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Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3


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18 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3

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Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 1

Consider the following statements [2011]
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 1

Suman is brilliant and dishonest if and only if Suman is rich is expressed as
Q ⇿ (P∧ ~R)
Negation of it will be ~ (Q ⇿ ( P∧ ~R))

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 2

If the mean deviation about the median of the numbers a, 2a,.......,50a is 50, then | a | equals [2011]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 2

Median is the mean of 25th and 26th observation

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 3

The negation of the statement [2012] "If I become a teacher, then I will open a school", is :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 3

Let p : I become a teacher.
q : I will open a school
Negation of p → q is ~ (p → q) = p ^ ~q
i.e. I will become a teacher and I will not open a school.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 4

Let x1 , x2,...., xn be n observation s, an d let be their arithmetic mean and σ2 be the variance. [2012]
Statement-1 : Variance of 2x1, 2x2, ..., 2xn is 4σ2.
Statement-2 : Arithmetic mean 2x1, 2x2, ..., 2xn is 4 .

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 4

KEY CONCEPT :  If each observation is multiplied by k, mean gets multiplied by k and variance gets multiplied by k2. Hence the new mean should be and new variance should be k2σ2.
So statement-1 is true and statement-2 is false.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 5

Let X ={1,2,3,4,5}. The number of different ordered pairs (Y,Z) that can formed such that Y ⊆ X , Z ⊆ X and Y ∩ Z is empty is : [2012]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 5

Let X = {1,2,3,4,5}
Total no. of elements = 5
Each element has 3 options.
Either set Y or set Z or none. (∵ Y ∩ Z = φ)
So, number of ordered pairs = 35

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 6

Let A and B two sets containing 2 elements and 4 elements respectively. The number of subsets of A × B having 3 or more elements is [JEE M 2013]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 6

Given n(A) = 2, n(B) = 4, n(A × B) = 8
Required number of subsets =

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 7

Consider
Statement-1 : (p ^ ~ q) ^ (~ p ^ q) is a fallacy.
Statement-2 : (p → q) ⇿ (~ q → ~ p) is a  tautology. [JEE M 2013]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 7

Statement-2 : (p → q) ⇿ (~q → ~p) ≡ (p → q) ⇿ (p → q)
which is always true.
So statement 2 is true
Statement-1 : (p ^ ~q) ^ (~p ^ q) = p ^ ~q ^ ~p ^ q = p ^ ~p ^ ~q ^ q = f ^ f = f
So statement-1 is true

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 8

All the studen ts of a class per for med poor ly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ? [JEE M 2013]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 8

If initially all marks were xi then

Now each is increased by 10

Hence, variance will not change even after the grace marks were given.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 9

 If X = {4n - 3n -1: n∈N} and Y = {9 (n -1) : n∈N}, where N is the set of natural numbers, then X ∪ Y is equal to:         [JEE M 2014]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 9


∴ X = {x : x is a multiple of 9}

Also,   {All multiples of 9}

Clearly X ⊂ Y.  ∴ X ∪ Y = Y

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 10

The variance of first 50 even natural numbers is [JEE M 2014]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 10

First 50 even natural numbers are 2, 4 , 6 ....., 100

Variance = 

= 3434 – 2601

⇒  σ2 = 833

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 11

The statement : ~ (p ⇿ ~ q) is :                 [JEE M 2014]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 11

             

Clearly equivalent to p ⇿q

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 12

Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is :           [JEE M 2015]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 12

n(A) = 4, n (B) = 2 ⇒ n(A × B) = 8
The number of subsets of A × B having atleast 3 = elements = 8C3 + 8C4 + ... + 8C8
= 28 – 8C08C1 8C2
= 256 – 1 – 8 – 28 = 219

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 13

The negation of ~ s ν (~ r ∧ s) is equivalent to : [JEE M 2015]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 13

~[~sν(~r ∧ s)]
= s∧~(~r ∧ s)
= s∧(r ν~s)
= (s ∧ r) ν (s ∧~ s)
= (s ∧ r) ν 0
= s ∧ r

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 14

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:           [JEE M 2015]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 14

Sum of 16 observations = 16 × 16 = 256

Sum of resultant 18 observations = 256 – 16 + (3 + 4+5) = 252

Mean of observations = 

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 15

If f(x) + and S = {x ∈ R : f(x) = f(–x)}; then S:                 [JEE M 2016]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 15

.......(1)

.......(2)

Adding (1) and (2) ⇒

Substracting (1) from (2) 

On adding the above equations

x2 = 2 or x = 

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 16

The Boolean Expression (p ∧ ~ q ) ν q ν (~ p∧ q) is equivalent to:                 [JEE M 2016]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 16




Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 17

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?                [JEE M 2016]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 17

⇒ 3a2 - 32a + 84= 0

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 18

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30°. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60°. Then the time taken (in minutes) by him, from B to reach the pillar, is:                [JEE M 2016]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 3 - Question 18

  ...(1)

tan 600

From (1) and (2)

3a = x + a ⇒ x = 2a

Here, the speed is uniform

So, time taken to cover x = 2 (time taken to cover a)

∴ Time taken to cover a = minutes = 5 minutes

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