Sets, Relations & Functions · JEE Main previous year questions

Let $f(x)=2^{10}\cdot x+1$ and $g(x)=3^{10}\cdot x-1$. If $(f\circ g)(x)=x$, then $x$ is equal to :

1. A.$\frac{3^{10}-1}{3^{10}-2^{-10}}$
2. B.$\frac{2^{10}-1}{2^{10}-3^{-10}}$
3. C.$\frac{1-3^{-10}}{2^{10}-3^{-10}}$
4. D.$\frac{1-2^{-10}}{3^{10}-2^{-10}}$

The set of all values of $\lambda$ for which the system of linear equations: $$2x_1-2x_2+x_3=\lambda x_1$$ $$2x_1-3x_2+2x_3=\lambda x_2$$ $$-x_1+2x_2=\lambda x_3$$ has a non-trivial solution

1. A.contains two elements
2. B.contains more than two elements
3. C.is an empty set
4. D.is a singleton

If the function. $$g(x)=\{\begin{array}{ll}k\sqrt{x+1},& 0\le x\le 3\\ mx+2,& 3<x\le 5\end{array}$$ is differentiable, then the value of $k+m$ is :

1. A.$\frac{10}{3}$
2. B.$4$
3. C.$2$
4. D.$\frac{16}{5}$

Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set $\mathrm{A}\times \mathrm{B}$, each having at least three elements are

1. A.$219$
2. B.$256$
3. C.$275$
4. D.$510$

If $g$ is the inverse of a function $f$ and $f^{\prime }(x)=\frac{1}{1+x^5}$, then $g^{\prime }(x)$ is equal to:

1. A.$\frac{1}{1+\{g(x){\}}^5}$
2. B.$1+\{g(x){\}}^5$
3. C.$1+x^5$
4. D.$5x^4$

If $X=\{4^n-3n-1:n\in N\}$ and $Y=\{9(n-1):n\in N\}$, where $N$ is the set of natural numbers, then $X\cup Y$ is equal to:

1. A.$X$
2. B.$Y$
3. C.$N$
4. D.$Y-X$

If $f$ and $g$ are differentiable functions in $[0,1]$ satisfying $f(0)=2=g(1),g(0)=0$ and $f(1)=6$, then for some $c\in ]0,1[$

1. A.$f^{\prime }(c)=g^{\prime }(c)$
2. B.$f^{\prime }(c)=2g^{\prime }(c)$
3. C.$2f^{\prime }(c)=g^{\prime }(c)$
4. D.$2f^{\prime }(c)=3g^{\prime }(c)$

Let $A$ and $B$ be non empty sets in $R$ and $f:A\to B$ is a bijective function. Statement 1: $f$ is an onto function. Statement 2: There exists a function $g:B\to A$ such that $fog=I_B$.

1. A.Statement 1 is true, Statement 2 is false.
2. B.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
3. C.Statement 1 is false, Statement 2 is true.
4. D.Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.

If $A=\{x\in z^{+}:x<10\text{ and }x\text{ is a multiple of 3 or 4}\}$, where $z^{+}$ is the set of positive integers, then the total number of symmetric relations on $A$ is

1. A.$2^5$
2. B.$2^{15}$
3. C.$2^{10}$
4. D.$2^{20}$

Statement 1: If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q>p$. Then the total number of functions from set $A$ to set $B$ is $q^p$. Statement 2: The total number of selections of $p$ different objects out of $q$ objects is ${}^q{C}_p$.

1. A.Statement 1 is true, Statement 2 is false.
2. B.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.
3. C.Statement 1 is false, Statement 2 is true.
4. D.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

If the number of 5-element subsets of the set $A=\{a_1,a_2,\dots ,a_{20}\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a_4$, then $k$ is

1. A.5
2. B.$\frac{20}{7}$
3. C.4
4. D.$\frac{10}{3}$

The range of the function $f(x)=\frac{x}{1+|x|},x\in R$, is

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

If $x+|y|=2y$, then $y$ as a function of $x$, at $x=0$ is

1. A.differentiable but not continuous
2. B.continuous but not differentiable
3. C.continuous as well as differentiable
4. D.neither continuous nor differentiable

Let $f:(-1,1)\to R$ be a differentiable function with $f(0)=-1$ and $f^{\prime }(0)=1$. Let $g(x)=[f(2f(x)+2){]}^2$. Then $g^{\prime }(0)=$

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

Let $p(x)$ be a function defined on $R$ such that $p^{\prime }(x)=p^{\prime }(1-x)$, for all $x\in [0,1],p(0)=1$ and $p(1)=41$. Then $\underset{0}{\overset{1}{\int }}p(x)dx$ equals :

1. A.$21$
2. B.$41$
3. C.$42$
4. D.$\sqrt{41}$

Consider the following relations $R=\{(x,y)\mid x,y\text{ are real numbers and }x=wy\text{ for some rational number }w\}$; $S=\left\{\left(\frac{m}{n},\frac{p}{q}\right)|m,n,p\text{ and }q\text{ are integers such that }n,q\ne 0\text{ and }qm=pn\right\}$. Then

1. A.$R$ is an equivalence relation but $S$ is not an equivalence relation
2. B.Neither $R$ nor $S$ is an equivalence relation
3. C.$S$ is an equivalence relation but $R$ is not an equivalence relation
4. D.$R$ and $S$ both are equivalence relations

Let $y$ be an implicit function of $x$ defined by $x^{2x}-2x^x\cot y-1=0$. Then $y^{\prime }(1)$ equals

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

$\underset{0}{\overset{\pi }{\int }}[\cot x]dx$, where $[.]$ denotes the greatest integer function, is equal to:

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

If $A,B$ and $C$ are three sets such that $A\cap B=A\cap C$ and $A\cup B=A\cup C$, then :

1. A.$A=C$
2. B.$B=C$
3. C.$A\cap B=\varphi$
4. D.$A=B$

If the expansion in powers of $x$ of the function $\frac{1}{(1-ax)(1-bx)}$ is $a_0+a_{1}x+a_2{x}^2+a_3{x}^3\dots$. then $a_n$ is

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

The set of points where $f(x)=\frac{x}{1+|x|}$ is differentiable is

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

The function $f(x)=\frac{x}{2}+\frac{2}{x}$ has a local minimum at

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

Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R=\{(x,y)\in W\times W\mid \text{the words }x\text{ and }y\text{ have at least one letter in common}\}$. Then, $R$ is

1. A.reflexive, symmetric and not transitive
2. B.reflexive, symmetric and transitive
3. C.reflexive, not symmetric and transitive
4. D.not reflexive, symmetric and transitive

ABC is a triangle, right angled at A. The resultant of the forces acting along $\overrightarrow{AB},\overrightarrow{AC}$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is the force along $\overrightarrow{AD}$, where D is the foot of the perpendicular from A onto BC. The magnitude of the resultant is

1. A.$\frac{A{B}^2+A{C}^2}{(AB{)}^2(AC{)}^2}$
2. B.$\frac{(AB)(AC)}{AB+AC}$
3. C.$\frac{1}{AB}+\frac{1}{AC}$
4. D.$\frac{1}{AD}$

If the coefficient of $x^7$ in ${\left[a{x}^2+\left(\frac{1}{bx}\right)\right]}^{11}$ equals the coefficient of $x^{-7}$ in ${\left[ax-\left(\frac{1}{b{x}^2}\right)\right]}^{11}$, then $a$ and $b$ satisfy the relation

1. A.$a-b=1$
2. B.$a+b=1$
3. C.$\frac{a}{b}=1$
4. D.$ab=1$