Limits, Continuity & Derivatives · NDA previous year questions

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2024 20 questions

Q372024 · 1 Sept 2024
What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as circle and lines)?
1. A.6
2. B.10
3. C.14
4. D.16
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Answer: C
Q732024 · 1 Sept 2024
What is the minimum value of the function $f (x) = lo g_{10} (x^{2} + 2 x + 11)$ ?
1. A.0
2. B.1
3. C.2
4. D.10
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Answer: B
Q742024 · 1 Sept 2024
Which one of the following is correct regarding $lim_{x \to 3} \frac{∣ x - 3∣}{x - 3}$ ?
1. A.Limit exists and is equal to 1
2. B.Limit exists and is equal to 0
3. C.Limit exists and is equal to -1
4. D.Limit does not exist
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Answer: D
Q752024 · 1 Sept 2024
What is the maximum value of a cos x + b sin x + c?
1. A. $a^{2} + b^{2} + c$
2. B. $a^{2} + b^{2} + c$
3. C. $a^{2} + b^{2} - c$
4. D. $a^{2} + b^{2}$
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Answer: B
Q762024 · 1 Sept 2024
If $f (2 x) = 4 x^{2} + 1$ , then for how many real values of x will f(2x) be the GM of f(x) and f(4x)?
1. A.Four
2. B.Two
3. C.One
4. D.None
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Answer: C
Q842024 · 1 Sept 2024
Direction : Consider the following for the items that follow : Let $f (x) = [x]^{2} - [x^{2}]$ , where [ . ] is the greatest integer function Consider the following statements : I. f(x) is continuous at x = 0. II. f(x) is continuous at x = 1 Which of the statements given above is/are correct?
1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II
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Answer: B
Q852024 · 1 Sept 2024
Directions
Direction : Consider the following for the items that follow : Let f(x) = cos2x + x on [-π/2, π/2].
What is the greatest value of f(x)?
1. A. $\frac{3}{2} - \frac{π}{12}$
2. B. $\frac{3}{2} + \frac{π}{12}$
3. C. $\frac{3}{2} + \frac{π}{9}$
4. D. $\frac{3}{2} + \frac{π}{6}$
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Answer: B
Q862024 · 1 Sept 2024
Directions
Direction : Consider the following for the items that follow : Let f(x) = cos2x + x on [-π/2, π/2].
What is the least value of f(x)?
1. A. $- (1 + \frac{π}{2})$
2. B. $- (\frac{1}{2} + \frac{π}{2})$
3. C. $- (1 + \frac{π}{4})$
4. D. $- 2 (\frac{1}{2} - \frac{π}{4})$
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Answer: A
Q952024 · 1 Sept 2024
Direction : Consider the following for the items that follow : Let $f : (- 1, 1) \to R$ be a differentiable function with $f (0) = - 1$ and $f^{'} (0) = 1$ Let $h (x) = f (2 f (x) + 2)$ and $g (x) = (h (x))^{2}$ . What is h'(0) equal to?
1. A.-2
2. B.-1
3. C.0
4. D.2
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Answer: D
Q962024 · 1 Sept 2024
Direction : Consider the following for the items that follow : Let $f : (- 1, 1) \to R$ be a differentiable function with $f (0) = - 1$ and $f^{'} (0) = 1$ Let $h (x) = f (2 f (x) + 2)$ and $g (x) = (h (x))^{2}$ . What is g'(0) equal to?
1. A.-4
2. B.-2
3. C.0
4. D.4
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Answer: A
Q102024 · 21 Apr 2024
If $26! = n 8^{k}$ , where $k$ and $n$ are positive integers, then what is the maximum value of $k$ ?
1. A.6
2. B.7
3. C.8
4. D.9
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Answer: B
Q792024 · 21 Apr 2024
The non-negative values of $b$ for which the function $\frac{16 x ^{3}}{3} - 4 b x^{2} + x$ has neither maximum nor minimum in the range $x > 0$ is
1. A. $0 < b < 1$
2. B. $1 < b < 2$
3. C. $b > 2$
4. D. $0 \leq b < 1$
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Answer: A
Q832024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: Let $f (x) = \frac{x}{l n x}; (x > 1)$ Consider the following statements : 1. $f (x)$ is increasing in the interval $(e, \infty)$ 2. $f (x)$ is decreasing in the interval $(1, e)$ 3. $9 ln 7 > 7 ln 9$ Which of the statements given 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: D
Q842024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: Let $f (x) = \frac{x}{l n x}; (x > 1)$ Consider the following statements : 1. $f^{''} (e) = \frac{1}{e}$ 2. $f (x)$ attains local minimum value at $x = e$ 3. A local minimum value of $f (x)$ is $e$ Which of the statements given 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: D
Q872024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: Let $f (x) = ∣ x ∣ + 1$ and $g (x) = [x] - 1$ , where [.] is the greatest integer function. Let $h (x) = \frac{f ( x )}{g ( x )}$ . Consider the following statements: 1. $f (x)$ is differentiable for all $x < 0$ 2. $g (x)$ is continuous at $x = 0.0001$ 3. The derivative of $g (x)$ at $x = 2.5$ is 1 Which of the statements given 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
Q882024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: Let $f (x) = ∣ x ∣ + 1$ and $g (x) = [x] - 1$ , where [.] is the greatest integer function. Let $h (x) = \frac{f ( x )}{g ( x )}$ . What is $lim_{x \to 0 -} h (x) + lim_{x \to 0 +} h (x)$ equal to?
1. A. $- \frac{3}{2}$
2. B. $- \frac{1}{2}$
3. C. $\frac{1}{2}$
4. D. $\frac{3}{2}$
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Answer: A
Q892024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: Let $φ (a) = \int_{a}^{a + 100 π} ∣ sin x ∣ d x$ What is $φ (a)$ equal to?
1. A.0
2. B. $a$
3. C. $100 a$
4. D.200
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Answer: D
Q902024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: Let $φ (a) = \int_{a}^{a + 100 π} ∣ sin x ∣ d x$ What is $φ (a)$ equal to? What is $φ^{'} (a)$ equal to?
1. A.0
2. B. $π$
3. C.100
4. D.200
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Answer: A
Q912024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: A differentiable function $f (x)$ has a local maximum at $x = 0$ . Let $y = 2 f (x) + a x - b$ . Which of the following is/are correct? 1. $f^{'} (0) = 0$ 2. $f^{''} (0) < 0$ Select the correct answer using the code given below:
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
Q922024 · 21 Apr 2024
Consider the following for the next two (02) items that follow: A differentiable function $f (x)$ has a local maximum at $x = 0$ . Let $y = 2 f (x) + a x - b$ . The function $y$ has a relative maxima at $x = 0$ for
1. A. $a > 0$ , $b = 0$
2. B.for all $b$ and $a = 0$
3. C.for all $b > 0$ only
4. D.for all $a$ and $b = 0$
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Answer: B