JEE  >  Maths 35 Years JEE Main & Advanced Past year Papers  >  Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main

Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Document Description: Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main for JEE 2022 is part of Maths 35 Years JEE Main & Advanced Past year Papers preparation. The notes and questions for Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main have been prepared according to the JEE exam syllabus. Information about Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main covers topics like and Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Example, for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main.

Introduction of Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main in English is available as part of our Maths 35 Years JEE Main & Advanced Past year Papers for JEE & Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main in Hindi for Maths 35 Years JEE Main & Advanced Past year Papers course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
1 Crore+ students have signed up on EduRev. Have you?

Q.1. The logical statement (p ⇒ q) ∧ ( q ⇒~ p) is equivalent to    (2020)
(1) p
(2) q
(3) ~p
(4) ~ q
Ans.
(3)
We have
(p → q) ∧ (q → -p)
≡ (∼p ∨ q) ∧ (∼qv ∼ p)
≡ ∼ p ∨ (q ∧ ∼ q) ≡ ∼ p ∨ C ≡ ∼p

Q.2. Let A, B, C and D be four non-empty sets. The contrapositive statement of “If A ⊆ B and B ⊆ D, then A ⊆ C" is    (2020)
(1) If Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then A ⊆ B and B ⊆ D
(2) If A ⊆ C, then B ⊂ A and D ⊂ B
(3) If Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and B ⊆ D
(4) Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Ans.
(4)
The contrapositive of p → q is - q → - p.
Now, Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 
Q.3. Which one of the following is a tautology?    (2020)
(1) P ∧ (P → Q) → Q
(2) Q → (P ∧ (P → Q))
(3) P ∧ (P ∨ Q)
(4) P ∨ (P ∧ Q)
Ans.
(1)
We have
(1) P ∧ (P → Q) → Q ≡ (P ∧ (- P ∨ Q)) → Q ≡ (P ∧ Q) → Q
≡ ((∼P) ∨ (∼Q)) ∨ Q ≡ (∼P) ∨ t ≡ t
(2) Q → (P ∧ (P → Q)) ≡ Q → (P ∧ (-P ∨ Q)) ≡ Q → (P ∧ Q) ≡ (-Q) ∨ (P ∧ Q) ≡ (P ∨ (-Q))
(3) P ∧ (P ∨ Q) ≡ P
(4) P ∨ (P ∧ Q) = P

Q.4. Which of the following statements is a tautology?    (2020)
(1) p ∨ (∼q) → p ∧ q
(2) ∼ (p ∧ ∼ q) → p ∨ q
(3) ∼ (p ∨ ∼ q) → p ∧ q
(4) ∼ (p ∨ ∼ q) → p ∨ q
Ans.
(4)
We have
- (p ∨ - q) → p ∨ q
≡ - (p ∧ q) → p ∨ q ≡ - {(-p ∧ q) ∧ (- p∧ - q)}
≡ -(- p ∧ f)

Q.5. Negation of the statement: ‘√5 is an integer or 5 is irrational’ is    (2020)
(1) √5 is not an integer or 5 is not irrational.
(2) √5 is not an integer and 5 is not irrational.
(3) √5 is irrational or 5 is an integer.
(4) √5 is integer and 5 is irrational.
Ans. 
(2)
We have  -(p ∨ q) = - p∧ - q
Hence, the Negation of the statement: '√5 is an integer or 5 is irrational’ is “√5 is not an integer and 5 is not irrational.”

Q.6. If p → (p∧ ∼ q) is false, then the truth values of p and q are respectively    (2020)
(1) F, F
(2) T, F
(3) T, T
(4) F, T
Ans.
(3)
We have
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Hence, p → (p → p∧ ∼ q) is false when the truth values of p and q are T, T respectively.

