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UPCET / UPSEE Quadratic Equations — practice questions — page 2

Explore 83 free Quadratic Equations questions in UPCET / UPSEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Quadratic Equations questions and exercises with answers

Showing questions 31–60 of 83.

If sin θ = 12/13 (acute θ in Q1), find tan(θ/2):Read the answer and worked solution →Minimum value of sin²θ + cos⁴θ:Read the answer and worked solution →If sin θ = 12/13 (acute), find tan(θ/2):Read the answer and worked solution →In triangle with sides 7, 8, 9, find cos of the angle opposite to side 9 (use cosine rule):Read the answer and worked solution →General solution of sin θ = sin α:Read the answer and worked solution →Sort each inverse function into its domainRead the answer and worked solution →Tap every function whose domain is [−1, 1]Read the answer and worked solution →Match each inverse function to its domainRead the answer and worked solution →Match each inverse function to its principal value branchRead the answer and worked solution →Flip and pair: function with its principal rangeRead the answer and worked solution →Which principal ranges sit around zero, and which run 0 to π?Read the answer and worked solution →Tap every function whose principal range has a point removedRead the answer and worked solution →Name the principal value branchRead the answer and worked solution →Domain checkRead the answer and worked solution →Put the reasoning in order: why an inverse needs a restricted domainRead the answer and worked solution →True or false: existence of inversesRead the answer and worked solution →True or false: reading the tableRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Complete the summary tableRead the answer and worked solution →Complete the rule about restricting a domainRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Principal values worked in the chapterRead the answer and worked solution →Order the steps for finding a principal valueRead the answer and worked solution →True or false: principal valuesRead the answer and worked solution →Match each identity to the condition it needsRead the answer and worked solution →Assertion and reason: identitiesRead the answer and worked solution →Match each mathematician to the year your syllabus givesRead the answer and worked solution →Trace how trigonometry travelledRead the answer and worked solution →

Worked practice questions

If sin θ = 12/13 (acute θ in Q1), find tan(θ/2):

  1. 1/3
  2. 3/4
  3. 2/3
  4. 1/5
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Correct answer: C. 2/3

cos θ = √(1 − sin²θ) = √(1 − 144/169) = √(25/169) = 5/13 (positive in Q1). We have tan(θ/2) = (1 − cos θ)/sin θ. That is tan(θ/2) = (1 − 5/13)/(12/13). That is tan(θ/2) = (8/13)/(12/13). That is tan(θ/2) = 8/12. So tan(θ/2) = 2/3.

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Minimum value of sin²θ + cos⁴θ:

  1. 1
  2. 3/4
  3. 1/2
  4. 1/4
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Correct answer: B. 3/4

Let x = sin²θ. Then cos²θ = 1 - x, cos⁴θ = (1-x)². Expression = x + (1-x)² = x + 1 - 2x + x² = x² - x + 1. Derivative: 2x - 1 = 0 → x = 1/2. Min value = 1/4 - 1/2 + 1 = 3/4.

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If sin θ = 12/13 (acute), find tan(θ/2):

  1. 5/12
  2. Cannot determine
  3. 1/5
  4. 2/3
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Correct answer: C. 1/5

For acute θ with sinθ = 12/13: cosθ = 5/13 (positive in Q1). tan(θ/2) = sinθ/(1 + cosθ) = (12/13)/(1 + 5/13) = (12/13)/(18/13) = 12/18 = 2/3. Hmm answer A is 1/5 my calc gives 2/3. Let me recheck. tan(θ/2) = (1 - cosθ)/sinθ = (1 - 5/13)/(12/13) = (8/13)/(12/13) = 8/12 = 2/3. So actually 2/3 is correct, which matches option C, not A. Setting answer to C.

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