The Empty Set — IISER Maths MCQs with Solutions
Free IISER Maths The Empty Set MCQs with step-by-step solutions (4 questions). Part of Sets. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — The Empty Set · medium · theory
If $A = \{x \in \mathbb{Q} : x^2 = 5\}$, then $A$ is:
A. infinite
B. empty, since $\sqrt5$ is irrational ✓ Correct
C. a singleton
D. a two-element set
Solution: $\sqrt5$ is irrational, so no rational number squares to 5; $A$ is empty.
Q2 — The Empty Set · hard · theory
The number of subsets of the empty set $\emptyset$ is:
A. $2$
B. $1$ ✓ Correct
C. $0$
D. infinite
Solution: The only subset of $\emptyset$ is $\emptyset$ itself, so it has $2^0 = 1$ subset.
Q3 — The Empty Set · hard · theory
Which of the following is a non-empty set?
A. $\{x \in \mathbb{R} : |x| = -1\}$
B. $\{x \in \mathbb{N} : 2 < x < 3\}$
C. $\{x \in \mathbb{R} : x^2 + 1 = 0\}$
D. $\{x \in \mathbb{R} : x^2 - 2x + 1 = 0\}$ ✓ Correct
Solution: $x^2-2x+1=(x-1)^2=0$ has the real root $x=1$, so that set is $\{1\}$ — non-empty.
Q4 — The Empty Set · hard · theory
For how many real values of $k$ is $\{x \in \mathbb{R} : x^2 - 2x + k = 0\}$ empty?
A. exactly two
B. none
C. exactly one
D. infinitely many (all $k > 1$) ✓ Correct
Solution: No real roots when discriminant $4 - 4k < 0$, i.e. $k > 1$ — infinitely many such $k$.