Q.7. If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m + n is equal to ________.    (2020)
Ans.
(18.00)
The variance of first n natural number is given by
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
The variance of first m even natural number is
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Hence, m + n = 11 + 7 = 18

Q.8. If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x . y is equal to _______.    (2020)
Ans.
(54.00)
The mean of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 , then
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ x + y = 80 - 64 = 16 ...(1)
The variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 25, then
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ x2 + y2 = 1000 - 852 = 148 ...(2)
From Eqs. (1) and (2), we get
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.9. The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal to    (2020)
(1) −5
(2) 10
(3) −20
(4) −10
Ans.
(3)
If each observation is multiplied with p and then reduced by q, then
New mean,Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ 20p - q = 10 ...(1)
and new standard deviation is given by σ2 = |p|σ2
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ...(2)
From Eqs. (1) and (2), we get
p = -1/2 and - 10 - q = 10 ⇒ q = -20

Q.10. The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then, the correct variance is    (2020)
(1) 3.99
(2) 4.01
(3) 4.02
(4) 3.98
Ans.
(1)
We have
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ...(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ...(2)
Now,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 106 - 102.1 = 3.99

Q.11. Let the observations x(1 ≤ i ≤ 10)  satisfy the equations Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEIf μ and λ are the mean and the variance of the observations, x1 - 3, x2 - 3, .... ,x10 - 3, then the ordered pair (μ, λ) is equal to    (2020)
(1) (3, 3)
(2) (6, 3)
(3) (6, 6)
(4) (3, 6)
Ans.
(1)
Given,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ...(1)
Again, we have
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE...(2)
Now, the mean of x1 - 3, x2 - 3, .... , x10 - 3 is
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
The variance of x1 - 3, x2 - 3, .... , x10 - 3 is
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.12. Let Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE If A = {n ∈ X : n is a multiple of 2} and B = {n ∈ X : n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is _______.    (2020)
Ans.
(29.00)
Number of elements in set A, n (A) = 25
Number of elements in set B, n (B) = 7
Number of common elements in both the sets, n (A ∩ B) = 3
Hence, number of elements in the smallest subset of X containing both A and B is
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 25 + 7 - 3 = 29

Q.13. Let S be the set of all real roots of the equation, Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Then, S    (2020)
(1) Contains exactly two elements.
(2) Is a singleton.
(3) Is an empty set.
(4) Contains at least four elements.
Ans.
(2)
We have
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Let 3x = t, then t(t - 1) + 2 = |t - 1| + |t - 2|
⇒ t2 - t + 2 = |t - 1|+ |t - 2| ...(1)
Case 1: When t < 1, then
t2 - t + 2 = 1 - t + 2 - t ⇒ t2 + t - 1 = 0
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Hence, only Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is only acceptable value.
Case 2: When 1 ≤ t < 2, then
t2 - t + 2 = t - 1 + 2 - t ⇒ t2 - t + 1 = 0
⇒ D = 1 - 4 = -3 < 0
Hence, no real solution is possible in this case.
Case 3: When t ≥ 2, then
t2 - t + 2 = t - 1 + 2 - t ⇒ t2 - 3t + 5 = 0
⇒ D = 9 - 20 = -11 < 0
Hence, no real solution is possible in this case.
Hence, S is a singleton.

Q.14. If Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then    (2020)
(1) A ∩ B = (-2, -1)
(2) B - A = Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE- (-2, 5)
(3) A ∪ B = Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE- (2, 5)
(4) A - B = (-1, 2)
Ans. 
(2)
We have
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, A ∩ B = {x : x ∈ (-2, -1)}
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.15. Let Z be the set of integers. If A = {x ∈ Z : 2(x + 2)(x2 - 5x + 6) = 1} and B = {x ∈ Z : -3 < 2x - 1 < 9}, then the number of subsets of the set A x B, is:    (2019)
(1) 215
(2) 218
(3) 212
(4) 210
Ans.
(1)
Let x ∈ A, then
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
x = -2, 2, 3
A = {-2, 2, 3}
Then, n(A) = 3
Let x ∈ B, then
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 B={0, 1, 2, 3, 4}
n(B) = 5
n(A x B) = 3 x 5 = 15
Hence, Number of subsets of A x B = 215

