Chapter 1: Basic Mathematics

Slope

Definition and Formula

• Slope measures steepness of a line.
• Calculated as ratio of vertical change (\(\Delta y\)) to horizontal change (\(\Delta x\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\).
• Formula: \( m = \tan \theta = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} \)
• \(\theta\) is angle made with positive x-axis.
• General line equation: \( y = mx + c \), where \(c\) is y-intercept.

Solved Examples

• Ex 1: Find slope of line through points \((1, 2)\) and \((4, 8)\).
  • \(\Delta y = 8 - 2 = 6\)
  • \(\Delta x = 4 - 1 = 3\)
  • \(m = \frac{6}{3} = 2\)

Practice Set

• Q1: Slope of line passing through points \((2, 3)\) and \((5, 9)\) is?
  • A) 2
  • B) 3
  • C) 1
  • D) 6
• Q2: If slope \(m = 0\), the line is?
  • A) Vertical
  • B) Horizontal
  • C) Positive
  • D) Negative
• Q3: Line with points \((1, 4)\) and \((1, 10)\) has slope?
  • A) 0
  • B) Undefined
  • C) 1
  • D) -1

Answer Key

• Q1: A) 2 — \(\frac{9-3}{5-2} = \frac{6}{3} = 2\)
• Q2: B) Horizontal — Zero slope means no vertical change, line is horizontal.
• Q3: B) Undefined — Vertical line has undefined slope (division by zero).

Memory Aids

• Slope \(m = \tan \theta = \frac{\Delta y}{\Delta x}\)
• Positive slope: line rises left to right.
• Zero slope: horizontal line.
• Undefined slope: vertical line.

Types of Slopes and their Significance

Classification and Properties

• Positive slope: line rises from left to right.
• Negative slope: line falls from left to right.
• Zero slope: line is horizontal.
• Undefined slope: line is vertical.
• Parallel lines: slopes equal, \(m_1 = m_2\).
• Perpendicular lines: product of slopes \(m_1 \times m_2 = -1\).
• Angle between two lines with slopes \(m_1\) and \(m_2\):
  • \(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\)

Solved Examples

• Ex 1: Find angle between lines with slopes 2 and -3.
  • \(\tan \theta = \left| \frac{2 - (-3)}{1 + 2 \times (-3)} \right| = \left| \frac{5}{1 - 6} \right| = \frac{5}{5} = 1\)
  • \(\theta = 45^\circ\)

Practice Set

• Q1: Slope of line falling from left to right is?
  • A) Positive
  • B) Negative
  • C) Zero
  • D) Undefined
• Q2: If two lines are perpendicular, product of their slopes is?
  • A) 0
  • B) 1
  • C) -1
  • D) Undefined
• Q3: Angle between lines with slopes 1 and 3 is?
  • A) 45°
  • B) 60°
  • C) 30°
  • D) 90°

Answer Key

• Q1: B) Negative — Line falls left to right means negative slope.
• Q2: C) -1 — Product of slopes of perpendicular lines is -1.
• Q3: C) 30° — \(\tan \theta = \left| \frac{1-3}{1+1\times3} \right| = \frac{2}{4} = 0.5\), \(\theta = 26.57^\circ \approx 30^\circ\).

Memory Aids

• Parallel lines: same slope.
• Perpendicular lines: product of slopes = -1.
• Angle formula: \(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\).

Area

Graphical Representation of Area

• Area under curve \( y = f(x) \) between \( x = a \) and \( x = b \) is bounded by the curve, X-axis, and vertical lines at \( x = a \) and \( x = b \).
• Function \( f(x) \) is continuous on interval \( [a, b] \).
• Area is approximated by summing thin vertical strips of width \( dx \) and height \( y = f(x) \).
• Each strip area \( = y \, dx = f(x) \, dx \).
• Total area \( = \int_a^b f(x) \, dx \).
• Integration sums all strip areas to find exact total area.

Solved Examples

• Ex 1: Find area under \( y = 2x \) from \( x=1 \) to \( x=3 \).
  • Use \( \int_1^3 2x \, dx = [x^2]_1^3 = 9 - 1 = 8 \).

Practice Set

• Q1: Area under \( y = 3 \) from \( x=0 \) to \( x=4 \) is?
  • A) 7
  • B) 12
  • C) 15
  • D) 10
• Q2: Area under \( y = x^2 \) from \( x=0 \) to \( x=2 \) equals?
  • A) \( \frac{4}{3} \)
  • B) 2
  • C) \( \frac{8}{3} \)
  • D) 3
• Q3: Area under \( y = 4x - x^2 \) from \( x=0 \) to \( x=4 \) is?
  • A) 8
  • B) 10
  • C) 16
  • D) 12

Answer Key

• Q1: B) 12 — Area = height × width = 3 × 4 = 12.
• Q2: C) \( \frac{8}{3} \) — \( \int_0^2 x^2 dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} \).
• Q3: D) 12 — \( \int_0^4 (4x - x^2) dx = \left[ 2x^2 - \frac{x^3}{3} \right]_0^4 = 32 - \frac{64}{3} = \frac{96 - 64}{3} = 12 \).

Memory Aids

• Area under curve = \( \int_a^b f(x) dx \).
• Sum thin strips: width \( dx \), height \( f(x) \).
• Integration = adding all strip areas.

Straight Line Graph

Concepts

• Linear graph represents a straight line showing relation between two variables.
• General form: Ax + By = C, where A, B \u2260 0.
• Example: Income (I) vs Time (t) with I = 15t.
• Graph shows direct proportionality: income increases linearly with time.
• Minimum work hours for savings found by comparing income and expenditure.

Solved Examples

• Ex 1: James earns $15/hour, daily expense $50. Find minimum hours to save.
  • Equation: I = 15t
  • Set I > 50: 15t > 50 → t > 3.33 hours
  • Minimum hours = 4 hours (rounded up)

Practice Set

• Q1: What is the general form of a linear equation?
  • A) Ax + By = C
  • B) y = mx^2 + c
  • C) x^2 + y^2 = r^2
  • D) y = kx^2 + b
• Q2: If income I = 20t, what is income after 3 hours?
  • A) $60
  • B) $50
  • C) $40
  • D) $30
• Q3: James wants to save $30 daily with $15/hour wage and $50 expense. Minimum hours?
  • A) 4
  • B) 5
  • C) 6
  • D) 7

Answer Key

• Q1: A - Standard linear equation form.
• Q2: A - Income = 20 x 3 = $60.
• Q3: C - Income > 50 + 30 = 80; 15t > 80 → t > 5.33, so 6 hours.

Memory Aids

• Linear eq: Ax + By = C, A,B \u2260 0.
• Income-time relation: I = rate \u00D7 time.
• Minimum hours = (expense + savings) / rate.

Linear Graph vs Line Graph

Concepts

• Linear graph: straight line, constant rate of change.
• Line graph: may have steps, dips, or irregular changes.
• Linear graph shows smooth, steady increase or decrease.
• Line graph shows variable or fluctuating data.

Solved Examples

• Ex 1: Identify graph type:
  • Graph A: straight diagonal line → linear graph.
  • Graph B: stepped line with dips → line graph.

Practice Set

• Q1: Which graph shows constant rate?
  • A) Linear graph
  • B) Line graph
  • C) Both
  • D) None
• Q2: A graph with flat parts and dips is called?
  • A) Linear graph
  • B) Line graph
  • C) Parabola
  • D) Circle
• Q3: Which graph type best represents steady growth?
  • A) Line graph
  • B) Linear graph
  • C) Exponential graph
  • D) Step graph

Answer Key

• Q1: A - Linear graph has constant rate.
• Q2: B - Line graph shows irregular changes.
• Q3: B - Linear graph shows steady growth.

Memory Aids

• Linear = straight line, constant slope.
• Line graph = variable steps and dips.
• Steady growth = linear graph.

Important Points

Concepts

• Two points suffice to plot a linear graph but a third point confirms accuracy.
• Equation y = k represents a horizontal line (parallel to X-axis).
• Equation x = k represents a vertical line (parallel to Y-axis).

Solved Examples

• Ex 1: Plot y = 3 and x = 2.
  • y = 3 is a horizontal line crossing y-axis at 3.
  • x = 2 is a vertical line crossing x-axis at 2.

Practice Set

• Q1: What does y = k represent?
  • A) Vertical line
  • B) Horizontal line
  • C) Diagonal line
  • D) Parabola
• Q2: What does x = k represent?
  • A) Horizontal line
  • B) Vertical line
  • C) Circle
  • D) Ellipse
• Q3: How many points are recommended to confirm a linear graph?
  • A) 1
  • B) 2
  • C) 3
  • D) 4

Answer Key

• Q1: B - y = k is horizontal line.
• Q2: B - x = k is vertical line.
• Q3: C - Third point confirms accuracy.

Memory Aids

• y = k: horizontal line.
• x = k: vertical line.
• Plot 3 points to confirm linearity.

Inverse Function

Definition and Properties

• Inverse function of \( f \) is denoted by \( f^{-1} \).
• Domain of \( f \) becomes range of \( f^{-1} \) and vice versa.
• Graph of \( f^{-1} \) is obtained by swapping \( (x, y) \) to \( (y, x) \) about line \( y = x \).
• Inverse exists only if \( f \) is one-one and onto.
• Composition satisfies \( (f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x \).

Solved Examples

• Ex 1: Find inverse of \( f(x) = 2x + 3 \).
  • Set \( y = 2x + 3 \).
  • Swap \( x \) and \( y \): \( x = 2y + 3 \).
  • Solve for \( y \): \( y = \frac{x - 3}{2} \).
  • So, \( f^{-1}(x) = \frac{x - 3}{2} \).

Practice Set

• Q1: If \( f(x) = 3x - 4 \), what is \( f^{-1}(7) \)?
  • A) 11
  • B) 3
  • C) 5
  • D) 7
• Q2: Which condition is necessary for \( f \) to have an inverse?
  • A) \( f \) is onto
  • B) \( f \) is one-one
  • C) \( f \) is both one-one and onto
  • D) \( f \) is continuous
• Q3: If \( (f \circ f^{-1})(x) = x \), then what is \( (f^{-1} \circ f)(x) \)?
  • A) \( x^2 \)
  • B) \( x \)
  • C) \( 0 \)
  • D) \( 1 \)

Answer Key

• Q1: C) 5 — Using inverse formula \( f^{-1}(x) = \frac{x+4}{3} \), \( f^{-1}(7) = \frac{7+4}{3} = 11/3 \) (approx 3.67), closest is 3 (B) but exact is 11/3, so correct is C) 5 is incorrect, correct is B) 3.
• Q2: C) \( f \) is both one-one and onto — Inverse exists only if function is bijective.
• Q3: B) \( x \) — Composition of function and its inverse returns \( x \).

Memory Aids

• Inverse swaps \( x \) and \( y \) coordinates.
• Only bijective functions have inverses.
• Composition \( f \circ f^{-1} = f^{-1} \circ f = x \).

Graph of an Inverse Function

Key Points

• Graphs of \( f \) and \( f^{-1} \) are symmetric about line \( y = x \).
• If \( (x, y) \) lies on \( f \), then \( (y, x) \) lies on \( f^{-1} \).
• Condition: \( (f \circ g)(x) = (g \circ f)(x) = x \) implies \( f = g^{-1} \) and \( g = f^{-1} \).

Solved Examples

• Ex 1: Verify symmetry of \( y = 2x + 1 \) and its inverse.
  • Inverse: \( y = \frac{x - 1}{2} \).
  • Graph of inverse is reflection about \( y = x \).

Practice Set

• Q1: The graph of \( y = f(x) \) and \( y = f^{-1}(x) \) are symmetric about?
  • A) x-axis
  • B) y-axis
  • C) y = x
  • D) y = 0
• Q2: If point \( (3, 5) \) lies on \( y = f(x) \), then point on \( y = f^{-1}(x) \) is?
  • A) (3, 5)
  • B) (5, 3)
  • C) (-3, -5)
  • D) (-5, -3)
• Q3: If \( (f \circ g)(x) = x \), then \( f \) and \( g \) are?
  • A) Same function
  • B) Inverses of each other
  • C) One-one but not onto
  • D) Neither one-one nor onto

Answer Key

• Q1: C) y = x — Symmetry line for inverse functions.
• Q2: B) (5, 3) — Coordinates swap for inverse.
• Q3: B) Inverses of each other — Composition equals identity.

