Sets, Relations & Functions · NDA previous year questions

If $f(x)=\frac{\sqrt{x-1}}{x-4}$ defines a function on $\mathbf{R}$, then what is its domain?

1. A.$(-\infty ,4)\cup (4,\infty )$
2. B.$[4,\infty )$
3. C.$(1,4)\cup (4,\infty )$
4. D.$[1,4)\cup (4,\infty )$

What is ${\int }_a^b[x]dx+{\int }_a^b[-x]dx$ equal to, where $[.]$ is the greatest integer function?

1. A.$b-a$
2. B.$a-b$
3. C.$0$
4. D.$2(b-a)$

What is ${\int }_{-1}^1\left\{\frac{d}{dx}\left({\tan }^{-1}\frac{1}{x}\right)\right\}dx$ equal to?

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

A function $f:A\to \mathbf{R}$ is defined by the equation $f(x)=x^2-4x+5$ where $A=(1,4)$. What is the range of the function?

1. A.$(2,5)$
2. B.$(1,5)$
3. C.$[1,5)$
4. D.$[1,5]$

What is ${\int }_2^8|x-5|dx$ equal to?

1. A.$2$
2. B.$3$
3. C.$4$
4. D.$9$

Let A and B be subsets of X and $C=(A\cap B^{\prime })\cup (A^{\prime }\cap B)$, where $A^{\prime }$ and $B^{\prime }$ are complements of A and B respectively in X. What is C equal to ?

1. A.$(A\cup B^{\prime })-(A\cap B^{\prime })$
2. B.$(A^{\prime }\cup B)-(A^{\prime }\cap B)$
3. C.$(A\cup B)-(A\cap B)$
4. D.$(A^{\prime }\cup B^{\prime })-(A^{\prime }\cap B^{\prime })$

What is the number of students who are good in either Hindi or Mathematics but not in English ?

1. A.99
2. B.107
3. C.125
4. D.130

What is the number of students who are good in Hindi and Mathematics but not in English ?

1. A.18
2. B.12
3. C.10
4. D.8

A survey of 850 students in a University yields that 680 students like music and 215 like dance. What is the least number of students who like both music and dance ?

1. A.40
2. B.45
3. C.50
4. D.55

Let $[x]$ denote the greatest integer function. What is the number of solutions of the equation $x^2-4x+[x]=0$ in the interval $[0,2]$ ?

1. A.Zero (No solution)
2. B.One
3. C.Two
4. D.Three

How many numbers between 100 and 1000 can be formed with the digits 5, 6, 7, 8, 9, if the repetition of digits is not allowed ?

1. A.$3^5$
2. B.$5^3$
3. C.120
4. D.60

How many four-digit numbers divisible by 10 can be formed using 1, 5, 0, 6, 7 without repetition of digits ?

1. A.24
2. B.36
3. C.44
4. D.64

What is the equation of the line passing through the point of intersection of the lines $x+2y-3=0$ and $2x-y+5=0$ and parallel to the line $y-x+10=0$?

1. A.$7x-7y+18=0$
2. B.$5x-7y+18=0$
3. C.$5x-5y+18=0$
4. D.$x-y+5=0$

Which one of the following is correct in respect of the function $f:\mathbb{R}\to {\mathbb{R}}^{+}$ defined as $f(x)=|x+1|$?

1. A.$f(x^2)=[f(x){]}^2$
2. B.$f(|x|)=|f(x)|$
3. C.$f(x+y)=f(x)+f(y)$
4. D.None of the above

Suppose $f:\mathbb{R}\to \mathbb{R}$ is defined by $f(x)=\frac{x^2}{1+x^2}$. What is the range of the function?

1. A.$[0,1)$
2. B.$[0,1]$
3. C.$(0,1]$
4. D.$(0,1)$

For f to be a function, what is the domain of f, if $f(x)=\frac{1}{\sqrt{|x|-x}}$?

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

What is the period of the function $f(x)=\sin x$?

1. A.$\pi /4$
2. B.$\pi /2$
3. C.$\pi$
4. D.$2\pi$

If $f(x)$ is an even function, where $f(x)\ne 0$, then which one of the following is correct?

1. A.$f^{\prime }(x)$ is an even function
2. B.$f^{\prime }(x)$ is an odd function
3. C.$f^{\prime }(x)$ may be an even or odd function depending on the type of function
4. D.$f^{\prime }(x)$ is a constant function

If $f:\mathbb{R}\to S$ defined by $f(x)=4\sin x-3\cos x+1$ is onto, then what is S equal to?

1. A.$[-5,5]$
2. B.$(-5,5)$
3. C.$(-4,6)$
4. D.$[-4,6]$

Consider the function $$f(x)=\{\begin{array}{ll}{x}^2\ln |x|& x\ne 0\\ 0& x=0\end{array}$$ What is $f^{\prime }(0)$ equal to?

1. A.$0$
2. B.$1$
3. C.$-1$
4. D.It does not exist

What is ${\int }_0^{\sqrt{2}}[x^2]dx$ equal to (where [.] is the greatest integer function)?

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

What is the value of ${\int }_{-\pi /4}^{\pi /4}(\sin x-\tan x)dx$?

1. A.$-\frac{1}{\sqrt{2}}+\ln \left(\frac{1}{\sqrt{2}}\right)$
2. B.$\frac{1}{\sqrt{2}}$
3. C.$0$
4. D.$\sqrt{2}$

If ${\int }_a^b{x}^{3}dx=0$ and ${\int }_a^b{x}^{2}dx=\frac{2}{3}$, then what are the values of a and b respectively?

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

What is ${\int }_0^{1}x(1-x{)}^{9}dx$ equal to?

1. A.1/110
2. B.1/132
3. C.1/148
4. D.1/240

If $E$ is the universal set and $A=B\cup C$, then the set $E-(E-(E-(E-(E-A))))$ is same as the set

1. A.$B^{\prime }\cup C^{\prime }$
2. B.$B\cup C$
3. C.$B^{\prime }\cap C^{\prime }$
4. D.$B\cap C$