Sets, Relations & Functions · NDA previous year questions

The angles of elevation of the top of a tower from two places situated at distances 21 m and $x$ m from the base of the tower are $45^{\circ }$ and $60^{\circ }$ respectively. What is the value of $x$?

1. A.$7\sqrt{3}$
2. B.$7-\sqrt{3}$
3. C.$7+\sqrt{3}$
4. D.14

What is $\frac{\cot 54^{\circ }}{\tan 36^{\circ }}+\frac{\tan 20^{\circ }}{\cot 70^{\circ }}$ equal to?

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

A person standing on the bank of a river observes that the angle subtended by a tree on the opposite of bank is $60^{\circ }$. When he retires 40 m from the bank, he finds the angle to be $30^{\circ }$. What is the breadth of the river?

1. A.60 m
2. B.40 m
3. C.30 m
4. D.20 m

If the angles of a triangle are $30^{\circ }$ and $45^{\circ }$ and the included side is $(\sqrt{3}+1)$, then what is the area of the triangle?

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

What is ${\sin }^{2}20^{\circ }+{\sin }^{2}70^{\circ }$ equal to?

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

If $A=\{1,3,5,7\}$, then what is the cardinality of the power set $P(A)$?

1. A.8
2. B.15
3. C.16
4. D.17

*Directions :* For the next six (06) questions that follow : In a state with a population of $75\times 10^6$, 45% of them know Hindi, 22% know English, 18% know Sanskrit, 12% know Hindi and English, 8% know English and Sanskrit, 10% know Hindi and Sanskrit and 5% know all the three languages. What is the number of people who do not know any of the above three languages ?

1. A.$3\times 10^6$
2. B.$4\times 10^6$
3. C.$3\times 10^7$
4. D.$4\times 10^7$

What is the number of people who know Hindi only ?

1. A.$21\times 10^6$
2. B.$25\times 10^6$
3. C.$28\times 10^6$
4. D.$3\times 10^7$

What is the number of people who know Sanskrit only ?

1. A.$5\times 10^6$
2. B.$45\times 10^5$
3. C.$4\times 10^6$
4. D.None of the above

What is the number of people who know English only ?

1. A.$5\times 10^6$
2. B.$45\times 10^5$
3. C.$4\times 10^6$
4. D.None of the above

What is the number of people who know only one language ?

1. A.$3\times 10^6$
2. B.$4\times 10^6$
3. C.$3\times 10^7$
4. D.$4\times 10^7$

What is the number of people who know only two languages ?

1. A.$11.25\times 10^5$
2. B.$11.25\times 10^6$
3. C.$12\times 10^5$
4. D.$12.5\times 10^5$

What is the degree of the equation $\frac{1}{x-3}=\frac{1}{x+2}-\frac{1}{2}$ ?

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

Which one of the following is a null set ?

1. A.$\{0\}$
2. B.$\{\{\{\}\}\}$
3. C.$\{\{\}\}$
4. D.$\{x\mid x^2+1=0,x\in R\}$

If $A=\{x,y\}$, $B=\{2,3\}$, $C=\{3,4\}$, then what is the number of elements in $A\times (B\cup C)$ ?

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

What is the value of ${\log }_y{x}^5{\log }_x{y}^2{\log }_z{z}^3$ ?

1. A.10
2. B.20
3. C.30
4. D.60

If A is a relation on a set R, then which one of the following is correct ?

1. A.$R\subseteq A$
2. B.$A\subseteq R$
3. C.$A\subseteq (R\times R)$
4. D.$R\subseteq (A\times A)$

Let N be the set of natural numbers and $f:N\to N$ be a function given by $f(x)=x+1$ for $x\in N$. Which one of the following is correct ?

1. A.f is one-one and onto
2. B.f is one-one but not onto
3. C.f is only onto
4. D.f is neither one-one nor onto

What is the range of the function $f(x)=\frac{|x|}{x},x\ne 0$ ?

1. A.Set of all real numbers
2. B.Set of all integers
3. C.$\{-1,1\}$
4. D.$\{-1,0,1\}$

If $y=x^x$, what is $\frac{dy}{dx}$ at $x=1$ equal to ?

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

The relation 'has the same father as' over the set of children is:

1. A.only reflexive
2. B.only symmetric
3. C.only transitive
4. D.an equivalence relation

Let $P=\{1,2,3\}$ and a relation on set P is given by the set $R=\{(1,2),(1,3),(2,1),(1,1),(2,2),(3,3),(2,3)\}$. Then R is:

1. A.Reflexive, transitive but not symmetric
2. B.Symmetric, transitive but not reflexive
3. C.Symmetric, reflexive but not transitive
4. D.None of the above

The value of the sum ${\sum }_{n=1}^{13}(i^n+i^{n+1})$ where $i=\sqrt{-1}$ is:

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

If a non-empty set A contains n elements, then its power set contains how many elements?

1. A.$n^2$
2. B.$2^n$
3. C.$2n$
4. D.$n+1$

Let $A=\{x\in W\text{, the set of whole numbers and }x<3\}$, $B=\{x\in N\text{, the set of natural numbers and }2\le x<4\}$ and $C=\{3,4\}$, then how many elements will $(A\cup B)\times C$ contain?

1. A.6
2. B.8
3. C.10
4. D.12