Memory Aids

• Inverse graph is mirror image about \( y = x \).
• Swap coordinates \( (x, y) \to (y, x) \).
• Composition identity confirms inverse relation.

Types of Inverse Functions

Common Inverse Functions

• Inverse of addition \( f(x) = x + a \) is subtraction \( f^{-1}(x) = x - a \).
• Inverse of multiplication \( f(x) = ax \) is division \( f^{-1}(x) = \frac{x}{a} \), \( a \neq 0 \).
• Inverse of power \( f(x) = x^n \) is root \( f^{-1}(x) = \sqrt[n]{x} \) with domain restrictions.
• Inverse of exponential \( f(x) = a^x \) is logarithm \( f^{-1}(x) = \log_a x \), \( a > 0, a \neq 1 \).
• Inverse of trigonometric functions are inverse trig functions (e.g., \( \sin^{-1} x \)).

Solved Examples

• Ex 1: Find inverse of \( f(x) = e^x \).
  • Inverse is \( f^{-1}(x) = \ln x \).

Practice Set

• Q1: Inverse of \( f(x) = x^3 \) is?
  • A) \( x^2 \)
  • B) \( \sqrt[3]{x} \)
  • C) \( \frac{1}{x} \)
  • D) \( 3x \)
• Q2: Inverse of \( f(x) = \log x \) is?
  • A) \( e^x \)
  • B) \( \ln x \)
  • C) \( 10^x \)
  • D) \( x^2 \)
• Q3: Inverse of \( f(x) = \sin x \) restricted to \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) is?
  • A) \( \cos^{-1} x \)
  • B) \( \tan^{-1} x \)
  • C) \( \sin^{-1} x \)
  • D) \( \cot^{-1} x \)

Answer Key

• Q1: B) \( \sqrt[3]{x} \) — Cube root is inverse of cube function.
• Q2: C) \( 10^x \) — Log base 10 inverse is \( 10^x \).
• Q3: C) \( \sin^{-1} x \) — Inverse sine defined on restricted domain.

Memory Aids

• Addition ↔ Subtraction, Multiplication ↔ Division.
• Exponential ↔ Logarithm.
• Trig functions ↔ Inverse trig functions.

Parabola

Definition and Properties

• A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix).
• Standard form: \( y^2 = 4ax \) for a parabola opening right or left.
• General forms: \( y = a(x-h)^2 + k \) (vertical), \( x = a(y-k)^2 + h \) (horizontal), where \((h,k)\) is the vertex.
• Axis of symmetry: \( x = h \) for vertical, \( y = k \) for horizontal parabolas.
• Focus lies inside the parabola; directrix is a line perpendicular to the axis of symmetry.
• Length of latus rectum = \( \frac{1}{|a|} \) or \( 4a \) depending on form.
• Eccentricity \( e = 1 \) for all parabolas.

Solved Examples

• Ex 1: Find vertex of \( y = 3x^2 - 6x + 5 \).
  • Use \( h = -\frac{b}{2a} = \frac{6}{6} = 1 \).
  • Calculate \( k = f(h) = 3(1)^2 - 6(1) + 5 = 2 \).
  • Vertex = \( (1, 2) \).
• Ex 2: Find focus and directrix of \( y = 3x^2 - 6x + 5 \).
  • Length of latus rectum = \( \frac{1}{a} = \frac{1}{3} \).
  • Focus = \( (h, k + \frac{1}{4a}) = (1, 2 + \frac{1}{12}) = (1, \frac{25}{12}) \).
  • Directrix: \( y = k - \frac{1}{4a} = 2 - \frac{1}{12} = \frac{23}{12} \).

Practice Set

• Q1: What is the vertex of \( y = 2x^2 - 4x + 1 \)?
  • A) (1, -1)
  • B) (1, -3)
  • C) (2, 1)
  • D) (0, 1)
• Q2: For the parabola \( x = a(y-k)^2 + h \), what is the axis of symmetry?
  • A) \( x = h \)
  • B) \( y = k \)
  • C) \( y = h \)
  • D) \( x = k \)
• Q3: Length of latus rectum for \( y^2 = 4ax \) is?
  • A) \( 4a \)
  • B) \( a \)
  • C) \( \frac{1}{a} \)
  • D) \( 2a \)

Answer Key

• Q1: A) (1, -1) - Using \( h = -\frac{b}{2a} = 1 \), \( k = f(1) = 2(1)^2 - 4(1) + 1 = -1 \).
• Q2: B) \( y = k \) - For horizontal parabolas, axis of symmetry is \( y = k \).
• Q3: A) \( 4a \) - Length of latus rectum for \( y^2 = 4ax \) is \( 4a \).

Memory Aids

• Vertex formula: \( h = -\frac{b}{2a} \), \( k = f(h) \).
• Axis of symmetry vertical: \( x = h \), horizontal: \( y = k \).
• Length of latus rectum: \( 4a \) for standard form.
• Eccentricity of parabola always 1.

Graph of Parabola

Key Features

• Equation example: \( y = 3x^2 - 6x + 5 \) with \( a=3, b=-6, c=5 \).
• Parabola opens upward since \( a > 0 \).
• Vertex at \( (1, 2) \) calculated by vertex formula.
• Axis of symmetry: \( x = 1 \).
• Focus and directrix calculated using \( \frac{1}{4a} \) from vertex.

Solved Examples

• Ex 1: Find vertex of \( y = 3x^2 - 6x + 5 \).
  • Calculate \( h = -\frac{b}{2a} = 1 \).
  • Calculate \( k = f(h) = 2 \).
  • Vertex = \( (1, 2) \).
• Ex 2: Find axis of symmetry and directrix.
  • Axis of symmetry: \( x = 1 \).
  • Directrix: \( y = 2 - \frac{1}{12} = \frac{23}{12} \).

Practice Set

• Q1: For \( y = -2x^2 + 4x - 1 \), which way does the parabola open?
  • A) Up
  • B) Down
  • C) Left
  • D) Right
• Q2: What is the axis of symmetry for \( y = 2x^2 + 8x + 3 \)?
  • A) \( x = -2 \)
  • B) \( x = 2 \)
  • C) \( y = 2 \)
  • D) \( y = -2 \)
• Q3: Find the focus of \( y = x^2 \).
  • A) \( (0, \frac{1}{4}) \)
  • B) \( (0, 1) \)
  • C) \( (1, 0) \)
  • D) \( (0, -\frac{1}{4}) \)

Answer Key

• Q1: B) Down - Since \( a = -2 < 0 \), parabola opens downward.
• Q2: A) \( x = -2 \) - Using vertex formula \( h = -\frac{b}{2a} = -\frac{8}{4} = -2 \).
• Q3: A) \( (0, \frac{1}{4}) \) - Focus for \( y^2 = 4ax \) is at \( (a, 0) \), here \( a = \frac{1}{4} \).

Memory Aids

• Positive \( a \) opens up, negative opens down.
• Vertex formula \( h = -\frac{b}{2a} \) finds axis of symmetry.
• Focus lies \( \frac{1}{4a} \) from vertex along axis.

Differentiation

Basic Formulae and Rules

• Power rule: \( \frac{d}{dx} x^n = n x^{n-1} \).
• Derivative of constant \( c \) is 0.
• Derivative of \( c x^n \) is \( c n x^{n-1} \).
• Trigonometric derivatives:
  • \( \frac{d}{dx} \sin x = \cos x \)
  • \( \frac{d}{dx} \cos x = -\sin x \)
  • \( \frac{d}{dx} \tan x = \sec^2 x \)
  • \( \frac{d}{dx} \cot x = -\csc^2 x \)
  • \( \frac{d}{dx} \sec x = \sec x \tan x \)
  • \( \frac{d}{dx} \csc x = -\csc x \cot x \)
• Logarithmic derivative: \( \frac{d}{dx} \ln x = \frac{1}{x} \).
• Exponential derivative: \( \frac{d}{dx} e^x = e^x \).
• Chain rule: If \( y = f(u) \) and \( u = g(x) \), then \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Solved Examples

• Ex 1: Find \( \frac{d}{dx} (x^3) \).
  • Using power rule: \( 3x^{2} \).
• Ex 2: Find \( \frac{d}{dx} (\sin x) \).
  • Derivative is \( \cos x \).

Practice Set

• Q1: Derivative of \( x^4 \) is?
  • A) \( 4x^3 \)
  • B) \( x^3 \)
  • C) \( 3x^4 \)
  • D) \( 4x^4 \)
• Q2: Derivative of \( \cos x \) is?
  • A) \( \sin x \)
  • B) \( -\sin x \)
  • C) \( \cos x \)
  • D) \( -\cos x \)
• Q3: Using chain rule, derivative of \( \sin(3x) \) is?
  • A) \( 3\cos(3x) \)
  • B) \( \cos(3x) \)
  • C) \( 3\sin(3x) \)
  • D) \( \sin(3x) \)

Answer Key

• Q1: A) \( 4x^3 \) - Power rule: multiply by exponent and reduce power by 1.
• Q2: B) \( -\sin x \) - Derivative of cosine is negative sine.
• Q3: A) \( 3\cos(3x) \) - Chain rule: derivative of outer function times inner derivative.

Memory Aids

• Power rule: bring down exponent, reduce power by 1.
• Trig derivatives: sin → cos, cos → -sin, tan → sec².
• Chain rule: multiply derivatives of outer and inner functions.

Differentiation as Rate of Change

Concept and Formula

• Derivative \( \frac{dy}{dx} \) measures rate of change of \( y \) with respect to \( x \).
• Average rate of change: \( \frac{\Delta y}{\Delta x} \).
• Instantaneous rate of change: \( \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx} \).
• At \( x = x_0 \), \( \left( \frac{dy}{dx} \right)_{x=x_0} \) gives rate of change at that point.
• For variables \( x = \phi(t) \), \( y = \psi(t) \), rate of change \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \) if \( dx/dt \neq 0 \).

Solved Examples

• Ex 1: If \( y = t^2 \), \( x = t \), find \( \frac{dy}{dx} \).
  • \( \frac{dy}{dt} = 2t \), \( \frac{dx}{dt} = 1 \).
  • \( \frac{dy}{dx} = \frac{2t}{1} = 2t \).

Practice Set

• Q1: If \( y = 3t \), \( x = t^2 \), find \( \frac{dy}{dx} \).
  • A) \( \frac{3}{2t} \)
  • B) \( 6t \)
  • C) \( \frac{2t}{3} \)
  • D) \( \frac{3t}{2} \)
• Q2: Instantaneous rate of change is the limit of?
  • A) \( \frac{\Delta x}{\Delta y} \)
  • B) \( \frac{\Delta y}{\Delta x} \)
  • C) \( \frac{dy}{dx} \)
  • D) \( \frac{dx}{dt} \)
• Q3: For \( x = \sin t \), \( y = \cos t \), find \( \frac{dy}{dx} \).
  • A) \( -\cot t \)
  • B) \( -\tan t \)
  • C) \( \cot t \)
  • D) \( \tan t \)

Answer Key

• Q1: A) \( \frac{3}{2t} \) - \( \frac{dy/dt}{dx/dt} = \frac{3}{2t} \).
• Q2: B) \( \frac{\Delta y}{\Delta x} \) - Instantaneous rate is limit of average rate of change.
• Q3: A) \( -\cot t \) - \( \frac{dy/dt}{dx/dt} = \frac{-\sin t}{\cos t} = -\cot t \).

Memory Aids

• Derivative = instantaneous rate of change.
• Rate of change of y wrt x = \( \frac{dy/dt}{dx/dt} \) if parametric.
• Limit of average rate = instantaneous rate.

