A dipeptide, “x”, on complete hydrolysis gives “y” and “z”; “y” on treatment with aqueous HNO$_2$, produces lactic acid. On the other hand, “z” on heating gives the following cyclic molecule. 
Based on the information given, the dipeptide X is:
Let’s break down the key information provided:
1. Hydrolysis of “x”:
- The dipeptide "x" undergoes complete hydrolysis to produce two amino acids: y and z.
- y is a compound that, when treated with aqueous HNO\(_2\), produces lactic acid. This strongly suggests that y is glycine, as glycine reacts with nitrous acid to form lactic acid.
Therefore, glycine must be one of the products after hydrolysis.
2. Heating of “z”:
- z on heating forms a cyclic molecule. This strongly indicates that z is proline, as proline is an amino acid that can form a cyclic structure under heating conditions.
3. Identifying the Dipeptide:
- The dipeptide must be one that hydrolyzes to give glycine (which produces lactic acid upon treatment with HNO\(_2\)) and proline (which forms a cyclic structure upon heating).
- The only dipeptide in the given options that fits this pattern is alanine-glycine (option 2), as alanine can undergo cyclization to form proline under heat.
Thus, the correct dipeptide x is alanine-glycine.
Let \( y^2 = 12x \) be the parabola and \( S \) its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ) = \frac{147}{4} \). Let \( C \) be the circle described by taking \( PQ \) as a diameter. If the equation of the circle \( C \) is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then \( \beta - \alpha \) is equal to:
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
Let $ A \in \mathbb{R} $ be a matrix of order 3x3 such that $$ \det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1 \\2 & 0 & 1 \\4 & 1 & 2 \end{array} \right] $$ where $ I $ is the identity matrix of order 3. If $ \det( (A + I) \cdot \text{adj}(A + I)) $ is $ 2^m $, then $ m $ is equal to: