Class 9 Mathematics Chapter 13: Introduction to Graphs – Important Questions & Answers

Introduction to Graphs is no longer treated as a 'bonus' chapter in CBSE Class 9 Mathematics. The 2024-25 rationalized syllabus emphasises graphical representation, data visualization, and coordinate geometry as core competencies. The CBSE board exam now allocates 12–15 marks specifically to graph-based questions across multiple question formats: MCQs test your recognition of coordinates and quadrants; short-answer questions assess your ability to plot and label points; while long-answer questions demand multi-step solutions involving line graphs, equations, and real-world scenarios (like tracking weather data or sales trends). Additionally, this chapter forms the foundation for Class 10 Linear Equations in Two Variables and Class 11 Coordinate Geometry—concepts that carry 20+ marks in senior secondary exams. The important questions in this guide reflect the exact difficulty levels and formats released by CBSE in recent sample papers and board examinations. Practising these ensures you're not just learning the chapter; you're training for examination success.

MCQs test quick recall of definitions, quadrant identification, and coordinate interpretation. These questions demand precision in 60 seconds or less. **Q1.** In the Cartesian plane, the point where the x-axis and y-axis intersect is called: (A) Ordinate (B) Origin (C) Abscissa (D) Quadrant **Answer: (B) Origin.** The origin is the point (0, 0) where both axes meet. **Q2.** The coordinates of a point are (−3, 5). Which quadrant does it lie in? (A) First (B) Second (C) Third (D) Fourth **Answer: (B) Second.** When x < 0 and y > 0, the point is in Quadrant II. **Q3.** On the y-axis, the x-coordinate of any point is always: (A) 1 (B) −1 (C) 0 (D) 2 **Answer: (C) 0.** Points on the y-axis have the form (0, y). **Q4.** The distance of the point (4, 3) from the x-axis is: (A) 4 units (B) 3 units (C) 7 units (D) 1 unit **Answer: (B) 3 units.** Distance from x-axis = |y-coordinate| = |3| = 3. **Q5.** Which point lies on the line y = 2x? (A) (1, 3) (B) (2, 4) (C) (3, 5) (D) (4, 7) **Answer: (B) (2, 4).** Substituting x = 2: y = 2(2) = 4. The point (2, 4) satisfies the equation.

These questions require brief explanations, simple calculations, or small diagrams. Answers should be 2–4 sentences with working shown. **Q1.** Plot the points A(2, 3), B(−2, 1), and C(0, −4) on the Cartesian plane. Write the quadrant or axis on which each point lies. **Solution:** - Point A(2, 3): x = 2 > 0, y = 3 > 0 → **Quadrant I** - Point B(−2, 1): x = −2 < 0, y = 1 > 0 → **Quadrant II** - Point C(0, −4): x = 0, y = −4 < 0 → Lies on the **negative y-axis** **Q2.** The point (p, −4) lies on the y-axis. Find the value of p. **Solution:** For any point on the y-axis, the x-coordinate is 0. Therefore, p = 0. The point is (0, −4). **Q3.** A line passes through the points (1, 2) and (3, 6). Check if the point (2, 4) also lies on this line. **Solution:** First, find the equation. Slope m = (6 − 2)/(3 − 1) = 4/2 = 2. Using point-slope form: y − 2 = 2(x − 1) → y = 2x. Checking (2, 4): 4 = 2(2) = 4 ✓ Yes, the point lies on the line. **Q4.** Write the coordinates of five points that satisfy the equation y = −x + 3. **Solution:** Choose any x-value and calculate y: - x = 0: y = 3 → (0, 3) - x = 1: y = 2 → (1, 2) - x = 2: y = 1 → (2, 1) - x = 3: y = 0 → (3, 0) - x = −1: y = 4 → (−1, 4) **Q5.** If a point is equidistant from both axes, what can you say about its coordinates? **Solution:** If a point (a, b) is equidistant from both axes, then |a| = |b|. This means the coordinates are either (a, a), (a, −a), (−a, a), or (−a, −a). Such points lie on the lines y = x or y = −x.

