Sets, Relations & Functions · JEE Main previous year questions

Let $f:(-1,1)\to B$, be a function defined by $$f(x)={\tan }^{-1}\frac{2x}{1-x^2},$$ then $f$ is both one-one and onto when B is the interval

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

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

1. A.Interval: $(-\infty ,\infty )$, Function: $x^3-3x^2+3x+3$
2. B.Interval: $[2,\infty )$, Function: $2x^3-3x^2-12x+6$
3. C.Interval: $\left(-\infty ,\frac{1}{3}\right]$, Function: $3x^2-2x+1$
4. D.Interval: $(-\infty ,-4)$, Function: $x^3+6x^2+6$

A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is a given constant and f(0) = 1, f(2a - x) is equal to

1. A.- f(x)
2. B.f(x)
3. C.f(a) + f(a - x)
4. D.f(- x)

If $f$ is a real valued differentiable function satisfying $$|f(x)-f(y)|\le (x-y{)}^2,x,y\in R$$ and $f(0)=0$, then $f(1)$ equals

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

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

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

The range of the function f(x) = ${}^{7-x}{P}_{x-3}$ is

1. A.{1, 2, 3, 4, 5}
2. B.{1, 2, 3, 4, 5, 6}
3. C.{1, 2, 3, 4}
4. D.{1, 2, 3}

The graph of the function y = f(x) is symmetrical about the line x = 2, then

1. A.$f(x)=-f(-x)$
2. B.$f(2+x)=f(2-x)$
3. C.$f(x)=f(-x)$
4. D.$f(x+2)=f(x-2)$

The domain of the function $f(x)=\frac{{\sin }^{-1}(x-3)}{\sqrt{9-x^2}}$

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

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :

1. A.a function
2. B.transitive
3. C.not symmetric
4. D.reflexive

A function $y=f(x)$ has a second order derivative $f^{\prime \prime }(x)=6(x-1)$. If its graph passes through the point $(2,1)$ and at that point the tangent to the graph is $y=3x-5$, then the function is :

1. A.$(x+1{)}^2$
2. B.$(x-1{)}^3$
3. C.$(x+1{)}^3$
4. D.$(x-1{)}^2$

The domain of ${\sin }^{-1}\left[{\log }_3\left(\frac{x}{3}\right)\right]$ is

1. A.[1, 9]
2. B.[-1, 9]
3. C.[9, 1]
4. D.[-9, -1]

f(x) and g(x) are two differentiable functions on [0, 2] such that $f^{\prime \prime }(x)-g^{\prime \prime }(x)=0,f^{\prime }(1)=2,g^{\prime }(1)=4,f(2)=3,g(2)=9$ then f(x) - g(x) at x = $\frac{3}{2}$ is

1. A.0
2. B.2
3. C.10
4. D.-5