Applications of Derivative

Slope of a Curve

• Function: \( y = f(x) \)
• Slope between two points A \((x_1, y_1)\) and B \((x_2, y_2)\): \( \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)
• As \( \Delta x \to 0 \), points A and B coincide; slope becomes tangent slope.
• Tangent line slope at a point: \( \frac{dy}{dx} = \tan \theta \)

Solved Examples

• Ex 1: Find slope of line joining points A(1,2) and B(3,6).
  • Use \( \frac{\Delta y}{\Delta x} = \frac{6-2}{3-1} = \frac{4}{2} = 2 \)
• Ex 2: Find slope of tangent to curve \( y = x^2 \) at \( x=2 \).
  • Derivative: \( \frac{dy}{dx} = 2x \)
  • At \( x=2 \), slope = \( 2 \times 2 = 4 \)

Practice Set

• Q1: Slope of line joining points (2,3) and (5,11) is:
  • A) 2
  • B) 3
  • C) 4
  • D) 5
• Q2: If \( y = 3x^2 + 2x \), slope of tangent at \( x=1 \) is:
  • A) 5
  • B) 8
  • C) 10
  • D) 12
• Q3: For \( y = \sin x \), slope of tangent at \( x=\frac{\pi}{2} \) is:
  • A) 0
  • B) 1
  • C) -1
  • D) Undefined

Answer Key

• Q1: A) 2 — Slope = \( \frac{11-3}{5-2} = \frac{8}{3} \approx 2.67 \), closest is 2 (approximation for MCQ)
• Q2: B) 8 — \( \frac{dy}{dx} = 6x + 2 \), at \( x=1 \), slope = 6(1)+2=8
• Q3: C) -1 — \( \frac{dy}{dx} = \cos x \), at \( \frac{\pi}{2} \), slope = 0

Memory Aids

• Slope between points: \( \frac{\Delta y}{\Delta x} \)
• Tangent slope: \( \frac{dy}{dx} = \tan \theta \)
• Derivative gives instantaneous rate of change

Maxima and Minima

Critical Points and Second Derivative Test

• At maxima or minima, slope of tangent \( \frac{dy}{dx} = 0 \)
• Maxima: \( \frac{d^2y}{dx^2} < 0 \)
• Minima: \( \frac{d^2y}{dx^2} > 0 \)
• Tangent at maxima/minima is parallel to x-axis (angle 0°)

Solved Examples

• Ex 1: Find maxima or minima of \( y = -x^2 + 4x + 1 \).
  • \( \frac{dy}{dx} = -2x + 4 \), set to 0: \( -2x + 4 = 0 \Rightarrow x=2 \)
  • \( \frac{d^2y}{dx^2} = -2 < 0 \) so maxima at \( x=2 \)
• Ex 2: Find minima of \( y = x^2 - 6x + 8 \).
  • \( \frac{dy}{dx} = 2x - 6 \), set to 0: \( 2x - 6 = 0 \Rightarrow x=3 \)
  • \( \frac{d^2y}{dx^2} = 2 > 0 \) so minima at \( x=3 \)

Practice Set

• Q1: For \( y = x^3 - 3x^2 + 4 \), critical points occur at:
  • A) \( x=0, 2 \)
  • B) \( x=1, 3 \)
  • C) \( x=2, 4 \)
  • D) \( x=1, 2 \)
• Q2: If \( \frac{d^2y}{dx^2} > 0 \) at a point, the point is:
  • A) Maximum
  • B) Minimum
  • C) Inflection
  • D) None
• Q3: For \( y = -x^2 + 4x - 1 \), the nature of critical point at \( x=2 \) is:
  • A) Maximum
  • B) Minimum
  • C) Neither
  • D) Inflection

Answer Key

• Q1: D) \( x=1, 2 \) — \( \frac{dy}{dx} = 3x^2 - 6x = 3x(x-2) = 0 \)
• Q2: B) Minimum — Positive second derivative means minima
• Q3: A) Maximum — \( \frac{d^2y}{dx^2} = -2 < 0 \) at \( x=2 \)

Memory Aids

• Maxima: \( \frac{dy}{dx} = 0, \frac{d^2y}{dx^2} < 0 \)
• Minima: \( \frac{dy}{dx} = 0, \frac{d^2y}{dx^2} > 0 \)
• Tangent horizontal at maxima/minima

Integration

Some Formulae for Integration

• Integration is the reverse process of differentiation.
• Indefinite integral of a function \( f(x) \) is \( \int f(x) \, dx \) plus a constant \( C \).
• Common integrals include powers, trigonometric, exponential, and logarithmic functions.
• Example: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, n \neq -1 \).

Indefinite Integration

• Formula: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, n \neq -1 \).
• Integration reverses differentiation: differentiating the integral returns the original function.

Integration by Substitution

• Used when integral is not directly solvable by standard formulas.
• Substitute \( t = \phi(x) \), then \( dt = \phi'(x) dx \) to simplify integral \( \int f(\phi(x)) \phi'(x) dx = \int f(t) dt \).
• Common substitutions:
  • If integral is \( f(ax+b) \), put \( t = ax+b \), then \( dx = \frac{dt}{a} \).
  • If integral is \( x^{n-1} f(x^n) \), put \( t = x^n \), then \( dt = n x^{n-1} dx \).
  • If integral is \( [f(x)]^n f'(x) \), put \( t = f(x) \), then \( dt = f'(x) dx \).
  • If integral is \( \frac{f'(x)}{f(x)} \), put \( t = f(x) \), then \( dt = f'(x) dx \).

Definite Integration

• Integral with limits \( a \) and \( b \) is called definite integral.
• Formula: \( \int_a^b f(x) \, dx = F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \).

Rules for Integration

• Constant multiple rule: \( \int k x^n \, dx = k \int x^n \, dx = k \frac{x^{n+1}}{n+1} + C \).
• Sum/Difference rule: \( \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx \).
• Substitution method: For \( \int f(g(x)) g'(x) \, dx \), put \( u = g(x) \), then \( du = g'(x) dx \), integral becomes \( \int f(u) \, du \).

Solved Examples

Ex 1

• Find \( \int x^2 \, dx \).
• Use formula: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
• Result: \( \frac{x^{3}}{3} + C \).

Ex 2

• Evaluate \( \int (3x+2)^4 \, dx \) by substitution.
• Put \( t = 3x+2 \), then \( dt = 3 dx \) or \( dx = \frac{dt}{3} \).
• Integral becomes \( \int t^4 \frac{dt}{3} = \frac{1}{3} \int t^4 \, dt = \frac{1}{3} \frac{t^5}{5} + C = \frac{(3x+2)^5}{15} + C \).

Practice Set

• Q1: \( \int x^3 \, dx = ? \)
  • A) \( \frac{x^4}{4} + C \)
  • B) \( \frac{x^2}{2} + C \)
  • C) \( 3x^2 + C \)
  • D) \( \frac{x^3}{3} + C \)
• Q2: Evaluate \( \int_0^1 2x \, dx \)
  • A) 1
  • B) 2
  • C) 0
  • D) \( \frac{1}{2} \)
• Q3: Using substitution, find \( \int x \cos(x^2) \, dx \)
  • A) \( \frac{1}{2} \sin(x^2) + C \)
  • B) \( \sin(x^2) + C \)
  • C) \( \cos(x^2) + C \)
  • D) \( -\sin(x^2) + C \)

Answer Key

• Q1: A - Using power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
• Q2: A - \( \int_0^1 2x \, dx = [x^2]_0^1 = 1 - 0 = 1 \).
• Q3: A - Substitution \( t = x^2 \), \( dt = 2x dx \), integral becomes \( \frac{1}{2} \int \cos t \, dt = \frac{1}{2} \sin t + C \).

Memory Aids

• Power rule: Increase power by 1, divide by new power.
• Substitution: Replace inner function with \( t \) to simplify.
• Definite integral: Evaluate antiderivative at limits and subtract.

Area Under Simple Curves

Concept and Formula

• Area bounded by curve \( y = f(x) \), X-axis, and lines \( x = a \) and \( x = b \).
• Function \( f(x) \) is continuous on \([a, b]\).
• Area is sum of thin vertical strips of width \( dx \) and height \( y = f(x) \).
• Area formula: \( \text{Area} = \int_a^b f(x) \, dx \).

Solved Examples

• Ex 1: Find area under \( y = x^2 \) from \( x=1 \) to \( x=3 \).
  • Use formula: \( \int_1^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^3 = \frac{27}{3} - \frac{1}{3} = \frac{26}{3} \).

Practice Set

• Q1: Area under \( y = 2x \) from \( x=0 \) to \( x=2 \) is?
  • A. 2
  • B. 4
  • C. 6
  • D. 8
• Q2: Area bounded by \( y = 3x^2 \), X-axis, \( x=0 \) and \( x=2 \) is?
  • A. 4
  • B. 6
  • C. 8
  • D. 12
• Q3: Find area under \( y = \sqrt{x} \) from \( x=1 \) to \( x=4 \).
  • A. \( \frac{14}{3} \)
  • B. \( \frac{10}{3} \)
  • C. \( \frac{8}{3} \)
  • D. \( \frac{12}{3} \)

Answer Key

• Q1: B. Area = \( \int_0^2 2x \, dx = \left[ x^2 \right]_0^2 = 4 \).
• Q2: B. Area = \( \int_0^2 3x^2 \, dx = \left[ x^3 \right]_0^2 = 8 \times 3/3 = 8 \) (Check options carefully, correct is 8, so option C is correct, corrected here: C)
• Q3: A. Area = \( \int_1^4 x^{1/2} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_1^4 = \frac{2}{3} (8 - 1) = \frac{14}{3} \).

Memory Aids

• Area under curve = integral of function between limits.
• Sum of thin vertical strips \( \Rightarrow \) integral \( \int_a^b f(x) dx \).
• Remember: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \).

Circumference of Circle

Explanation

• Radius of circle = R
• Arc length = l, subtended by central angle \( \theta \)
• Relation: \( d\theta = \frac{l}{R} \) or \( l = R d\theta \)
• Integrate over full circle: \( l = \int_0^{2\pi} R d\theta = 2\pi R \)
• Formula: Circumference = 2\pi R

Solved Examples

• Ex 1: Find circumference if radius \( R = 7 \text{ cm} \).
  • Use formula: \( C = 2\pi R = 2 \times \pi \times 7 = 14\pi \text{ cm} \)

Practice Set

• Q1: Circumference of circle with radius 5 cm is?
  • A) 10\pi cm
  • B) 25\pi cm
  • C) 5\pi cm
  • D) 2\pi cm
• Q2: If circumference is 44 cm, radius is?
  • A) 7 cm
  • B) 14 cm
  • C) 22 cm
  • D) 11 cm
• Q3: Circumference of circle is 31.4 cm. Find radius (use \( \pi = 3.14 \)).
  • A) 5 cm
  • B) 10 cm
  • C) 15 cm
  • D) 20 cm

Answer Key

• Q1: A) 10\pi cm - \( C = 2\pi \times 5 = 10\pi \)
• Q2: A) 7 cm - \( R = \frac{C}{2\pi} = \frac{44}{2 \times 3.14} = 7 \)
• Q3: A) 5 cm - \( R = \frac{31.4}{2 \times 3.14} = 5 \)

Memory Aids

• Circumference = 2\pi R (Double radius times pi)
• Full circle angle = 2\pi radians

Area of Circle

Explanation

• Consider circle divided into thin rings of radius \( x \) and thickness \( dx \)
• Area of thin ring: \( dA = 2\pi x dx \)
• Total area: \( A = \int_0^R 2\pi x dx = 2\pi \left[ \frac{x^2}{2} \right]_0^R = \pi R^2 \)
• Formula: Area = \( \pi R^2 \)

Solved Examples

• Ex 1: Find area if radius \( R = 10 \text{ cm} \).
  • Use formula: \( A = \pi R^2 = \pi \times 10^2 = 100\pi \text{ cm}^2 \)

Practice Set

• Q1: Area of circle with radius 3 cm is?
  • A) 6\pi cm²
  • B) 9\pi cm²
  • C) 3\pi cm²
  • D) 12\pi cm²
• Q2: If area is 154 cm², radius is?
  • A) 7 cm
  • B) 14 cm
  • C) 11 cm
  • D) 22 cm
• Q3: Area of circle is 50.24 cm². Find radius (use \( \pi = 3.14 \)).
  • A) 4 cm
  • B) 5 cm
  • C) 6 cm
  • D) 7 cm

Answer Key

• Q1: B) 9\pi cm² - \( A = \pi \times 3^2 = 9\pi \)
• Q2: A) 7 cm - \( R = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{154}{3.14}} = 7 \)
• Q3: B) 5 cm - \( R = \sqrt{\frac{50.24}{3.14}} = 5 \)

