Pair of Linear Equations in Two Variables Class 10 Previous Year Questions: 1 Mark (Chapter 3)

1-mark questions on Pair of Linear Equations in Two Variables

This page has 29 1-mark questions on Pair of Linear Equations in Two Variables from CBSE Class 10 board papers of 2023, 2024, 2025 and 2026. Of these, 18 appeared only in Standard papers and 11 appeared only in Basic papers. Question formats: 29 MCQs. 12 of them were asked in two or more paper sets.

0 of 29 practised

All 29 Pair of Linear Equations in Two Variables 1-mark questions

Q1. If the pair of linear equations x - y = 1, x + ky = 5 has a unique solution x = 2, y = 1, then the value of k is :

Mark practised

(A) -2
(B) -3
(C) 3
(D) 4

Marks: 1
Level: Basic
Asked in: 3 paper sets
Years: 2023
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Basic, Set 430/4/2, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 3, because substituting (2, 1) into the second equation gives k = 5 − 2 = 3.
Why? A solution of a pair of linear equations is an ordered pair that satisfies both equations, so the point (2, 1) must satisfy the second equation; substituting it turns that equation into one for k.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/2: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the unique-solution condition a₁/a₂ ≠ b₁/b₂ gives k ≠ −1, which only rules out one option and does not determine k.
Substituting (2, 1) only into the first equation, seeing it is satisfied, and then guessing the option instead of using the second equation.
Making an arithmetic slip in 5 − 2 and taking k as 2.

Q2. The value of \(k\) for which the pair of equations \(kx = y + 2\) and \(6x = 2y + 3\) has infinitely many solutions,

Mark practised

(A) is \(k = 3\)
(B) does not exist
(C) is \(k = -3\)
(D) is \(k = 4\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Standard, Set 30/1/1, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) does not exist, because infinitely many solutions require the three ratios k/6, (−1)/(−2), (−2)/(−3) to be equal, but (−1)/(−2) = 1/2 and (−2)/(−3) = 2/3 are unequal.
Why? For coincident lines, the ratios of all corresponding coefficients must be equal; since the y-coefficient ratio and the constant ratio are already unequal, no k can make the pair dependent.

The full 1 mark is for the correct option.

Answer:

(B) does not exist

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/1: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 3 for k because k/6 and (−1)/(−2) both equal 1/2, without checking that (−2)/(−3) equals 2/3.
Choosing 4 for k because k/6 matches the constant ratio 2/3, without checking that the y-coefficient ratio is 1/2.
Assuming that when k is 3 the second equation becomes a multiple of the first; actually the constant term would need to be 4, not 3.

Why this works

Write both equations in the general form, with all terms on one side. The first equation has coefficients k, −1, −2; the second has 6, −2, −3. A pair of linear equations has infinitely many solutions when the lines are coincident, and the condition is that the three ratios of corresponding coefficients are equal. Here the y-coefficient ratio is (−1)/(−2), which is 1/2, and the constant ratio is (−2)/(−3), which is 2/3. Since 1/2 and 2/3 are different, the three ratios can never be equal for any value of k. So the value of k does not exist. A quick check by substitution: substituting the expression for y from the first equation into the second gives (6 − 2k)x = −1. This has exactly one solution when 6 − 2k is not zero, and no solution when 6 − 2k is zero; it never has infinitely many solutions.

Key idea to remember

Dependent pair of linear equations: see Key ideas used on this page.

Explanation

A dependent pair of linear equations is always consistent and has infinitely many distinct common solutions; its graph is a pair of coincident lines. For a pair of linear equations in two variables, this happens exactly when the three ratios of corresponding coefficients are equal. Here the y-coefficient ratio is 1/2 and the constant ratio is 2/3, so the equality can never hold for any k.

Important formulas

a₁/a₂ = b₁/b₂ = c₁/c₂ · Condition for infinitely many solutions (coincident lines) for a pair of linear equations in two variables.

Q3. The lines represented by linear equations \(x = a\) and \(y = b\) ( \(a \neq b\) ) are

Mark practised

(A) intersecting at (a, b).
(B) intersecting at (b, a).
(C) parallel.
(D) coincident.

Marks: 1
Level: Basic
Asked in: 3 paper sets
Years: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Basic, Set 430/3/2, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) intersecting at (a, b), because the line x = a is vertical and the line y = b is horizontal, so they meet at (a, b).
Why? A vertical line and a horizontal line always intersect at exactly one point; on x = a every point has x-coordinate a, and on y = b every point has y-coordinate b.

The full 1 mark is for the correct option.

Answer:

(A) intersecting at (a, b).

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/3/2: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing (B) intersecting at (b, a), because the coordinates are written in the order the equations appear, instead of as (x, y).
Choosing (C) parallel, because both equations contain only one variable, without noticing that one line is vertical and the other horizontal.
Thinking a ≠ b means the lines do not meet, although a and b are just the fixed x- and y-coordinates of the meeting point.

Why this works

The point (a, b) satisfies both equations: substituting x = a into x = a gives a = a, and substituting y = b into y = b gives b = b. Since a vertical line and a horizontal line cannot be parallel or coincident, they intersect only at (a, b). The condition a ≠ b only tells us the two fixed numbers are different; it does not stop the lines from meeting.

Key idea to remember

Graphical Method of Solution of a Pair of Linear Equations: see Key ideas used on this page.

Explanation

In the graphical method, each linear equation is represented by a line. Intersecting lines give a unique solution, parallel lines give no solution, and coincident lines give infinitely many solutions. Here x = a is a vertical line and y = b is a horizontal line, so they intersect at the single point (a, b).

Important formulas

x = a · A line parallel to the y-axis, passing through (a, 0).
y = b · A line parallel to the x-axis, passing through (0, b).

Q4. The value of \(k\) for which the system of equations \(3x - y + 8 = 0\) and \(6x - ky + 16 = 0\) has infinitely many solutions, is

Mark practised

(A) \(-2\)
(B) \(2\)
(C) \(\frac{1}{2}\)
(D) \(-\frac{1}{2}\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Standard, Set 30/2/1, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 2, because for infinitely many solutions the ratios must be equal: 3/6 = (−1)/(−k) = 8/16, which gives k = 2.
Why? Coincident lines have infinitely many solutions, and for them a1/a2 = b1/b2 = c1/c2.

The full 1 mark is for the correct option.

Answer:

(B) 2

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/1: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing b1/b2 as (−1)/k instead of (−1)/(−k) = 1/k; this gives the wrong option −2.
Choosing 1/2 because 3/6 and 8/16 are both half, without also checking that (−1)/(−k) is half.

Why this works

Write both equations in the general form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Here a1 is 3, b1 is −1, c1 is 8, a2 is 6, b2 is −k, c2 is 16. For infinitely many solutions, the lines are coincident, so the NCERT condition is a1/a2 = b1/b2 = c1/c2. Now a1/a2 = 3/6 = 1/2 and c1/c2 = 8/16 = 1/2. So b1/b2 must also be 1/2. Since b1/b2 = (−1)/(−k) = 1/k, we get 1/k = 1/2, hence k = 2. Quick check: with k = 2, the second equation is exactly double the first equation, so both lines coincide and have infinitely many solutions.

Key idea to remember

Dependent pair of linear equations: see Key ideas used on this page.

Explanation

When two linear equations represent the same line, their coefficients are proportional: a1/a2 = b1/b2 = c1/c2. Such a pair is called a dependent pair, and it is always consistent because the lines are coincident, giving infinitely many common solutions. This solution uses that ratio condition to find k.

Important formulas

For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, infinitely many solutions ⇔ a1/a2 = b1/b2 = c1/c2. · This is the dependent (consistent) case; the lines are coincident.

