JEE MAIN 2023 ONLINE 11TH APRIL MORNING SHIFT

Two radioactive elements A and B initially have same number of atoms. The half life of A is same as the average life of B. If \(\lambda_{A}\) and \(\lambda_{B}\) are decay constants of A and B respectively, then choose the correct relation from the given options.

In the extraction process of copper, the product obtained after carrying out the reactions

(i) \(2 \mathrm{Cu}_{2} \mathrm{~S}+3 \mathrm{O}_{2} \rightarrow 2 \mathrm{Cu}_{2} \mathrm{O}+2 \mathrm{SO}_{2}\)

(ii) \(2 \mathrm{Cu}_{2} \mathrm{O}+\mathrm{Cu}_{2} \mathrm{~S} \rightarrow 6 \mathrm{Cu}+\mathrm{SO}_{2}\) is called

The number of elements in the set

\(S=\left\{\theta \in[0,2 \pi]: 3 \cos ^{4} \theta-5 \cos ^{2} \theta-2 \sin ^{6} \theta+2=0\right\}\) is :

A transmitting antenna is kept on the surface of the earth. The minimum height of receiving antenna required to receive the signal in line of sight at \(4 \mathrm{~km}\) distance from it is \(x \times 10^{-2} \mathrm{~m}\). The value of \(x\) is __________.

(Let, radius of earth \(\mathrm{R}=6400 \mathrm{~km}\) )

Given below are two statements:

Statement-I : Methane and steam passed over a heated \(\mathrm{Ni}\) catalyst produces hydrogen gas.

Statement-II : Sodium nitrite reacts with \(\mathrm{NH}_{4} \mathrm{Cl}\) to give \(\mathrm{H}_{2} \mathrm{O}, \mathrm{N}_{2}\) and \(\mathrm{NaCl}\).

In the light of the above statements, choose the most appropriate answer from the options given below :

Let \(f:[2,4] \rightarrow \mathbb{R}\) be a differentiable function such that \(\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]\) with \(f(2)=\frac{1}{2}\) and \(f(4)=\frac{1}{4}\).

Consider the following two statements :

(A) : \(f(x) \leq 1\), for all \(x \in[2,4]\)

(B) : \(f(x) \geq \frac{1}{8}\), for all \(x \in[2,4]\)

Then,