JEE Advanced 2025 · Paper 1 · Mathematics Q15

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MathematicsList match+4 / -1Question 15 of 48

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

P
The minimum value of $n$ for which the function $f(x) = \left[\dfrac{10x^3 - 45x^2 + 60x + 35}{n}\right]$ 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 $l(x) = \displaystyle\sum_{k=0}^{4}\left(\sin|x - k| + \cos\left|x - k + \frac12\right|\right)$ , $x \in \mathbb{R}$ , is NOT differentiable at $x_0$ , is

List-II

1
$8$
2
$9$
3
$5$
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

Answer and worked solution

Official answer: B

(P) Let $g(x)=10x^3-45x^2+60x+35$ . Then $g'(x)=30(x-1)(x-2)\le0$ on $[1,2]$ , so $g$ falls from 60 to 55.

$[g/n]$ is constant on $[1,2]$ exactly when no multiple of $n$ lies in $(55,60]$ :

$n=1,\dots,8$ each have a multiple there ( $n=6$ : 60; $n=7$ : 56; $n=8$ : 56).
$n=9$ has none (its multiples nearby are 54 and 63).

So the minimum is $n=9$ → (2).

(Q) $x^3+3x$ is increasing, so we need $2n^2-13n-15=(2n-15)(n+1)>0$ , i.e. $n\ge8$ → (1).

(R) Near $x=3$ , $h(x)=(x-3)^n\cdot(\text{positive})$ . This has a local minimum at 3 exactly when $n$ is even. The smallest even $n>5$ is 6 → (4).

(S) $\cos|x-k+\frac12|=\cos(x-k+\frac12)$ is smooth. Each $\sin|x-k|$ has a corner at $x=k$ , where the one-sided derivatives are $\mp1$ . These corners are at distinct points, so they cannot cancel. That gives 5 points ( $x=0,\dots,4$ ) → (3).

The match is P→2, Q→1, R→4, S→3, which is option (B).

Topics:Monotonicity Continuity Maxima and minima Differentiability

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