Sets, Relations & Functions · NDA previous year questions

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2015 27 questions

Q12015 · 27 Sept 2015
Let $X$ be the set of all persons living in Delhi. The persons $a$ and $b$ in $X$ are said to be related if the difference in their ages is at most 5 years. The relation is
1. A.an equivalence relation
2. B.reflexive and transitive but not symmetric
3. C.symmetric and transitive but not reflexive
4. D.reflexive and symmetric but not transitive
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Answer: D
Q22015 · 27 Sept 2015
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. $(A \cap B) = (- 2, 1)$ 2. $(A ∖ B) = (- 7, - 2)$ 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
Q602015 · 27 Sept 2015
The domain of the function $f (x) = \frac{1}{∣ x ∣ - x}$ is
1. A. $[0, \infty)$
2. B. $(- \infty, 0)$
3. C. $[1, \infty)$
4. D. $(- \infty, 0]$
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Answer: B
Q612015 · 27 Sept 2015
If $f : R \to R, g : R \to R$ be two functions given by $f (x) = 2 x - 3 and g (x) = x^{3} + 5,$ then $(f \circ g)^{- 1} (x)$ is equal to
1. A. $(\frac{x + 7}{2})^{\frac{1}{3}}$
2. B. $(\frac{x - 7}{2})^{\frac{1}{3}}$
3. C. $(x - \frac{7}{2})^{\frac{1}{3}}$
4. D. $(x + \frac{7}{2})^{\frac{1}{3}}$
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Answer: B
Q632015 · 27 Sept 2015
Consider the following statements: Statement 1: The function $f : R \to R$ such that $f (x) = x^{3}$ for all $x \in R$ is one-one. Statement 2: $f (a) = f (b) \Rightarrow a = b$ for all $a, b \in R$ if the function $f$ is one-one. Which one of the following is correct in respect of the above statements?
1. A.Both the statements are true and Statement 2 is the correct explanation of Statement 1.
2. B.Both the statements are true and Statement 2 is \textbf{\textit{not}} the correct explanation of Statement 1.
3. C.Statement 1 is true but Statement 2 is false.
4. D.Statement 1 is false but Statement 2 is true.
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Answer: A
Q642015 · 27 Sept 2015
If $f (x) = 25 - x^{2}$ , then what is $x \to 1 lim \frac{f ( x ) - f ( 1 )}{x - 1}$ equal to?
1. A. $\frac{1}{5}$
2. B. $\frac{1}{24}$
3. C. $24$
4. D. $- \frac{1}{24}$
Show answer
Answer: D
Q672015 · 27 Sept 2015
Consider the following functions: 1. $f (x) = {\frac{1}{x} 0 if x \neq = 0 if x = 0$ 2. $f (x) = {2 x + 5 x^{2} + 2 x + 5 if x > 0 if x \leq 0$ Which of the above functions is/are derivable at $x = 0$ ?
1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2
Show answer
Answer: B
Q682015 · 27 Sept 2015
If $f (x) = \frac{s i n ( e ^{x - 2} - 1 )}{l n ( x - 1 )}$ , then $x \to 2 lim f (x)$ is equal to
1. A. $- 2$
2. B. $- 1$
3. C.0
4. D.1
Show answer
Answer: D
Q702015 · 27 Sept 2015
Passage
For the next two (02) items that follow: Consider the function
1. A.1
2. B.0
3. C. $- 1$
4. D. $- 2$
Show answer
Answer: A
Q12015 · 19 Apr 2015
Let $X$ be the set of all persons living in a city. Persons $x, y$ in $X$ are said to be related as $x < y$ if $y$ is at least 5 years older than $x$ . Which one of the following is correct?
1. A.The relation is an equivalence relation on $X$
2. B.The relation is transitive but neither reflexive nor symmetric
3. C.The relation is reflexive but neither transitive nor symmetric
4. D.The relation is symmetric but neither transitive nor reflexive
Show answer
Answer: B
Q22015 · 19 Apr 2015
In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is
1. A.35
2. B.40
3. C.50
4. D.55
Show answer
Answer: B
Q32015 · 19 Apr 2015
Let $Z$ be the set of integers and $a R b$ , where $a, b \in Z$ if and only if $(a - b)$ is divisible by 5. Consider the following statements : 1. The relation $R$ partitions $Z$ into five equivalent classes. 2. Any two equivalent classes are either equal or disjoint. 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
Show answer
Answer: C
Q42015 · 19 Apr 2015
If $A = {x : x is a multiple of 3}$ and $B = {x : x is a multiple of 4}$ and $C = {x : x is a multiple of 12}$ , then which one of the following is a null set?