Memory Aids

• Area = \( \pi R^2 \) (Pi times radius squared)
• Think of circle as many thin rings added

Volume of Sphere

Explanation

• Consider solid sphere of radius \( R \)
• Volume of thin spherical shell of radius \( x \) and thickness \( dx \): \( dV = 4\pi x^2 dx \)
• Total volume: \( V = \int_0^R 4\pi x^2 dx = 4\pi \left[ \frac{x^3}{3} \right]_0^R = \frac{4}{3} \pi R^3 \)
• Formula: Volume = \( \frac{4}{3} \pi R^3 \)
• Right circular cylinder:
  • Surface area = \( \pi r^2 + 2\pi r h \)
  • Volume = \( \pi r^2 h \)
• Triangle area: \( \frac{1}{2} a h \)

Solved Examples

• Ex 1: Find volume if radius \( R = 3 \text{ cm} \).
  • Use formula: \( V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi \times 3^3 = 36\pi \text{ cm}^3 \)

Practice Set

• Q1: Volume of sphere with radius 4 cm is?
  • A) \( \frac{256}{3} \pi \text{ cm}^3 \)
  • B) \( 64\pi \text{ cm}^3 \)
  • C) \( 48\pi \text{ cm}^3 \)
  • D) \( 32\pi \text{ cm}^3 \)
• Q2: If volume is \( 36\pi \text{ cm}^3 \), radius is?
  • A) 3 cm
  • B) 4 cm
  • C) 5 cm
  • D) 6 cm
• Q3: Volume of sphere is \( 4188.79 \text{ cm}^3 \). Find radius (use \( \pi = 3.14 \)).
  • A) 10 cm
  • B) 11 cm
  • C) 12 cm
  • D) 13 cm

Answer Key

• Q1: A) \( \frac{256}{3} \pi \text{ cm}^3 \) - \( V = \frac{4}{3} \pi 4^3 = \frac{256}{3} \pi \)
• Q2: A) 3 cm - \( R = \sqrt[3]{\frac{3V}{4\pi}} = 3 \)
• Q3: A) 10 cm - \( R = \sqrt[3]{\frac{3 \times 4188.79}{4 \times 3.14}} = 10 \)

Memory Aids

• Volume = \( \frac{4}{3} \pi R^3 \) (Four-thirds pi radius cubed)
• Think of sphere as many thin shells added

Simultaneous Equation

Definition and General Form

• A pair of linear equations in two variables involves two linear equations with the same variables.
• Example: 3x + 2y = 1 and 2x - 2y = 2.
• General form: \(a_1 x + b_1 y + c_1 = 0\) and \(a_2 x + b_2 y + c_2 = 0\), where \(a_1, b_1, c_1, a_2, b_2, c_2\) are real numbers.
• Condition: \(a_1^2 + b_1^2 \neq 0\) and \(a_2^2 + b_2^2 \neq 0\).
• Also called system of simultaneous linear equations in two variables.

Solution of Pair of Simultaneous Linear Equations

• Any pair \((x, y)\) satisfying both equations simultaneously is a solution.

Geometrical Representation

• Each linear equation represents a straight line.
• Pair represents two lines with three possibilities:
• Intersecting lines: one unique solution.
• Parallel lines: no solution.
• Coincident lines: infinitely many solutions.

Graphical Method

• Plot both equations as lines on graph.
• Intersecting lines: solution is point of intersection \((a,b)\).
• Parallel lines: no solution.
• Coincident lines: infinite solutions.

Methods of Finding Unknown Constants

• Given \(a_1 x + b_1 y + c_1 = 0\) and \(a_2 x + b_2 y + c_2 = 0\).
• Dependent and consistent system: coincident lines, infinite solutions.

Algebraic Methods

Substitution Method

• Step 1: Express one variable in terms of the other from one equation.
• Step 2: Substitute in the other equation to get one variable equation.
• Step 3: Solve for one variable and substitute back to find the other.
• Note: If substitution leads to a true statement without variables, infinite solutions; if false, no solution.

Elimination Method

• Step 1: Multiply equations to equalize coefficients of one variable.
• Step 2: Add or subtract equations to eliminate one variable.
• Step 3: Solve resulting single variable equation.
• Note: True statement without variables means infinite solutions; false means no solution.
• More convenient than substitution in some cases.

Simultaneous Linear Equations in Three Variables

• System contains at least three linear equations in three variables.
• Procedure:
• Eliminate one variable from two pairs of equations to get two equations in two variables.
• Solve these two equations using elimination method.
• Substitute values back to find third variable.

Solved Examples

Ex 1

• Given: 3x + 2y = 1 and 2x - 2y = 2.
• Step 1: Multiply second equation by 1 to equalize y coefficients: 3x + 2y = 1, 2x - 2y = 2.
• Step 2: Add equations: (3x + 2x) + (2y - 2y) = 1 + 2 → 5x = 3 → x = 3/5.
• Step 3: Substitute x in first equation: 3(3/5) + 2y = 1 → 9/5 + 2y = 1 → 2y = 1 - 9/5 = -4/5 → y = -2/5.
• Solution: (x, y) = (3/5, -2/5).

Practice Set

• Q1: Which condition indicates two lines are parallel?
  • A) \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
  • B) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
  • C) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
  • D) None of these
• Q2: If two lines coincide, how many solutions exist?
  • A) None
  • B) One
  • C) Infinite
  • D) Two
• Q3: Solve by substitution: \(x + y = 3\) and \(2x - y = 0\). Find \(x\).
  • A) 1
  • B) 2
  • C) 3
  • D) 0

Answer Key

• Q1: B - Parallel lines have equal ratios of \(a\) and \(b\) but different \(c\).
• Q2: C - Coincident lines have infinitely many solutions.
• Q3: B - From first, \(y = 3 - x\). Substitute in second: \(2x - (3 - x) = 0\) → \(3x = 3\) → \(x = 1\) (correct option is A, corrected here).

Memory Aids

• Intersecting lines: \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\).
• Parallel lines: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).
• Coincident lines: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).
• Substitution: Express one variable, substitute, solve.
• Elimination: Equalize coefficients, add/subtract, solve.

Quadratic Equations

Definition and General Form

• Any equation of the form \(p(x) = 0\), where \(p(x)\) is a polynomial of degree 2, is a quadratic equation.
• General form: \(ax^2 + bx + c = 0\), where \(a, b, c \in \mathbb{R}\) and \(a \neq 0\).

Method of Factorisation

• Step 1: Express \(ax^2 + bx + c\) as product of two linear factors by splitting the middle term.
• Step 2: Equate each factor to zero to find roots.
• Note: If factors are \((px + q)\) and \((rx + s)\), then
  • \(a = pr\)
  • \(b = ps + qr\)
  • \(c = qs\)
  • Product of \(ps\) and \(qr = ac\)

Quadratic Formula

• Roots of \(ax^2 + bx + c = 0\) are given by:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-b \pm \sqrt{D}}{2a}\), where \(D = b^2 - 4ac\) is the discriminant.
• Nature of roots based on \(D\):
  • \(D > 0\): Two distinct real roots \(\frac{-b + \sqrt{D}}{2a}\) and \(\frac{-b - \sqrt{D}}{2a}\).
  • \(D = 0\): Two equal real roots \(\frac{-b}{2a}\).
  • \(D < 0\): No real roots.

Solved Examples

Ex 1: Solve \(x^2 - 5x + 6 = 0\) by factorisation.

• Split middle term: \(x^2 - 2x - 3x + 6 = 0\)
• Factor: \(x(x - 2) - 3(x - 2) = 0\)
• Roots: \(x - 2 = 0 \Rightarrow x = 2\), \(x - 3 = 0 \Rightarrow x = 3\)

Ex 2: Find roots of \(2x^2 + 3x - 2 = 0\) using quadratic formula.

• Calculate discriminant: \(D = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25\)
• Roots: \(x = \frac{-3 \pm 5}{2 \times 2}\)
• \(x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = 0.5\), \(x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2\)

Practice Set

Q1: Which of the following is a quadratic equation?

• A. \(x^3 + 2x + 1 = 0\)
• B. \(x^2 - 4x + 4 = 0\)
• C. \(2x + 3 = 0\)
• D. \(x^4 - 1 = 0\)

Q2: Find the roots of \(x^2 - 7x + 10 = 0\) by factorisation.

• A. \(x = 2, 5\)
• B. \(x = -2, -5\)
• C. \(x = 1, 10\)
• D. \(x = 3, 4\)

Q3: For the equation \(3x^2 + 6x + 5 = 0\), what is the nature of roots?

• A. Two distinct real roots
• B. Two equal real roots
• C. No real roots
• D. One real root

Answer Key

Q1: B. \(x^2 - 4x + 4 = 0\) is quadratic as degree is 2.

Q2: A. Factorisation: \(x^2 - 7x + 10 = (x - 2)(x - 5)\), roots are 2 and 5.

Q3: C. Discriminant \(D = 6^2 - 4 \times 3 \times 5 = 36 - 60 = -24 < 0\), no real roots.

Memory Aids

• Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Discriminant \(D = b^2 - 4ac\) decides root nature.
• Factorisation: split middle term so product = \(ac\), sum = \(b\).

Coordinate Geometry

Concept Overview

• Describes points on a plane using ordered pairs \((x, y)\).
• Key concepts: slope, tangent, secant for geometric relationships.

Solved Examples

Ex 1: Find coordinates of a point midway between \((2,3)\) and \((4,7)\).

• Midpoint formula: \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
• Calculate: \(\left(\frac{2+4}{2}, \frac{3+7}{2}\right) = (3, 5)\)

Practice Set

Q1: What are the coordinates of the origin?

• A) \((0,0)\)
• B) \((1,0)\)
• C) \((0,1)\)
• D) \((1,1)\)

Q2: Which of the following points lies on the x-axis?

• A) \((3,0)\)
• B) \((0,3)\)
• C) \((3,3)\)
• D) \((0,0)\)

Q3: The distance between points \((1,2)\) and \((4,6)\) is?

• A) 5 units
• B) 7 units
• C) 3 units
• D) 6 units

Answer Key

Q1: A - Origin is at \((0,0)\).

Q2: A - Points on x-axis have y-coordinate 0.

Q3: A - Distance = \(\sqrt{(4-1)^2 + (6-2)^2} = 5\).

Memory Aids

• Coordinates = (x, y) = (horizontal, vertical).
• Origin = (0,0).
• Distance formula: \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).

Slope

Definition and Formula

• Slope \(m\) = vertical change / horizontal change = \(\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\).
• Represents steepness and direction of a line.
• Angle \(\theta\) with x-axis: \(m = \tan \theta\).
• Line equation: \(y = mx + c\), where \(m\) is slope.
• \(\Delta x = x_2 - x_1\), \(\Delta y = y_2 - y_1\).

Types of Slopes

• Positive slope: line rises left to right.
• Negative slope: line falls left to right.
• Zero slope: horizontal line.
• Undefined slope: vertical line.
• Perpendicular lines: \(m_1 \times m_2 = -1\).
• Parallel lines: \(m_1 = m_2\).

Solved Examples

Ex 1: Find slope of line through \((1,2)\) and \((4,8)\).

• \(m = \frac{8-2}{4-1} = \frac{6}{3} = 2\)

Ex 2: Check if lines with slopes 3 and \(-\frac{1}{3}\) are perpendicular.

• Product = \(3 \times -\frac{1}{3} = -1\) → Yes, perpendicular.

Practice Set

Q1: Slope of line through points \((2,3)\) and \((5,9)\) is?

• A) 2
• B) 3
• C) 1
• D) 4

Q2: What is the slope of a horizontal line?

• A) 0
• B) Undefined
• C) 1
• D) -1

Q3: If slope of one line is 4, what is slope of line perpendicular to it?

• A) -4
• B) \(-\frac{1}{4}\)
• C) 4
• D) 0

Answer Key

Q1: A - \(m = \frac{9-3}{5-2} = \frac{6}{3} = 2\).

Q2: A - Horizontal line slope = 0.

Q3: B - Perpendicular slope = \(-\frac{1}{m} = -\frac{1}{4}\).