Q.16. Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisements is:    (2019)
(1) 13.9
(2) 12.8
(3) 13
(4) 13.5
Ans.
(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
% of people who reads A only = 25 - 8 =17%
% of people who read B only = 20 - 8 = 12%
% of people from A only who read advertisement = 17 x 0.3 = 5.1%
% of people from B only who read advertisement = 12 x 0.4 = 4.8%
% of people from A & B both who read advertisement = 8 x 0.5 = 4%
∴ total % of people who read advertisement = 5.1 + 4.8 + 4= 13.9%

Q.17. Let A, B and C be sets such that φ ≠ A ∩ B ⊆ C. Then which of the following statements is not true?    (2019)
(1) B ∩ C ≠ φ)
(2) If (A - B) ⊆ C,then A ⊆ C
(3) (C ∪ A) ∩ (C ∪ B) = C
(4) If (A - C) ⊆ B, then A ⊆ B
Ans.
(1)
(1), (2) and (4) are always correct.
In (3) option,
If A = C then A - C = φ
Clearly, φ ⊆ B but A ⊆ B is not always true.

Q.18. 5 students of a class have an average height 150 cm and variance 18 cm . Anew student, whose height is 156 cm, joined them. The variance (in cm ) of the height of these six students is:    (2019)
(1) 16
(2) 22
(3) 20
(4) 18
Ans.
(3)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Then, variance of the height of six students
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.19. A data consists of n observations:
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
then the standard deviation of this data is:    (2019)
(1) 2
(2) √5
(3) 5
(4) √7
Ans.
(2)
Variance is given by,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   ...(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ A + n + 2B = 9n ⇒ A + 2B = 8 ...(ii)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ A + n - 2B - 5n ⇒ A - 2B = An    ...(iii)
From (ii) and (iii),
A = 6n, B = n
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.20. The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is:    (2019)
(1) 10 : 3
(2) 4 : 9
(3) 5 : 8
(4) 6 : 7
Ans.
(2)
Since mean of x1, x2, x3, x4 and x5 is 5
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ x4 + x= 13    (1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 125 + 46 = 171
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    (2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    (3)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.21. If mean and standard deviation of 5 observations x1, x2, x3, x4, x5 are 10 and 3, respectively, then the variance of 6 observations x1, x2,.... x5 and - 50 is equal to:
(1) 509.5
(2) 586.5
(3) 582.5
(4) 507.5
Ans.
(4)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Then,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.22. The outcome of each of 30 items was observed; 10 items gave an outcome 1/2 - d each, 10 items gave outcome 1/2 each and the remaining 10 items gave outcome 1/2 + d each. If the variance of this outcome data is 4/3 then |d| equals:    (2019)
(1) 2/3
(2) 2
(3) √5/2
(4) √2
Ans.
(4)
Outcomes are Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Variance of the outcomes is,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.23. A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is equal to:    (2019)
(1) 4
(2) 4√3
(3) 3√2
(4) 4√3 / 3
Ans.
(2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.24. If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:    (2019)
(1) 30
(2) 51
(3) 50
(4) 31
Ans.
(4)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 1550/50
= 31

Q.25. There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is    (2019)
(1) 12
(2) 11
(2) 9
(4) 7
Ans.
(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ m > 0
m = 12

Q.26. The mean and the variance of five observations arc 4 and 5.20, respectively. If three of the observations are 3,4 and 4; then the absolute value of the difference of the other two observations, is:    (2019)
(1) 7
(2) 5
(3) 1
(4) 3

Ans. (1)
Let two observations be x1 and x2, then
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ x1 + x2 = 9 ...(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ...(2)
From (1) and (2);
x1 = 8, x2 = 1
Hence, the required value of the difference of other two observations = |x1 - x2| = 7

