Chapter 3: Pair of Linear Equations in Two Variables

Class 10 Mathematics · MCQs with answers

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What is the solution of the pair x + y = 14 and x − y = 4?
1. (A) x = 5, y = 9
2. (B) x = 10, y = 4
3. (C) x = 9, y = 5
4. (D) x = 7, y = 7
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Correct answer: (C) x = 9, y = 5

Explanation: Adding the equations gives 2x = 18, so x = 9. Then y = 14 − 9 = 5.

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What is the solution of the pair 2x + 3y = 11 and 2x − 4y = −24?
1. (A) x = 2, y = 5
2. (B) x = −2, y = 5
3. (C) x = 5, y = −2
4. (D) x = −2, y = −5
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Correct answer: (B) x = −2, y = 5

Explanation: Subtracting the second equation from the first gives 7y = 35, so y = 5. Then 2x + 15 = 11, so x = −2.

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If the lines of two linear equations intersect at one point, how many solutions does the pair have?
1. (A) No solution
2. (B) Infinitely many solutions
3. (C) Exactly two solutions
4. (D) Exactly one solution
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Correct answer: (D) Exactly one solution

Explanation: The point of intersection is the only point on both lines, so there is a unique solution and the pair is consistent.

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If the lines of two linear equations are parallel, how many solutions does the pair have?
1. (A) Exactly one solution
2. (B) Infinitely many solutions
3. (C) No solution
4. (D) Exactly two solutions
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Correct answer: (C) No solution

Explanation: Parallel lines never meet, so there is no common point. Such a pair is called inconsistent.

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If the lines of the two equations are coincident (lie exactly on top of each other), how many solutions does the pair have?
1. (A) Infinitely many solutions
2. (B) Exactly one solution
3. (C) No solution
4. (D) Exactly two solutions
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Correct answer: (A) Infinitely many solutions

Explanation: Every point of coincident lines lies on both lines, so there are infinitely many solutions. The pair is dependent (and consistent).

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What is the condition for the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have a unique solution?
1. (A) a₁/a₂ = b₁/b₂ ≠ c₁/c₂
2. (B) a₁/a₂ = b₁/b₂ = c₁/c₂
3. (C) b₁/b₂ = c₁/c₂
4. (D) a₁/a₂ ≠ b₁/b₂
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Correct answer: (D) a₁/a₂ ≠ b₁/b₂

Explanation: When a₁/a₂ ≠ b₁/b₂, the lines intersect and there is a unique solution.

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What is the condition for a pair of linear equations to have no solution (be inconsistent)?
1. (A) a₁/a₂ ≠ b₁/b₂
2. (B) a₁/a₂ = b₁/b₂ ≠ c₁/c₂
3. (C) a₁/a₂ = b₁/b₂ = c₁/c₂
4. (D) a₁/a₂ = c₁/c₂ ≠ b₁/b₂
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Correct answer: (B) a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Explanation: When a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and there is no solution.

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How many solutions does the pair 3x + 2y = 5 and 2x − 3y = 7 have?
1. (A) Exactly one solution
2. (B) No solution
3. (C) Infinitely many solutions
4. (D) Exactly two solutions
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Correct answer: (A) Exactly one solution

Explanation: a₁/a₂ = 3/2 and b₁/b₂ = 2/(−3) = −2/3. They are not equal, so the lines intersect and there is exactly one solution.

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How many solutions does the pair 2x − 3y = 8 and 4x − 6y = 9 have?
1. (A) Exactly one solution
2. (B) Infinitely many solutions
3. (C) No solution
4. (D) Exactly two solutions
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Correct answer: (C) No solution

Explanation: a₁/a₂ = 2/4 = 1/2 and b₁/b₂ = −3/−6 = 1/2, but c₁/c₂ = 8/9. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and there is no solution.

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How many solutions does the pair 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0 have?
1. (A) Exactly one solution
2. (B) Infinitely many solutions
3. (C) No solution
4. (D) Exactly two solutions
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Correct answer: (B) Infinitely many solutions

Explanation: a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2. The second equation is just twice the first, so the lines coincide and there are infinitely many solutions.

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For which value of k does the pair x + 2y = 3 and 5x + ky + 7 = 0 have no solution?
1. (A) 10
2. (B) 5
3. (C) 2
4. (D) −10
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Correct answer: (A) 10

Explanation: For x + 2y − 3 = 0 and 5x + ky + 7 = 0, a₁/a₂ = 1/5, b₁/b₂ = 2/k and c₁/c₂ = −3/7. For no solution, 1/5 = 2/k, so k = 10; and 1/5 ≠ −3/7 holds.

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For which value of k does the pair 2x + 3y = 7 and (k − 1)x + (k + 2)y = 3k have infinitely many solutions?
1. (A) 1
2. (B) 3
3. (C) 5
4. (D) 7
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Correct answer: (D) 7

Explanation: Condition: 2/(k − 1) = 3/(k + 2) = 7/(3k). The first equality gives 2k + 4 = 3k − 3, so k = 7. Check: 2/6 = 3/9 = 7/21 = 1/3.

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10 students of Class 10 took part in a maths quiz. The number of girls is 4 more than the number of boys. How many boys and girls are there?
1. (A) 3 boys, 7 girls
2. (B) 7 boys, 3 girls
3. (C) 4 boys, 6 girls
4. (D) 5 boys, 5 girls
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Correct answer: (A) 3 boys, 7 girls

Explanation: Let there be x boys and y girls: x + y = 10 and y = x + 4. Solving, 2x + 4 = 10, so x = 3 and y = 7.

