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These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.

These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.

These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.

These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.

 

01 Hours

 

55 Marks

 

Calculator is allowed

IB MATH AA HL, Paper 3, May, 2022, TZ1, Solved Past Paper

Master the 2022 IB May for Paper 3 Mathematics AA HL with examiner tailored solutions and comments for TZ1

Question 1 [Explained]

The question invites us to delve into the characteristics of polygonal numbers and to uncover and establish notable relationships among them.

 

Polygonal numbers are integers that can be visually represented through dot arrangements forming regular polygons. This includes forms like triangular, square, and pentagonal numbers.

 

For instance, triangular numbers are those that can be organized into an equilateral triangle. The sequence starts with 1, 3, 6, 10, and 15.

 

The initial number in each polygonal series is typically a solitary dot. To mathematically define these numbers for any r-sided polygon, where (r) is a positive integer starting from 3, the nth polygonal number (Pr (n)) is calculated using:

 

(Pr(n)=(r−2)n2−(r−4)n2), where (n) is a positive integer.

Question 1 (Figure)
Question 1 [a] [i] [Explanation]

For square numbers, the polynomial P4 (n) is expressed as ((4−2)n2−(4−4)n2=n2). This equation simplifies to represent square numbers directly as (n2), which is the definition of a square number.

 

For triangular numbers, we are tasked to confirm that the polynomial P3 (n) correctly calculates as (n(n+1)2). This formula is widely recognized for generating triangular numbers, which sequentially adds the natural numbers.

Question 1 [a] [ii] [Explanation]

Identify the rank of the number 351 within the sequence of triangular numbers, recognizing it as the sum of all natural numbers up to a certain limit.

Video Solution by an IB Examiner - Coming soon

Question 1 [b] [i] [Explanation]

To demonstrate that the sum of (P3(n)) and (P3(n+1)) equals ((n+1)2), where (P3) represents a specific polynomial function applied to the variable (n).