1. A. $(A ∖ B) \cup C$
2. B. $(A ∖ B) ∖ C$
3. C. $(A \cap B) \cap C$
4. D. $(A \cap B) ∖ C$
Show answer
Answer: D
Q52015 · 19 Apr 2015
Let $A = {x, y, z}$ and $B = {p, q, r, s}$ . What is the number of distinct relations from $B$ to $A$ ?
1. A.4096
2. B.4094
3. C.128
4. D.126
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Answer: A
Q162015 · 19 Apr 2015
Let $A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ . Then the number of subsets of $A$ containing exactly two elements is
1. A.20
2. B.40
3. C.45
4. D.90
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Answer: C
Q172015 · 19 Apr 2015
How many words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together?
1. A.354
2. B.348
3. C.288
4. D.None of the above
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Answer: C
Q182015 · 19 Apr 2015
What is $\sum_{r = 0}^{1}^{n + r} C_{n}$ equal to?
1. A. $^{n + 2} C_{1}$
2. B. $^{n + 2} C_{n}$
3. C. $^{n + 3} C_{n}$
4. D. $^{n + 2} C_{n + 1}$
Show answer
Answer: A, D
Q562015 · 19 Apr 2015
For each non-zero real number $x$ , let $f (x) = \frac{x}{∣ x ∣}$ . The range of $f$ is
1. A.a null set
2. B.a set consisting of only one element
3. C.a set consisting of two elements
4. D.a set consisting of infinitely many elements
Show answer
Answer: C
Q572015 · 19 Apr 2015
If $f (x) = lo g_{e} (\frac{1 + x}{1 - x})$ , $g (x) = \frac{3 x + x ^{3}}{1 + 3 x ^{2}}$ and $g \circ f (t) = g (f (t))$ , then what is $g \circ f (\frac{e - 1}{e + 1})$ equal to?
1. A. $2$
2. B. $1$
3. C. $0$
4. D. $1/2$
Show answer
Answer: B
Q582015 · 19 Apr 2015
Consider the following functions : 1. $f (x) = x^{3}, x \in R$ 2. $f (x) = sin x, 0 < x < 2 π$ 3. $f (x) = e^{x}, x \in R$ Which of the above functions have inverse defined on their ranges?
1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3
Show answer
Answer: C
Q592015 · 19 Apr 2015
If $G (x) = 25 - x^{2}$ , then what is $lim_{x \to 1} \frac{G ( x ) - G ( 1 )}{x - 1}$ equal to?
1. A. $- \frac{1}{2 6}$
2. B. $\frac{1}{5}$
3. C. $- \frac{1}{6}$
4. D. $\frac{1}{6}$
Show answer
Answer: A
Q622015 · 19 Apr 2015
Passage
For the next two (2) items that follow : Given a function
Consider the function $f (x) = ⎩ ⎨ ⎧ - 1 a x + b 1 if x \leq 0 if 0 < x < 1 if x \geq 1$ where $a, b$ are constants. The function is continuous everywhere. What is the value of $a$ ?
1. A. $- 1$
2. B. $0$
3. C. $1$
4. D. $2$
Show answer
Answer: D
Q642015 · 19 Apr 2015
Passage
For the next two (2) items that follow : Given a function
Consider the function $f (x) = {\frac{α c o s x}{π - 2 x} 3 if x \neq = \frac{π}{2} if x = \frac{π}{2}$ which is continuous at $x = \frac{π}{2}$ , where $α$ is a constant. What is $lim_{x \to 0} f (x)$ equal to?
1. A. $0$
2. B. $3$
3. C. $\frac{3}{π}$
4. D. $\frac{6}{π}$
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Answer: D
Q652015 · 19 Apr 2015
For the next two (2) items that follow : Given that $lim_{x \to \infty} (\frac{2 + x ^{2}}{1 + x} - A x - B) = 3$ . What is the value of $A$ ?
1. A.-1
2. B.1
3. C.2
4. D.3
Show answer
Answer: B
Q662015 · 19 Apr 2015
Given that $lim_{x \to \infty} (\frac{2 + x ^{2}}{1 + x} - A x - B) = 3$ . What is the value of $B$ ?
1. A.-1
2. B.3
3. C.-4
4. D.-3
Show answer
Answer: C