Comprehensive Basic Mathematics for NEET and Physics Study Notes

Rules of Power and Exponents

• Zero Power Rule: For any non-zero number $x$, if the power is zero, the result is always one.

  • $x^0 = 1$

  • $1^0 = 1$

  • $18^0 = 1$

  • $100^0 = 1$

  • $e^0 = (2.71)^0 = 1$

  • $10^0 = 1$

  • $4^0 = 1$

  • $2^0 = 1$

• Infinity Power Rule: If the power of a non-zero number (greater than 1) is infinity, the result is infinity.

  • $2^\text{∞} = \text{∞}$

  • $e^\text{∞} = \text{∞}$

  • $(2.5)^\text{∞} = \text{∞}$

  • Note: If the base is less than 1, e.g., $(0.2)^\text{∞}$, the result tends toward 0.

• Negative Property of Exponents: A negative power indicates the reciprocal of the base with a positive power.

  • $x^{-n} = \frac{1}{x^n}$

  • $10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001$

  • $0.004 = 4 \times 10^{-3} = \frac{4}{1000}$

  • $e^{-a} = \frac{1}{e^a}$

• Product Property: If the bases are the same, powers are added during multiplication.

  • $x^n \times x^m = x^{n+m}$

  • $10^3 \times 10^4 = 10^7$

  • Note: In addition, powers do NOT add: $10^2 + 10^3 = 100 + 1000 = 1100$ (not $10^5$).

• Division Property: If the bases are the same, the power of the denominator is subtracted from the power of the numerator.

  • $\frac{x^n}{x^m} = x^{n-m}$

  • Example: $\frac{0.002}{0.4} = \frac{2 \times 10^{-3}}{4 \times 10^{-1}} = 0.5 \times 10^{-2} = 5 \times 10^{-3}$

Metric Prefixes and Unit Conversions

• Large Units:

  • Mega (M) = $10^6$ (e.g., $1 \text{ mega coulomb} = 10^6 \text{ C}$)

  • Kilo (k) = $10^3$ (e.g., $1 \text{ kilo coulomb} = 10^3 \text{ C}$)

• Small Units:

  • Deci (d) = $10^{-1}$ (e.g., $1 \text{ deci coulomb} = 10^{-1} \text{ C}$)

  • Centi (c) = $10^{-2}$ (e.g., $1 \text{ centi coulomb} = 10^{-2} \text{ C}$)

  • Milli (m) = $10^{-3}$

  • Micro (μ) = $10^{-6}$

  • Nano (n) = $10^{-9}$

  • Pico (p) = $10^{-12}$

  • Fermi (f) = $10^{-15}$ (Radius of nucleus ≈ $1 \text{ fm}$)

• Specific Length Units:

  • Angstrom (Å) = $10^{-10} \text{ m}$

  • Radius of atom ≈ $1 \text{ Å} = 10^{-10} \text{ m}$

• Conversion Scale:

  • $1 \text{ Å} = 0.1 \text{ nm}$

  • $1 \text{ nm} = 10 \text{ Å}$

  • $5 \text{ Å} = 5 \times 10^{-10} \text{ m} = 0.5 \times 10^{-9} \text{ m} = 0.5 \text{ nm}$

Decimal Manipulation and Scientific Notation

• Rule of Ten: When the decimal point moves forward (left) by one digit, you multiply by $10$. When it moves backward (right), you divide by $10$ (or multiply by a negative power of 10).

  • $245.0 = 24.5 \times 10^1 = 2.45 \times 10^2$

  • $0.16 \times 10^{-18} = 1.6 \times 10^{-19}$

  • $36.45 = 3.645 \times 10^1$

  • $437.52 = 43.75 \times 10^1 = 437.52 \times 10^0$

• Physics Application: Mass-Energy Equivalence:

  • Find the energy equivalent of $0.5 \text{ g}$ of a substance ($E = mc^2$, NEET 2020).

