Sets, Relations & Functions · NDA previous year questions

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2016 39 questions

Q22016 · 18 Sept 2016
Let S be a set of all distinct numbers of the form $\frac{p}{q}$ , where $p, q \in {1, 2, 3, 4, 5, 6}$ . What is the cardinality of the set S ?
1. A.21
2. B.23
3. C.32
4. D.36
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Answer: B
Q32016 · 18 Sept 2016
If $A = {x \in R : x^{2} + 6 x - 7 < 0}$ and $B = {x \in R : x^{2} + 9 x + 14 > 0}$ , then which of the following is/are correct ? 1. $A \cap B = {x \in R : - 2 < x < 1}$ 2. $A ∖ B = {x \in R : - 7 < x < - 2}$ 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: A
Q42016 · 18 Sept 2016
Let R be a relation from $A = {1, 2, 3, 4}$ to $B = {1, 3, 5}$ such that $R = {(a, b) : a < b$ , where $a \in A$ and $b \in B}$ . What is $R o R^{- 1}$ equal to ?
1. A.\{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)\}
2. B.\{(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)\}
3. C.\{(3, 3), (3, 5), (5, 3), (5, 5)\}
4. D.\{(3, 3), (3, 4), (4, 5)\}
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Answer: C
Q672016 · 18 Sept 2016
Consider the following for the next three (03) items that follow: Let $f (x) = [x]$ , where $[.]$ is the greatest integer function and $g (x) = sin x$ be two real valued functions over $R$ . Which one of the following statements is correct?
1. A. $lim_{x \to 0} (fog) (x)$ exists.
2. B. $lim_{x \to 0} (gof) (x)$ exists.
3. C. $lim_{x \to 0^{-}} (fog) (x) = lim_{x \to 0^{-}} (gof) (x)$
4. D. $lim_{x \to 0^{+}} (fog) (x) = lim_{x \to 0^{+}} (gof) (x)$
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Answer: D
Q682016 · 18 Sept 2016
Consider the following for the next three (03) items that follow: Let $f (x) = [x]$ , where $[.]$ is the greatest integer function and $g (x) = sin x$ be two real valued functions over $R$ . Which of the following statements are correct? 1. $(f of) (x) = f (x)$ . 2. $(g og) (x) = g (x)$ only when $x = 0$ . 3. $(g o (f og)) (x)$ can take only three values. Select the correct answer using the code given below:
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: C
Q822016 · 18 Sept 2016
Let $f (x)$ be a function such that $f^{'} (\frac{1}{x}) + x^{3} f^{'} (x) = 0$ . What is $\int_{- 1}^{1} f (x) d x$ equal to?
1. A. $2 f (1)$
2. B.0
3. C. $2 f (- 1)$
4. D. $4 f (1)$
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Answer: Dropped in the final answer key (full marks to everyone)
Q832016 · 18 Sept 2016
If $\int_{0}^{π /2} \frac{d x}{3 c o s x + 5} = k cot^{- 1} 2$ , then what is the value of k?
1. A.1/4
2. B.1/2
3. C.1
4. D.2
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Answer: B
Q842016 · 18 Sept 2016
What is $\int_{1}^{3} ∣1 - x^{4} ∣ d x$ equal to?
1. A.-232/5
2. B.-116/5
3. C.116/5
4. D.232/5
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Answer: D
Q1202016 · 18 Sept 2016
What is the domain of the function $f (x) = \frac{1}{∣ x ∣ - x}$ ?
1. A. $(- \infty, 0)$
2. B. $(0, \infty)$
3. C. $0 < x < 1$
4. D. $x > 1$
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Answer: A
Q12016 · 17 Apr 2016
What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?
1. A.860
2. B.854
3. C.840
4. D.824
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Answer: B
Q22016 · 17 Apr 2016
Suppose there is a relation $*$ between the positive numbers $x$ and $y$ given by $x * y$ if and only if $x \leq y^{2}$ . Then which one of the following is correct?
1. A. $*$ is reflexive but not transitive and symmetric
2. B. $*$ is transitive but not reflexive and symmetric
3. C. $*$ is symmetric and reflexive but not transitive
4. D. $*$ is symmetric but not reflexive and transitive
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Answer: Dropped in the final answer key (full marks to everyone)
Q32016 · 17 Apr 2016
Let $R$ be a relation on the set $N$ of natural numbers defined by ' $n R m ⟺ n$ is a factor of $m$ '. Then which one of the following is correct?
