Pair of Linear Equations in Two Variables: MCQs with Answers

Official CBSE answer · Solution by Team TutorVertex

Parallel lines: $\frac{\text{a}}{\text{3}}$ = $\frac{\text{2}}{\text{b}}$ ≠ $\frac{\text{9}}{\text{18}}$

$\frac{\text{a}}{\text{3}}$ = $\frac{\text{2}}{\text{b}}$ ⇒ ab = 6

Coincidence would require a = $\frac{\text{3}}{\text{2}}$, which is not an integer.

Correct answer: D — ab = 6

Official CBSE answer · Solution by Team TutorVertex

Coincident lines: $\frac{\text{3}}{\text{6}}$ = $\frac{\text{(−1)}}{\text{(−k)}}$ = $\frac{\text{8}}{\text{16}}$

$\frac{\text{1}}{\text{2}}$ = $\frac{\text{1}}{\text{k}}$

k = 2

Correct answer: C — 2

Official CBSE answer · Solution by Team TutorVertex

First equation × (−3): −3x − 6y − 15 = 0

Second equation: −3x − 6y + 1 = 0

Same x and y coefficients, different constants.

The lines are distinct and parallel: no solution.

Correct answer: D — None

Official CBSE answer · Solution by Team TutorVertex

Parallel lines: $\frac{\text{3}}{\text{2}}$ = 2m/5

15 = 4m

m = $\frac{\text{15}}{\text{4}}$

The constant ratio $\frac{\text{−2}}{\text{1}}$ differs from $\frac{\text{3}}{\text{2}}$.

Correct answer: D — $\frac{\text{15}}{\text{4}}$

Official CBSE answer · Solution by Team TutorVertex

Unique solution: $\frac{\text{(2k)}}{\text{6}}$ ≠ $\frac{\text{5}}{\text{5}}$

$\frac{\text{k}}{\text{3}}$ ≠ 1

k ≠ 3

Correct answer: A — k ≠ 3

Official CBSE answer · Solution by Team TutorVertex

First equation × 3: 9x + 15y = 6

For coincident lines: 2k = 6

k = 3

Correct answer: A — 3

Official CBSE answer · Solution by Team TutorVertex

No solution: $\frac{\text{k}}{\text{2}}$ = $\frac{\text{2}}{\text{1}}$ ≠ $\frac{\text{3}}{\text{5}}$

$\frac{\text{k}}{\text{2}}$ = 2 ⇒ k = 4

Then the equations give 4x + 2y = 3 and 4x + 2y = 10.

These cannot both hold.

Correct answer: D — k = 4

Official CBSE answer · Solution by Team TutorVertex

No solution requires proportional x and y coefficients.

3(k − 1) = 2k − 1

3k − 3 = 2k − 1 ⇒ k = 2

Then: 3x + y = 1 and 3x + y = 5.

Therefore, there is no solution.

Correct answer: D — 2

Official CBSE answer · Solution by Team TutorVertex

Distinct parallel lines have no common solution.

Their coefficient ratios satisfy:

$\frac{\text{a₁}}{\text{a₂}}$ = $\frac{\text{b₁}}{\text{b₂}}$ ≠ $\frac{\text{c₁}}{\text{c₂}}$

These ratio expressions assume non-zero denominators.

Correct answer: B — $\frac{\text{a₁}}{\text{a₂}}$ = $\frac{\text{b₁}}{\text{b₂}}$ ≠ $\frac{\text{c₁}}{\text{c₂}}$

Official CBSE answer · Solution by Team TutorVertex

No solution: $\frac{\text{3}}{\text{6}}$ = $\frac{\text{(−k)}}{\text{10}}$ ≠ $\frac{\text{7}}{\text{3}}$

$\frac{\text{1}}{\text{2}}$ = −$\frac{\text{k}}{\text{10}}$

−k = 5 ⇒ k = −5

Correct answer: B — −5

Official CBSE solution · typesetting normalised

$\frac{\text{3}}{\text{2}}$ ≠ 2k/5

Hence k ≠ $\frac{\text{15}}{\text{4}}$.

Correct answer: B — k ≠ $\frac{\text{15}}{\text{4}}$

Official CBSE answer · Solution by Team TutorVertex

First equation: 2x − 5y = 6

Multiply by 3: 6x − 15y = 18

Second equation: 6x − 15y = 18

Both equations represent the same line.

Correct answer: C — Coincident

Official CBSE answer · Solution by Team TutorVertex

On x = a, the x-coordinate is a.

On y = b, the y-coordinate is b.

Common point = (a, b).

Correct answer: D — At (a, b)

Official CBSE answer · Solution by Team TutorVertex

Multiply the given equation by −3:

−15x + 9y = −6

−15x + 9y = 5 has proportional x and y coefficients, but a different constant.

Hence the two lines are parallel.

Correct answer: D — −15x + 9y = 5

Official CBSE answer · Solution by Team TutorVertex

$\frac{\text{3}}{\text{6}}$ = $\frac{\text{4}}{\text{8}}$ = $\frac{\text{1}}{\text{2}}$

$\frac{\text{5}}{\text{7}}$ ≠ $\frac{\text{1}}{\text{2}}$

Equal coefficient ratios but a different constant ratio ⇒ parallel lines.

Correct answer: A — Parallel

Official CBSE answer · Solution by Team TutorVertex

The same x-coordinate cannot equal two different numbers.

The two vertical lines are parallel.

Number of common solutions = 0.

Correct answer: A — 0

Official CBSE answer · Solution by Team TutorVertex

$\frac{\text{6}}{\text{2}}$ = $\frac{\text{(−3)}}{\text{(−1)}}$ = 3

$\frac{\text{10}}{\text{9}}$ ≠ 3

The lines are distinct and parallel.

Correct answer: B — Parallel

Official CBSE answer · Solution by Team TutorVertex

A common solution would require y to equal both 0 and −7.

This is impossible.

The two horizontal lines are parallel: no solution.

Correct answer: D — None

Official CBSE answer · Solution by Team TutorVertex

The two lines meet at (0, 2).

There is exactly one common point.

Therefore, the system is consistent with a unique solution.

Correct answer: A — Consistent with a unique solution

Official CBSE answer · Solution by Team TutorVertex

Given: x = 2, y = 1

Substitute in x + ky = 5:

2 + k(1) = 5

k = 3

Correct answer: C — 3

Official CBSE answer · Solution by Team TutorVertex

Both conditions hold only at (3, 1).

Number of common solutions = 1.

Correct answer: B — 1

Official CBSE solution · typesetting normalised

On subtracting first equation from second equation, we get

2025x + 2024y − 2024x − 2025y = −1 − 1

⇒ x − y = −2

Correct answer: B — −2