Q5. Which out of the following type of straight lines will be represented by the system of equations \(3x + 4y = 5\) and \(6x + 8y = 7\) ?

Mark practised

(A) Parallel
(B) Intersecting
(C) Coincident
(D) Perpendicular to each other

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Standard, Set 30/5/1, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) Parallel, because a1/a2 = 3/6 = 1/2, b1/b2 = 4/8 = 1/2 and c1/c2 = (−5)/(−7) = 5/7, so a1/a2 = b1/b2 ≠ c1/c2.
Why? When a1/a2 = b1/b2 ≠ c1/c2, the lines represented by the pair of linear equations are parallel.

The full 1 mark is for the correct option.

Answer:

(A) Parallel

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/5/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing Coincident because the ratios 3/6 and 4/8 are equal, without checking that the constant term ratio 5/7 is different.
Choosing Intersecting because the right sides 5 and 7 are different, without comparing the ratios.
Choosing Perpendicular because the slopes look like −3/4 and −6/8, without noticing that these slopes are equal.

Why this works

Rewrite each equation with all terms on the left side. The coefficient pairs are (3, 4, −5) for the first equation and (6, 8, −7) for the second, where the third number is the constant term. Compare the ratios a1/a2 = 3/6 = 1/2, b1/b2 = 4/8 = 1/2 and c1/c2 = (−5)/(−7) = 5/7. Since the first two ratios are equal but the third is different, the lines are parallel. Quick check: multiplying the first equation by 2 makes its right side 10, while the second has right side 7; the same left side cannot equal two different numbers, so there is no common solution. Another correct method also earns full marks.

Key idea to remember

Comparing the ratios a1/a2, b1/b2 and c1/c2: see Key ideas used on this page.

Explanation

The comparison rule for a pair of linear equations says that when the ratio of the x-coefficients equals the ratio of the y-coefficients but not the ratio of the constant terms, the lines are parallel and the pair is inconsistent. This solution compares those three ratios for the given equations.

Important formulas

a1/a2 = b1/b2 ≠ c1/c2 · Parallel lines; no solution.
a1/a2 = b1/b2 = c1/c2 · Coincident lines; infinitely many solutions.
a1/a2 ≠ b1/b2 · Intersecting lines; unique solution.

Q6. The solution of the pair of linear equations \(3x + 4y = 5\) ; \(4x + 3y = 9\) is :

Mark practised

(A) \(x = -3, y = 1\)
(B) \(x = 3, y = 1\)
(C) \(x = 3, y = -1\)
(D) \(x = -3, y = -1\)

Marks: 1
Level: Basic
Asked in: 3 paper sets
Compartment: 2025
Topic: 3.3 Algebraic Methods of Solving a Pair of Linear Equations
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) x = 3, y = −1, because adding the equations gives x + y = 2 and subtracting the first from the second gives x − y = 4, so x = 3 and y = −1.
Why? Adding or subtracting the two equations eliminates one variable, leaving two simple equations in x and y that give the solution directly.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/1: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing option (B) without checking it in the first equation: 3 times the x-value plus 4 times the y-value gives 13, not 5.
Subtracting the equations but keeping the sign of y wrong: the second minus the first gives x − y = 4, and 3y − 4y is −y, not +y.
Solving only the sum x + y = 2 and forgetting the difference x − y = 4, so any pair that adds to 2 seems correct even though it does not satisfy the original equations.

Why this works

Add the two equations: (3x + 4y) + (4x + 3y) = 5 + 9, so 7x + 7y = 14, which gives x + y = 2. Subtract the first from the second: (4x + 3y) − (3x + 4y) = 9 − 5, so x − y = 4. Now x = (2 + 4)/2 = 3 and y = (2 − 4)/2 = −1. Check: 3(3) + 4(−1) = 5 and 4(3) + 3(−1) = 9, so both equations are satisfied. Hence the solution is x = 3, y = −1.

Key idea to remember

Elimination Method: see Key ideas used on this page.

Explanation

In the elimination method, we add or subtract the two equations so that one variable gets eliminated. Here adding gives x + y = 2 and subtracting gives x − y = 4; solving these two gives x = 3 and y = −1.

Important formulas

If x + y = S and x − y = D, then x = (S + D)/2 and y = (S − D)/2. · Used after adding and subtracting the given equations.

Q7. The value of \(m\) for which lines \(14x + my = 20\) and \(-3x + 2y = 16\) are parallel, is :

Mark practised

(A) \(-\frac{3}{14}\)
(B) \(-\frac{7}{3}\)
(C) \(-\frac{28}{3}\)
(D) \(-\frac{3}{28}\)

Marks: 1
Level: Basic
Asked in: 3 paper sets
Years: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Mathematics Basic, Set 430/4/2, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) -28/3, because parallel lines have a₁/a₂ = b₁/b₂ ≠ c₁/c₂; here 14/(−3) = m/2, so m = −28/3, and c₁/c₂ = 5/4 is different.
Why? For parallel lines the ratios of the coefficients of x and y are equal, but the ratio of the constant terms is different; equating these ratios gives m.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/4/2: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using 20 and 16 as the constant terms without changing signs; the constants are −20 and −16.
Dropping the minus sign of −3, which leads to 28/3 instead of −28/3.
Equating the ratio of the x-coefficients to the ratio of the constant terms, which cannot determine m.

Why this works

Move the constant term to the left side; then the first equation has x-coefficient 14, y-coefficient m and constant term −20, and the second has x-coefficient −3, y-coefficient 2 and constant term −16. For parallel lines, the ratio of the x-coefficients must equal the ratio of the y-coefficients, while the ratio of the constant terms must be different. So 14/(−3) = m/2. Multiplying by 2 gives m = −28/3. Also, the ratio of the constant terms is (−20)/(−16) = 5/4, which is different from −14/3. Thus the condition for parallel lines holds.

Key idea to remember

Condition for parallel lines: see Key ideas used on this page.

Explanation

For two linear equations, when the ratio of the coefficients of x equals the ratio of the coefficients of y, but the ratio of the constant terms is different, the lines are parallel and the pair has no solution. This solution uses that condition to find m, then verifies the constant ratio is different.

Important formulas

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ · Condition for two lines to be parallel.

Q8. If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has :

Mark practised

(A) a unique solution
(B) two solutions
(C) no solution
(D) an infinite number of solutions

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2026
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2026 Mathematics Standard, Set 30/1/1, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) an infinite number of solutions, because coincident lines satisfy a₁/a₂ = b₁/b₂ = c₁/c₂, so every point on the common line is a solution.
Why? When two lines are coincident, each point on the common line satisfies both equations, and a line contains infinitely many points.

The full 1 mark is for the correct option.

Answer:

(D) an infinite number of solutions

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (D).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing a unique solution because intersecting lines meet at one point; coincident lines share every point, not just one.
Choosing no solution because parallel lines have no common point; coincident lines overlap completely and have infinitely many common points.
Thinking the pair has exactly two solutions because a line has two ends; a line has infinitely many points, so the common solutions are infinite.

Why this works

Coincident lines lie exactly on top of each other. A line contains infinitely many points, and every one of those points satisfies both equations. So the pair of linear equations has infinitely many solutions. This matches the NCERT statement that a dependent pair of linear equations is always consistent and has infinitely many distinct common solutions.

Key idea to remember

Dependent pair of linear equations in two variables: see Key ideas used on this page.

Explanation

A dependent pair of linear equations is one whose two equations represent the same line. When the lines are coincident, the equations are equivalent, so every solution of one equation is also a solution of the other. Therefore the pair has infinitely many solutions.

Important formulas

a₁/a₂ = b₁/b₂ = c₁/c₂ ⇒ coincident lines ⇒ infinitely many solutions · This is the condition for a dependent and consistent pair.

