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Must Do Problems of Sets for NDA

Must do Problems of NDA

Must do Problems of NDA
Q 2374812756

Let `A = {1, 2, 3, 4}, B = {2, 3, 4, 5, 6}`, then
`A ∩ B` is equal to :
BITSAT Mock
(A)

`{1}`

(B)

`{5, 6}`

(C)

`{1, 2, 3}`

(D)

`{2, 3, 4}`

Solution:

`A ∩ B = {1, 2, 3, 4} ∩ {2, 3, 4, 5, 6}`

`= {2, 3, 4}`.
Correct Answer is `=>` (D) `{2, 3, 4}`
Q 1551545424

The set `A = {x : x in R, x^2 =16 and 2x = 6}`
equals
BITSAT 2011
(A)

`phi`

(B)

`{14, 3, 4}`

(C)

`{3}`

(D)

`{4}`

Solution:

Since, `x^2 =16 => x = ± 4`

and `2x = 6 => x = 3`

Hence, no value of `x` is satisfied.

` :. A = phi`
Correct Answer is `=>` (A) `phi`
Q 2863167945

If `A = { x : x^2 - 3x + 2 = 0}` and `B = { x : x^2 + 2x - 8 = 0}`, then `(A- B)` is

(A)

`{1, 2}`

(B)

`{2}`

(C)

`{1}`

(D)

`{4, 3}`

Solution:

Given, `A = { x : x^2 - 3x + 2 = 0 }`

`= { x : (x- 1) (x- 2) = 0 }`

`= {1, 2}`

and `B = { x : x^2 + 2x - 8 = {1}`

`= { x : (x+4) (x - 2) = 0 }`

`= {2, - 4}`

` (A - B) = {1 , 2} - { 2 , -4 } = { 1 }`
Correct Answer is `=>` (C) `{1}`
Q 2813867749

If `x = { (4^n - 3n - 1) | n in N }`and `Y = { 9 (n - 1) | n in N }`. then what Is `X cup Y` equal to ?

(A)

`X`

(B)

`Y`

(C)

`N`

(D)

A null set

Solution:

`X = {( 4^n - 3n - 1) | n in N }`

and `Y = { 9 ( n - 1) | n in N}`

`=> X = { 0 , 9 , 54 , ... }`

and `Y = (0, 9, 18, 27 , 36 , 54, ... ]`

`:. X cupY = { 0, 9, 18, 27, 36 ,54, ... ] = Y`
Correct Answer is `=>` (B) `Y`
Q 2873145946

If `A = { 4n + 2 | n`. is a natural number and `B = { 3 n | n` is a natural number } , then what is `(A cap B)` equal to ?

(A)

`{ 12 n^2 + 6n | , n` is a natural number}

(B)

`{ 24 n - 12| , n` is a natural number}

(C)

`{ 60 n + 30 | , n` is a natural number}

(D)

`{ 12 n - 6 | , n` is a natural number}

Solution:

`A = { 4n + 2 | n in N }`

`= { 6 , 10, 14, 18, 22, 26 , 30 ... }`

and `B = { 3 n | n in N}`

`= {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... }`

`=> A cap B = { 6, 18, 30, ... }`

or `A cap B = { 6 + (n - 1) 12 | n in N}`

`= { 12 n - 6 | n in N}`
Correct Answer is `=>` (D) `{ 12 n - 6 | , n` is a natural number}
Q 2833178042

Let `A = { x : x in R , | x | < 1 } , B = { x : x in R , |x - 1| >= 1 }` and `A cup B = R - D`, then the set `D` is

(A)

`{x : 1 < x <= 2}`

(B)

`{ x : 1 <= x < 2}`

(C)

`{x : 1 <= x <= 2}`

(D)

None of these

Solution:

We have,

`A = { x : x in R, - 1 < x < 1 }`

and `B = { x : x in R , x - 1 <= -1`

or `x - 1 >= 1}`

` = { x : x in R , x <= 0` or `x >= 2}`

`:. A cup B = {x : x in R , x < 1` or `x >= 2}`

`= R - D`

where, `D = {x : x in R , 1 <= x < 2}`
Correct Answer is `=>` (B) `{ x : 1 <= x < 2}`
Q 2324712651

Consider the set `A` of all determinants (of square matrices) of order `3` with entries
`0` or `1` only. Let `B` be the subset of `A` consisting of the determinants with value `1`.

