📘 Unit 4: Functions and Graphs – Exercise 4.3 (Solved)
Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska. Based on PECTAA 2026 syllabus (National Curriculum 2023).
📖 What's Inside: This exercise covers linear function applications (balance, fare, manufacturing cost, travel time, printing charges) and absolute value inequalities (temperature deviation, rod tolerance, machine alignment). Perfect for Punjab Boards exam preparation.
📚 Related Resources – Unit 4: Functions and Graphs
📑 Problems (1–8)
1 Balance after t months
\( B(t) = 5000 + 200t \) represents total balance (in rupees) after \( t \) months. Balance after 6 months?
\[
\begin{aligned}
B(t) &= 5000 + 200t \\
B(6) &= 5000 + 200(6) \\
B(6) &= 5000 + 1200 = 6200
\end{aligned}
\]
✅ The balance after 6 months will be Rs. 6200.
2 Fare for k kilometres
\( f(k) = 150 + 20k \) represents total fare (in rupees) for \( k \) km. Fare for 12 km ride?
\[
\begin{aligned}
f(k) &= 150 + 20k \\
f(12) &= 150 + 20(12) = 150 + 240 = 390
\end{aligned}
\]
✅ Total fare = Rs. 390.
3 Cost of sofa sets
\( f(n) = 5500n \) models cost of \( n \) sofa sets. Cost of 50 sofa sets?
\[
\begin{aligned}
f(n) &= 5500n \\
f(50) &= 5500 \times 50 = 275000
\end{aligned}
\]
✅ Cost = Rs. 275,000.
4 Travel time
\( T(d) = \frac{d}{60} \) represents time in hours for distance \( d \) km. Time for 180 km?
\[
\begin{aligned}
T(d) &= \frac{d}{60} \\
T(180) &= \frac{180}{60} = 3
\end{aligned}
\]
✅ It will take 3 hours.
5 Encoding and printing charges
\( f(x) = 100 + 5x \) where \( x \) = number of pages. Charge for 55 pages?
\[
\begin{aligned}
f(x) &= 100 + 5x \\
f(55) &= 100 + 5(55) = 100 + 275 = 375
\end{aligned}
\]
✅ Company will charge Rs. 375.
6 Chemical reaction temperature deviation
Stable at 37°C, stop if \( |T - 37| > 2.5 \). Find temperature values where process is stopped.
\[
\begin{aligned}
|T - 37| &> 2.5 \\
\Rightarrow T - 37 &> 2.5 \quad \text{or} \quad -(T - 37) > 2.5 \\
T &> 39.5 \quad \text{or} \quad -T + 37 > 2.5 \\
& \quad \Rightarrow -T > -34.5 \Rightarrow T < 34.5
\end{aligned}
\]
✅ Process must be stopped for \( T < 34.5^\circ \) or \( T > 39.5^\circ \).
7 Metal rod tolerance
Length must be \( 2.5 \pm 0.04 \) metres: \( |x - 2.5| \le 0.04 \). Range of acceptable lengths?
\[
\begin{aligned}
|x - 2.5| &\le 0.04 \\
-0.04 &\le x - 2.5 \le 0.04 \\
2.46 &\le x \le 2.54
\end{aligned}
\]
✅ Acceptable lengths: between 2.46 m and 2.54 m inclusive.
8 Machine alignment rejection
Model \( |x| > 0.1 \). Positions of centre (in mm) that cause rejection?
\[
\begin{aligned}
|x| &> 0.1 \\
x &> 0.1 \quad \text{or} \quad -x > 0.1 \\
& \Rightarrow x < -0.1
\end{aligned}
\]
✅ Rejected if \( x < -0.1 \, \text{mm} \) or \( x > 0.1 \, \text{mm} \).
📐 Key Concepts – Linear & Absolute Value Models
Linear Function: \( f(x) = mx + b \) where \( m \) is rate, \( b \) is fixed/initial value.
Absolute Value Inequality: \( |X| < a \Rightarrow -a < X < a \) ; \( |X| > a \Rightarrow X < -a \text{ or } X > a \) (for \( a>0 \)).
Tolerance Interpretation: \( |x - \text{target}| \le \text{tolerance} \) gives acceptable range: \( \text{target} - \text{tolerance} \le x \le \text{target} + \text{tolerance} \).
Unit 1✨ Complex Numbers
Unit 2Quadratic Equations
Unit 3Matrices & Determinants
Unit 4📗 Functions (Ex 4.1)
Unit 4📊 Absolute Value (Ex 4.2)
Unit 4📈 Linear Models (Current)
Unit 5Algebraic Fractions
Unit 6Vectors in Plane
Unit 7Trigonometry
Unit 8Chords & Arcs
Unit 9Tangent & Angles
Unit 10Practical Geometry
Unit 11Information Handling
Unit 12Probability
Created by Hira Science Academy | Aligned with PECTAA 2026 Syllabus