Q9. The pair of equations \(x = a\) and \(y = b\) graphically represents lines which are :

Mark practised

(A) parallel
(B) intersecting at (b, a)
(C) coincident
(D) intersecting at (a, b)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2023
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Standard, Set 30/2/1, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) intersecting at (a, b), because the common solution of the pair is x = a, y = b, so the lines meet at the point (a, b).
Why? For a pair of linear equations in two variables, the point where the two lines intersect is exactly the common solution of the equations. Here the common solution is x = a, y = b, so the lines intersect at (a, b).

The full 1 mark is for the correct option.

Answer:

(D) intersecting at (a, b)

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/2/1: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing (B) intersecting at (b, a), which swaps the coordinates; on the line x = a the x-coordinate of every point is a, so the intersection cannot have x-coordinate b.
Choosing (A) parallel, thinking that because x = a is parallel to the y-axis and y = b is parallel to the x-axis the two lines are parallel; a vertical line and a horizontal line always meet.
Choosing (C) coincident, thinking both equations give simple straight lines; coincident lines are the same line, but x = a and y = b are different lines.

Why this works

The equation x = a means every point on the line has x-coordinate a, so the line is vertical. The equation y = b means every point on the line has y-coordinate b, so the line is horizontal. A point that lies on both lines must have x-coordinate a and y-coordinate b, so the only common point is (a, b). NCERT Example 3 shows two lines intersecting at the common solution point. Here the common solution is x = a, y = b, so the lines intersect at (a, b).

Key idea to remember

Graphical Method of Solution of a Pair of Linear Equations: see Key ideas used on this page.

Explanation

In the graphical method, each linear equation in two variables is drawn as a line. For a pair of equations, the two lines may intersect at one point, may be parallel, or may coincide. If they intersect at one point, that point is the unique solution of the pair. The equations x = a and y = b have the common solution x = a, y = b, so the lines intersect at (a, b). Parallel lines mean no solution, and coincident lines mean infinitely many solutions.

Q10. The value of 'k' for which the pair of linear equations \(x + y - 4 = 0\) and \(2x + ky - 8 = 0\) has infinitely many solutions, is

Mark practised

(A) \(k \neq 2\)
(B) \(k \neq -2\)
(C) \(k = 2\)
(D) \(k = -2\)

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Basic, Set 430/2/1, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) k = 2, because for infinitely many solutions, a1/a2 = b1/b2 = c1/c2, so 1/2 = 1/k = (−4)/(−8), giving k = 2.
Why? A dependent pair of linear equations has infinitely many solutions exactly when its lines coincide, and that happens when the three coefficient ratios are all equal.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/2/1: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing −2 because you think 1/k = 1/2 needs a minus sign; actually 1/(−2) = −1/2, which is not equal to 1/2.
Using only a1/a2 = b1/b2 and not checking c1/c2; here c1/c2 = (−4)/(−8) = 1/2, so the value 2 gives identical lines, not parallel lines.

Why this works

Infinitely many solutions mean the lines are coincident, so a1/a2 = b1/b2 = c1/c2. Here a1/a2 = 1/2 and c1/c2 = (−4)/(−8), which also equals 1/2. Put b1/b2 = 1/k equal to these, so 1/k = 1/2, giving k = 2. Check: with k = 2, the second equation is 2x + 2y − 8, and dividing by 2 gives x + y − 4, exactly the first equation. If k is −2, then 1/k = −1/2, which is not equal to 1/2, so the lines would meet in one point, not infinitely many.

Key idea to remember

Dependent pair of linear equations: see Key ideas used on this page.

Explanation

A dependent pair is a pair whose two equations are equivalent, so their graphs are the same line and every point on it is a common solution. This happens when a1/a2 = b1/b2 = c1/c2. In this question the first and third ratios already equal 1/2, so k is chosen to make the second ratio 1/2 as well.

Important formulas

a1/a2 = b1/b2 = c1/c2 · Condition for infinitely many solutions (coincident lines) of a pair of linear equations.

Q11. If \(x = 1\) and \(y = 2\) is a solution of the pair of linear equations \(2x - 3y + a = 0\) and \(2x + 3y - b = 0\), then :

Mark practised

(A) \(a = 2b\)
(B) \(2a = b\)
(C) \(a + 2b = 0\)
(D) \(2a + b = 0\)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 3.3 Algebraic Methods of Solving a Pair of Linear Equations
Source: (CBSE 2025 Mathematics Standard, Set 30/1/1, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 2a = b, because substituting x = 1, y = 2 gives a = 4 and b = 8, so 2a = b.
Why? A solution of a pair of linear equations satisfies both equations, so substituting the given values turns each equation into an equation in one unknown.

The full 1 mark is for the correct option.

Answer:

(B) 2a = b

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Finding a = 4 from the first equation only, and picking an option without finding b.
Misreading the sign before b in 2x + 3y − b = 0, so b is taken as −8, which leads to option D.
Swapping the two values, so a and b change places and option A is chosen.

Q12. The value of 'k' for which the system of linear equations 6x + y = 3k and 36x + 6y = 3 have infinitely many solutions is :

Mark practised

(A) 6
(B) 1/6
(C) 1/2
(D) 1/3

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Mathematics Standard, Set 30/3/1, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 1/6, because for infinitely many solutions a1/a2 = b1/b2 = c1/c2; here a1/a2 = 6/36 = 1/6 and c1/c2 = (-3k)/(-3) = k, so k = 1/6.
Why? For a dependent pair of linear equations the lines are coincident, so the three ratios of corresponding coefficients are equal.

The full 1 mark is for the correct option.

Answer:

(B) 1/6

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Comparing the right-hand sides 3k and 3 directly, without writing the equations with all terms on one side, and concluding k is one instead of one-sixth.
Stopping after checking that the x-ratios and y-ratios are equal, without checking the constant ratio; equal x- and y-ratios also occur for parallel lines, which have no solution.
Dividing the second equation by 6 to get 6x + y equals one-half, and then choosing that value as k, instead of equating 3k with one-half.

Why this works

Write each equation with all terms on one side: the first is 6x + y - 3k and the second is 36x + 6y - 3. For a pair of linear equations to have infinitely many solutions, NCERT gives the condition a1/a2 = b1/b2 = c1/c2. Here a1/a2 = 6/36, which reduces to 1/6, b1/b2 = 1/6, and c1/c2 = (-3k)/(-3), which equals k. Therefore k = 1/6. Quick check: substituting k = 1/6, the first equation becomes 6x + y = 3 × (1/6) = 1/2. Dividing the second equation by 6 gives 6x + y = 3/6 = 1/2. Both equations represent the same line, so they have infinitely many common solutions.

Key idea to remember

Dependent pair of linear equations: see Key ideas used on this page.

Explanation

NCERT states that for two linear equations written as a1x + b1y + c1 and a2x + b2y + c2 (all terms on one side), the pair has infinitely many solutions when a1/a2 = b1/b2 = c1/c2. In this question, a1/a2 = 6/36 = 1/6, so c1/c2 = (-3k)/(-3) = k must also be 1/6. Hence k = 1/6.

Important formulas

a1/a2 = b1/b2 = c1/c2 · Condition for coincident lines, i.e. infinitely many solutions.
a1/a2 = b1/b2 ≠ c1/c2 · Condition for parallel lines, i.e. no solution.

Q13. The larger of two supplementary angles exceeds the smaller by 18 degrees. What is the measure of larger angle ?

Mark practised

(A) 81°
(B) 99°
(C) 36°
(D) 54°

Marks: 1
Level: Basic
Years: 2023
Topic: 3.3 Algebraic Methods of Solving a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Basic, Set 430/1/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 99°, because the larger angle = (180° + 18°)/2 = 99°.
Why? Supplementary angles add up to 180°, so the two angles have sum 180° and difference 18°. The larger angle is half the sum of these two values.