Let `C` be the subset of `A` consisting of all determinants with value `- 1`. Then
BITSAT Mock
(A)

`C` is empty

(B)

`B` has as many elements as `C`

(C)

`A = B cup C`

(D)

`B` has twice as many elements as `C`

Solution:

From a determinant of value `1`, we

can obtain determinant of values

`- 1` in as many ways as we can do

the reverse. So `B` and `C` contain

the same number of elements.
Correct Answer is `=>` (B) `B` has as many elements as `C`
Q 2823845741

If `A` is the set of the divisions of the number `15`, `B` is the set of prime numbers smaller than `10` and `C` is the set of even numbers smaller than `9`, then `(A cup C) cap B` is the set

(A)

`{1, 3, 5}`

(B)

`{1, 2. 3}`

(C)

`{2}`

(D)

`{2, 5}`

Solution:

Givcn, `A = [15 ,30, 45, 60 ,75, ... }`

`B = {2, 3, 5, 7} ; C = { 2, 4, 6, 8}`

Now, `A cup C = { 15, 30, 45, 60, 75, ... }`

`cup {2, 4, 6, 8}`

`= [2, 4, 6, 8, 15, 30, 45, 60, ... }`

Now, `(A cup C) cap B`

`= { 2 , 4 , 6 , 8, 15 , 30 , 45, ... } cap { 2 , 3 ,5 , 7}`

`= {2}`
Correct Answer is `=>` (C) `{2}`
Q 2513178040

The set `(A uu B uu C) nn (A nn B' nn C') nn C'` is equal to
UPSEE 2013
(A)

`B nn C'`

(B)

`A nn C`

(C)

`B' nn C ' `

(D)

None of these

Solution:

`( A uu B uu C) nn ( A nn B' nn C') nn C'`

` = ( A uu B uu C ) nn (A' uu B uu C) nn C'`

` = [ ( A nn A') uu ( B uu C )] nn C'`

` = (phi uu B C) nn C' = ( B uu C) nn C'`


` = ( B nn C' ) uu ( C nn C')`


` = ( B nn C') uuphi = B nn C'`
Correct Answer is `=>` (A) `B nn C'`
Q 2873067846

Let `N` denotes the set of natural numbers and `A = {n^2 : n in N }` and `B = { n^3 : n in N }`. Which one of the following is correct ?

(A)

`A cup B = N`

(B)

The complement of `(A cup B)` is an infinite set

(C)

`A cap B` must be a finite set

(D)

`A cap B` must be a proper subset of {`m^6 : m in N`}

Solution:

`∵ A = { n^2 : n in N}`

and `B = { n^3 : n in N}`

So, ` A cap B` must be a proper subset of

`{ m^6 : m in N}`.
Correct Answer is `=>` (D) `A cap B` must be a proper subset of {`m^6 : m in N`}
Q 2883645547

Consider the following Venn diagram. If `n(E) = 42, n(A) == 15, n(B) = 12` and `n(A cup B) = 22`, then the area represented by shaded portion in the above Venn diagram, is

(A)

`25`

(B)

`27`

(C)

`32`

(D)

`37`

Solution:

Shaded region

`= n (E) - n (A cup B) + n (A cap B)`

`= n(E) - n (A cup B) + n (A) + n (B)`

`-n(A cup B)`

`= 42 - 22 + 15 + 12 -22 = 25`
Correct Answer is `=>` (A) `25`
Q 2853480344

Which one of the following is correct?

(A)

`A xx (B - C) = (A - B ) xx (A - C)`

(B)

`A xx (B - C) = (A xx B ) - (A xx C)`

(C)

`A cap (B cup C) = (A cap B ) cup C`

(D)

`A cup (B cap C) = (A cup B ) cap C`

Solution:

Shaded region

`= n (E) - n (A cup B) + n (A cap B)`

`= n(E) - n (A cup B) + n (A) + n (B)`

`-n(A cup B)`

`= 42 - 22 + 15 + 12 -22 = 25`
Correct Answer is `=>` (B) `A xx (B - C) = (A xx B ) - (A xx C)`
Q 2833678542

A relation between three sets is established using two expressions, `(A cup B) = (A cup C)` and `(A cap B) = (A cap C)`, which stays valid if and only if
I. B = C II. A = B = C III. A = C
Which of the above statement(s) is / are correct?