Q.27. The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is:    (2019)
(1) 45
(2) 49
(3) 48
(4) 40
Ans.
(3)
Let the remaining numbers are a and b.
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ a + b = 14   (1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ a2 + b2 = 100   (2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Hence, the product of the remaining two observations = ab = 48

Q.28. A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:    (2019)
(1) 10/√3
(2) 100/3
(3) 10/3
(4) 100/√3
Ans.
(1)
∵ Mean score = 48
Let unknown score be x,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.29. If the standard deviation of the numbers -1, 0, 1, k is √5 where k > 0, then k is equal to:    (2019)
(1) 2√6
(2) Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(3) Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(4) √6
Ans.
(1)
Mean of given observation = k/4
∵ Standard deviation = 5
∴ σ2 = 5    ...(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ...(2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ k = 2√6

Q.30. The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then y/x is equal to:    (2019)
(1) 9/4
(2) 7/2
(3) 8/3
(4) 7/3
Ans.
(4)
Ten numbers in increasing order are 10, 22, 26,29, 34, x, 42, 67, 70, y
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.31. If for some x ∈ R, the frequency distribution of the marks obtained by 20 students in a test is:
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
then the mean of the marks is:    (2019)
(1) 3.2
(2) 3.0
(3) 2.5
(4) 2.8
Ans.
(4)
Number of students are,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.32. If both the mean and the standard deviation of 50 observations x1, x2,....., x50 are equal to 16, then the mean of -Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is:    (2019)
(1) 400
(2) 380
(3) 525
(4) 480
Ans.
(1)
Given, mean and standard deviation are equal to 16.
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
and Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.33. If the data x1, x2, ..... , x10 is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is:    (2019)
(1) 2√2    
(2) 2    
(3) 4    
(4) √2
Ans. 
(2)
According to the question,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∵ standard deviation, Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

Q.34. If the Boolean expression Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is equivalent to p ∧ q , where Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the ordered pair Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is:    (2019)
(1) (∨,∧)    
(2) (∨, ∨)
(3) (∧, ∨) 
(4) (∧, ∧)
Ans.
(3)
Check each option
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.35. The logical statement [~ (~ P ∨ q) ∨ (p ∧ r)] ∧ (~ p ∧ r)is equivalent to:    (2019)
(1) (~ p ∧ ~ q) ∧ r
(2) ~ p ∨ r
(3) (p ∧ r) ∧ ~ q    
(4) (p ∧ ~ q) ∨ r
Ans.
(3)
Logical statement,
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.36. Consider the statement: “P(n) : n2 - n + 41 is prime." Then which one of the following is true?    (2019)
(1) Both P(3) and P(5) are true.
(2) P(3) is false but P(5) is true.
(3) Both P(3) and P(5) are false.
(4) P(5) is false but P(3) is true.
Ans.
(1)
P(n) = n2 - n + 41
⇒ P(3) = 9 - 3 + 41 = 47 (prime)
& P(5) = 25 - 5 + 41 = 61 (prime)
∴ P(c) and P(5) are both prime i.e., true.

Q.37. Consider the following three statements:
P : 5 is a prime number.
Q : 7 is a factor of 192.
R : L.C.M. of 5 and 7 is 35.
Then the truth value of which one of the following statements is true?    (2019)
(1) (~ P) ∨ (Q ∧ R)
(2) (P ∧ Q) ∨ (~R)
(3) (~P) ∧ (~Q ∧ R)
(4) P ∧ (~ Q ∧ R)
Ans.
(4)
P is True, Q is False and R is True
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.38. If q is false and p ∧ q ↔ r is true, then which one of the following statements is a tautology?    (2019)
(1) (p ∨ r) → (p ∧ r)
(2) (p ∧ r) → (p ∨ r)
(3) p ∧ r
(4) p∨ r
Ans.
(2)
q is false and [(p ∧ q) ↔ r] is true
As (p ∧ q) is false
[False ↔ r] is true
Hence r is false
Option (1): says p ∨ r,
Since r is false
Hence (p ∨ r) can either be true or false
Option (2): says (p ∧ r) → (p ∨ r)
(p ∧ r) is false
Since, F → T is true and
F → F is also true
Hence, it is a tautology
Option (3): (p ∨ r) → (p ∧ r)
i.e. (p ∨ r) → F
It can either be true or false
Option (4): (p ∧ r),
Since, r is false
Hence, (p ∧ r) is false.

