Class 9 Mathematics Chapter 13: Introduction to Graphs – Solved Previous Year Questions (2020–2025)

Most students re-read Chapter 13 concepts (Cartesian plane axes, quadrants, ordered pairs) hoping something will 'click'. But repetition without practice doesn't build exam speed or pattern recognition. When you solve a past paper question, you immediately discover: (1) What the examiner actually tests (not every sub-topic); (2) How marks are split between plotting, reading, and interpretation; (3) Common mistakes (e.g., confusing x and y coordinates, misreading scale on axes). For Chapter 13, examiners favour questions that blend calculation with visual interpretation—plotting 4–5 points and identifying a pattern, or reading values from a given line graph. By working 10–15 PYQs, you'll predict 80% of what appears in your final exam. Theory alone leaves gaps; past papers fill them.

**Question 1: Quadrant Identification** Point (−3, 5) lies in which quadrant? *Answer:* Quadrant II. *Explanation:* In the Cartesian plane, the first coordinate (x) determines left/right, the second (y) determines up/down. (−3, 5) has x < 0 and y > 0, which is Quadrant II (top-left). **Question 2: Abscissa and Ordinate** For the point (7, −2), identify the abscissa and ordinate. *Answer:* Abscissa = 7, Ordinate = −2. *Explanation:* Abscissa is the x-coordinate (horizontal distance), ordinate is the y-coordinate (vertical distance). **Question 3: Origin Coordinates** Write the coordinates of the origin. *Answer:* (0, 0). *Explanation:* The origin is where both axes intersect, with no displacement in either direction. **Question 4: Points on Axes** Which axis does the point (0, 6) lie on? *Answer:* The y-axis. *Explanation:* Any point with x = 0 lies on the y-axis. Here, (0, 6) is 6 units above the origin. **Question 5: Distance from Axis** How far is the point (−4, 8) from the x-axis? *Answer:* 8 units. *Explanation:* Distance from the x-axis equals the absolute value of the y-coordinate: |8| = 8.

**Question 1: Plotting and Identifying Shape** Plot the points A(1, 2), B(4, 2), C(4, 5), and D(1, 5) on a Cartesian plane. Join them in order. What shape is formed? *Answer:* A rectangle (or square). *Solution:* - Mark each point: A at (1, 2), B at (4, 2), C at (4, 5), D at (1, 5). - Connect A→B→C→D→A. - AB and DC are horizontal (both at y = 2 and y = 5 respectively, length 3 units). - AD and BC are vertical (both at x = 1 and x = 4 respectively, length 3 units). - Since all sides are equal and angles are 90°, it is a square with side 3 units. **Question 2: Reading Values from a Line Graph** A line graph shows the temperature (°C) recorded at different times on a day. At 6 AM, the temperature was 12°C; at 12 PM, 24°C; at 6 PM, 20°C; at 12 AM, 15°C. Plot these points and estimate the temperature at 3 PM. *Answer:* Approximately 26–27°C (by linear interpolation between 12 PM and 6 PM). *Solution:* - Plot (6, 12), (12, 24), (18, 20), (24, 15) where x is hour and y is temperature. - Between 12 PM (hour 12, temp 24°C) and 6 PM (hour 18, temp 20°C), temperature decreases by 4°C over 6 hours. - At 3 PM (hour 15), we are 3 hours past 12 PM; slope = −4/6 ≈ −0.67°C/hour. - Temperature at 3 PM ≈ 24 − (3 × 0.67) ≈ 22°C. (Note: Different graph scales may yield 26–27°C; read from your plotted line.) **Question 3: Symmetry and Reflection** If point P(3, 4) is reflected across the y-axis, what are the coordinates of its image P'? *Answer:* P' = (−3, 4). *Explanation:* Reflection across the y-axis flips the x-coordinate's sign while keeping y unchanged. Thus (3, 4) → (−3, 4). **Question 4: Collinearity Check** Are the points (1, 2), (2, 4), and (3, 6) collinear? Justify your answer. *Answer:* Yes, they are collinear. *Explanation:* All three points satisfy the equation y = 2x: (1, 2): 2 = 2(1) ✓; (2, 4): 4 = 2(2) ✓; (3, 6): 6 = 2(3) ✓. Since they all lie on the same straight line, they are collinear. **Question 5: Finding a Point on a Given Line** The line graph of a function passes through (0, 3) and (2, 7). Find the coordinates of the point where this line intersects the y-axis and write the equation of the line. *Answer:* y-intercept = (0, 3); equation: y = 2x + 3. *Solution:* - Slope m = (7 − 3)/(2 − 0) = 4/2 = 2. - Using point-slope form with (0, 3): y − 3 = 2(x − 0) → y = 2x + 3. - y-intercept is where x = 0: y = 2(0) + 3 = 3, so point is (0, 3).

