Sets, Relations & Functions · NDA previous year questions

For the next four (4) items that follow : Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $$f(x)=x^3+x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3)$$ for $x\in \mathbb{R}$. What is $f(1)$ equal to?

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

For the next four (4) items that follow : Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $$f(x)=x^3+x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3)$$ for $x\in \mathbb{R}$. What is $f^{\prime }(1)$ equal to?

1. A.-6
2. B.-5
3. C.1
4. D.0

For the next four (4) items that follow : Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $$f(x)=x^3+x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3)$$ for $x\in \mathbb{R}$. What is $f^{\prime \prime \prime }(10)$ equal to?

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

For the next two (2) items that follow: Consider the function $$f(x)=|x^2-5x+6|$$ What is $f^{\prime }(4)$ equal to?

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

For the next two (2) items that follow: Consider the function $$f(x)=|x^2-5x+6|$$ What is $f^{\prime \prime }(2.5)$ equal to?

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

For the next two (2) items that follow : Consider the functions $f(x)=xg(x)$ and $g(x)=\left[\frac{1}{x}\right]$ where $[\cdot ]$ is the greatest integer function. What is ${\int }_{1/3}^{1/2}g(x)dx$ equal to?

1. A.$\frac{1}{6}$
2. B.$\frac{1}{3}$
3. C.$\frac{5}{18}$
4. D.$\frac{5}{36}$

For the next two (2) items that follow : Consider the functions $f(x)=xg(x)$ and $g(x)=\left[\frac{1}{x}\right]$ where $[\cdot ]$ is the greatest integer function. What is ${\int }_{1/3}^{1}f(x)dx$ equal to?

1. A.$\frac{37}{72}$
2. B.$\frac{2}{3}$
3. C.$\frac{17}{72}$
4. D.$\frac{37}{144}$

What is $${\int }_{-2}^{2}xdx-{\int }_{-2}^2[x]dx$$ equal to, where $[\cdot ]$ is the greatest integer function?

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

If $${\int }_{-2}^{5}f(x)dx=4\text{and}{\int }_0^5\{1+f(x)\}dx=7$$ then what is ${\int }_{-2}^{0}f(x)dx$ equal to?

1. A.$-3$
2. B.2
3. C.3
4. D.5

What is ${\int }_0^{4\pi }|\cos x|dx$ equal to?

1. A.0
2. B.2
3. C.4
4. D.8

What is the number of points of local minima of the function $f(x)$?

1. A.None
2. B.One
3. C.Two
4. D.Three

What is the number of points of local maxima of the function $f(x)$?

1. A.None
2. B.One
3. C.Two
4. D.Three

For the next two (2) items that follow : Consider the curves $y=|x-1|$ and $|x|=2$ What is/are the point(s) of intersection of the curves?

1. A.$(-2,3)$ only
2. B.$(2,1)$ only
3. C.$(-2,3)$ and $(2,1)$
4. D.Neither $(-2,3)$ nor $(2,1)$

For the next four (4) items that follow: Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x)=x^3+x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3)$ for $x\in \mathbb{R}$. Consider the following: 1. $f(2)=f(1)-f(0)$ 2. $f^{\prime \prime }(2)-2f^{\prime }(1)=12$ 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 $X$ be the set of all persons living in Delhi. The persons $a$ and $b$ in $X$ are said to be related if the difference in their ages is at most 5 years. The relation is

1. A.an equivalence relation
2. B.reflexive and transitive but not symmetric
3. C.symmetric and transitive but not reflexive
4. D.reflexive and symmetric but not transitive

If $A=\{x\in \mathbb{R}:x^2+6x-7<0\}$ and $B=\{x\in \mathbb{R}:x^2+9x+14>0\}$, then which of the following is/are correct? 1. $(A\cap B)=(-2,1)$ 2. $(A\setminus B)=(-7,-2)$ 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

The domain of the function $$f(x)=\frac{1}{\sqrt{|x|-x}}$$ is

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

If $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ be two functions given by $$f(x)=2x-3\text{ and }g(x)=x^3+5,$$ then $(f\circ g{)}^{-1}(x)$ is equal to

1. A.${\left(\frac{x+7}{2}\right)}^{\frac{1}{3}}$
2. B.${\left(\frac{x-7}{2}\right)}^{\frac{1}{3}}$
3. C.${\left(x-\frac{7}{2}\right)}^{\frac{1}{3}}$
4. D.${\left(x+\frac{7}{2}\right)}^{\frac{1}{3}}$

Consider the following statements: Statement 1: The function $f:\mathbb{R}\to \mathbb{R}$ such that $f(x)=x^3$ for all $x\in \mathbb{R}$ is one-one. Statement 2: $f(a)=f(b)\Rightarrow a=b$ for all $a,b\in \mathbb{R}$ if the function $f$ is one-one. Which one of the following is correct in respect of the above statements?

1. A.Both the statements are true and Statement 2 is the correct explanation of Statement 1.
2. B.Both the statements are true and Statement 2 is \textbf{\textit{not}} the correct explanation of Statement 1.
3. C.Statement 1 is true but Statement 2 is false.
4. D.Statement 1 is false but Statement 2 is true.

If $f(x)=\sqrt{25-x^2}$, then what is $$\underset{x\to 1}{\lim }\frac{f(x)-f(1)}{x-1}$$ equal to?

1. A.$\frac{1}{5}$
2. B.$\frac{1}{24}$
3. C.$\sqrt{24}$
4. D.$-\frac{1}{\sqrt{24}}$

Consider the following functions: 1. $f(x)=\{\begin{array}{ll}\frac{1}{x}& \text{if }x\ne 0\\ 0& \text{if }x=0\end{array}$ 2. $f(x)=\{\begin{array}{ll}2x+5& \text{if }x>0\\ x^2+2x+5& \text{if }x\le 0\end{array}$ Which of the above functions is/are derivable at $x=0$?

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

If $f(x)=\frac{\sin (e^{x-2}-1)}{\ln (x-1)}$, then $$\underset{x\to 2}{\lim }f(x)$$ is equal to

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

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

Let $X$ be the set of all persons living in a city. Persons $x,y$ in $X$ are said to be related as $x<y$ if $y$ is at least 5 years older than $x$. Which one of the following is correct?

1. A.The relation is an equivalence relation on $X$
2. B.The relation is transitive but neither reflexive nor symmetric
3. C.The relation is reflexive but neither transitive nor symmetric
4. D.The relation is symmetric but neither transitive nor reflexive

In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is

1. A.35
2. B.40
3. C.50
4. D.55