Limits, Continuity & Derivatives · NDA previous year questions

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2017 49 questions

Q52017 · 10 Sept 2017
If $z - \frac{4}{z} = 2$ , then the maximum value of $∣ z ∣$ is equal to
1. A. $1 + 3$
2. B. $1 + 5$
3. C. $1 - 5$
4. D. $5 - 1$
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Answer: B
Q152017 · 10 Sept 2017
It is given that the roots of the equation $x^{2} - 4 x - lo g_{3} P = 0$ are real. For this, the minimum value of $P$ is
1. A. $\frac{1}{27}$
2. B. $\frac{1}{64}$
3. C. $\frac{1}{81}$
4. D.1
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Answer: C
Q352017 · 10 Sept 2017
Match List-I with List-II and select the correct answer using the code given below the lists : | List-I (Function) | List-II (Maximum value) | | :--- | :--- | | A. $sin x + cos x$ | 1. $10$ | | B. $3 sin x + 4 cos x$ | 2. $2$ | | C. $2 sin x + cos x$ | 3. $5$ | | D. $sin x + 3 cos x$ | 4. $5$ | Code :
1. A.A 2, B 3, C 1, D 4
2. B.A 2, B 3, C 4, D 1
3. C.A 3, B 2, C 1, D 4
4. D.A 3, B 2, C 4, D 1
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Answer: B
Q362017 · 10 Sept 2017
The value of $sin^{- 1} (\frac{3}{5}) + tan^{- 1} (\frac{1}{7})$ is equal to
1. A.0
2. B. $\frac{π}{4}$
3. C. $\frac{π}{3}$
4. D. $\frac{π}{2}$
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Answer: B
Q372017 · 10 Sept 2017
The principal value of $sin^{- 1} x$ lies in the interval
1. A. $(- \frac{π}{2}, \frac{π}{2})$
2. B. $[- \frac{π}{2}, \frac{π}{2}]$
3. C. $[0, \frac{π}{2}]$
4. D. $[0, π]$
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Answer: B
Q592017 · 10 Sept 2017
If $f (x) = \frac{x}{2} - 1$ , then on the interval $[0, π]$ which one of the following is correct?
1. A. $tan [f (x)]$ , where $[\cdot]$ is the greatest integer function, and $\frac{1}{f ( x )}$ are both continuous
2. B. $tan [f (x)]$ , where $[\cdot]$ is the greatest integer function, and $f^{- 1} (x)$ are both continuous
3. C. $tan [f (x)]$ , where $[\cdot]$ is the greatest integer function, and $\frac{1}{f ( x )}$ are both discontinuous
4. D. $tan [f (x)]$ , where $[\cdot]$ is the greatest integer function, is discontinuous but $\frac{1}{f ( x )}$ is continuous
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Answer: C
Q632017 · 10 Sept 2017
A function is defined as follows : $f (x) = {- \frac{x}{x ^{2}}, 0, x \neq = 0 x = 0$ Which one of the following is correct in respect of the above function?
1. A. $f (x)$ is continuous at $x = 0$ but not differentiable at $x = 0$
2. B. $f (x)$ is continuous as well as differentiable at $x = 0$
3. C. $f (x)$ is discontinuous at $x = 0$
4. D.None of the above
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Answer: C
Q642017 · 10 Sept 2017
Consider the following : 1. $x + x^{2}$ is continuous at $x = 0$ 2. $x + cos \frac{1}{x}$ is discontinuous at $x = 0$ 3. $x^{2} + cos \frac{1}{x}$ is continuous at $x = 0$ Which of the above are correct?
1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3
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Answer: A
Q652017 · 10 Sept 2017
A function is defined in $(0, \infty)$ by $f (x) = ⎩ ⎨ ⎧ 1 - x^{2} ln x ln 2 - 1 + 0 \cdot 5 x for 0 < x \leq 1 for 1 < x \leq 2 for 2 < x < \infty$ Which one of the following is correct in respect of the derivative of the function, i.e., $f^{'} (x)$ ?
1. A. $f^{'} (x) = 2 x$ for $0 < x \leq 1$
2. B. $f^{'} (x) = - 2 x$ for $0 < x \leq 1$
3. C. $f^{'} (x) = - 2 x$ for $0 < x < 1$
4. D. $f^{'} (x) = 0$ for $0 < x < \infty$
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Answer: C
Q662017 · 10 Sept 2017
If $lim_{x \to \frac{π}{2}} \frac{s i n x}{x} = l$ and $lim_{x \to \infty} \frac{c o s x}{x} = m$ , then which one of the following is correct?
