M23-5-MATHX-HP1-ENG-TZ1-XX

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Keycap Digit One

Paper 1

Question 1 - MaximumMarks - 7

Consider a function f which is constructed using the formula \(( \frac{7x+7}{2x-4} )\). This formula is applicable for all real numbers (x), except when (x) equals 2.

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In essence, we have a mathematical rule that assigns to every eligible number (x), a corresponding value generated by the formula, with the sole exclusion of the number 2 where this rule is undefined.

Question 1 - A

Determine the value of (x) for which the function&nbsp;f&nbsp;(x)&nbsp;equals zero. This inquiry involves identifying the point at which the function intersects the horizontal axis, a fundamental concept in algebra.

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Question 1 - A - Video

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Question 1 - A - Answer

To find the zero of f (x), we need to solve for (x) when (&nbsp;f&nbsp;(x) = 0).

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Starting with the equation:

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\([0 = \frac{7x + 7}{2x - 4}]\)

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We can multiply both sides of the equation by the denominator (2x - 4) to eliminate the fraction:

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\([0 \cdot (2x - 4) = 7x + 7]\)

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Simplifying, we get:

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[0 = 7x + 7]

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Now, to find the value of (x), we subtract 7 from both sides of the equation:

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[-7 = 7x ]

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Dividing both sides by 7 gives us:

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[x = -1]

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Thus, the zero of the function f&nbsp;(x) is (x = -1).

Question 1 - A - Answer - EC

The candidate correctly identified the need to set the function equal to zero and proceeded to find the value of (x) that satisfies this condition. The steps involved multiplying both sides of the equation by the denominator to remove the fraction, which is a valid algebraic operation, as it does not change the equality (provided \((x \neq 2 )\), which is given in the function&#39;s domain). The student then correctly applied basic algebraic techniques to isolate (x) and solve the equation. This approach is methodical and shows a clear understanding of the algebraic processes required to solve for zeros of a rational function.

Question 1 - B

Consider the function y = f&nbsp;(x). Could you formulate the corresponding equation for this function&#39;s graph?

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Question 1 - B - Video

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Question 1 - B - i

Consider a rational function where its graph approaches but never reaches a certain line on the x-axis. What is the equation of this line that the graph never touches?

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This line represents a boundary that the function will never cross, no matter how far the graph is extended along the x-axis.

Question 1 - B - i - Answer

The vertical asymptote of a rational function occurs where the denominator is zero while the numerator is nonzero, leading to undefined values for the function. For the given function \(f (x) = \frac{7x+7}{2x-4}\), the denominator (2x - 4) is zero when (x = 2). However, the numerator (7x + 7) is not zero at (x = 2) (since (7(2)&nbsp;+ 7 = 21) which is not equal to zero). Therefore, the equation of the vertical asymptote is (x = 2). It is important to note that the value (x = -2) is excluded from the domain, but it does not affect the vertical asymptote since the denominator is not zero at (x = -2).

Question 1 - B - i - Answer - EC

The candidate correctly identified the location of the vertical asymptote by solving the equation (2x - 4 = 0), which is where the denominator of the function f&nbsp;(x) equals zero. The student also rightly checked the numerator to ensure it is not zero at (x = 2), which is a crucial step in determining the existence of a vertical asymptote. The response is clear, concise, and directly addresses the given function, with the correct notation and methodology. The answer provided is complete and would be awarded the full mark available for this part of the question.

Question 1 - B - ii

Consider a function with a line that the graph approaches but never touches or crosses. What would you call this particular line? It&#39;s constant and extends across the graph horizontally.

Question 1 - B - ii - Answer

The horizontal asymptote of the function \(f(x) = \frac{7x+7}{2x-4}\) is determined by examining the leading coefficients of the numerator and denominator since both are polynomials of the same degree. In this case, both the numerator and the denominator are first-degree polynomials.

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To find the horizontal asymptote, we divide the leading coefficient of the numerator by the leading coefficient of the denominator. For the given function, the leading coefficient of the numerator (7) divided by the leading coefficient of the denominator (2) gives us:

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\(( y = \frac{7}{2})\)

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Therefore, the equation of the horizontal asymptote is (y = 3.5).

