Mathematics NDA Questions on Set

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NDA Questions on Set

Questions

Q 2721145021

Let S be a set of all distinct number of the form `p/q`, where p,q in [ 1,2,3,4,5,6 ] . what is the cardinality of the set S ?
NDA Paper 1 2016

(A)

(B)

(C)

(D)

Solution:

`1/1 , 1/2 , 1/3 , 1/4 , 1/5 , 1/6 => 6` elements

`2/1 , 2/3 , 2/5=> 3` elements

`3/1 , 3/2 , 3/4, 3/5=> 4` elements

`4/1 , 4/3 , 4/5=> 3` elements

`5/1 , 5/2 , 5/3, 5/4, 5/6 => 5` elements

`6/1, 6/5 => 2` elements

Total 23 elements , Hence cardinaly `=23`

Correct Answer is `=>` (B) 23

Q 2781291127

In an examination , 70% students passed in physics , 80 % student passe in chemistry, 75% students passed in Mathematics and 85 % students passed in Biology , and x % students failed in all four subjects . What is the minimum value of x ?
NDA Paper 1 2016

(A)

(B)

(C)

(D)

None of the above

Solution:

`n(AcupB) = n(A) + n(B) - n(AcapB)`

Maximum student who pass can be 100%

If all student pass `n(A cup B) = 100`

n(A) = biology = 85%
n(B) = chemistry = 80%

`100 = 80 + 85 - n(AcapB)`

`n(AcapB) = 65` (who pass in bio and chem)

Taking a case when all maths student have passed in bio and all physics student also passed in chemistry as shown in the figure above.

So union of students who passed in the four exams is 100 in this case so `x = 0%`

ANS [4]

Correct Answer is `=>` (D) None of the above

Q 2771345226

If `A = { x in RR : x^2 + 6x - 7 < 0 } ` and `B = { x in RR : x^2 + 9x + 14 > 0 }` , then which of the following is/are correct ?

1. `A nn B = { x in RR : - 2 < x < 1 }`
2. `A \\ B = { x in RR : - 7 < x < -2 }`

Select the correct answer using the code given below :
NDA Paper 1 2016

(A)

1 only

(B)

2 only

(C)

Both 1 and 2

(D)

Neither 1 nor 2

Solution:

`x^2 +6x -7 < 0 => (x+7)(x-1) < 0`

`A-> -7 < x < 1`

`B->x^2 +9x +14 > 0 => (x+2) (x+7) > 0`

`=> x < -7 ` and `x > -2`

`=>A cap B`

`={ x in R ; -2 < x < 1}`

`A | B ` i.e. ` A-B = { x in R : -7 < x < -2}`

`=> ` Both `1` and `2` are correct.

Correct Answer is `=>` (C) Both 1 and 2

Q 2280180917

If `A={x in R: x^(2) +6x - 7 < 0 }`and `B = {x in R: x^( 2) + 9x + 14 > 0}`, then which of the following is/are correct?

1. `(A cap B) = (- 2, 1)`

2. `(A -B) =(-7, -2)`

Select the correct answer using the code given below
NDA Paper 1 2015

(A)

Only `1`

(B)

Only `2`

(C)

Both `1` and `2`

(D)

Neither `1` nor `2`

Solution:

We have, `A = { x in R : x^(2) +6x - 7 < 0 }`

and `B = {x in R: x^(2) +9x+ 14 > 0 }`

`:. A= { x in R: (x-1)(x+ 7) < 0 }`

and `B = {x in R : (x + 2) (x + 7) > 0}`

`A = (- 7, 1)` and `B = (- oo ,- 7) cup (- 2, oo)`

`A cap B = (- 2, 1)`

`A- B = (- 7, -2)`

Hence, both `1` and `2` are correct.

Correct Answer is `=>` (C) Both `1` and `2`

Q 1648423303

In a class of `60` students, `45` students like music,
`50` students like dancing, `5` students like neither.
Then, the number of students in the class who like
both music and dancing, is
NDA Paper 1 2015

(A)

`35`

(B)

`40`

(C)

`50`

(D)

`55`

Solution:

Let the number of students in the class be `x`, who

like both music and dancing.

