Limits, Continuity & Derivatives · NDA previous year questions

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2018 35 questions

Q682018 · 9 Sept 2018
If $y = sin (ln x)$ , then which one of the following is correct?
1. A. $\frac{d ^{2} y}{d x ^{2}} + y = 0$
2. B. $\frac{d ^{2} y}{d x ^{2}} = 0$
3. C. $x^{2} \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} + y = 0$
4. D. $x^{2} \frac{d ^{2} y}{d x ^{2}} - x \frac{d y}{d x} + y = 0$
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Answer: C
Q692018 · 9 Sept 2018
If $y = ∣ sin x ∣^{∣ x ∣}$ , then what is the value of $\frac{d y}{d x}$ at $x = - \frac{π}{6}$ ?
1. A. $\frac{2 ^{- \frac{π}{6}} ( 6 l n 2 - 3 π )}{6}$
2. B. $\frac{2 ^{\frac{π}{6}} ( 6 l n 2 + 3 π )}{6}$
3. C. $\frac{2 ^{- \frac{π}{6}} ( 6 l n 2 + 3 π )}{6}$
4. D. $\frac{2 ^{\frac{π}{6}} ( 6 l n 2 - 3 π )}{6}$
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Answer: A
Q702018 · 9 Sept 2018
What is $\frac{d 1 - s i n 2 x}{d x}$ equal to, where $\frac{π}{4} < x < \frac{π}{2}$ ?
1. A. $cos x + sin x$
2. B. $- (cos x + sin x)$
3. C. $\pm (cos x + sin x)$
4. D.None of the above
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Answer: A
Q722018 · 9 Sept 2018
Which one of the following is correct in respect of the function $f (x) = x sin x + cos x + \frac{1}{2} cos^{2} x$ ?
1. A.It is increasing in the interval $(0, \frac{π}{2})$
2. B.It remains constant in the interval $(0, \frac{π}{2})$
3. C.It is decreasing in the interval $(0, \frac{π}{2})$
4. D.It is decreasing in the interval $(\frac{π}{4}, \frac{π}{2})$
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Answer: A
Q732018 · 9 Sept 2018
In which one of the following intervals is the function $f (x) = x^{2} - 5 x + 6$ decreasing?
1. A. $(- \infty, 2]$
2. B. $[3, \infty)$
3. C. $(- \infty, \infty)$
4. D. $(2, 3)$
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Answer: A
Q742018 · 9 Sept 2018
A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector?
1. A.25 m
2. B.20 m
3. C.10 m
4. D.5 m
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Answer: C
Q752018 · 9 Sept 2018
What is the minimum value of $[x (x - 1) + 1]^{\frac{1}{3}}$ , where $0 \leq x \leq 1$ ?
1. A. $(\frac{3}{4})^{\frac{1}{3}}$
2. B. $1$
3. C. $\frac{1}{2}$
4. D. $(\frac{3}{8})^{\frac{1}{3}}$
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Answer: A
Q392018 · 22 Apr 2018
The maximum value of $sin (x + \frac{π}{5}) + cos (x + \frac{π}{5})$ , where $x \in (0, \frac{π}{2})$ , is attained at
1. A. $\frac{π}{20}$
2. B. $\frac{π}{15}$
3. C. $\frac{π}{10}$
4. D. $\frac{π}{2}$
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Answer: A
Q402018 · 22 Apr 2018
What is the maximum value of $16 sin θ - 12 sin^{2} θ$ ?
1. A. $3/4$
2. B. $4/3$
3. C. $16/3$
4. D. $4$
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Answer: C
Q412018 · 22 Apr 2018
If $sin x = \frac{1}{5}, sin y = \frac{1}{10}$ , where $0 < x < \frac{π}{2}, 0 < y < \frac{π}{2}$ , then what is $(x + y)$ equal to?
1. A. $π$
2. B. $π /2$
3. C. $π /4$
4. D. $0$
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Answer: C
Q422018 · 22 Apr 2018
What is the principal value of $sin^{- 1} (sin \frac{2 π}{3})$ ?
1. A. $π /4$
2. B. $π /2$
3. C. $π /3$
4. D. $2 π /3$
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Answer: C
Q432018 · 22 Apr 2018
What is $tan^{- 1} (\frac{1}{4}) + tan^{- 1} (\frac{3}{5})$ equal to?
1. A. $0$
2. B. $π /4$
3. C. $π /3$
4. D. $π /2$
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Answer: B
Q442018 · 22 Apr 2018
If a flag-staff of 6 m height placed on the top of a tower throws a shadow of $23$ m along the ground, then what is the angle that the sun makes with the ground?
