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Learn/ 9th Class Mathematics/ Unit 10 /Numericals

Unit 10: Unit 10 — Exercise 10 2

9th Class Mathematics · Unit 10: Unit 10

1.Plot the graph of y = 2x² - 4x + 3 from -1 to 3. Draw tangent at (2,3) and find gradient.
Given
Curve y = 2x2 - 4x + 3
Tangent point (2, 3)
Points on tangent line A(2, 3) and B(3, 7)
Formula
Gradient formula m = y2 - y1x2 - x1
Working
Substituting m = 7 - 33 - 2 = 41 = 4
Result
Gradient of tangent line m = 4
2.Plot the graph of y = 3x² + x + 1. Draw tangent at (1, 5). Also find gradient of the tangent line at this point.
Given
Curve y = 3x2 + x + 1
Tangent point (1, 5)
Points on tangent line A(1, 5) and B(2, 12)
Formula
Gradient formula m = y2 - y1x2 - x1
Working
Substituting m = 12 - 52 - 1 = 71 = 7
Result
Gradient of tangent line m = 7
3.The strength of students in a school was 1000 in 2016. If the strength decay according to the equation S = 1000e^(-t), where S is the number of students at time t. (a) Graph the given equation for t = 0 (in 2016) to t = 9 (in 2025). (b) From the graph, estimate the student's strength in 2019 and in 2023.
Given
Initial strength S0 = 1000 in 2016
Decay equation S = 1000e-t
Time range t = 0 to t = 9 (2016 to 2025)
Working
For 2019 (t = 3) S = 1000e-3 ≈ 49.8 ≈ 50
For 2023 (t = 7) S = 1000e-7 ≈ 0.91 ≈ 1
Result
Student's strength in 2019 S ≈ 50
Student's strength in 2023 S ≈ 1
4.The demand and supply functions for a product are given by the equations Pd = 400 - 5Q, Ps = 3Q + 24. Plot the graph of each function over the interval Q = 0 to Q = 300.
Given
Demand function Pd = 400 - 5Q
Supply function Ps = 3Q + 24
Quantity range Q = 0 to Q = 300
Demand function values calculated for various Q Q = 0: Pd = 400; quad Q = 50: Pd = 150; quad Q = 100: Pd = -100; quad Q = 150: Pd = -350
Supply function values calculated for various Q Q = 0: Ps = 24; quad Q = 50: Ps = 174; quad Q = 100: Ps = 324; quad Q = 150: Ps = 474
Result
Graphs plotted Table of values provided; graphs shown
5.Shahid's salary S(x) in rupees is based on the following formula: S(x) = 45000 + 4500x, where x is the number of years he has been with the company. Sketch and interpret the graph of salary function for 0 ≤ x ≤ 5.
Given
Salary function S(x) = 45000 + 4500x
Year range 0 le x le 5
Calculate table values x = 0: S = 45000; quad x = 1: S = 49500; quad x = 2: S = 54000; quad x = 3: S = 58500; quad x = 4: S = 63000; quad x = 5: S = 67500
Result
Starting salary S(0) = Rs. 45,000
Annual increase Rs. 4,500 per year
6.A company manufactures school bags. The cost function of producing x bags is C(x) = 1200 + 20x and the revenue from selling x bags is R(x) = 50x. (a) Find the break-even point. (b) Determine the profit or loss when 250 bags are sold. (c) Plot graphs of both the functions and identify the break-even point.
Given
Cost function C(x) = 1200 + 20x
Revenue function R(x) = 50x
(a) Break-even condition R(x) = C(x)
Working
Setting equal 50x = 1200 + 20x
Solving for x 50x - 20x = 1200 ⇒ 30x = 1200 ⇒ x = 40
Result
Break-even point x = 40 bags
(b) Profit function Profit = R(x) - C(x) = 50x - (1200 + 20x) = 30x - 1200
Working
At x = 250 Profit = 30(250) - 1200 = 7500 - 1200 = 6300
Result
Profit when 250 bags sold Profit = Rs. 6,300
7.A newspaper agency fixed cost of Rs.70 per edition and marginal printing and distribution costs of Rs. 40 per copy. Profit function is p(x) = 10x - 70, where x is the number of newspapers. Plot the graph and find profit for 500 newspaper.
Given
Profit function p(x) = 10x - 70
Number of newspapers x = 500
Working
Finding the profit p(500) = 10(500) - 70 = 5000 - 70
Calculating p(500) = 4930
Result
Profit for 500 newspapers p(500) = Rs. 4,930
8.Ali manufacturers expensive shirts for sale to a school. Its cost (in rupees) for x shirts is C(x) = 1500 + 10x + 0.2x², 0 ≤ x ≤ 150. Plot the graph and find the cost of 200 shirts.
Given
Cost function C(x) = 1500 + 10x + 0.2x2
Number of shirts x = 200
Working
Substituting x = 200 C(200) = 1500 + 10(200) + 0.2(200)2
Calculating constant term = 1500
Calculating linear term + 10(200) = 1500 + 2000 = 3500
Calculating quadratic term + 0.2(40000) = 3500 + 8000 = 11500
Result
Cost of 200 shirts C(200) = Rs. 11,500