Let f: \mathbb{R} \rightarrow \mathbb{R} and g: \mathbb{R} \rightarrow \mathbb{R} be functions defined by f(x)={\\begin{array}{c}x| x|\end{array} \sin (\frac{1}{x}) {,} , x \neq 0{,} ; 0{,} , x=0{,} } {\text{ and } g(x)= {\\begin{array}{cc}1-2 x{ &\end{array} , 0 \leq x \leq \frac{1}{2}{, };0{,}, \otherwise} Let a, b, c, d \in \mathbb{R} . Define the function h: \mathbb{R} \rightarrow \mathbb{R} by h(x)=a f(x)+b (g(x)+g (\frac{1}{2}-x) ) +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 List-II (P) If a = 0, b = 1, c = 0 and d = 0, then (1) h is one-one (Q) If a = 1, b = 0, c = 0 and d = 0, then (2) h is onto. (R) If a = 0, b = 0, c = 1 and d = 0, then (3) h is differentiable on ℝ . (S) If a = 0, b = 0, c = 0 and d = 1, then (4) the range of h is [0, 1] (5) the range of h is {0, 1} The correct option is [JEE Adv 2024 P1]

Correct Answer is : (P) → (5) (Q) → (3) (R) → (2) (S) → (4)

(P) → (4) (Q) → (3) (R) → (1) (S) → (2)

(P) → (5) (Q) → (2) (R) → (4) (S) → (3)

(P) → (5) (Q) → (3) (R) → (2) (S) → (4)

(P) → (4) (Q) → (2) (R) → (1) (S) → (3)

Limits, Continuity and Differentiability

Section: Mathematics

Question : 8 of 51

Marks: +1, -0

Let f:ℝ⟶ℝ and g:ℝ⟶ℝ be functions defined by
f(x)={

x|x|sin‌(‌

x≠0,

x=0,

‌ and ‌g(x)={

1−2x,

0≤x≤‌

otherwise

Let a,b,c,d∈ℝ . Define the function h:ℝ⟶ℝ by
h(x)=af(x)+b(g(x)+g(‌

−x)) +c(x−g(x))+dg(x),x∈ℝ
Match each entry in List-I to the correct entry in List-II.

The correct option is

[JEE Adv 2024 P1]

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