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5 pencils and 7 pens cost ₹50 together, while 7 pencils and 5 pens cost ₹46. What is the cost of one pencil and one pen?
1. (A) Pencil ₹5, pen ₹3
2. (B) Pencil ₹4, pen ₹4
3. (C) Pencil ₹2, pen ₹6
4. (D) Pencil ₹3, pen ₹5
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Correct answer: (D) Pencil ₹3, pen ₹5

Explanation: 5x + 7y = 50 and 7x + 5y = 46. Adding gives x + y = 8 and subtracting gives y − x = 2, so y = 5 and x = 3. Check: 15 + 35 = 50 and 21 + 25 = 46.

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If 2 is added to both the numerator and the denominator of a fraction, it becomes 9/11. If 3 is added to both, it becomes 5/6. What is the fraction?
1. (A) 9/11
2. (B) 7/9
3. (C) 3/4
4. (D) 5/7
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Correct answer: (B) 7/9

Explanation: Let the fraction be x/y: 11(x + 2) = 9(y + 2) and 6(x + 3) = 5(y + 3), i.e. 11x − 9y = −4 and 6x − 5y = −3. Solving gives x = 7, y = 9. Check: 9/11 and 10/12 = 5/6.

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The sum of the ages of a father and his son is 50 years. The father is 4 times as old as the son. What are their ages?
1. (A) Son 10, father 40
2. (B) Son 12, father 38
3. (C) Son 8, father 42
4. (D) Son 15, father 35
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Correct answer: (A) Son 10, father 40

Explanation: Let the son's age be x and the father's 4x: x + 4x = 50, so x = 10. The father is 40.

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What is the solution of the pair x = 2y and x + y = 9?
1. (A) x = 3, y = 6
2. (B) x = 4, y = 5
3. (C) x = 6, y = 3
4. (D) x = 2, y = 7
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Correct answer: (C) x = 6, y = 3

Explanation: Substituting x = 2y into the second equation gives 2y + y = 9, so y = 3 and x = 6.

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At which point does the line 2x + y = 6 cut the y-axis?
1. (A) (3, 0)
2. (B) (0, 6)
3. (C) (6, 0)
4. (D) (0, 3)
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Correct answer: (B) (0, 6)

Explanation: On the y-axis, x = 0. Putting x = 0 gives y = 6, so the point is (0, 6). (It cuts the x-axis at (3, 0).)

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At which point do the lines x = 2 and y = 3 intersect?
1. (A) (3, 2)
2. (B) (0, 0)
3. (C) (2, 0)
4. (D) (2, 3)
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Correct answer: (D) (2, 3)

Explanation: x = 2 is a vertical line and y = 3 is a horizontal line. The point on both is (2, 3).

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Which pair of equations has the solution x = 2, y = −3?
1. (A) x + y = 1 and 2x + 3y = 5
2. (B) x − y = 1 and x + y = −1
3. (C) x + y = −1 and 2x − 3y = 13
4. (D) 2x + y = 7 and x − y = 5
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Correct answer: (C) x + y = −1 and 2x − 3y = 13

Explanation: Putting x = 2, y = −3: 2 + (−3) = −1 and 4 + 9 = 13, so both equations hold. In the other pairs, at least one equation fails.

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Using the cross-multiplication method, what is the solution of 2x + y − 5 = 0 and 3x + 2y − 8 = 0?
1. (A) x = 2, y = 1
2. (B) x = 1, y = 2
3. (C) x = 1, y = 3
4. (D) x = 3, y = 1
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Correct answer: (A) x = 2, y = 1

Explanation: Using x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁): x/(−8 + 10) = y/(−15 + 16) = 1/(4 − 3), so x = 2 and y = 1.

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What is the solution of the pair 2/x + 3/y = 13 and 5/x − 4/y = −2?
1. (A) x = 2, y = 3
2. (B) x = 1/3, y = 1/2
3. (C) x = 3, y = 2
4. (D) x = 1/2, y = 1/3
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Correct answer: (D) x = 1/2, y = 1/3

Explanation: Putting 1/x = p and 1/y = q gives 2p + 3q = 13 and 5p − 4q = −2. Solving gives p = 2 and q = 3, so x = 1/2 and y = 1/3.

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What is the solution of the pair 3x + 4y = 10 and 2x − 2y = 2?
1. (A) x = 1, y = 2
2. (B) x = 2, y = 1
3. (C) x = 2, y = 2
4. (D) x = 3, y = 1
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Correct answer: (B) x = 2, y = 1

Explanation: Multiply the second equation by 2: 4x − 4y = 4. Adding it to the first gives 7x = 14, so x = 2 and y = 1.

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For which value of k are the lines 3x + 2ky = 2 and 2x + 5y + 1 = 0 parallel?
1. (A) 4/15
2. (B) 3/5
3. (C) 5/3
4. (D) 15/4
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Correct answer: (D) 15/4

Explanation: For parallel lines, a₁/a₂ = b₁/b₂: 3/2 = 2k/5, so k = 15/4. Then c₁/c₂ = −2/1 ≠ 3/2, so the lines are parallel, not coincident.

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Assertion (A): The pair x + 2y = 4 and 2x + 4y = 12 has no solution. Reason (R): For this pair, a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
1. (A) Both A and R are true, and R is the correct explanation of A.
2. (B) Both A and R are true, but R is not the correct explanation of A.
3. (C) A is true, but R is false.
4. (D) A is false, but R is true.
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Correct answer: (A) Both A and R are true, and R is the correct explanation of A.

Explanation: a₁/a₂ = 1/2, b₁/b₂ = 2/4 = 1/2 and c₁/c₂ = 4/12 = 1/3. The lines are parallel, so there is no solution, and R correctly explains A.

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