JEE Advance - Mathematics (2025 - Paper 1 Online - No. 15)
Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] 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 | List–II |
|---|---|
| (P) The minimum value of $n$ for which the function $$ f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right] $$ is continuous on the interval $[1,2]$, is | (1) 8 |
| (Q) The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right)$, $x \in \mathbb{R}$, is an increasing function on $\mathbb{R}$, is | (2) 9 |
| (R) The smallest natural number $n$ which is greater than 5 , such that $x=3$ is a point of local minima of $$ h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right) $$ is | (3) 5 |
| (S) Number of $x_0 \in \mathbb{R}$ such that
$$ l(x)=\sum\limits_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right) $$ $x \in \mathbb{R}$, is NOT differentiable at $x_0$, is |
(4) 6 |
| (5) 10 |
(P) → (1) (Q) → (3) (R) → (2) (S) → (5)
(P) → (2) (Q) → (1) (R) → (4) (S) → (3)
(P) → (5) (Q) → (1) (R) → (4) (S) → (3)
(P) → (2) (Q) → (3) (R) → (1) (S) → (5)
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