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UPCET / UPSEE Application of Integrals — practice questions

Explore 82 free Application of Integrals questions in UPCET / UPSEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Application of Integrals questions and exercises with answers

Showing questions 1–30 of 82.

A fair coin is tossed twice. How many outcomes are there in the sample space?Read the answer and worked solution →A fair six-sided die is rolled once. What is the probability of getting an even number?Read the answer and worked solution →$\sin(180° - \theta)$ equals:Read the answer and worked solution →$\sin^2\theta + \cos^2\theta$ equals:Read the answer and worked solution →$\sin(A + B) = $Read the answer and worked solution →$\cos 2\theta$ equals:Read the answer and worked solution →If $\sin\theta + \cos\theta = 1$, then $\sin\theta \cdot \cos\theta$ equals:Read the answer and worked solution →$\cos 3\theta$ equals:Read the answer and worked solution →sin(0°) equals:Read the answer and worked solution →cos(90°) equals:Read the answer and worked solution →sin²θ + cos²θ equals:Read the answer and worked solution →tan(45°) equals:Read the answer and worked solution →In which quadrant is sin θ positive and cos θ negative?Read the answer and worked solution →sec θ is defined as:Read the answer and worked solution →sin(A + B) equals:Read the answer and worked solution →cos(A + B) equals:Read the answer and worked solution →sin(60°) cos(30°) + cos(60°) sin(30°) equals:Read the answer and worked solution →If sin θ = 3/5 (θ in Q1), find cos θ:Read the answer and worked solution →Find cos(75°) using cos(A + B):Read the answer and worked solution →sin(2θ) equals:Read the answer and worked solution →Find sin(15°):Read the answer and worked solution →If tan θ = 1/2 (acute), find sin 2θ:Read the answer and worked solution →For 0 ≤ θ < 2π, solutions of sin θ = 1/2:Read the answer and worked solution →1 - cos²θ equals:Read the answer and worked solution →sin(A) + sin(B) equals (sum-to-product):Read the answer and worked solution →In a right triangle, if hypotenuse = 13 and one leg = 5, sin θ for angle θ opposite to that leg:Read the answer and worked solution →tan(A + B) equals:Read the answer and worked solution →sin(180° - θ) equals:Read the answer and worked solution →In triangle ABC with sides a, b, c opposite to angles A, B, C: sine rule states:Read the answer and worked solution →If sin θ + cos θ = 1, then sin θ × cos θ equals:Read the answer and worked solution →

Worked practice questions

A fair coin is tossed twice. How many outcomes are there in the sample space?

  1. $2$
  2. $4$
  3. $6$
  4. $8$
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Correct answer: B. $4$

Each toss has 2 possible outcomes (H or T), and the tosses are independent. By the multiplication principle, the sample space has $2 \times 2 = 4$ outcomes: {HH, HT, TH, TT}.

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A fair six-sided die is rolled once. What is the probability of getting an even number?

  1. $\dfrac{1}{6}$
  2. $\dfrac{1}{2}$
  3. $\dfrac{1}{3}$
  4. $\dfrac{2}{3}$
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Correct answer: B. $\dfrac{1}{2}$

The even outcomes are {2, 4, 6}, so the number of favourable outcomes is 3. The total sample space has 6 outcomes. $P(\text{even}) = \dfrac{3}{6} = \dfrac{1}{2}$.

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$\sin(180° - \theta)$ equals:

  1. $-\cos\theta$
  2. $-\sin\theta$
  3. $\sin\theta$
  4. $\cos\theta$
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Correct answer: C. $\sin\theta$

Supplementary-angle identity: $\sin(180° - \theta) = \sin\theta$. Reason: sine is positive in the second quadrant (where $180° - \theta$ lies for $0° < \theta < 180°$), and the reference angle is $\theta$. Companion identities: - $\cos(180° - \theta) = -\cos\theta$ - $\tan(180° - \theta) = -\tan\theta$

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