`:. (45 - x) + x + (50 - x) + 5 = 60`

`=> 100 x = 60`

`=> x = 100 - 60 = 40`

Correct Answer is `=>` (B) `40`

Q 1668823705

If `A = {x : x` is a multiple of `3}` and `B = {x : x` is a
multiple of `4}` and `C = {x :x` is a multiple of `12}`, then
which one of the following is a null set?
NDA Paper 1 2015

(A)

`(A // B) cup C`

(B)

`(A // B) // C`

(C)

`(A cap B) cap C`

(D)

`(A cap B) // C`

Solution:

Here, `A = {3, 6, 9, 12, 15,....... }`

`B = { 4, 8, 12, 16, 20, ....... }`

`C = {12, 24, 36, 48, ...... }`

`:. A cap B = {12, 24, 36, 48, .......... }`

`=>(A cap B)//C = {12, 24, 36, ...... } - {12, 24, 36, ......... }`

Which is a null set.

Correct Answer is `=>` (D) `(A cap B) // C`

Q 1722167931

In a survey of `25` students, it was found that `15` had taken
Mathematics, `12` had taken Physics and `11` had taken
Chemistry, `5` had taken Mathematics and Chemistry, `9` had
taken Mathematics and Physics, `4` had taken Physics and
Chemistry and `3` had taken all the three subjects.

Consider the following statements

I. The number of students who had taken only one
subject is equal to the number of students who had
taken only two subjects.

II. The number of students who had taken at least two
subjects is four times the number of students who had
taken all the three subjects.

Which of the above statements is/are correct?

NDA Paper 1 2014

(A)

Only 1

(B)

Only 2

(C)

Both 1 and 2

(D)

Neither 1 nor 2

Solution:

Given that,

Total number of surveyed students `= 25`

Number of students, who had taken all three subjects `= 3`

Number of students, who had taken Physics and Chemistry
` = 4`

Number of students, who had taken Mathematics and

Physics `= 9`

Number of students who had taken Mathematics and

Chemistry `= 5`

Number of students, who had taken Chemistry `= 11`

Number of students, who had taken Physics `= 12`

Number of students, who had taken Mathematics `= 15`

I. The number of students, who had taken only

one subject `= 2 + 5 + 4 = 11`

and the number of students, who had taken only two

subjects `= 6 + 2 + 1 = 9`

`=> 11 != 9`

II. The number of students who had taken atleast two

subjects `= 1 + 2 + 6 + 3 = 12`.

The number of students who had taken all three subjects `= 3`

` => 12 = 4 xx 3 = 12`

Correct Answer is `=>` (B) Only 2

Q 1609591418

`A` and `B` are two sets having `3` elements in common. If
`n( A) = 5` and `n(B) = 4`, then what is `n( A xx B)` equal to?
NDA Paper 1 2014

(A)

`0`

(B)

`9`

(C)

`15`

(D)

`20`

Solution:

Here,` n(A) = 5` and `n(8) = 4`

`:. n(A xx B) = 5 xx 4 = 20`

`[∵ n(A) = m, n(B) = n => n(A xx B ) = mn]`

Correct Answer is `=>` (D) `20`

Q 1712867739

(A)

`2`

(B)

`3`

(C)

`5`

(D)

`6`

Correct Answer is `=>` (A) `2`

Q 1762067835

(A)

`7`

(B)

`8`

(C)

`9`

(D)

`10`

Correct Answer is `=>` (C) `9`

Q 2345012863

Which one of the following is a null set?
NDA Paper 1 2013

(A)

{0}

(B)

{ { {}}}

(C)

{{}}

(D)

`{x| x^ 2 + 1|= 0, x in R}`

Solution:

`{0} ->` Singleton set

and `x^2 + 1 = 0`

`=> x = ± l`

` x in c`

while, `{x| x^2 + 1| = 0, x in R}`.

So, it is a null set.

Hence, option (d) is correct

Correct Answer is `=>` (D) `{x| x^ 2 + 1|= 0, x in R}`

Q 2305323268

Out of `500` first year students, `260` passed in the
first semester and `210` passed in the second
semester. If `170` did not pass in either semester,
how many passed in both semesters?
NDA Paper 1 2013

(A)

`30`

(B)

`40`

(C)

`70`

(D)

`140`

Solution:

Total number of students `= 500`

Passed student in `1`st semester `= 260`

Passed student in lind semester `= 210`

Fail student in both semester `= 170`

Now. the number of passed students atleast one of these

semester `= 500 - 170 = 330`

`:.` The number of students passed in both semester = (Passed

student in 1st semester + Passed student in lind semester)

- (Passed students atleast one of these semester)

`= (260 + 210)- 330= 470- 330 = 140`

Correct Answer is `=>` (D) `140`

Q 2375523466

If A and B are two non-empty sets having n
elements in common, then what is the number of
common elements in the sets `A xx B` and `B xx A`?
NDA Paper 1 2013

(A)

`n`

(B)

`n^2`

(C)

`2n`

(D)

Zero

Solution:

Let us consider an example.

