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QM [ Mathematical equations and coordinate system ]

Understanding Mathematical Equations and the Coordinate System

Mathematics is often viewed as a complex and abstract field, but at its core lies a powerful language used to describe the world around us. Among the fundamental concepts in mathematics are equations and coordinate systems, which serve as essential tools for graphing and analyzing relationships between variables. This article explores mathematical equations and the coordinate system, providing insights into their significance and applications.

What Are Mathematical Equations?

Mathematical equations are statements that assert the equality of two expressions. They can take various forms, such as linear equations, quadratic equations, and polynomial equations, each representing different relationships between variables.

Linear Equations

Linear equations are perhaps the most fundamental type of equation. A linear equation in two variables represents a straight line. For example, the equation that represents a straight line can be expressed in various formats, and its graph will always form a straight line.

Quadratic Equations

Quadratic equations are another important class of equations, characterized by the highest exponent of the variable being two. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or using the quadratic formula. The graph of a quadratic equation produces a parabola, which opens either upward or downward depending on the coefficient of the variable.

The Coordinate System

The coordinate system provides a framework for graphing mathematical equations and visualizing relationships between variables. The most commonly used coordinate system is the Cartesian coordinate system, named after the mathematician René Descartes.

Cartesian Coordinate System

In a Cartesian coordinate system, the plane is divided into four quadrants by two perpendicular lines known as the x-axis (horizontal) and the y-axis (vertical). Each point in the plane can be represented by an ordered pair, where the first number indicates the horizontal position and the second number indicates the vertical position.

Plotting Points

To plot a point in the Cartesian plane:

Start at the origin (0, 0).
Move horizontally to the right or left based on the x-coordinate.
Move vertically up or down based on the y-coordinate.

For example, to plot the point (3, 2), you would move 3 units to the right on the x-axis and 2 units up on the y-axis.

Graphing Linear Equations

To graph a linear equation, you can find its intercepts. The y-intercept occurs when the horizontal coordinate is zero, while the x-intercept occurs when the vertical coordinate is zero. For instance, consider the equation that defines a straight line:

Find the y-intercept: Set the horizontal coordinate to zero and solve for the vertical coordinate.
Find the x-intercept: Set the vertical coordinate to zero and solve for the horizontal coordinate.

By plotting these two points and drawing a straight line through them, you can graph the equation.

The Importance of Mathematical Equations and the Coordinate System

Understanding mathematical equations and the coordinate system is crucial for various fields, including science, engineering, economics, and social sciences. These tools help model real-world phenomena, enabling predictions and analyses.

Real-World Applications

Physics: Equations describe motion, forces, and energy relationships. For example, relationships between force, mass, and acceleration can be expressed mathematically.
Economics: Mathematical equations model supply and demand relationships. Graphing these equations allows economists to visualize market equilibrium.
Engineering: Engineers use equations to design structures and systems. The coordinate system helps in visualizing dimensions and relationships between components.

Problem-Solving Skills

Mastering equations and the coordinate system enhances problem-solving skills. Students learn to analyze situations, formulate equations, and interpret graphical data, which are valuable skills in various professions.

Conclusion

Mathematical equations and the coordinate system are foundational elements of mathematics that provide powerful tools for understanding and analyzing relationships between variables. From simple linear equations to complex quadratic equations, these concepts allow us to model real-world phenomena and make informed decisions. By mastering these principles, students can develop critical thinking and problem-solving skills that will serve them well in their academic and professional pursuits.

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1. Total Revenue earned by Mr. X is Rs. 550,000 and variable cost and Fixed Cost incurred by
Mr. X is Rs. 270,000 and Rs. 100,000 respectively. Calculate the total contribution earned by
Mr. X is?

a) Rs. 270000

b) Rs 370000

c) Rs. 280,000

d) Rs. 180,000

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2. Which of the following is irrational?

a) √(4/9)

b) 4/5

c) √7

d) √(81)

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3. The total cost of producing 12 units is Rs. 1,500 and the total cost of producing 18 units is
Rs. 2,700. Find the variable cost per unit?

a) 125

b) 140

c) 130

d) 200

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4. 6 years ago the age of father was 3 times of his son’s age. After 9 years it will be twice the
age of his son. The present ages of father and son are:

a) 48 and 20

b) 51 and 21

c) 54 and 22

d) 57 and 23

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5. Seven years ago the age of a father was thrice of his son. After 7 yean the age of the father
will be twice that of his son. The present ages of the father and the son are?

a) (69,23)

b) (85,39)

c) (78,32)

d) none

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6. A line passes through a point (2,3) and cuts at x-intercept of 3. Find equation of that line.

a) Y= 3x+9

b) Y= 3x- 9

c) Y= - 9x+3

d) Y= - 3x+9

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7. 5 years ago the age of father was 3 times of his son’s age. After 7 years it will be twice the
age of his son. The present age of father and son is

a) 53 and 21

b) 41 and 17

c) 50 and 20

d) 47 and 19

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8. The equation representing a straight line is 7y=11x+3. The y-intercept is:

a) 3/7

b) 11/3

c) 7/3

d) 11/7

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9. The equation representing a straight line is 4y=5x+8. The slope of line is:

a) 1.25

b) 0.8

c) 0.5

d) 5

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10. Which of the following equations is not linear?

a) y=2x^2

b) X+9=0

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11. Kamran and Salman invested in a business. The sum of investment of Kamran and seven
times the investment of Salman amounts to Rs 18 million. Difference between thrice the
investment of Kamran and twice the investment of Salman is Rs 8 million. Amount invested
by Kamran and Salman is:

a) Rs 11 and 1 million respectively

b) Rs 7.5 and 1.5 million respectively

c) Rs 4 and 2 million respectively

d) Rs 8 million each.

