Sets, Relations & Functions · NDA previous year questions

For the following two (02) items: Let $u=\int e^x\cos xdx$ and $v=\int e^x\sin xdx$. Consider the following: I. $\frac{du}{dx}=-v$ II. $\frac{dv}{dx}=-u$ Which of the above is/are correct?

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

For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5]. What is $\frac{dy}{dx}$ at x = 3.5 equal to?

1. A.0
2. B.1
3. C.2
4. D.3.5

For the following two (02) items: Consider the function $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ What is $f\circ f\circ f\circ f\circ f(0)$ equal to?

1. A.0
2. B.1
3. C.5
4. D.10

For the following two (02) items: Consider the function $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ What is the inverse of the function?

1. A.${\log }_{10}(2x-1)$
2. B.$\frac{1}{2}{\log }_{10}(2x-1)$
3. C.$\frac{1}{4}{\log }_{10}(\frac{2x}{2-x})$
4. D.$\frac{1}{2}{\log }_{10}(\frac{1+x}{1-x})$

What is ${\int }_n^{n+1}(x-[x])dx$, where [.] is the greatest integer function and n is natural number?

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

Consider the following statements: I. $y=x{e}^{2x}$ is the solution of $\frac{dy}{dx}=y(2+\frac{1}{x})$ II. $y=x\ln |x|+cx$ is the solution of $\frac{dy}{dx}=\frac{x+y}{x}$ Which of the statements given above is/are correct?

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

What is the domain of the function (f(x))?

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

Consider the following statements: I. f is one-one function. II. f is onto function if the codomain is the set of natural numbers. Which of the statements given above is/are correct?

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

Consider the following for the two (02) items that follow : Let the function $f(x)=x^2-1$ What is ${\lim }_{x\to 1}\{f\circ f(x)\}$ equal to?

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

Consider the following for the two (02) items that follow: Let the function $f(x)=x^2+9$ What is ${\lim }_{x\to 0}\frac{\sqrt{f(x)}-3}{\sqrt{f(x)+7}-4}$ equal to?

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

Consider the following for the two (02) items that follow: The function f(x) satisfies $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all positive real values of x and y, and f(2) = 3 What is $f(16)$ equal to?

1. A.18
2. B.27
3. C.54
4. D.81

Consider the following for the two (02) items that follow: The function f(x) satisfies $f(x),f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$. for all positive real values of x and y, and f(2) = 3 What is $f(1)f(4)$ equal to?

1. A.4
2. B.8
3. C.9
4. D.18

Consider the following for the two (02) items that follow: A function f is such that f(xy) = f(x + y) for all real values of x and y, and f(5) = 10 What is f(0) equal to?

1. A.0
2. B.1
3. C.5
4. D.10

What is f(20)+f(-20) equal to?

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

Consider the following for the two (02) items that follow: Let $f(x)=[x^2]$ where [.] is the greatest integer function. What ${\int }_{\sqrt{2}}^{\sqrt{3}}f(x)dx$ equal to?

1. A.$\sqrt{3}-\sqrt{2}$
2. B.$2(\sqrt{3}-\sqrt{2})$
3. C.$3-\sqrt{2}$
4. D.1

Consider the following for the two (02) items that follow: Let $f(x)=[x^2]$ where [.] is the greatest integer function. ${\int }_{\sqrt{2}}^{2}f(x)dx$ is equal to ?

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

Consider the following statements: I. The set of all irrational numbers between $\sqrt{12}$ and $\sqrt{15}$ is an infinite set. II. The set of all odd integers less than 1000 is a finite set. Which of the statements given above are correct?

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

How many 4-digit numbers are there having all digits as odd?

1. A.625
2. B.400
3. C.196
4. D.120

Let P and Q be two non-void relations on a set A. Which of the following statements are correct? I. P and Q are reflexive ⇒ P ∩ Q is reflexive. II. P and Q are symmetric ⇒ P ∪ Q is symmetric. III. P and Q are transitive ⇒ P ∩ Q is transitive. Select the answer using the code given below.

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

If A and B are two non-empty sets having 10 elements in common, then how many elements do A x B and B x A have in common?

1. A.10
2. B.20
3. C.40
4. D.100

What is the remainder when $7^n-6n$ is divided by 36 for $n=100$?

1. A.0
2. B.1
3. C.2
4. D.6

What is the point of intersection of the diagonals of the trapezium?

1. A.(3, 7/2)
2. B.(3, 7/3)
3. C.(7/2, 2)
4. D.(5/2, 2)

What is the point of intersection of L and P?

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

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 f o g(x) = g o f(x), then what is the value of k?

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