Maths Patterns

Consolidated Question Patterns & Solutions: Maths Weeks 1-4

This document is your master guide to the types of problems you will encounter in the first four weeks of Mathematics for Data Science I. Each section provides a summary table, an illustrative example with a detailed solution for each key pattern, and a mental algorithm to help you quickly assess and solve problems.

📚 Table of Contents

1. Week 1: Sets, Relations, and Functions
2. Week 2: Coordinate Geometry and Straight Lines
3. Week 3: Quadratic Functions
4. Week 4: Polynomial Functions

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Week 1: Sets, Relations, and Functions

This week is about the fundamental building blocks of mathematical language: collections, their properties, and the rules that map one collection to another.

Pattern #	Pattern Name	Frequency	Difficulty	Description & Core Skill
1.1	Irrational Number Identification	Medium	Easy	Simplify radical expressions to check if they reduce to a rational number (p/q) or remain irrational.
1.2	Function Domain & Cardinality	High	Medium	Find integers excluded from a domain due to division by zero or square roots of negative numbers.
1.3	Cardinality of Set Relations	Medium	Medium	Calculate the size of set operations (union, difference) on relations defined by mathematical rules.
1.4	Venn Diagram Word Problems	High	Medium	Use the Principle of Inclusion-Exclusion to solve word problems with overlapping categories.
1.5	Identifying Relation Properties	High	Medium	Test a relation for reflexivity, symmetry, and transitivity.
1.6	Classifying Functions	High	Medium	Determine if a function is injective (one-to-one), surjective (onto), or bijective.

📝 Pattern Examples & Solutions

Pattern 1.2: Function Domain & Cardinality

Example: Suppose $f:D⟶\mathbb{R}$ is a function defined by $f(x)=\frac{\sqrt{x^2-16}}{x+4}$, where $D\subset \mathbb{Z}$. Let $A$ be the set of integers not in the domain of $f$. Find $|A|$.

Click for Step-by-Step Solution

1. Identify Restrictions: The domain is limited by two rules:

   • The denominator cannot be zero: $x+4\ne 0⟹x\ne -4$.
   • The expression inside the square root must be non-negative: $x^2-16\ge 0$.

2. Solve the Inequality: $x^2\ge 16$. This is true when $x\ge 4$ or $x\le -4$.

3. Determine the Valid Domain (D): The integers in the domain must satisfy both conditions. So, the domain consists of integers $x\le -5$ and integers $x\ge 4$. $D=\{...,-6,-5\}\cup \{4,5,6,...\}$

4. Find the Excluded Integers (A): The set A contains all integers that are not in D.

   • The integers from -3 to 3 are excluded by the square root condition.
   • The integer -4 is excluded by the denominator condition.
   • $A=\{-4,-3,-2,-1,0,1,2,3\}$.

5. Calculate Cardinality: Count the elements in A. $|A|=8$.

Pattern 1.4: Venn Diagram Word Problems

Example: In a Zoo, there are 6 Bengal white tigers and 6 Bengal royal tigers. Out of these tigers, 5 are males and 10 are either Bengal royal tigers or males. Find the number of female Bengal white tigers.

Click for Step-by-Step Solution

1. Define Sets:

   • $W$: Bengal white tigers, $|W|=6$.
   • $R$: Bengal royal tigers, $|R|=6$.
   • $M$: Male tigers, $|M|=5$.
   • Total tigers = $6+6=12$.
   • Female tigers $|F|=12-5=7$.

2. Use the Inclusion-Exclusion Principle: We are given that “10 are either Bengal royal tigers or males”. This translates to $|R\cup M|=10$. The formula is: $|R\cup M|=|R|+|M|-|R\cap M|$.

3. Solve for the Intersection: We can find the number of tigers that are both Royal and Male ($|R\cap M|$). $10=6+5-|R\cap M|$ $10=11-|R\cap M|$ $|R\cap M|=1$. So, there is 1 male Bengal royal tiger.

4. Find Other Categories:

   • Number of male Bengal white tigers: Total Males - Male Royal Tigers = $|M|-|R\cap M|=5-1=4$.
   • Number of female Bengal white tigers: Total White Tigers - Male White Tigers = $|W|-4=6-4=2$.

5. Final Answer: There are 2 female Bengal white tigers.

🧠 Week 1 Mental Algorithm

1. See a radical or fraction? $\to$ Think Domain Restrictions. Check for division by zero and square roots of negatives.
2. See a word problem with overlapping groups? $\to$ Think Venn Diagrams. Use the Inclusion-Exclusion formula: $|A\cup B|=|A|+|B|-|A\cap B|$.
3. See a relation with pairs $(a,b)$? $\to$ Think Properties. Check Reflexive (all a,a?), Symmetric (if a,b then b,a?), and Transitive (if a,b and b,c then a,c?).
4. See a function $f:A\to B$? $\to$ Think Classification. Check Injective (is it one-to-one?) and Surjective (is the range equal to B?).

