IB MATH AA HL, Paper 1, May, 2021, TZ2, Solved Past Paper

Let's explore a simple property of consecutive numbers. Suppose we have an integer n and the next integer, which is n + 1. We want to investigate the relationship between their squares.

 

The task is to demonstrate that subtracting the square of the first number from the square of the second yields a result that is notably equal to the sum of these two integers.

 

It's a neat characteristic of consecutive numbers that reveals a direct connection between algebraic expressions and arithmetic operations.

Consider the task of determining the angle, denoted as (x), that solves a particular trigonometric puzzle. The puzzle involves balancing the square of a cosine function multiplied by two, and adding to that five times a sine function. The sum is set equal to four, and (x) must be found within the bounds of a full circular rotation, meaning anywhere from zero to \(( 2\pi )\) radians. What are the valid angles for (x) that resolve this equation?

Consider the algebraic expression ((x + k)⁷), where (k) is a real number. When we multiply this out, or 'expand' it, there is a term that includes (x⁵). The number in front of (x⁵) in this expanded form is called its coefficient. We're told that this coefficient is 63.

 

What we want to know is: What are the different numbers that (k) could be to make this happen?

Let's consider the mathematical function (f ) defined by the expression \((f(x) = \ln(x^2 - 16))\) which is valid for (x) values greater than 4.

 

On the graph of this function, there is a specific point labeled A where the function intersects the horizontal axis, indicating that (f (a) = 0). Additionally, at another point B on this graph, a line (L) serves as the tangent to the curve at B.