Maths W1 to W4 Triage

Ultimate Math Problem-Solver’s Arsenal (Weeks 1-4)

This guide is the culmination of analyzing all provided assignments and PYQs. It identifies every unique question pattern and provides a detailed, solved example for each one. For each pattern, we will focus on the Abstraction—the core idea you need to recognize—and the Execution, the step-by-step process to solve it.

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Week 1: Sets, Relations, and Functions

• TL;DR Concepts: Week 1 is about structure and rules. Sets group things. Relations link things. Functions are special relations with a strict “one input, one output” rule. Domain is what you can put in; Range is what you actually get out.

Identified Patterns & Solved Examples

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Pattern 1: Irrational Number Simplification

• Abstraction: An expression is irrational if, after full simplification (including combining radicals and rationalizing denominators), a square root of a non-perfect square remains.
• Example (from PYQ): Is the expression $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$ rational or irrational?

Click for Solution

1. **Simplify Radicals First:** This is the crucial first step. * $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$ * $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ 2. **Substitute and Combine:** * $(2\sqrt{2}+\sqrt{2})(2\sqrt{3}-\sqrt{3})$ * $= (3\sqrt{2})(1\sqrt{3})$ 3. **Multiply:** * $= 3\sqrt{2 \times 3} = 3\sqrt{6}$ 4. **Conclusion:** Since 6 is not a perfect square, $\sqrt{6}$ is irrational. The expression is **irrational**.

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Pattern 2: Function Domain Restrictions & Cardinality

• Abstraction: A function’s domain is restricted by two cardinal sins of mathematics: 1. Never divide by zero. 2. Never take the square root of a negative number. The set of excluded numbers is the set that commits either of these sins.
• Example (from PYQ): Find the cardinality of the set of integers not in the domain of $f(x)=\frac{\sqrt{x^2-16}}{x+4}$.

Click for Solution

1. **Check for Sin #1 (Division by Zero):** * $x+4 \neq 0 \implies x \neq -4$. The integer `-4` is excluded. 2. **Check for Sin #2 (Square Root of Negative):** * $x^2 - 16 \geq 0 \implies x^2 \geq 16$. * This is true for $x \geq 4$ or $x \leq -4$. * The integers that violate this are those *between* -4 and 4: $\{-3, -2, -1, 0, 1, 2, 3\}$. 3. **Combine Exclusions:** The set of all excluded integers is the union of the results from both checks. * $A = \{-4\} \cup \{-3, -2, -1, 0, 1, 2, 3\} = \{-4, -3, -2, -1, 0, 1, 2, 3\}$. 4. **Find Cardinality:** The number of elements in A is **8**.

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Pattern 3: Venn Diagrams & Inclusion-Exclusion

• Abstraction: When groups overlap, simply adding them up double-counts the overlap. The Inclusion-Exclusion Principle is the formal way to correct this: $|A\cup B|=|A|+|B|-|A\cap B|$.
• Example (from PYQ): A zoo has 6 white tigers (W) and 6 royal tigers (R). There are 5 males (M) in total. 10 tigers are either royal or male. Find the number of female white tigers.

Click for Solution

1. **Translate to Set Notation:** * $|W|=6$, $|R|=6$, Total=12. * $|M|=5$, so $|F| = 12-5=7$. * "Either royal or male" means $|R \cup M| = 10$. 2. **Use Inclusion-Exclusion to Find the Overlap:** * $|R \cup M| = |R| + |M| - |R \cap M|$ * $10 = 6 + 5 - |R \cap M| \implies |R \cap M| = 1$. * This means there is **1 male royal tiger**. 3. **Use Logic to Find Sub-categories:** * Total Males = (Male Royals) + (Male Whites) * $5 = 1 + (\text{Male Whites}) \implies \text{Male Whites} = 4$. * Total Whites = (Male Whites) + (Female Whites) * $6 = 4 + (\text{Female Whites}) \implies \text{Female Whites} = 2$.

Final Answer: There are 2 female white tigers.

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Pattern 4: Relation Properties (Reflexive, Symmetric, Transitive)

• Abstraction: Test the definition for each property. A relation “shares a property” is always symmetric and transitive. It’s only reflexive if all elements have that property.
• Example (from PYQ): A relation on 525 ponds is defined as: two ponds are related if both are polluted by Fertiliser (F) and Pharma (Ph). Is this relation reflexive, symmetric, transitive?

