As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
Chemical Reactions go with the breaking and bonding of covalent bonds which involve of exchange of electrons. The functional groups of Organic compounds play a consequential role in the process. Based on the above theory, reactions can be classified into five main groups:
Rearrangement Reactions are the type of reactions in which products get formed simply by the rearrangement of atoms and electrons in the reactant molecules.
O
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NH4CNO → NH2 –C – NH2
Substitution Reactions are the reactions in which an atom or group of atoms is replaced by some other atom or group of atoms without any change in the structure of the remaining part of the molecule.
CH3Br + KOH (aqueous) → CH3OH + KBr
Addition Reactions are the reactions in which products get formed by the addition of some reagent to an unsaturated compound.
CH2 = CH2 + HCl → CH5Cl
Elimination Reactions are the reactions in which the products get formed by the loss of simple molecules like HX from the reactant molecules.
C2H5OH → C2H4
A polymerization Reaction is the union of two or more molecules of a substance that form a single molecule with higher molecular weight.
n (CH = CH2) → (-CH2 – CH2 -) n