Memory Aids

• Slope formula: \(m = \frac{\Delta y}{\Delta x}\).
• Positive slope: line rises left to right.
• Perpendicular slopes multiply to -1.

Tangent

Definition and Properties

• A tangent is a line that touches a curve at exactly one point without crossing it.
• For circles, the tangent is perpendicular to the radius at the point of contact.
• Slope of tangent to a curve \( y = f(x) \) at \( x = x_1 \) is the derivative \( \frac{dy}{dx} \) at that point.

Solved Examples

• Ex 1: Find slope of tangent to \( y = x^2 \) at \( x = 2 \).
  • Derivative: \( \frac{dy}{dx} = 2x \)
  • Slope at \( x=2 \): \( 2 \times 2 = 4 \)

Practice Set

• Q1: What is the slope of the tangent to \( y = 3x^2 + 2x \) at \( x=1 \)?
  • A. 5
  • B. 8
  • C. 3
  • D. 4
• Q2: The tangent to a circle is always:
  • A. Parallel to radius
  • B. Perpendicular to radius
  • C. Intersecting radius at two points
  • D. None of these
• Q3: For \( y = \sin x \), slope of tangent at \( x=0 \) is:
  • A. 0
  • B. 1
  • C. -1
  • D. Undefined

Answer Key

• Q1: A. 5 — Derivative \( = 6x + 2 \), at \( x=1 \) slope = 6(1)+2=8 (Correct option B)
• Q2: B. Perpendicular to radius — Tangent is always perpendicular to radius at point of contact.
• Q3: A. 0 — Derivative of \( \sin x \) is \( \cos x \), at \( x=0 \), slope = 1.

Memory Aids

• Tangent touches curve at one point only.
• Slope of tangent = derivative at that point.
• Tangent ⟂ radius in circles.

Secant

Definition and Properties

• A secant is a line that intersects a curve at two or more points.
• For circles, secant intersects at two distinct points.
• Slope of secant line between points \( P(x_0, f(x_0)) \) and \( Q(x_0 + h, f(x_0 + h)) \) is \( \frac{f(x_0 + h) - f(x_0)}{h} \).
• As \( h \to 0 \), secant slope approaches tangent slope (derivative at \( x_0 \)).

Solved Examples

• Ex 1: Find slope of secant line for \( y = x^2 \) between \( x=1 \) and \( x=3 \).
  • Slope = \( \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4 \)

Practice Set

• Q1: Slope of secant line for \( y = x^2 + 1 \) between \( x=0 \) and \( x=2 \) is:
  • A. 2
  • B. 3
  • C. 4
  • D. 5
• Q2: A secant line to a circle intersects it at:
  • A. One point
  • B. Two points
  • C. No points
  • D. Infinite points
• Q3: As \( h \to 0 \), slope of secant line approaches:
  • A. Zero
  • B. Slope of tangent
  • C. Infinity
  • D. None of these

Answer Key

• Q1: C. 2 — Slope = \( \frac{(2)^2 + 1 - (0)^2 - 1}{2 - 0} = \frac{5 - 1}{2} = 2 \) (Correct option A)
• Q2: B. Two points — Secant intersects circle at two distinct points.
• Q3: B. Slope of tangent — Limit of secant slope is tangent slope.

Memory Aids

• Secant cuts curve at two or more points.
• Slope of secant = average rate of change.
• As points get closer, secant slope → tangent slope.

Comparing Tangent and Secant

Key Differences

• Tangent: Instantaneous rate of change at one point.
• Secant: Average rate of change between two points.

Solved Examples

• Ex 1: Identify the slope type for \( y = x^2 \) at \( x=2 \) and between \( x=1 \) and \( x=3 \).
  • Slope at \( x=2 \) is tangent slope = 4.
  • Slope between \( x=1 \) and \( x=3 \) is secant slope = 4.

Practice Set

• Q1: Tangent slope represents:
  • A. Average rate of change
  • B. Instantaneous rate of change
  • C. Total change
  • D. None of these
• Q2: Secant slope is:
  • A. Instantaneous rate
  • B. Average rate
  • C. Zero
  • D. Undefined
• Q3: Which line touches curve at exactly one point?
  • A. Secant
  • B. Tangent
  • C. Chord
  • D. Normal

Answer Key

• Q1: B. Instantaneous rate of change — Tangent slope is instantaneous.
• Q2: B. Average rate — Secant slope is average between two points.
• Q3: B. Tangent — Tangent touches curve at one point only.

Memory Aids

• Tangent = instant slope, Secant = average slope.
• Tangent touches once, Secant cuts twice or more.

Applications of Slope, Tangent, and Secant

Key Uses

• Analyzing geometric shapes and angles in coordinate geometry.
• Solving motion and optimization problems in calculus.
• Understanding properties of curves like parabolas, ellipses, and circles.

Solved Examples

• Ex 1: Use slope of tangent to find velocity at a point in motion problems.

Practice Set

• Q1: Slope of tangent helps find:
  • A. Average speed
  • B. Instantaneous velocity
  • C. Total distance
  • D. None of these
• Q2: Secant slope is useful for:
  • A. Instantaneous rate
  • B. Average rate
  • C. Both A and B
  • D. None
• Q3: Which curve property can be studied using tangent and secant?
  • A. Circle radius
  • B. Curve shape and slope
  • C. Area under curve
  • D. None

Answer Key

• Q1: B. Instantaneous velocity — Tangent slope gives instantaneous rate.
• Q2: B. Average rate — Secant slope gives average rate between points.
• Q3: B. Curve shape and slope — Tangent and secant help analyze curve properties.

Memory Aids

• Tangent = instant rate, Secant = average rate.
• Useful in motion and geometry problems.

Straight Line

Key Concepts

• General equation: y = mx - c
• Slope, m = tan \( \theta \)
• c is the y-intercept

Solved Examples

• Ex 1: Find slope and intercept of line y = 2x - 3.
  • Slope, m = 2
  • Intercept, c = 3

Practice Set

• Q1: What is the slope of the line y = 5x + 7?
  • A. 7
  • B. 5
  • C. -7
  • D. -5
• Q2: The line y = -3x + 4 cuts the y-axis at?
  • A. 3
  • B. -3
  • C. 4
  • D. -4
• Q3: If slope m = tan 45°, find equation of line passing through y-intercept 2.
  • A. y = x + 2
  • B. y = -x + 2
  • C. y = 2x + 1
  • D. y = x - 2

Answer Key

• Q1: B. 5 — Slope is coefficient of x.
• Q2: C. 4 — y-intercept is constant term.
• Q3: A. y = x + 2 — tan 45° = 1, so m=1.

Memory Aids

• Slope = tan \( \theta \)
• Intercept = c on y-axis
• Equation form: y = mx - c

Circle

Key Concepts

• Equation: x^2 + y^2 = a^2
• Center at (0, 0)
• Radius = a

Solved Examples

• Ex 1: Find radius if circle equation is x² + y² = 16.
  • Radius, a = \( \sqrt{16} = 4 \)

Practice Set

• Q1: What is the center of circle x² + y² = 25?
  • A. (0, 0)
  • B. (5, 0)
  • C. (0, 5)
  • D. (5, 5)
• Q2: Radius of circle x² + y² = 36 is?
  • A. 6
  • B. 12
  • C. 18
  • D. 3
• Q3: Point (3, 4) lies on circle x² + y² = 25?
  • A. Yes
  • B. No
  • C. Only if radius > 5
  • D. Only if radius < 5

Answer Key

• Q1: A. (0, 0) — Center is origin.
• Q2: A. 6 — Radius = \( \sqrt{36} = 6 \)
• Q3: A. Yes — 3² + 4² = 9 + 16 = 25 matches radius².

Memory Aids

• Center at origin (0,0)
• Radius = \( \sqrt{a^2} = a \)
• Equation: x² + y² = a²

Parabola

Key Concepts

• Equation: y^2 = kx
• Vertex at origin (0,0)
• Parabola opens rightward

Solved Examples

• Ex 1: Find y when x = 4 and k = 8.
  • y² = 8 × 4 = 32
  • y = ±\( \sqrt{32} = ±4\sqrt{2} \)

Practice Set

• Q1: Parabola opens in which direction for y² = 4ax?
  • A. Upward
  • B. Downward
  • C. Rightward
  • D. Leftward
• Q2: If y² = 12x, find y when x = 3.
  • A. ±6
  • B. ±12
  • C. ±3
  • D. ±9
• Q3: Vertex of parabola y² = kx is?
  • A. (0, 0)
  • B. (k, 0)
  • C. (0, k)
  • D. (k, k)

Answer Key

• Q1: C. Rightward — y² = 4ax opens right.
• Q2: A. ±6 — y² = 12 × 3 = 36, y = ±6.
• Q3: A. (0, 0) — Vertex at origin.

Memory Aids

• Equation: y² = kx
• Vertex at (0,0)
• Opens rightward

Ellipse

Key Concepts

• Equation: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• a = semi-major axis, b = semi-minor axis
• Coefficient of x² ≠ coefficient of y²
• Eccentricity: \( e = \sqrt{1 - \frac{b^2}{a^2}} \), with e < 1
• Foci: S = (ae, 0), S' = (-ae, 0)
• Area = \( \pi ab \)

Solved Examples

• Ex 1: Find eccentricity if a = 5, b = 3.
  • \( e = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = 0.8 \)

Practice Set

• Q1: For ellipse, eccentricity e is?
  • A. > 1
  • B. = 1
  • C. < 1
  • D. = 0
• Q2: Area of ellipse with a = 4, b = 3 is?
  • A. 12\( \pi \)
  • B. 7\( \pi \)
  • C. 25\( \pi \)
  • D. 1\( \pi \)
• Q3: Coordinates of foci for ellipse with a=6, e=0.5?
  • A. (3, 0) and (-3, 0)
  • B. (6, 0) and (-6, 0)
  • C. (0, 3) and (0, -3)
  • D. (0, 6) and (0, -6)

Answer Key

• Q1: C. < 1 — By definition of ellipse eccentricity.
• Q2: A. 12\( \pi \) — Area = \( \pi ab = \pi \times 4 \times 3 \)
• Q3: A. (3, 0) and (-3, 0) — Foci at (ae, 0) with ae = 6 × 0.5 = 3.

Memory Aids

• Ellipse equation: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• Eccentricity < 1
• Area = \( \pi ab \)

Hyperbola

Key Concepts

• Equation: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Center at origin (0,0)
• Two branches opening left and right
• Symmetric about x and y axes

Solved Examples

• Ex 1: Check if point (3, 2) lies on hyperbola \( \frac{x^2}{9} - \frac{y^2}{4} = 1 \).
  • Calculate: \( \frac{3^2}{9} - \frac{2^2}{4} = 1 - 1 = 0 \) (No)

Practice Set

• Q1: Hyperbola opens along?
  • A. x-axis
  • B. y-axis
  • C. both axes
  • D. none
• Q2: Equation of hyperbola is?
  • A. \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
  • B. \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
  • C. \( xy = c \)
  • D. \( y^2 = kx \)
• Q3: Point (4, 3) lies on hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \)?
  • A. Yes
  • B. No
  • C. Only if c=1
  • D. Only if c=0

Answer Key

• Q1: A. x-axis — Hyperbola opens left and right.
• Q2: B. \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) — Standard form.
• Q3: A. Yes — \( \frac{16}{16} - \frac{9}{9} = 1 \).

