Question 10 - End-of-Chapter Exercises - Chapter 2 Class 9 - Introduction to Linear Polynomials (Ganita Manjari

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Question 10 (i) The graph of a linear polynomial p(x) passes through the points (1, 5) and (3, 11). (i) Find the polynomial p(x). Since it is a linear polynomial, its equation is y = ax + b Or p(x) = ax + b Given that p(x) passes through (1, 5) Putting x = 1, y = 5 5 = a × 1 + b 5 = a + b a + b = 5 Also, Given that p(x) passes through (3, 11) Putting x = 3, y = 1 11 = a × 3 + b 11 = 3a + b 3a + b = 11 Now, our equations are a + b = 5 …(1) 3a + b = 11 …(2) Doing (2) – (1) (3a + b) – (a + b) = 11 – 5 3a + b – a – b = 6 (3a – a) + (b – b) = 6 2a = 6 a = 6/2 a = 3 Putting a = 3 in (1) a + b = 5 3 + b = 5 b = 5 – 3 b = 2 Now, putting values of a & b in our expression y = ax + b y = 3x + 2 Thus, p(x) = 3x + 2 Question 10 (ii) (ii) Find the coordinates where the graph of p(x) cuts the axes. Now, our equation is y = 3x + 2 Coordinates when p(x) cuts x-axis In x-axis, y = 0 Putting y = 0 in our equation 0 = 3x + 2 0 – 2 = 3x –2 = 3x 3x = –2 x = (−𝟐)/𝟑 Thus, ((−𝟐)/𝟑, 0) is point where p(x) cuts x-axis Coordinates when p(x) cuts y-axis In y-axis, x = 0 Putting x = 0 in our equation y = 3 × 0 + 2 y = 0 + 2 y = 2 Thus, (0, 2) is point where p(x) cuts x-axis Question 10 (iii) (iii) Draw the graph of p(x) and verify your answers. We need to draw y = 3x + 2 Finding points on the line Putting x = 0 y = 3 × 0 + 2 y = 0 + 2 y = 2 So, x = 0, y = 2 lie on the line i.e. (0, 2) lies on the line Putting x = 1 y = 3 × 1 + 2 y = 3 + 2 y = 5 So, x = 1, y = 5 lie on the line i.e. (1, 5) lies on the line We plot all the points in the graph and join the points y = 3x + 2