The full 1 mark is for the correct option.

Answer:

(B) 99°

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/1/1: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 81°, which is the smaller angle, because the question asks for the larger angle.
Solving x + (x + 18°) = 180° as 2x = 180° and giving 90°, which forgets the 18°.
Taking the angles as x and 90° − x, which ignores the 18° difference.

Why this works

Let the smaller angle be x and the larger angle be x + 18°. Since supplementary angles add to 180°, x + (x + 18°) = 180°, so 2x = 162°, giving x = 81°. Therefore the larger angle is 81° + 18° = 99°. Quick check: 99° + 81° = 180° and 99° − 81° = 18°, so the answer fits the question.

Key idea to remember

Supplementary angles: see Key ideas used on this page.

Explanation

Supplementary angles are two angles whose sum is 180°. If the smaller angle is x, the larger angle is x + 18°. Then x + (x + 18°) = 180°, which gives x = 81°; hence the larger angle is 99°.

Important formulas

x + (x + 18°) = 180° · The smaller and larger angles together make 180°.
larger angle = (sum + difference)/2 · When two quantities have a fixed sum and a fixed difference, the larger is half of (sum + difference).

Q14. The value of k for which the equations \(3x - y + 8 = 0\) and \(6x - ky + 16 = 0\) represent coincident lines is :

Mark practised

(A) \(\frac{1}{2}\)
(B) \(-\frac{1}{2}\)
(C) 2
(D) -2

Marks: 1
Level: Basic
Years: 2023
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Basic, Set 430/4/3, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2, because for coincident lines a1/a2 = b1/b2 = c1/c2, and 3/6 = 8/16 = 1/2, so (-1)/(-k) = 1/2 gives k = 2.
Why? For coincident lines, the ratios of the corresponding coefficients are equal: a1/a2 = b1/b2 = c1/c2. Equating the b-ratio to the common value 1/2 gives k.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/3: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1/2 because 3/6 = 8/16 = 1/2, without checking that (-1)/(-k) must also equal 1/2.
Getting the sign of k wrong by mishandling the negative signs in (-1)/(-k); the ratio is 1/k, not -1/k.

Why this works

The equations are already in the general form. The coefficients are 3, -1, 8 for the first equation and 6, -k, 16 for the second. For coincident lines, a1/a2 = b1/b2 = c1/c2. Now 3/6 = 1/2 and 8/16 = 1/2, so b1/b2 must also be 1/2: (-1)/(-k) = 1/k = 1/2, hence k = 2. Check: with k = 2, the second equation is exactly twice the first equation, so the two lines are the same line.

Key idea to remember

Coincident lines: see Key ideas used on this page.

Explanation

Two lines are coincident when the ratios of the corresponding coefficients are equal: a1/a2 = b1/b2 = c1/c2. Then every point on one line lies on the other, so the pair has infinitely many solutions. This solution uses that condition to find k.

Important formulas

a1/a2 = b1/b2 = c1/c2 · Condition for coincident lines, giving infinitely many solutions.

Q15. The solution of the pair of equations \(x + y = a + b\) and \(ax - by = a^2 - b^2\) is :

Mark practised

(A) \(x = b, y = a\)
(B) \(x = -a, y = b\)
(C) \(x = a, y = b\)
(D) \(x = a, y = -b\)

Marks: 1
Level: Standard
Years: 2023
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Standard, Set 30/5/3, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) x = a, y = b, because substituting x = a and y = b satisfies both x + y = a + b and ax − by = a² − b².
Why? A solution of a pair of linear equations is a pair (x, y) that satisfies both equations; substituting the option into both equations is the test.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/5/3: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A (x = b, y = a) because it satisfies x + y = a + b, without checking that the second equation gives ab − ba, not a² − b².
Choosing D (x = a, y = −b) because x = a seems to match the a² term, without checking that ax − by becomes a² + b², not a² − b².
Writing x = a, y = b by comparing ax with a² and by with b², without substituting into both equations to verify.

Why this works

Use the substitution method: from one equation write one variable in terms of the other, then substitute into the other equation. From x + y = a + b, write y = a + b − x. Substitute into ax − by = a² − b²: ax − b(a + b − x) = a² − b² ax − ab − b² + bx = a² − b² (a + b)x − ab − b² = a² − b² (a + b)x = a² + ab = a(a + b) So x = a (when a + b is not 0). Then y = a + b − a = b. Hence the solution is x = a, y = b, which is option C. A quick check: putting x = a and y = b in the first equation gives a + b = a + b, and in the second gives a × a − b × b = a² − b². The derivation divides by a + b, so it assumes a + b is not 0. If a + b is 0, the system reduces to the single condition that x + y is 0, so it has infinitely many solutions; option C is still one of them.

Key idea to remember

Solution of a pair of linear equations: see Key ideas used on this page.

Explanation

A solution of a pair of linear equations in two variables is an ordered pair (x, y) — a value of x and a value of y together — that satisfies both equations. Here we substitute each option into both equations; the option that makes both sides equal is the solution.

Q16. 3 chairs and 1 table cost ₹ 900; whereas 5 chairs and 3 tables cost ₹ 2,100. If the cost of 1 chair is ₹ \(x\) and the cost of 1 table is ₹ \(y\), then the situation can be represented algebraically as

Mark practised

(A) \(3x + y = 900\), \(3x + 5y = 2100\)
(B) \(x + 3y = 900\), \(3x + 5y = 2100\)
(C) \(3x + y = 900\), \(5x + 3y = 2100\)
(D) \(x + 3y = 900\), \(5x + 3y = 2100\)

Marks: 1
Level: Standard
Years: 2023
Topic: 3.3 Algebraic Methods of Solving a Pair of Linear Equations
Source: (CBSE 2023 Mathematics Standard, Set 30/6/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 3x + y = 900, 5x + 3y = 2100, because 3 chairs and 1 table cost ₹900, and 5 chairs and 3 tables cost ₹2100.
Why? The total cost of a group of items is the sum of the costs of the items, so each statement translates directly into a linear equation in x and y.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/6/1: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the second equation with 5y instead of 3y, because 5 chairs and 3 tables must give 5x + 3y.
Writing the first equation with x + 3y instead of 3x + y, because 3 chairs and 1 table must give 3x + y.
Choosing option A because its first equation looks correct, without checking the second equation.

Why this works

Let x be the cost of one chair and y be the cost of one table. Three chairs cost 3x, and one table costs y, so the first statement gives 3x + y = 900. Five chairs cost 5x, and three tables cost 3y, so the second statement gives 5x + 3y = 2100. To check the equations, solve them: from 3x + y = 900, y = 900 − 3x. Substitute into 5x + 3y = 2100: 5x + 3(900 − 3x) = 2100, so 5x + 2700 − 9x = 2100, so −4x = −600, so x = 150. Then y = 900 − 3(150) = 450. Verify: 3(150) + 450 = 900 and 5(150) + 3(450) = 750 + 1350 = 2100. Option A uses 3x + 5y for the second equation, which would mean 3 chairs and 5 tables. Option B uses x + 3y for the first equation, which would mean 1 chair and 3 tables. Option D has the same wrong first equation. Only option C matches both statements.

Key idea to remember

Pair of Linear Equations in Two Variables: see Key ideas used on this page.

Explanation

A situation with two unknown quantities can be written as two linear equations. Here x is the cost of one chair and y is the cost of one table; each sentence about total cost becomes one equation.

Important formulas

cost of n items at ₹x each = nx · Used to turn '3 chairs' into 3x and '5 chairs' into 5x.
total cost = cost of chairs + cost of tables · The two statements give the two equations.

