MATHEMATICS CLASS- 10
MCQ- CH-3(Pair of Linear Equations in Two Variables)
Class 10 Mathematics
Chapter 3: Pair of Linear Equations in Two Variables
MCQs with Answers and Solutions
Q1. Which of the following is a pair of linear equations in two variables?
(a) x² + y = 5, x + y = 3
(b) x + y = 5, 2x − y = 3
(c) x³ + y = 1, x + y = 2
(d) x + y² = 4, x − y = 2
Answer: (b) x + y = 5, 2x − y = 3
Solution: Both equations are linear and contain two variables.
Q2. The solution of x + y = 5 and x − y = 1 is
(a) (2,3)
(b) (3,2)
(c) (1,4)
(d) (4,1)
Answer: (b) (3,2)
Solution: Adding equations gives 2x = 6 ⇒ x = 3, y = 2.
Q3. The graphical solution of a pair of linear equations is the
(a) Origin
(b) Slope
(c) Point of intersection
(d) x-intercept
Answer: (c) Point of intersection
Solution: The coordinates of the intersection point satisfy both equations.
Q4. Two intersecting lines represent
(a) No solution
(b) One unique solution
(c) Infinitely many solutions
(d) Two solutions
Answer: (b) One unique solution
Solution: Intersecting lines meet at one point only.
Q5. Two parallel lines represent
(a) One solution
(b) Two solutions
(c) No solution
(d) Infinite solutions
Answer: (c) No solution
Solution: Parallel lines never intersect.
Q6. Coincident lines represent
(a) No solution
(b) One solution
(c) Infinite solutions
(d) Two solutions
Answer: (c) Infinite solutions
Solution: Every point on one line lies on the other.
Q7. The pair x + y = 2 and 2x + 2y = 4 has
(a) No solution
(b) Unique solution
(c) Infinite solutions
(d) Two solutions
Answer: (c) Infinite solutions
Solution: Second equation is a multiple of the first.
Q8. The pair x + y = 2 and x + y = 5 has
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Two solutions
Answer: (b) No solution
Solution: Same slope, different intercepts.
Q9. If a₁/a₂ ≠ b₁/b₂, then the pair has
(a) No solution
(b) Infinite solutions
(c) Unique solution
(d) Cannot be determined
Answer: (c) Unique solution
Solution: The lines intersect at one point.
Q10. The solution of 2x + y = 7 and x − y = 2 is
(a) (3,1)
(b) (2,3)
(c) (1,3)
(d) (3,2)
Answer: (a) (3,1)
Solution: Adding equations gives 3x = 9 ⇒ x = 3, y = 1.
Q11. The pair 3x + 2y = 5 and 6x + 4y = 10 represents
(a) Intersecting lines
(b) Parallel lines
(c) Coincident lines
(d) Perpendicular lines
Answer: (c) Coincident lines
Solution: Both equations represent the same line.
Q12. The pair 3x + 2y = 5 and 6x + 4y = 12 has
(a) Unique solution
(b) No solution
(c) Infinite solutions
(d) Two solutions
Answer: (b) No solution
Solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
Q13. The coordinates (2,3) satisfy
(a) x + y = 5
(b) x − y = 5
(c) 2x + y = 10
(d) x + y = 7
Answer: (a) x + y = 5
Solution: 2 + 3 = 5.
Q14. Which method eliminates y from x + y = 5 and x − y = 1 directly?
(a) Addition
(b) Subtraction
(c) Multiplication
(d) Graphing
Answer: (a) Addition
Solution: Adding cancels y.
Q15. The equations x = 3 and y = 4 intersect at
(a) (0,0)
(b) (3,4)
(c) (4,3)
(d) (3,0)
Answer: (b) (3,4)
Solution: Both conditions are satisfied at (3,4).
Q16. The pair 4x + 2y = 8 and 2x + y = 4 has
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Two solutions
Answer: (c) Infinite solutions
Solution: First equation is twice the second.
Q17. The solution of x + y = 9 and x − y = 3 is
(a) (6,3)
(b) (3,6)
(c) (5,4)
(d) (4,5)
Answer: (a) (6,3)
Solution: Adding gives 2x = 12 ⇒ x = 6, y = 3.
Q18. The pair 2x + 3y = 6 and 4x + 6y = 12 has
(a) No solution
(b) One solution
(c) Infinite solutions
(d) Two solutions
Answer: (c) Infinite solutions
Solution: One equation is a multiple of the other.
Q19. The pair x + 2y = 5 and 2x + 4y = 9 has
(a) Unique solution
(b) No solution
(c) Infinite solutions
(d) One variable
Answer: (b) No solution
Solution: Ratios of coefficients are equal but constants differ.
Q20. The graphical solution of a pair of equations is obtained by
(a) Factorisation
(b) Plotting lines
(c) Prime factorisation
(d) Division
Answer: (b) Plotting lines
Solution: Graphical method uses graphs of both equations.
Q21. If a₁/a₂ = b₁/b₂ = c₁/c₂, then the pair has
(a) Unique solution
(b) No solution
(c) Infinite solutions
(d) Two solutions
Answer: (c) Infinite solutions
Solution: The lines are coincident.
Q22. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, then the pair has
(a) Unique solution
(b) No solution
(c) Infinite solutions
(d) Two solutions
Answer: (b) No solution
Solution: The lines are parallel.
Q23. The solution of 3x + y = 11 and x + y = 7 is
(a) (1,6)
(b) (2,5)
(c) (3,4)
(d) (4,3)
Answer: (b) (2,5)
Solution: Subtract equations: 2x = 4 ⇒ x = 2.
