Enhanced Curriculum Support
This is a comprehensive educational resource designed to provide students with the tools and guidance necessary to excel. This support system is structured to cater to various aspects of learning, ensuring that students are well-prepared for academic challenges and practical applications of mathematical concepts. Some are the key benefits are mentioned below:
Comprehensive Learning: This holistic approach helps students gain a thorough understanding of the subject. Practical Application: The resources encourage students to apply mathematical concepts to real-life scenarios, enhancing their practical understanding and problem-solving skills.
Critical Thinking and Reasoning: Value-Based and HOTS questions promote critical thinking and reasoning abilities. These skills are crucial for students to tackle complex problems and make informed decisions.
Exam Preparedness: Sample Question Papers and NCERT Exemplar Solutions provide ample practice for exams. They help students familiarize themselves with the exam format and types of questions, reducing exam anxiety.
Ethical and Moral Development: Value-Based Questions integrate ethical and moral lessons into the learning process, helping in the overall development of students' character and social responsibility. By incorporating these diverse elements, Enhanced Curriculum Support aims to provide a robust and well-rounded knowledge, preparing students for both academic success and real-world challenges.
Sample Questions
Sec A
1. The linear equation 2x – 5y = 7 has:
(a) No solution (b) Infinitely many solutions
(c) A unique solution (d) Two solutions
2. If (4, 19) is a solution of the equation y = ax + 3, then a = ? :
(a) 4 (b) 6 (d) 5
(c) 3 (d) 5
3. If the line represented by the equation 3x + ky = 9 passes through the points (2, 3), then the value of k is:
(a) 2 (b) 1
(c) 3 (d) 4
4. x = 2, y = 5 is a solution of the linear equation:
(a) 5 x + y = 7 (b) x + y = 7
(c) 5x + 2y = 7 (d) x + 2y = 7
5. Express y in terms of x in the equation 5y - 3x - 10 = 0:
(a) y =
(c) y =
6. Assertion (A): For all values of k, (
Reason (R): The linear equation ax + b = 0 can be expressed as a linear equation in two variables as ax + y + b = 0.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
7. The line represented by the equation x + y = 16 passes through (2, 14). How many more lines pass through the point (2, 14):
(a) 10 (b) 2
(c) many (d) 100
8. Which of the following point does not lie on the line y = 2x + 3?
(a) (-5, -7) (b) (-1, 1)
(c) (3, 9) (d) (3, 7)
9. If (-2, 5) is a solution of 2x + my = 11, then the value of ‘m’ is:
(a) -2 (b) 2
(c) 3 (d) -3
10. The graph of the linear equation 2x + 3y = 6 is a line which meets the x-axis at the point:
(a) (0,3) (b) (3,0)
(c) (2, 0) (d) (0,2)
11. Any solution of the linear equation 2x + 0y + 9 = 0 in two variables is of the form:
(a) (−
(c) (0,−
12. If (x, y) = (y, x), then:
(a) x – y = 0 (b) x + y = 0
(c) x ÷ y = 0 (d) xy = 0
13. If a linear equation has solutions (1, 2), (-1, -16) and (0, -7), then it is of the form:
(a) y = 9x - 7 (b) 9x - y + 7 = 0
(c) x - 9y = 7 (d) x = 9y - 7
14. Write the linear equation such that each point on its graph has an ordinate 5 times its abscissa.
(a) y = 5x (b) none of these
(c) 5x + y = 2 (d) x = 5y
Sec B
1. Find whether the given equation have x = 2, y = 1 as a solution: 2x + 5y = 9
(OR)
The following values of x and y are thought to satisfy a linear equation :
| variable | values | values |
|---|---|---|
| x | 1 | 2 |
| y | 1 | 3 |
2. Express the linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in 2x = -5y.
3. The sum of two numbers is 45. If one number is 5 more than three times the other, form a pair of linear equations and solve them to find the two numbers.
4. A train travels 150 km in the same time that a car travels 120 km. If the speed of the train is 20 km/h more than the speed of the car, form and solve a pair of linear equations to find the speeds of the train and the car.
5. Solve the system of equations by substitution: x + 2y = 7 , 3x − y = 4.
Sec C
1. Find the solution of the linear equation x + 2y = 8 which represents a point on:
(i) The x-axis (ii) The y-axis
2. Find solutions of the form x = a, y = 0 and x = 0, y = b for the following pairs of equations. Do they have any common such solution? 5x + 3y = 15 and 5x + 2y = 10
3. For what value of c, the linear equation 2x + cy = 8 has equal values of x and y for its solution?
4. Draw the graphs of y = x and y = -x in the same graph. Also find the co-ordinates of the point where the two lines intersect.
5. Find at least 3 solutions for the following linear equation in two variables: 5x + 3y = 4.
Sec D
1. Read the text carefully and answer the questions: Peter, Kevin James, Reeta and Veena were students of Class 9th B at Govt Sr Sec School, Sector 5, Gurgaon. Once the teacher told Peter to think a number x and to Kevin to think another number y so that the difference of the numbers is 10 (x > y). Now the teacher asked James to add double of Peter's number and that three times of Kevin's number, the total was found 120. Reeta just entered in the class, she did not know any number. The teacher said Reeta to form the 1st equation with two variables x and y. Now Veena just entered the class so the teacher told her to form 2nd equation with two variables x and y. Now teacher Told Reeta to find the values of x and y. Peter and kelvin were told to verify the numbers x and y.
