Sets, Relations & Functions · NDA previous year questions

Consider the following statements 1. A = (A ∪ B) ∪ (A - B), 2. A ∪ (B - A) = (A ∪ B) 3. B = (A ∪ B) - (A - B) Which of the statements given above 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

A function satisfies $f(x-y)=\frac{f(x)}{f(y)}$, where $f(y)\ne 0$. If $f(1)=0.5$, then what is $f(2)+f(3)+f(4)+f(5)+f(6)$ equal to ?

1. A.$\frac{15}{32}$
2. B.$\frac{17}{32}$
3. C.$\frac{29}{64}$
4. D.$\frac{31}{64}$

Consider the following for the next items that follow: Let a function f be defined on $\mathbb{R}-\{0\}$ and $2f(x)+f\left(\frac{1}{x}\right)=x+3$. What is f(0.5) equal to ?

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

Consider the following for the next items that follow: A function is defined by $f(x)=\pi +{\sin }^{2}x$. What is the range of the function?

1. A.$[0,1]$
2. B.$[\pi ,\pi +1]$
3. C.$[\pi -1,\pi +1]$
4. D.$[\pi -1,\pi -1]$

Consider the following for the next items that follow: A function is defined by $f(x)=\pi +{\sin }^{2}x$. What is the period of the function?

1. A.$2\pi$
2. B.$\pi$
3. C.$\frac{\pi }{2}$
4. D.The function is non-periodic

Consider the following for the next items that follow: Let $f(x)=\sqrt{2-x}+\sqrt{2+x}$. What is the domain of the function?

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

Consider the following for the next items that follow: Let $f(x)=\sqrt{2-x}+\sqrt{2+x}$. What is the greatest value of the function?

1. A.$\sqrt{3}$
2. B.$\sqrt{6}$
3. C.$\sqrt{8}$
4. D.4

Consider the following for the next items that follow: Let $f(x)=|x|$ and $g(x)=[x]-1$, where [.] is the greatest integer function. Let $h(x)=\frac{f(g(x))}{g(f(x))}$. What is ${\lim }_{x\to 0+}h(x)$ equal to ?

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

Consider the following for the next items that follow: Let $f(x)=|x|$ and $g(x)=[x]-1$, where [.] is the greatest integer function. Let $h(x)=\frac{f(g(x))}{g(f(x))}$. What is ${\lim }_{x\to 0-}h(x)$ equal to ?

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

What is the range of the function f(x) = x + |x| if the domain is the set of real numbers ?

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

If $f(x)=x(4x^2-3)$, then what is $f(\sin \theta )$ equal to ?

1. A.$-\sin 3\theta$
2. B.$-\cos 3\theta$
3. C.$\sin 3\theta$
4. D.$-\sin 4\theta$

The central angles p, q, r and s (in degrees) of four sectors in a Pie Chart satisfy the relation 9p = 3q = 2r = 6s. What is the value of 4p - q ?

1. A.12
2. B.24
3. C.30
4. D.36

Consider the following statements : 1. If f is the subset of Z × Z defined by f = {(xy, x − y); x, y ∈ Z}, then f is a function from Z to Z. 2. If f is the subset of N × N defined by f = {(xy, x + y); x, y ∈ N}, then f is a function from N to N. Which of the statements given above 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 set of all irrational numbers between $\sqrt{2}$ and $\sqrt{5}$ is an infinite set. 2. The set of all odd integers less than 100 is a finite set. Which of the statements given above is/are correct?

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

Let A = {7, 8, 9, 10, 11, 12, 13; 14, 15, 16} and let f ∶ A → N be defined by f(x) = the highest prime factor of x. How many elements are there in the range of f?

1. A.4
2. B.5
3. C.6
4. D.7

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 the following : 1. A ∩ B = A ∩ C ⇒ B = C 2. A ∪ B = A ∪ C ⇒ B = C 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

Let $f(x)$ be a function satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in N$ such that $f(1)=2$ : If ${\sum }_{x=2}^{n}f(x)=2044$, then what is the value of $n$ ?

1. A.8
2. B.9
3. C.10
4. D.11

1. A.341
2. B.682
3. C.1023
4. D.1364

Let $f(x)$ be a function satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in N$ such that $f(1)=2$ : If ${\sum }_{x=2}^{n}f(x)=2044$, then what is the value of $n$ ? What is ${\sum }_{x=1}^n{2}^{x}f(x)$ equal to ?

1. A.1365
2. B.2730
3. C.4024
4. D.5460

How many received medals in exactly two of the three sports ?

1. A.30 − 4x
2. B.35 − 7x
3. C.40 − 7x
4. D.45 − 5x

How many received medals in at least two of three sports ?

1. A.30 − 6x
2. B.35 − 6x
3. C.40 − 5x
4. D.40 − 6x

How many received medals in exactly one of three sports ?

1. A.21x − 40
2. B.21x − 35
3. C.20x − 35
4. D.20x − 25

Let z = [y] and y = [x] − x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?

1. A.−1
2. B.0
3. C.1
4. D.2

​If f(x) = 4x + 1 and g(x) = kx + 2 such that fog(x) = gof(x), then what is the value of k ?

1. A.7
2. B.5
3. C.4
4. D.3