Chapter-3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES — Online MCQ Test
MATHS · CLASS 10th · Karnataka State Board
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Chapter-3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES — Important Questions & Answers
A pair of linear equations in two variables is of the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. What condition must be satisfied for the system to have a unique solution?
- A. a₁/a₂ = b₁/b₂
- B. a₁/a₂ ≠ b₁/b₂
- C. a₁/a₂ = b₁/b₂ = c₁/c₂
- D. a₁ × a₂ = b₁ × b₂
Answer: B. a₁/a₂ ≠ b₁/b₂
For a unique solution, the lines must intersect at exactly one point, which occurs when a₁/a₂ ≠ b₁/b₂ (the lines are not parallel).
For a unique solution, the lines must intersect at exactly one point, which occurs when a₁/a₂ ≠ b₁/b₂ (the lines are not parallel).
Which of the following is the correct definition of consistent equations?
- A. Equations that have no solution
- B. Equations that have at least one solution
- C. Equations that have exactly two solutions
- D. Equations that are always contradictory
Answer: B. Equations that have at least one solution
Consistent equations are those that have at least one common solution, either unique or infinitely many.
Consistent equations are those that have at least one common solution, either unique or infinitely many.
Solve the system: x + y = 5 and x - y = 3. What is the value of x?
- A. 1
- B. 2
- C. 3
- D. 4
Answer: D. 4
Adding both equations: 2x = 8, so x = 4. Verification: 4 + 1 = 5 and 4 - 1 = 3 ✓
Adding both equations: 2x = 8, so x = 4. Verification: 4 + 1 = 5 and 4 - 1 = 3 ✓
Analyze the system: 4x + 6y = 8 and 2x + 3y = 4. The nature of this system is:
- A. Inconsistent (parallel lines)
- B. Consistent with unique solution
- C. Consistent with infinitely many solutions
- D. Degenerate
Answer: C. Consistent with infinitely many solutions
The first equation simplifies to 2x + 3y = 4 when divided by 2, which is identical to the second equation, so infinitely many solutions exist.
The first equation simplifies to 2x + 3y = 4 when divided by 2, which is identical to the second equation, so infinitely many solutions exist.
If the pair of equations px + qy = m and rx + sy = n represent two perpendicular lines, which statement is always true?
- A. pr + qs = 0
- B. p/r = q/s
- C. ps = qr
- D. p + r = q + s
Answer: A. pr + qs = 0
For perpendicular lines, the product of their slopes equals -1. The slopes are -p/q and -r/s, so (-p/q)(-r/s) = -1, which simplifies to pr + qs = 0.
For perpendicular lines, the product of their slopes equals -1. The slopes are -p/q and -r/s, so (-p/q)(-r/s) = -1, which simplifies to pr + qs = 0.