JEE ADVANCED 2024 PAPER 1 ONLINE

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. $\mathrm{NaOH}$ solution, gives $\mathbf{P}$ and $\mathbf{Q}$. Compound $\mathbf{P}$ does not give Tollens' test, whereas $\mathbf{Q}$ on acidification gives positive Tollens' test. Treatment of $\mathbf{P}$ with excess cyclohexanone in the presence of catalytic amount of $p$-toluenesulfonic acid (PTSA) gives product $\mathbf{R}$.

Sum of the number of methylene groups $\left(-\mathrm{CH}_2-\right)$ and oxygen atoms in $\mathbf{R}$ is __________.

Let $f(x)=x^4+a x^3+b x^2+c$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $i \sqrt{3}$ is a root of the equation $4 x^3+3 a x^2+2 b x=0$, where $i=\sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x)=0$, then $\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2$ is equal to ____________.

Consider the following volume-temperature $(\mathrm{V}-\mathrm{T})$ diagram for the expansion of 5 moles of an ideal monoatomic gas.

Considering only $\mathrm{P}-\mathrm{V}$ work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence $\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}$ is ____________.

[Use the given data: Molar heat capacity of the gas for the given temperature range, $\mathrm{C}_{\mathrm{V}, \mathrm{m}}=12 \mathrm{~J} \mathrm{~K}^{-1}$ $\mathrm{mol}^{-1}$ and gas constant, $\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]$

Let $S=\left\{A=\left(\begin{array}{lll}0 & 1 & c \\ 1 & a & d \\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.$ and $\left.|A| \in\{-1,1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is __________.