JEE Mathematics Sets MCQs Set 02

Answer: (d) not a well-defined collection

Explanation:
1. A set must be well defined: for every object we must be able to say definitely whether it belongs or not.
2. 'Most intelligent' is a matter of opinion, so membership cannot be decided objectively.
3. Hence the collection is not a well-defined set.

Teacher's Note:
1. Words such as intelligent, tall, beautiful or best never define a set.
2. Numbers, names or a clear rule always do.

Answer: (d) 32

Explanation:
1. \( A=\{-2,-1,0,1,2\} \), so \( n(A)=5 \).
2. Number of subsets \( =2^5=32 \).

Teacher's Note:
1. A set with n elements has \( 2^n \) subsets, including the empty set and the set itself.
2. Note that the integers strictly between -3 and 3 exclude both -3 and 3.

Answer: (a) \( [3,7]\subset(2,10) \)

Explanation:
1. Every number from 3 to 7 lies strictly between 2 and 10, so \( [3,7]\subset(2,10) \) is true.
2. In (b), \( 1 \) is in \( (0,\infty) \) but not in \( (4,\infty) \).
3. In (c), \( 7 \) is in \( (5,7] \) but not in \( [5,7) \); in (d), \( 2.5 \) lies in \( [2,7] \) but not in \( (2.9,8) \).

Teacher's Note:
1. To test A is a subset of B, find one element of A missing from B; if you find it the statement is false.
2. Pay special attention to the end points of intervals.

Answer: (b) The set of all real numbers lying between 1 and 2

Explanation:
1. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24: finite.
2. Human beings living in India and three digit natural numbers (900 of them) are finite sets.
3. Between any two real numbers there are infinitely many real numbers, so the set in (b) is infinite.

Teacher's Note:
1. Intervals of real numbers are always infinite, however short.
2. Large but countable collections such as the people of a country are still finite.

Answer: (b) \( \{x : 1\le x\lt 2\} \)

Explanation:
1. \( A\cup B=(-\infty,1)\cup[2,\infty) \).
2. The real numbers missing from it are those from 1 up to but not including 2.
3. So \( D=\{x : 1\le x\lt 2\} \).

Teacher's Note:
1. Draw the number line and shade A and B; the unshaded part is D.
2. Watch which end points are included.

Answer: (b) \( A \)

Explanation:
1. \( B'=\{1,2,3,4,5,8,9,10\} \).
2. Every element of \( A \) is in \( B' \), since \( A \) and \( B \) are disjoint.
3. So \( A\cap B'=A \).

Teacher's Note:
1. If A and B are disjoint then \( A\subset B' \), so \( A\cap B'=A \).
2. This is a quick shortcut for such questions.

Answer: (a) \( \{9,21\} \)

Explanation:
1. \( A=\{5,9,13,17,21\} \) and \( B=\{3,6,9,\dots,24\} \).
2. We know \( A-(A-B)=A\cap B \).
3. Multiples of 3 in \( A \) are 9 and 21, so the answer is \( \{9,21\} \).

Teacher's Note:
1. The identity \( A-(A-B)=A\cap B \) is worth remembering.
2. List both sets explicitly when they are small.

Answer: (a) 1 only

Explanation:
1. (1): \( \mathbb{N}\cup(B\cap\mathbb{Z})=(\mathbb{N}\cup B)\cap(\mathbb{N}\cup\mathbb{Z})=(\mathbb{N}\cup B)\cap\mathbb{Z} \) because \( \mathbb{N}\subset\mathbb{Z} \). So (1) is true.
2. (2): \( A=\{3,6,\dots,24\} \) has 8 elements, and \( B=\{1,2,\dots,8\}\subset\mathbb{N} \) has 8 elements too. So (2) is false.

Teacher's Note:
1. Distributive laws plus \( \mathbb{N}\subset\mathbb{Z} \) settle statement 1.
2. A finite set always has subsets of the same size inside a bigger set like N.

Answer: (b) \( n(A\cap B)\le n(A\cup B) \)

Explanation:
1. \( A\cap B\subseteq A\cup B \), and a subset never has more elements than the set containing it.
2. So \( n(A\cap B)\le n(A\cup B) \).

Teacher's Note:
1. Equality holds only when A = B.
2. Think of the intersection as the overlap and the union as everything.

Answer: (c) 60

Explanation:
1. Mathematics alone \( =n(M)-n(M\cap P)-n(M\cap C)+n(M\cap P\cap C) \).
2. Value \( =100-30-28+18=60 \).

Teacher's Note:
1. The triple intersection is subtracted twice and so must be added back once.
2. Draw a Venn diagram and fill from the centre outwards.

Answer: (b) 15

Explanation:
1. Let the numbers passing exactly English and Hindi, English and Sanskrit, Hindi and Sanskrit be \( x,y,z \).
2. English: \( 3+x+y+2=15 \) so \( x+y=10 \); Hindi: \( 4+x+z+2=15 \) so \( x+z=9 \); Sanskrit: \( 7+y+z+2=20 \) so \( y+z=11 \).
3. Adding, \( 2(x+y+z)=30 \), so exactly two subjects: \( 15 \). (Check: \( 3+4+7+2+15=31 \).)

