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UPCET / UPSEE Properties — practice questions

Explore 3 free Properties questions in UPCET / UPSEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Properties questions and exercises with answers

Showing questions 1–3 of 3.

The value of $\int_a^a f(x)\,dx$ is:Read the answer and worked solution →For a definite integral with limits swapped, which identity holds?Read the answer and worked solution →For $a < c < b$, the integral $\int_a^b f(x)\,dx$ equals:Read the answer and worked solution →

Worked practice questions

The value of $\int_a^a f(x)\,dx$ is:

  1. $f(a)$
  2. $0$
  3. $2f(a)$
  4. Undefined
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Correct answer: B. $0$

By the Fundamental Theorem, $\int_a^a f(x)\,dx = g(a) - g(a) = 0$ for any antiderivative $g$. The interval has zero width, so the signed area is zero.

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For a definite integral with limits swapped, which identity holds?

  1. $\int_a^b f\,dx = \int_b^a f\,dx$
  2. $\int_a^b f\,dx = -\int_b^a f\,dx$
  3. $\int_a^b f\,dx = \int_b^a f\,dx + c$
  4. $\int_a^b f\,dx = 1/\int_b^a f\,dx$
Show answer and solution

Correct answer: B. $\int_a^b f\,dx = -\int_b^a f\,dx$

**Property II:** swapping limits flips the sign. $\int_a^b f\,dx = g(b)-g(a) = -[g(a)-g(b)] = -\int_b^a f\,dx$.

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For $a < c < b$, the integral $\int_a^b f(x)\,dx$ equals:

  1. $\int_a^c f(x)\,dx - \int_c^b f(x)\,dx$
  2. $\int_a^c f(x)\,dx + \int_c^b f(x)\,dx$
  3. $\int_a^c f(x)\,dx \cdot \int_c^b f(x)\,dx$
  4. $\int_c^b f(x)\,dx - \int_a^c f(x)\,dx$
Show answer and solution

Correct answer: B. $\int_a^c f(x)\,dx + \int_c^b f(x)\,dx$

Property IV (additivity): the integral can be split at any interior point $c$: $\int_a^b = \int_a^c + \int_c^b$.

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More Mathematics topics

SequencesProbability & StatisticsBinomial TheoremSequences and SeriesGeometryLinear ProgrammingLimits and ContinuityPermutations and Combinations