Sets, Relations & Functions · NDA previous year questions

Let $f:R\to R$ be a function whose inverse is $\frac{x+5}{3}$. What is f(x) equal to?

1. A.f(x) = 3x + 5
2. B.f(x) = 3x - 5
3. C.f(x) = 5x - 3
4. D.f(x) does not exist

Consider the following statements : 1. If $y=\ln (\sec x+\tan x)$, then $\frac{dy}{dx}=\sec x$. 2. If $y=\ln (\mathrm{cosec}x-\cot x)$, then $\frac{dy}{dx}=\mathrm{cosec}x$. 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 $A=\{x\in R\mid x\ge 0\}$. A function $f:A\to A$ is defined by $f(x)=x^2$. Which one of the following is correct?

1. A.The function does not have inverse
2. B.f is its own inverse
3. C.The function has an inverse but f is not its own inverse
4. D.None of the above

If $y=\ln (e^{mx}+e^{-mx})$, then what is $\frac{dy}{dx}$ at $x=0$ equal to?

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

Let $U=\{x\in N:1\le x\le 10\}$ be the universal set, N being the set of natural numbers. If $A=\{1,2,3,4\}$ and $B=\{2,3,6,10\}$ then what is the complement of $(A-B)$?

1. A.$\{6,10\}$
2. B.$\{1,4\}$
3. C.$\{2,3,5,6,7,8,9,10\}$
4. D.$\{5,6,7,8,9,10\}$

The solution of the simultaneous linear equations $2x+y=6$ and $3y=8+4x$ will also be satisfied by which one of the following linear equations?

1. A.$x+y=5$
2. B.$2x+y=9$
3. C.$2x-3y=10$
4. D.$2x+3y=6$

Let $A=\{x:x\text{ is a square of a natural number and }x\text{ is less than }100\}$ and B is a set of even natural numbers. What is the cardinality of $A\cap B$?

1. A.4
2. B.5
3. C.9
4. D.None of the above

If A and B are two non-empty sets having n elements in common, then what is the number of common elements in the sets $A\times B$ and $B\times A$?

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

If A and B are any two sets, then what is $A\cap (A\cup B)$ equal to?

1. A.Complement of A
2. B.Complement of B
3. C.B
4. D.A

The set $A=\{x:x+4=4\}$ can also be represented by :

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

If a line makes the angles $\alpha ,\beta ,\gamma$ with the axes, then what is the value of $1+\cos 2\alpha +\cos 2\beta +\cos 2\gamma$ equal to?

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

If $A=\{a,b,c\}$, then what is the number of proper subsets of A ?

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

If $\Phi$ is a null set, then which one of the following is correct?

1. A.$\Phi =0$
2. B.$\Phi =\{0\}$
3. C.$\Phi =\{\Phi \}$
4. D.$\Phi =\{\}$

If $P,Q,R$ are three non-collinear points, then what is $PQ\cap PR$ equal to?

1. A.Null set
2. B.$\{P\}$
3. C.$\{P,Q,R\}$
4. D.$\{Q,R\}$

Let $M$ be the set of men and $R$ is a relation ‘is son of’ defined on $M$. Then $R$ is

1. A.an equivalence relation
2. B.a symmetric relation only
3. C.a transitive relation only
4. D.None of the above

If $A,B,C$ are non-empty sets such that $A\cap C=\Phi$, then what is $(A\times B)\cap (C\times B)$ equal to?

1. A.$A\times C$
2. B.$A\times B$
3. C.$B\times C$
4. D.$\Phi$

If $A=\{4n+2\mid n\text{ is a natural number}\}$ and $B=\{3n\mid n\text{ is a natural number}\}$, then what is $(A\cap B)$ equal to?

1. A.$\{12n^2+6n\mid n\text{ is a natural number}\}$
2. B.$\{24n-12\mid n\text{ is a natural number}\}$
3. C.$\{60n+30\mid n\text{ is a natural number}\}$
4. D.$\{12n-6\mid n\text{ is a natural number}\}$

If $f(x)$ is an even function, then what is ${\int }_0^{\pi }f(\cos x)dx$ equal to?

1. A.0
2. B.${\int }_0^{\pi /2}f(\cos x)dx$
3. C.$2{\int }_0^{\pi /2}f(\cos x)dx$
4. D.1

What is the value of $k$ for which the system of equations $kx+2y=5$ and $3x+y=1$ has no solution?

1. A.0
2. B.3
3. C.6
4. D.15

Consider the following with regard to a relation $R$ on a set of real numbers defined by $xRy$ if and only if $3x+4y=5$: 1. $0R1$ 2. $1R\frac{1}{2}$ 3. $\frac{2}{3}R\frac{3}{4}$ Which of the above are correct?

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

What is the set of points $(x,y)$ satisfying the equations $x^2+y^2=4$ and $x+y=2$?

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

What is the solution set for the equation $x^4-26x^2+25=0$?

1. A.$\{-5,-1,1,5\}$
2. B.$\{-5,-1\}$
3. C.$\{1,5\}$
4. D.$\{-5,0,1,5\}$

For the next four (4) items that follow : $ABC$ is a triangle right angled at $B$. The hypotenuse $(AC)$ is four times the perpendicular $(BD)$ drawn to it from the opposite vertex and $AD<DC$. What is one of the acute angles of the triangle?

1. A.$15^{\circ }$
2. B.$30^{\circ }$
3. C.$45^{\circ }$
4. D.None of the above

For the next four (4) items that follow : $ABC$ is a triangle right angled at $B$. The hypotenuse $(AC)$ is four times the perpendicular $(BD)$ drawn to it from the opposite vertex and $AD<DC$. What is angle $ABD$?

1. A.$15^{\circ }$
2. B.$30^{\circ }$
3. C.$45^{\circ }$
4. D.None of the above

For the next four (4) items that follow : $ABC$ is a triangle right angled at $B$. The hypotenuse $(AC)$ is four times the perpendicular $(BD)$ drawn to it from the opposite vertex and $AD<DC$. What is $AD:DC$ equal to?

1. A.$(7-2\sqrt{3}):1$
2. B.$(7-4\sqrt{3}):1$
3. C.$1:2$
4. D.None of the above