Q17. If the system of equations \(2x + 3y = 5\) and \(4x + ky = 10\) has infinitely many solutions, then the value of 'k' is :

Mark practised

(A) 1
(B) \(\frac{1}{2}\)
(C) \(\frac{3}{2}\)
(D) 6

Marks: 1
Level: Basic
Compartment: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430/S/3, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 6, because for infinitely many solutions a₁/a₂ = b₁/b₂ = c₁/c₂, so 2/4 = 3/k = -5/-10, and 3/k = 1/2 gives k = 6.
Why? For a dependent pair of linear equations, the lines coincide and the ratios of all corresponding coefficients are equal; putting the y-ratio equal to the common ratio gives k.

The full 1 mark is for the correct option.

Answer:

(D) 6

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430/S/3: option (D).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing option B, 1/2, because 1/2 is the common ratio; k is the coefficient of y, so you must find k from 3/k equal to the common ratio.
Checking only that 2/4 and 3/k are equal, and skipping the constant ratio; infinitely many solutions require a₁/a₂, b₁/b₂ and c₁/c₂ all to be equal.

Why this works

Write both equations in the general form: the first has coefficients 2 and 3 with constant -5; the second has coefficients 4 and k with constant -10. For infinitely many solutions the three ratios must be equal. Here a₁/a₂ = 2/4 = 1/2 and c₁/c₂ = -5/-10 = 1/2, so the common ratio is 1/2. The y-ratio must also equal this common ratio: 3/k = 1/2, which gives k = 6. Check: with k = 6, the second equation has coefficients 4 and 6 and its right side is double 5, so it is exactly twice the first equation; the two lines coincide and have infinitely many common solutions. NCERT Example 2 uses the same idea: one equation becomes the other after multiplying by a constant.

Key idea to remember

Dependent pair of linear equations (coincident lines): see Key ideas used on this page.

Explanation

For the pair written in the general form, if a₁/a₂ = b₁/b₂ = c₁/c₂, then the two lines coincide. Coincident lines have infinitely many common solutions. This solution uses that condition: since a₁/a₂ and c₁/c₂ are both 1/2, we set b₁/b₂ equal to 1/2 to find k.

Important formulas

a₁/a₂ = b₁/b₂ = c₁/c₂ · Condition for coincident lines, which give infinitely many solutions.
3/k = 1/2 ⇒ k = 6 · Applying the condition to find k.

Q18. The value of 'k' for which the lines represented by the equations \(3x + 5y = 2\) and \(9x + 15y = 2k\) coincide, is :

Mark practised

(A) 3
(B) -3
(C) 6
(D) -6

Marks: 1
Level: Basic
Years: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Basic, Set 430(B), Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 3, because a1/a2 = 3/9 = 1/3, b1/b2 = 5/15 = 1/3, and c1/c2 = (−2)/(−2k) = 1/k, so coincident lines require 1/k = 1/3, giving k = 3.
Why? For coincident lines, the ratios of the coefficients of x, y and the constant term are all equal.

The full 1 mark is for the correct option.

Answer:

(A) 3

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430(B): option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Giving 6, because 3 times 2 is 6, without dividing by the 2 in 2k.
Choosing −3, because the second equation would then have a negative right side, which does not match the first equation's positive right side, so the lines are parallel, not coincident.
Checking only a1/a2 and b1/b2 and ignoring c1/c2, so any value of k is accepted.

Why this works

Write each equation in the general form. The first equation has coefficients 3, 5 and −2; the second has coefficients 9, 15 and −2k. For coincident lines, a1/a2 = b1/b2 = c1/c2. Here a1/a2 = 3/9 = 1/3 and b1/b2 = 5/15 = 1/3. So c1/c2 = (−2)/(−2k) = 1/k must also equal 1/3, which gives k = 3. Quick check: for k = 3, the second equation is exactly 3 times the first equation, so the two lines are the same.

Key idea to remember

Coincident lines: see Key ideas used on this page.

Explanation

For two lines written in the general form, coincident lines occur when a1/a2 = b1/b2 = c1/c2. Then the pair has infinitely many solutions and is called a dependent (consistent) pair. This solution uses that condition to find k.

Important formulas

a1/a2 = b1/b2 = c1/c2 · Condition for two lines to be coincident.

Q19. If 3 chairs and 1 table cost ₹ 900 and 5 chairs and 3 tables ₹ 2100, then the cost of one chair is :

Mark practised

(A) ₹ 100
(B) ₹ 110
(C) ₹ 150
(D) ₹ 450

Marks: 1
Level: Basic
Years: 2024
Topic: 3.3 Algebraic Methods of Solving a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Basic, Set 430(B), Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) ₹ 150, because taking a chair as ₹ x and a table as ₹ y, 3x + y = 900 gives y = 900 − 3x, and 5x + 3(900 − 3x) = 2100 gives x = 150.
Why? Substitution Method: from one equation write one variable in terms of the other, substitute it into the other equation, and solve the resulting linear equation in one variable.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430(B): option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing ₹ 450, which is the cost of one table, instead of ₹ 150 for one chair.
Expanding 3(900 − 3x) as 2700 − 3x instead of 2700 − 9x, which gives the wrong value of x.

Why this works

Let the cost of one chair be ₹ x and the cost of one table be ₹ y. The conditions give 3x + y = 900 and 5x + 3y = 2100. From the first equation, y = 900 − 3x. Substituting in the second: 5x + 3(900 − 3x) = 2100, so 5x + 2700 − 9x = 2100, i.e. −4x = −600, so x = 150. Then y = 900 − 3(150) = 450. Check: 3 × 150 + 450 = 900 and 5 × 150 + 3 × 450 = 750 + 1350 = 2100. So the cost of one chair is ₹ 150. Another correct method also earns full marks.

Key idea to remember

Substitution Method: see Key ideas used on this page.

Explanation

In the substitution method, we express one variable in terms of the other from one equation, substitute this expression into the other equation, and solve the resulting linear equation in one variable. Then we substitute the value back to find the second variable. Here, substituting y = 900 − 3x into 5x + 3y = 2100 gives x = 150.

Q20. If a pair of linear equations in two variables is consistent, then the lines represented by the two equations are :

Mark practised

(A) always intersecting
(B) parallel
(C) always coincident
(D) intersecting or coincident

Marks: 1
Level: Standard
Years: 2024
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2024 Mathematics Standard, Set 30/3/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) intersecting or coincident, because a consistent pair of linear equations has a solution, and two lines with a common solution either intersect at exactly one point or coincide.
Why? The word 'consistent' is defined as having a solution. The graph of a linear equation is a straight line, and two straight lines share at least one point only when they are intersecting or coincident.

The full 1 mark is for the correct option.

Answer:

(D) intersecting or coincident

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/3/2: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A (always intersecting), because a consistent pair can also have coincident lines, which give infinitely many solutions.
Choosing C (always coincident), because a consistent pair can also have intersecting lines, which give exactly one solution.
Choosing B (parallel), because parallel lines never meet and therefore the pair has no solution, so it is inconsistent, not consistent.

Why this works

Consistent means the pair has at least one solution. On the graph, a solution is a point that lies on both lines. Two straight lines can have a common point in two ways: they cut each other at one point, so they are intersecting lines, or they lie on top of each other, so they are coincident lines. Parallel lines never meet, so they give no common solution and the pair is inconsistent. Therefore the correct choice is D. Quick check from the NCERT summary: if a1/a2 is not equal to b1/b2 the lines intersect, and if a1/a2 = b1/b2 = c1/c2 they coincide; both cases are consistent.

Key idea to remember

Consistent pair of linear equations: see Key ideas used on this page.

Explanation

A pair of linear equations in two variables which has a solution is called a consistent pair. The graph of such a pair is two lines that have at least one point in common. The common point is unique when the lines intersect, and there are infinitely many common points when the lines are coincident. In both cases the pair is consistent. Parallel lines have no common point, so that pair is inconsistent.