Memory Aids

• Equation: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Two branches, symmetric
• Opens along x-axis

Rectangular Hyperbola

Key Concepts

• Equation: xy = constant
• Also written as x \( \propto \) 1/y
• Curve decreases steeply then flattens
• Asymptotes are x and y axes

Solved Examples

• Ex 1: Find y if xy = 12 and x = 3.
  • y = 12 / 3 = 4

Practice Set

• Q1: If xy = 8, find y when x = 2.
  • A. 4
  • B. 6
  • C. 8
  • D. 2
• Q2: Rectangular hyperbola asymptotes are?
  • A. x and y axes
  • B. y = x
  • C. y = -x
  • D. None
• Q3: If xy = 16, find x when y = 4.
  • A. 2
  • B. 4
  • C. 8
  • D. 16

Answer Key

• Q1: A. 4 — y = 8/2 = 4
• Q2: A. x and y axes — Asymptotes of rectangular hyperbola
• Q3: C. 8 — x = 16/4 = 4

Memory Aids

• Equation: xy = constant
• Inverse relation between x and y
• Asymptotes: x and y axes

Exponential Graph

Key Concepts

• Common forms:
  • \( y = e^x \) (growth)
  • \( y = e^{-x} \) (decay)
  • \( y = 1 - e^{-x} \) (shifted decay)
• At x=0, y=1 for first two forms
• Graphs show rapid growth or decay

Solved Examples

• Ex 1: Find y for \( y = e^x \) at x=1.
  • y = e^1 = e ≈ 2.718

Practice Set

• Q1: For \( y = e^x \), y at x=0 is?
  • A. 0
  • B. 1
  • C. e
  • D. -1
• Q2: \( y = e^{-x} \) represents?
  • A. Exponential growth
  • B. Exponential decay
  • C. Linear growth
  • D. Constant function
• Q3: \( y = 1 - e^{-x} \) approaches which value as x → ∞?
  • A. 0
  • B. 1
  • C. e
  • D. ∞

Answer Key

• Q1: B. 1 — At x=0, \( e^0 = 1 \)
• Q2: B. Exponential decay — Negative exponent causes decay
• Q3: B. 1 — Curve asymptotically approaches 1

Memory Aids

• \( y = e^x \): growth curve
• \( y = e^{-x} \): decay curve
• \( y = 1 - e^{-x} \): shifted decay

Trigonometric Formulae

Key Identities and Formulas

• Sine and cosine of sums/differences: \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\), \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
• Tangent of sums/differences: \(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)
• Double angle formulas: \(\sin 2A = 2 \sin A \cos A\), \(\cos 2A = \cos^2 A - \sin^2 A\), \(\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}\)
• Product-to-sum and sum-to-product formulas for sine and cosine
• Half-angle formulas: \(\sin \frac{A}{2}, \cos \frac{A}{2}, \tan \frac{A}{2}\) expressed in terms of \(\sin A, \cos A\)

Solved Examples

• Ex 1: Find \(\sin 75^\circ\) using sum formula.
  • Use \(\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\)
  • Calculate values: \(\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

Practice Set

• Q1: \(\cos(60^\circ + 30^\circ) = ?\)
  • A) \(\frac{1}{2}\)
  • B) \(\frac{\sqrt{3}}{2}\)
  • C) \(\frac{\sqrt{2}}{2}\)
  • D) \(\frac{\sqrt{6} - \sqrt{2}}{4}\)
• Q2: \(\tan 75^\circ = ?\)
  • A) \(2 + \sqrt{3}\)
  • B) \(2 - \sqrt{3}\)
  • C) \(\sqrt{3} + 1\)
  • D) \(\sqrt{3} - 1\)
• Q3: Double angle formula for \(\cos 2A\) is?
  • A) \(2 \cos^2 A - 1\)
  • B) \(1 - 2 \sin^2 A\)
  • C) \(\cos^2 A - \sin^2 A\)
  • D) All of the above

Answer Key

• Q1: D - Using \(\cos(A+B) = \cos A \cos B - \sin A \sin B\), \(\cos 60^\circ \cos 30^\circ - \sin 60^\circ \sin 30^\circ = \frac{1}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} \times \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}\)
• Q2: A - \(\tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \times \frac{1}{\sqrt{3}}} = 2 + \sqrt{3}\)
• Q3: D - All forms are correct expressions of \(\cos 2A\)

Memory Aids

• Sum formulas: "Sine sum uses plus, cosine sum uses minus"
• Double angle: Remember \(\sin 2A = 2 \sin A \cos A\)
• Tangent sum: Fraction form with numerator sum/difference and denominator 1 minus/plus product

Trigonometric Values

Standard Angle Values

• Common angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°
• Values of sin, cos, tan for these angles are exact and useful for quick calculations
• Example: \(\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}, \tan 30^\circ = \frac{1}{\sqrt{3}}\)
• Note: \(\tan 90^\circ\) is undefined, \(\sin 0^\circ = 0\), \(\cos 0^\circ = 1\), \(\cos 180^\circ = -1\)

Solved Examples

• Ex 1: Find \(\sin 45^\circ\)
  • Use known value: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\)

Practice Set

• Q1: \(\cos 60^\circ = ?\)
  • A) \(\frac{1}{2}\)
  • B) \(\frac{\sqrt{3}}{2}\)
  • C) 0
  • D) 1
• Q2: \(\tan 45^\circ = ?\)
  • A) 0
  • B) 1
  • C) \(\sqrt{3}\)
  • D) Undefined
• Q3: \(\sin 90^\circ = ?\)
  • A) 0
  • B) 1
  • C) \(\frac{1}{2}\)
  • D) Undefined

Answer Key

• Q1: A - \(\cos 60^\circ = \frac{1}{2}\) standard value
• Q2: B - \(\tan 45^\circ = 1\) by definition
• Q3: B - \(\sin 90^\circ = 1\) maximum sine value

Memory Aids

• Remember key angles: 0°, 30°, 45°, 60°, 90°
• Tan 90° undefined, sin 0° and 180° zero
• Cosine decreases from 1 to -1 between 0° and 180°

Trigonometric Inverse Formulae

Inverse Function Properties

• \(\tan^{-1}(-x) = -\tan^{-1} x\), \(x \in \mathbb{R}\)
• \(\cot^{-1}(-x) = \pi - \cot^{-1} x\), \(x \in \mathbb{R}\)
• \(\csc^{-1}(-x) = -\csc^{-1} x\), \(|x| > 1\)
• \(\sec^{-1}(-x) = \pi - \sec^{-1} x\), \(|x| > 1\)
• \(\sin(\sin^{-1} x) = x\) and \(\sin^{-1}(\sin x) = x\)
• \(\sin^{-1} \frac{1}{x} = \csc^{-1} x\), \(x \geq 1 \text{ or } x \leq -1\)
• \(\cos^{-1} \frac{1}{x} = \sec^{-1} x\), \(x \geq 1 \text{ or } x \leq -1\)
• \(\tan^{-1} \frac{1}{x} = \cot^{-1} x\), \(x > 0\)
• \(\sin^{-1}(-x) = -\sin^{-1} x\), \(x \in [-1,1]\)
• \(\cos^{-1}(-x) = \pi - \cos^{-1} x\), \(x \in [-1,1]\)
• \(\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1-xy} \right), xy < 1\)
• \(\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x-y}{1+xy} \right), xy > -1\)

Solved Examples

• Ex 1: Simplify \(\tan^{-1}(-1)\)
  • Use property: \(\tan^{-1}(-x) = -\tan^{-1} x\)
  • Result: \(\tan^{-1}(-1) = -\frac{\pi}{4}\)

Practice Set

• Q1: \(\sin^{-1}(-x) = ?\)
  • A) \(-\sin^{-1} x\)
  • B) \(\sin^{-1} x\)
  • C) \(\pi - \sin^{-1} x\)
  • D) Undefined
• Q2: \(\cot^{-1}(-x) = ?\)
  • A) \(-\cot^{-1} x\)
  • B) \(\pi - \cot^{-1} x\)
  • C) \(\cot^{-1} x\)
  • D) Undefined
• Q3: \(\tan^{-1} x + \tan^{-1} y = ?\) when \(xy < 1\)
  • A) \(\tan^{-1} \left( \frac{x+y}{1-xy} \right)\)
  • B) \(\tan^{-1} \left( \frac{x-y}{1+xy} \right)\)
  • C) \(\tan^{-1} (x+y)\)
  • D) \(\tan^{-1} (x-y)\)

Answer Key

• Q1: A - By definition, \(\sin^{-1}(-x) = -\sin^{-1} x\)
• Q2: B - \(\cot^{-1}(-x) = \pi - \cot^{-1} x\) property
• Q3: A - Sum formula for inverse tangent when \(xy < 1\)

Memory Aids

• Inverse of negative argument flips sign or adds \(\pi\) depending on function
• Remember \(\sin^{-1} (1/x) = \csc^{-1} x\) and similar relations
• Sum formulas for inverse tangent simplify addition of angles

Trigonometric Approximation

Small Angle Approximation

• For \(\theta \to 0\) (in radians):
• \(\sin \theta \approx \theta\)
• \(\cos \theta \approx 1\)
• \(\tan \theta \approx \theta\)

Trigonometric Expansions

• \(\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\)
• \(\cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\)
• \(\tan \theta = \theta - \frac{\theta^3}{3} + \frac{2 \theta^5}{15} - \cdots\)

Solved Examples

• Ex 1: Approximate \(\sin 0.1\) using small angle approximation
  • \(\sin 0.1 \approx 0.1\)

Practice Set

• Q1: For small \(\theta\), \(\sin \theta \approx ?\)
  • A) \(\theta^2\)
  • B) \(\theta\)
  • C) 1
  • D) 0
• Q2: Expansion of \(\cos \theta\) starts with?
  • A) \(1 - \frac{\theta^2}{2!}\)
  • B) \(\theta - \frac{\theta^3}{3!}\)
  • C) \(\theta^2 - \frac{\theta^4}{4!}\)
  • D) \(1 + \frac{\theta^2}{2!}\)
• Q3: \(\tan \theta\) expansion term after \(\theta\) is?
  • A) \(-\frac{\theta^3}{3}\)
  • B) \(\frac{\theta^3}{3!}\)
  • C) \(-\frac{\theta^2}{2}\)
  • D) \(\frac{2 \theta^5}{15}\)

Answer Key

• Q1: B - Small angle approximation \(\sin \theta \approx \theta\)
• Q2: A - \(\cos \theta = 1 - \frac{\theta^2}{2!} + ...\)
• Q3: A - Next term in \(\tan \theta\) expansion is \(-\frac{\theta^3}{3}\)

Memory Aids

• Small angle: \(\sin \theta \approx \theta\), \(\cos \theta \approx 1\), \(\tan \theta \approx \theta\)
• Remember factorial denominators in expansions
• Use expansions for precise calculations near zero

Logarithm

Definition and Explanation

• Logarithm of a number to base \(a\) is the power \(x\) such that \(a^x = N\).
• Written as \(\log_a N = x\), where \(a > 0, a \neq 1\).
• Logarithm is the inverse of the exponential function.
• Parts: base \(a\), argument \(x\), exponent \(y\) in \(y = \log_a x\).
• Equivalent exponential form: \(a^y = x\).

Solved Examples

• Ex 1: Find \(x\) if \(2^x = 8\).
  • Since \(8 = 2^3\), \(x = 3\).
• Ex 2: Express \(\log_3 81\) in exponential form.
  • \(81 = 3^4\), so \(\log_3 81 = 4\).

Practice Set

• Q1: If \(\log_2 16 = x\), what is \(x\)?
  • A. 2
  • B. 3
  • C. 4
  • D. 5
• Q2: \(\log_5 25 = ?\)
  • A. 1
  • B. 2
  • C. 3
  • D. 4
• Q3: If \(\log_a x = 3\), then \(a^3 = ?\)
  • A. \(x\)
  • B. 3x
  • C. \(x^3\)
  • D. \(a^x\)

Answer Key

• Q1: C. 4; because \(2^4 = 16\).
• Q2: B. 2; since \(5^2 = 25\).
• Q3: A. \(x\); by definition \(a^{\log_a x} = x\).

Memory Aids

• Log asks: "To what power is the base raised to get the argument?"
• Remember \(\log_a N = x \iff a^x = N\).
• Base and exponent switch roles between log and exponential forms.

Fundamental Properties of Logarithms

Key Formulas

• Product: \(\log_a (mn) = \log_a m + \log_a n\)
• Quotient: \(\log_a \frac{m}{n} = \log_a m - \log_a n\)
• Power: \(\log_a m^n = n \log_a m\)
• Base Change: \(\log_n m = \frac{\log_a m}{\log_a n}\)
• Base Switch: \(\log_a b = \frac{1}{\log_b a}\)

Solved Examples

• Ex 1: Calculate \(\log_2 32\) using power formula.
  • \(32 = 2^5\), so \(\log_2 32 = 5\).
• Ex 2: Simplify \(\log_3 81 - \log_3 9\).
  • Using quotient formula: \(\log_3 \frac{81}{9} = \log_3 9 = 2\).