  • $m = 0.5 \text{ g} = 0.5 \times 10^{-3} \text{ kg}$

  • $c = 3 \times 10^8 \text{ m/s}$

  • $E = 0.5 \times 10^{-3} \times (3 \times 10^8)^2 = 0.5 \times 10^{-3} \times 9 \times 10^{16} = 4.5 \times 10^{13} \text{ J}$

Concept of Roots and Fractional Exponents

• Square Roots and Cube Roots:

  • $\sqrt{x} = x^{1/2}$

  • $\sqrt[3]{x} = x^{1/3}$

  • $\sqrt{x} \times \sqrt{x} = x$

  • $\sqrt{2} \times \sqrt{2} = 2$

• Fractional Power Simplification:

  • $x^{3/2} = x^1 \times x^{1/2} = x\sqrt{x}$

  • $x^{5/2} = x^2 \times x^{1/2} = x^2\sqrt{x}$

  • $(x^n)^m = x^{n \times m}$

  • $(8)^{2/3} = (2^3)^{2/3} = 2^2 = 4$

  • $(4)^{3/2} = (2^2)^{3/2} = 2^3 = 8$

• Important Numerical Root Values:

  • $\sqrt{2} \approx 1.41$

  • $\sqrt{3} \approx 1.73$

  • $\sqrt{5} \approx 2.23$

  • $\sqrt{6} = \sqrt{2} \times \sqrt{3} \approx 2.45$

  • $\sqrt{10} \approx 3.16 \approx \pi (3.14)$

  • $\pi^2 \approx 10$

  • $\sqrt{11} \approx 3.31$

  • $\sqrt{169} = 13$

  • $\sqrt{196} = 14$

  • $\sqrt{225} = 15$

  • $\sqrt{256} = 16$

Calculations with Fractions and Significant Figures

• Fractional Conversions:

  • $0.75 = ¾$

  • $0.5 = ½$

  • $0.25 = ¼$

  • $0.125 = 1/8$

  • $0.67 = 2/3$

  • $0.33 = 1/3$

  • $1.33 = 4/3$

• Rounding and Significant Figures (Addition/Subtraction):

  • Rule: The result should have the same number of decimal places as the quantity with the least decimal places.

  • Example: $9.99 \text{ m} - 0.0099 \text{ m}$

  • Calculated value: $9.9801$

  • Correct significant figure answer: $9.98 \text{ m}$ (since $9.99$ has only 2 decimal places).

Basic Algebra and Series/Parallel Operations

• Common Fraction Sums:

  • $\frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}$

  • Example for Resistance ($R_p$) or Capacitance ($C_s$):

  • $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \implies C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$

• Componendo and Dividendo:

  • If $\frac{x+1}{x-1} = 5$, then using C&D: $\frac{x}{1} = \frac{5+1}{5-1} = \frac{6}{4} = 1.5$

  • Used in Wave Optics for intensity ratios: $\frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}}$

• Solving Simultaneous Equations:

  • $x + 2y = 3$

  • $3x - 2y = 5$

  • Adding equations: $4x = 8 \implies x = 2$. Substituting results in $y = 0.5$.

Proportionality and Basic Graphs

• Linear Dependency: $y \propto x$

  • Graph: Straight line through origin: $y = mx$.

  • Example: IDEAL Transformer turns ratio $\frac{V_s}{V_p} = \frac{N_s}{N_p}$.

• Inverse Dependency: $y \propto \frac{1}{x}$

  • Graph: Rectangular Hyperbola.

  • Example: Magnetic field from a wire $B \propto \frac{1}{r}$.

  • Example: Boyle's Law $P \propto \frac{1}{V}$ (at constant $T$).

• Square Dependency: $y \propto x^2$

  • Graph: Parabola.

  • Example: Kinetic Energy $K.E. = \frac{1}{2}mv^2$. If $v$ doubles, $K.E.$ becomes 4 times.

  • Example: Power dissipated in resistor $P = I^2 R$.

• Root Dependency: $y \propto \sqrt{x}$

  • Example: Wave speed on a string $v = \sqrt{\frac{T}{\mu}}$. If Tension ($T$) is doubled, speed increases by factor $\sqrt{2}$.

• Inverse Square Dependency: $y \propto \frac{1}{x^2}$

  • Graph: Steeper decay than rectangular hyperbola.

  • Example: Coulomb's Law $F \propto \frac{1}{r^2}$.

Percentage Change in Physics

• Definitions:

  • "Increased to $200\%$": Final value is $2 \times \text{Initial}$.

