Quant · free practice

Set Theory and Venn Diagrams questions for IPMAT, with solutions

10 free Set Theory and Venn Diagrams practice questions for IPMAT Indore, Rohtak and JIPMAT, each with the answer and a full worked solution. No sign-in. Try each one first, then open the solution. The lessons for this chapter are in Set Theory and Venn Diagrams on Learn.

1. Question 1Multiple choiceWritten by IPMATwaySet Algebra and Power Sets

   Consider the following statements.

   Statement I: The empty set is a subset of every set.

   Statement II: $\{\emptyset\}$ is an empty set.

   Which of the following is correct?

   1. A.Statement I is false but Statement II is true
   2. B.Both Statement I and Statement II are false
   3. C.Statement I is true but Statement II is false
   4. D.Both Statement I and Statement II are true
   Show the answer and solution

   Answer: C. Statement I is true but Statement II is false

   Every element of $\emptyset$ (there are none) lies in any set, so $\emptyset\subseteq A$. $\{\emptyset\}$ contains one element, so $n=1$.

2. Question 2Multiple choiceWritten by IPMATwaySet Algebra and Power Sets

   The number of proper subsets of a set with $5$ elements is

   1. A.$30$
   2. B.$31$
   3. C.$25$
   4. D.$32$
   Show the answer and solution

   Answer: B. $31$

   All subsets $2^5=32$; a proper subset excludes the set itself: $31$.

3. Question 3Multiple choiceWritten by IPMATwayTwo-Set Venn Problems

   Consider the following statements.

   Statement I: If $n(A)=10$, $n(B)=15$ and $n(A\cup B)=20$, then $n(A\cap B)=5$.

   Statement II: If $A\subset B$, then $n(A\cup B)=n(A)$.

   Which of the following is correct?

   1. A.Statement I is true but Statement II is false
   2. B.Statement I is false but Statement II is true
   3. C.Both Statement I and Statement II are false
   4. D.Both Statement I and Statement II are true
   Show the answer and solution

   Answer: A. Statement I is true but Statement II is false

   Statement I follows from $n(A\cap B)=n(A)+n(B)-n(A\cup B)$. Statement II fails whenever $B$ is strictly larger than $A$.

4. Question 4Type the answerWritten by IPMATwayThree-Set Venn Problems

   In a survey of $200$ students, $120$ study Mathematics, $90$ study Physics and $70$ study Chemistry; $40$ study Mathematics and Physics, $30$ study Physics and Chemistry, $50$ study Chemistry and Mathematics, and $20$ study none of these. The number of students who study all three is

   Show the answer and solution

   Answer: 20

   At least one $=200-20=180=120+90+70-(40+30+50)+t$, so $t=20$.

5. Question 5Multiple choiceWritten by IPMATwayTwo-Set Venn Problems

   If $n(U)=40$, $n(A)=25$ and $n(B)=20$, the greatest possible value of $n(A\cup B)$ is

   1. A.$25$
   2. B.$45$
   3. C.$40$
   4. D.$5$
   Show the answer and solution

   Answer: C. $40$

   The union cannot exceed $U$; $25+20=45>40$, so the greatest union is $40$ (with overlap $5$).

6. Question 6Type the answerWritten by IPMATwayThree-Set Venn Problems

   The number of natural numbers less than or equal to $1000$ that are divisible by at least one of $3$, $7$ and $11$ is

   Show the answer and solution

   Answer: 480

   Singles $\lfloor\frac{1000}{3}\rfloor=333$, $142$, $90$; pairs (multiples of $21$, $33$, $77$) $47$, $30$, $12$; all three (multiples of $231$) $4$. Total $480$.

7. Question 7Multiple choiceWritten by IPMATwayThree-Set Venn Problems

   The number of natural numbers not exceeding $1000$ that are divisible by none of $10$, $15$ and $25$ is

   1. A.$854$
   2. B.$146$
   3. C.$806$
   4. D.$794$
   Show the answer and solution

   Answer: A. $854$

   Overlaps are multiples of the LCMs: $30$ ($33$), $75$ ($13$), $50$ ($20$); all three: $150$ ($6$). So $146$ are divisible by at least one and $854$ by none.

8. Question 8Multiple choiceWritten by IPMATwayTwo-Set Venn Problems

   A survey shows that $63\%$ of the people watch a news channel and $76\%$ watch another channel. If $x\%$ watch both, then

   1. A.$x=39$
   2. B.$x\le 39$
   3. C.$39\le x\le 63$
   4. D.$63\le x\le 76$
   Show the answer and solution

   Answer: C. $39\le x\le 63$

   Overlap at least $139-100=39$; at most the smaller group, $63$.

9. Question 9Multiple choiceWritten by IPMATwaySet Algebra and Power Sets

   How many subsets of $\{1,2,\dots,8\}$ contain at least one odd number and at least one even number?

   1. A.$224$
   2. B.$225$
   3. C.$240$
   4. D.$192$
   Show the answer and solution

   Answer: B. $225$

   Total subsets: $2^8=256$. Subsets with no odd number are subsets of $\{2,4,6,8\}$: $2^4=16$ (including the empty set). Subsets with no even number: also $16$. Both bad families contain the empty set, so the union has $16+16-1=31$ subsets. Required count $=256-31=225$. Equivalently: non-empty subsets of the odds ($15$) times non-empty subsets of the evens ($15$) $=225$.

10. Question 10Type the answerWritten by IPMATwaySet Algebra and Power Sets

    How many subsets of $\{1,2,\dots,10\}$ (including the empty set) contain no two consecutive integers?

    Show the answer and solution

    Answer: 144

    Let $a_n$ count valid subsets of $\{1,\dots,n\}$. If $n$ is not chosen, the rest is any valid subset of $\{1..n-1\}$: $a_{n-1}$. If $n$ is chosen, $n-1$ is not, leaving a valid subset of $\{1..n-2\}$: $a_{n-2}$. So $a_n=a_{n-1}+a_{n-2}$, with $a_1=2$ ($\emptyset,\{1\}$) and $a_2=3$. Continuing: $a_3=5,a_4=8,a_5=13,a_6=21,a_7=34,a_8=55,a_9=89,a_{10}=144$.