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Only III

Solution:

We have, `B = B cup (A cap B)`

`= B cup (A cap C) [ ∵ A cap B = A cap C]`

`= (B cup A ) cap (B cup C)`

`= (A cup C) cap (B cup C)`

`=(A cap B) cup C`

`= (A cap C) cap C = C`

Hence, only statement I is correct
Correct Answer is `=>` (A) Only I
Q 2843478343

lf A is any set and `P(A)` is its power set. the which of the following is/are correct?
I. `P(A) cap P(B) = P(A cap B)`
II. `P(A) cup P(B) = P(A cup B)`
Select the correct answer using the code given below.

(A)

Only II

(B)

Only I

(C)

Both I and II

(D)

Neither I nor II

Solution:

Let `x in P(A cap B)`

<=> ` x subseteq ( A cap B)`

<=> `x subseteq A` and `x subseteq B`

<=> ` x in P(A)` and `x in P(B)`

<=> `x in P(A) cap P( B)`

`:. P(A cap B) subseteq P(A) cap P(B)`

and `P(A) cap P(B) subseteq P( A cap B)`

Hence , `P(A) cap P(B) = P(A cap B)`

Now, consider sets `A = { 1}, B = {2}`

`=> A cup B = {1,2}`

`:. P(A) = { phi , { 1} } ,P(B) = { phi . {2}}`

and `P(A cup B)`

`= { phi , {1} ,{2} , { 1,2} }`

`!= P(A) cup P(B)`

Hence, Statement I is true hut Statement II is false.
Correct Answer is `=>` (B) Only I
Q 2833445342

Total numher of elements in the power set of `A` containing `15` elements is

(A)

`2^(15)`

(B)

`(15)^2`

(C)

`2^(15 - 1)`

(D)

`2^(15) - 1`

Solution:

If a set `A` has `n` elements, then its

power set will contain `2"` elements.

`:.` Total number of elements in is power

set of `A = 2^(15)`
Correct Answer is `=>` (A) `2^(15)`
Q 2539780612

If `n(A) = 10, n(B) = 6` and `n(C) = 5` for three disjoint sets `A, B, C,` then `n (A cup B cup C)` equals
BCECE Mains 2015
(A)

`21`

(B)

`11`

(C)

`1`

(D)

`9`

Solution:

Since, `A, B, C` are disjoint sets.

`:. n(A cup B cup C) = n(A) + n(B) + n(C)`

`= 10 + 6+ 5 = 21`
Correct Answer is `=>` (A) `21`
Q 2610223110

Let `X` be the universal set for sets `A` and `B`. If `n(A) = 200, n(B) = 300` and `n(A cap B) = 100`, then `n(A' cap B')` is equal to `300` provided `n(X)` is equal to
BCECE Mains 2015
(A)

`600`

(B)

`700`

(C)

`800`

(D)

`900`

Solution:

We know that,

`n (A cup B) = n (A) + n(B) - n (A cap B)`

`:. n(A cup B) = 200 + 300 - 100 = 400`

Also, `n(A' cap B') = n ((A cup B)')`

`= n(X) - n(A cup B)`

`:. 300 = n(X) - 400 => n(X) = 700`
Correct Answer is `=>` (B) `700`
Q 2843778643

In a city, `25%` of the families have phone, `15%` of the families have car, `65%` of the families have neither phone nor car and `2000` families have both phone and car.
Percentage of families having either phone or car, is

(A)

`10%`

(B)

`30%`

(C)

`35%`

(D)

`40%`

Solution:

Let `P` and `C` be the set of

families having phone and car,

respectively.

Given, `n (P cap C) = 65%`

`=> n(U) - n (P cup C) = 65 %`

`=> n(P cup C) = 100 - 65 = 35%`

Percentage of families having either

phone or car, `n(P cup C)= 35%`
Correct Answer is `=>` (C) `35%`
Q 2883278147

If two sets A and B having `3` and `6` elements respectively, then which of the following is/are correct'?
I. The minimum number of elements of `(A cup B) = 6`.
II. The maximum number of elements of `(A cap B) = 3`.
Select the correct answer using the code given below.