Q21. If the pair of linear equations

Mark practised

\[2x + 3y = 5 \text{ and } 4ky - (1 - 3k)x = 6k + 2\]

represents coincident lines, then the value of 'k' is :

(A) \(-\frac{1}{3}\)
(B) 3
(C) \(\frac{1}{4}\)
(D) 4

Marks: 1
Level: Standard
Compartment: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/2, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 3, because for coincident lines a1/a2 = b1/b2 = c1/c2, and 2/(3k - 1) = 3/(4k) gives 8k = 9k - 3, so k = 3.
Why? For coincident lines, the ratios of the corresponding coefficients are equal: a1/a2 = b1/b2 = c1/c2.

The full 1 mark is for the correct option.

Answer:

(B) 3

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/2: option (B).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the coefficient of x in the second equation as 1 - 3k instead of 3k - 1; the negative sign in front of (1 - 3k) changes the value of k.
Checking only 2/(3k - 1) = 3/(4k) and not verifying that the ratio of the constant terms is also equal; coincident lines need all three ratios equal.
Giving 1/4, the common ratio of the coefficients, instead of the value of k.

Why this works

Coincident lines lie exactly on top of each other, so the pair has infinitely many solutions. The condition is that the ratios of the corresponding coefficients are equal: a1/a2 = b1/b2 = c1/c2.

Rearrange each equation so that the x-term, y-term and constant term are on one side. From the first equation the coefficient of x is 2 and the coefficient of y is 3. From the second equation the coefficient of x is 3k - 1 and the coefficient of y is 4k. Equating the first two ratios gives 2/(3k - 1) = 3/(4k). Cross-multiplying gives 8k = 9k - 3, so k = 3. With k = 3, the three ratios are 2/8, 3/12 and 5/20, which are equal, so the condition is satisfied.

Key idea to remember

Coincident lines: see Key ideas used on this page.

Explanation

Two lines are coincident when they lie exactly on top of each other, so the pair has infinitely many solutions. For a pair of linear equations in two variables, this happens exactly when the ratios of the corresponding coefficients are equal. This solution rewrites both equations in that form, compares the first two ratios to find k, and then verifies that the third ratio also matches.

Important formulas

a1/a2 = b1/b2 = c1/c2 · Condition for coincident lines; the pair is dependent and consistent.

Q22. The pair of linear equations

Mark practised

\[9x - 15y + 19 = 0 \text{ and } 5y - 3x - 9 = 0\]

represents two lines which are :

(A) intersecting exactly at one point.
(B) intersecting exactly at two points.
(C) parallel.
(D) coincident.

Marks: 1
Level: Standard
Compartment: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/3, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) parallel, because a₁/a₂ = b₁/b₂ = -3, while c₁/c₂ = -19/9.
Why? When a₁/a₂ = b₁/b₂ but c₁/c₂ is different, the two lines are parallel and the pair of equations has no solution.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/3: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Reading the second equation without writing it in the same variable order as the first, so the x- and y-coefficients get swapped.
Choosing coincident because the x-coefficient ratio equals the y-coefficient ratio, without checking the constant-term ratio.
Thinking two distinct straight lines can meet at exactly two points; distinct lines either meet at one point, are parallel, or coincide.

Why this works

Put both equations in the standard form a₁x + b₁y + c₁ = 0. The first equation is already in this form. Rearrange the second equation so the x-term comes first. Now compare the ratios. The ratio of the x-coefficients equals the ratio of the y-coefficients, so the lines cannot meet at exactly one point. The ratio of the constant terms is different, so the lines are not the same line. Lines with equal x- and y-coefficient ratios but a different constant-term ratio are parallel; they have no common point. A quick check is to multiply the second equation by the factor that makes its x-coefficient equal to the first equation's x-coefficient: the variable terms become the same, but the constant term remains different, so the lines never meet.

Key idea to remember

Comparison of ratios a₁/a₂, b₁/b₂ and c₁/c₂: see Key ideas used on this page.

Explanation

For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the values of a₁/a₂, b₁/b₂ and c₁/c₂ decide the position of the lines. If a₁/a₂ ≠ b₁/b₂, the lines intersect at exactly one point. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and the pair is inconsistent. If all three ratios are equal, the lines are coincident. Here the first two ratios are equal and the third is different, so the lines are parallel.

Important formulas

a₁/a₂ ≠ b₁/b₂ ⇒ lines intersect at exactly one point · The pair is consistent and has a unique solution.
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ⇒ lines are parallel · The pair is inconsistent and has no solution.
a₁/a₂ = b₁/b₂ = c₁/c₂ ⇒ lines are coincident · The pair is dependent and consistent, with infinitely many solutions.

Q23. A system of two linear equations in two variables is inconsistent, if the lines in the graph are :

Mark practised

(A) coincident
(B) parallel
(C) intersecting at one point
(D) intersecting at right angles

Marks: 1
Level: Standard
Years: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Mathematics Standard, Set 30/3/2, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) parallel, because parallel lines never meet, so the pair has no solution.
Why? A pair of linear equations with no solution is called an inconsistent pair, and parallel lines have no common point.

The full 1 mark is for the correct option.

Answer:

(B) parallel

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/2: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing coincident, because coincident lines share every point; that gives infinitely many solutions, so the pair is consistent.
Choosing intersecting at one point, because the lines meet, giving a unique solution.
Choosing intersecting at right angles, because perpendicular lines still intersect at one point, so the pair is consistent.

Why this works

The graph of each linear equation is a straight line. A solution of the pair is a point that lies on both lines. Parallel lines never meet, so there is no common point and no solution. NCERT calls a pair with no solution an inconsistent pair. Coincident lines meet at every point, giving infinitely many solutions; intersecting lines meet at one point, giving a unique solution. Both of those are consistent.

Key idea to remember

Inconsistent pair of linear equations: see Key ideas used on this page.

Explanation

The NCERT text defines an inconsistent pair of linear equations as a pair which has no solution. In the graphical method, the lines representing the equations are parallel exactly when there is no solution.

Important formulas

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ · Condition for parallel lines, so the pair has no solution and is inconsistent.
a₁/a₂ = b₁/b₂ = c₁/c₂ · Condition for coincident lines, so the pair has infinitely many solutions and is consistent.

Q24. The system of equations 2x + 1 = 0 and 3y - 5 = 0 has

Mark practised

(A) unique solution
(B) two solutions
(C) no solution
(D) infinite number of solutions

Marks: 1
Level: Standard
Years: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Mathematics Standard, Set 30/6/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) unique solution, because 2x + 1 = 0 gives x = −1/2 and 3y − 5 = 0 gives y = 5/3, so the two lines intersect at exactly one point.
Why? A pair of linear equations has a unique solution when the lines represented by them intersect at exactly one point.

The full 1 mark is for the correct option.

Answer:

(A) unique solution

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/1: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 'no solution' because 2x + 1 = 0 has no y and 3y − 5 = 0 has no x; actually one line is vertical and the other horizontal, so they intersect.
Choosing 'two solutions' because there are two equations; two distinct straight lines can have only one common point.
Choosing 'infinite number of solutions' because both equations look simple; infinitely many solutions need the two lines to coincide, which is not the case here.

Why this works

Each equation represents a straight line. 2x + 1 = 0 means x = −1/2, so every point on this line has the same x-coordinate; the line is vertical. 3y − 5 = 0 means y = 5/3, so every point on this line has the same y-coordinate; the line is horizontal. A vertical line and a horizontal line meet at exactly one point, (−1/2, 5/3). So the pair is consistent and has a unique solution. Check: 2(−1/2) + 1 = 0 and 3(5/3) − 5 = 0. Another correct method also earns full marks.

