Exhaustive Physics, Chemistry, and Mathematics Study Notes

Mechanics, Kinematics, and Measurement

• Slopes of Motion Graphs: The slope of a time-speed graph is defined as the acceleration of the body ($a = \frac{dv}{dt}$).
• Relative Velocity: For two bodies A and B moving in the same direction with velocities $\mathbf{V_A} = 20\hat{i}\,m/s$ and $\mathbf{V_B} = 15\hat{i}\,m/s$, the relative velocity of A with respect to B ($\mathbf{V_{AB}}$) is calculated as $\mathbf{V_A} - \mathbf{V_B} = 5\hat{i}\,m/s$.
• Kinematic Equations: For a bicycle starting from rest ($u = 0$) and accelerating at $1.5\,m/s^2$ for $4$ seconds, the total distance covered is found using $s = ut + \frac{1}{2}at^2$. Calculation: $s = 0 + \frac{1}{2}(1.5)(4)^2 = 12\,m$.
• Work Done by Gravity: A man weighing $70\,kg$ carries a $30\,kg$ box to the top of a building of height $20\,m$. The total mass moved is $100\,kg$. The work done is $W = mgh = 100\,kg \times 9.8\,m/s^2 \times 20\,m = 19600\,J$.
• Dimensional Analysis: To convert a velocity of $72\,kmh^{-1}$ into $ms^{-1}$: $72 \times \frac{5}{18} = 20\,ms^{-1}$.
• Unit Analysis: In the equation $S = a + bt + ct^2$, where $S$ is in metres and $t$ is in seconds, the unit of the constant $c$ must be $ms^{-2}$ to satisfy dimensional homogeneity.
• Errors in Measurement:     * Maximum Error in Density: Density $\rho = \frac{M}{L^3}$. The maximum relative error is $\frac{\Delta \rho}{\rho} = \frac{\Delta M}{M} + 3\frac{\Delta L}{L}$. Given mass error $1.5\%$ and length error $1\%$, the maximum error is $1.5 + 3(1) = 4.5\%$.     * Percentage Error in Length: A rod measured as $5.6\,cm$ with a ruler having a smallest division (least count) of $0.1\,cm$ has a percentage error of $\frac{0.1}{5.6} \times 100 \approx 1.79\%$.
• Screw Gauge Calculations: Diameter is given by $\text{Main scale reading} + (\text{Circular scale reading} \times \text{Least Count})$. If $1\,mm = 100$ divisions, $\text{Least Count} = 0.01\,mm$. For a reading of $0\,mm$ and $52$ divisions: $0 + 52(0.01) = 0.52\,mm = 0.052\,cm$.
• Circular Motion: For a body moving in a circular path with constant speed, the work done by the net force (centripetal force) is zero because the force is always perpendicular to the displacement.
• Projectile and Path Equations: A particle with coordinates $x = a\sin(\omega t)$ and $y = a\cos(\omega t)$ follows a circular path because $x^2 + y^2 = a^2$.
• Frame of Reference: A coin dropped in an elevator reaches the floor in time $t_1$ at rest and $t_2$ moving uniformly. In both cases, the acceleration relative to the elevator is $g$, so $t_1 = t_2$.
• Relative Motion Crossing: A train $150\,m$ long moving North at $10\,m/s$ and a parrot flying South at $5\,m/s$. Relative speed is $10 + 5 = 15\,m/s$. Time to cross = $\frac{150}{15} = 10\,s$.
• Contact Forces: Frictional force is a contact force, unlike gravitational, magnetic, or electrostatic forces.

