
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): 
Acid is any hydrogen-containing substance that is capable of donating a proton (hydrogen ion) to another substance. Base is an ion or molecule capable of accepting a hydrogen ion from acid.
| Physical Properties | ACIDS | BASES |
| Taste | Sour | Bitter |
| Colour on Litmus paper | Turns blue litmus red | Turns red litmus blue |
| Ions produced on dissociation | H+ | OH- |
| pH | <7 (less than 7) | >7 (more than 7) |
| Strong acids | HCl, HNO3, H2SO4 | NaOH, KOH |
| Weak Acids | CH3COOH, H3PO4, H2CO3 | NH4OH |
| Type of Reaction | Acid | Bases |
| Reaction with Metals | Acid + Metal → Salt + Hydrogen gas (H2) E.g., Zn(s)+ dil. H2SO4 → ZnSO4 (Zinc Sulphate) + H2 | Base + Metal → Salt + Hydrogen gas (H2) E.g., 2NaOH +Zn → Na2ZnO2 (Sodium zincate) + H2 |
| Reaction with hydrogen carbonates (bicarbonate) and carbonates | Metal carbonate/Metal hydrogen carbonate + Acid → Salt + Carbon dioxide + Water E.g., HCl+NaOH → NaCl+ H2O 2. Na2CO3+ 2 HCl(aq) →2NaCl(aq)+ H2O(l) + CO2(g) 3. Na2CO3+ 2H2SO4(aq) →2Na2SO4(aq)+ H2O(l) + CO2(g) 4. NaHCO3+ HCl → NaCl+ H2O+ CO2 | Base+ Carbonate/ bicarbonate → No reaction |
| Neutralisation Reaction | Base + Acid → Salt + Water E.g., NaOH(aq) + HCl(aq) → NaCl(aq) + H2O(l) | Base + Acid → Salt + Water E.g., CaO+ HCl (l) → CaCl2 (aq)+ H2O (l) |
| Reaction with Oxides | Metal oxide + Acid → Salt + Water E.g., CaO+ HCl (l) → CaCl2 (aq)+ H2O (l) | Non- Metallic oxide + Base → Salt + Water E.g., Ca(OH)2+ CO2 → CaCO3+ H2O |
| Dissolution in Water | Acid gives H+ ions in water. E.g., HCl → H+ + Cl- HCl + H2O → H3O+ + Cl– | Base gives OH- ions in water. |
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