JEE (Advanced) 2016 Paper-2
For the following questions answer them individually
Among the following, reaction(s) which gives(give) tert-butyl benzene as the major product is(are)
PARAGRAPH 1
Thermal decomposition of gaseous $$X_2$$ to gaseous X at 298 K takes place according to the following equation :
$$ X_2(g) \rightleftharpoons 2X(g)$$
The standard reaction Gibbs energy, $$\triangle_rG^\circ$$, of this reaction is positive. At the start of the reaction, there is one mole of $$X_2$$ and no $$X$$ As the reaction proceeds, the number of moles of X formed is given by $$\beta$$. Thus, $$\beta_{equilibrium}$$ is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2bar. Consider the gases to behave ideally. (Given: $$R = 0.083 L bar K^{-1} mol^{-1})
The equilibrium constant $$K_p$$ for this reaction at 298 K, in terms of $$\beta_{equilibrium}$$, is
PARAGRAPH 2
Treatment of compound O with $$KMnO_4/H^+$$ gave P, which on heating with ammonia gave Q. The compound Q on treatment with $$Br_2/NaOH$$ produced R. On strong heating, Q gave S, which on
further treatment with ethyl 2-bromopropanoate in the presence of KOH followed by acidification, gave a compound T.
For the following questions answer them individually
Let $$\begin{bmatrix}1 & 0 & 0 \\4 & 1 & 0 \\16 & 4 & 1 \end{bmatrix}$$ and I be the identity matrix of order 3. If $$Q = [q_{ij}]$$ is a matrix such that $$P^{50} - Q = I$$, then $$\frac{q_{31} + q_{32}}{q_{21}}$$ equals
Let $$b_i > 1$$ for i = 1, 2, ......, 101. Suppose $$\log_e b_1, \log_e b_2, ......., \log_e b_{101}$$ are in Arithmetic Progression (A.P.) with the common difference $$\log_e 2$$. Suppose $$a_1, a_2, ........, a_{101}$$ are in A.P. such that $$a_1 = b_1$$ and $$a_{51} = b_{51}$$. If $$t = b_1 + b_2 + ....... + b_{51}$$ and $$s = a_1 + a_2 + ..... + a_{51}$$, then
The value of $$\sum_{k = 1}^{13} \frac{1}{\sin\left(\frac{\pi}{4} + \frac{(k - 1)\pi}{6}\right)\sin\left(\frac{\pi}{4} + \frac{k \pi}{6}\right)}$$ is equal to
The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^2 \cos x}{1 + e^x}dx$$ is equal to
