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UPCET / UPSEE Sets Basics — practice questions

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

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Sets Basics questions and exercises with answers

Showing questions 1–30 of 82.

$\sin(180° - \theta)$ 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 →General solution of cos θ = 0:Read the answer and worked solution →If tan θ + cot θ = 2, find sin 2θ:Read the answer and worked solution →

Worked practice questions

$\sin(180° - \theta)$ 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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$\sin^2\theta + \cos^2\theta$ equals:

  1. $2$
  2. $0$
  3. $\tan\theta$
  4. $1$
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Correct answer: D. $1$

The **Pythagorean identity** $\sin^2\theta + \cos^2\theta = 1$ holds for every real $\theta$. It comes from Pythagoras applied to the unit circle: any point on it has coordinates $(\cos\theta, \sin\theta)$ with distance 1 from origin so the sum of squared coordinates is 1. Companion identities: - Dividing by $\cos^2\theta$: $1 + \tan^2\theta = \sec^2\theta$. - Dividing by $\sin^2\theta$: $\cot^2\theta + 1 = \csc^2\theta$.

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$\sin(A + B) = $

  1. $\sin A + \sin B$
  2. $\sin A \cos B - \cos A \sin B$
  3. $\sin A \cos B + \cos A \sin B$
  4. $\cos A \cos B - \sin A \sin B$
Show answer and solution

Correct answer: C. $\sin A \cos B + \cos A \sin B$

Angle-sum formula for sine: $\sin(A + B) = \sin A \cos B + \cos A \sin B$. Companion: $\sin(A - B) = \sin A \cos B - \cos A \sin B$ (option C — common trap, the sine of the *difference*). For cosine: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ (sign flips inside vs outside, option D is $\cos(A+B)$). Sine is not linear in its argument: $\sin(A+B) \neq \sin A + \sin B$ generally.

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SequencesProbability & StatisticsBinomial TheoremSequences and SeriesGeometryLinear ProgrammingLimits and ContinuityPermutations and Combinations