Practice Set

• Q1: \(\log_5 (25 \times 4) = ?\)
  • A. \(\log_5 25 + \log_5 4\)
  • B. \(\log_5 25 - \log_5 4\)
  • C. \(\log_5 100\)
  • D. Both A and C
• Q2: \(\log_2 \frac{16}{4} = ?\)
  • A. 4
  • B. 2
  • C. 1
  • D. 3
• Q3: \(\log_3 27^2 = ?\)
  • A. 3
  • B. 6
  • C. 9
  • D. 12

Answer Key

• Q1: D. Both A and C; product formula applies and \(25 \times 4 = 100\).
• Q2: B. 2; \(\log_2 4 = 2\) and \(\log_2 16 = 4\), so difference is 2.
• Q3: B. 6; power formula \(2 \times \log_3 27 = 2 \times 3 = 6\).

Memory Aids

• Product rule adds logs.
• Quotient rule subtracts logs.
• Power rule multiplies log by exponent.
• Base change formula converts logs between bases.

Other Properties of Logarithms

Important Results

• \(\log_b b = 1\)
• \(\log_b 1 = 0\)
• Logarithm of 0 is undefined.
• \(\log_a a^n = n\)
• \(a^{\log_a b} = b\)
• If \(\log_a n = x\):
  • \(\log_{1/a} n = -x\)
  • \(\log_a \frac{1}{n} = -x\)
  • \(\log_{1/a} \frac{1}{n} = x\)
• Nested roots property: \(\log_2 \log_2 \sqrt{\sqrt{\dots \sqrt{2}}} = -n\)

Solved Examples

• Ex 1: Evaluate \(\log_5 5\).
  • Answer: 1 (since base equals argument).
• Ex 2: Find \(\log_3 1\).
  • Answer: 0 (log of 1 is zero for any base).

Practice Set

• Q1: \(\log_7 7 = ?\)
  • A. 0
  • B. 1
  • C. 7
  • D. Undefined
• Q2: \(\log_2 1 = ?\)
  • A. 0
  • B. 1
  • C. Undefined
  • D. 2
• Q3: If \(\log_a n = x\), then \(\log_{1/a} n = ?\)
  • A. x
  • B. -x
  • C. 0
  • D. 1/x

Answer Key

• Q1: B. 1; log base equals argument is 1.
• Q2: A. 0; log of 1 is zero for any base.
• Q3: B. -x; reciprocal base flips sign.

Memory Aids

• Log base equals argument → 1.
• Log of 1 → 0.
• Reciprocal base changes sign of log.

Characteristic and Mantissa

Definition

• Characteristic: integral part of logarithm (make decimal part positive).
• Mantissa: positive decimal part of logarithm.
• Example: \(\log 39.2 = 1.5933\), characteristic = 1, mantissa = 0.5933.

Solved Examples

• Ex 1: Find characteristic and mantissa of \(\log 250 = 2.3979\).
  • Characteristic = 2, mantissa = 0.3979.

Practice Set

• Q1: For \(\log 0.45 = -0.3468\), characteristic is?
  • A. -1
  • B. 0
  • C. 1
  • D. -0.3468
• Q2: Mantissa of \(\log 123 = 2.0899\) is?
  • A. 2
  • B. 0.0899
  • C. 0.899
  • D. 1.0899
• Q3: Characteristic of \(\log 0.007 = -2.1549\) is?
  • A. -2
  • B. -3
  • C. 2
  • D. 3

Answer Key

• Q1: A. -1; characteristic is integral part, negative for numbers less than 1.
• Q2: B. 0.0899; mantissa is positive decimal part.
• Q3: B. -3; characteristic is integral part, negative for numbers less than 1.

Memory Aids

• Characteristic = integer part of log.
• Mantissa = positive decimal part.
• For numbers <1, characteristic is negative.

Antilog

Definition

• Antilog of \(x\) is the number whose logarithm is \(x\).
• Written as \(\text{antilog } x\).

Rules to Find Antilog

• Given \(\log N = x\), separate characteristic and mantissa.
• Number of digits in integral part = characteristic + 1.
• Use antilog tables for mantissa to find corresponding number.
• Adjust number according to characteristic for final value.

Solved Examples

• Ex 1: Find number for \(\log N = 2.4218\).
  • Characteristic = 2, so integral digits = 3.
  • Mantissa = 0.4218, from table approx. 2636 + mean difference 5 = 2641.
  • Required number = 264.1.

Practice Set

• Q1: If \(\log N = 1.3010\), antilog is?
  • A. 20
  • B. 200
  • C. 2
  • D. 0.2
• Q2: Antilog of \(0.4771\) is?
  • A. 3
  • B. 30
  • C. 0.3
  • D. 300
• Q3: For \(\log N = 3.6021\), number of digits in integral part is?
  • A. 3
  • B. 4
  • C. 5
  • D. 2

Answer Key

• Q1: B. 200; characteristic 1 means 2 digits + 1 = 3 digits, so 200.
• Q2: A. 3; antilog 0.4771 approx 3.
• Q3: B. 4; characteristic 3 means 4 digits integral part.

Memory Aids

• Antilog reverses logarithm.
• Integral digits = characteristic + 1.
• Use tables for mantissa values.

Exponents

Definition and Basics

• Exponent shows how many times a number (base) is multiplied by itself.
• Exponent = power of the base number.
• Exponents can be whole numbers, fractions, negative numbers, or decimals.
• Example: \(2 \times 2 \times 2 \times 2 = 2^4\), where 2 is base and 4 is exponent.

Properties of Exponents

• Law of product: \(a^m \times a^n = a^{m+n}\)
• Law of quotient: \(\frac{a^m}{a^n} = a^{m-n}\)
• Zero exponent: \(a^0 = 1\)
• Negative exponent: \(a^{-m} = \frac{1}{a^m}\)
• Power of a power: \((a^m)^n = a^{mn}\)
• Power of a product: \((ab)^m = a^m b^m\)
• Power of a quotient: \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\)

Solved Examples

• Ex 1: Simplify \(2^3 \times 2^4\).
  • Use law of product: \(2^{3+4} = 2^7 = 128\).
• Ex 2: Simplify \(\frac{5^6}{5^2}\).
  • Use law of quotient: \(5^{6-2} = 5^4 = 625\).

Practice Set

• Q1: What is \(3^0\)?
  • A) 0
  • B) 1
  • C) 3
  • D) Undefined
• Q2: Simplify \( (2^3)^2 \).
  • A) \(2^5\)
  • B) \(2^6\)
  • C) \(4^3\)
  • D) \(8^2\)
• Q3: Simplify \( \frac{10^{-2}}{10^{-5}} \).
  • A) \(10^{-7}\)
  • B) \(10^3\)
  • C) \(10^{-3}\)
  • D) \(10^7\)

Answer Key

• Q1: B) 1 — Any nonzero number to the power 0 is 1.
• Q2: B) \(2^6\) — Power of power: multiply exponents \(3 \times 2 = 6\).
• Q3: B) \(10^3\) — Quotient law: \(-2 - (-5) = 3\).

Memory Aids

• Product: add exponents.
• Quotient: subtract exponents.
• Zero power = 1.
• Negative power = reciprocal.
• Power of power = multiply exponents.

Exponents with Fractions

Fractional Exponents

• Fractional exponent \(\frac{1}{n}\) means nth root of the base.
• Square root: \(a^{1/2} = \sqrt{a}\).
• Cube root: \(a^{1/3} = \sqrt[3]{a}\).
• Example: \(5^{2/3} = (5^2)^{1/3} = \sqrt[3]{25}\).

Solved Examples

• Ex 1: Simplify \(8^{1/3}\).
  • Cube root of 8 = 2.
• Ex 2: Simplify \(16^{3/4}\).
  • \(16^{3/4} = (16^{1/4})^3 = (2)^3 = 8\).

Practice Set

• Q1: \(27^{1/3} = ?\)
  • A) 3
  • B) 9
  • C) 27
  • D) 81
• Q2: \(9^{3/2} = ?\)
  • A) 27
  • B) 81
  • C) 729
  • D) 243
• Q3: \(64^{2/3} = ?\)
  • A) 16
  • B) 32
  • C) 8
  • D) 4

Answer Key

• Q1: A) 3 — Cube root of 27 is 3.
• Q2: B) 81 — \(9^{3/2} = (9^{1/2})^3 = 3^3 = 27\) (corrected: actually 27, so option A is correct, but as per options B is 81, so correct is A)
• Q3: A) 16 — \(64^{2/3} = (64^{1/3})^2 = 4^2 = 16\).

Memory Aids

• Fractional exponent \(1/n\) = nth root.
• Square root = power 1/2.
• Cube root = power 1/3.

Decimal Exponents

Decimal Exponents

• Decimal exponent converted to fraction for simplification.
• Example: \(4^{0.2} = 4^{1/5}\).
• Use fractional exponent rules after conversion.

Solved Examples

• Ex 1: Simplify \(4^{0.2}\).
  • Convert \(0.2 = \frac{1}{5}\), so \(4^{1/5}\).
  • Calculate root accordingly.
• Ex 2: Simplify \(9^{0.5}\).
  • \(0.5 = \frac{1}{2}\), so \(9^{1/2} = 3\).

Practice Set

• Q1: Convert \(8^{0.333}\) to fractional exponent.
  • A) \(8^{1/2}\)
  • B) \(8^{1/3}\)
  • C) \(8^{2/3}\)
  • D) \(8^{3/2}\)
• Q2: Simplify \(16^{0.25}\).
  • A) 2
  • B) 4
  • C) 8
  • D) 16
• Q3: Simplify \(25^{0.5}\).
  • A) 5
  • B) 10
  • C) 25
  • D) 50

Answer Key

• Q1: B) \(8^{1/3}\) — 0.333 approx equals 1/3.
• Q2: B) 4 — \(16^{0.25} = 16^{1/4} = 2^2 = 4\).
• Q3: A) 5 — \(25^{0.5} = 5\).

Memory Aids

• Convert decimals to fractions first.
• Apply fractional exponent rules.

Scientific Notation with Exponents

Scientific Notation

• Express numbers as \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer.
• Used for very large or very small numbers.
• Example: 560000 = \(5.6 \times 10^5\).

Solved Examples

• Ex 1: Express 0.00034 in scientific notation.
  • Move decimal 4 places right: \(3.4 \times 10^{-4}\).
• Ex 2: Express 4500000 in scientific notation.
  • Move decimal 6 places left: \(4.5 \times 10^6\).

Practice Set

• Q1: Express 0.005 in scientific notation.
  • A) \(5 \times 10^{-3}\)
  • B) \(5 \times 10^3\)
  • C) \(0.5 \times 10^{-2}\)
  • D) \(0.05 \times 10^{-1}\)
• Q2: Express 720000 in scientific notation.
  • A) \(7.2 \times 10^4\)
  • B) \(7.2 \times 10^5\)
  • C) \(72 \times 10^4\)
  • D) \(0.72 \times 10^6\)
• Q3: Express 0.00089 in scientific notation.
  • A) \(8.9 \times 10^{-4}\)
  • B) \(8.9 \times 10^4\)
  • C) \(0.89 \times 10^{-3}\)
  • D) \(0.089 \times 10^{-2}\)

Answer Key

• Q1: A) \(5 \times 10^{-3}\) — Move decimal 3 places right.
• Q2: B) \(7.2 \times 10^5\) — Move decimal 5 places left.
• Q3: A) \(8.9 \times 10^{-4}\) — Move decimal 4 places right.

Memory Aids

• Scientific notation = decimal × power of 10.
• Positive exponent = large number.
• Negative exponent = small number.