Thermodynamics and Kinetic Theory

• Specific Heat Calculations: To raise the temperature of $2$ moles of ideal gas at constant pressure ($\Delta T = 5^\circ C$) requires $70\,cal$. Since $Q_p = nC_p\Delta T$, $70 = 2(C_p)(5) \Rightarrow C_p = 7\,cal\,mol^{-1}K^{-1}$. Using Mayer's Relation $C_p - C_v = R$ (where $R \approx 2\,cal\,mol^{-1}K^{-1}$), $C_v = 5\,cal\,mol^{-1}K^{-1}$. Heat needed at constant volume ($Q_v = nC_v\Delta T$) is $2(5)(5) = 50\,cal$.
• First Law of Thermodynamics: Along path $iaf$, $Q = 50\,cal$, $W = 20\,cal \Rightarrow \Delta U = 30\,cal$. Along path $ibf$, if $Q = 36\,cal$, then $W = Q - \Delta U = 36 - 30 = 6\,cal$.
• Kinetic Theory of Gases: The root mean square (rms) speed of a gas molecule of mass $m$ at temperature $T$ is given by $v_{rms} = \sqrt{\frac{3kT}{m}}$.
• Sound Propagation: When sound waves travel in a gaseous medium, the process is considered adiabatic because the compressions and rarefactions happen too rapidly for heat exchange.
• Adiabatic Relations: For a monoatomic gas (where $\gamma = 5/3$), the adiabatic relation is $PV^{5/3} = \text{constant}$.

Optics and Waves

• Lens Maker's Formula: For a double convex lens with radii $R_1 = 15\,cm$ and $R_2 = -30\,cm$ and $n = 1.5$: $\frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2}) = (1.5 - 1)(\frac{1}{15} - \frac{1}{-30}) = 0.5(\frac{3}{30}) = \frac{1}{20}$. Thus, $f = 20\,cm$.
• Young’s Double Slit Experiment (YDSE): The number of fringes $n$ and wavelength $\lambda$ in a set segment are related by $n_1\lambda_1 = n_2\lambda_2$. Given $n_1 = 12$, $\lambda_1 = 600\,nm$, $\lambda_2 = 400\,nm$: $12(600) = n_2(400) \Rightarrow n_2 = 18$.
• Interference Intensity: If the ratio of maximum to minimum intensity is $\frac{I_{max}}{I_{min}} = 16$, then the ratio of amplitudes $\frac{a_1+a_2}{a_1-a_2} = \sqrt{16} = 4$. Solving gives $\frac{a_1}{a_2} = \frac{5}{3}$. The intensity ratio is $\frac{I_1}{I_2} = (\frac{a_1}{a_2})^2 = 25:9$.
• Vibrations and Sound:     * Tuning Fork: A vibrating tuning fork on a table creates forced vibrations in the table board.     * Phase Difference: For a wave with $T = 0.05\,s$ and $v = 300\,m/s$, $\lambda = vT = 15\,m$. The path difference $\Delta x = 15 - 10 = 5\,m$. Phase difference $\Delta \phi = \frac{2\pi}{\lambda}\Delta x = \frac{2\pi}{15}(5) = \frac{2\pi}{3}$.
• Wave Equations: For $Y = 0.5\sin[\pi(400t - x)]$, the velocity is $v = \frac{\omega}{k} = \frac{400\pi}{\pi} = 400\,m/s$.
• Resolving Power: The resolving power of an optical microscope is inversely proportional to wavelength. Ratio $RP_1 : RP_2 = \lambda_2 : \lambda_1 = 6000:4000 = 3:2$.

