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Indian Navy MR Vectors & 3D Geometry — practice questions

Explore 61 free Vectors & 3D Geometry questions in Indian Navy MR Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Vectors & 3D Geometry questions and exercises with answers

Showing questions 1–30 of 61.

$\sin(180° - \theta)$ equals:Read the answer and worked solution →For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:Read the answer and worked solution →sin(A + B) equals:Read the answer and worked solution →If tan θ = 1/2 (acute), find sin 2θ:Read the answer and worked solution →If sin θ = 12/13 (acute θ in Q1), find tan(θ/2):Read the answer and worked solution →Two non-zero vectors a and b are PERPENDICULAR if and only if: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 →

Worked practice questions

$\sin(180° - \theta)$ equals:

  1. $-\cos\theta$
  2. $-\sin\theta$
  3. $\sin\theta$
  4. $\cos\theta$
Show answer and solution

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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For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:

  1. $1$
  2. $0$
  3. $2.0$
  4. $1.25$
Show answer and solution

Correct answer: D. $1.25$

$F(0.5) = 3(0.5) - 2(0.5)^3 = 1.5 - 2(0.125) = 1.5 - 0.25 = 1.25$. (Note: this exceeds 1, which signals a problem with the source p.d.f. — but the question is purely about evaluating the given $F$ formula.)

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sin(A + B) equals:

  1. cosA cosB - sinA sinB
  2. sinA cosB - cosA sinB
  3. sinA cosB + cosA sinB
  4. sinA + sinB
Show answer and solution

Correct answer: C. sinA cosB + cosA sinB

**Compound angle formula:** sin(A + B) = sinA cosB + cosA sinB. (And sin(A - B) = sinA cosB - cosA sinB.)

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