B.Sc. FY Sem 1 - MSU Vadodara

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Mathematics: Calculus & Algebra Differential Calculus Limits & Continuity: Definition of limit: $\lim_{x \to a} f(x) = L$ Properties of limits, L'Hopital's Rule Continuity at a point and on an interval Differentiation: First principles: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ Rules of differentiation: Chain rule, Product rule, Quotient rule Higher-order derivatives Applications: Maxima/Minima, Tangents & Normals, Rate of Change Partial Differentiation: First and second order partial derivatives Euler's theorem for homogeneous functions: $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = nz$ Integral Calculus Indefinite Integrals: Standard integration formulas Methods of integration: Substitution, By parts ($\int u dv = uv - \int v du$), Partial fractions Definite Integrals: Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$ Properties of definite integrals Applications: Area under curve, Volume of revolution Linear Algebra (Basic) Matrices: Types of matrices, Matrix operations (addition, scalar multiplication, multiplication) Determinants: Properties, Cofactor expansion Inverse of a matrix: $A^{-1} = \frac{1}{|A|} \text{adj}(A)$ Systems of Linear Equations: Cramer's Rule Gaussian Elimination, Rank of a matrix Homogeneous and non-homogeneous systems Physics: Mechanics & Oscillations Units and Measurements Physical quantities, SI units, Dimensional analysis Error analysis: propagation of errors Newtonian Mechanics Kinematics: Displacement, Velocity, Acceleration, Equations of motion Dynamics: Newton's Laws of Motion, Inertial frames Work, Energy & Power: Work-Energy Theorem: $W = \Delta K$ Conservation of Mechanical Energy: $E = K+U = \text{constant}$ Power: $P = \vec{F} \cdot \vec{v}$ Rotational Motion: Angular displacement, velocity, acceleration Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ Moment of Inertia, Parallel & Perpendicular Axes Theorems Angular Momentum: $\vec{L} = I \vec{\omega}$, Conservation of Angular Momentum Gravitation: Newton's Law of Gravitation, Gravitational potential energy, Kepler's Laws Oscillations and Waves Simple Harmonic Motion (SHM): Equation of SHM: $x(t) = A \cos(\omega t + \phi)$ Energy in SHM, Simple Pendulum, Spring-Mass system Damped and Forced Oscillations: Resonance Wave Motion: Transverse and Longitudinal waves Wave equation, Superposition principle, Standing waves Chemistry: Physical & Inorganic Atomic Structure Bohr's Model: Postulates, Limitations, Energy of electron in H-atom Quantum Mechanical Model: De Broglie hypothesis: $\lambda = h/p$ Heisenberg's Uncertainty Principle: $\Delta x \Delta p \ge h/(4\pi)$ Schrödinger Equation (qualitative), Atomic orbitals (s, p, d, f) Quantum numbers: n, l, m_l, m_s Electronic Configuration: Aufbau principle, Pauli's exclusion principle, Hund's rule Chemical Bonding Ionic Bonding: Lattice energy, Born-Haber cycle Covalent Bonding: Lewis structures, VSEPR theory (shapes of molecules) Valence Bond Theory (VBT): Hybridization (sp, sp2, sp3, etc.) Molecular Orbital Theory (MOT): Bonding and anti-bonding orbitals, Bond order, Magnetic properties Hydrogen bonding, Van der Waals forces Gaseous State Gas Laws: Boyle's, Charles', Gay-Lussac's, Avogadro's laws Ideal Gas Equation: $PV = nRT$ Dalton's Law of Partial Pressures, Graham's Law of Diffusion Kinetic Theory of Gases: Postulates, Kinetic energy of gas molecules Deviation from ideal behavior: Van der Waals equation Basic Organic Chemistry Nomenclature: IUPAC naming of simple organic compounds Isomerism: Structural, Geometrical, Optical Reaction Intermediates: Carbocations, carbanions, free radicals Reaction Mechanisms: Electrophiles, Nucleophiles Inductive effect, Mesomeric effect, Hyperconjugation