Electricity and Magnetism

• Electrostatic Work and Potential:     * Work to carry $6\,\mu C$ charge through a $9\,V$ battery is $W = qV = 6 \times 10^{-6} \times 9 = 54 \times 10^{-6}\,J$.     * Potential from field $\mathbf{E} = 25\hat{i} + 30\hat{j}\,N/C$: $V = -\int \mathbf{E} \cdot d\mathbf{r} = -(25x + 30y)$. At $(2,2)$, $V = -(25(2) + 30(2)) = -110\,V$.
• Circuits and Resistors:     * Potentiometer: $\frac{E_1}{E_2} = \frac{l_1}{l_2}$. $\frac{1.5}{E_2} = \frac{27}{54} \Rightarrow E_2 = 3\,V$.     * Color Code: For a resistor with code Yellow (4), Violet (7), Brown ($10^1$), Gold ($5\%$), the value is $470\,\Omega \pm 5\%$.     * Specific Resistance: Increases with an increase in temperature for conductors.     * Parallel resistors: Thermal energy ratio for $R$ and $3R$ in parallel is $P = \frac{V^2}{R}$. Ratio $P_1:P_2 = \frac{V^2/R}{V^2/3R} = 3:1$.     * Capacitor Heat: Energy stored $U = \frac{1}{2}CV^2 = \frac{1}{2}(4 \times 10^{-6})(400)^2 = 0.32\,J$. This energy is released as heat in the resistor.
• Magnetism:     * Magnetic Force: $\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$. If a particle moves along a magnetic field line, $\mathbf{v}$ is parallel to $\mathbf{B}$, so the force is zero.     * Stored Energy: For a coil in a steady state DC circuit, the magnetic energy is $U = \frac{1}{2}LI^2$. In Question 34, calculation leads to $25\,J$.     * Magnetic Field at Center: $B_0 = \frac{\mu_0 n I}{2R}$. If rewound to 3 turns ($n=3$, $R' = R/3$), the new field $B = \frac{\mu_0(3)I}{2(R/3)} = 9B_0$.
• Galvanometer: Torque $\tau = NIAB = C\theta$. For $N=175$, $A=10^{-4}\,m^2$, $C=10^{-6}\,N\text{-}m/rad$, $I=10^{-3}\,A$, and $\theta = \frac{\pi}{180}$, the magnetic field $B$ is approximately $10^{-3}\,T$.

Modern Physics and Nuclear Chemistry

• Duality: Kinetic energy ratio for electron and proton with same de Broglie wavelength: $K = \frac{h^2}{2m\lambda^2} \Rightarrow \frac{K_e}{K_p} = \frac{m_p}{m_e} \approx 1836$. Ratio is $1: 1836$ (A).
• Bohr Model:     * Energy in $n$-th orbit $E_n = -13.6 \frac{Z^2}{n^2}\,eV$. For helium ($Z=2$), $E_{He} = 4 E_n$.     * Spectral lines for transition to $n=4$: $\binom{4}{2} = 6$.     * Wave number for $n=4 \rightarrow n=2$: $\bar{\nu} = R(\frac{1}{2^2} - \frac{1}{4^2}) = R(\frac{1}{4} - \frac{1}{16}) = \frac{3R}{16}$.
• Nuclear Decay: Process $A(180, 72) \xrightarrow{\alpha} A_1(176, 70) \xrightarrow{\beta} A_2(176, 71) \xrightarrow{\alpha} A_3(172, 69) \xrightarrow{\gamma} A_4(172, 69)$.
• Uncertainty Principle: $\Delta E \Delta t \geq \frac{h}{4\pi}$. High certainty in measurement of energy implies high uncertainty in time.
• X-Rays: Moseley's Law $\frac{1}{\lambda} \propto (Z-1)^2$. For Molybdenum ($Z=42$) and Zinc ($Z=30$), the wavelength of Zinc is determined as $1.3872\,\text{…}$.