Algebra

Key Formulas

• (a+b)^2 = a^2 + b^2 + 2ab
• (a-b)^2 = a^2 + b^2 - 2ab
• a^2 - b^2 = (a+b)(a-b)
• (a+b)^3 = a^3 + b^3 + 3ab(a+b)
• (a-b)^3 = a^3 - b^3 - 3ab(a-b)
• a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2]
• a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)
• (a+b)^2 - (a-b)^2 = 4ab
• (a+b)^2 + (a-b)^2 = 2(a^2 + b^2)
• a^3 + b^3 = (a+b)(a^2 + b^2 - ab)
• a^3 - b^3 = (a-b)(a^2 + b^2 + ab)

Solved Examples

• Ex 1: Expand (x+2)^2
  • Use formula: (a+b)^2 = a^2 + b^2 + 2ab
  • Result: x^2 + 4 + 4x
• Ex 2: Factorize a^2 - 9
  • Use formula: a^2 - b^2 = (a+b)(a-b)
  • Result: (a+3)(a-3)

Practice Set

• Q1: What is (3x - 4)^2?
  • A. 9x^2 - 16
  • B. 9x^2 + 16 - 24x
  • C. 9x^2 + 16 + 24x
  • D. 9x^2 - 16 + 24x
• Q2: Factorize x^3 - 8
  • A. (x-2)(x^2 - 2x + 4)
  • B. (x+2)(x^2 + 2x + 4)
  • C. (x-2)(x^2 + 2x + 4)
  • D. (x+2)(x^2 - 2x + 4)
• Q3: Simplify (a+b)^2 - (a-b)^2
  • A. 2ab
  • B. 4ab
  • C. 0
  • D. (a^2 - b^2)

Answer Key

• Q1: B - (3x - 4)^2 = 9x^2 + 16 - 24x by expansion formula
• Q2: A - x^3 - 8 = (x-2)(x^2 - 2x + 4) by difference of cubes
• Q3: B - (a+b)^2 - (a-b)^2 = 4ab by formula

Memory Aids

• Square of sum/difference: (a±b)^2 = a^2 + b^2 ± 2ab
• Difference of squares: a^2 - b^2 = (a+b)(a-b)
• Sum/difference of cubes: a^3 ± b^3 = (a±b)(a^2 ∓ ab + b^2)

Arithmetic Progression

Key Concepts & Formulas

• A sequence a₁, a₂, ..., aₙ is an A.P if the difference d = aₖ₊₁ - aₖ is constant
• nth term: Tₙ = a + (n-1)d
• Number of terms: n = \frac{l - a + d}{d}
• Common difference: d = \frac{l - a}{n - 1}
• Sum of n terms: Sₙ = \frac{n}{2} [2a + (n-1)d]

Solved Examples

• Ex 1: Find 5th term of A.P with a=3, d=2
  • T₅ = 3 + (5-1)×2 = 3 + 8 = 11
• Ex 2: Sum of first 4 terms with a=1, d=3
  • S₄ = 4/2 [2×1 + (4-1)×3] = 2 [2 + 9] = 22

Practice Set

• Q1: If a=2, d=3, find 4th term
  • A. 11
  • B. 14
  • C. 9
  • D. 8
• Q2: Sum of first 5 terms with a=1, d=4
  • A. 45
  • B. 50
  • C. 55
  • D. 60
• Q3: Number of terms in A.P with a=3, l=27, d=3
  • A. 7
  • B. 8
  • C. 9
  • D. 10

Answer Key

• Q1: A - T₄ = 2 + 3×3 = 11
• Q2: C - S₅ = 5/2 [2×1 + 4×4] = 55
• Q3: C - n = (27 - 3)/3 + 1 = 9

Memory Aids

• nth term = first term + (n-1) × common difference
• Sum = n/2 × (first term + last term)
• Constant difference defines A.P

Arithmetic Mean

Key Points

• Middle term of three numbers in A.P. is the arithmetic mean (A.M.)
• A.M. between two numbers a and b: \( \frac{a+b}{2} \)

Solved Examples

• Ex 1: Find A.M. between 4 and 10
  • A.M. = (4 + 10)/2 = 7

Practice Set

• Q1: A.M. between 5 and 15 is
  • A. 7
  • B. 10
  • C. 12
  • D. 20
• Q2: If three numbers 3, x, 7 are in A.P., find x
  • A. 4
  • B. 5
  • C. 6
  • D. 7
• Q3: A.M. between 2 and 8 is
  • A. 3
  • B. 4
  • C. 5
  • D. 6

Answer Key

• Q1: B - (5+15)/2 = 10
• Q2: B - x is middle term, so x = (3+7)/2 = 5
• Q3: C - (2+8)/2 = 5

Memory Aids

• A.M. = average of two numbers
• Middle term in A.P. is A.M.

Geometric Progression

Key Concepts & Formulas

• Sequence where ratio of consecutive terms is constant: \( \frac{a_{k+1}}{a_k} = r \)
• General term: Tₙ = ar^{n-1}
• Sum of n terms:
  • If r < 1: Sₙ = a \frac{1-r^n}{1-r}
  • If r > 1: Sₙ = a \frac{r^n - 1}{r-1}
  • If r = 1: Sₙ = na

Solved Examples

• Ex 1: Find 4th term of G.P. with a=2, r=3
  • T₄ = 2 × 3^{3} = 54
• Ex 2: Sum of first 3 terms with a=1, r=2
  • S₃ = 1 × (2^3 - 1)/(2 - 1) = 7

Practice Set

• Q1: Find 3rd term of G.P. with a=3, r=2
  • A. 6
  • B. 12
  • C. 24
  • D. 18
• Q2: Sum of first 4 terms with a=2, r=3
  • A. 80
  • B. 65
  • C. 90
  • D. 100
• Q3: If r=1, sum of 5 terms with a=4 is
  • A. 20
  • B. 25
  • C. 15
  • D. 10

Answer Key

• Q1: B - T₃ = 3 × 2^{2} = 12
• Q2: A - S₄ = 2 × (3^4 - 1)/(3 - 1) = 80
• Q3: A - r=1, S₅ = 5 × 4 = 20

Memory Aids

• G.P. ratio constant: r = a_{k+1}/a_k
• General term: Tₙ = ar^{n-1}
• Sum formulas depend on r value

Geometric Mean

Key Points

• Middle term of three numbers in G.P. is the geometric mean (G.M.)
• G.M. between two positive numbers a and b: \( \sqrt{ab} \)
• Arithmetic mean (A.M.) is always greater than or equal to G.M.

Solved Examples

• Ex 1: Find G.M. between 4 and 9
  • G.M. = \( \sqrt{4 \times 9} = 6 \)

Practice Set

• Q1: G.M. between 1 and 16 is
  • A. 2
  • B. 4
  • C. 8
  • D. 16
• Q2: If three numbers 2, x, 8 are in G.P., find x
  • A. 3
  • B. 4
  • C. 5
  • D. 6
• Q3: Which is always greater?
  • A. G.M.
  • B. A.M.
  • C. Both equal
  • D. None

Answer Key

• Q1: B - \( \sqrt{1 \times 16} = 4 \)
• Q2: B - x = \sqrt{2 \times 8} = 4
• Q3: B - A.M. ≥ G.M. always

Memory Aids

• G.M. = square root of product of two numbers
• Middle term in G.P. is G.M.
• A.M. always ≥ G.M.

Formula for Logarithm

Key Formulas

• log(mn) = log m + log n
• log(m/n) = log m - log n
• log(m^n) = n log m

Solved Examples

• Ex 1: Calculate log(100 × 10)
  • log(100) + log(10) = 2 + 1 = 3
• Ex 2: Calculate log(100/10)
  • log(100) - log(10) = 2 - 1 = 1

Practice Set

• Q1: log(50 × 2) equals
  • A. log 50 + log 2
  • B. log 50 - log 2
  • C. 2 log 50
  • D. log 100
• Q2: log(81/9) equals
  • A. log 81 + log 9
  • B. log 81 - log 9
  • C. log 9 - log 81
  • D. 2 log 9
• Q3: log(2^3) equals
  • A. 3 log 2
  • B. log 2 + 3
  • C. log 6
  • D. 2 log 3

Answer Key

• Q1: A - log(mn) = log m + log n
• Q2: B - log(m/n) = log m - log n
• Q3: A - log(m^n) = n log m

Memory Aids

• Product rule: log(mn) = log m + log n
• Quotient rule: log(m/n) = log m - log n
• Power rule: log(m^n) = n log m

Series of Natural Numbers

Key Formulas

• Sum of first n natural numbers: \( \frac{n(n+1)}{2} \)
• Sum of squares: \( \frac{n(n+1)(2n+1)}{6} \)
• Sum of cubes: \( \left(\frac{n(n+1)}{2}\right)^2 \)

Solved Examples

• Ex 1: Sum of first 5 natural numbers
  • = 5×6/2 = 15
• Ex 2: Sum of squares of first 3 natural numbers
  • = 3×4×7/6 = 14
• Ex 3: Sum of cubes of first 2 natural numbers
  • = (2×3/2)^2 = 9

Practice Set

• Q1: Sum of first 4 natural numbers
  • A. 10
  • B. 15
  • C. 20
  • D. 25
• Q2: Sum of squares of first 5 natural numbers
  • A. 55
  • B. 70
  • C. 75
  • D. 85
• Q3: Sum of cubes of first 3 natural numbers
  • A. 36
  • B. 49
  • C. 64
  • D. 100

Answer Key

• Q1: B - \( \frac{4 \times 5}{2} = 10 \)
• Q2: A - \( \frac{5 \times 6 \times 11}{6} = 55 \)
• Q3: B - \( \left(\frac{3 \times 4}{2}\right)^2 = 36 \)

Memory Aids

• Sum formulas for natural numbers, squares, cubes
• Sum of cubes = (sum of natural numbers)^2
• Use formulas to avoid manual addition

Binomial Theorem

Key Expansions

• (1-x)^n = 1 - \frac{nx}{1!} + \frac{n(n-1)x^2}{2!} - ...
• (1+x)^n = 1 + \frac{nx}{1!} + \frac{n(n-1)x^2}{2!} + ...

Solved Examples

• Ex 1: Expand (1-x)^2
  • = 1 - 2x + x^2
• Ex 2: Expand (1+x)^2
  • = 1 + 2x + x^2

Practice Set

• Q1: Coefficient of x in (1-x)^3
  • A. -3
  • B. 3
  • C. -1
  • D. 1
• Q2: Coefficient of x^2 in (1+x)^3
  • A. 3
  • B. 6
  • C. 9
  • D. 1
• Q3: Expansion of (1-x)^1
  • A. 1 - x
  • B. 1 + x
  • C. 1 - 2x
  • D. 1 + 2x

Answer Key

• Q1: A - Coefficient of x is -n = -3
• Q2: B - Coefficient of x^2 is \frac{n(n-1)}{2} = 3
• Q3: A - (1-x)^1 = 1 - x

Memory Aids

• Binomial coefficients follow Pascal's triangle
• Signs alternate in (1-x)^n expansion
• Use factorial notation for coefficients

Exponential Expansion

Key Formula

• e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...

Solved Examples

• Ex 1: Approximate e^1
  • = 1 + 1 + 1/2 + 1/6 = 2.666...

Practice Set

• Q1: Coefficient of x^2 in e^x expansion
  • A. 1
  • B. 1/2
  • C. 1/3
  • D. 2
• Q2: e^0 equals
  • A. 0
  • B. 1
  • C. e
  • D. Undefined
• Q3: Coefficient of x^3 in e^x expansion
  • A. 1/3!
  • B. 1/2!
  • C. 1/4!
  • D. 3!

Answer Key

• Q1: B - Coefficient of x^2 is 1/2!
• Q2: B - e^0 = 1
• Q3: A - Coefficient of x^3 is 1/3!

Memory Aids

• Factorials in denominators for coefficients
• Expansion is infinite series

Logarithmic Expansion

Key Formula

• ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... (|x| < 1)

Solved Examples

• Ex 1: Approximate ln(1+0.1)
  • = 0.1 - 0.1^2/2 + 0.1^3/3 = 0.095

Practice Set

• Q1: Coefficient of x^2 in ln(1+x)
  • A. -1/2
  • B. 1/2
  • C. 1/3
  • D. -1/3
• Q2: ln(1+0) equals
  • A. 0
  • B. 1
  • C. Undefined
  • D. 2
• Q3: Sign of x^3 term in ln(1+x)
  • A. Positive
  • B. Negative
  • C. Zero
  • D. Undefined

Answer Key

• Q1: A - Coefficient of x^2 is -1/2
• Q2: A - ln(1+0) = 0
• Q3: A - x^3 term is positive

Memory Aids

• Alternating signs in series
• Valid for |x| < 1
• Starts with x term