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Week 2: Coordinate Geometry and Straight Lines

This week is about using equations to describe and analyze lines and points on a plane.

Pattern #	Pattern Name	Frequency	Difficulty	Description & Core Skill
2.1	Line Intersection & Collision	High	Easy	Find where two lines meet by solving their equations simultaneously.
2.2	Geometric Properties & Formulas	High	Medium	Apply formulas like distance, midpoint, section, and area to solve problems about geometric shapes.
2.3	Data Fitting & Sum of Squared Errors (SSE)	High	Medium	Calculate $\sum (y_{actual}-y_{predicted}{)}^2$ to measure how well a line fits a dataset.
2.4	Shortest Distance Problems	Medium	Medium	Use the formula $d = \frac{
2.5	Reflection Geometry	Medium	Hard	Use the concept of a reflected point to determine the path of a ray bouncing off an axis.

📝 Pattern Examples & Solutions

Pattern 2.1: Line Intersection & Collision

Example: A bird flies along $2y-6x=6$. An airplane flies with a slope of 2 and passes through $(4,8)$. Find the sum of the coordinates of their collision point $(\alpha ,\beta )$.

Click for Step-by-Step Solution

1. Find the Bird’s Line Equation: $2y=6x+6⟹y=3x+3$.
2. Find the Airplane’s Line Equation: Using point-slope form $y-y_1=m(x-x_1)$: $y-8=2(x-4)⟹y-8=2x-8⟹y=2x$.
3. Solve the System: Set the two equations equal to find the intersection point: $3x+3=2x⟹x=-3$. Substitute $x=-3$ into the second equation: $y=2(-3)=-6$.
4. Calculate the Sum: The collision point $(\alpha ,\beta )$ is $(-3,-6)$. $\alpha +\beta =-3+(-6)=-9$.

Pattern 2.3: Sum of Squared Errors (SSE)

Example: Radhika fits a line $y=4x+2$ to her data. Calculate the SSE. Data: (1, 6), (3, 14), (5, 24), (7, 29), (9, 39), (11, 45).

Click for Step-by-Step Solution

1. Calculate Predicted Values and Squared Errors: Create a table to track calculations. | x | y (Actual) | y (Predicted) = 4x+2 | Error (A-P) | Error² | |---|---|---|---|---| | 1 | 6 | $4(1)+2=6$ | 0 | 0 | | 3 | 14 | $4(3)+2=14$ | 0 | 0 | | 5 | 24 | $4(5)+2=22$ | 2 | 4 | | 7 | 29 | $4(7)+2=30$ | -1 | 1 | | 9 | 39 | $4(9)+2=38$ | 1 | 1 | | 11 | 45 | $4(11)+2=46$ | -1 | 1 |

2. Sum the Squared Errors: $\text{SSE}=0+0+4+1+1+1=7$.

🧠 Week 2 Mental Algorithm

1. See two lines and the word “intersect” or “collide”? $\to$ Think System of Equations. Get both into $y=mx+c$ form and set them equal.
2. See geometric shapes, midpoints, or divided lines? $\to$ Think Coordinate Formulas. Use distance, midpoint, or section formulas.
3. See a line, data points, and the term “SSE” or “best fit”? $\to$ Think SSE Table. Calculate $(y_{actual}-y_{predicted}{)}^2$ for each point and sum them.
4. See “shortest distance from a point to a line”? $\to$ Think Point-to-Line Formula. Get the line into $Ax+By+C=0$ form first.

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Week 3: Quadratic Functions

This week focuses on parabolas, their properties, and their applications.

Pattern #	Pattern Name	Frequency	Difficulty	Description & Core Skill
3.1	Finding Coefficients from Slope	High	Medium	Use the slope formula ($f^{\prime }(x)=2ax+b$) at given points to create and solve for the parabola’s coefficients.
3.2	Real-World Quadratic Modeling	High	Easy-Medium	Translate word problems about area, motion, or products into a quadratic equation and solve it.
3.3	Intersection of Curves	Medium	Medium	Find intersection points by setting the equations of a parabola and another curve (line or parabola) equal.
3.4	Finding Maximum/Minimum Values	High	Easy	Find the vertex of the parabola, as its y-coordinate represents the function’s maximum or minimum value.

📝 Pattern Examples & Solutions

Pattern 3.1: Finding Coefficients from Slope

Example: The slope of $y=a{x}^2+bx+c$ at $(3,2)$ is 32 and at $(2,3)$ is 2. Find $a$.

Click for Step-by-Step Solution

1. Find the Slope Formula: The slope is the derivative: $y^{\prime }=2ax+b$.
2. Form Equations from the Given Data:
   • At $x=3$, slope is 32: $2a(3)+b=32⟹6a+b=32$.
   • At $x=2$, slope is 2: $2a(2)+b=2⟹4a+b=2$.
3. Solve the System: Subtract the second equation from the first: $(6a+b)-(4a+b)=32-2$ $2a=30⟹a=15$.

