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 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)= [\frac{10 x^3-45 x^2+60 x+35}{n}] is continuous on the interval [1,2], is (1) 8 (Q) The minimum value of n for which g(x)= (2 n^2-13 n. -15) (x^3+3 x) , 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)= (x^2-9) ^n (x^2+2 x+3), is (3) 5 (S) Number of x_0 \in \mathbb{R} such that I(x)=\sum_{k=0}^{4} ( \sin |x-k|+\cos {| x-k+\frac{1}{2} |}) , x \in \mathbb{R}, is NOT differentiable at x_0, is (4) 6 (5) 10 [JEE Adv 2025 P1]

Correct Answer is : (P) ⟶ (2) (Q) ⟶ (1) (R) ⟶ (4) (S) ⟶ (3)

(P) \rightarrow (1) (Q) \rightarrow (3) (R) \rightarrow (2) (S) \rightarrow (5)

(P) \rightarrow (2) (Q) \rightarrow (1) (R) \rightarrow (4) (S) \rightarrow (3)

( {\P}) \rightarrow(5); (Q) \rightarrow (1); (R) \rightarrow (4); (S) \rightarrow (3)

(P) \rightarrow (2); (Q) \rightarrow (3); (R) \rightarrow (1); (S) \rightarrow (5)

Limits, Continuity and Differentiability

Section: Mathematics

Question : 3 of 51

Marks: +1, -0

Let ℝ denote the set of all real numbers. For a real number x, let [x] denote the greatest integer less than or equal to 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)=[‌

10x3−45x2+60x+35

] is continuous on the interval [1,2], is

(1)

(Q)

The minimum value of n for which g(x)=(2n2−13n. −15)(x3+3x),x∈ℝ, is an increasing function on ℝ, is

(2)

(R)

The smallest natural number n which is greater than 5, such that x=3 is a point of local minima of h(x)=(x2−9)n(x2+2x+3), is

(3)

(S)

Number of x0∈ℝ such that I(x)=

∑

k=0

(sin‌|x−k|+cos‌|x−k+‌

|),x∈ℝ, is NOT differentiable at x0, is

(4)

(5)

[JEE Adv 2025 P1]

(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)

Validate

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