Click for Solution

1. **Reflexive Test:** Is (A, A) in the relation for *every* pond A? * The condition is "A is polluted by F & Ph AND A is polluted by F & Ph". This simplifies to "A is polluted by F & Ph". * But what if a pond is only polluted by Pesticides? Then this condition is false for that pond. Since it's not true for *every* pond, the relation is **not reflexive**. 2. **Symmetric Test:** If (A, B) is in the relation, is (B, A)? * If (A,B) is in, it means "A is polluted by F&Ph AND B is polluted by F&Ph". * Since "AND" is commutative, this is the same as "B is polluted by F&Ph AND A is polluted by F&Ph". * This is the condition for (B,A). So, it **is symmetric**. 3. **Transitive Test:** If (A, B) and (B, C) are in, is (A, C)? * If (A,B) is in, we know A has the property. If (B,C) is in, we know C has the property. * The condition for (A,C) is "A has the property AND C has the property", which we know is true. * So, it **is transitive**.

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Week 2: Coordinate Geometry

• TL;DR Concepts: Week 2 is about lines. The slope ($m$) is their personality. We can write their life story (equation) in different ways (point-slope, slope-intercept). We can measure the distance between points or from a point to a line. SSE measures how well a line’s story fits a set of data points.

Identified Patterns & Solved Examples

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Pattern 5: Line Intersection (Collision)

• Abstraction: Two lines intersect at a point where their $(x,y)$ coordinates are identical. The solution is to set their equations equal to each other.
• Example (from PYQ): A bird is on path $2y-6x=6$. A plane passes through $(4,8)$ with slope 2. Find the sum of coordinates $(\alpha +\beta )$ of their collision point.

Click for Solution

1. **Find Both Equations in `y=mx+c` form:** * Bird: $2y=6x+6 \implies y=3x+3$. * Plane (Point-Slope): $y-8 = 2(x-4) \implies y=2x$. 2. **Set `y` values equal:** * $3x+3=2x \implies x=-3$. 3. **Find `y`:** * Substitute $x=-3$ into the simpler equation: $y=2(-3)=-6$. 4. **Calculate Sum:** * The point is $(\alpha, \beta) = (-3, -6)$. * $\alpha + \beta = -3 + (-6) = -9$.

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Pattern 6: Sum of Squared Errors (SSE)

• Abstraction: SSE is a “total unhappiness” score. For each data point, the “unhappiness” is the vertical distance to the line, squared. The SSE is the sum of all these squared distances.
• Example (from PYQ): For the data points (5, 24) and (7, 29), calculate their contribution to the SSE for the line $y=4x+2$.

Click for Solution

1. **Create a Calculation Table:**

x (Input)	y (Actual)	y (Predicted) = 4x+2	Error (A - P)	Error²
5	24	$4(5)+2=22$	$24-22=2$	4
7	29	$4(7)+2=30$	$29-30=-1$	1

3. Sum the Contributions: The total contribution from these two points is $4+1=5$.

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Pattern 7: Reflection Geometry

• Abstraction: The key is collinearity. The reflected source point (A’), the point on the mirror (B), and the destination point (C) all lie on a single straight line.
• Example (from PYQ): A ray from A(1,2) reflects off the X-axis at B and goes to (5,3). Find the equation of the line AB.

Click for Solution

1. **Find the Reflected Source (A'):** Reflecting A(1,2) across the x-axis gives A'(1, -2). 2. **Find the line containing B:** B lies on the line connecting A'(1,-2) and (5,3). 3. **Find the coordinates of B:** B is the x-intercept of the line A'C. * Slope of A'C = $\frac{3 - (-2)}{5 - 1} = \frac{5}{4}$. * Equation: $y - (-2) = \frac{5}{4}(x-1)$. * Set $y=0$ to find B: $2 = \frac{5}{4}(x-1) \implies 8 = 5x-5 \implies x = 13/5$. So, B is $(13/5, 0)$. 4. **Find the Equation of AB:** Now find the line through A(1,2) and B(13/5, 0). * Slope = $\frac{2-0}{1 - 13/5} = \frac{2}{-8/5} = -5/4$. * Equation: $y - 0 = -5/4(x - 13/5) \implies 4y = -5x + 13 \implies 5x+4y=13$.