Pattern 3.4: Finding Maximum/Minimum Values

Example: A water fountain’s stream is modeled by $h(t)=-0.5t^2+4t+1$. Find the time it takes to reach maximum height.

Click for Step-by-Step Solution

1. Identify the Goal: The “time to reach maximum height” is the x-coordinate (t-coordinate) of the vertex of the parabola.
2. Use the Vertex Formula: For a quadratic $a{t}^2+bt+c$, the vertex occurs at $t=-\frac{b}{2a}$.
3. Calculate: Here, $a=-0.5$ and $b=4$. $t=-\frac{4}{2(-0.5)}=-\frac{4}{-1}=4$. Answer: 4 seconds.

🧠 Week 3 Mental Algorithm

1. See “slope of a parabola”? $\to$ Think Derivative. Use the formula $f^{\prime }(x)=2ax+b$.
2. See “maximum” or “minimum” value? $\to$ Think Vertex. Calculate the coordinates using $x=-b/(2a)$.
3. See a word problem with “product”, “area”, or projectile motion? $\to$ Think Quadratic Equation. Formulate an equation and solve it (factor or use the quadratic formula).
4. See “intersection” of two curves? $\to$ Think Set Equations Equal. Solve the resulting equation to find the x-coordinates of the intersection.

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Week 4: Polynomial Functions

This week expands on quadratics to cover polynomials of any degree, with a focus on their graphical behavior.

Pattern #	Pattern Name	Frequency	Difficulty	Description & Core Skill
4.1	Graph Behavior from Factored Form	High	Medium	Analyze a polynomial’s factored form to determine its behavior at the roots (cross vs. touch) and the sign of the function in intervals between roots.
4.2	End Behavior & Turning Points	High	Medium	Use the leading term ($a_n{x}^n$) to determine the graph’s end behavior and know that the number of turning points is at most $n-1$.
4.3	Finding and Using Roots	Medium	Easy	Find the x-intercepts by setting the polynomial to zero, often from a factored form, and perform calculations with them.
4.4	Polynomial Algebra	Medium	Medium	Perform addition, subtraction, and multiplication of polynomials, and construct a polynomial from given properties.

📝 Pattern Examples & Solutions

Pattern 4.1: Graph Behavior from Factored Form

Example: A roller coaster’s height is modeled by $h(t)=(\dots )(t+1)(\dots )$. Analyze the behavior in the interval $(-5,-1)$. From Q10, we see other factors are positive in this interval, and the factor $(t+1)$ has an odd multiplicity. The leading coefficient is negative overall.

Click for Step-by-Step Solution

1. Analyze End Behavior for the Interval:
   • Let’s check the sign of $h(t)$ just to the left of $-1$, say at $t=-1.1$. The factor $(t+1)$ becomes negative. Let’s assume other factors result in a positive value. With an overall negative leading coefficient, $h(t)$ would be positive.
   • Let’s check the sign of $h(t)$ just to the right of $-5$. The factor $(t+5{)}^2$ is positive.
   • At $t=-5$, the graph touches the x-axis (even multiplicity). At $t=-1$, it crosses (odd multiplicity).
2. Determine Path:
   • Let’s check a point inside, e.g., $t=-3$. $h(t)=(-0.01t^3\dots )(t+5{)}^2(t-5)(t+1)(2-t{)}^3$. $h(-3)=(\text{pos})(\text{pos})(\text{neg})(\text{neg})(\text{pos})=\text{positive}$.
   • The graph starts at the x-axis at $t=-5$, goes up into positive territory, and then must come down to cross the axis at $t=-1$.
3. Conclusion: The roller coaster will first go up and then go down in the interval $(-5,-1)$.

Pattern 4.2: End Behavior & Turning Points

Example: Consider $p(x)=-x^5+5x^4-7x-2$. What is its end behavior as $x\to \infty$ and what is the maximum number of turning points?

Click for Step-by-Step Solution

1. Identify the Leading Term: The term with the highest power is $-x^5$.
2. Analyze End Behavior:
   • Degree: 5 (Odd).
   • Leading Coefficient: -1 (Negative).
   • For an odd degree, negative coefficient polynomial, the graph goes up on the left and down on the right.
   • Therefore, as $x\to \infty$ (moves to the right), $p(x)\to -\infty$.
3. Determine Max Turning Points:
   • For a polynomial of degree $n=5$, the maximum number of turning points is $n-1=4$.

🧠 Week 4 Mental Algorithm

1. See a polynomial in factored form? $\to$ Think Roots and Multiplicity.
   • Odd multiplicity $⟹$ Crosses x-axis.
   • Even multiplicity $⟹$ Touches x-axis (this is a turning point).
2. See “end behavior” or “as x approaches infinity”? $\to$ Think Leading Term. Look only at $a_n{x}^n$. Use the even/odd degree and pos/neg coefficient rules.
3. See “turning points”? $\to$ Think Degree minus 1. A polynomial of degree $n$ has at most $n-1$ turning points.
4. See “intersection of polynomials”? $\to$ Set Equations Equal. This is the same strategy as with lines and parabolas.