


1 gram of sodium hydroxide was treated with 25 ml. of 0.75 M HCI solution, the mass of sodium hydroxide left unreacted is equal to :
Identify Z in the following reaction sequence.

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
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
Organic chemistry is the branch of chemistry that involves the scientific study of organic compounds. Organic chemistry primarily deals with the structure and chemical composition of organic compounds, the physical and chemical properties of organic compounds, and the chemical reactions undergone by these compounds.
Intermediates can be understood as the first product of a consecutive reaction. For example, in a chemical reaction, if A→B and B→C, then, B can be said to be the intermediate for reaction A→C. The reactions in organic chemistry occur via the formation of these intermediates.
Reagents are the chemicals that we add to bring about a specific change to an organic molecule. Any general reaction in organic chemistry can be written as:
Substrate + Reagent → Product
Where the substrate is an organic molecule to which we add the reagent. Based on the ability to either donate or abstract electrons, the reagents can be classified as: