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Sets and their Representations — JEE Main Mathematics MCQs with Solutions

Free JEE Main Mathematics Sets and their Representations MCQs with step-by-step solutions (20 questions). Part of Sets. Practise online on Prepizo — no login needed.

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Questions with solutions

Q1 — Sets and their Representations · easy · numerical
The set $\{x : x \text{ is a prime number and } x < 20\}$ in roster form is:
A. $\{2,3,5,7,9,11,13,17,19\}$
B. $\{2,3,5,7,11,13,17,19\}$  ✓ Correct
C. $\{1,2,3,5,7,11,13,17,19\}$
D. $\{3,5,7,11,13,17,19\}$
Solution: Primes below 20: 2, 3, 5, 7, 11, 13, 17, 19 (1 is not prime, 9 is composite).
Q2 — Sets and their Representations · medium · numerical
The number of elements in the set $\{x \in \mathbb{Z} : x^2 < 30\}$ is:
A. $12$
B. $11$  ✓ Correct
C. $10$
D. $9$
Solution: $x^2<30$ for $x=-5,\dots,5$ (since $5^2=25<30$ but $6^2=36>30$): that is 11 integers.
Q3 — Sets and their Representations · medium · numerical
The set of letters in the word "MATHEMATICS", written in roster form, has how many elements?
A. $7$
B. $11$
C. $9$
D. $8$  ✓ Correct
Solution: Distinct letters: M, A, T, H, E, I, C, S — 8 elements (repeats counted once).
Q4 — Sets and their Representations · easy · numerical
The roster form of $\left\{ \dfrac{n}{n+1} : n \in \mathbb{N},\ n \le 4 \right\}$ is:
A. $\left\{\tfrac11, \tfrac12, \tfrac13, \tfrac14\right\}$
B. $\left\{\tfrac12, \tfrac23, \tfrac34, \tfrac45\right\}$  ✓ Correct
C. $\left\{\tfrac12, \tfrac23, \tfrac34, \tfrac45, \tfrac56\right\}$
D. $\left\{1, 2, 3, 4\right\}$
Solution: For $n=1,2,3,4$: $\tfrac12,\tfrac23,\tfrac34,\tfrac45$.
Q5 — Sets and their Representations · medium · theory
The set-builder form of $\{1, 4, 9, 16, 25\}$ is:
A. $\{x : x = n^2,\ n \in \mathbb{N},\ 1 \le n \le 5\}$  ✓ Correct
B. $\{x : x = n^2,\ n \in \mathbb{Z}\}$
C. $\{x : x = 2n,\ n \in \mathbb{N}\}$
D. $\{x : x \text{ is a multiple of } 5\}$
Solution: These are the squares of the first five natural numbers.
Q6 — Sets and their Representations · medium · numerical
The number of two-digit natural numbers whose digits add up to $8$ (the set $\{x : 10 \le x \le 99,\ \text{digit sum} = 8\}$) is:
A. $7$
B. $9$
C. $10$
D. $8$  ✓ Correct
Solution: 17, 26, 35, 44, 53, 62, 71, 80 — eight numbers.
Q7 — Sets and their Representations · easy · numerical
The roster form of $\{x \in \mathbb{Z} : -3 < x \le 2\}$ is:
A. $\{-3, -2, -1, 0, 1\}$
B. $\{-2, -1, 0, 1, 2\}$  ✓ Correct
C. $\{-3, -2, -1, 0, 1, 2\}$
D. $\{-2, -1, 0, 1\}$
Solution: Integers strictly greater than $-3$ and at most $2$: $-2,-1,0,1,2$.
Q8 — Sets and their Representations · medium · numerical
The set $\{x \in \mathbb{N} : x \text{ is a divisor of } 24\}$ has how many elements?
A. $6$
B. $7$
C. $8$  ✓ Correct
D. $9$
Solution: Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24 — eight elements.
Q9 — Sets and their Representations · medium · numerical
The number of elements in $\{x \in \mathbb{N} : x^3 < 100\}$ is:
A. $5$
B. $4$  ✓ Correct
C. $10$
D. $3$
Solution: $1^3,2^3,3^3,4^3 = 1,8,27,64 < 100$ but $5^3=125>100$: four elements.
Q10 — Sets and their Representations · hard · numerical
