UGCNET Questions

For a conditional statement p → q, the contrapositive ¬q → ¬p is logically equivalent to the original. Here p is “the number is divisible by 4” and q is “the number is even.” The contrapositive is “if a number is not even, then it is not divisible by 4.” Both the original statement and its contrapositive are true or false together in all possible cases, so they are logically equivalent.

Option A:
Option A reverses the direction of the implication to q → p, which is the converse. The truth of the original statement does not guarantee the truth of its converse; being even does not ensure divisibility by 4.

Option B:
Option B states “If a number is not divisible by 4, then it is not even,” which is the inverse. The inverse is not logically equivalent to the original because many numbers are not divisible by 4 but are still even.

Option C:
Option C correctly expresses the contrapositive ¬q → ¬p: if a number is not even, then it is not divisible by 4. This preserves the logical content of the original implication and is therefore equivalent to it.

Option D:
Option D, “If a number is not divisible by 4, then it is even,” contradicts the original pattern and misstates the relationship between being even and being divisible by 4.

The contrapositive of a conditional “If P then Q” is “If not Q then not P.” It is logically equivalent to the original statement. Here, P is “it is a bird” and Q is “it can fly.” Negating Q gives “it cannot fly,” and negating P gives “it is not a bird.” Therefore, the contrapositive is “If it cannot fly, then it is not a bird,” which preserves the truth conditions of the original conditional.

Option A:
Option A follows the correct procedure of reversing and negating both the antecedent and consequent. It captures the idea that anything that fails the flight condition cannot belong to the bird category under this rule.

Option B:
Option B is the converse, “If Q then P,” which is not guaranteed to be equivalent; many non birds can fly.

Option C:
Option C is the inverse, “If not P then not Q,” which is also not equivalent and can be false even if the original is true.

Option D:
Option D contradicts the original conditional by stating that non birds can fly as a rule, which is unrelated to the given statement.

From the first statement, we know that every square belongs to the set of rectangles. The second statement says that no rectangles belong to the set of circles, so the sets of rectangles and circles are disjoint. Since squares are a subset of rectangles, and rectangles do not overlap with circles, squares also cannot overlap with circles. Therefore, the conclusion that no squares are circles is logically valid.

Option A:
Option A, “Some squares are circles,” contradicts the second premise. If any square were also a circle, it would simultaneously be a rectangle and a circle, which is impossible when no rectangles are circles.

Option B:
Option B, “No squares are circles,” correctly applies the idea that if rectangles and circles are disjoint sets, then any subset of rectangles (such as squares) must also be disjoint from circles. Hence, no square can be a circle.

Option C:
Option C, “All circles are squares,” reverses and overextends the relationship. We are told only that all squares are rectangles, not that all circles are contained in the set of squares.

Option D:
Option D, “Some rectangles are circles,” is directly inconsistent with the second statement, which asserts that no rectangles are circles.

The conditions state that A must come before B and B must come before C. This implies a strict ordering: A first, B second, C third. Any schedule that violates either of these precedence constraints is unacceptable. Among the options, only the sequence A, B, C respects both requirements, placing A before B and B before C.

Option A:
Option A puts C before B, contradicting the condition that B must be before C.

Option B:
Option B puts B before A, violating the requirement that A must be before B.

Option C:
Option C satisfies A < B and B < C simultaneously and thus fully respects the given conditions. Option D: Option D reverses the required order entirely and fails both constraints.

Given a conditional “If P then Q,” the inverse is formed by negating both the antecedent and the consequent while keeping their order: “If not P then not Q.” Here, P is “the student is sincere” and Q is “the student passes the exam.” The inverse is therefore “If a student is not sincere, then the student does not pass the exam.”

Option A:
Option A is the converse, reversing the roles of antecedent and consequent to “If Q then P.”

Option B:
Option B correctly negates both antecedent and consequent without swapping them, matching the standard definition of the inverse.

Option C:
Option C negates only the antecedent while leaving the consequent positive, which does not correspond to any standard paired form (it is neither converse, inverse, nor contrapositive of the original).

Option D:
Option D is the contrapositive, “If not Q then not P,” which is logically equivalent to the original conditional, not its inverse.

Statement (1) presents a solvable linear equation, 3x + 2 = 20, which has a unique solution. Solving it gives x = 6, so statement (1) by itself fully determines the value of x. Statement (2) only tells us that x is an integer but gives no equation or relation to narrow down x further. Therefore, statement (2) alone does not allow us to find a unique x. In data sufficiency terms, (1) alone is sufficient while (2) alone is not.

Option A:
Option A correctly reflects that statement (1) yields a unique solution for x, and that statement (2) adds no new restrictive information when used alone.

Option B:
Option B reverses the roles of the statements and incorrectly claims that simply knowing x is an integer is sufficient to find a unique value, which is impossible without an equation.

Option C:
Option C assumes that neither statement can independently identify x, which is incorrect because statement (1) clearly does.

Option D:
Option D states that each statement alone is sufficient, but this overstates the power of statement (2), which does not determine any specific numerical value.