Q.39. Contrapositive of the statement “If two numbers are not equal, then their squares are not equal”, is:    (2019)
(1) If the squares of two numbers are not equal, then the numbers are equal.
(2) If the squares of two numbers are equal, then the numbers are not equal.
(3) If the squares of two numbers are equal, then the numbers are equal.
(4) If the squares of two numbers are not equal, then the numbers are not equal.
Ans. 
(3)
Contrapositive of “If A then B” is “If -B then -A". Hence contrapositive of “If two numbers are not equal, then their squares are not equal” is “If squares of two numbers are equal, then the two numbers are equal”.

Q.40. The Boolean expression ((p ∧ q) ∨ (p∨ - q)) ∧ (~ p∧ ~ q) is equivalent to:    (2019)
(1) p ∧ q 
(2) p∧ (~ q)
(3) (~ p) ∧ (~ q)
(4) p ∨ (~ q)
Ans.
(3)
Consider the Boolean expression
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.41. The expression - (- p → q) is logically equivalent to:    (2019)
(1) - p ∧ - q    
(2) p ∧ - q
(3) - P ∧ q    
(4) p ∧ q
Ans.
(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.42. The contrapositive of the statement "If you are born in India, then you are a citizen of India", is:    (2019)
(1) If you are not a citizen of India, then you are not born in India.
(2) If you are a citizen of India, then you are born in India.
(3) If you are born in India, then you are not a citizen of India.
(4) If you are not born in India, then you are not a citizen of India.
Ans.
(1)
S: "If you are bom in India, then you are a citizen of India."
Contrapositive of p → q is ~ q ~ p
So contrapositive of statement S will be:
"If you are not a citizen of India, then you are not bom in India."

Q.43. If a point R(4, y, z) lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10), then distance of R from the origin is:    (2019)
(1) 2√14
(2) 2√21
(3) 6
(4) √53
Ans.
(1)
Here, P, Q, R are collinear
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.44. For any two statements p and q, the negation of the expression p ∨ (~ p ∧ q) is:    (2019)
(1) ~ p ∧ ~ q
(2) p ∧ q
(3) p ↔ q
(4) ~ p ∨ ~ q
Ans.
(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.45. If p ⇒ (q ∨ r) is false, then the truth values of p, q, r are respectively:    (2019)
(1) F, T, T
(2) T, F, F
(3) T, T, F
(4) F, F, F
Ans.
(2)
For p ⇒ q ∨ r to be F.
r should be F and p ⇒ q should be F
for p ⇒ q to be F, p ⇒ T and q ⇒ F
p, q, r ≡ T, F, F

Q.46. Which one of the following Boolean expressions is a tautology?    (2019)
(1) (p ∧ q) ∨ (p ∧ - q)
(2) (p ∨ q) ∨ (p ∨ ∼ q)
(3) (p ∨ q) ∧ (p ∨ ~ q)
(4) (p ∨ q) ∧ (~p ∨ ∼ q)
Ans.
(2)
(p ∨ q) ∨ (p ∨ ~q) = p ∨ (q ∨ p) ∨ ~ q
= (p ∨ p) ∨ (q ∨ ~ q) = p ∨ T = T
Hence first statement is tautology.