Key idea to remember

Consistent pair of linear equations: see Key ideas used on this page.

Explanation

A pair of linear equations is consistent when it has at least one solution. Graphically, this happens when the two lines intersect, giving a unique solution, or coincide, giving infinitely many solutions. Here the vertical line x = −1/2 and the horizontal line y = 5/3 intersect at one point, so the pair is consistent with a unique solution.

Q25. The system of equations \(x + 5 = 0\) and \(2x - 1 = 0\), has

Mark practised

(A) No solution
(B) Unique solution
(C) Two solutions
(D) Infinite solutions

Marks: 1
Level: Standard
Years: 2025
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2025 Mathematics Standard, Set 30/6/2, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) No solution, because the first equation forces x = −5 and the second forces x = 1/2, and no value of x can be both.
Why? A solution of a pair of equations must satisfy both equations at the same time; here the two equations force different values of x, so no common solution exists.

The full 1 mark is for the correct option.

Answer:

(A) No solution

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/2: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 'Unique solution' because each equation separately has one solution, without checking that a system solution must satisfy both equations at the same time.
Choosing 'Two solutions' because there are two equations, forgetting that a solution is a common value of x, not one value per equation.
Thinking x = −5 and x = 1/2 are the two solutions of the system, when actually the system has no common solution.

Why this works

Each equation fixes one value of x: the first line gives x = −5 and the second gives x = 1/2. A solution of the pair would need both at once, which is impossible. Check: putting x = −5 in the second equation gives 2(−5) − 1 = −11, not 0; putting x = 1/2 in the first gives 1/2 + 5 = 11/2, not 0. So neither value satisfies both equations. The two lines are vertical and parallel, so the pair is inconsistent and has no solution.

Key idea to remember

Inconsistent pair of linear equations: see Key ideas used on this page.

Explanation

A pair of linear equations is called inconsistent if it has no solution. In the graphical method, this happens when the lines are parallel. Here the two given equations are vertical lines x = −5 and x = 1/2; being parallel, they never meet, so the pair has no solution.

Q26. If pair of linear equations given by \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) has infinitely many solutions, then which of the following is definitely true ?

Mark practised

(A) \(\frac{a_1}{a_2} = \frac{c_2}{c_1}\)
(B) \(a_1a_2 \neq b_1b_2\)
(C) \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
(D) \(a_1b_2 = a_2b_1\)

Marks: 1
Level: Standard
Years: 2026
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2026 Mathematics Standard, Set 30(B), Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) \(a_1b_2 = a_2b_1\), because infinitely many solutions means the lines are coincident, so \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), and cross-multiplying the first equality gives \(a_1b_2 = a_2b_1\).
Why? Coincident lines lie exactly on top of each other, and for such a dependent pair the NCERT condition is \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\). Equality of the first two ratios is exactly \(a_1b_2 = a_2b_1\).

The full 1 mark is for the correct option.

Answer:

(D) a₁b₂ = a₂b₁

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30(B): option (D).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A because it shows equal ratios, without checking that the third ratio must be \(\frac{c_1}{c_2}\), not \(\frac{c_2}{c_1}\).
Choosing B or C because they contain 'not equal', forgetting that infinitely many solutions mean the ratios are equal.
Confusing parallel lines with infinitely many solutions; parallel lines have no solution, not infinitely many.

Why this works

Infinitely many solutions happen only when the two lines are the same line. Then one equation is a constant multiple of the other, so \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\). Cross-multiplying the first two ratios gives \(a_1b_2 = a_2b_1\), so option D is true. Option A would need \(\frac{a_1}{a_2} = \frac{c_1}{c_2}\), not \(\frac{c_2}{c_1}\). Options B and C say the ratios are unequal, which is false for coincident lines.

Key idea to remember

Dependent pair of linear equations in two variables: see Key ideas used on this page.

Explanation

A dependent pair of linear equations is a pair whose two equations represent the same line. Such a pair is always consistent and has infinitely many solutions, because every point on the line satisfies both equations. The NCERT condition for this case is \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).

Important formulas

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) · Condition for infinitely many solutions, when the lines are coincident.
\(a_1b_2 = a_2b_1\) · Obtained by cross-multiplying the first two equal ratios.

Q27. The value of \(k\) for which the pair of linear equations \(kx - 3y = 5\), \(4x - 6y = 10\) has infinitely many solutions, is :

Mark practised

(A) 4
(B) 3
(C) 2
(D) 1

Marks: 1
Level: Basic
Years: 2026
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2026 Mathematics Basic, Set 430(B), Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2, because for infinitely many solutions a₁/a₂ = b₁/b₂ = c₁/c₂, so k/4 = (−3)/(−6) = (−5)/(−10) = 1/2, giving k = 2.
Why? For a dependent (consistent) pair of linear equations the lines coincide, and the ratio condition a₁/a₂ = b₁/b₂ = c₁/c₂ holds; equating k/4 to the common ratio gives k.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 430(B): option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 4 because the second equation has coefficient 4 for x, without checking whether k/4 equals the common ratio 1/2.
Using the condition for parallel lines a₁/a₂ = b₁/b₂ ≠ c₁/c₂, even though here the constant-term ratio (−5)/(−10) is also 1/2, so the lines coincide.
Using the wrong ratio 1 instead of 1/2, which leads to the wrong value 4 for k.

Why this works

For the ratio test, write the equations in the general form with coefficients a₁, b₁, c₁ for the first equation and a₂, b₂, c₂ for the second. A pair of linear equations has infinitely many solutions exactly when the two lines are coincident, and then a₁/a₂ = b₁/b₂ = c₁/c₂. In this question b₁/b₂ = (−3)/(−6) = 1/2 and c₁/c₂ = (−5)/(−10) = 1/2, so k/4 must also be 1/2. Hence k = 2. Quick check: when k = 2, the first equation is half of the second, so the two lines are the same line. Another correct method also earns full marks.

Key idea to remember

Dependent pair of linear equations: see Key ideas used on this page.

Explanation

A dependent pair of linear equations is one in which the two equations are equivalent, so their graphs are coincident lines. Such a pair has infinitely many common solutions. For the general equations, the condition is a₁/a₂ = b₁/b₂ = c₁/c₂. Here the y-coefficients and the constant terms both give the ratio 1/2, so k/4 must equal 1/2, giving k = 2.

Important formulas

a₁/a₂ = b₁/b₂ = c₁/c₂ · Condition for coincident lines, which give infinitely many solutions.

Q28. The pair of linear equations \(\frac{3x}{2} + \frac{5y}{3} = 7\) and \(9x + 10y = 14\), is :

Mark practised

(A) consistent
(B) inconsistent
(C) consistent with one solution
(D) consistent with many solutions

Marks: 1
Level: Standard
Years: 2026
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2026 Mathematics Standard, Set 30/2/2, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) inconsistent, because a₁/a₂ = (3/2)/9 = 1/6, b₁/b₂ = (5/3)/10 = 1/6, but c₁/c₂ = 7/14 = 1/2, so the lines are parallel and have no solution.
Why? When a₁/a₂ equals b₁/b₂ but the constant ratio is different, the two lines are parallel; parallel lines have no common point, so the pair is called inconsistent.

The full 1 mark is for the correct option.

Answer:

(B) inconsistent

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/2/2: option (B).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing C because the two equations look different, without comparing all three ratios.
Multiplying the first equation by 6 and then thinking 9x + 10y = 42 and the second equation are the same line; the constants are different, so the lines are parallel.
Comparing only a₁/a₂ and b₁/b₂ and calling the pair consistent even though the constant ratio is different.

