Year 1 — Term 3
16 study notes · 223 practice questions · Aligned with NIE syllabus
Counted in the Idasara Academy app, which you open with a free account. The free notes, revision, practice and library pages for each lesson are linked below.
Lessons in This Term
Equation of a straight line
Evolution of the Number System: The historical development of numerical concepts and symbols, starting from simple counting to complex number systems.
Line through the intersection of two lines
Understanding Functions: A function is a special type of relation where each element in the domain (input set) is mapped to exactly one element in the codomain (output set).
Position of two points relative to a line
Angular Measure (Degrees and Radians): Angular measure quantifies the amount of rotation or the extent of an angle, using units such as degrees and radians.
Want access to all Combined Mathematics content?
Create Free AccountCartesian coordinates and the straight line
Angle Between Two Straight Lines: When two non-parallel straight lines intersect, they form two pairs of vertically opposite angles.
Perpendicular distance from a point to a line
Trigonometric Ratios in the Cartesian Plane: Trigonometric ratios (sine, cosine, tangent) are defined for any angle using a point P(x,y) on the terminal arm of the angle, with 'r' being the distance from the origin to P.
Cartesian equation of a circle
A circle is the locus of a point P(x, y) moving in a plane at a constant distance r (radius) from a fixed point C(a, b) (centre).
Want access to all Combined Mathematics content?
Create Free AccountPosition of a straight line relative to a circle
The relative geometric position of a straight line ax + by + k = 0 with respect to a circle (x - a)² + (y - b)² = r² is determined either by the perpendicular distance d from the centre to the line or by substituting the linear equation into the circle equation to form a quadratic.
Tangents to a circle from an external point
From an external point P(x₁, y₁) lying outside a circle S ≡ x² + y² + 2gx + 2fy + c = 0, exactly two real tangents can be drawn to the circle.
Circle through the intersection of two circles
The equation of any circle passing through the points of intersection of a given circle S ≡ x² + y² + 2gx + 2fy + c = 0 and a straight line U ≡ ax + by + k = 0 is represented by the family S + λU = 0, where λ is a real parameter.
Want access to all Combined Mathematics content?
Create Free AccountPosition of two circles
The relative geometric position of two circles with centres C₁, C₂ and radii r₁, r₂ is governed by comparing the distance between their centres d = C₁C₂ against the sum (r₁ + r₂) and difference |r₁ - r₂| of their radii.
Condition for two circles to intersect orthogonally
Two circles are said to intersect orthogonally if the tangents at either point of intersection are mutually perpendicular.
Mathematical induction
The Principle of Mathematical Induction is a rigorous deductive proof method for statements P(n) asserted for all positive integers n ∈ ℤ⁺.
Want access to all Combined Mathematics content?
Create Free AccountFinite series
A finite series is the sum of terms of a sequence, denoted using sigma notation as S_n = ∑_{r=1}^n u_r = u₁ + u₂ + ...
Sums of elementary series
Evaluating polynomial series requires standard summation identities for powers of natural numbers: (1) Sum of the first n natural numbers: ∑_{r=1}^n r = n(n + 1)/2; (2) Sum of squares: ∑_{r=1}^n r² = n(n + 1)(2n + 1)/6; (3) Sum of cubes: ∑_{r=1}^n r³ = [n(n + 1)/2]² = (∑ r)².
Summation of series
The Method of Differences (telescoping series) is a powerful algebraic technique used to sum series where standard power formulas do not apply.
Want access to all Combined Mathematics content?
Create Free AccountConvergence and divergence of series
An infinite series ∑_{r=1}^∞ u_r is investigated by examining the sequence of its partial sums {S_n}, where S_n = ∑_{r=1}^n u_r.