General Chemistry

• Thermochemistry:     * Enthalpy of formation for $CH_4$ calculated via Hess's Law using provided combustion data: $-75\,kJ/mol$.     * Heat evolved for toluene ($46\,g$) combustion: Toluene molar mass is $92\,g/mol$. Since $46\,g$ is $0.5\,mol$, heat evolved = $0.5 \times 3910.3 = 1955.15\,kJ$. Converting to kCal: $\frac{1955.15}{4.1} = 477\,kCal$.
• Chemical Equilibrium:     * If reaction proceeds as $2aA + 2bB \rightleftharpoons 2cC + 2dD$, the new equilibrium constant is $K^2$.     * Le Chatelier's: In endothermic water splitting ($2H_2O \rightleftharpoons 2H_2 + O_2$), increasing temperature at constant pressure shifts equilibrium forward, decreasing water.
• Solutions:     * Mole fraction of glucose ($10\,moles$) = $0.50$. $\frac{10}{10 + n_{\text{water}}} = 0.5 \Rightarrow n_{\text{water}} = 10\,moles$. Mass of water = $10 \times 18 = 180\,g$ (or $180\,mL$).     * Henry's Law: $p = K_H X \Rightarrow 760 = 4.27 \times 10^5 X \Rightarrow X \approx 1.78 \times 10^{-3}$.
• Electrochemistry:     * Molar conductivity of $NH_4OH$ at infinite dilution is calculated as $\Lambda^0(NH_4Cl) + \Lambda^0(NaOH) - \Lambda^0(NaCl) = 149.7 + 248.1 - 126.5 = 271.3\,\Omega^{-1}cm^2mol^{-1}$.
• Kinetics:     * For first-order reaction $t_{75\%} = 2 \times t_{50\%}$. Given $t_{75\%} = 60\,min$, then $t_{50\%} = 30\,min$.     * Radioactive disintegration is always a first-order process.

Organic Chemistry

• Reactions:     * Hoffmann Elimination: 2-bromopentane with sodium tert-butoxide favors the less substituted alkene (1-pentene).     * Iodoform Test: Acetophenone reacts with $I_2/NaOH$ to give yellow precipitate of iodoform ($CHI_3$) and sodium benzoate (which yields benzoic acid $PhCO_2H$ upon acidification).     * Tollens Negative: Ketones (like cyclohexanone) and certain alcohols test negative.     * Hydrolysis: Sucrose yields glucose and fructose.
• Functional Groups and Polymers:     * Schiff Base: Formed by the reaction of primary amines with carbonyl compounds (imine).     * Polyurethane: Monomer is diisocyanate.     * Amino Acids: Cysteine contains a sulfur atom.     * Formaldehyde: Cyclic trimer is 1,3,5-trioxane.

Mathematics

• Functions and Calculus:     * Function $f(x) = x^2$ on $[0, \infty)$ is one-one but not onto if range is not specified. Range of $f(x) = \frac{x^2}{1+x^2}$ is $[0, 1)$.     * $f(x) = \frac{x}{|x|}$ is not continuous at $x=0$.     * Integral of $\sec^{4/3}x \csc^{2/3}x \,dx = 3\tan^{1/3}x + c$.     * Limit: $\lim_{n\rightarrow \infty} [\frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n}] = \ln(2)$.
• Probability and Stats:     * Probability of getting a 4 given an even number on a die: $P(4|\text{even}) = \frac{1}{3}$.     * Mean of defective bolts ($n=400, p=0.1$) is $np = 40$. Standard deviation $\sqrt{npq} = \sqrt{400(0.1)(0.9)} = 6$.
• Algebra and Matrices:     * Permutations of 'EULER': Total letters $= 5$, duplicate 'E's $= 2$. Ways $= \frac{5!}{2!} = 60$.     * Zeros in $100!$: Calculated by $\lfloor 100/5 \rfloor + \lfloor 100/25 \rfloor = 20 + 4 = 24$.     * The sum of roots $\alpha^5 + \beta^5$: For $x^2 - x + 2 = 0$, use recurrence or power sums to find value $64$.
• Geometry:     * Distance between parallel lines $2x + y + 4 = 0$ and $2x + y + 8 = 0$: $d = \frac{|8-4|}{\sqrt{2^2+1^2}} = \frac{4}{\sqrt{5}}$.     * Shortest distance between $y-x=1$ and $y=x^2$ occurs where slope of tangent to $y=x^2$is$1$.$2x=1 \Rightarrow x=0.5$. Distance is$\frac{3\sqrt{2}}{8}$$.

Answer Key Reference

• Question Summary: Total of 225 questions across basic and advanced Physics, Chemistry, and Mathematics.
• Key Trends: 1-75 (Physics focus), 76-135 (Chemistry focus), 136-225 (Mathematics focus).