Limits, Continuity & Derivatives · NDA previous year questions

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2013 28 questions

Q142013 · 11 Aug 2013
In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. What is the minimum number of people who were having diabetes and high blood pressure?
1. A.0
2. B.10
3. C.20
4. D.30
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Answer: C
Q712013 · 11 Aug 2013
Consider the following functions : 1. $f (x) = e^{x}$ , where $x > 0$ 2. $g (x) = ∣ x - 3∣$ Which of the above functions is/are continuous?
1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2
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Answer: C
Q722013 · 11 Aug 2013
What is the derivative of $x^{3}$ with respect to $x^{2}$ ?
1. A. $3 x^{2}$
2. B. $\frac{3 x}{2}$
3. C. $x$
4. D. $\frac{3}{2}$
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Answer: B
Q732013 · 11 Aug 2013
What is $lim_{x \to 2} \frac{2 - x}{x ^{3} - 8}$ equal to?
1. A. $\frac{1}{8}$
2. B. $- \frac{1}{8}$
3. C. $\frac{1}{12}$
4. D. $- \frac{1}{12}$
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Answer: D
Q742013 · 11 Aug 2013
A function $f : R \to R$ is defined as $f (x) = x^{2}$ for $x \geq 0$ and $f (x) = - x$ for $x < 0$ . Consider the following statements in respect of the above function : 1. The function is continuous at $x = 0$ . 2. The function is differentiable at $x = 0$ . Which of the above statements is/are correct?
1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2
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Answer: A
Q752013 · 11 Aug 2013
If $f (x) = 2 x^{2} + 3 x - 5$ , then what is $f^{'} (0) + 3 f^{'} (- 1)$ equal to?
1. A. $- 1$
2. B.0
3. C.1
4. D.2
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Answer: B
Q762013 · 11 Aug 2013
What is $lim_{x \to 0} \frac{1 - c o s x}{x}$ equal to?
1. A.0
2. B. $\frac{1}{2}$
3. C.1
4. D.2
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Answer: A
Q772013 · 11 Aug 2013
What is $lim_{x \to 0} \frac{c o s x}{π - x}$ equal to?
1. A.0
2. B. $π$
3. C. $\frac{1}{π}$
4. D.1
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Answer: C
Q782013 · 11 Aug 2013
What is $lim_{x \to 0} \frac{s i n 2 x + 4 x}{2 x + s i n 4 x}$ equal to?
1. A.0
2. B. $\frac{1}{2}$
3. C.1
4. D.2
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Answer: C
Q792013 · 11 Aug 2013
The maximum value of the function $f (x) = x^{3} + 2 x^{2} - 4 x + 6$ exists at
1. A. $x = - 2$
2. B. $x = 1$
3. C. $x = 2$
4. D. $x = - 1$
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Answer: A
Q802013 · 11 Aug 2013
The minimum value of the function $f (x) = ∣ x - 4∣$ exists at
1. A. $x = 0$
2. B. $x = 2$
3. C. $x = 4$
4. D. $x = - 4$
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Answer: C
Q812013 · 11 Aug 2013
What is $\int_{1}^{2} ln x d x$ equal to?
1. A. $ln 2$
2. B.1
3. C. $ln (\frac{4}{e})$
4. D. $ln (\frac{e}{4})$
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Answer: C
Q822013 · 11 Aug 2013
What is $\int \frac{d x}{4 + x ^{2}}$ equal to? where $c$ is an arbitrary constant.
1. A. $ln 4 + x^{2} + x + c$
2. B. $ln 4 + x^{2} - x + c$
3. C. $sin^{- 1} (\frac{x}{2}) + c$
4. D.None of the above
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Answer: A
Q892013 · 11 Aug 2013
What is the derivative of $sin (sin x)$ ?
1. A. $cos (cos x)$
2. B. $cos (sin x)$
3. C. $cos (sin x) cos x$
4. D. $cos (cos x) cos x$
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Answer: C
Q902013 · 11 Aug 2013
What is the derivative of $∣ x - 1∣$ at $x = 2$ ?
1. A. $- 1$
2. B.0
3. C.1
4. D.Derivative does not exist
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Answer: C
Q662013 · 14 Apr 2013
Consider the following statements in respect of a function f(x): 1. $f (x)$ is continuous at $x = a$ iff $lim_{x \to a} f (x)$ exists. 2. If $f (x)$ is continuous at a point, then $\frac{1}{f ( x )}$ is also continuous at that point. Which of the above statements is/are correct?
1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2
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Answer: D
Q672013 · 14 Apr 2013
If $2 x^{3} - 3 y^{2} = 7$ , what is $\frac{d y}{d x}$ equal to $(y \neq = 0)$ ?
1. A. $\frac{x ^{2}}{2 y}$
2. B. $\frac{x}{2 y}$
3. C. $\frac{x ^{2}}{y}$
4. D.None of the above
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Answer: C
Q682013 · 14 Apr 2013
The derivative of $∣ x ∣$ at $x = 0$
1. A.is 1
2. B.is $- 1$
3. C.is 0
4. D.does not exist
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Answer: D
Q692013 · 14 Apr 2013
Consider the function $f (x) = {x^{2}, 3 x - 2, x > 2 x \leq 2$ Which one of the following statements is correct in respect of the above function ?
1. A. $f (x)$ is derivable but not continuous at $x = 2$
2. B. $f (x)$ is continuous but not derivable at $x = 2$
3. C. $f (x)$ is neither continuous nor derivable at $x = 2$
4. D. $f (x)$ is continuous as well as derivable at $x = 2$
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Answer: B
Q702013 · 14 Apr 2013
Consider the following statements : 1. $lim_{x \to 0} sin \frac{1}{x}$ does not exist. 2. $lim_{x \to 0} x sin \frac{1}{x}$ exists. Which of the above statements is/are correct ?
1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2
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Answer: C
Q712013 · 14 Apr 2013
What is $lim_{x \to 0} \frac{s i n x - t a n x}{x}$ equal to ?
1. A.0
2. B.1
3. C. $- 1$
4. D.1/2
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Answer: A
Q722013 · 14 Apr 2013
What is $lim_{x \to 0} \frac{1 - 1 + x}{x}$ equal to ?
1. A.1/2
2. B. $- 1/2$
3. C.1
4. D. $- 1$
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Answer: B
Q732013 · 14 Apr 2013
If $y = sin (a x + b)$ , then what is $\frac{d ^{2} y}{d x ^{2}}$ at $x = - \frac{b}{a}$ , where $a, b$ are constants and $a \neq = 0$ ?
1. A.0
2. B. $- 1$
3. C. $sin (a - b)$
4. D. $sin (a + b)$
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Answer: A
Q762013 · 14 Apr 2013
The function $f (x) = x^{2} - 4 x, x \in [0, 4]$ attains minimum value at
1. A. $x = 0$
2. B. $x = 1$
3. C. $x = 2$
4. D. $x = 4$
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Answer: C
Q772013 · 14 Apr 2013
The curve $y = x e^{x}$ has minimum value equal to
1. A. $- \frac{1}{e}$
2. B. $\frac{1}{e}$
3. C. $- e$
4. D.e
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Answer: A