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12. Zara and Bilal invested in a business. The sum of the investment of Zara and five times the investment of Bilal amounts to Rs 30 million. The difference between twice the investment of Zara and three times the investment of Bilal is Rs 6 million. How much did Zara and Bilal invest?

a) Rs 11 and 10 million respectively

b) Rs 20 and 15 million respectively

c) Rs 12 and 6 million respectively

d) Rs 8 million each.

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13. Sum of thrice of Arif and twice of Ali is 18,000 and difference of thrice of Arif and twice of
Ali is 15,000. How much amount Arif and Ali have?

a) 5500, 750

b) 3000,750

c) 4500,750

d) 2500,750

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14. Hamid and Sajid invested in a business. The sum of Hamid's investment and seven times Sajid's investment amounts to Rs 25 million. The difference between thrice Hamid's investment and twice Sajid's investment is Rs 10 million. How much did Hamid and Sajid invest?

a) Hamid: Rs 5.22 million, Sajid: Rs 2.83 million

b) Hamid: Rs 15 million, Sajid: Rs 10 million

c) Hamid: Rs 20 million, Sajid: Rs 5 million

d) Hamid: Rs 12 million, Sajid: Rs 8 million

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15. A two-digit number is equal to “4 times the sum of its digits” and “12 times equal to the
difference of its digits.” If xy represents the required number, solving which of the following
simultaneous equations could lead to that number?

a) 4x+xy=4y and 12x – xy=12y

b) Xy – 4y =4x and 12y – xy=12x

c) Xy – 4x = 4y and xy +12y =12x

d) Xy – 4x = 4y and xy – 12x =12y

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16. A line passes through (3,2). It has an X-intercept value which is thrice the y-intercept value.

a) y=x+1

b) y=−3x+11

c) y=−x+5

d) y=3x−7

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17. Which of the following equation is not linear?

a) Y=2x-7

b) x-y/5=20

c) y=5x^2

d) None of these

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18. Which of the following statements is correct in respect of the equation? X+2Y+3=0

a) The value of the intercept on the y-axis is 3

b) The value of the intercept on x-axis is –3

c) The slope of the line is 1

d) The degree of the equation is 0

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19. If a negative slope line passes through a point (4,8) and x intercept is ¼ of the y-intercept.
Find equation of straight line

a) 4x – y=24

b) X+4y=20

c) X – 4y=47

d) 4x + y = 32.

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20. Which of the following statements is correct in respect of the equation? 5X+2Y-10=0

a) The value of the intercept on the y-axis is 2

b) The value of the intercept on x-axis is –5

c) The slope of the line is 2.5

d) The degree of the equation is 1

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21.

A company has Revenue function R=9x and cost function c= -0.09x^2+2x+5000 you are required to determine Break-even point (in units) of this company.

a) 200 units

b) 400 units

c) 500 units

d) 700 units

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22. For the equation $y - 2 = 3 (x - 1)$ , what is the slope of the line?

a) 3

b) -3

c) 1

d) 2

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23. If the slope of a line is 0, which of the following statements is true?

a) The line is vertical.

b) The line is horizontal.

c) The line intersects the y-axis at 0.

d) The line has no slope.

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24. Consider the following equations:

$2 x + 3 y = 6$
$4 x - y = 11$

What is the value of $x + y$ ?

a) 5

b) 6

c) 7

d) 8

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25. Which of the following equations represents a line that is parallel to the x-axis?

a) y=2

b) x=3

c) 3y−6=0

d) y=−x+5

Your score is

The average score is 55%

CH 1 Mathematical equations and coordinate system.

Understanding Mathematical Equations and the Coordinate System

What Are Mathematical Equations?

Linear Equations

Quadratic Equations

The Coordinate System

Cartesian Coordinate System

Plotting Points

Graphing Linear Equations

The Importance of Mathematical Equations and the Coordinate System

Real-World Applications

Problem-Solving Skills

Conclusion

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Understanding Mathematical Equations and the Coordinate System

What Are Mathematical Equations?

Linear Equations

Quadratic Equations

The Coordinate System

Cartesian Coordinate System

Plotting Points

Graphing Linear Equations

The Importance of Mathematical Equations and the Coordinate System

Real-World Applications

Problem-Solving Skills

Conclusion

Leave a ReplyCancel Reply

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