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): 
Polarity, in chemical bonding, distribution of electrical charge over the atoms joined by the bond. Specifically, while bonds between identical atoms, as in H2, are electrically uniform in the sense that both hydrogen atoms are electrically neutral, bonds between atoms of different elements are electrically inequivalent.
A non-polar covalent bond is a type of chemical bond that is formed when electrons are shared equally between two atoms. Thus, in an atom, the number of electrons shared by the adjacent atoms will be the same.
The covalent bond is also termed as nonpolar because the difference in electronegativity is mostly negligible.
A polar covalent bond exists when atoms with different electronegativities share electrons in a covalent bond. Consider the hydrogen chloride (HCl) molecule. Each atom in HCl requires one more electron to form an inert gas electron configuration.
Dipole moment is defined as the product of the magnitude of charge and the distance of separation between the centres of positive and negative charge.
Dipole moment is a vector quantity and it is denoted by µ.
µ = charge (Q) * Distance of separation(r)
The dipole moment is expressed in Debye units (D).