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Indian Navy AA (Artificer Apprentice) Limits and Continuity — practice questions

Explore 54 free Limits and Continuity 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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Limits and Continuity questions and exercises with answers

Showing questions 1–30 of 54.

$\sin^2\theta + \cos^2\theta$ 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 →lim_(x→0) [(1 + x)^(1/x)] =Read the answer and worked solution →What does the sign of f ′(x) tell you?Read the answer and worked solution →Complete the definition of a critical pointRead the answer and worked solution →Order the first derivative testRead the answer and worked solution →Match each sign change to its verdictRead the answer and worked solution →The second derivative testRead the answer and worked solution →True or false: maxima and minimaRead the answer and worked solution →Order the working rule for absolute maxima and minimaRead the answer and worked solution →Tap every point you must test for an absolute maximumRead the answer and worked solution →Local or absolute?Read the answer and worked solution →Complete the rate of change statementRead the answer and worked solution →The chain rule for related ratesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →True or false: increasing and decreasingRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Match each situation to the right toolRead the answer and worked solution →Tap every statement true of a local maximum at cRead the answer and worked solution →Reading the tests backRead the answer and worked solution →Which test is being described?Read the answer and worked solution →Assertion and reason: rates of changeRead the answer and worked solution →Spot the odd oneRead the answer and worked solution →Complete the second derivative testRead the answer and worked solution →True or false: scope of this chapterRead the answer and worked solution →Match each kind of matrix to its conditionRead 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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