`Let A= {a, b,c}` and `B ={a, b,c,d}`

Here, `3` elements are common in `A` and `B`.

Now,

`A xx B = {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b),(b, c}(b, d)`,

`(c, a), (c, b), (c, c), (c, d)}`

`B xx A= {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b),`

`(c, c), (d, a), (d. b), (d, c)}`

Here, common element in `(A xx B)` and `(B xx A)` is `9` i.e., `(3)^`

So, in general, If `A` and `B` are two non-empty sets having 'n'

elements in common, then (n)2 is the common elements in the

sets `A xx B` and `B xx A`.

Correct Answer is `=>` (B) `n^2`

Q 2385623567

If `A` and `B` are any two sets, then what is the value
of `A cap (A cup B)`?
NDA Paper 1 2013

(A)

Complement of `A`

(B)

Complement of `B`

(C)

`B`

(D)

`A`

Solution:

Here, `A` and `B` are any two set and U be `U` be universal

set.

Correct Answer is `=>` (D) `A`

Q 2362545435

Which one of the following is an example of
non-empty set?
NDA Paper 1 2013

(A)

Set of all even prime numbers

(B)

(`x : x^2 - 2 = 0` and `x` is rational)

(C)

{`x : x` is a natural number, `x < 8` and simultaneously `x > 12`}

(D)

{`x : x` is a point common to any two parallel lines}

Solution:

Let `S` be the set of all even prime numbers.

`:. S = {2} =` Non-empty set

Correct Answer is `=>` (A) Set of all even prime numbers

Q 2302545438

In a group of `50` people, two tests were conducted,
one for diabetes and one for blood pressure. `30`
people were diagnosed with diabetes and `40`
people were diagnosed with high blood pressure.
What is the minimum number of people who were
having diabetes and high blood pressure?
NDA Paper 1 2013

(A)

`0`

(B)

`10`

(C)

`20`

(D)

`30`

Solution:

Given,

U = 50,

The number of people who diagnosed with diabetes n(A) = 30,

The number of people who diagnosed with high blood pressure n(B) = 40,

`n(AcupB) = n(A)+n(B)-n(AcapB)`

`n(AcapB) = n(A)+n(B)-n(AcupB)`

`= 30 + 40 - n(AcupB)`

`M i n {n(AcapB)} = 70 - Max{n(AcupB)}
= 70 -50
= 20`

Correct Answer is `=>` (C) `20`

Q 2332645532

If A is a subset of B, then which one of the
following is correct?
NDA Paper 1 2013

(A)

`A^c subseteq B^c`

(B)

`B^c subseteq A^c`

(C)

`A^c = B^c`

(D)

`A subseteq A cap B`

Solution:

Let `U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}`

`A = {1, 2, 3, 4, 5}`

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

Given, `A subseteq B`

Now, `A^c = {6, 7, 8, 9, 10}`

`B^c = {7, 8, 9, 10}`

Here, `B^c subseteq A^c`

Now, we have `A cap B = {1, 2, 3, 4, 5}`

Here, `A = (A cap B)`

Correct Answer is `=>` (B) `B^c subseteq A^c`

Q 2315723669

The set `A = {x: x + 4 = 4}` can also be represented
by
NDA Paper 1 2013

(A)

`0`

(B)

`phi`

(C)

`{phi}`

(D)

`{0}`

Solution:

Given that, `A = {x: x + 4 = 4}`

`=> x + 4 = 4`

`=> x = 0`

`:. A= {0}`

Correct Answer is `=>` (D) `{0}`

Q 2305023868

In a city, three daily newspapers A, B and C are published, `42%` read A, `51%`
read B, `68%` read C, `30%` read A and B, `28%` read B and C,
`36%` read A and C, `8` do not read any of the three newspapers.
What is the percentage of persons who read all the
three papers?
NDA Paper 1 2013

(A)

20%

(B)

25%

(C)

30%

(D)

40%

Solution:

Let the percentage of newspapers A, B

and `C` in a city `= 100%`

Given that,

Not read any of the newspapers `= 8%`

Now, given

`n(A cup B cup C) = n(cup)- n(A cup B cup C)`'

