List of top Mathematics Questions on Functions asked in IISER

फलन f : R $\rightarrow$ R को f(x) = sin2(7x) − sin2(5x) से परिभाषित करें। निम्न में से कौन सा कथन सत्य नहीं है?

एक तल में स्थित सभी वृत्तों के समुच्चय को C से निरूपित कीजिए। उपसमुच्चय R $\subseteq$ C $\times$ C को निम्न द्वारा परिभाषित कीजिए:
R = {(C1, C2) $\in$ C $\times$ C $|$ C1 व C2 एक दूसरे को प्रतिच्छेद करते हैं}.
निम्न में से कौन सा कथन सत्य है?

For real numbers $a$ and $b$, consider the function $f : \mathbb{R} \to \mathbb{R}$ given by \[ f(x) = \begin{cases} -ax - b ;& \text{if } x \\ 5x + 1 ;& \text{if } -1 \le x \le 1, \\ a^2x + 3b ;& \text{if } x > 1 . \end{cases} \] How many pairs $(a, b)$ are there for which $f$ is continuous at every point of $\mathbb{R}$?

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin^2(7x) - \sin^2(5x)$. Which of the following statements is NOT TRUE?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by
\[ f(x) = \begin{cases} x^2 - 4x - 5 & \text{if } x \ge 1 2x & \text{if } x < 1 \end{cases} \]
Which one of the following statements is TRUE?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = |x^3 - 3x|[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which one of the following statements is TRUE?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function given by $f(x) = \cos(\tan^{-1} x)$. Which one of the following statements is TRUE?

Let $f : \mathbb{R} \to \mathbb{R}$ be a strictly decreasing function with $|f(t)| < \pi/2$ for all $t \in \mathbb{R}$. Let $g : [0, \pi] \to \mathbb{R}$ be a function defined by $g(t) = \sin(f(t))$. Which one of the following statements is Correct?

Let $f, g : \mathbb{R} \to \mathbb{R}$ be functions. If $g$ is continuous, then which one of the following cases implies that $f$ is continuous?

Let $f : \mathbb{Q} \to \mathbb{Q}$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{Q}$, and $f(1) = 10$. Which one of the following statements is Correct ?