These require working, reasoning, or construction. Show all steps clearly. **Q1.** Plot the points P(1, 1), Q(4, 1), R(4, 4), and S(1, 4) on the Cartesian plane. Join them in order. What shape is formed? Calculate its perimeter. **Solution:** Plotting the points reveals a square (all sides are 3 units and all angles are 90°). - PQ = 4 − 1 = 3 units (horizontal) - QR = 4 − 1 = 3 units (vertical) - RS = 4 − 1 = 3 units (horizontal) - SP = 4 − 1 = 3 units (vertical) Perimeter = 3 + 3 + 3 + 3 = **12 units** **Q2.** A line passes through the origin and the point (2, 3). (i) Find the equation of the line. (ii) Does the point (4, 6) lie on this line? **Solution:** (i) Line passes through (0, 0) and (2, 3). Slope m = 3/2. Equation: **y = (3/2)x** (ii) Check (4, 6): 6 = (3/2)(4) = 6 ✓ **Yes, it lies on the line.** **Q3.** The following table shows the temperature (in °C) recorded at different times of a day. Plot this data on a line graph and answer: (a) At what time was the temperature highest? (b) What was the temperature at 2 PM? | Time | 6 AM | 9 AM | 12 PM | 3 PM | 6 PM | |------|------|------|-------|------|------| | Temp (°C) | 15 | 20 | 28 | 32 | 24 | **Solution:** Plot points (6, 15), (9, 20), (12, 28), (3, 32), (6, 24) on a graph with time on x-axis and temperature on y-axis. (a) The **highest temperature was 32°C at 3 PM**. (b) From the graph, **temperature at 2 PM ≈ 30°C** (interpolated between 12 PM and 3 PM). **Q4.** Verify that the three points A(0, −1), B(1, 1), and C(2, 3) are collinear (lie on the same line). **Solution:** Check if slope between A and B equals slope between B and C. Slope AB = (1 − (−1))/(1 − 0) = 2/1 = 2 Slope BC = (3 − 1)/(2 − 1) = 2/1 = 2 Since slopes are equal, **the three points are collinear**.

These are comprehensive, multi-part questions requiring detailed reasoning, graphs, or extended calculations. **Q1.** A transport company tracks the distance covered by a delivery van at different times. The data is given below: | Time (hours) | 0 | 1 | 2 | 3 | 4 | 5 | |--------------|---|---|---|---|---|---| | Distance (km) | 0 | 20 | 40 | 60 | 80 | 100 | (i) Plot the line graph for this data. (ii) Find the equation of the line. (iii) How much distance will the van cover in 6 hours? (iv) At what time will the van cover 150 km? **Solution:** (i) Plot points (0, 0), (1, 20), (2, 40), (3, 60), (4, 80), (5, 100) and join them with a straight line. (ii) From the graph, slope m = Δy/Δx = 100/5 = 20 km/h. Since it passes through origin, equation is **y = 20x** (where y = distance in km, x = time in hours). (iii) At x = 6: y = 20(6) = **120 km** (iv) At y = 150: 150 = 20x → x = 150/20 = **7.5 hours** **Q2.** A shop sells two types of notebooks: Type A and Type B. The profit (in ₹) from selling x notebooks of each type is given by: Type A: y = 5x + 50 Type B: y = 4x + 100 (i) Plot both lines on the same graph. (ii) For what value of x do both types give the same profit? (iii) If x = 30, which type is more profitable? **Solution:** (i) For Type A (y = 5x + 50): Points (0, 50), (10, 100), (20, 150), (30, 200) For Type B (y = 4x + 100): Points (0, 100), (10, 140), (20, 180), (30, 220) [Plot both lines on the same coordinate system] (ii) Set 5x + 50 = 4x + 100 → x = 50. Both give equal profit when x = 50 notebooks. (iii) At x = 30: Type A: y = 5(30) + 50 = 200 ₹ Type B: y = 4(30) + 100 = 220 ₹ **Type B is more profitable by ₹20.** **Q3.** Three vertices of a rectangle are A(1, 2), B(5, 2), and C(5, 6). (i) Find the coordinates of the fourth vertex D. (ii) Calculate the area and perimeter of the rectangle. (iii) Find the coordinates of the intersection point of the diagonals. **Solution:** (i) For a rectangle, opposite sides are parallel and equal. Since A(1, 2) and B(5, 2) have the same y-coordinate, AB is horizontal. C(5, 6) is directly above B. The fourth vertex D must be at **(1, 6)**. (ii) Length AB = 5 − 1 = 4 units. Width BC = 6 − 2 = 4 units. **Area = 4 × 4 = 16 square units** **Perimeter = 2(4 + 4) = 16 units** (iii) The diagonals of a rectangle intersect at their midpoints. Midpoint of AC = ((1+5)/2, (2+6)/2) = **(3, 4)** This is also the midpoint of BD: ((5+1)/2, (2+6)/2) = (3, 4) ✓

