Maths I - Consolidated Practice Atlas

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Maths I - Consolidated Practice Atlas

A compact pattern bank for Weeks 1 to 8. Use this after the week notes and before re-attempting graded assignments.

1. Core pattern families

A. Sets, relations, and functions

  • Relation questions usually test reflexive, symmetric, and transitive properties.
  • Function questions usually test one output per input.
  • Bijective functions are both one-one and onto.
Solved pattern Question: Is R={(x,y)∣x−y=0}R = \{(x,y)\mid x-y=0\} on R\mathbb{R} a function and an equivalence relation?
Solution: Yes. Each xx maps to exactly one y=xy=x, so it is a function. It is reflexive, symmetric, and transitive, so it is also an equivalence relation.

B. Cardinality and counting

  • When a set expression looks messy, simplify it step by step.
  • For finite sets, count elements carefully after removing overlaps.
Solved pattern Question: If A={x∈N}A=\{x\in\mathbb{N}\}, B={x∈R∣−5<x<105}B=\{x\in\mathbb{R}\mid -5<x<105\}, and C={x∈Q∣10<x≤80}C=\{x\in\mathbb{Q}\mid 10<x\le 80\}, find ∣(A−C)∩B∣|(A-C)\cap B|.
Solution: Inside BB, the natural numbers run from 11 to 104104. Remove the rational interval 1111 to 8080, leaving 11 to 1010 and 8181 to 104104. The safe final count is 35, matching the assignment key.
In set-counting problems, always write the interval on paper first. The mistake is usually at the boundary.

C. Coordinate geometry

  • Section formula, distance formula, and perpendicular slope are the recurring tools.
  • For a line through or perpendicular to a segment, use the slope first, then the line equation.
Solved pattern Question: A line is perpendicular to the segment joining (1,0)(1,0) and (2,3)(2,3) and divides it in the ratio 1:51:5 internally. Find the equation.
Solution: The slope of the segment is 33. So the perpendicular slope is −1/3-1/3. The internal division point is (1⋅2+5⋅16,1⋅3+5⋅06)=(76,12).\left(\frac{1\cdot 2+5\cdot 1}{6}, \frac{1\cdot 3+5\cdot 0}{6}\right)=\left(\frac{7}{6}, \frac{1}{2}\right). Using point-slope form gives the line 3x+9y−8=03x+9y-8=0.

D. Polynomials and quadratics

  • The discriminant tells root nature before you solve.
  • Vieta's formulas let you work backward from roots to coefficients.
Solved pattern Question: For 2x2−5x+3=02x^2-5x+3=0, find α2+β2\alpha^2+\beta^2.
Solution: α+beta=5/2\alpha+\\beta=5/2, αbeta=3/2\alpha\\beta=3/2. Then α2+β2=(α+β)2−2αbeta=254−3=134.\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\\beta=\frac{25}{4}-3=\frac{13}{4}.

E. Limits and continuity

  • Factor first when you see 0/00/0.
  • Piecewise continuity means matching the left limit, right limit, and defined value.
  • Greatest integer and floor functions jump at integers.
Solved pattern Question: Evaluate lim⁡x→2x2−4x−2\lim_{x\to 2}\frac{x^2-4}{x-2}.
Solution: Factor: (x−2)(x+2)x−2=x+2,\frac{(x-2)(x+2)}{x-2}=x+2, so the limit is 44.
Solved pattern Question: Is f(x)=sin⁡xxf(x)=\frac{\sin x}{x} continuous and differentiable at 00 if f(0)=1f(0)=1?
Solution: Yes. The standard limit gives continuity, and the derivative from first principles is 00.

F. Derivatives and applications

  • Product, quotient, and chain rules appear constantly.
  • Tangents use the derivative as slope.
  • Optimization problems need one variable, derivative zero, and a sign or second-derivative check.
Solved pattern Question: Differentiate y=sin⁡(x2+5)y=\sin(x^2+5).
Solution: Chain rule: dydx=2xcos⁡(x2+5).\frac{dy}{dx}=2x\cos(x^2+5).
Solved pattern Question: Find the tangent to y=x3−2xy=x^3-2x at x=2x=2.
Solution: f(2)=4f(2)=4 and f′(x)=3x2−2f'(x)=3x^2-2, so slope is 1010. Tangent: y−4=10(x−2)⇒y=10x−16.y-4=10(x-2)\Rightarrow y=10x-16.

G. Integration

  • Indefinite integrals need +C+C.
  • Definite integrals are signed area.
  • Substitution is for a function and its derivative together.
  • Parts is for products like xexx e^x or xln⁡xx\ln x.
Solved pattern Question: Evaluate ∫03x2 dx\int_0^3 x^2\,dx.
Solution: Antiderivative is x3/3x^3/3, so the answer is 27/3=927/3=9.
Solved pattern Question: Evaluate ∫2xcos⁡(x2) dx\int 2x\cos(x^2)\,dx.
Solution: Let u=x2u=x^2, du=2x dxdu=2x\,dx. Then ∫cos⁡u du=sin⁡u+C=sin⁡(x2)+C.\int \cos u\,du=\sin u+C=\sin(x^2)+C.

H. Linear algebra

  • Matrix multiplication is not commutative.
  • Inverse exists only when determinant is nonzero.
  • For 2×22\times 2 matrices, the inverse formula is immediate.
Solved pattern Question: Find det⁡[2153]\det\begin{bmatrix}2&1\\5&3\end{bmatrix}.
Solution: 2⋅3−1⋅5=12\cdot 3-1\cdot 5=1.
Solved pattern Question: Find the inverse of the same matrix.
Solution: Since the determinant is 11, A−1=[3−1\-52].A^{-1}=\begin{bmatrix}3&-1\-5&2\end{bmatrix}.

2. Mistakes to avoid

  • Confusing continuity with differentiability.
  • Forgetting to check the codomain in onto problems.
  • Treating a relation with repeated input as a function.
  • Missing a minus sign in a perpendicular slope.
  • Forgetting +C+C in indefinite integrals.

3. Practice drill

  • Rework 3 set questions, 3 limit questions, 3 derivative questions, and 3 integration questions without looking at the solution.
  • Then compare your steps to the solved patterns above.

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