Question
Assertion: $\int_{-1}^1 \frac{x^3}{\cos x} dx = 0$
Reason: If $f(x)$ is a continuous function defined on $[0, a]$, then $\int_0^a f(x) dx = \int_0^a f(a-x) dx$
- aBoth Assertion and Reason are true, and Reason is the correct explanation of Assertion.
- bBoth Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
- cAssertion is true and Reason is false.
- dAssertion is false and Reason is true.
Answer
Answer
AIWritten by AI - it can contain mistakes.
Correct option: b
(b) Both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.
Assertion: Let $f(x) = \frac{x^3}{\cos x}$. Then $f(-x) = \frac{(-x)^3}{\cos(-x)} = -\frac{x^3}{\cos x} = -f(x)$, so $f(x)$ is an odd function. Hence $\int_{-1}^1 \frac{x^3}{\cos x} dx = 0$. The assertion is true.
Reason: The property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ is true for any continuous function on $[0, a]$, but the integral in the assertion is evaluated using the property of odd functions on $[-a, a]$, not this property.
Therefore, both are true, but Reason is not the correct explanation of Assertion.
From ISC 2027 Specimen Mathematics Paper 1, question 1(xvii).