Limits, Continuity & Derivatives · NDA previous year questions

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2017 49 questions

Q1172017 · 10 Sept 2017
Consider the following statements : Statement I : $x > sin x$ for all $x > 0$ Statement II : $f (x) = x - sin x$ is an increasing function for all $x > 0$ Which one of the following is correct in respect of the above statements?
1. A.Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I
2. B.Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true
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Answer: A
Q72017 · 23 Apr 2017
If $∣ z + 4∣ \leq 3$ , then the maximum value of $∣ z + 1∣$ is
1. A.0
2. B.4
3. C.6
4. D.10
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Answer: C
Q392017 · 23 Apr 2017
The maximum value of $sin (x + \frac{π}{6}) + cos (x + \frac{π}{6})$ in the interval $(0, \frac{π}{2})$ is attained at
1. A. $\frac{π}{12}$
2. B. $\frac{π}{6}$
3. C. $\frac{π}{3}$
4. D. $\frac{π}{2}$
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Answer: A
Q402017 · 23 Apr 2017
If $K = sin (\frac{π}{18}) sin (\frac{5 π}{18}) sin (\frac{7 π}{18})$ , then what is the value of $K$ ?
1. A. $\frac{1}{2}$
2. B. $\frac{1}{4}$
3. C. $\frac{1}{8}$
4. D. $\frac{1}{16}$
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Answer: C
Q412017 · 23 Apr 2017
The expression $\frac{s i n α + s i n β}{c o s α + c o s β}$ is equal to
1. A. $tan (\frac{α + β}{2})$
2. B. $cot (\frac{α + β}{2})$
3. C. $sin (\frac{α + β}{2})$
4. D. $cos (\frac{α + β}{2})$
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Answer: A
Q422017 · 23 Apr 2017
If $sin θ = 3 sin (θ + 2 α)$ , then the value of $tan (θ + α) + 2 tan α$ is equal to
1. A. $- 1$
2. B.0
3. C.1
4. D.2
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Answer: B
Q432017 · 23 Apr 2017
What is the value of $tan 1 8^{\circ}$ ?
1. A. $\frac{5 - 1}{10 + 2 5}$
2. B. $\frac{5 - 1}{10 + 5}$
3. C. $\frac{10 + 2 5}{5 - 1}$
4. D. $\frac{10 + 5}{5 - 1}$
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Answer: A
Q442017 · 23 Apr 2017
If $tan (α + β) = 2$ and $tan (α - β) = 1$ , then $tan (2 α)$ is equal to
1. A. $- 3$
2. B. $- 2$
3. C. $- \frac{1}{3}$
4. D.1
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Answer: A
Q452017 · 23 Apr 2017
Consider the following for triangle $A B C$ : 1. $sin (\frac{B + C}{2}) = cos (\frac{A}{2})$ 2. $tan (\frac{B + C}{2}) = cot (\frac{A}{2})$ 3. $sin (B + C) = cos A$ 4. $tan (B + C) = - cot A$ Which of the above are correct?
1. A.1 and 3
2. B.1 and 2
3. C.1 and 4
4. D.2 and 3
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Answer: B
Q462017 · 23 Apr 2017
If $sec θ - cosec θ = \frac{4}{3}$ , then what is $(sin θ - cos θ)$ equal to?
1. A. $- 2$ only
2. B. $\frac{1}{2}$ only
3. C.Both $- 2$ and $\frac{1}{2}$
4. D.Neither $\frac{1}{2}$ nor $- 2$
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Answer: B
Q472017 · 23 Apr 2017
What is the maximum value of the function $f (x) = 4 sin^{2} x + 1$ ?
1. A.5
2. B.3
3. C.2
4. D.1
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Answer: A
Q732017 · 23 Apr 2017
Let $f (x + y) = f (x) f (y)$ for all $x$ and $y$ . Then what is $f^{'} (5)$ equal to [where $f^{'} (x)$ is the derivative of $f (x)$ ]?
1. A. $f (5) f^{'} (0)$
2. B. $f (5) - f^{'} (0)$
3. C. $f (5) f (0)$
4. D. $f (5) + f^{'} (0)$
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Answer: A
Q742017 · 23 Apr 2017
Let $f (x)$ be defined as follows: $f (x) = ⎩ ⎨ ⎧ 2 x + 1, x - 1, x + 2, - 3 < x < - 2 - 2 \leq x < 0 0 \leq x < 1$ Which one of the following statements is correct in respect of the above function?