`= 100% - 8% = 92%`

Also, `n (A)= 42%, n (B)= 51%, n (C)= 30%`

`n = (A cap B) = 30%, n = (B cap C) = 28%` ,

`n (A cap C) = 36%`

By formula, `n (A cup B cup C) = n (A)+ n (B)+ n (C)`

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

`92% = 42% +51%+ 68%- 30%-36%`

`- 28% + n (A cap B cap C)`

`=> n (A cap B cap C) = 92% - 67%`

`=> n (A cap B cap C) = 25%`

Percentage of persons who read all the three papers `= 25%`

Correct Answer is `=>` (B) 25%

Q 2305023868

(A)

19%

(B)

31%

(C)

44%

(D)

None of these

Solution:

Let the percentage of newspapers A, B

and `C` in a city `= 100%`

Given that,

Not read any of the newspapers `= 8%`

Now, given

`n(A cup B cup C) = n(cup)- n(A cup B cup C)`'

`= 100% - 8% = 92%`

Also, `n (A)= 42%, n (B)= 51%, n (C)= 30%`

`n = (A cap B) = 30%, n = (B cap C) = 28%` ,

`n (A cap C) = 36%`

By formula, `n (A cup B cup C) = n (A)+ n (B)+ n (C)`

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

`92% = 42% +51%+ 68%- 30%-36%`

`- 28% + n (A cap B cap C)`

`=> n (A cap B cap C) = 92% - 67%`

`=> n (A cap B cap C) = 25%`

The percentage of persons who read only two papers

`= (30% + 36% + 28%) - (25% + 25% + 25%)`

`= 94%- 75%= 19%`

Correct Answer is `=>` (A) 19%

Q 2305023868

(A)

38%

(B)

48%

(C)

51%

(D)

None of these

Solution:

Let the percentage of newspapers A, B

and `C` in a city `= 100%`

Given that,

Not read any of the newspapers `= 8%`

Now, given

`n(A cup B cup C) = n(cup)- n(A cup B cup C)`'

`= 100% - 8% = 92%`

Also, `n (A)= 42%, n (B)= 51%, n (C)= 30%`

`n = (A cap B) = 30%, n = (B cap C) = 28%` ,

`n (A cap C) = 36%`

By formula, `n (A cup B cup C) = n (A)+ n (B)+ n (C)`

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

`92% = 42% +51%+ 68%- 30%-36%`

`- 28% + n (A cap B cap C)`

`=> n (A cap B cap C) = 92% - 67%`

`=> n (A cap B cap C) = 25%`

The percentage of persons who read only one paper

`= (42 - 5 - 25 - 11)% + (51 - 5 - 25 - 3)%`

`+ (68- 11-25- 3)%`

`= 1% + 18% + 29% = 48%`

Correct Answer is `=>` (B) 48%

Q 2305023868

(A)

(B)

(C)

(D)

None of these

Solution:

Let the percentage of newspapers A, B

and `C` in a city `= 100%`

Given that,

Not read any of the newspapers `= 8%`

Now, given

`n(A cup B cup C) = n(cup)- n(A cup B cup C)`'

`= 100% - 8% = 92%`

Also, `n (A)= 42%, n (B)= 51%, n (C)= 30%`

`n = (A cap B) = 30%, n = (B cap C) = 28%` ,

`n (A cap C) = 36%`

By formula, `n (A cup B cup C) = n (A)+ n (B)+ n (C)`

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

`92% = 42% +51%+ 68%- 30%-36%`

`- 28% + n (A cap B cap C)`

`=> n (A cap B cap C) = 92% - 67%`

`=> n (A cap B cap C) = 25%`

The percentage of persons who read only A but neither B

nor `C = (42 - 5 - 25- 11)% = 1%`

Correct Answer is `=>` (C) 1%

Q 2385001867

If `A= {1, 3, 5, 7}`, then what is the cardinality of
the power set `P(A)`?
NDA Paper 1 2013

(A)

`8`

(B)

`15`

(C)

`16`

(D)

`17`

Solution:

Given that, `A= {1, 3, 5, 7}`

Here, `n (A)= 4`

`:.` Number of elements on power set of `A`

`= P(A) = 2^(n (A))`

`= 2^4 = 16`

`:.` Cardinality of the power set `P(A) = 16`

Correct Answer is `=>` (C) `16`

Q 2315001869

Consider the following
I. `A cup (B cap C)= (A cap B) cup (A cap C)`
II. `A cap (B cup C)= (A cup B) cap (A cup C)`
Which of the above statements is/are correct'?
NDA Paper 1 2013

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

We know that,

In set theory 'Distribution law'

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

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

So, the given both statements are incorrect.

