1
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

If $$f(x) = {{(\tan 1^\circ )x + {{\log }_e}(123)} \over {x{{\log }_e}(1234) - (\tan 1^\circ )}},x > 0$$, then the least value of $$f(f(x)) + f\left( {f\left( {{4 \over x}} \right)} \right)$$ is :

A
2
B
4
C
0
D
8
2
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$$, where $$\lceil x\rceil$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements

(S1) : $$A \cap B=(1, \infty)-\mathbb{N}$$ and

(S2) : $$A \cup B=(1, \infty)$$

A
only $$(\mathrm{S} 2)$$ is true
B
only (S1) is true
C
neither (S1) nor (S2) is true
D
both (S1) and (S2) are true
3
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$f:\mathbb{R}-{0,1}\to \mathbb{R}$$ be a function such that $$f(x)+f\left(\frac{1}{1-x}\right)=1+x$$. Then $$f(2)$$ is equal to

A
$$\frac{9}{4}$$
B
$$\frac{7}{4}$$
C
$$\frac{7}{3}$$
D
$$\frac{9}{2}$$
4
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$f(x) = \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr } } \right|,\,x \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$. If $$\alpha$$ and $$\beta$$ respectively are the maximum and the minimum values of $$f$$, then

A
$${\alpha ^2} - {\beta ^2} = 4\sqrt 3$$
B
$${\beta ^2} - 2\sqrt \alpha = {{19} \over 4}$$
C
$${\beta ^2} + 2\sqrt \alpha = {{19} \over 4}$$
D
$${\alpha ^2} + {\beta ^2} = {9 \over 2}$$
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