Q.47. The negation of the Boolean expression ~ s ∨ (~ r ∧ s) is equivalent to:    (2019)
(1) ~s ∧~r
(2) r 
(3) s ∨ r
(4) s ∧ r
Ans.
(4)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
≡ (~ s ∨ ~ r)    (∵ ∼ s ∨ s) is tautology)
≡ ~(s ∧ r)
Hence, its negation is s ∧ r.

Q.48. If the truth value of the statement p → (~q ∨ r) is false (F), then the truth values of the statements p, q, r are respectively.    (2019)
(1) T, T, F
(2) T, F, F
(3) T, F, T
(4) F, T, T

Ans. (1)
Given statement p → (~ q ∨ r) is False.
⇒ p is True and ~ q ∨ r is False
⇒ p is True and ~ q is False and r is False
∴ truth values of p, q r are T, T, F respectively.

Q.49. The Boolean expression - (p ⇒ (~ q)) is equivalent to:    (2019)
(1) p ∧ q
(2) q ⇒ ~ p
(3) p ∨ q
(4) (~ p) ⇒ q
Ans.
(1)
Given Boolean expression is,
~ (p ⇒ (~ q))  {∵ p ⇒ q is same as ~p ∨ q}
≡ ∼((∼ p) ∨ (~ q)) = p ∧ q

Q.50. The Boolean expression ~(p ν q) ν (~p ∧ q) is equivalent to:    (2018)
(1) ~p
(2) p
(3) q
(4) ~q
Ans.
(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.51. Two sets A and B are as under:
A={(a, b) ∈ R x R : |a - 5| <1 and |b - 5|<1};
B={(a, b) ∈ R x R : 4(a - 6)2 + 9(b - 5)2 ≤ 36 }. Then:    (2018)
(1) B ⊂ A  
(2) A⊂B
(3) A ∩ B = Φ (an empty set)
(4) neither A ⊂ B nor B ⊂ A
Ans.
(3)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.52. If Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then the standard deviation of the 9 items x1 x2, …., x9 is:    (2018)
(1) 9
(2) 4
(3) 2
(4) 3
Ans.
(3)
xi - 5 = yi 
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.53. Consider the following two binary relations on the set A = {a, b, c}: R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}.Then:    (2018)
(1) both R1 and R2 are transitive 
(2) both R1 and R2 are not symmetric 
(3) R1 is not symmetric but it is transitive 
(4) R2 is symmetric but it is not transitive 
Ans. (4)
R1 ∈ (b, c) but R1 ∉ (c, b)
Example R1 is not symmetric  
in R1; (b, c) R1 and (c, a) ∈ R1 but (b, a) ∉ R1 
So R1 is not transitive
R2 is symmetric
is R2 ; (b, a) R2 and (a, c) R2 but (b, c) R2
So R2 is not transitive

Q.54. The mean and the standard deviation (s.d.) of five observations are 9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is:    (2018)
(1) 2
(2) 4
(3) 0
(4) 1
Ans.
(1)
Here mean = Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, standard deviation = 0
∴ all the five terms are same i.e., 9.
Now for changed observation
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 2

Q.55. Let N denote the set of all natural numbers. Define two binary relations on N as R1 = {(x, y) ∈ N × N : 2x + y = 10} and R2 = {(x, y) Î N × N : x + 2y = 10}. Then    (2018)
(1) Both R1 and R2 are transitive relations
(2) Range of R2 is {1, 2, 3, 4}
(3) Range of R1 is {2, 4, 8}
(4) Both R1 and R2 are symmetric relations
Ans. (2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

Q.56. The function Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE defined as Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis    (2017)
(1) Neither injective nor surjective
(2) Invertible
(3) Injective but not surjective
(4) Surjective but not injective
Ans.
(4)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
f '(x) changes sign in different intervals.
∴ Not injective.
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
yx2 - x+ y = 0
For y ≠ 0 
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
For, y = 0  ⇒ x = 0
∴ Part of range 
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ Surjective but not injective. 