  • "Increased by $30\%$": Final value is $1.3 \times \text{Initial}$.

  • "Decreased by $30\%$": Final value is $0.70 \times \text{Initial}$.

• Small Changes (< 5\%):

  • Use Error Method (differentiation/power rule).

  • If $T = 2\pi\sqrt{\frac{L}{g}}$, then $\frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L}$.

  • Example: If $L$ increases by $2\%$, $T$ increases by $1\%$.

• Large Changes ($\ge 10\%$):

  • Use Ratio Method: $\% \text{ change} = \frac{\text{Final} - \text{Initial}}{\text{Initial}} \times 100\%$

  • Example: If $K.E.$ increases by $300\%$, Final $K.E.$ becomes $4$ times. Momentum $P \propto \sqrt{K.E.}$, so $P$ becomes $\sqrt{4} = 2$ times ($100\%$ change).

Series in Mathematics

• Arithmetic Progression (A.P.):

  • Terms: $a, a+d, a+2d, \dots$

  • $n$-th term: $a_n = a + (n-1)d$

  • Sum of $n$ terms: $S_n = \frac{n}{2} [2a + (n-1)d]$

  • Sum of first $n$ natural numbers: $S = \frac{n(n+1)}{2}$

• Geometric Progression (G.P.):

  • Terms: $a, ar, ar^2, \dots$

  • Common ratio ($r$): ratio of successive terms.

  • Sum of an infinite G.P. (only if |r| < 1): $S_\infty = \frac{a}{1-r}$

  • Example: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \dots \implies a=1, r=1/2 \implies S_\infty = \frac{1}{1 - 0.5} = 2$.

Binomial Theorem and Approximations

• General Formula for approximation: $(1+x)^n \approx 1 + nx$ (valid ONLY if $x \ll 1$).

• Applications:

  • $(1-x)^n \approx 1 - nx$

  • $(1+x)^{-n} \approx 1 - nx$

  • $(1-x)^{-n} \approx 1 + nx$

• Physics Example: Acceleration due to gravity at height $h$

  • $g_h = g_0(1 + \frac{h}{R})^{-2} \approx g_0(1 - \frac{2h}{R})$ (for $h \ll R$).

Quadratic Equations

• Standard Form: $ax^2 + bx + c = 0$

• Root Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant ($D = b^2 - 4ac$):

  • If D > 0: Two distinct real roots.

  • If $D = 0$: One real root (equal roots).

  • If D < 0: No real roots (imaginary roots).

• Example: $x^2 - 5x + 6 = 0$

  • Factoring: $(x-2)(x-3) = 0 \implies x=2, 3$.

Master Class on Trigonometry

• Basics: $\sin\theta = P/H$, $\cos\theta = B/H$, $\tan\theta = P/B$.

• The ASTC Rule (All-Silver-Tea-Cups):

  • Quadrant I ($0 \text{ to } 90^\text{∘}$): All positive.

  • Quadrant II ($90 \text{ to } 180^\text{∘}$): Only Sine and Cosec positive.

  • Quadrant III ($180 \text{ to } 270^\text{∘}$): Only Tan and Cot positive.

  • Quadrant IV ($270 \text{ to } 360^\text{∘}$): Only Cos and Sec positive.

• Important Triangle ($37^\text{∘}, 53^\text{∘}$):

  • $\sin(37^\text{∘}) = 3/5$, $\cos(37^\text{∘}) = 4/5$, $\tan(37^\text{∘}) = 3/4$

  • $\sin(53^\text{∘}) = 4/5$, $\cos(53^\text{∘}) = 3/5$, $\tan(53^\text{∘}) = 4/3$

• Conversion and Signs:

  • $\sin(-\theta) = -\sin\theta$

  • $\cos(-\theta) = \cos\theta$ (Even function)

  • $\sin(90 + \theta) = \cos\theta$

  • $\cos(90 + \theta) = -\sin\theta$

• Small Angle Approximation: If $\theta \le 5^\text{∘}$, then:

  • $\sin\theta \approx \theta$ (must be in Radians)

  • $\tan\theta \approx \theta$

  • $\cos\theta \approx 1$

• Phasor Diagram Rule:

  • Standard Sine is at $0^\text{∘}$. Cosine is $90^\text{∘}$ ahead of Sine.