Question 1 - B - ii - Answer - EC

The candidate correctly identified that for a rational function where the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The calculation was straightforward and correctly executed, resulting in \((y = \frac{7}{2})\) or (y = 3.5), which is the expected answer according to the standard rules for determining horizontal asymptotes for rational functions. The response is concise, accurate, and displays a clear understanding of the concepts involved.

Question 1 - C

Let&#39;s explore how to construct the counterpart of a given function, f&nbsp;(x). In other words, determine the function that will reverse the effect of f&nbsp;(x). This reverse function is often denoted as f&nbsp;-1(x).

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This is akin to finding a two-way street on a map. If f&nbsp;(x) represents the route from A to B, then f&nbsp;&minus;1(x) would represent the path from B back to A. It&#39;s essential to figure out the original input based on the output.

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Question 1 - C - Video

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Question 1 - C - Answer

To determine the inverse of the function \(f(x) = \frac{7x + 7}{2x - 4}\), we start by replacing f&nbsp;(x) with (y) and then interchange the roles of (x) and (y). This yields the equation \((x = \frac{7y + 7}{2y - 4})\).

 
By cross multiplying, we obtain (2xy - 4x = 7y + 7). We then group all terms containing (y) on one side to get (2xy - 7y = 4x + 7) and factor out (y) as (&nbsp;y(2x - 7) = 4x + 7).

 
Finally, we isolate (y) to find the inverse function:\(( f^{-1}(x) = \frac{4x + 7}{2x - 7} )\).

Question 1 - C - Answer - EC

The candidate&#39;s response correctly follows the steps for finding the inverse of a function. The interchanging of (x) and (y) was done properly, and the subsequent algebraic manipulations were accurately performed. The solution shows clear understanding of the concept of an inverse function and the necessary algebraic skills to isolate (y). The final expression for&nbsp;\( f^{-1}(x) \) is correctly simplified and presented.

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It is important to note that the domain restriction\( (x \neq 2) \)in the original function translates to a restriction in the inverse function&#39;s domain, which the candidate has implicitly acknowledged by solving for the inverse correctly. This detail, although not explicitly mentioned, is inherent in the correct application of the inverse function formula.

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Overall, the response demonstrates a high level of mathematical proficiency consistent with the expectations of an IB Mathematics AA HL examination.

Question 2 - MaximumMarks - 6

Imagine that on a particular Monday, we decided to conduct a survey at an amusement park. To get a fair representation, we selected 40 visitors at random as they were exiting the park. We were curious about the popularity of a specific ride, so we asked these visitors how many times they had ridden The Dragon that day. Their responses were neatly summarized in a frequency table.

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This survey wasn&#39;t just for our curiosity&mdash;it was conducted under the assumption that these 40 visitors&#39; experiences would be reflective of the general crowd for the next day. In other words, we expected Tuesday&#39;s visitors to have similar ride experiences on The Dragon.

Question 2 - Table

Question 2 - A

For the ensuing Tuesday, calculate the probabilistic forecast considering the given parameters.

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Simply put, the task is to use the provided details to predict the likelihood of an event occurring on Tuesday.

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Higher Level

Mathematics AA HL

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Standard Level

Mathematics AA SL

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Business Management HL

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Business Management SL

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Core

Theory of Knowledge

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Ab Initio

Spanish Ab Initio

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Biology SL

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Physics HL

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Chemistry HL

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Chemistry SL

Economics HL

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Economics SL

History HL

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History SL

Geography SL

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Geography HL

Mathematics AI HL

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Mathematics AI SL

Biology HL

Psychology SL

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Psychology HL

Design Technology SL

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Design Technology HL

Physics SL

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Environmental Systems & Societies SL

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Sports, Exercise & Health Science HL

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Sports, Exercise & Health Science SL

English B SL

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English B HL

English A Language & Literature HL

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English Literature SL

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English A Language & Literature SL

English Literature HL

Spanish B SL

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Question 3 - MaximumMarks - 6

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Question 5 - MaximumMarks - 6

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Question 8 - MaximumMarks - 7

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Question 10 - MaximumMarks - 14

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Question 11 - MaximumMarks - 19

IB MATH AA HL, Paper 1, May, 2023, TZ1, Solved Past Paper