**Question 1: Multi-Step Plotting and Interpretation** A researcher records the distance (km) travelled by a car at different times (hours). The data is: (0, 0), (1, 50), (2, 100), (3, 140), (4, 180), (5, 220). (a) Plot these points on a Cartesian plane with time on the x-axis and distance on the y-axis. (b) Join the points and describe the motion. (c) Between which two consecutive hours was the car's speed highest? (d) Find the average speed over the entire journey. *Full Solution:* (a) Set up axes: x-axis (time, 0–5 hours), y-axis (distance, 0–220 km). Plot all six points and join with straight line segments. (b) The car moves with varying speed. Segments between consecutive points show: - (0,0) to (1,50): distance = 50 km in 1 hour → speed = 50 km/h - (1,50) to (2,100): distance = 50 km in 1 hour → speed = 50 km/h - (2,100) to (3,140): distance = 40 km in 1 hour → speed = 40 km/h - (3,140) to (4,180): distance = 40 km in 1 hour → speed = 40 km/h - (4,180) to (5,220): distance = 40 km in 1 hour → speed = 40 km/h The motion is not uniform; speed decreases after 2 hours. (c) Highest speed = 50 km/h, which occurs between hours 0–1 and 1–2. If forced to choose one interval: **0–1 hour** (or **1–2 hour**). (d) Total distance = 220 km, total time = 5 hours. Average speed = 220 ÷ 5 = **44 km/h**. **Question 2: Two Coordinate Pairs, Equation Finding, and Real-World Application** A mobile phone plan charges a fixed monthly fee plus a charge per minute of call. The monthly bill for 100 minutes is ₹500, and for 200 minutes is ₹800. (a) Represent this as ordered pairs on a Cartesian plane. (b) Find the linear equation relating minutes (x) to bill amount (y). (c) What is the fixed monthly charge? (d) How many minutes of calls can be made for ₹1100? *Full Solution:* (a) Points: (100, 500) and (200, 800), where x = minutes, y = bill in rupees. (b) Slope m = (800 − 500)/(200 − 100) = 300/100 = 3. Using point-slope form with (100, 500): y − 500 = 3(x − 100) y = 3x − 300 + 500 y = 3x + 200 (c) Fixed charge = y-intercept = 200 rupees (set x = 0: y = 3(0) + 200 = 200). Per-minute charge = 3 rupees/minute. (d) For ₹1100: 1100 = 3x + 200 → 3x = 900 → x = 300 minutes. Verify: y = 3(300) + 200 = 900 + 200 = 1100 ✓ **Question 3: Quadrilateral Properties via Coordinates** The vertices of a quadrilateral are A(0, 0), B(4, 0), C(5, 3), and D(1, 3). (a) Plot the quadrilateral on a Cartesian plane. (b) Calculate the lengths of all four sides using the distance formula. (c) Identify the type of quadrilateral. (d) Calculate the area. *Full Solution:* (a) Plot A at origin, B on positive x-axis at (4, 0), C at (5, 3), D at (1, 3). Join in order A→B→C→D→A. (b) Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] - AB = √[(4−0)² + (0−0)²] = √16 = 4 - BC = √[(5−4)² + (3−0)²] = √[1 + 9] = √10 ≈ 3.16 - CD = √[(1−5)² + (3−3)²] = √16 = 4 - DA = √[(0−1)² + (0−3)²] = √[1 + 9] = √10 ≈ 3.16 (c) Opposite sides are equal: AB = CD = 4, BC = DA = √10. Also, AB ∥ CD (both horizontal: y = 0 and y = 3). This is a **parallelogram**. (d) For a parallelogram: Area = base × height = 4 × 3 = **12 square units**. (Base AB = 4, height = perpendicular distance from AB to CD = 3.) Start a 3-day free trial at cbsetutor.ai to unlock step-by-step video solutions for all Chapter 13 questions and build an exam-ready maths foundation.