Teacher's Note:
1. Always verify the total with the given number of candidates.
2. Exactly-two regions do not include the centre of the Venn diagram.

Answer: (b) 2 only

Explanation:
1. (1): \( ((A\cap B)\cup C)\cap B'=C\cap B' \) because \( A\cap B\subset B \). Its complement is \( C'\cup B \), which is not \( B\cap C \). So (1) is false.
2. (2): \( A'\cap B'=(A\cup B)' \) lies outside \( A\cup B \), so intersecting it with \( A\cup B\cup C' \) leaves only the part inside \( C' \).
3. Left side \( =A'\cap B'\cap C'=(A\cup B\cup C)' \), which equals the right side. So (2) is true.

Teacher's Note:
1. Use De Morgan's laws: \( (X\cup Y)'=X'\cap Y' \).
2. Test a suspicious identity with a single element in a Venn diagram.

Answer: (b) 7%

Explanation:
1. Percentages passing are at most 30, 28, 20 and 15 respectively.
2. Students who passed at least one subject are at most \( 30+28+20+15=93 \)%.
3. So at least \( 100-93=7 \)% failed in all four.

Teacher's Note:
1. Work with the complements: those who passed.
2. Union of the passing groups is at most the sum of their sizes.

Answer: (a) 2.20

Explanation:
1. \( n(A'\cap B')=2009-280=1729 \).
2. The number 1729 is the smallest number that is a sum of two cubes in two ways: \( 1^3+12^3=9^3+10^3 \).
3. So \( x_1=1,x_2=12,y_1=9,y_2=10 \) and the ratio is \( \frac{22}{10}=2.20 \).

Teacher's Note:
1. Complement of a union: \( n(A'\cap B')=n(U)-n(A\cup B) \).
2. 1729 is the famous Hardy-Ramanujan number.

Answer: (c) 12.29

Explanation:
1. \( n(A-B)=1681-1075=606 \).
2. For \( m=1 \): \( 605=5\cdot11^2 \) is not a product of three distinct primes. For \( m=2 \): \( 602=2\cdot7\cdot43 \) works.
3. Then \( \frac{p_1p_3}{p_2}=\frac{2\cdot43}{7}\approx12.29 \).

Teacher's Note:
1. Test the smallest values of m one by one.
2. Factorise the remaining number completely.

Answer: (b) 22

Explanation:
1. Mathematics only \( =23-12-9+4=6 \); Physics only \( =24-12-7+4=9 \); Chemistry only \( =19-9-7+4=7 \).
2. Exactly one subject: \( 6+9+7=22 \).

Teacher's Note:
1. Union check: \( 23+24+19-12-9-7+4=42 \), so everybody studies at least one subject.
2. Each 'only' region needs the triple overlap added back.

Answer: (b) 600

Explanation:
1. Both languages: \( 750+400-1000=150 \).
2. Hindi only \( =750-150=600 \).

Teacher's Note:
1. Use \( n(H\cup B)=n(H)+n(B)-n(H\cap B) \) with the union equal to 1000.
2. Subtract the overlap from each language for 'only' counts.

Answer: (d) \( Y \)

Explanation:
1. \( 4^n-3n-1=(1+3)^n-3n-1=9\left[\binom n2+3\binom n3+\dots\right] \), a multiple of 9.
2. Hence every element of \( X \) is a multiple of 9, i.e. \( X\subseteq Y \) (and \( Y \) contains 0 and all positive multiples of 9).
3. So \( X\cup Y=Y \).

Teacher's Note:
1. Expand \( (1+3)^n \) with the binomial theorem.
2. If X is a subset of Y then the union is simply Y.

Answer: (d) \( 3^5 \)

Explanation:
1. Each element of \( X \) can go into \( Y \) only, \( Z \) only, or neither.
2. That gives 3 choices for each of 5 elements: \( 3^5 \).

Teacher's Note:
1. Disjoint subsets: each element has three options.
2. Compare with the 2 options per element for counting subsets.

Answer: (c) 29

Explanation:
1. The smallest such subset is \( A\cup B \).
2. Multiples of 2: 25; multiples of 7: 7; multiples of 14: 3.
3. So \( n(A\cup B)=25+7-3=29 \).

Teacher's Note:
1. 'Smallest subset containing both' translates to the union.
2. Common multiples of 2 and 7 are the multiples of 14.

Answer: (c) \( B=C \)

Explanation:
1. Take \( x\in B \). If \( x\in A \) then \( x\in A\cap B=A\cap C \), so \( x\in C \).
2. If \( x\notin A \) then \( x\in A\cup B=A\cup C \) gives \( x\in C \).
3. So \( B\subseteq C \) and similarly \( C\subseteq B \); hence \( B=C \).

Teacher's Note:
1. Both the union and the intersection determine the set uniquely together.
2. This is the cancellation law for sets.