(i) What are the equation formed by Reeta and Veena ?
(ii) What was the equation formed by Veena ?
(iii) Which number did Peter think ?
(or)
Which number did Kelvin think?
2. Read the text carefully and answer the questions: Ajay lives in Delhi, The city of Ajay's father in laws residence is at Jaipur is 600 km from Delhi. Ajay used to travel this 600 km partly by train and partly by car. He used to buy cheap items from Delhi and sale at Jaipur and also buying cheap items from Jaipur and sale at Delhi. Once From Delhi to Jaipur in forward journey he covered 2x km by train and the rest y km by taxi. But, while returning he did not get a reservation from Jaipur in the train. So first 2y km he had to travel by taxi and the rest x km by Train. From Delhi to Jaipur he took 8 hrs but in returning it took 10 hrs.
(i) Write the above information in terms of equation.
(ii) Find the value of x and y?
(iii) Find the speed of Taxi?
(or)
Find the speed of Train ?
3. A father is three times as old as his son. Five years ago, the father was four times as old as his son. Form a pair of linear equations and solve them to find their present ages.
4. Solve the system of linear equations graphically: x+ y = 5 , 2x − y = 4 Plot both lines and find the solution graphically.
5. A company manufactures pens and pencils. Each pen requires 2 hours of work and each pencil requires 3 hours of work. If the total working hours available are 120 hours, and the total number of pens and pencils produced is 40, form a system of equations and solve it to determine how many pens and pencils were made.
Value Based Questions
Problem 1
A company uses a formula to determine the salary of its employees based on years of experience:
Problem 2
A city plans to reduce water consumption by implementing a policy where the water bill increases linearly with usage. The equation representing the bill is y =
HOTS
Q1
Explain why the graph of a linear equation in two variables is always a straight line. What does the slope of this line represent?
Q2
A person travels from town A to town B at a constant speed. Represent the relationship between the time taken and the distance traveled as a linear equation. What do the slope and intercept of this line represent in real-world terms?
Q3
Consider the equations:
Q4
If the equation of a line is
NCERT Exemplar Solutions
Questions
1. The equation 2x + 5y = 7 has a unique solution, if x, y are :
(A) Natural numbers (B) Positive real numbers
(C) Real numbers (D) Rational numbers
2. The equation x = 7, in two variables, can be written as:
(A) 1 . x + 1 . y = 7 (B) 1. x + 0. y = 7
(C) 0 . x + 1 . y = 7 (D) 0 . x + 0 . y = 7
3. The positive solutions of the equation ax + by + c = 0 always lie in the:
(A) 1st quadrant (B) 2nd quadrant
(C) 3rd quadrant (D) 4th quadrant
4. If we multiply or divide both sides of a linear equation with a non-zero number, then the solution of the linear equation :
(A) Changes (B) Remains the same
(C) Changes in case of multiplication only (D) Changes in case of division only
5. How many linear equations in x and y can be satisfied by x = 1 and y = 2?
(A) Only one (B) Two
(C) Infinitely many (D) Three
6. Write whether the following statements are True or False? Justify your answers.
(i) ax + by + c = 0, where a, b and c are real numbers, is a linear equation in two variables.
(ii) A linear equation 2x + 3y = 5 has a unique solution.
(iii) All the points (2, 0), (–3, 0), (4, 2) and (0, 5) lie on the x-axis.
(iv) The line parallel to the y-axis at a distance 4 units to the left of y-axis is given by the equation x = – 4.
(v) The graph of the equation y = mx + c passes through the origin.
7. At what point does the graph of the linear equation x + y = 5 meet a line which is parallel to the y-axis, at a distance 2 units from the origin and in the positive direction of x-axis.
8. The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation of two variables and draw the graph of the same by taking the constant mass equal to 6 kg. Read from the graph, the force required when the acceleration produced is (i) 5
9. If the temperature of a liquid can be measured in Kelvin units as x°K or in Fahrenheit units as y°F, the relation between the two systems of measurement of temperature is given by the linear equation:
y =
(i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313°K.
(ii) If the temperature is 158° F, then find the temperature in Kelvin.
10. The Autorikshaw fare in a city is charged Rs 10 for the first kilometer and @ Rs 4 per kilometer for subsequent distance covered. Write the linear equation to express the above statement. Draw the graph of the linear equation.
11. The work done by a body on application of a constant force is the product of the constant force and the distance travelled by the body in the direction of force. Express this in the form of a linear equation in two variables and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units. Verify it by plotting the graph.
Case Based Questions
Q1
A school is organizing a picnic. The cost of booking a bus for the picnic is ₹1200. The food cost for each student is ₹150. The total cost for 20 students is ₹3000, including the bus cost.
1. Write the pair of linear equations for the given situation.
2. Find the number of students and the cost of booking the bus.
Sol 1
1. Let the number of students be x and the cost of booking the bus be ₹y.
The total cost of food for x students is 150x.
The total cost is the sum of the bus cost and the food cost, which is ₹3000.
Thus, the equations are:
150x + y = 3000(1)
y = 1200 (since the bus cost is constant) (2)
2. Substitute
y = 1200 from equation (2) into equation (1):
150x + 1200 = 3000 ⟹ 150x = 1800 ⟹x=
Answer: The number of students is 12, and the cost of booking the bus is ₹1200.