Why this works

Write the ratios directly. a₁/a₂ = (3/2)/9 = 1/6, b₁/b₂ = (5/3)/10 = 1/6 and c₁/c₂ = 7/14 = 1/2. Since a₁/a₂ = b₁/b₂ but c₁/c₂ is different, the lines are parallel and have no common point, so the pair is inconsistent. A quick check: multiplying the first equation by 6 gives 9x + 10y = 42, while the second equation has the same left side but a different constant term, so the lines are parallel, not coincident. (Note: The printed question has the term +10y in the second equation; the NCERT text version has −10y, so the two pairs are different.)

Key idea to remember

Comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂: see Key ideas used on this page.

Explanation

For a pair of linear equations, compare the three ratios a₁/a₂, b₁/b₂ and the ratio of the constant terms. If a₁/a₂ ≠ b₁/b₂, the lines intersect at one point and the pair has a unique solution. If a₁/a₂ = b₁/b₂ but the constant ratio is different, the lines are parallel and the pair has no solution, so it is inconsistent. If all three ratios are equal, the lines coincide and there are infinitely many solutions. Here the first two ratios are 1/6 and the constant ratio is 1/2, which is the parallel case.

Important formulas

a₁/a₂ ≠ b₁/b₂ · Lines intersect; the pair is consistent with a unique solution.
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ · Lines are parallel; the pair is inconsistent.
a₁/a₂ = b₁/b₂ = c₁/c₂ · Lines coincide; the pair is dependent and consistent.

Q29. Equation of a line coincident with \(2.5x - 2y = 3\) is :

Mark practised

(A) \(5x - 4y = 3\)
(B) \(5x - 4y + 6 = 0\)
(C) \(15x - 12y - 3 = 0\)
(D) \(5x - 4y - 6 = 0\)

Marks: 1
Level: Standard
Years: 2026
Topic: 3.2 Graphical Method of Solution of a Pair of Linear Equations
Source: (CBSE 2026 Mathematics Standard, Set 30/3/3, Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 5x − 4y − 6 = 0, because multiplying the given equation by 2 gives 5x − 4y = 6, which is the same line.
Why? Multiplying both sides of an equation by the same non-zero number does not change its graph, so the new equation represents the same line.

The full 1 mark is for the correct option.

Answer:

(D) 5x − 4y − 6 = 0

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/3/3: option (D).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A because 5x − 4y appears in both equations, without checking that the constants 3 and 6 are different.
Choosing C because 15x − 12y − 3 divided by 3 gives 5x − 4y − 1, which is parallel to the given line, not the same line.
Choosing B because 5x − 4y + 6 has the same x and y coefficients, but its constant has the opposite sign, so the lines are parallel.

Why this works

Multiplying the given equation by 2 turns it into 5x − 4y = 6. Bringing 6 to the left side gives 5x − 4y − 6 = 0, which is option D. A quick check: the point (0, −1.5) lies on both lines. Comparing the ratios of the coefficients gives the same conclusion.

Key idea to remember

Coincident lines: see Key ideas used on this page.

Explanation

Two linear equations represent coincident lines when one equation is a constant multiple of the other. Then every point on one line is also on the other, so the pair has infinitely many solutions. Here option D is exactly the given equation multiplied by 2.

Important formulas

a₁/a₂ = b₁/b₂ = c₁/c₂ · Condition for coincident lines, where a₁, b₁, c₁ and a₂, b₂, c₂ are the coefficients of the two equations in general form.

1 / 29

Key ideas used on this page

Dependent pair of linear equations
Used in questions 2 4 10 12 27
Coincident lines
Used in questions 14 18 21 29
Solution of a pair of linear equations
Used in questions 1 11 15
Graphical Method of Solution of a Pair of Linear Equations
Used in questions 3 9
Dependent pair of linear equations in two variables
Used in questions 8 26
Consistent pair of linear equations
Used in questions 20 24
Inconsistent pair of linear equations
Used in questions 23 25
Comparing the ratios a1/a2, b1/b2 and c1/c2
Used in question 5
Elimination Method
Used in question 6
Condition for parallel lines
Used in question 7
Supplementary angles
Used in question 13
Pair of Linear Equations in Two Variables
Used in question 16
Dependent pair of linear equations (coincident lines)
Used in question 17
Substitution Method
Used in question 19
Comparison of ratios a₁/a₂, b₁/b₂ and c₁/c₂
Used in question 22
Comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂
Used in question 28

How to use these questions

Try each question on paper first. Then read the steps: the mark beside each step shows what the examiner looks for. If a step is new to you, revise Section 3.2 and Section 3.3 of the NCERT chapter.

To revise the lessons behind these questions, read the NCERT chapter, exactly as NCERT publishes it on its own website.

Official papers behind this page

Every question on this page comes from an official CBSE Class 10 Mathematics board paper. The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.

Year	Exam	Course	Question paper	Marking scheme
2026	Main	Standard (041)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given.
2026	Main	Basic (241)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 29.1 MB, Class 10 Maths Basic (241)) to see how marks are given.
2025	Main	Standard (041)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Main	Basic (241)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Compartment	Standard (041)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 3.4 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 8.1 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Compartment	Basic (241)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Main	Standard (041)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Main	Basic (241)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Compartment	Basic (241)	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Main	Standard (041)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Main	Basic (241)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 49.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 26 MB, Class 10 Maths Basic (241)) to see how marks are given.

Sources and session

Each question is reproduced as printed in its CBSE paper, with its year, paper code and question number; only the layout is tidied. The same question printed in several paper sets counts once, and the card says how often it was printed in 2023–2026. These counts describe past papers; they are not predictions.

Every solution is original and follows NCERT Mathematics Class X (reprint 2026-27). The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.

Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.

1-mark questions on Pair of Linear Equations in Two Variables

All 29 Pair of Linear Equations in Two Variables 1-mark questions

Q1. If the pair of linear equations x - y = 1, x + ky = 5 has a unique solution x = 2, y = 1, then the value of k is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q2. The value of \(k\) for which the pair of equations \(kx = y + 2\) and \(6x = 2y + 3\) has infinitely many solutions,

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q3. The lines represented by linear equations \(x = a\) and \(y = b\) ( \(a \neq b\) ) are

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q4. The value of \(k\) for which the system of equations \(3x - y + 8 = 0\) and \(6x - ky + 16 = 0\) has infinitely many solutions, is

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q5. Which out of the following type of straight lines will be represented by the system of equations \(3x + 4y = 5\) and \(6x + 8y = 7\) ?

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q6. The solution of the pair of linear equations \(3x + 4y = 5\) ; \(4x + 3y = 9\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q7. The value of \(m\) for which lines \(14x + my = 20\) and \(-3x + 2y = 16\) are parallel, is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q8. If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q9. The pair of equations \(x = a\) and \(y = b\) graphically represents lines which are :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q10. The value of 'k' for which the pair of linear equations \(x + y - 4 = 0\) and \(2x + ky - 8 = 0\) has infinitely many solutions, is

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q11. If \(x = 1\) and \(y = 2\) is a solution of the pair of linear equations \(2x - 3y + a = 0\) and \(2x + 3y - b = 0\), then :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q12. The value of 'k' for which the system of linear equations 6x + y = 3k and 36x + 6y = 3 have infinitely many solutions is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q13. The larger of two supplementary angles exceeds the smaller by 18 degrees. What is the measure of larger angle ?

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q14. The value of k for which the equations \(3x - y + 8 = 0\) and \(6x - ky + 16 = 0\) represent coincident lines is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q15. The solution of the pair of equations \(x + y = a + b\) and \(ax - by = a^2 - b^2\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q16. 3 chairs and 1 table cost ₹ 900; whereas 5 chairs and 3 tables cost ₹ 2,100. If the cost of 1 chair is ₹ \(x\) and the cost of 1 table is ₹ \(y\), then the situation can be represented algebraically as

Step-by-step solution

Step 1: Choose the option