Limits, Continuity & Derivatives · NDA previous year questions

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2021 33 questions

Q332021 · 14 Nov 2021
Let p, q and 3 be respectively the first, third and fifth terms of an AP. Let d be the common difference. If the product (pq) is minimum, then what is the value of d?
1. A.1
2. B. $\frac{3}{8}$
3. C. $\frac{9}{8}$
4. D. $\frac{9}{4}$
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Answer: C
Q352021 · 14 Nov 2021
Let the equation sec x.cosec x = p have a solution, where p is a positive real number. What should be the smallest value of p?
1. A. $\frac{1}{2}$
2. B.1
3. C.2
4. D.Minimum does not exist
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Answer: C
Q362021 · 14 Nov 2021
For what value of $θ$ , where $0 < θ < \frac{π}{2}$ , does $sin θ + sin θ cos θ$ maximum value?
1. A. $\frac{π}{2}$
2. B. $\frac{π}{3}$
3. C. $\frac{π}{4}$
4. D. $\frac{π}{6}$
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Answer: B
Q592021 · 14 Nov 2021
Directions
Direction: Consider the following for the next two (02) items that follow. The two ends of the latus rectum of a parabola are (-2, 4) and (-2, -4).
What is the maximum number of parabolas that can be drawn through these two points as end points of latus rectum?
1. A.Only one
2. B.Two
3. C.Four
4. D.Infinite
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Answer: B
Q742021 · 14 Nov 2021
What is $lim_{n \to \infty} \frac{a ^{n} + b ^{n}}{a ^{n} - b ^{n}}$ where $a > b > 1$ , equal to?
1. A.-1
2. B.0
3. C.1
4. D.Limit does not exist
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Answer: C
Q752021 · 14 Nov 2021
1. A.-2
2. B.-1
3. C.0
4. D.1
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Answer: D
Q762021 · 14 Nov 2021
Consider the following statements in respect of f(x) = |x| - 1 1. f(x) is continuous at x = 1. 2. f(x) 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
Q772021 · 14 Nov 2021
If $f (x) = \frac{[ x ]}{∣ x ∣}$ , $x \neq = 0$ , where $[\cdot]$ denotes the greatest integer function, then what is the right-hand limit of f(x) at $x = 1$ ?
1. A.-1
2. B.0
3. C.1
4. D.Right-hand limit of f(x) at x = 1 does not exist.
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Answer: C
Q782021 · 14 Nov 2021
Consider the following statements in respect of the function $f (x) = sin (\frac{1}{x ^{2}}), x \neq = 0$ : 1. It is continuous at $x = 0$ , if $f (0) = 0$ . 2. It is continuous at $x = \frac{2}{π}$ . 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: B
Q822021 · 14 Nov 2021
Consider the following statements in respect of the function $f (x) = x^{2} + 1$ in the interval $[1, 2]$ : 1. The maximum value of the function is 5. 2. The minimum value of the function is 2. 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
Q832021 · 14 Nov 2021
If f(x) satisfies $f (1) = f (4)$ , then what is $\int_{1}^{4} f^{'} (x) d x$ equal to?
1. A.-1
2. B.0
3. C.1
4. D.2
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Answer: B
Q842021 · 14 Nov 2021
What is $\int_{0}^{\frac{π}{2}} e^{l n (c o s x)} d x$ equal to?
1. A.-1
2. B.0
3. C.1
4. D.2
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Answer: C
Q852021 · 14 Nov 2021
If $\int 1 - sin 2 x d x = A sin x + B cos x + C$ , where $0 < x < \frac{π}{4}$ , then which one of the following is correct?
1. A. $A + B = 0$
2. B. $A + B - 2 = 0$
3. C. $A + B + 2 = 0$
4. D. $A + B - 1 = 0$
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Answer: B
Q902021 · 14 Nov 2021
Where does the tangent to the curve $y = e^{x}$ at the point $(0, 1)$ meet x-axis?
