Mathematics: Algebra and Calculus

Mathematics: Algebra and Calculus

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Algebra in NAT-I centres on solving linear and quadratic equations, manipulating polynomials, applying the binomial theorem, working with matrices/determinants, and analysing sequences (AP/GP). The must-know formula is the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a, where the discriminant Δ = b² − 4ac tells you whether roots are real and distinct, real and equal, or complex.

Calculus is built on three pillars: limits, differentiation, and integration. Memorise the standard results d/dx (xⁿ) = n·xⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, and their integration counterparts ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, ∫ (1/x) dx = ln|x| + C. Exam pointers: (1) Roots of ax² + bx + c = 0 satisfy Vieta’s relations α + β = −b/a, αβ = c/a. (2) AP nth term: aₙ = a + (n−1)d; GP nth term: aₙ = a·rⁿ⁻¹. (3) Use L’Hôpital’s rule for 0/0 or ∞/∞ limits by differentiating top and bottom separately.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Equations and Polynomials

A quadratic equation ax² + bx + c = 0 is solved by factorisation, completing the square (rewrite as a(x + b/2a)² = (b² − 4ac)/4a), or the quadratic formula. The discriminant Δ = b² − 4ac governs root nature: Δ > 0 (two real distinct roots), Δ = 0 (repeated root), Δ < 0 (complex conjugate pair). Vieta’s formulas let you build a quadratic from known roots.

Sequences and Series

An arithmetic progression (AP) has constant difference d; the nth term is aₙ = a + (n−1)d and the sum of n terms is Sₙ = n/2 · [2a + (n−1)d]. A geometric progression (GP) has constant ratio r; nth term aₙ = a·rⁿ⁻¹, and sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Watch for the r = 1 trap where the GP sum degenerates to Sₙ = na.

Functions

A function f: A → B assigns exactly one output to each input. Recognise linear (f(x) = mx + c), quadratic (parabola), polynomial, trigonometric (sin, cos, tan), exponential (eˣ, aˣ), and logarithmic (ln x, log x) types. The domain is the set of permissible x-values; the range is the resulting y-values. Composition f(g(x)) feeds one function into another; the inverse f⁻¹ reverses the mapping (defined only when f is one-to-one).

Matrices and Determinants

A matrix is a rectangular array; two matrices add element-wise and multiply row-by-column when inner dimensions match. For a 2×2 matrix [[a,b],[c,d]], the determinant is |A| = ad − bc. Cramer’s rule solves a 2×2 linear system using x = |Aₓ|/|A|, y = |Aᵧ|/|A|, provided |A| ≠ 0.

Limits and Continuity

lim(x→a) f(x) evaluates the value f approaches as x nears a. Standard results include lim(x→0) (sin x)/x = 1 and lim(x→0) (1 − cos x)/x = 0. For indeterminate forms 0/0 or ∞/∞, L’Hôpital’s rule permits lim f(x)/g(x) = lim f′(x)/g′(x) when conditions hold.

Differentiation

The derivative f′(x) is the instantaneous rate of change, geometrically the slope of the tangent. Master these rules: product (fg)′ = f′g + fg′, quotient (f/g)′ = (f′g − fg′)/g², and chain (f(g(x)))′ = f′(g(x)) · g′(x). Apply derivatives to find maxima/minima by setting f′(x) = 0 and using the second-derivative test.

Integration

Indefinite integration recovers a family of antiderivatives: ∫ f(x) dx = F(x) + C. Definite integration ∫ₐᵇ f(x) dx = F(b) − F(a) computes net signed area under the curve. Standard forms include ∫ sin x dx = −cos x + C and ∫ cos x dx = sin x + C — sign slips here are the most common trap.

Key Facts Table

Topic	Essential Formula	Typical NAT-I MCQ
Quadratic	x = (−b ± √Δ)/2a	Find roots / discriminant
AP sum	Sₙ = n/2 [2a + (n−1)d]	Sum of first n terms
GP sum	Sₙ = a(1−rⁿ)/(1−r)	Sum when r ≠ 1
Limit	lim (sin x)/x = 1	Standard trigonometric limit
Derivative	d/dx (xⁿ) = n xⁿ⁻¹	Differentiate polynomials
Integral	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C	Area under curve

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Common Mistakes

Sign errors plague completing the square: always keep the coefficient of x² equal to 1 before halving the linear coefficient. In GP problems, when r = −1 the sum Sₙ oscillates between a and 0 — the formula a(1 − rⁿ)/(1 − r) still applies because 1 − r = 2 ≠ 0, but the closed form a(1−(−1)ⁿ)/2 clarifies the alternation. With chain rule, students frequently differentiate only the outer function and forget g′(x); a habit of writing the derivative as f′(u) · du/dx prevents this.

For limits, do not split fractions or cancel factors inside an indeterminate form before resolving the 0/0 status. L’Hôpital’s rule applies only when the original limit is of the form 0/0 or ∞/∞ and both derivatives exist near the point. A common trap presents lim(x→0) (sin 3x)/(tan 2x); the correct answer is 3/2 (not 1), found by multiplying and dividing by 3x and 2x separately.

In integration, dropping + C is half-marks lost on indefinite integrals. Sign confusion between ∫ sin x dx = −cos x + C and ∫ cos x dx = sin x + C is the most-tested trap in the calculus block.

Connections to Adjacent Topics

Quadratics feed directly into conic sections (parabola opens from the squared term) and optimisation problems solved via f′(x) = 0. Matrices connect to linear transformations and systems of equations — eigenvalues of 2×2 matrices satisfy λ² − tr(A)λ + |A| = 0. The Fundamental Theorem of Calculus links the two halves of calculus: differentiation and integration are inverse operations.

Worked Micro-Example

Find the area under y = x² from x = 0 to x = 3.

1. Set up: A = ∫₀³ x² dx.
2. Integrate: = [x³/3]₀³.
3. Evaluate: = 27/3 − 0 = 9 square units.

Practice Prompts

1. If α and β are roots of 2x² − 5x + 3 = 0, find α² + β² without solving for α, β individually. (Hint: use (α + β)² − 2αβ.)
2. Evaluate lim(x→0) (eˣ − 1)/x. (Apply L’Hôpital or the standard series expansion.)

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