Correct Answer is `=>` (D) Neither I nor II

Q 2305101968

In a state with a population of `75 xx 10^6 , 45%` of them know
Hindi, `22%` know English, `18%` know Sanskrit, `12%` know
Hindi and English `8%` know English and Sanskrit, `10%`
know Hindi and Sanskrit and `5%` known all the three
languages.

What is the number of people who do not know
any of the above three languages?

NDA Paper 1 2013

(A)

`3 xx 10^6`

(B)

`4 xx 10^6`

(C)

`3 xx 10^7`

(D)

`4 xx 10^7`

Solution:

n(A) = 45% (Hindi)
n(B) = 22% (English)
n(C) = 18% (Sanskrit)

The number of people who knows any of language given By formula,

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

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

`= (45 +22 +18 -12-8-10+5) % = 60 %`

`"Now, The number of people who do not know
any of the above three languages "`
`"= 100 % - (who knows any of language)
= 40 %"`

`"No. of people = 40% of"``\ \ 75xx10^6 = 3 xx 10^7`

Correct Answer is `=>` (C) `3 xx 10^7`

Q 2305312268

(A)

`5 xx 10^6`

(B)

`45 xx 10^5`

(C)

`4 xx 10^6`

(D)

None of these

Solution:

Given that, `P = 75 xx 10^6`

`45%` of population know Hindi,

i.e., `a + f + Q + d = 45%` of `P`

`= (45)/(100) xx 75 xx 10^6`

`=> a + f + g + d = 33.75 xx 10^6` .........(i)

`22%` of population know English,

i.e., `f + b + g + e = 22%` of `P`

`= (22)/(100) xx 75 xx 10^6`

`=> f + b + g + e = 16.50 xx 10^6` .......(ii)

`18%` of population know Sanskrit,

i.e., `d + g + e + c = 18%` of `P`

`= (18)/(100) xx 75 xx 10^6`

`=> d + g + e + c = 13.5 xx 10^6` ... (iii)

`12%` of population know Hindi and English,

i.e, `g + f = 12%` of `P= (12)/(100) xx 75 xx 10^6`

`=> g + f = 9 xx 10^6` ... (iv)

`8%` know English and Sanskrit,

i.e., `g + e = 8%` of `P = 8/(100) xx 75 xx 10^6`

`=> g + e = 6 xx 10^6` ... (v)

`10%` of population know Hindi and Sanskrit,

i.e., `g + d= 10%` of `P= (10)/(100) xx 75 xx 10^6`

`=> g + d = 7.5 xx 10^6` ... (vi)

and `5%` of population know all the three languages,

i.e., `g = 5%` of `P= 5/(100) xx 75 xx 10^6`

`=> g = 3.75 xx 10^6` ... (vii)

From Eqs. (vi) and (vii),

`d=3.75 xx 10^6`

From Eqs. (v) and (vii),

`e = 225 xx 10^6`

From Eqs. (iv) and (vii),

`f = 5.25 xx 10^6`

From Eq. (ii),

`b = 10.5 xx 10^6 - (f + g + e)`

`= 16.5 xx 10^6 - 11 .25 xx 10^6`

`= 5.25 xx 10^6`

From Eq. (iii),

`c = 13.5 xx 10^6 - (d + g + e)`

`= 13.5 xx 10^6 - 9.75 xx 10^6 = 3.75 xx 10^6`

From Eq. (i),

`a = 33.75 xx 10^6 - (f + g + d)`

`= 33.75 xx 10^6 - 12.75 xx 10^6`

`= 21 xx 10^6`

The number of people who know only English

`= b = 5.25 xx 10^6`

Correct Answer is `=>` (D) None of these

Q 2355545464

If `A, B` and `C` are non-empty sets such that
`A cap C = phi`, then what is `(A xx B) cap (C xx B)` equal
to?
NDA Paper 1 2011

(A)

`A xx C`

(B)

`A xx B`

(C)

`B xx C`

(D)

`phi`

Solution:

Since, `A, B` and `C` are non-empty sets such that

`A cap C = phi`.