1. A.It is discontinuous at $x = - 2$ but continuous at every other point.
2. B.It is continuous only in the interval $(- 3, - 2)$ .
3. C.It is discontinuous at $x = 0$ but continuous at every other point.
4. D.It is discontinuous at every point.
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Answer: C
Q752017 · 23 Apr 2017
Consider the following statements: 1. If $lim_{x \to a} f (x)$ and $lim_{x \to a} g (x)$ both exist, then $lim_{x \to a} {f (x) g (x)}$ exists. 2. If $lim_{x \to a} {f (x) g (x)}$ exists, then both $lim_{x \to a} f (x)$ and $lim_{x \to a} g (x)$ must exist. 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
Q762017 · 23 Apr 2017
What is $\frac{d ^{2} x}{d y ^{2}}$ equal to?
1. A. $- (\frac{d ^{2} y}{d x ^{2}})^{- 1} (\frac{d y}{d x})^{- 3}$
2. B. $(\frac{d ^{2} y}{d x ^{2}})^{- 1} (\frac{d y}{d x})^{- 2}$
3. C. $- (\frac{d ^{2} y}{d x ^{2}}) (\frac{d y}{d x})^{- 3}$
4. D. $(\frac{d ^{2} y}{d x ^{2}})^{- 1}$
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Answer: C
Q772017 · 23 Apr 2017
What is the derivative of $lo g_{10} (5 x^{2} + 3)$ with respect to $x$ ?
1. A. $\frac{x l o g _{10} e}{5 x ^{2} + 3}$
2. B. $\frac{2 x l o g _{10} e}{5 x ^{2} + 3}$
3. C. $\frac{10 x l o g _{10} e}{5 x ^{2} + 3}$
4. D. $\frac{10 x l o g _{e} 10}{5 x ^{2} + 3}$
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Answer: C
Q782017 · 23 Apr 2017
Suppose the function $f (x) = x^{n}$ , $n \neq = 0$ is differentiable for all $x$ . Then $n$ can be any element of the interval
1. A. $[1, \infty)$
2. B. $(0, \infty)$
3. C. $(\frac{1}{2}, \infty)$
4. D.None of the above
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Answer: A
Q792017 · 23 Apr 2017
What is the length of the longest interval in which the function $f (x) = 3 sin x - 4 sin^{3} x$ is increasing?
1. A. $\frac{π}{3}$
2. B. $\frac{π}{2}$
3. C. $\frac{3 π}{2}$
4. D. $π$
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Answer: A
Q802017 · 23 Apr 2017
Let $f (x) = x + \frac{1}{x}$ , where $x \in (0, 1)$ . Then which one of the following is correct?
1. A. $f (x)$ fluctuates in the interval
2. B. $f (x)$ increases in the interval
3. C. $f (x)$ decreases in the interval
4. D.None of the above
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Answer: C
Q812017 · 23 Apr 2017
What is $\int \frac{d x}{x ( x ^{7} + 1 )}$ equal to?
1. A. $\frac{1}{2} ln \frac{x ^{7} - 1}{x ^{7} + 1} + c$
2. B. $\frac{1}{7} ln \frac{x ^{7} + 1}{x ^{7}} + c$
3. C. $ln \frac{x ^{7} - 1}{7 x} + c$
4. D. $\frac{1}{7} ln \frac{x ^{7}}{x ^{7} + 1} + c$
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Answer: D
Q822017 · 23 Apr 2017
What is $\int \frac{( x ^{e - 1} + e ^{x - 1} ) d x}{x ^{e} + e ^{x}}$ equal to?
1. A. $\frac{x ^{2}}{2} + c$
2. B. $ln (x + e) + c$
3. C. $ln (x^{e} + e^{x}) + c$
4. D. $\frac{1}{e} ln (x^{e} + e^{x}) + c$
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Answer: D
Q852017 · 23 Apr 2017
If $f (x)$ and $g (x)$ are continuous functions satisfying $f (x) = f (a - x)$ and $g (x) + g (a - x) = 2$ , then what is $\int_{0}^{a} f (x) g (x) d x$ equal to?
1. A. $\int_{0}^{a} g (x) d x$
2. B. $\int_{0}^{a} f (x) d x$
3. C. $2 \int_{0}^{a} f (x) d x$
4. D.0
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Answer: B
Q862017 · 23 Apr 2017
What is $\int_{e^{- 1}}^{e^{2}} \frac{l n x}{x} d x$ equal to?
1. A. $\frac{3}{2}$
2. B. $\frac{5}{2}$
3. C.3
4. D.4
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Answer: B
Q872017 · 23 Apr 2017
If $x d y = y (d x + y d y); y (1) = 1$ and $y (x) > 0$ , then what is $y (- 3)$ equal to?
1. A.3
2. B.2
3. C.1
4. D.0
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Answer: A
Q1192017 · 23 Apr 2017
What is the maximum area of a triangle that can be inscribed in a circle of radius $a$ ?
1. A. $\frac{3 a ^{2}}{4}$
2. B. $\frac{a ^{2}}{2}$
3. C. $\frac{3 3 a ^{2}}{4}$
4. D. $\frac{3 a ^{2}}{4}$
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Answer: C