1. A. $6 0^{\circ}$
2. B. $4 5^{\circ}$
3. C. $3 0^{\circ}$
4. D. $1 5^{\circ}$
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Answer: A
Q682018 · 22 Apr 2018
If $f (x) = \frac{x ^{2} - 9}{x ^{2} - 2 x - 3}$ , $x \neq = 3$ is continuous at $x = 3$ , then which one of the following is correct?
1. A. $f (3) = 0$
2. B. $f (3) = 1.5$
3. C. $f (3) = 3$
4. D. $f (3) = - 1.5$
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Answer: B
Q692018 · 22 Apr 2018
What is $lim_{x \to 0} \frac{t a n x}{s i n 2 x}$ equal to?
1. A.1/2
2. B.1
3. C.2
4. D.Limit does not exist
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Answer: A
Q702018 · 22 Apr 2018
What is $lim_{h \to 0} \frac{2 x + 3 h - 2 x}{2 h}$ equal to?
1. A. $\frac{1}{2 2 x}$
2. B. $\frac{3}{2 x}$
3. C. $\frac{3}{2 2 x}$
4. D. $\frac{3}{4 2 x}$
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Answer: D
Q712018 · 22 Apr 2018
If $y = e^{x^{2}} sin 2 x$ , then what is $\frac{d y}{d x}$ at $x = π$ equal to?
1. A. $(1 + π) e^{π^{2}}$
2. B. $2 π e^{π^{2}}$
3. C. $2 e^{π^{2}}$
4. D. $e^{π^{2}}$
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Answer: C
Q722018 · 22 Apr 2018
Consider the following information for the next three (03) items that follow: Three sides of a trapezium are each equal to 6 cm. Let $α \in (0, \frac{π}{2})$ be the angle between a pair of adjacent sides. If the area of the trapezium is the maximum possible, then what is $α$ equal to?
1. A. $\frac{π}{6}$
2. B. $\frac{π}{4}$
3. C. $\frac{π}{3}$
4. D. $\frac{2 π}{5}$
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Answer: C
Q732018 · 22 Apr 2018
Consider the following information for the next three (03) items that follow: Three sides of a trapezium are each equal to 6 cm. Let $α \in (0, \frac{π}{2})$ be the angle between a pair of adjacent sides. If the area of the trapezium is maximum, what is the length of the fourth side?
1. A.8 cm
2. B.9 cm
3. C.10 cm
4. D.12 cm
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Answer: D
Q742018 · 22 Apr 2018
Consider the following information for the next three (03) items that follow: Three sides of a trapezium are each equal to 6 cm. Let $α \in (0, \frac{π}{2})$ be the angle between a pair of adjacent sides. What is the maximum area of the trapezium?
1. A. $363 c m^{2}$
2. B. $303 c m^{2}$
3. C. $273 c m^{2}$
4. D. $243 c m^{2}$
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Answer: C
Q752018 · 22 Apr 2018
What is $\int \frac{d x}{2 ^{x} - 1}$ equal to?
1. A. $ln (2^{x} - 1) + c$
2. B. $\frac{l n ( 1 - 2 ^{- x} )}{l n 2} + c$
3. C. $\frac{l n ( 2 ^{- x} - 1 )}{2 l n 2} + c$
4. D. $\frac{l n ( 1 + 2 ^{- x} )}{l n 2} + c$
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Answer: B
Q762018 · 22 Apr 2018
What is $\int_{0}^{π} e^{x} sin x d x$ equal to?
1. A. $\frac{e ^{π} + 1}{2}$
2. B. $\frac{e ^{π} - 1}{2}$
3. C. $e^{π} + 1$
4. D. $\frac{e ^{π} + 1}{4}$
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Answer: A
Q772018 · 22 Apr 2018
What is $\int_{1}^{e} x ln x d x$ equal to?
1. A. $\frac{e + 1}{4}$
2. B. $\frac{e ^{2} + 1}{4}$
3. C. $\frac{e - 1}{4}$
4. D. $\frac{e ^{2} - 1}{4}$
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Answer: B
Q1182018 · 22 Apr 2018
If $f (x) = ∣ x ∣ + ∣ x - 1∣$ , then which one of the following is correct?
1. A. $f (x)$ is continuous at $x = 0$ and $x = 1$
2. B. $f (x)$ is continuous at $x = 0$ but not at $x = 1$
3. C. $f (x)$ is continuous at $x = 1$ but not at $x = 0$
4. D. $f (x)$ is neither continuous at $x = 0$ nor at $x = 1$
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Answer: A
Q1192018 · 22 Apr 2018
What is the derivative of the function $f (x) = e^{t a n x} + ln (sec x) - e^{l n x}$ at $x = \frac{π}{4}$ ?
1. A. $e /2$
2. B. $e$
3. C. $2 e$
4. D. $4 e$
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Answer: C