`:. (A xx B) cap (C xx B)= (A cap C) xx B = phi xx B = phi`

Correct Answer is `=>` (D) `phi`

Q 2325745661

If `A= {4n + 2| n` is a natural number} and
`B = {3n | n` is a natural number}, then what is
`(A cap B)` equal to?
NDA Paper 1 2011

(A)

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

(B)

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

(C)

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

(D)

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

Solution:

`∵ A={4n + 2 : n in N}`

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

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

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

`:. A cap B = { 6, 18, 30, ... } = { 6 + (n - 1)12| n in N}`

`={12n - 6 | n in N}`

Correct Answer is `=>` (D) {`12n - 6 | n` is a natural number}

Q 2305745668

If P, Q and R are the non-collinear Points, then
what is `PQ cap PR` equal to
NDA Paper 1 2011

(A)

Null set

(B)

`{P}`

(C)

`{P,Q,R}`

(D)

`{Q,R}`

Solution:

Since `P, Q` and `R` are three non-collinear points, then

`PQ cap PR = {P}`

Correct Answer is `=>` (B) `{P}`

Q 2345234163

If `phi` is a null set, then which one of the following is
correct?
NDA Paper 1 2011

(A)

`phi = 0`

(B)

`phi = {0}`

(C)

`phi = {phi}`

(D)

`phi = {}`

Solution:

If `phi` is a null set, then its other representation is `{ }`.

i.e., `phi ={}`.

Correct Answer is `=>` (D) `phi = {}`

Q 2355334264

If `A = {a, b, c}`, then what is the number of proper
subsets of `A`?
NDA Paper 1 2011

(A)

`5`

(B)

`6`

(C)

`7`

(D)

`8`

Solution:

Given, `A= {a, b, c}`

Number of subsets of `A = 2n`

`= 23 = 8`

{where, `n =` number of elements in `A, n = 3}`

`:.` Proper subset of `A = 2n - 1 = 8 - 1 = 7`

Correct Answer is `=>` (C) `7`

Q 2335145062

Read the following passage and give answer.

The students of a class are offered three languages (Hindi,
English and French). 15 students learn all the three
languages whereas 28 students do not learn any language.
The number of students learning Hindi and English but not
French is twice the number of students learning Hindi and
French but not English. The number of students learning
English and French but not Hindi is thrice the number of
students learning Hindi and French but not English. 23
students learn only Hindi and 17 students learn only
English. The total number of students learning French is 46
and the tatal number of students learning only French is 11.
How many students learn precisely two
languages?
NDA Paper 1 2011

(A)

`55`

(B)

`40`

(C)

`30`

(D)

`13`

Solution:

The total number of students learning French

`= 46`

`=> 15 + 11 + x + 3x = 46`

`=> 4x =20 => x = 5`

The number of students learning precisely two languages

`= x + 2x + 3x`

`= 6x = 6 xx 5 = 30`

Correct Answer is `=>` (C) `30`

Q 2335145062

(A)

`15`

(B)

`30`

(C)

`45`

(D)

`55`

Solution:

The total number of students learning French

`= 46`

`=> 15 + 11 + x + 3x = 46`

`=> 4x =20 => x = 5`

The number of students learning atleast two languages

`= x + 2x + 3x + 15`

`= 6x + 15 = 30 + 15 = 45`

Correct Answer is `=>` (C) `45`

Q 2335145062

Read the following passage and give answer.

The students of a class are offered three languages (Hindi,
English and French). 15 students learn all the three
languages whereas 28 students do not learn any language.
The number of students learning Hindi and English but not
French is twice the number of students learning Hindi and
French but not English. The number of students learning
English and French but not Hindi is thrice the number of
students learning Hindi and French but not English. 23
students learn only Hindi and 17 students learn only
English. The total number of students learning French is 46
and the tatal number of students learning only French is 11.
What is the total strength of the class?
NDA Paper 1 2011

(A)

`124`

(B)

`100`

(C)

`96`

(D)

`66`

Solution:

The total number of students learning French

`= 46`

`=> 15 + 11 + x + 3x = 46`

`=> 4x =20 => x = 5`

The total strength of the class

`= 28 + 23 + 17 + 11 + x + 2x + 3x + 15`

`= 79 + 6x + 15 = 79 + 30 + 15`

`= 124`

Correct Answer is `=>` (A) `124`

Q 2335145062

(A)

`30`

(B)

`43`

(C)

`45`

(D)

`73`

Solution:

The total number of students learning French

`= 46`

`=> 15 + 11 + x + 3x = 46`

`=> 4x =20 => x = 5`

The number of students learn English and French

`= 15 + 3x`

`=15+15 = 30`

Correct Answer is `=>` (A) `30`

Q 2335145062

(A)

