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Indian Navy AA (Artificer Apprentice) Vectors & 3D Geometry — practice questions

Explore 77 free Vectors & 3D Geometry questions in Indian Navy AA (Artificer Apprentice) 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 77.

$\sin^2\theta + \cos^2\theta$ equals:Read the answer and worked solution →A **discrete** random variable is one whose possible values:Read the answer and worked solution →The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:Read the answer and worked solution →The midpoint of the segment joining $(2,3)$ and $(4,7)$ is: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 →Trace how trigonometry travelledRead the answer and worked solution →

Worked practice questions

$\sin^2\theta + \cos^2\theta$ 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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A **discrete** random variable is one whose possible values:

  1. Are finite or countably infinite
  2. Form an interval of real numbers (uncountably infinite)
  3. Are always negative
  4. Are obtained only by measurement
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Correct answer: A. Are finite or countably infinite

**Discrete RV:** finite or countably infinite values (typically non-negative integers — number of heads, defective items, etc.). Continuous RV has uncountably infinite values forming an interval.

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The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:

  1. $F(x) = P[X = x]$
  2. $F(x) = 1 - p$
  3. $F(x) = P[X \leq x] = \sum_{x_i \leq x} p_i$
  4. $F(x) = P[X > x]$
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Correct answer: C. $F(x) = P[X \leq x] = \sum_{x_i \leq x} p_i$

c.d.f. accumulates: $F(x) = P[X \leq x] = \sum_{x_i \leq x} p_i$. It is non-decreasing, starts at 0 (for $x < \min$), reaches 1 (at $x \geq \max$). For discrete RV it's a step function.

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