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

` n (A cup B) = n(A) + n(B) - n(A cap B)`

`= 3 + 6 - n (A cap B)`

`= 9 - n ( A cap B)`

As maximum number of element in

`( A cap B) =` minimum of `n (A)` and

`n(B) = 3`

`:.` minimum number of elements in

`(A cup B) = 9 - 3 = 6`

Hence, both statements are correct.
Correct Answer is `=>` (C) Both I and II
Q 2544634553

A survey shows that `63%` of the Americans like cheese whereas `76%` like apples. If `x%` of the
Americans like both cheese and apples, then
UPSEE 2008
(A)

`x=39`

(B)

`x= 63`

(C)

`39 le x

(D)

`1/4`

Solution:

Given, `n(C) = 63, n(A) = 76`, and `n(C cap A) = x`

We know that,

`n(C cup A)= n(C) + n(A)- n(C cap A)`

`100 = 63 + 76- x`

`=> x=139-100=39`.

and `n(C cap A) le P(C)`

`=> x le 63`

`:. 39 le x le 63`
Correct Answer is `=>` (C) `39 le x
Q 2580391217

Let `S= {(a,b,c) in N xx N xx N : a+ b + c = 21, a le b le c}` and

` T = {(a,b,c) in N xx N xx N : a,b,c` are in AP},
where `N` is the set of all natural numbers.
Then, the number of elements in the set
`S cap T` is
WBJEE 2015
(A)

`6`

(B)

`7`

(C)

`13`

(D)

`14`

Solution:

We have, `a+ b + c = 21` and n`(a+b)/2 = b`

`=> a+c + (a+c)/2 =21`

`=> a+c =14 => (a+c)/2 =7`

`=> b =7`

So, a can take values from `1` to `6`,

`c` can have values from `8` to `13`

or `a= b = c = 7` `[ :. a le b le c ]`

So, there are `7` such triplets
Correct Answer is `=>` (B) `7`
Q 2589591417

A college awarded `38` medals in football, `15` in basketball and `20` in cricket. If these medals went to a total of `58` men and only three men got medals in all the three sports. Then the number of students who received medals in exactly two of the three sports, is
BCECE Mains 2015
(A)

`18`

(B)

`15`

(C)

`9`

(D)

`6`

Solution:

Let `F, B` and `C` denote the sets of students who

received medals in football, basketball and cricket

respectively. Then,

`n(F) = 37, n(B) = 15, n(C) = 20`

`n(F cup B cup C) = 58` and `n (F cap B cap C) = 3`

Now, `n(F cup B cup C) = n(F) + n(B) + n(C) - n(F cap B)`

`- n(B cap C) - n(C cap F) + n(F cap B cap C)`

`=> 58= 38 + 15 + 20 - {n(F cap B) + n(B cap C)`

`+ n(C cap F)} + 3`

`=> n(F cap B) + (B cap C)+ n(C cap F) = 18`

Hence, the number of students who received

medals in exactly two of three sports

`= n(F cap B) + n(B cap C) + n(C cap F) - 3n(F cap B cap C)`

`= 18 - 3 xx 3 = 9`
Correct Answer is `=>` (C) `9`
Q 2503201148

There is a group of 265 persons who like either singing or dancing or painting. In this group 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number of persons who like only dancing and painting is
WBJEE 2014
(A)

`10`

(B)

`20`

(C)

`30`

(D)

`40`

Solution:

Let D denotes dancing, P denotes painting and S denotes singing.

`:. n(D cup P cup S) =265`

`n(S)=200, n(D)=60, n(S cap P)-30`

and `n(D cap P cap S)=10`

`n(D cup P cup S)=n(D)+n(P)+n(S)-n(D cap P)-n(P cap S)-n(S cap D)+n(D cap P cap S)`

`:. 265 =110+55+200-n(D cap P)-30-60+10`

`=> 265=285-n(D cap P)`

`=> n (D cap P)=20`

`:.` only person who like dancing painting

`=n(D cap P)-n(D cap P cap S)`

`=20-10=10`
Correct Answer is `=>` (A) `10`
Q 2853578444

Consider the following statements
I. All poets (P) are learned (L).
II. All learned (L) are happy (H).
Which one of the following Venn diagrams correctly represents both the above statements taken together?

Solution:

We have three categories, i.e. poets,

learned and happy for which we can

Venn diagram as follows.

Statement I represents

Thus,`P subseteq L`

Statement II represents

Thus, `L subseteq H`

On combining both statements, we get

`P subseteq L subseteq H` and the Venn diagram of

both statements taken together is given

below.
Correct Answer is `=>` (D)
 
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