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Week 3: Quadratic Functions

• TL;DR Concepts: Week 3 is about parabolas. The vertex is their most important feature (max/min point). Their slope isn’t constant; it’s a line given by the derivative ($2ax+b$). Most problems boil down to either finding the vertex or solving a quadratic equation.

Identified Patterns & Solved Examples

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Pattern 8: Finding Parabola Coefficients from Slopes

• Abstraction: The slope of $a{x}^2+bx+c$ at a point x is 2ax+b. Each given slope at a given point creates one linear equation involving a and b. Two such conditions are enough to solve for both.
• Example (from PYQ): The slope of $y=a{x}^2+bx+c$ at $x=3$ is 32 and at $x=2$ is 2. Find $a$.

Click for Solution

1. **Form Equations with the Derivative:** * Slope at $x=3$: $2a(3)+b = 32 \implies 6a+b=32$. * Slope at $x=2$: $2a(2)+b = 2 \implies 4a+b=2$. 2. **Solve the System:** Subtract the second equation from the first. * $(6a+b) - (4a+b) = 32-2 \implies 2a=30 \implies a=15$.

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Pattern 9: Finding Maximum/Minimum Values (Vertex)

• Abstraction: “Maximum/minimum value/height” is a keyword for the y-coordinate of the vertex. The “time/point where it occurs” is the x-coordinate of the vertex ($x=-b/2a$).
• Example (from PYQ): The path of a missile is $h(t)=-2t^2+12t+54$. Find the maximum height attained.

Click for Solution

1. **Find the Time of Max Height (t-coordinate of vertex):** * $a=-2, b=12$. * $t = -b/(2a) = -12/(2 \times -2) = -12/(-4) = 3$ seconds. 2. **Find the Max Height (h-coordinate of vertex):** * Substitute $t=3$ back into the equation: * $h(3) = -2(3)^2 + 12(3) + 54 = -18 + 36 + 54 = 72$.

Final Answer: The maximum height is 72 meters.

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Week 4: Polynomials

• TL;DR Concepts: Week 4 is about wiggles. The leading term ($a_n{x}^n$) dictates the ultimate fate of the graph (end behavior). The roots (x-intercepts) and their multiplicity (the exponent on the factor) dictate the journey along the x-axis. Odd multiplicity = crosses, Even multiplicity = touches/bounces.

Identified Patterns & Solved Examples

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Pattern 10: End Behavior & Turning Points

• Abstraction: End behavior depends only on the degree (Even/Odd) and the leading coefficient (Positive/Negative). The number of turning points is at most the degree minus one.
• Example (from PYQ): Analyze $p(x)=-x^5+5x^4-7x-2$.

Click for Solution

1. **Identify Leading Term:** $-x^5$. 2. **Analyze End Behavior:** * Degree = 5 (**Odd**). * Leading Coefficient = -1 (**Negative**). * The rule for Odd/Negative is: Starts Up, Ends Down ($\nwarrow \dots \searrow$). * So, as $x \to \infty$, $p(x) \to -\infty$. 3. **Analyze Turning Points:** * Degree is $n=5$. * Maximum turning points = $n-1 = 4$.

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Pattern 11: Graph Behavior from Factored Form

• Abstraction: At a root from a factor $(x-r{)}^m$: if m is odd, the sign of the function flips; if m is even, the sign stays the same on both sides of the root.
• Example (from PYQ): Analyze the behavior of $p(x)=-(x-2{)}^2(x-3{)}^2$ in the interval $(2,3)$.

Click for Solution

1. **Analyze Roots:** The roots are at $x=2$ and $x=3$, both with even multiplicity (2). This means the graph touches the x-axis at both points. 2. **Test a Point in the Interval:** Let's check $x=2.5$. * $p(2.5) = -(2.5-2)^2(2.5-3)^2 = -(0.5)^2(-0.5)^2 = -(\text{positive})(\text{positive}) = \text{negative}$. 3. **Sketch the Path:** The graph is at $y=0$ at $x=2$. It dips into negative territory between 2 and 3, then comes back up to touch the axis at $y=0$ at $x=3$. This path means it **first decreases, then increases**.

