The Ultimate Strategist’s Arsenal: Mathematics I (Weeks 1-4) - The Complete Edition

This guide covers every identified question pattern from your assignments, leaving no stone unturned. Each pattern is broken down with the TAA Framework (Triage, Abstract, Act) to make the logic explicit and easy to follow.

Week 1: Sets, Relations, & Functions

Pattern 1.1: Irrational Number Simplification

Triage: “Is the expression made of radicals ( $\dots$ ), and does the question ask if it’s rational or irrational?”
Abstract: The goal is to simplify completely. An expression is irrational if a $n$ (where n is not a perfect square) remains after all possible operations.
Act (Execution):

Problem: Is $(8 + 2) (12 - 3)$ rational?
1. Simplify Radicals: $8 = 22$ and $12 = 23$ .
2. Substitute: $(22 + 2) (23 - 3)$ .
3. Combine: $(32) (3) = 36$ .
4. Conclusion: $6$ is irrational, so the expression is irrational.

Pattern 1.2: Function Domain Restrictions

Triage: “Does a function have a fraction with a variable in the denominator OR a square root?”
Abstract: Find all inputs that are “illegal.” The two illegal acts are: 1. Division by Zero. 2. Square Root of a Negative.
Act (Execution):

Problem: Find the cardinality of integers not in the domain of $f (x) = \frac{x ^{2} - 16}{x + 4}$ .
1. Rule 1 (Denominator $\neq = 0$ ): $x + 4 \neq = 0 ⟹ x \neq = - 4$ . (Excludes: -4)
2. Rule 2 ( $\dots \geq 0$ ): $x^{2} - 16 \geq 0 ⟹ x^{2} \geq 16$ . This is true for $x \geq 4$ or $x \leq - 4$ .
3. Find Violators: The integers that violate Rule 2 are those between -4 and 4: ${- 3, - 2, - 1, 0, 1, 2, 3}$ .
4. Combine Exclusions: ${- 4} \cup {- 3, - 2, - 1, 0, 1, 2, 3}$ . Final Answer: Cardinality is 8.

Pattern 1.3: Set Operations with Defined Relations

Triage: “Are sets or relations defined by a mathematical rule (e.g., $y = 2 x$ , $y = x^{2}$ ) and the question asks for the cardinality of a set operation?”
Abstract: First, explicitly list the elements of each relation. Then, perform the required set operation (union, intersection, difference) on these lists of elements.
Act (Execution):

Problem: $S = {n \in N, n \leq 33}$ . $R_{1} = {(x, y) ∣ y = 3 x}$ , $R_{2} = {(x, y) ∣ y = x^{2}}$ . Find $∣ R_{1} ∖ R_{2} ∣$ .
1. List Elements of $R_{1}$ :
  - $R_{1} = {(1, 3), (2, 6), (3, 9), ..., (11, 33)}$ .
  - $∣ R_{1} ∣ = 11$ .
2. List Elements of $R_{2}$ :
  - $R_{2} = {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}$ .
  - $∣ R_{2} ∣ = 5$ .
3. Find the Intersection ( $R_{1} \cap R_{2}$ ): Find pairs that are in both lists. The only common element is $(3, 9)$ .
4. Apply Set Difference:
  - Formula: $∣ R_{1} ∖ R_{2} ∣ = ∣ R_{1} ∣ - ∣ R_{1} \cap R_{2} ∣$ .
  - $∣ R_{1} ∖ R_{2} ∣ = 11 - 1 = 10$ . Final Answer: 10

Week 2: Coordinate Geometry

Pattern 2.1: Line Intersection (Collision)

Triage: “Does it ask where two lines ‘intersect’, ‘meet’, or ‘collide’?”
Abstract: The intersection is the $(x, y)$ point that satisfies both equations. The tool is solving a system of equations.
Act (Execution):

Problem: Path 1 is $y = 3 x + 3$ . Path 2 is $y = 2 x$ . Find the collision point.
1. Set Equations Equal: $3 x + 3 = 2 x$ .
2. Solve: $x = - 3$ .
3. Find y: $y = 2 (- 3) = - 6$ . Final Answer: $(- 3, - 6)$ .

