Consider the following subset of ℝ: U = {x ∈ ℝ : x² − 9x + 18 ≤ 0, x² − 7x + 12 ≤ 0}. Which one of the following statements is true?

C. x² − 9x + 18 ≤ 0 ⇔ x ∈ [3, 6]; x² − 7x + 12 ≤ 0 ⇔ x ∈ [3, 4]. So U = [3, 4] and inf U = 3.

Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy — CSIR NET December 2023 Part B, solved

Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Consider the following subset of $\mathbb{R}: U =$ { $x \in \mathbb{R} : x^{2} - 9x + 18 \le 0, x^{2} - 7x + 12 \le 0$ }. Which one of the following statements is true?

A.inf U = 5.
B.inf U = 4.
C.inf U = 3.✓ Correct
D.inf U = 2.

The idea, in plain words

Two conditions have to hold at once, so solve each one separately and keep only the overlap. Each gives a stretch of the line, and the answer is the left-hand end of the part they share.

Solution

$x^{2} - 9x + 18 \le 0 \Leftrightarrow x \in [3, 6]; x^{2} - 7x + 12 \le 0 \Leftrightarrow x \in [3, 4]$ .

So U = [3, 4] and inf U = 3.

Why each option is right or wrong

Checked against 2 symbolic computations

A.WrongHypothesis dropped
False.

5 lies in [3, 6] but not in [3, 4]

so it is not even an element of U.

Choosing 5 almost always means solving $x^{2} - 9x + 18 \le 0$

reading off its interval, and never imposing the second constraint.
B.WrongBoundary and endpoint
False — 4 is the supremum of U, not the infimum.

SymPy gives the roots as {3, 6} and {3, 4}

so U = [3, 4]: 3 is the left endpoint and 4 the right.

This option is what correct algebra plus a misread of which endpoint was asked for produces

which is why it is the most-picked wrong answer.
C.Correct
True. $x^{2} - 9x + 18 \le 0$ has roots 3 and 6

so it holds exactly on $[3, 6]. x^{2} - 7x + 12 \le 0$ has roots 3 and 4

so it holds on [3, 4].

Both factorisations verified by SymPy.

The intersection is [3, 4]

so inf U = 3.
D.WrongExecution slip
False.

2 satisfies neither inequality.

This is the factorisation-slip option: reading $x^{2} - 7x + 12$ as (x − 2)(x − 5) — which is actually $x^{2} - 7x + 10$ — plants a spurious root at 2.

3 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.

Last revised 11 Sept 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

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