IB Mathematics SL 2.4 Key features of graphs AA SL Paper 1- Exam Style Questions- New Syllabus
The functions \( f \) and \( g \) are defined by:
\( f(x) = \ln(2x – 9) \), where \( x > \frac{9}{2} \)
\( g(x) = 2\ln x – \ln d \), where \( x > 0 \), \( d \in \mathbb{R}^+ \)
Part (a):
State the equation of the vertical asymptote to the graph of \( y = g(x) \).
Part (b):
The graphs of \( y = f(x) \) and \( y = g(x) \) intersect at two distinct points.
(i) Show that, at the points of intersection, \( x^2 – 2dx + 9d = 0 \).
(ii) Hence show that \( d^2 – 9d > 0 \).
(iii) Find the range of possible values of \( d \).
Part (c):
In the case where \( d = 10 \), find the value of \( q – p \), where the graphs intersect at \( x = p \) and \( x = q \), \( p < q \). Express your answer in the form \( a\sqrt{b} \), where \( a, b \in \mathbb{Z}^+ \).
The following diagram shows part of the graphs of \( y = f(x) \) and \( y = g(x) \).
▶️ Answer/Explanation
Part (a)
For \( g(x) = 2\ln x – \ln d \), rewrite using logarithmic properties:
\[ g(x) = \ln(x^2) – \ln d = \ln\left(\frac{x^2}{d}\right) \]
The vertical asymptote occurs when \( \frac{x^2}{d} = 0 \), so \( x^2 = 0 \), \( x = 0 \). As \( x \to 0^+ \), \( g(x) \to -\infty \).
Answer: \( x = 0 \)
Part (b)
(i) Set \( f(x) = g(x) \):
\[ \ln(2x – 9) = 2\ln x – \ln d \]
Rewrite: \( \ln(2x – 9) = \ln\left(\frac{x^2}{d}\right) \).
\[ 2x – 9 = \frac{x^2}{d} \]
Multiply by \( d \): \( d(2x – 9) = x^2 \).
\[ x^2 – 2dx + 9d = 0 \]
(ii) For two real roots, discriminant of \( x^2 – 2dx + 9d = 0 \):
\[ \Delta = (-2d)^2 – 4 \cdot 1 \cdot 9d = 4d^2 – 36d \]
\[ 4d(d – 9) > 0 \]
Since \( d > 0 \), \( d > 9 \). Thus, \( d^2 – 9d > 0 \).
(iii) From (ii), \( d > 9 \).
Answer: \( d > 9 \)
Part (c)
For \( d = 10 \), solve: \( \ln(2x – 9) = 2\ln x – \ln 10 \).
\[ \ln(2x – 9) = \ln\left(\frac{x^2}{10}\right) \]
\[ 2x – 9 = \frac{x^2}{10} \]
\[ x^2 – 20x + 90 = 0 \]
Discriminant: \( \Delta = (-20)^2 – 4 \cdot 1 \cdot 90 = 400 – 360 = 40 \).
\[ x = \frac{20 \pm \sqrt{40}}{2} = 10 \pm \sqrt{10} \]
\[ p = 10 – \sqrt{10}, \quad q = 10 + \sqrt{10} \]
\[ q – p = (10 + \sqrt{10}) – (10 – \sqrt{10}) = 2\sqrt{10} \]
Answer: \( 2\sqrt{10} \)