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Limits Continuity and Differentiability question

2025 · Shift 1 · Q31
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  5. /2025 · Shift 1 · Q31

Limits Continuity and Differentiability question

2025 · Shift 1 · Q31

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityMCQ+4 / −1

Let R\mathbb{R}R denote the set of all real numbers. For a real number xxx, let [ x ] denote the greatest integer less than or equal to xxx. Let nnn 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 nnn for which the function f(x)=[10x3−45x2+60x+35n]f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]f(x)=[n10x3−45x2+60x+35​] is continuous on the interval [1,2][1,2][1,2], is (1) 8
(Q) The minimum value of nnn for which g(x)=(2n2−13n−15)(x3+3x)g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right)g(x)=(2n2−13n−15)(x3+3x), x∈Rx \in \mathbb{R}x∈R, is an increasing function on R\mathbb{R}R, is (2) 9
(R) The smallest natural number nnn which is greater than 5 , such that x=3x=3x=3 is a point of local minima of h(x)=(x2−9)n(x2+2x+3)h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right)h(x)=(x2−9)n(x2+2x+3) is (3) 5
(S) Number of x0∈Rx_0 \in \mathbb{R}x0​∈R such that

l(x)=∑k=04(sin⁡∣x−k∣+cos⁡∣x−k+12∣)l(x)=\sum\limits_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right)l(x)=k=0∑4​(sin∣x−k∣+cos​x−k+21​​)

x∈Rx \in \mathbb{R}x∈R, is NOT differentiable at x0x_0x0​, is		 (4) 6
(5) 10
  1. A
    (P) → (1) (Q) → (3) (R) → (2) (S) → (5)
  2. B
    (P) → (2) (Q) → (1) (R) → (4) (S) → (3)
  3. C
    (P) → (5) (Q) → (1) (R) → (4) (S) → (3)
  4. D
    (P) → (2) (Q) → (3) (R) → (1) (S) → (5)
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