Pattern 2.2: Geometric Property Formulas

Triage: “Does it involve shapes (triangles, parallelograms), ‘midpoints’, or a point that ‘divides a line in a ratio’?”
Abstract: This is a direct application of a specific formula. You must choose the right one for the job.
Act (Execution):

Problem: Find the point M that divides the line from $A (- 3, 3)$ to $B (1, 7)$ in the ratio $1 : 3$ .
1. Identify Tool: “divides in ratio” $\to$ Section Formula.
2. Assign Variables: $(x_{1}, y_{1}) = (- 3, 3)$ , $(x_{2}, y_{2}) = (1, 7)$ , $m = 1, n = 3$ .
3. Apply Formula:
  - $x_{M} = \frac{m x _{2} + n x _{1}}{m + n} = \frac{1 ( 1 ) + 3 ( - 3 )}{1 + 3} = \frac{- 8}{4} = - 2$ .
  - $y_{M} = \frac{m y _{2} + n y _{1}}{m + n} = \frac{1 ( 7 ) + 3 ( 3 )}{1 + 3} = \frac{16}{4} = 4$ . Final Answer: $(- 2, 4)$ .

Pattern 2.3: Reflection Geometry

Triage: “Does it involve a ‘reflection’, ‘ray of light’, or a ‘mirror’ placed on an axis?”
Abstract: The key is collinearity. The reflected source point, the point on the mirror, and the destination point all lie on a single straight line.
Act (Execution):

Problem: A ray from A(1,2) reflects off the X-axis at B and goes to (5,3). Find the equation of the original line segment AB.
1. Find Reflected Source A’: Reflecting A(1,2) across the x-axis gives A’(1, -2).
2. Find Point B: B is the x-intercept of the line connecting A’(1,-2) and (5,3).
  - Slope of line A’C = $\frac{3 - ( - 2 )}{5 - 1} = \frac{5}{4}$ .
  - Equation: $y - 3 = \frac{5}{4} (x - 5)$ . Set $y = 0 ⟹ - 3 = \frac{5}{4} (x - 5) ⟹ - 12 = 5 x - 25 ⟹ x = 13/5$ . So, B is $(13/5, 0)$ .
3. Find Equation of AB: Find the line through A(1,2) and B(13/5, 0).
  - Slope = $\frac{2 - 0}{1 - 13/5} = - 5/4$ .
  - Equation: $y - 2 = - 5/4 (x - 1) ⟹ 5 x + 4 y = 13$ . Final Answer: $5 x + 4 y = 13$ .

Week 3: Quadratic Functions

Pattern 3.1: Quadratic Max/Min (Vertex Problems)

Triage: “Does it ask for a ‘maximum/minimum value/height’ of a parabola?”
Abstract: The max/min of a parabola is its vertex.
Act (Execution):

Problem: Find the maximum height for the path $h (t) = - 2 t^{2} + 12 t + 54$ .
1. Find When (t-coordinate of vertex): Use Formula: $t = - \frac{b}{2 a}$ .
  - $a = - 2, b = 12$ .
  - $t = - \frac{12}{2 ( - 2 )} = 3$ .
2. Find What (h-coordinate of vertex): Substitute $t = 3$ back into the equation.
  - $h (3) = - 2 (3)^{2} + 12 (3) + 54 = 72$ . Final Answer: 72.

Pattern 3.2: Finding Coefficients from Slope

Triage: “Does it give the ‘slope of a parabola’ at specific points and ask for coefficients a, b, or c?”
Abstract: The slope of $a x^{2} + b x + c$ is given by its derivative. Each slope condition creates a linear equation.
Act (Execution):

Problem: The slope of $y = a x^{2} + b x + c$ at $x = 3$ is 32 and at $x = 2$ is 2. Find $a$ .
1. Find the Slope Formula: Slope = $2 a x + b$ .
2. Create Equations:
  - At $x = 3$ : $2 a (3) + b = 32 ⟹ 6 a + b = 32$ .
  - At $x = 2$ : $2 a (2) + b = 2 ⟹ 4 a + b = 2$ .
3. Solve the System: $(6 a + b) - (4 a + b) = 32 - 2 ⟹ 2 a = 30$ . Final Answer: a = 15.

