Sets, Relations & Functions · NDA previous year questions

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If $y={\ln }^2\left(\frac{x^2-x+1}{x^2+x+1}\right)$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $x=0$ equal to ?

1. A.-2
2. B.0
3. C.1
4. D.2

If $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1+x^4+x^8}{1-x^2+x^4}\right)=ax+b{x}^3$, then which one of the following is correct ?

1. A.$a=b$
2. B.$a=2b$
3. C.$a+b=0$
4. D.$2a=b$

If A = {{1, 2, 3}}, then how many elements are there in the power set of A?

1. A.1
2. B.2
3. C.4
4. D.8

Consider all the subsets of the set A = {1, 2, 3, 4}. How many of them are supersets of the set {4}?

1. A.6
2. B.7
3. C.8
4. D.9

Consider the following statements in respect of two non-empty sets A and B : 1. x ∉ (A ∪ B) ⇒ x ∉ A or x ∉ B 2. x ∉ (A ∩ B) ⇒ x ∉ A and x ∉ B 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

Consider the following statements in respect of two non-empty sets A and B : 1. A ∪ B = A ∩ B if A = B 2. A Δ B = ϕ  if A = B 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

Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if $x^2-5xy+4y^2=0$ : 1. R is reflexive 2. R is symmetric 3. R is transitive Which of the above statements is /are correct ?

1. A.1 only
2. B.2 only
3. C.1 and 2 only
4. D.1, 2 and 3

Consider the following statements in respect of any relation R on a set A : 1. If R is reflexive, then $R^{-1}$ is also reflexive 2. If R is symmetric, then $R^{-1}$ is also symmetric 3. If R is transitive, then $R^{-1}$ is also transitive Which of the above statements 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

What is the domain of the function $f(x)=\sqrt{1-(x-1{)}^2}$?

1. A.(0, 1)
2. B.[-1, 1]
3. C.(0, 2)
4. D.[0, 2]

Consider the following for the next three (03) items that follow : The algebraic sum of the deviations of a set of values $x_1,x_2,x_3,\cdots ,x_n$ measured from 100 is -20 and the algebraic sum of the deviations of the same set of values measured from 92 is 140. What is the algebraic sum of the deviations of the same set of values measured from 99?

1. A.0
2. B.10
3. C.20
4. D.40

Consider the following for the next three (03) items that follow : The algebraic sum of the deviations of a set of values $x_1,x_2,x_3,\cdots ,x_n$ measured from 100 is -20 and the algebraic sum of the deviations of the same set of values measured from 92 is 140. If the algebraic sum of the deviations of the same set of values measured from y is 180, then what is the value of y ?

1. A.80
2. B.85
3. C.90
4. D.95

What is the period of the function $f(x)=\ln (2+{\sin }^{2}x)$?

1. A.$\frac{\pi }{2}$
2. B.$\pi$
3. C.$2\pi$
4. D.$3\pi$

Consider the following statements in respect of sets: 1. The union over the intersection of sets is distributive. 2. The complement of the union of two sets is equal to the intersection of their complements. 3. If the difference between the two sets is equal to the empty set, then the two sets must be equal. Which of the above statements 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

Consider three sets X, Y and Z having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let S = (X - Y) ∪ Z. How many proper subsets does S have?

1. A.255
2. B.256
3. C.1023
4. D.1024

Consider the following statements in respect of relations and functions: 1. All relations are functions but all functions are not relations. 2. A relation from A to B is a subset of Cartesian product A × B. 3. A relation in A is a subset of Cartesian product A × A. Which of the above statements 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

Suppose set A consists of first 250 natural numbers that are multiple of 3 and set B consists of first 200 even natural numbers. How many elements does A ∪ B have?

1. A.342
2. B.364
3. C.384
4. D.400

Let f(x) be a polynomial function such that $f\circ f(x)=x^4$. What is f'(1) equal to ?

1. A.0
2. B.1
3. C.2
4. D.4

What is the range of the function f(x) = 1 - sinx defined on entire real line?

1. A.(0, 2)
2. B.[0, 2]
3. C.(-1, 1)
4. D.[-1, 1]

Consider the following statements: 1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets. 2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets. 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

Consider the following statements: 1. The null set is a subset of every set. 2. Every set is a subset of itself. 3. If a set has 10 elements, then its power set will have 1024 elements. Which of the above statements 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

Let R be a relation defined as xRy if and only if 2x + 3y = 20, where x, y ∈ N. How many elements of the form (x, y) are there in R?

1. A.2
2. B.3
3. C.4
4. D.6

Consider the following statements: 1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto. 2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto. 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

The average of a set of 15 observations is recorded, but later it is found that for one observation, the digit in the tens place was wrongly recorded as 8 instead of 3. After correcting the observation, the average is

1. A.reduced by $\frac{1}{3}$
2. B.increased by $\frac{10}{3}$
3. C.reduced by $\frac{10}{3}$
4. D.reduced by 50

What is the domain of the function $f(x)=3^x$?

1. A.$(-\infty ,\infty )$
2. B.$(0,\infty )$
3. C.$[0,\infty )$
4. D.$(-\infty ,\infty )-\{0\}$