**Case Study:** A weather research centre monitors rainfall patterns across a region. The following table shows monthly rainfall (in mm) over 12 months: | Month | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | |-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----| | Rainfall (mm) | 10 | 15 | 20 | 40 | 80 | 150 | 200 | 180 | 100 | 50 | 25 | 12 | **Questions:** (a) Plot a line graph of the rainfall data on a coordinate system where x-axis represents months (1–12) and y-axis represents rainfall in mm. (b) Identify the month with maximum rainfall and the month with minimum rainfall. (c) Calculate the average rainfall for the year. (d) In which months was the rainfall below the average? Identify the coordinates of these points on your graph. (e) If the centre predicts next year's rainfall follows the pattern y = 1.1x + 5 (where x = month, y = rainfall in mm), will the rainfall in month 8 exceed this year's August rainfall? **Solution:** (a) Plot 12 points: (1, 10), (2, 15), (3, 20), (4, 40), (5, 80), (6, 150), (7, 200), (8, 180), (9, 100), (10, 50), (11, 25), (12, 12). Join with a smooth curve showing the seasonal trend. (b) **Maximum rainfall:** July (month 7) with **200 mm** **Minimum rainfall:** January (month 1) with **10 mm** (c) Average = (10 + 15 + 20 + 40 + 80 + 150 + 200 + 180 + 100 + 50 + 25 + 12) ÷ 12 = 882 ÷ 12 = **73.5 mm** (d) Months with rainfall below average (< 73.5 mm): January (10), February (15), March (20), April (40), May (80 is close—actually above), October (50), November (25), December (12). **Coordinates: (1, 10), (2, 15), (3, 20), (4, 40), (10, 50), (11, 25), (12, 12)** (e) Predicted August rainfall (month 8): y = 1.1(8) + 5 = 8.8 + 5 = 13.8 mm This year's August rainfall: 180 mm **No, the predicted rainfall (13.8 mm) is much less than this year's August rainfall (180 mm). The model significantly underestimates monsoon months and is unsuitable for this climate pattern.**

The questions above represent the full spectrum of Chapter 13 asked in CBSE board exams and competitive assessments. However, one-time practice isn't enough—conceptual clarity in graphs requires consistent, adaptive learning. That's where cbsetutor.ai's AI tutor comes in. Our platform generates unlimited, randomized practice questions across all difficulty levels in Introduction to Graphs. Every time you attempt a question, the AI tracks your exact weak point: Do you struggle with quadrant identification? Does plotting coordinates confuse you? Are you weak at reading data from line graphs? Our adaptive algorithm adjusts question difficulty and provides instant, personalised feedback—not generic solutions. You'll drill MCQs, then progress to short answers, then long answers, all within the same topic. The AI tutor mimics your exam hall experience with timed practice sessions and board-pattern question sequences. Plus, if you're stuck, our AI explains every step in simple, Hindi-friendly language. Start a 3-day free trial at cbsetutor.ai and experience graph mastery through personalized AI coaching—the same method used by 50,000+ CBSE students across India.

**Cartesian Plane:** A 2D coordinate system formed by two perpendicular number lines (x-axis and y-axis) intersecting at the origin (0, 0). **Coordinates & Quadrants:** Every point is represented as (x, y) where x is the abscissa and y is the ordinate. The plane is divided into 4 quadrants: - Quadrant I: x > 0, y > 0 - Quadrant II: x < 0, y > 0 - Quadrant III: x < 0, y < 0 - Quadrant IV: x > 0, y < 0 **Plotting Points:** Start from origin, move x units horizontally (right if positive, left if negative), then y units vertically (up if positive, down if negative). **Line Equations:** A linear equation y = mx + c represents a straight line where m is slope and c is y-intercept. Points (x, y) lying on the line satisfy this equation. **Reading Line Graphs:** Extract data using interpolation (between plotted points) and extrapolation (beyond plotted points). Always check axis labels and scale. **Collinearity:** Three points are collinear if they have equal slopes between consecutive pairs of points. These concepts build the foundation for higher mathematics and real-world data analysis.