`45`

(B)

`51`

(C)

`96`

(D)

None of these

Solution:

The total number of students learning French

`= 46`

`=> 15 + 11 + x + 3x = 46`

`=> 4x =20 => x = 5`

The number of students learn atleast one of the languages

`= 124 - 28 = 96`

Correct Answer is `=>` (C) `96`

Q 2355145964

If A and B are two disjoint sets, then which one of
the following is correct?
NDA Paper 1 2010

(A)

`A - B = A - (A cap B)`

(B)

`B - A'= B cap A`

(C)

`A cap B = (A - B) cap B`

(D)

All of these

Solution:

`∵ A cap B = phi`

`:. A - B = A - (A cap B)`

Now, `B - A' = phi`

and ` B cap A = phi`

or `B - A' = B cap A`

and `(A - B) cap B = A cap B`

So, option (d) is correct.

Correct Answer is `=>` (D) All of these

Q 2315156069

Let N denote the set of natural numbers and
`A= {n^2 : n in N }` and `B = {n^(3//2) : n in N }`.
Which one of the following is correct?
NDA Paper 1 2010

(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}`

`A= {1, 4, 9, 16, 25, 49, 64, 81, ... }`

`B = {1, 8,27, 64,125, ... }`

`A cap B = {1, 64, ... }`

Hence, `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 2385756667

Let `U = {1, 2, 3, ... , 20}.` If `A, B, C` be the subsets of
`U`, then `A` be the set of all numbers, which are
perfect squares, `B` be the set of all numbers which
are multiples of `5` and `C` be the set of all numbers,
which are divisible by `2` and `3`.
Consider the following statements
I. `A, B, C` are mutually exclusive.
II. `A, B, C` are mutually exhaustive.
III. The number of elements in the complement set
of `A cup B` is `12`.
Which of the statements given above are correct?
NDA Paper 1 2010

(A)

I and II

(B)

I and III

(C)

II and III

(D)

I, II and III

Solution:

`U = {1, 2, 3, ... , 20}`

`A` = Set of all natural numbers which are perfect square

`= {1, 4, 9, 16}`

`B =` Set of all natural numbers which are multiples of `5`

`= {5, 10, 15,20}`

`C` = Set of all natural numbers which are divisible by `2` and `3`

`= {6, 12, 18}`

`∵ A cap B cap C = phi`

and `A cup B = {1, 4, 9, 16, 5, 10, 15, 20}`

`=> n (A cup B) = 8`

`=> n (A cup B)' = 20 - 8= 12`

So, `A, B` and `C` are mutually exclusive and the number of

elements in the complement set of `A cup B` is `12`.

Correct Answer is `=>` (B) I and III

Q 2385856767

If the cardinality of a set `A` is `4` and that of a set `B`
is `3`, then what is the cardinality of the set `A Delta B`?
NDA Paper 1 2010

(A)

`1`

(B)

`5`

(C)

`7`

(D)

Cannot be determined

Solution:

Since, the sets `A` and `B` are not known, then

cardinality of the set `A Delta B` cannot be determined.

Correct Answer is `=>` (D) Cannot be determined

Q 2345267163

For a set `A`, consider the following statements
I. `A cup P(A) = P(A)`
II. `{A} cap P(A) = A`
III. `P(A)-{A}=P(A)`
where `P` denotes power set
Which of the statements given above is/are correct?
NDA Paper 1 2010

(A)

Only I

(B)

Only II

(C)

Only III

(D)

I, II and III

Solution:

Let `A {1, 2}` and `{A}= { {1, 2}}`

`=> P(A) = { {1}, {2}, {1, 2}, phi},`

then, `A cup P(A) != P(A)`

and `{A} cap P(A) = {{1,2}} cap {{1}, {2}, {1,2}, phi }`

`= {1, 2} = A`

Correct Answer is `=>` (B) Only II

Q 2305467368

If `A = P{1, 2}`, where `P` denotes the power set, then
which one of the following is correct?
NDA Paper 1 2010

(A)

`{1, 2} subset A`

(B)

`1 in A`

(C)

`phi notin A`

(D)

`{1, 2} in A`

Solution:

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

From above, it is clear that

`{1, 2} in A`

Correct Answer is `=>` (D) `{1, 2} in A`

Q 2315767660

Out of `32` persons, `30` invest in National Savings
Certificates and `17` invest in shares. What is the
number of persons who invest in both?
NDA Paper 1 2010

(A)

`13`

(B)