1. A. $l = 1, m = 1$
2. B. $l = \frac{2}{π}, m = \infty$
3. C. $l = \frac{2}{π}, m = 0$
4. D. $l = 1, m = \infty$
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Answer: C
Q672017 · 10 Sept 2017
1. A. $(- 1)^{k} (k - 1) π$
2. B. $(- 1)^{k - 1} (k - 1) π$
3. C. $(- 1)^{k} k π$
4. D. $(- 1)^{k - 1} k π$
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Answer: A
Q682017 · 10 Sept 2017
The set of all points, where the function $f (x) = 1 - e^{- x^{2}}$ is differentiable, is
1. A. $(0, \infty)$
2. B. $(- \infty, \infty)$
3. C. $(- \infty, 0) \cup (0, \infty)$
4. D. $(- 1, \infty)$
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Answer: C
Q692017 · 10 Sept 2017
If $f (x) = x (x - x + 1)$ , then $f (x)$ is
1. A.continuous but not differentiable at $x = 0$
2. B.differentiable at $x = 0$
3. C.not continuous at $x = 0$
4. D.None of the above
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Answer: B
Q702017 · 10 Sept 2017
Let $g$ be the greatest integer function. Then the function $f (x) = (g (x))^{2} - g (x^{2})$ is discontinuous at
1. A.all integers
2. B.all integers except 0 and 1
3. C.all integers except 0
4. D.all integers except 1
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Answer: D
Q712017 · 10 Sept 2017
If $y = (cos x)^{(c o s x)^{(c o s x)^{\cdot^{\cdot^{\cdot}}}}}$ , then $\frac{d y}{d x}$ is equal to
1. A. $- \frac{y ^{2} t a n x}{1 - y l n ( c o s x )}$
2. B. $\frac{y ^{2} t a n x}{1 + y l n ( c o s x )}$
3. C. $\frac{y ^{2} t a n x}{1 - y l n ( s i n x )}$
4. D. $\frac{y ^{2} s i n x}{1 + y l n ( s i n x )}$
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Answer: A
Q722017 · 10 Sept 2017
Consider the following statements : 1. Derivative of $f (x)$ may not exist at some point. 2. Derivative of $f (x)$ may exist finitely at some point. 3. Derivative of $f (x)$ may be infinite (geometrically) at some point. Which of the above statements are correct?
1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3
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Answer: D
Q732017 · 10 Sept 2017
If $y = cos^{- 1} (\frac{2 x}{1 + x ^{2}})$ , then $\frac{d y}{d x}$ is equal to
1. A. $- \frac{2}{1 + x ^{2}}$ for all $∣ x ∣ < 1$
2. B. $- \frac{2}{1 + x ^{2}}$ for all $∣ x ∣ > 1$
3. C. $\frac{2}{1 + x ^{2}}$ for all $∣ x ∣ < 1$
4. D.None of the above
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Answer: A
Q742017 · 10 Sept 2017
Which one of the following is correct in respect of the function $f (x) = x (x - 1) (x + 1)$ ?
1. A.The local maximum value is larger than local minimum value
2. B.The local maximum value is smaller than local minimum value
3. C.The function has no local maximum
4. D.The function has no local minimum
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Answer: A
Q752017 · 10 Sept 2017
The maximum value of $\frac{l n x}{x}$ is
1. A. $e$
2. B. $\frac{1}{e}$
3. C. $\frac{2}{e}$
4. D.1
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Answer: B
Q762017 · 10 Sept 2017
A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be $k$ times the height of the jar. The value of $k$ is
1. A.1
2. B.2
3. C.3
4. D.4
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Answer: B
Q772017 · 10 Sept 2017
What is $\int tan^{- 1} (sec x + tan x) d x$ equal to?
1. A. $\frac{π x}{4} + \frac{x ^{2}}{4} + c$
2. B. $\frac{π x}{2} + \frac{x ^{2}}{4} + c$
3. C. $\frac{π x}{4} + \frac{π x ^{2}}{4} + c$
4. D. $\frac{π x}{4} - \frac{x ^{2}}{4} + c$
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Answer: A
Q782017 · 10 Sept 2017
$\int (ln x)^{- 1} d x - \int (ln x)^{- 2} d x$ is equal to
1. A. $x (ln x)^{- 1} + c$
2. B. $x (ln x)^{- 2} + c$
3. C. $x (ln x) + c$
4. D. $x (ln x)^{2} + c$
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Answer: A
Q792017 · 10 Sept 2017
What is $\int_{0}^{2 π} 1 + sin \frac{x}{2} d x$ equal to?
1. A.8
2. B.4
3. C.2
4. D.0
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Answer: A
Q802017 · 10 Sept 2017
The value of $\int_{0}^{\frac{π}{4}} tan x d x + \int_{0}^{\frac{π}{4}} cot x d x$ is equal to
1. A. $\frac{π}{4}$
2. B. $\frac{π}{2}$
3. C. $\frac{π}{2 2}$
4. D. $\frac{π}{2}$
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Answer: D
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