Q24. The pair x = 2 and x = 5 represents
(a) Coincident lines
(b) Intersecting lines
(c) Parallel lines
(d) Perpendicular lines
Answer: (c) Parallel lines
Solution: Both are vertical parallel lines.
Q25. Which method is NOT used to solve a pair of linear equations?
(a) Graphical Method
(b) Substitution Method
(c) Elimination Method
(d) Completing the Square
Answer: (d) Completing the Square
Solution: It is mainly used for quadratic equations.
Q26. (CBSE PYQ) The pair x + y = 10 and 2x + 2y = 20 has
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Exactly two solutions
Answer: (c) Infinite solutions
Q27. A father is 30 years older than his son. If their ages add up to 50 years, the son's age is
(a) 10 years
(b) 15 years
(c) 20 years
(d) 25 years
Answer: (a) 10 years
Solution: x + (x+30)=50 ⇒ x=10.
Q28. Two numbers differ by 4 and their sum is 20. The smaller number is
(a) 8
(b) 10
(c) 12
(d) 14
Answer: (a) 8
Solution: x+y=20, y−x=4 ⇒ x=8.
Q29. The solution of 5x − y = 9 and x + y = 3 is
(a) (2,1)
(b) (1,2)
(c) (3,0)
(d) (0,3)
Answer: (a) (2,1)
Q30. A pair of equations having exactly one solution is called
(a) Inconsistent
(b) Dependent
(c) Consistent Independent
(d) Impossible
Answer: (c) Consistent Independent
Q31. (CBSE PYQ) If 2x + 3y = 11 and x − y = 1, then x equals
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (b) 2
Solution: x=y+1 ⇒ 2(y+1)+3y=11 ⇒ y=1.8? Wait.
Actually: 2x+3y=11, x−y=1 ⇒ x=y+1.
2(y+1)+3y=11 ⇒ 5y=9 ⇒ y=9/5, x=14/5.
None match.
Corrected Answer: x = 14/5.
Q32. The pair 7x + 5y = 15 and 14x + 10y = 30 has
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Cannot be determined
Answer: (c) Infinite solutions
Q33. The point (1,2) satisfies
(a) x + y = 3
(b) x + y = 4
(c) x − y = 3
(d) 2x + y = 5
Answer: (a) x + y = 3
Q34. If x + y = 12 and x − y = 4, then y =
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (b) 4
Q35. The equations 3x + y = 7 and 6x + 2y = 15 are
(a) Parallel
(b) Coincident
(c) Intersecting
(d) Perpendicular
Answer: (a) Parallel
Q36. The pair x + 3y = 10 and x + 3y = 10 has
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Three solutions
Answer: (c) Infinite solutions
Q37. A pen and a pencil cost ₹25 together. If pen costs ₹5 more than pencil, the pencil costs
(a) ₹8
(b) ₹10
(c) ₹12
(d) ₹15
Answer: (b) ₹10
Q38. The pair y = 2x + 1 and y = 2x − 3 represents
(a) Coincident lines
(b) Parallel lines
(c) Intersecting lines
(d) Perpendicular lines
Answer: (b) Parallel lines
Q39. The equations x = 0 and y = 0 intersect at
(a) (1,1)
(b) (0,1)
(c) (1,0)
(d) (0,0)
Answer: (d) (0,0)
Q40. If x = 4 and y = 3, then 2x + 3y equals
(a) 15
(b) 16
(c) 17
(d) 18
Answer: (c) 17
Q41. (CBSE PYQ) The pair 4x + 5y = 20 and 8x + 10y = 40 has
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Two solutions
Answer: (c) Infinite solutions
Q42. Which pair has a unique solution?
(a) x+y=3, 2x+2y=6
(b) x+y=3, x+y=5
(c) x+y=3, x−y=1
(d) 2x+4y=6, x+2y=3
Answer: (c) x+y=3, x−y=1
Q43. A taxi fare consists of fixed charge and distance charge. Such problems are generally solved using
(a) One linear equation
(b) Pair of linear equations
(c) Quadratic equation
(d) Polynomial theorem
Answer: (b) Pair of linear equations
Q44. The substitution method involves
(a) Drawing graphs
(b) Replacing one variable using another equation
(c) Multiplying equations only
(d) Factorising equations
Answer: (b)
Q45. The elimination method aims to
(a) Remove one variable
(b) Remove both variables
(c) Remove constants
(d) Draw graphs
Answer: (a)
Q46. If a₁/a₂ ≠ b₁/b₂, then lines are
(a) Parallel
(b) Coincident
(c) Intersecting
(d) Vertical
Answer: (c)
Q47. The pair x+y=6 and 3x+3y=18 is
(a) Consistent Independent
(b) Consistent Dependent
(c) Inconsistent
(d) Impossible
Answer: (b)
Q48. A shop sells 2 notebooks and 3 pens for ₹36. Another customer buys 4 notebooks and 6 pens for ₹72. The equations have
(a) One solution
(b) No solution
(c) Infinite solutions
(d) Three solutions
Answer: (c)
Q49. The pair x+y=8 and 2x+2y=20 is
(a) Parallel
(b) Coincident
(c) Intersecting
(d) Perpendicular
Answer: (a) Parallel
Q50. Which statement is true?
(a) Parallel lines have one solution.
(b) Coincident lines have no solution.
(c) Intersecting lines have a unique solution.
(d) Every pair of linear equations has infinite solutions.
Answer: (c) Intersecting lines have a unique solution.
Solution: Intersecting lines meet at exactly one point.