Limits, Continuity & Derivatives · NDA previous year questions

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2015 22 questions

Q652015 · 27 Sept 2015
Consider the function $f (x) = ⎩ ⎨ ⎧ a x - 2 - 1 a + 2 (x - 1)^{2} for - 2 < x < - 1 for - 1 \leq x \leq 1 for 1 < x < 2$ What is the value of $a$ for which $f (x)$ is continuous at $x = - 1$ and $x = 1$ ?
1. A. $- 1$
2. B.1
3. C.0
4. D.2
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Answer: A
Q662015 · 27 Sept 2015
The function $f (x) = \frac{1 - s i n x + c o s x}{1 + s i n x + c o s x}$ is not defined at $x = π$ . The value of $f (π)$ so that $f (x)$ is continuous at $x = π$ is
1. A. $- \frac{1}{2}$
2. B. $\frac{1}{2}$
3. C. $- 1$
4. D.1
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Answer: C
Q692015 · 27 Sept 2015
Passage
For the next two (02) items that follow: Consider the function
1. A.1
2. B.0
3. C. $- 1$
4. D. $- 2$
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Answer: C
Q712015 · 27 Sept 2015
The derivative of $ln (x + sin x)$ with respect to $(x + cos x)$ is
1. A. $\frac{1 + c o s x}{( x + s i n x ) ( 1 - s i n x )}$
2. B. $\frac{1 - c o s x}{( x + s i n x ) ( 1 + s i n x )}$
3. C. $\frac{1 - c o s x}{( x - s i n x ) ( 1 + c o s x )}$
4. D. $\frac{1 + c o s x}{( x - s i n x ) ( 1 - c o s x )}$
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Answer: A
Q722015 · 27 Sept 2015
If $y = cot^{- 1} [\frac{1 + s i n x + 1 - s i n x}{1 + s i n x - 1 - s i n x}]$ , where $0 < x < \frac{π}{2}$ , then $\frac{d y}{d x}$ is equal to
1. A. $\frac{1}{2}$
2. B.2
3. C. $sin x + cos x$
4. D. $sin x - cos x$
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Answer: A
Q732015 · 27 Sept 2015
The function $f (x) = \frac{x ^{2}}{e ^{x}}$ is monotonically increasing if
1. A. $x < 0 only$
2. B. $x > 2 only$
3. C. $0 < x < 2$
4. D. $x \in (- \infty, 0) \cup (2, \infty)$
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Answer: C
Q742015 · 27 Sept 2015
If $x^{a} y^{b} = (x - y)^{a + b}$ , then the value of $\frac{d y}{d x} - \frac{y}{x}$ is equal to
1. A. $\frac{a}{b}$
2. B. $\frac{b}{a}$
3. C.1
4. D.0
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Answer: D
Q752015 · 27 Sept 2015
If $s = t^{2} + 1$ , then $\frac{d ^{2} s}{d t ^{2}}$ is equal to
1. A. $\frac{1}{s}$
2. B. $\frac{1}{s ^{2}}$
3. C. $\frac{1}{s ^{3}}$
4. D. $\frac{1}{s ^{4}}$
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Answer: C
Q762015 · 27 Sept 2015
Consider the following statements: 1. The function $f (x) = x^{2} + 2 cos x$ is increasing in the interval $(0, π)$ 2. The function $f (x) = ln (1 + x^{2} - x)$ is decreasing in the interval $(- \infty, \infty)$ 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
Q772015 · 27 Sept 2015
Consider the following statements: 1. $f (x) = ln x$ is an increasing function on $(0, \infty)$ . 2. $f (x) = e^{x} - x (ln x)$ is an increasing function on $(1, \infty)$ . 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
Q782015 · 27 Sept 2015
For the next two (02) items that follow: Consider the function $f (x) = (\frac{1}{x})^{2 x^{2}}, where x > 0$ At what value of $x$ does the function attain maximum value?