Pattern 3.3: Intersection of a Parabola and a Line

Triage: “Does it ask for the intersection points of a parabola and a straight line?”
Abstract: The intersection points are where the y values are equal. Set the equations equal to each other to form a new quadratic equation to solve.
Act (Execution):

Problem: An air defense system follows the line $h (t) = 10 t$ . A missile follows the parabola $h (t) = - 2 t^{2} + 12 t + 54$ . At what height do they intersect?
1. Set Equations Equal: $10 t = - 2 t^{2} + 12 t + 54$ .
2. Form a Standard Quadratic: $2 t^{2} - 2 t - 54 = 0 ⟹ t^{2} - t - 27 = 0$ .
3. Solve for t (using Quadratic Formula):
  - Formula: $t = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
  - $t = \frac{1 \pm ( - 1 ) ^{2} - 4 ( 1 ) ( - 27 )}{2 ( 1 )} = \frac{1 \pm 109}{2}$ .
  - Since time cannot be negative, we would take the positive root. However, re-checking the assignment question reveals a different missile equation was used for that part, let’s use the one from Q13: $h (t) = - t^{2} + 14 t$ .
4. Re-solve with Correct Equation: $10 t = - t^{2} + 14 t ⟹ t^{2} - 4 t = 0 ⟹ t (t - 4) = 0$ .
  - Solutions are $t = 0$ (launch) and $t = 4$ .
5. Find the Height: Substitute $t = 4$ into the simple line equation: $h (4) = 10 (4) = 40$ . Final Answer: 40 m.

Week 4: Polynomials

Pattern 4.1: Polynomial End Behavior

Triage: “Does it ask about ‘end behavior’ or what happens ‘as x → ∞’?”
Abstract: End behavior is dictated only by the leading term, $a_{n} x^{n}$ .
Act (Execution):

Problem: For $p (x) = - x^{5} + 5 x^{4} - 2$ , describe its end behavior.
1. Identify Leading Term: $- x^{5}$ .
2. Analyze: Degree is 5 (Odd). Leading Coefficient is -1 (Negative).
3. Apply Rule: Odd/Negative means Up on the left, Down on the right ( $↖ \dots ↘$ ). Final Answer: As $x \to \infty$ , $p (x) \to - \infty$ .

Pattern 4.2: Behavior from Factored Form (Roots & Multiplicity)

Triage: “Is the polynomial given in factored form (e.g., (x-a)²(x-b))?”
Abstract: The exponent on a factor is its multiplicity. Odd multiplicity means the graph crosses the x-axis. Even multiplicity means it touches and bounces.
Act (Execution):

Problem: A polynomial has a factor of $(x - 3)^{2}$ . What happens at $x = 3$ ?
1. Identify Root and Multiplicity: The root is $x = 3$ . The multiplicity is 2 (Even).
2. Apply Rule: An even multiplicity means the graph touches the x-axis and turns around. Final Answer: The graph has a turning point on the x-axis at x=3.

Pattern 4.3: Finding Roots from Complex Factors

Triage: “Does the question ask for the ‘x-intercepts’ or ‘roots’ of a polynomial given as a product of complex factors?”
Abstract: If a product of factors equals zero, then at least one of the individual factors must be zero. Set each factor equal to zero and solve.
Act (Execution):

Problem: Find the sum of the roots of $f (x) = (x^{2} - 23) ((x - 10)^{3} - 1)$ .
1. Set Factor 1 to Zero: $x^{2} - 23 = 0 ⟹ x^{2} = 23 ⟹ x = 23, - 23$ .
2. Set Factor 2 to Zero: $(x - 10)^{3} - 1 = 0 ⟹ (x - 10)^{3} = 1$ .
  - Taking the cube root of both sides gives $x - 10 = 1 ⟹ x = 11$ .
3. Sum the Roots: Sum = $(23) + (- 23) + 11 = 11$ . Final Answer: 11.

🧠 IIT Madras Notes

Explorer

Maths TLDR