Integral Calculus & Area · NDA previous year questions

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2015 20 questions

Q822015 · 27 Sept 2015
$\int \frac{d x}{1 + e ^{- x}}$ is equal to where $c$ is the constant of integration
1. A. $1 + e^{x} + c$
2. B. $ln (1 + e^{- x}) + c$
3. C. $ln (1 + e^{x}) + c$
4. D. $2 ln (1 + e^{- x}) + c$
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Answer: C
Q832015 · 27 Sept 2015
The value of $\int_{a}^{b} \frac{x ^{7} + s i n x}{c o s x} d x$ where $a + b = 0$ is
1. A. $2 b - a sin (b - a)$
2. B. $a + 3 b cos (b - a)$
3. C. $sin a - (b - a) cos b$
4. D.0
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Answer: D
Q842015 · 27 Sept 2015
If $0 < a < b$ , then $\int_{a}^{b} \frac{∣ x ∣}{x} d x$ is equal to
1. A. $∣ b ∣ - ∣ a ∣$
2. B. $∣ a ∣ - ∣ b ∣$
3. C. $\frac{∣ b ∣}{∣ a ∣}$
4. D.0
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Answer: A
Q852015 · 27 Sept 2015
$\int_{0}^{2 π} sin^{5} (\frac{x}{4}) d x$ is equal to
1. A. $\frac{8}{15}$
2. B. $\frac{16}{15}$
3. C. $\frac{32}{15}$
4. D.0
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Answer: C
Q862015 · 27 Sept 2015
$\int_{- 1}^{1} x ∣ x ∣ d x$ is equal to
1. A.0
2. B. $\frac{2}{3}$
3. C.2
4. D. $- 2$
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Answer: A
Q872015 · 27 Sept 2015
For the next two (02) items that follow: Consider the integrals $A = \int_{0}^{π} \frac{s i n x d x}{s i n x + c o s x}$ and $B = \int_{0}^{π} \frac{s i n x d x}{s i n x - c o s x}$ Which one of the following is correct?
1. A. $A = 2 B$
2. B. $B = 2 A$
3. C. $A = B$
4. D. $A = 3 B$
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Answer: Dropped in the final answer key (full marks to everyone)
Q882015 · 27 Sept 2015
For the next two (02) items that follow: Consider the integrals $A = \int_{0}^{π} \frac{s i n x d x}{s i n x + c o s x}$ and $B = \int_{0}^{π} \frac{s i n x d x}{s i n x - c o s x}$ What is the value of $B$ ?
1. A. $\frac{π}{4}$
2. B. $\frac{π}{2}$
3. C. $\frac{3 π}{4}$
4. D. $π$
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Answer: Dropped in the final answer key (full marks to everyone)
Q892015 · 27 Sept 2015
The area bounded by the coordinate axes and the curve $x + y = 1$ , is
1. A.1 square unit
2. B. $\frac{1}{2}$ square unit
3. C. $\frac{1}{3}$ square unit
4. D. $\frac{1}{6}$ square unit
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Answer: D
Q782015 · 19 Apr 2015
The integral $\int \frac{d x}{a c o s x + b s i n x}$ is of the form $\frac{1}{r} ln [tan (\frac{x + α}{2})]$ . What is $r$ equal to?
1. A. $a^{2} + b^{2}$
2. B. $a^{2} + b^{2}$
3. C. $a + b$
4. D. $a^{2} - b^{2}$
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Answer: B
Q802015 · 19 Apr 2015
What is $\int \frac{d x}{x ^{2} + a ^{2}}$ equal to? where $c$ is the constant of integration.
1. A. $ln \frac{x + x ^{2} + a ^{2}}{a} + c$
2. B. $ln \frac{x - x ^{2} + a ^{2}}{a} + c$
3. C. $ln \frac{x ^{2} + x ^{2} + a ^{2}}{a} + c$
4. D.None of the above
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Answer: A
Q812015 · 19 Apr 2015
What is $\int \frac{x e ^{x} d x}{( x + 1 ) ^{2}}$ equal to? where $c$ is the constant of integration.
1. A. $(x + 1)^{2} e^{x} + c$
2. B. $(x + 1) e^{x} + c$
3. C. $\frac{e ^{x}}{x + 1} + c$
4. D. $\frac{e ^{x}}{( x + 1 ) ^{2}} + c$
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Answer: C
Q822015 · 19 Apr 2015
For the next four (4) items that follow : Consider the integral $I_{m} = \int_{0}^{π} \frac{s i n 2 m x}{s i n x} d x$ , where $m$ is a positive integer. What is $I_{1}$ equal to?
1. A.0
2. B.1/2
3. C.1
4. D.2
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Answer: A
Q832015 · 19 Apr 2015
Consider the integral $I_{m} = \int_{0}^{π} \frac{s i n 2 m x}{s i n x} d x$ , where $m$ is a positive integer. What is $I_{2} + I_{3}$ equal to?
1. A.4
2. B.2
3. C.1
4. D.0
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Answer: D
Q842015 · 19 Apr 2015
Consider the integral $I_{m} = \int_{0}^{π} \frac{s i n 2 m x}{s i n x} d x$ , where $m$ is a positive integer. What is $I_{m}$ equal to?
1. A.0
2. B.1
3. C. $m$
4. D. $2 m$
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Answer: A
Q852015 · 19 Apr 2015
Consider the integral $I_{m} = \int_{0}^{π} \frac{s i n 2 m x}{s i n x} d x$ , where $m$ is a positive integer. Consider the following : 1. $I_{m} - I_{m - 1}$ is equal to 0. 2. $I_{2 m} > I_{m}$ Which of the above 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
Q862015 · 19 Apr 2015
For the next two (2) items that follow : Consider the line $x = 3 y$ and the circle $x^{2} + y^{2} = 4$ . What is the area of the region in the first quadrant enclosed by the $x$ -axis, the line $x = 3$ and the circle?
1. A. $\frac{π}{3} - \frac{3}{2}$
2. B. $\frac{π}{2} - \frac{3}{2}$
3. C. $\frac{π}{3} - \frac{1}{2}$
4. D.None of the above
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Answer: A
Q872015 · 19 Apr 2015
Consider the line $x = 3 y$ and the circle $x^{2} + y^{2} = 4$ . What is the area of the region in the first quadrant enclosed by the $x$ -axis, the line $x = 3 y$ and the circle?
1. A. $\frac{π}{3}$
2. B. $\frac{π}{6}$
3. C. $\frac{π}{3} - \frac{3}{2}$
4. D.None of the above
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Answer: A
Q882015 · 19 Apr 2015
Consider the curves $y = sin x$ and $y = cos x$ . What is the area of the region bounded by the above two curves and the lines $x = 0$ and $x = \frac{π}{4}$ ?
1. A. $2 - 1$
2. B. $2 + 1$
3. C. $2$
4. D.2
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Answer: A
Q892015 · 19 Apr 2015
Consider the curves $y = sin x$ and $y = cos x$ . What is the area of the region bounded by the above two curves and the lines $x = \frac{π}{4}$ and $x = \frac{π}{2}$ ?
1. A. $2 - 1$
2. B. $2 + 1$
3. C. $22$
4. D.2
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Answer: A
Q902015 · 19 Apr 2015
Consider the following statements : 1. The general solution of $\frac{d y}{d x} = f (x) + x$ is of the form $y = g (x) + c$ , where $c$ is an arbitrary constant. 2. The degree of $(\frac{d y}{d x})^{2} = f (x)$ 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