1. A. $e$
2. B. $e$
3. C. $\frac{1}{e}$
4. D. $\frac{1}{e}$
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Answer: C
Q792015 · 27 Sept 2015
For the next two (02) items that follow: Consider the function $f (x) = (\frac{1}{x})^{2 x^{2}}, where x > 0$ The maximum value of the function is
1. A. $e$
2. B. $e^{\frac{2}{e}}$
3. C. $e^{\frac{1}{e}}$
4. D. $\frac{1}{e}$
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Answer: C
Q802015 · 27 Sept 2015
Consider the function $f (x) = - 2 x^{3} - 9 x^{2} - 12 x + 1$ The function $f (x)$ is an increasing function in the interval
1. A. $(- 2, - 1)$
2. B. $(- \infty, - 2)$
3. C. $(- 1, 2)$
4. D. $(- 1, \infty)$
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Answer: A
Q812015 · 27 Sept 2015
Consider the function $f (x) = - 2 x^{3} - 9 x^{2} - 12 x + 1$ The function $f (x)$ is a decreasing function in the interval
1. A. $(- 2, - 1)$
2. B. $(- \infty, - 2) only$
3. C. $(- 1, \infty) only$
4. D. $(- \infty, - 2) \cup (- 1, \infty)$
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Answer: D
Q602015 · 19 Apr 2015
Consider the following statements : 1. $f (x) = [x]$ , where $[.]$ is the greatest integer function, is discontinuous at $x = n$ , where $n \in Z$ . 2. $f (x) = cot x$ is discontinuous at $x = nπ$ , where $n \in Z$ . 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
Q612015 · 19 Apr 2015
Passage
For the next two (2) items that follow : Given a function
Consider the function $f (x) = ⎩ ⎨ ⎧ - 1 a x + b 1 if x \leq 0 if 0 < x < 1 if x \geq 1$ where $a, b$ are constants. The function is continuous everywhere. What is the value of $b$ ?
1. A. $- 1$
2. B. $1$
3. C. $0$
4. D. $2$
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Answer: A
Q632015 · 19 Apr 2015
Passage
For the next two (2) items that follow : Given a function
Consider the function $f (x) = {\frac{α c o s x}{π - 2 x} 3 if x \neq = \frac{π}{2} if x = \frac{π}{2}$ which is continuous at $x = \frac{π}{2}$ , where $α$ is a constant. What is the value of $α$ ?
1. A. $6$
2. B. $3$
3. C. $2$
4. D. $1$
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Answer: A
Q672015 · 19 Apr 2015
What is the derivative of $tan^{- 1} (\frac{1 + x ^{2} - 1}{x})$ with respect to $tan^{- 1} x$ ?
1. A. $0$
2. B. $1/2$
3. C. $1$
4. D. $x$
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Answer: B
Q732015 · 19 Apr 2015
Consider the following statements: 1. $y = \frac{e ^{x} + e ^{- x}}{2}$ is an increasing function on $[0, \infty)$ . 2. $y = \frac{e ^{x} - e ^{- x}}{2}$ is an increasing function on $(- \infty, \infty)$ . 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
Q742015 · 19 Apr 2015
For the next two (2) items that follow : Consider the function $f (x) = \frac{x ^{2} - 1}{x ^{2} + 1}$ , where $x \in R$ . At what value of $x$ does $f (x)$ attain minimum value?
1. A. $- 1$
2. B. $0$
3. C. $1$
4. D. $2$
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Answer: B
Q752015 · 19 Apr 2015
Consider the function $f (x) = \frac{x ^{2} - 1}{x ^{2} + 1}$ , where $x \in R$ . What is the minimum value of $f (x)$ ?
1. A. $0$
2. B. $1/2$
3. C. $- 1$
4. D. $2$
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Answer: C
Q762015 · 19 Apr 2015
Consider the function $f (x) = 0.75 x^{4} - x^{3} - 9 x^{2} + 7$ What is the maximum value of the function?
1. A.1
2. B.3
3. C.7
4. D.9
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Answer: C