In the following asymmetric transformation, the key aldol reaction involves the attack of

The reaction shown is an asymmetric aldol reaction using a chiral auxiliary derived from proline. The stereochemistry of the newly formed chiral centers in the aldol product is determined by the chiral auxiliary and the facial selectivity of the enolate and aldehyde.
Step 1: Enolate Formation
Step 2: Aldol Addition to Benzaldehyde (PhCHO)
Step 3: Oxidative Cleavage of the Chiral Auxiliary
The final step releases the chiral aldol product with the stereochemistry established in the previous step, preserving the relative and absolute configuration defined by the selective facial attack in the Zimmerman–Traxler transition state.
Two positively charged particles \(m_1\) and \(m_2\) have been accelerated across the same potential difference of 200 keV. Given mass of \(m_1 = 1 \,\text{amu}\) and \(m_2 = 4 \,\text{amu}\). The de Broglie wavelength of \(m_1\) will be \(x\) times that of \(m_2\). The value of \(x\) is _______ (nearest integer). 
Structures of four disaccharides are given below. Among the given disaccharides, the non-reducing sugar is: 
The ratio of the fundamental vibrational frequencies \( \left( \nu_{^{13}C^{16}O} / \nu_{^{12}C^{16}O} \right) \) of two diatomic molecules \( ^{13}C^{16}O \) and \( ^{12}C^{16}O \), considering their force constants to be the same, is ___________ (rounded off to two decimal places).
Courage : Bravery :: Yearning :
Select the most appropriate option to complete the analogy.
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