  • Phase difference (e.g., between $\sin(\omega t + 0.5)$ and $\cos(\omega t)$) is found by expressing both in Sine and subtracting angles.

Calculus: Differentiation

• Concept: Differentiation measures the rate of change or the slope of a curve ($dy/dx$).

• Fundamental Rules:

  • $\frac{d}{dx}(\text{constant}) = 0$

  • $\frac{d}{dx}(x^n) = nx^{n-1}$

  • $\frac{d}{dx}(\sin x) = \cos x$

  • $\frac{d}{dx}(\cos x) = -\sin x$

  • $\frac{d}{dx}(e^x) = e^x$

  • $\frac{d}{dx}(\ln x) = 1/x$

• Advanced Rules:

  • Product Rule: $\frac{d}{dx}(AB) = A \frac{dB}{dx} + B \frac{dA}{dx}$

  • Quotient Rule: $\frac{d}{dx}(\frac{A}{B}) = \frac{B(dA/dx) - A(dB/dx)}{B^2}$

  • Chain Rule (Outside-Inside Rule): Differentiate the outer function, then multiply by the derivative of the inner function.

    • $\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \times 2x = 2x\cos(x^2)$

• Maxima and Minima:

  • At both maxima and minima, the first derivative is zero: $\frac{dy}{dx} = 0$.

  • To distinguish:

    • Maxima: Second derivative is negative (\frac{d^2y}{dx^2} < 0).

    • Minima: Second derivative is positive (\frac{d^2y}{dx^2} > 0).

Calculus: Integration

• Concept: Integration is the reverse of differentiation and represents the area under a curve.

• Rules:

  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (except $n = -1$)

  • $\int \frac{1}{x} dx = \ln x + C$

  • $\int e^x dx = e^x + C$

  • $\int \sin x dx = -\cos x + C$

  • $\int \cos x dx = \sin x + C$

• Definite Integration: Finding area between limits.

  • $\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)$.

• Average Value Application: $y_{avg} = \frac{\int y dx}{\int dx}$.

Analytical Geometry and Graphical Shapes

• Circle: $x^2 + y^2 = R^2$ (Center at origin)

• Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

• Parabola: $y = kx^2$ (Upward) or $x = ky^2$ (Rightward)

• Slope Visuals (Ramlal Analogy):

  • "Laughing Ramlal": Curve opening upwards (Increasing slope, d^2y/dx^2 > 0).

  • "Crying Ramlal": Curve opening downwards (Decreasing slope, d^2y/dx^2 < 0).

• Tangents: Slope of Distance-Time graph = Velocity. Slope of Velocity-Time graph = Acceleration.

Logarithms

• Base Change: $\ln(x) = \log_e(x) = 2.303 \log_{10}(x)$.

• Properties:

  • $\log(ab) = \log a + \log b$

  • $\log(a/b) = \log a - \log b$

  • $\log(a^n) = n\log a$

• Values to Remember (Base 10):

  • $\log_{10}(2) \approx 0.301$

  • $\log_{10}(3) \approx 0.477$

  • $\log_{10}(10) = 1$

  • $\log_{10}(1) = 0$

  • $\ln(2) \approx 0.693$

Questions & Discussion

• Question: Finding phase difference between $y_1 = a \sin(\omega t + kx + 0.57)$ and $y_2 = a \cos(\omega t + kx)$.

  • Answer: Convert $y_2$ to Sine: $y_2 = a \sin(\omega t + kx + \pi/2)$. Phase difference = $(\pi/2) - 0.57 = 1.57 - 0.57 = 1.0 \text{ radian}$.

• Question: Why is the value of sine never greater than 1?

  • Answer: Because in a right-angled triangle, the perpendicular is always shorter than or equal to the hypotenuse ($\sin\theta = P/H$).

• Question: Calculate work done by a force $F = \alpha + \beta x^2$ over displacement $1m$.

  • Answer: $W = \int F dx = [\alpha x + \frac{\beta x^3}{3}]_0^1$. If $W=5J$ and $\alpha=1$, then $1 + \beta/3 = 5 \implies \beta = 12 \text{ N/m}^2$.