Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) Write one linear equation in two variables whose graph passes through the origin.
Perfect! Any equation of the form ax + by = 0 passes through origin (0,0).
(2) Write the y-intercept of the equation 3x + 4y = 12. y-intercept =
Excellent! The y-intercept is 3 (point: (0, 3)).
(3) Find the x-intercept of the equation 4x - 5y = 20. x-intercept =
Correct! The x-intercept is 5 (point: (5, 0)).
(4) Write one point lying on the line x + y = 10.
Great! Any point whose coordinates sum to 10 lies on this line.
(5) Write the equation of a line parallel to x-axis and passing through (0, -5).
Perfect! Lines parallel to x-axis have the form y = constant.
Short Answer Questions (2 Marks Each)
Note: Answer each question with complete graph plotting, detailed calculations, and step-by-step solutions. Write down the answers on sheet and submit to the school subject teacher.
(1) Verify whether the point (2, -1) lies on the line 3x - y = 7.
Excellent verification!
(2) Find the value of k if the point (k, 2) lies on the line 2x + 3y = 12. k =
Perfect! The value of k is 3.
(3) Determine the coordinates of the points where the line 3x + 2y = 12 cuts the x-axis and y-axis.
x-intercept:
y-intercept:
Excellent! The line cuts axes at (4, 0) and (0, 6).
(4) Write the equation of a line passing through (3, 4) and parallel to the x-axis.
Perfect! All points on this line have y-coordinate = 4.
(5) Write the equation of a line passing through (0, -2) and parallel to the y-axis.
Great! This is actually the y-axis itself.
Long Answer Questions (4 Marks Each)
Note: Answer each question with complete graph plotting, detailed calculations, and step-by-step solutions. Write down the answers on sheet and submit to the school subject teacher.
(1) Draw the graphs of the equations 2x + y = 6 and x - y = 2 on the same graph paper. Find their point of intersection.
Point of intersection: x=
(2) Solve graphically: x + y = 5 and x - y = 1.
Graphical solution: x =
The lines intersect at (3, 2).
(3) The sum of two numbers is 10. Represent this situation algebraically and draw the graph of the equation. Also, find two solutions from the graph. Assume the two numbers be x and y.
Algebraic representation:
Two solutions:
Any point on the line represents two numbers that sum to 10.
(4) A person has ₹ 60 to buy pens and pencils. A pen costs ₹ 5 and a pencil costs ₹ 2. Represent the situation by a linear equation and draw its graph. Assume x = number of pens, y = number of pencils.
Linear equation:
x-intercept: x =
y-intercept: y =
Plot these points and draw the line representing spending combinations.
(5) Two numbers are such that twice the first plus thrice the second is 12. Draw the graph of the equation representing the situation and write three solutions. Assume the numbers be x and y.
Equation:
Three solutions:
Perfect! These represent different pairs of numbers satisfying the condition.
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) The equation of a line parallel to x-axis at a distance 4 units above it is:
(a) y = -4 (b) x = 4 (c) y = 4 (d) x = -4
Correct! Lines parallel to x-axis have form y = constant. 4 units above means y = 4.
(2) The equation of a line parallel to y-axis at a distance 3 units to the left of it is:
(a) x = 3 (b) y = -3 (c) x = -3 (d) y = 3
Correct! Lines parallel to y-axis have form x = constant. 3 units left means x = -3.
(3) The point of intersection of x + y = 6 and x - y = 2 is:
(a) (4, 2) (b) (2, 4) (c) (1, 5) (d) (5, 1)
Correct! Adding equations: 2x = 8, so x = 4. Then y = 2.
(4) If the x-intercept of a line is 5, then its equation (parallel to y-axis) is:
(a) x = 5 (b) y = 5 (c) y = -5 (d) x = -5
Correct! A line parallel to y-axis passing through x-intercept 5 has equation x = 5.
(5) Which equation represents a line parallel to x-axis?
(a) y = 2 (b) x = 2 (c) 2x + 3y = 6 (d) x - y = 0
Correct! Lines parallel to x-axis have the form y = constant.
(6) Which equation represents a line parallel to y-axis?
(a) x = 2 (b) y = 2 (c) x + y = 0 (d) y = x
Correct! Lines parallel to y-axis have the form x = constant.
(7) The solution of 2x + 3y = 12 when x = 0 is:
(a) (0, 4) (b) (0, 6) (c) (4, 0) (d) (6, 0)
Correct! When x = 0: 2(0) + 3y = 12, so y = 4.
(8) The solution of 4x - y = 8 when y = 0 is:
(a) (0, -8) (b) (2, 0) (c) (4, 0) (d) (8, 0)
Correct! When y = 0: 4x - 0 = 8, so x = 2.
(9) Which of the following points lies on the line y = 2x + 3?
(a) (1, 5) (b) (2, 7) (c) (3, 9) (d) All of these
Correct!
(10) If two linear equations in two variables have the same graph, then they have:
(a) No solution (b) Exactly one solution (c) Infinitely many solutions (d) Two solutions
Correct! Same graph means the equations are equivalent with infinitely many common solutions.
Let's classify different types of lines!!!
Excellent! You understand different orientations of lines.
True or False: System Solutions
Determine whether these intersection statements are True or False:
Comprehensive Hard Quiz
🎉 You Did It! What You've Mastered:
By completing this advanced worksheet, you now have expert knowledge of:
(1) Advanced Equation Types: Lines parallel to axes, through origin, and general forms
(2) Intersection Analysis: Finding where two lines meet using algebraic and graphical methods
(3) System Classification: Understanding unique, no solution, and infinite solution cases
(4) Real-World Modeling: Converting complex scenarios into linear equation systems
(5) Graphical Problem Solving: Using graphs to solve practical problems visually
(6) Advanced Verification: Checking solutions for complex multi-step problems
(7) Multiple Equation Forms: Standard, slope-intercept, intercept, and point-slope forms
(8) Parametric Analysis: Understanding how changing parameters affects line properties
(9) Complex Applications: Taxi fares, mixture problems, work rate problems
(10) System Properties: Parallel, intersecting, and coincident line relationships
(11) Advanced Calculations: Working with fractions, decimals, and exact solutions
(12) Geometric Interpretation: Understanding area calculations using linear equations
(13) Problem Strategy: Systematic approaches to complex multi-variable problems
(14) Verification Techniques: Multiple methods to confirm solution accuracy
(15) Critical Analysis: Distinguishing between different types of solution sets
Outstanding mastery of advanced linear equations in two variables achieved!