1. A. $R$ is reflexive, symmetric but not transitive
2. B. $R$ is transitive, symmetric but not reflexive
3. C. $R$ is reflexive, transitive but not symmetric
4. D. $R$ is an equivalence relation
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Answer: C
Q222016 · 17 Apr 2016
Passage
For the next three (3) items that follow : Consider the function
For the next three (3) items that follow : Consider the function $f (θ) = 4 (sin^{2} θ + cos^{4} θ)$ . Consider the following statements : 1. $f (θ) = 2$ has no solution. 2. $f (θ) = \frac{7}{2}$ has a solution. 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
Q232016 · 17 Apr 2016
For the next two (2) items that follow : Consider a triangle $A B C$ in which $cos A + cos B + cos C = 3 sin \frac{π}{3}$ . What is the value of $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}$ ?
1. A. $\frac{1}{2}$
2. B. $\frac{1}{4}$
3. C. $\frac{1}{8}$
4. D. $\frac{1}{16}$
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Answer: C
Q242016 · 17 Apr 2016
For the next two (2) items that follow : Consider a triangle $A B C$ in which $cos A + cos B + cos C = 3 sin \frac{π}{3}$ . What is the value of $cos (\frac{A + B}{2}) cos (\frac{B + C}{2}) cos (\frac{C + A}{2})$ ?
1. A. $\frac{1}{4}$
2. B. $\frac{1}{2}$
3. C. $\frac{1}{16}$
4. D.None of the above
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Answer: D
Q372016 · 17 Apr 2016
For the next two (2) items that follow : Consider the two lines $x + y + 1 = 0$ and $3 x + 2 y + 1 = 0$ What is the equation of the line passing through the point of intersection of the given lines and parallel to $x$ -axis?
1. A. $y + 1 = 0$
2. B. $y - 1 = 0$
3. C. $y - 2 = 0$
4. D. $y + 2 = 0$
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Answer: D
Q382016 · 17 Apr 2016
For the next two (2) items that follow : Consider the two lines $x + y + 1 = 0$ and $3 x + 2 y + 1 = 0$ What is the equation of the line passing through the point of intersection of the given lines and parallel to $y$ -axis?
1. A. $x + 1 = 0$
2. B. $x - 1 = 0$
3. C. $x - 2 = 0$
4. D. $x + 2 = 0$
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Answer: B
Q392016 · 17 Apr 2016
$(a, 2 b)$ is the mid-point of the line segment joining the points $(10, - 6)$ and $(k, 4)$ . If $a - 2 b = 7$ , then what is the value of $k$ ?
1. A.2
2. B.3
3. C.4
4. D.5
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Answer: A
Q402016 · 17 Apr 2016
What is the acute angle between the lines represented by the equations $y - 3 x - 5 = 0$ and $3 y - x + 6 = 0$ ?
1. A. $3 0^{\circ}$
2. B. $4 5^{\circ}$
3. C. $6 0^{\circ}$
4. D. $7 5^{\circ}$
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Answer: A
Q492016 · 17 Apr 2016
What is the range of the function $y = \frac{x ^{2}}{1 + x ^{2}}$ where $x \in R$ ?
1. A. $[0, 1)$
2. B. $[0, 1]$
3. C. $(0, 1)$
4. D. $(0, 1]$
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Answer: A
Q502016 · 17 Apr 2016
For the next two (2) items that follow : Consider the function $f (x) = \frac{27 ( x ^{2/3} - x )}{4}$ How many solutions does the function $f (x) = 1$ have?
1. A.One
2. B.Two
3. C.Three
4. D.Four
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Answer: B
Q512016 · 17 Apr 2016
For the next two (2) items that follow : Consider the function $f (x) = \frac{27 ( x ^{2/3} - x )}{4}$ How many solutions does the function $f (x) = - 1$ have?
1. A.One
2. B.Two
3. C.Three
4. D.Four
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Answer: A
Q612016 · 17 Apr 2016
For the next two (2) items that follow : Consider the function $f (x) = x^{3} 6 p sin x - 1 p^{2} cos x 0 p^{3}$ where $p$ is a constant. What is the value of $f^{'} (0)$ ?
1. A. $p^{3}$
2. B. $3 p^{3}$
3. C. $6 p^{3}$
4. D. $- 6 p^{3}$
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Answer: D
Q622016 · 17 Apr 2016
For the next two (2) items that follow : Consider the function $f (x) = x^{3} 6 p sin x - 1 p^{2} cos x 0 p^{3}$ where $p$ is a constant. What is the value of $p$ for which $f^{''} (0) = 0$ ?
1. A. $- \frac{1}{6}$ or 0
2. B. $- 1$ or 0
3. C. $- \frac{1}{6}$ or 1
4. D. $- 1$ or 1
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Answer: A
Q632016 · 17 Apr 2016
If then what is equal to?
1. A.10
2. B.2
3. C.1
4. D.0
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Answer: D