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Pattern 12: Finding Roots from Complex Factors

• Abstraction: For a product of factors to be zero, at least one of the factors must be zero. Set each factor equal to zero and solve independently.
• Example (from PYQ): Find the sum of the roots of $f(x)=\left(x^2-23\right)\left({\left(x-10\right)}^3-1\right)$.

Click for Solution

1. **Set Each Factor to Zero:** * Case 1: $x^2 - 23 = 0 \implies x^2 = 23 \implies x = \sqrt{23}, -\sqrt{23}$. * Case 2: $(x-10)^3 - 1 = 0 \implies (x-10)^3 = 1 \implies x-10 = 1 \implies x = 11$. 2. **Sum the Roots:** * Sum = $(\sqrt{23}) + (-\sqrt{23}) + 11 = 0 + 11 = 11$.

Of course. You’re looking for the final piece of the arsenal: a concise, powerful reference sheet of all the core formulas and concepts, combined with a practical guide on how to choose the right tool for the job.

Here is your Ultimate Math Formulas & Concepts Arsenal for Weeks 1-4, designed for quick recall and strategic application.

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The Strategist’s Math Arsenal: Formulas, Concepts, and Application Guide (Weeks 1-4)

This guide is not just a list of formulas; it’s a manual for your mental toolkit. It’s divided into two parts:

1. The Arsenal: All the core formulas and concepts in one place for quick reference.
2. The Strategist’s Guide: A “how-to” for assessing any problem and choosing the right weapon from your arsenal.

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Part 1: The Arsenal (Core Formulas & Concepts)

Week 1: Sets, Relations, & Functions

Concept	Core Formula / Rule	TL;DR (The Core Idea)
Set Union (Inclusion-Exclusion)	$A\cup B=A+B-A\cap B$	To find the total, add the groups and subtract the overlap you counted twice.
Set Difference	$A\setminus B$	”Everything in A that is NOT in B.”
Domain Restrictions	1. Denominator $\ne 0$ 2. Inside a square root $\ge 0$	Don’t divide by zero or take the square root of a negative.
Relation: Reflexive	For ALL $a\in A$, the pair $(a,a)$ must be in the relation.	Everyone must be related to themselves.
Relation: Symmetric	If $(a,b)$ is in the relation, then $(b,a)$ must also be in.	If A is related to B, then B must be related to A. The relationship is a two-way street.
Relation: Transitive	If $(a,b)$ and $(b,c)$ are in, then $(a,c)$ must also be in.	If A is a friend of B, and B is a friend of C, then A is a friend of C.
Function: Injective (1-to-1)	$f(x_1)=f(x_2)⟹x_1=x_2$	Different inputs MUST give different outputs. No two inputs share the same output.
Function: Surjective (Onto)	Range = Codomain	Every possible output in the codomain is actually used by some input.

Week 2: Coordinate Geometry & Straight Lines

Concept	Core Formula / Rule	TL;DR (The Core Idea)
Slope (m)	$m=\frac{y_2-y_1}{x_2-x_1}$	Rise over Run. The “steepness” of the line.
Point-Slope Form	$y-y_1=m(x-x_1)$	The most useful way to build a line’s equation. You need a point and the slope.
Slope-Intercept Form	$y=mx+c$	Tells you the slope and where the line crosses the y-axis.
Parallel Lines	$m_1=m_2$	Same slope, same direction. They never meet.
Perpendicular Lines	$m_1\times m_2=-1$	Slopes are negative reciprocals. They meet at a 90° angle.
Distance Formula	$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$	Just the Pythagorean theorem in disguise.
Point-to-Line Distance	$d = \frac{	Ax_0 + By_0 + C
Section Formula	$(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$	Finds the point that splits a line segment at a specific ratio m:n.
Sum of Squared Errors (SSE)	$\text{SSE}=\sum (y_{actual}-y_{predicted}{)}^2$	A score for how badly a line fits the data. Lower is better.

Week 3: Quadratic Functions (Parabolas)

Concept	Core Formula / Rule	TL;DR (The Core Idea)
Standard Form	$f(x)=a{x}^2+bx+c$	The sign of ‘a’ tells you if the parabola opens up ($a>0$) or down ($a<0$).
Vertex x-coordinate	$x=-\frac{b}{2a}$	Finds the line of symmetry and the location of the max/min point.
Max/Min Value	$f(-\frac{b}{2a})$	The actual highest or lowest value the function can reach.
Slope of Parabola	$f^{\prime }(x)=2ax+b$	The slope is not constant. This formula tells you the slope at any point x.
Quadratic Formula	$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$	The ultimate weapon to find the roots (x-intercepts) of any quadratic.

