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Indian Navy SSR (Sailor) Mathematics — practice by topic

Browse 372 questions across 80 Mathematics topics. Choose what you want to revise, then open a question to check its answer and worked solution.

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Find the length of the latus rectum of the parabola $y^2 = 8x$.Read the answer and worked solution →The number of ways to arrange $n$ distinct objects in a row is:Read the answer and worked solution →If $\tan^{-1}\left(\dfrac{1}{2}\right) + \tan^{-1}\left(\dfrac{1}{3}\right) = ?$Read the answer and worked solution →The magnitude of the zero (null) vector $\vec 0$ is:Read the answer and worked solution →

Worked practice questions

Find the length of the latus rectum of the parabola $y^2 = 8x$.

  1. $2$
  2. $4$
  3. $8$
  4. $16$
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Correct answer: C. $8$

Compare $y^2 = 8x$ with the standard form $y^2 = 4ax$: $4a = 8 \Rightarrow a = 2$. Latus rectum length = $4a = \mathbf{8}$. **Geometric meaning.** The latus rectum is the chord through the focus, perpendicular to the axis. For $y^2 = 4ax$ the focus is $(a, 0)$ plugging $x = a$ gives $y^2 = 4a^2$ ⇒ $y = \pm 2a$ ⇒ chord length $4a$.

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The number of ways to arrange $n$ distinct objects in a row is:

  1. $2^n$
  2. $n$
  3. $n!$
  4. $n^2$
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Correct answer: C. $n!$

Arrangements of $n$ distinct objects in a row: $n$ choices for the first position then $n-1$ for the second, $n-2$ for the third, etc. By the multiplication principle: $n \times (n-1) \times \ldots \times 1 = n!$ For 5 books on a shelf: $5! = 120$ arrangements. For 10 students in a line: $10! = 3,628,800$.

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If $\tan^{-1}\left(\dfrac{1}{2}\right) + \tan^{-1}\left(\dfrac{1}{3}\right) = ?$

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

Use the identity $\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\dfrac{a + b}{1 - ab}\right)$, valid when $ab < 1$. With $a = 1/2$, $b = 1/3$: $ab = 1/6 < 1$, so the identity applies cleanly. $\dfrac{a + b}{1 - ab} = \dfrac{1/2 + 1/3}{1 - 1/6} = \dfrac{5/6}{5/6} = 1$. So the sum equals $\tan^{-1}(1) = \dfrac{\pi}{4}$.

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