SSC CGL 2024 Tier-I · 10 September 2024, Shift 3 · Quantitative Aptitude · Question 20 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 6 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

For what value of k will the lines 2x + 7ky – 8 = 0 and x + y – 9 = 0 have no solution?

Answer

Why

Correct — C. No solution means the lines are parallel and distinct, so the x and y coefficients share a ratio that the constants do not.

Condition: a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂
Here a₁ = 2, b₁ = 7k, c₁ = −8 and a₂ = 1, b₂ = 1, c₂ = −9.

2⁄1 = 7k⁄1
7k = 2
k = 2⁄7

Check the second half: c₁⁄c₂ = −8⁄−9 = 8⁄9, which is not 2, so the lines never meet → option (c).

Why the others are wrong

(a)
8⁄9 is the constant ratio, −8⁄−9. That is the ratio the condition requires to be different from 2, not the one k is solved from.
(b)
The coefficient ratio inverted: a₂⁄a₁ = 1⁄2 instead of a₁⁄a₂ = 2. With k = 1⁄2 the y coefficient is 3.5, the ratios 2 and 3.5 differ, and the lines cross at one point.
(d)
Solves 7k = 9, borrowing the 9 from the second constant term. The y coefficient ratio then becomes 9, nowhere near the 2 the x coefficients force.

Concept

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, two ratio tests decide everything.

Unique solution: a₁⁄a₂ ≠ b₁⁄b₂ — the lines cross.
No solution: a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂ — parallel and distinct.
Infinitely many: a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂ — one line written twice.

No solution is therefore a two-part condition, and the second part is the only thing separating it from infinitely many solutions.

Both equations must be in the same form — everything on the left, equal to zero — before you read off coefficients.

The stem already is, so c₁ = −8 and c₂ = −9 keep their minus signs. Their ratio is positive 8⁄9.

Key facts

No solution requires a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂.
Here a₁⁄a₂ = 2⁄1 = 2, so 7k must equal 2 and k = 2⁄7.
At k = 2⁄7 the constant ratio is 8⁄9, which differs from 2 and confirms the lines are parallel.

Study next

The condition for a pair of linear equations to have infinitely many solutionsReading the same test as slopes — equal slopes with different interceptsFinding k so that two given lines are perpendicular

Common traps

Solving a₁⁄a₂ = b₁⁄b₂ and never checking that c₁⁄c₂ is different.
Flipping the ratio and setting 7k = 1⁄2.
Dropping the minus signs on the constants when forming c₁⁄c₂.

SSC states the consistency test as a fill-in-the-parameter question: one coefficient is left as k and you decide the case from the ratios.

The wording rotates between no solution, unique solution and infinitely many solutions, and only the constant ratio separates the first from the last.

Related PYQs

No directly related past PYQ was found.