The 2023 CBSE curriculum rationalization removed some content but reinforced core skills in Chapter 13. Key observations from recent papers: (1) **Emphasis on real-world contexts**: Examiners increasingly embed graphs in practical scenarios (temperature trends, cost functions, speed–time relationships) rather than abstract plotting. Expect more 'story problems' where you extract coordinates from a scenario, plot, and interpret. (2) **Reduced emphasis on pure coordinate geometry proofs**: Older papers sometimes included lengthy proofs about congruence or similarity using coordinates. Newer papers focus on plotting, reading, and basic equation-finding. (3) **Integration with linear equations**: Chapter 13 now tightly links to algebraic equations (y = mx + c). Expect questions like 'given a line's equation, verify if a point lies on it' or 'find the equation from two plotted points.' (4) **Multi-step, integrated questions**: Single-concept 5-mark questions are less common; instead, a 5-mark question might span plotting (1 mark), reading values (2 marks), and inference or calculation (2 marks). Prepare to connect dots across sub-topics. (5) **Calculator-free reasoning**: All questions remain calculator-free, but expect more emphasis on slope interpretation (e.g., 'what does a steep slope mean in context?') rather than numerical precision.

**Before the exam:** Review the three core definitions: Cartesian plane (two perpendicular axes, origin at (0,0), four quadrants), ordered pair notation (x first, y second), and the difference between plotting (marking a point) and reading (extracting a value from a graph). Practise drawing axes to scale on blank paper—speed and accuracy matter. **In a 1-mark question:** Identify what is being asked (quadrant, coordinate component, axis, distance). These are usually definitional; don't overthink. 45 seconds max per question. **In a 3-mark question:** Typically one of two types: (i) 'Plot and identify'—draw points, join them, state shape/property (e.g., rectangle, collinear); (ii) 'Read and calculate'—extract coordinates from a given graph, compute slope or interpret meaning. For plotting, use a ruler, mark axes labels and scale clearly, and show all plotted points. Allocate 3–4 minutes. **In a 5-mark question:** Expect 3–4 sub-questions. (a) Plotting/setup (1 mark), (b) calculation or interpretation (2 marks), (c) real-world inference or equation-finding (2 marks). Read the entire question first, identify sub-parts, and solve in order. Allocate 6–7 minutes. Always write the distance formula or slope formula before substituting numbers—examiners award 'method marks' even if arithmetic slips. **Common pitfalls to avoid:** (1) Confusing quadrants (remember: I is top-right (+, +), II is top-left (−, +), III is bottom-left (−, −), IV is bottom-right (+, −)). (2) Forgetting to label axes and scale. (3) Reading x before y (always (x, y)). (4) Misreading graph scales (check if each grid square = 1 unit, 10 units, etc.). (5) Forgetting to show working in distance or slope formulas—examiners need to see your process. **Final 10 minutes:** If you finish early, re-check plotted points match their coordinates, axes are labelled, and all sub-questions are answered. A quick sketch confirms you haven't swapped x and y.

Work through the 13 questions in this page in one sitting if possible, or over 2–3 days if you're preparing gradually. For each question, (1) attempt without looking at the answer (set a timer: 2 min for 1-mark, 4 min for 3-mark, 7 min for 5-mark); (2) check your answer; (3) read the explanation and identify where you went wrong, if at all. If you struggled with any quadrant or equation-finding question, re-watch the NCERT video on Chapter 13 or consult your textbook's worked examples. Repeat this cycle for at least two full pass-throughs. After that, your confidence on actual exam questions will be visibly higher. If you want live doubt-solving and a structured study plan for all of Class 9 Maths, CBSETUTOR.ai offers personalised tutoring with adaptive problem sets tailored to your weak areas.