Sets, Relations & Functions · NDA previous year questions

If $f(x)=[x{]}^2-30[x]+221=0$, where [x] is the greatest integer function, then what is the sum of all integer solutions?

1. A.13
2. B.17
3. C.27
4. D.30

If f(x) = 9x - 8√x such that g(x) = f(x) - 1, then which one of the following is correct?

1. A.g(x) = 0 has no real roots
2. B.g(x) = 0 has only one real root which is an integer
3. C.g(x) = 0 has two real roots which are integers
4. D.g(x) = 0 has only one real root which is not an integer

What is ${\lim }_{\theta \to \frac{\pi }{2}}(\sec \theta -\tan \theta )$ equal to?

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

Let f(x) f(y) = f(xy) for all real x, y. If f(2) = 4, then what is the value of f(1/2)?

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

Let $A=\{x\in \mathbb{R}:-1<x<1\}$. Which of the following is/are bijective functions from A to itself? 1. $f(x)=x|x|$ 2. $g(x)=\cos (\pi x)$ 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

Let $R$ be a relation on the open interval $(-1,1)$ and is given by $R=\{(x,y):|x+y|<2\}$. Then which one of the following is correct?

1. A.$R$ is reflexive but neither symmetric nor transitive
2. B.$R$ is reflexive and symmetric but not transitive
3. C.$R$ is reflexive and transitive but not symmetric
4. D.$R$ is an equivalence relation

For any three non-empty sets $A,B,C$, what is $(A\cup B-\{(A-B)\cup (B-A)\cup (A\cap B)\})$ equal to?

1. A.Null set
2. B.$A$
3. C.$B$
4. D.$(A\cup B)-(A\cap B)$

If $a,b,c$ are the sides of a triangle $ABC$, then what is $\left|\begin{array}{ccc}b\sin A& 1& \cos A\\ c\sin A& \cos A& 1\end{array}\right|$ equal to?

1. A.Zero
2. B.Area of triangle
3. C.Perimeter of triangle
4. D.$a^2+b^2+c^2$

Let $A=\{1,2,3,4,5\}$ and $B=\{6,7\}$. What is the number of onto functions from $A$ to $B$?

1. A.10
2. B.20
3. C.30
4. D.32

What is $\frac{\sqrt{3}\cos 10^{\circ }-\sin 10^{\circ }}{\sin 25^{\circ }\cos 25^{\circ }}$ equal to ?

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

What is $\sin 9^{\circ }-\cos 9^{\circ }$ equal to?

1. A.$-\frac{\sqrt{5-\sqrt{5}}}{2}$
2. B.$-\frac{\sqrt{5-\sqrt{3}}}{2}$
3. C.$\frac{\sqrt{5-\sqrt{5}}}{2}$
4. D.$\frac{\sqrt{5-\sqrt{5}}}{4}$

Which one of the following is correct in respect of $f(x)=\frac{1}{\sqrt{|x|-x}}$ and $g(x)=\frac{1}{\sqrt{x-|x|}}$ ?

1. A.$f(x)$ has some domain and $g(x)$ has no domain
2. B.$f(x)$ has no domain and $g(x)$ has some domain
3. C.$f(x)$ and $g(x)$ have the same domain
4. D.$f(x)$ and $g(x)$ do not have any domain

Consider the following for the next two (02) items that follow: Given that $\int \frac{3\cos x+4\sin x}{2\cos x+5\sin x}dx=\frac{\alpha x}{29}+\frac{\beta }{29}\ln |2\cos x+5\sin x|+c$ What is the value of $\alpha$ ?

1. A.7
2. B.13
3. C.17
4. D.26

Consider the following for the next two (02) items that follow: Given that $\int \frac{3\cos x+4\sin x}{2\cos x+5\sin x}dx=\frac{\alpha x}{29}+\frac{\beta }{29}\ln |2\cos x+5\sin x|+c$ What is the value of $\beta$ ?

1. A.7
2. B.13
3. C.17
4. D.26

Consider the following for the next two (02) items that follow: Let $f(x)$ and $g(x)$ be two functions such that $g(x)=x-\frac{1}{x}$ and $f\circ g(x)=x^3-\frac{1}{x^3}$. What is $g[f(x)-3x]$ equal to?

1. A.$x^3-\frac{1}{x^3}$
2. B.$x^3+\frac{1}{x^3}$
3. C.$x^2-\frac{1}{x^2}$
4. D.$x^2+\frac{1}{x^2}$

Consider the following for the next two (02) items that follow: Let $f(x)$ and $g(x)$ be two functions such that $g(x)=x-\frac{1}{x}$ and $f\circ g(x)=x^3-\frac{1}{x^3}$. What is $f^{\prime \prime }(x)$ equal to?

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

Let $f(x)=|x-1|,g(x)=[x]$ and $h(x)=f(x)g(x)$ where [.] is greatest integer function. What is ${\int }_{-1}^{0}h(x)dx$ equal to?

1. A.$-\frac{3}{2}$
2. B.-1
3. C.0
4. D.$\frac{1}{2}$

Let $f(x)=|x-1|,g(x)=[x]$ and $h(x)=f(x)g(x)$ where [.] is greatest integer function. What is ${\int }_0^{2}h(x)dx$ equal to?

1. A.$-\frac{3}{2}$
2. B.-1
3. C.0
4. D.$\frac{1}{2}$

Let $\int \frac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha (x+1{)}^{\frac{3}{2}}+\beta (x-1{)}^{\frac{3}{2}}+c$ What is the value of $\alpha$?

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

Let $\int \frac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha (x+1{)}^{\frac{3}{2}}+\beta (x-1{)}^{\frac{3}{2}}+c$ What is the value of $\beta$?

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

A set S contains (2n + 1) elements. There are 4096 subsets of S which contain at most n elements. What is n equal to?

1. A.5
2. B.6
3. C.7
4. D.8

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

The Cartesian product A × A has 16 elements among which are (0, 2) and (1, 3). Which of the following statements is/are correct? 1. It is possible to determine set A. 2. A × A contains the element (3, 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

Let A = {1, 2, 3, ..., 20}. Define a relation R from A to A by R = {(x, y) : 4x - 3y = 1}, where x, y ∈ A. Which of the following statements is/are correct? 1. The domain of R is {1, 4, 7, 10, 13, 16). 2. The range of R is {1, 5, 9, 13, 17). 3. The range of R is equal to codomain of R. Select the correct answer using the code given below:

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

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2 1 2 3 100