Year 2 — Term 3
14 study notes · 45 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
Velocity and acceleration in circular motion
The Natural Exponential Function (e^x): The natural exponential function, denoted as e^x, is a mathematical function where 'e' is Euler's number (an irrational constant approximately 2.71828) and 'x' is the exponent.
Motion in a horizontal circle
Natural Logarithmic Function (ln x): The natural logarithmic function, denoted as ln x, is the inverse function of the natural exponential function e^x.
Motion in a vertical circle
Motion of a particle of mass m in a vertical circle of radius a involves non-uniform speed due to gravity.
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Create Free AccountSimple harmonic motion
Simple Harmonic Motion (SHM) is rectilinear oscillatory motion where the acceleration ẍ of a particle is directly proportional to its displacement x from a fixed equilibrium centre O and always directed towards O: ẍ = -ω² x (where ω is a positive constant).
Simple harmonic motion on a horizontal line
Horizontal Simple Harmonic Motion arises when a particle of mass m moves on a smooth horizontal table attached to the ends of elastic strings or springs governed by Hooke's Law: T = λx / l (where λ is modulus of elasticity, l is natural length, and x is extension).
Simple harmonic motion on a vertical line
Vertical Simple Harmonic Motion (SHM): Vertical Simple Harmonic Motion occurs when a particle oscillates along a vertical line under the influence of a restoring force that is directly proportional to its displacement from a fixed equilibrium position and always directed towards that position.
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Create Free AccountEvents of a random experiment
A random experiment is an observational procedure whose outcome cannot be predicted with certainty, but whose set of all possible outcomes—the sample space S—is known beforehand.
Probability models for random events
Classical probability models define the likelihood of an event A in an equiprobable sample space S as P(A) = n(A) / n(S), satisfying Kolmogorov's axioms: (1) 0 ≤ P(A) ≤ 1 for all A; (2) P(S) = 1 and P(∅) = 0; (3) Complement Rule: P(A') = 1 - P(A); (4) The Addition Theorem: For any two events A and B: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Conditional probability
Conditional probability measures the probability of an event A occurring given that another event B has already occurred with P(B) > 0, defined as: P(A | B) = P(A ∩ B) / P(B).
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Bayes' Theorem provides a mathematical method for computing posterior probabilities (reversing cause and effect) given observed evidence.
The nature of statistics
Statistics is the mathematical discipline concerned with collecting, organizing, presenting, analyzing, and interpreting numerical data to draw valid conclusions.
Measures of central tendency
Measures of central tendency summarize a data distribution by identifying a single representative central location: (1) Mean (Arithmetic Average): x̄ = (∑ f_i x_i) / N, or using an assumed mean A with deviation d_i = x_i - A: x̄ = A + (∑ f_i d_i) / N (or with step-deviation u_i = d_i / c: x̄ = A + c (∑ f_i u_i) / N); (2) Median: The middle observation dividing sorted data into two equal halves.
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Create Free AccountMeasures of relative position
Measures of relative position (partition values) divide a sorted frequency distribution into specified fractional proportions: (1) Quartiles divide data into four equal quarters: Lower Quartile Q₁ (25th percentile, position N/4), Median Q₂ (50th percentile, position N/2), and Upper Quartile Q₃ (75th percentile, position 3N/4).
Measures of dispersion
Measures of dispersion quantify the extent of spread, scatter, or variability of data values around their central mean: (1) Range: Difference between maximum and minimum values (X_max - X_min); (2) Mean Absolute Deviation: (∑ f_i |x_i - x̄|) / N; (3) Variance: Mean squared deviation from arithmetic mean: σ² = [∑ f_i (x_i - x̄)²] / N = [ (∑ f_i x_i²) / N ] - x̄²; (4) Standard Deviation: Positive square root of variance: σ = √Variance; (5) Coefficient of Variation (relative dispersion for comparing datasets with different units or means): CV = (σ / x̄) × 100%.