The number of ordered pairs in the set $\{(x, y) : x, y \in \mathbb{Z},\ x^2 + y^2 = 25\}$ is:
A. $4$
B. $8$
C. $16$
D. $12$  ✓ Correct
Solution: Solutions: $(\pm5,0),(0,\pm5),(\pm3,\pm4),(\pm4,\pm3)$ = $2+2+4+4 = 12$.
Q11 — Sets and their Representations · medium · numerical
The roster form of $\{x : x \text{ is a solution of } x^2 - 5x + 6 = 0\}$ is:
A. $\{1, 6\}$
B. $\{2, -3\}$
C. $\{-2, -3\}$
D. $\{2, 3\}$  ✓ Correct
Solution: $x^2-5x+6=(x-2)(x-3)=0$, giving roots 2 and 3.
Q12 — Sets and their Representations · medium · theory
The set $\{3, 6, 9, 12, \dots, 99\}$ in set-builder form is:
A. $\{x : x = 3n,\ n \in \mathbb{N},\ 1 \le n \le 33\}$  ✓ Correct
B. $\{x : x = 3n,\ n \in \mathbb{N}\}$
C. $\{x : x \text{ is odd},\ x < 100\}$
D. $\{x : x = 3n,\ 1 \le n \le 99\}$
Solution: Multiples of 3 from $3(1)$ to $3(33)=99$.
Q13 — Sets and their Representations · medium · numerical
The number of elements in $\{x : x \text{ is a perfect square},\ 50 < x < 200\}$ is:
A. $6$
B. $7$  ✓ Correct
C. $5$
D. $8$
Solution: $64, 81, 100, 121, 144, 169, 196$ — seven perfect squares.
Q14 — Sets and their Representations · hard · numerical
The number of elements in the set $\{x \in \mathbb{Z} : |x - 3| + |x + 2| = 5\}$ is:
A. $5$
B. $6$  ✓ Correct
C. infinite
D. $2$
Solution: $|x-3|+|x+2|=5$ holds for all $x$ in $[-2,3]$ (the distance between $-2$ and $3$). Integers there: $-2,-1,0,1,2,3$ = 6.
Q15 — Sets and their Representations · hard · numerical
If $A = \{x \in \mathbb{R} : x = \tfrac{2t}{1+t^2},\ t \in \mathbb{R}\}$, then $A$ equals:
A. $(-1, 1)$
B. $\mathbb{R}$
C. $[-1, 1]$  ✓ Correct
D. $[0, 1]$
Solution: With $x=\sin(2\theta)$ substitution ($t=\tan\theta$), $x$ ranges over $[-1,1]$; endpoints are attained at $t=\pm1$.
Q16 — Sets and their Representations · medium · numerical
The roster form of $\{x \in \mathbb{R} : x^2 = 16\}$ is:
A. $\{4\}$
B. $\{-4\}$
C. $\{-4, 4\}$  ✓ Correct
D. $\{16\}$
Solution: $x^2=16 \Rightarrow x=\pm4$.
Q17 — Sets and their Representations · medium · theory
Which of the following is the correct set-builder description of $\{-1, 1\}$?
A. $\{x \in \mathbb{R} : x^2 = -1\}$
B. $\{x \in \mathbb{R} : |x| = 1\}$  ✓ Correct
C. $\{x \in \mathbb{N} : x \le 1\}$
D. $\{x : x = 1\}$
Solution: $|x|=1 \Rightarrow x=\pm1$, giving exactly $\{-1,1\}$.
Q18 — Sets and their Representations · hard · numerical
The number of elements in $\{x \in \mathbb{Z} : \tfrac{x-2}{x+3} \in \mathbb{Z}\}$ is:
A. $2$
B. $5$
C. infinite
D. $4$  ✓ Correct
Solution: $\tfrac{x-2}{x+3}=1-\tfrac{5}{x+3}$; integer iff $(x+3)\mid5$, so $x+3\in\{\pm1,\pm5\}$, i.e. $x=-2,-4,2,-8$: 4 values.
Q19 — Sets and their Representations · hard · numerical
The number of elements in $\{x \in \mathbb{Z} : x^2 - 5|x| + 6 = 0\}$ is:
A. $2$
B. $1$
C. $4$  ✓ Correct
D. $3$
Solution: Let $t=|x|\ge0$: $t^2-5t+6=0 \Rightarrow t=2,3$, so $|x|=2$ or $3$, giving $x=\pm2,\pm3$: 4 values.
Q20 — Sets and their Representations · medium · numerical
The set $\{x \in \mathbb{N} : x \text{ and } 18 \text{ are coprime},\ x < 18\}$ has how many elements?
A. $8$
B. $9$
C. $6$  ✓ Correct
D. $5$
Solution: Numbers below 18 coprime to $18=2\cdot3^2$: 1,5,7,11,13,17 = 6 (this is $\varphi(18)=6$).