C. | If \(f(2a-x) = -f(x)\), then \(\int_{0}^{2a}\displaystyle f(x)\,dx =\) | III. | \(2\int_{0}^{a}\displaystyle f(x)\,dx\) |
| D. | An odd function | IV. | \(x^3 + \sin x\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-I, B-III, C-II, D-IV
(A) An even function
This matches (I) \(x^2 + \cos x\).
\(f(-x) = (-x)^2 + \cos(-x) = x^2 + \cos x = f(x)\)
(B) For an even function, \(\int_{-a}^{a} f(x)\,dx =\)
This matches (III) \(2\int_{0}^{a} f(x)\,dx\). For an even function the areas on the two sides of the y-axis are equal, so the integral is twice the integral over [0, a].
(C) If \(f(2a-x) = -f(x)\), then \(\int_{0}^{2a} f(x)\,dx =\)
This matches (II) 0. By the property
\( \displaystyle \int_{0}^{2a} f(x)\,dx = \int_{0}^{a} f(x)\,dx + \int_{0}^{a} f(2a-x)\,dx \)
\( \displaystyle = \int_{0}^{a} f(x)\,dx - \int_{0}^{a} f(x)\,dx = 0 \)
(D) An odd function
This matches (IV) \(x^3 + \sin x\).
\(f(-x) = (-x)^3 + \sin(-x) = -x^3 - \sin x = -f(x)\)
So the matching is A-I, B-III, C-II, D-IV.