Week 4: Polynomial Functions

Concept	Core Formula / Rule	TL;DR (The Core Idea)
Leading Term	$a_n{x}^n$	The boss of the polynomial. Its degree (n) and sign (a_n) dictate the end behavior.
End Behavior Rules	Even Degree: Arms go together (up/up or down/down). Odd Degree: Arms go opposite (down/up or up/down).	The “dance moves” of the graph at the far left and right.
Root Multiplicity	Exponent m on a factor (x-r)^m.	Odd m: Graph crosses the x-axis at r. Even m: Graph touches (bounces off) the x-axis at r.
Turning Points	A polynomial of degree n has at most n-1 turning points.	The number of “hills and valleys” the graph can have.

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Part 2: The Strategist’s Guide (How to Assess and Apply)

This is the TAA (Triage, Abstract, Act) framework in a compact, actionable format.

The Nifty Keyword-to-Tool Assessor

Scan any problem for these keywords. The moment you see one, your brain should immediately jump to the associated “Tool” and “Action Plan”.

IF YOU SEE THE KEYWORD(S)…	THEN THE CATEGORY IS…	AND YOUR IMMEDIATE ACTION PLAN IS…
“survey”, “how many like both”, ”%“	Venn Diagram (W1)	Draw circles. Use the formula: $A\cup B=A+B-A\cap B$. Work from the inside out.
”domain”, 1/(...), sqrt(...)	Domain Restrictions (W1)	List the two rules: 1. Denominator $\ne 0$. 2. Inside sqrt $\ge 0$. Find the integers that break these rules.
”relation”, “reflexive”, “symmetric”	Relation Properties (W1)	Systematically test the three definitions. Remember: “shares a property” is always S and T.
”collision”, “intersection of lines”	System of Equations (W2)	Get both equations into y = ... form. Set them equal. Solve for x, then find y.
”shortest distance from point to line”	Point-to-Line Distance (W2)	1. Get the line into $Ax+By+C=0$ form. 2. Plug the point and A,B,C into the formula. Remember the absolute value!
”SSE”, “best fit”, “data points”	SSE Calculation (W2)	Create a table with columns: Actual, Predicted, Error, Error². Sum the last column.
”maximum/minimum height/value”	Quadratic Vertex (W3)	This is a vertex problem. Use $x=-b/(2a)$ to find where it happens. Plug that x back in to find the value.
”slope of a parabola”	Quadratic Derivative (W3)	The tool is Slope = 2ax + b. Use this to create linear equations to solve for a and b.
”end behavior”, “as x → infinity”	Polynomial Leading Term (W4)	Find the term with the highest power, $a_n{x}^n$. Apply the “dance moves” rules based on n and a_n.
(x-a)²(x-b)³, “factored form”	Polynomial Roots & Multiplicity (W4)	Identify the roots (a and b). Check the exponents: Even exponent = Touches/Bounces. Odd exponent = Crosses.

The “Memory Palace” Nifty Tricks

• Happy vs. Sad Parabola (W3):

  • y = +ax²: + is positive, so it’s a “happy” parabola U with a minimum.
  • y = -ax²: - is negative, so it’s a “sad” parabola ∩ with a maximum.

• The End Behavior Dance (W4):

  • Even Degree (Like a Parabola): Arms move together. Positive a_n $\to$ Arms UP. Negative a_n $\to$ Arms DOWN.
  • Odd Degree (Like a Line): Arms move opposite. Positive a_n $\to$ Finishes UP (like y=x). Negative a_n $\to$ Finishes DOWN (like y=-x).

• The Bouncer at Club X-Axis (W4):

  • A root with Odd Multiplicity is a regular guest. They cross the rope (the x-axis).
  • A root with Even Multiplicity is a troublemaker. They get to the rope, touch it, and get bounced back.

By combining this quick-reference arsenal with the assessment guide, you can transform any problem from an unknown threat into a familiar scenario with a clear, step-by-step solution path.