1. A. $(1, 0)$
2. B. $(- 1, 0)$
3. C. $(2, 0)$
4. D. $(- \frac{1}{2}, 0)$
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Answer: B
Q912021 · 14 Nov 2021
Consider the following statements in respect of the function $f (x) = x + \frac{1}{x}$ : 1. The local maximum value of f(x) is less than its local minimum value. 2. The local maximum value of f(x) occurs at $x = 1$ . 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
Q922021 · 14 Nov 2021
What is the maximum area of a rectangle that can be inscribed in a circle of radius 2 units?
1. A.4 square units
2. B.6 square units
3. C.8 square units
4. D.16 square units
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Answer: C
Q932021 · 14 Nov 2021
What is $\int \frac{d x}{x ( x ^{2} + 1 )}$ equal to?
1. A. $\frac{1}{2} ln (\frac{x ^{2}}{x ^{2} + 1}) + C$
2. B. $ln (\frac{x ^{2}}{x ^{2} + 1}) + C$
3. C. $\frac{3}{2} ln (\frac{x ^{2}}{x ^{2} + 1}) + C$
4. D. $\frac{1}{2} ln (\frac{x ^{2} + 1}{x ^{2}}) + C$
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Answer: A
Q942021 · 14 Nov 2021
What is the derivative of $e^{e^{x}}$ with respect to $e^{x}$ ?
1. A. $e^{e^{x}}$
2. B. $e^{x}$
3. C. $e^{e^{x}} e^{x}$
4. D. $e e^{x}$
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Answer: A
Q952021 · 14 Nov 2021
What is the condition that $f (x) = x^{3} + x^{2} + k x$ has no local extremum?
1. A. $4 k < 1$
2. B. $3 k > 1$
3. C. $3 k < 1$
4. D. $3 k \leq 1$
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Answer: B
Q962021 · 14 Nov 2021
If $f (x) = 2^{x}$ , then what is $\int_{2}^{10} \frac{f ^{'} ( x )}{f ( x )} d x$ equal to?
1. A. $4 ln 2$
2. B. $ln 4$
3. C. $ln 5$
4. D. $8 ln 2$
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Answer: D
Q972021 · 14 Nov 2021
If $\int_{- 2}^{0} f (x) d x = k$ , then $\int_{- 2}^{0} ∣ f (x) ∣ d x$ is
1. A.less than k
2. B.greater than k
3. C.less than or equal to k
4. D.greater than or equal to k
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Answer: D
Q982021 · 14 Nov 2021
If the function $f (x) = x^{2} - k x$ is monotonically increasing in the interval $(1, \infty)$ , then which one of the following is correct?
1. A. $k < 2$
2. B. $2 < k < 3$
3. C. $3 < k < 4$
4. D. $k > 4$
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Answer: A
Q1002021 · 14 Nov 2021
The tangent to the curve $x^{2} = y$ at $(1, 1)$ makes an angle $θ$ with the positive direction of x-axis. Which one of the following is correct?
1. A. $θ < \frac{π}{6}$
2. B. $\frac{π}{6} < θ < \frac{π}{4}$
3. C. $\frac{π}{4} < θ < \frac{π}{3}$
4. D. $\frac{π}{3} < θ < \frac{π}{2}$
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Answer: D
Q762021 · 18 Apr 2021
What is the derivative of $e^{x}$ with respect to $x^{e}$ ?
1. A. $\frac{x e ^{x}}{e x ^{e}}$
2. B. $\frac{e ^{x}}{x ^{e}}$
3. C. $\frac{x e ^{x}}{x ^{e}}$
4. D. $\frac{e ^{x}}{e x ^{e}}$
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Answer: A
Q772021 · 18 Apr 2021
If a differentiable function f(x) satisfies $lim_{x \to - 1} \frac{f ( x ) + 1}{x ^{2} - 1} = - \frac{3}{2}$ then what is $lim_{x \to - 1} f (x)$ equal to?
1. A. $- \frac{3}{2}$
2. B.-1
3. C.0
4. D.1
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Answer: B