JEE MAIN 2026 ONLINE 28TH JANUARY MORNING SHIFT

The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t)=A \sin \omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t=T / 2 \beta$. The value of $\beta$ is $\_\_\_\_$ .

$$\text { Given below are the four isomeric compounds }(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{~S})$$

Identify correct statements from below.

A. $\mathrm{Q}, \mathrm{R}$ and S will give precipitate with 2, 4-DNP.

B. $P$ and $Q$ will give positive Bayer's test.

C. Q and R will give sooty flame.

D. R and S will give yellow precipitate with $\mathrm{I}_2 / \mathrm{NaOH}$.

E. Q alone will deposit silver with Tollen's reagent

Choose the correct option.

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2-y^2 \sec ^2 \theta=8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2 \sec ^2 \theta+y^2=6$ be $e_2$ and $l_2$, respectively. If $e_1^2=e_2^2\left(\sec ^2 \theta+1\right)$, then $\left(\frac{l_1 l_2}{e_1 e_2}\right) \tan ^2 \theta$ is equal to

A convex lens of refractive index 1.5 and focal length $f=18 \mathrm{~cm}$ is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha \times \mathrm{f}$. The value of $\alpha$ is $\_\_\_\_$ .

(refractive index of water $=4 / 3$ )

$$\text { Given below are two statements for the following reaction sequence. }$$

Statement I : Compound ' $Z$ ' will give yellow precipitate with NaOI .

Statement II : Compound ' Q ' has two different types of ' H ' atoms (aromatic : aliphatic) in the ratio $1: 3$.

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

If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then

the number of solutions of the equation $\sin ^{-1}(k x-1)=\sin ^{-1} x-\cos ^{-1} x$ is $\_\_\_\_$.