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JEE Advanced 2024 · Paper 1

JEE Advanced 2024 Mathematics question on Limits Continuity and Differentiability

Let f:R→Rf : \mathbb{R} \to \mathbb{R} and g:R→Rg : \mathbb{R} \to \mathbb{R} be functions defined by
f(x)={x ∣x∣ sin⁡(1x),x≠0,0,x=0,andg(x)={1−2x,0≤x≤12,0,otherwise.f(x) = \begin{cases} x\,|x|\,\sin\left(\dfrac{1}{x}\right), & x \neq 0, \\ 0, & x = 0, \end{cases} \quad \text{and} \quad g(x) = \begin{cases} 1-2x, & 0 \leq x \leq \dfrac{1}{2}, \\ 0, & \text{otherwise}. \end{cases}
Let a,b,c,d∈Ra,b,c,d \in \mathbb{R}. Define the function h:R→Rh : \mathbb{R} \to \mathbb{R} by
h(x)=a f(x)+b(g(x)+g(12−x))+c (x−g(x))+d g(x), x∈R.h(x) = a\,f(x) + b\left( g(x) + g\left(\frac{1}{2} - x\right) \right) + c\,(x - g(x)) + d\,g(x), \ x \in \mathbb{R}.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) If a=0a=0, b=1b=1, c=0c=0, and d=0d=0, then
(Q) If a=1a=1, b=0b=0, c=0c=0, and d=0d=0, then
(R) If a=0a=0, b=0b=0, c=1c=1, and d=0d=0, then
(S) If a=0a=0, b=0b=0, c=0c=0, and d=1d=1, then
List-II
(1) hh is one-one.
(2) hh is onto.
(3) hh is differentiable on R\mathbb{R}.
(4) the range of hh is [0,1][0,1].
(5) the range of hh is {0,1}\{0,1\}.
  1. A.
    (P)→(4)(Q)→(3)(R)→(1)(S)→(2)(P)\to(4)\quad (Q)\to(3)\quad (R)\to(1)\quad (S)\to(2)
  2. B.
    (P)→(5)(Q)→(2)(R)→(4)(S)→(3)(P)\to(5)\quad (Q)\to(2)\quad (R)\to(4)\quad (S)\to(3)
  3. C.
    (P)→(5)(Q)→(3)(R)→(2)(S)→(4)(P)\to(5)\quad (Q)\to(3)\quad (R)\to(2)\quad (S)\to(4)
  4. D.
    (P)→(4)(Q)→(2)(R)→(1)(S)→(3)(P)\to(4)\quad (Q)\to(2)\quad (R)\to(1)\quad (S)\to(3)
Show answer

Correct option: C

[JEE Advanced 2024 · Paper 1]

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