`15`

(C)

`17`

(D)

`19`

Solution:

`n(N cup S)= 32, n(N) = 30, n(S) = 17`

We know that,

`n(N cup S)= n(N) + n(S) - n(N cap S)`

`=> 32 = 30 + 17 - n (N cap S)`

`=> n(N cap S) = 47 - 32 = 15`

Correct Answer is `=>` (B) `15`

Q 2325391261

In an examination out of `100` students, `75` passed
in English, `60` passed in Mathematics and
`45` passed in both English and Mathematics.
What is the number of students passed in exactly
one of the two subjects'?
NDA Paper 1 2009

(A)

`45`

(B)

`60`

(C)

`75`

(D)

`90`

Solution:

Here, `n(E) = 75, n(M) = 60` and `n(E cap M) = 45`

`:. n(E cup M) = n(E) + n(M) - n(E cap M)`

`= 75 + 60 - 45 = 90`

Required number of students `= 90 - 45 = 45`

Correct Answer is `=>` (A) `45`

Q 2315478369

Let `A = { x| x <= 9 , x in N}`. If `B = {a,b,c}` be the
subset of `A` where `(a + b +c)` is a multiple of `3`.
What is the largest possible number of subsets like
`B`?
NDA Paper 1 2009

(A)

`12`

(B)

`21`

(C)

`27`

(D)

`30`

Solution:

`A = {x : x <= 9, x in N} = {1, 2, 3, 4, 5, 6, 7, 8, 9}`

Total possible multiple of `3` are

`3, 6, 9, 12, 15, 18, 21, 24,27`

But ` 3` and `27` are not possible.

`6 -> 1 + 2 + 3`

`9 -> 2 + 3 + 4, 5 + 3 + 1, 6 + 2 + 1`

`12 -> 9 + 2 + 1,8 + 3 + 1,7 + 1 + 4,7 + 2 + 3,`

`6 + 5 + 1, 5 + 4 + 3, 6 + 4 + 2`,

`15 -> 9 + 4 + 2,9 + 5 + 1,8 + 6 + 1,8 + 5 + 2,`

`7 + 6 + 2,7 + 5 + 3, 6 + 5 + 4, 8 + 4 + 3`,

`18 -> 9 + 8 + 1, 9 + 7 + 2' 9 + 6 + 3, 9 + 5 + 4`,

`7 + 6 + 5, 8 + 7 + 3`,

`21 -> 9 + 8 + 4, 9 + 7 + 5, 8 + 7 + 6`

`24 -> 9 + 8 + 7`

Hence, total number of largest possible subsets are `30`.

Correct Answer is `=>` (D) `30`

Q 2335180062

If `n(A) = 115, n(B) =326` and `n(A - B) =47`, then
what is `n(A cup B)` equal to?
NDA Paper 1 2009

(A)

`373`

(B)

`165`

(C)

`370`

(D)

`394`

Solution:

Now, `n(A - B) = n (A) - n (A cap B)`

`=> 47 = 115 - n (A cap B)`

`=> n (A cap B) = 68`

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

`= 115 + 326 - 68 = 373`

Correct Answer is `=>` (A) `373`

Q 2315280160

If `P(A)` denotes the power set of `A` and `A` is the
void set, then what is number of elements in
`P{P{P{P(A)} }}`?
NDA Paper 1 2009

(A)

`0`

(B)

`1`

(C)

`4`

(D)

`16`

Solution:

The number of elements in power set of `A` is `1` .

`:. P{P(A)} = 2^1 = 2`

`=> P{P{P(A)}} = 2^2 = 4`

`=> P{P{P{P(A)}}} = 2^4 = 16`

Correct Answer is `=>` (D) `16`

Q 2315380260

If `N_a = {ax | x in N}`,then what is `N_(12) cap N_8` equal
to?
NDA Paper 1 2009

(A)

`N_(12)`

(B)

`N_(20)`

(C)

`N_(24)`

(D)

`N_(48)`

Solution:

Given, `N_8 = {ax | x in N}`

`:. N_(12) = {12, 24, 36, 48, ... }`

and `N_8 = {8, 16, 24, ... }`

`:. N_8 cap N_(12) = {24, 48, ... } = N_(24)`

Correct Answer is `=>` (C) `N_(24)`

Q 2315480360

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?
NDA Paper 1 2009

(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 cup Y = {0, 9, 18, 27, 36, 54, ... } = Y`

Correct Answer is `=>` (B) `Y`

Q 2325580461