List of top Mathematics Questions on Functions asked in JEE Advanced

Let $ \mathbb{N} $ denote the set of all natural numbers, and $ \mathbb{Z} $ denote the set of all integers. Consider the functions $ f : \mathbb{N} \to \mathbb{Z} $ and $ g : \mathbb{Z} \to \mathbb{N} $ defined by $$ f(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{4 - n}{2}, & \text{if } n \text{ is even} \end{cases} \quad \text{and} \quad g(n) = \begin{cases} 3 + 2n, & \text{if } n \ge 0 \\ -2n, & \text{if } n<0 \end{cases} $$ Define $ (g \circ f)(n) = g(f(n)) $ for all $ n \in \mathbb{N} $, and $ (f \circ g)(n) = f(g(n)) $ for all $ n \in \mathbb{Z} $. Then which of the following statements is (are) TRUE?

Let $ \mathbb{R} $ denote the set of all real numbers. For a real number $ x $, let $ \lfloor x \rfloor $ denote the greatest integer less than or equal to $ x $. Let $ n $ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I

• [(P)] The minimum value of $ n $ for which the function $$ f(x) = \left\lfloor \frac{10x^3 - 45x^2 + 60x + 35}{n} \right\rfloor $$ is continuous on the interval $ [1, 2] $, is
• [(Q)] The minimum value of $ n $ for which $$ g(x) = (2n^2 - 13n - 15)(x^3 + 3x), $$ $ x \in \mathbb{R} $, is an increasing function on $ \mathbb{R} $, is
• [(R)] The smallest natural number $ n $ which is greater than 5, such that $ x = 3 $ is a point of local minima of $$ h(x) = (x^2 - 9)^n (x^2 + 2x + 3), $$ is
• [(S)] Number of $ x_0 \in \mathbb{R} $ such that $$ l(x) = \sum_{k=0}^{4} \left( \sin |x - k| + \cos \left| x - k + \frac{1}{2} \right| \right) $$ is not differentiable at $ x_0 $, is

List-II

• [(1)] 8
• [(2)] 9
• [(3)] 5
• [(4)] 6
• [(5)] 10

Options:

Let \(f:\R→\R\) be a function defined by
\(f(x) = \begin{cases}     x^2\sin(\frac{\pi}{x^2}), & \text{if } x \ne 0, \\     0, & \text{if } x = 0. \end{cases}\)
Then which of the following statements is TRUE ?

Let \(f:\R→\R\) be a function such that f(x + y) = f(x) + f(y) for all x, y ∈ \(\R\), and \(g:\R→(0,\infin)\) be a function such that g(x + y) = g(x)g(y) for all x, y ∈ \(\R\). If \(f(\frac{-3}{5})=12\) and \(g(\frac{-1}{3})=2\), then the value of \((f(\frac{1}{4})+g(-2)-8)g(0)\) is ______.

Let the function \(f:\R→\R\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}}\frac{(x)^{2023}+2024x+2025}{(x^2-x+3)}+\frac{2}{e^{\pi x}}\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}\)
Then the number of solutions of f (x) = 0 in \(\R\) is __.

Let \(f:[0,\frac{\pi}{2}]→[0,1]\) be the function defined by \(f(x)=\sin^2x\) and let \(g:[0,\frac{\pi}{2}]→[0,\infin)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}=x^2}\).

Let the functions $f :(-1,1) \rightarrow R$ and $g :(-1,1) \rightarrow(-1,1)$ be defined by
$f(x)=|2 x-1|+|2 x+1|$ and $g(x)=x-[x] \text {, }$
where $[ x ]$ denotes the greatest integer less than or equal to $x$. Let fog : $(-1,1) \rightarrow R$ be the composite function defined by $(f o g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which fog is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which fog is NOT differentiable. Then the value of $c + d$ is ____

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying
$f(x+y)=f(x)+f(y)+f(x) f(y)$ and $f(x)=x g(x)$
for all $x, y \in R $. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?

Let $S=\{1,2,3, \ldots \ldots, 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_{k}$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=$

Let $f\left(x\right) = sin\left(\frac{\pi}{6}sin\left(\frac{\pi}{2}sin\,x\right)\right)$ for all $x \in R$ and $g\left(x\right) = \frac{\pi}{2}$ $sin\, x$ for all $x \in R$. Let $(fog) (x)$ denote $f(g(x))$ and $(gof) (x)$ denote $g(f(x))$. Then which of the following is (are) true ?

The function f (x) = $ [ x]^2 - [ x]^2 $ (where, [x] is the greatest integer less than or equal to x), is discontinuous at