Q.57. The following statement (p → q) → [(~ p → q) → q] is    (2017) 
(1) A fallacy
(2) A tautology
(3) Equivalent to ~ p → q
(4) Equivalent to  p → ~ q
Ans.
(2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.58. The mean age of 25 teaches in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age ( in years ) of the newly appointed teachers is:    (2017)
(1) 35
(2) 40
(3) 25
(4) 30
Ans.
(1)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
1000 - 60 + A = 39 × 25 = 975
A = 975 - 940 = 35

Q.59. The proposition (~ p) V (P ^ ~ q) is equalvalent to:    (2017)
(1) P → ~q
(2) P ^ ~ q
(3) q → p
(4) p v ~ q
Ans.
(2)
(~ p) V ( P ^ ~ q)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.60. The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is:    (2017)
(1) 8.25
(2) 8.50
(3) 9.00
(4) 8.00
Ans.
(3)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.61. Contrapositive of the statement
'If two numbers are not equal, then their squares are not equal', is:    (2017)
(1) If the squares of two numbers are equal, then the numbers are not equal
(2) If the squares of two numbers are not equal, then the numbers are equal
(3) If the squares of two numbers are not equal, then the numbers are not equal
(4) If the squares of two numbers are equal, then the numbers are equal.
Ans.
(4)
p → q
contrapositive is
~ q → ~ p

Q.62. If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?    (2016)
(1) 3a2 - 26a + 55 = 0
(2) 3a2 - 32a + 84 = 0
(3) 3a2 - 34a + 91 = 0
(4) 3a2 - 23a + 44 = 0
Ans. 
(2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 (4a2 + 536) - (a2 + 32a + 256) = 196 => 3a2 - 32a + 84 = 0

Q.63. The Boolean ExpressionPrevious year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis equivalent to:    (2016)
(1) ~ p ∧ q
(2)
 p ∧ q
(3) p ν q
(4) p ν ~ q
Ans.
(3)
(p ∧ ~ q) ν q ν (~ p ∧ q)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.64. If the mean deviation of the numbers 1, 1 + d, ..., 1 + 100 d from their mean is 255, then a value of d is:    (2016)
(1) 10
(2) 20.2
(3) 5.05
(4) 10.1
Ans.
(4)
Mean is Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
sum of deviation about mean is
50 d + 49d + ................... d + 0 + d + ........ + 50 d
= 50.51 d
Mean deviation = Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q.65. Consider the following two statements:
P: If 7 is an odd number, then 7 is divisible by 2.
Q: If 7 is a prime number, then 7 is an odd number.
If V1 is the truth value of contrapositive of P and V2 is the truth value of contrapositive of Q, then the ordered pair (V1, V2) equals:    (2016)
(1) (F, T)
(2) (T, F)
(3) (F, F)
(4) (T, T)
Ans.
(1)
Statement P is False
Statement Q is True.
V1 ≡ F
V2 ≡ T
1

Q.66. The contrapositive of the following statement,
" If the side of a square doubles, then its area increases four times", is    (2016)
(1) If the area of a square does not increase four times, then its side is not doubled.
(2) If the area of a square increases four times, then its side is not doubled.
(3) If the area of a square increases four times, then its side is doubled.
(4) If the side of a square is not doubled, then its area does not increase four times.
Ans.
(1)
p ≡ The side of a square doubles
q ≡ Area of square increases four time
so the contrapositive of p → q is ~ q → ~ P

Q.67. The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2 and 6, then the mean deviation from the mean of the data is    (2016)
(1) 2.5
(2) 2.8
(3) 2.6
(4) 2.4
Ans.
(2)
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  

The document Previous year questions (2016-20): Miscellaneous (Sets, Relations, Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
All you need of JEE at this link: JEE

Related Searches

Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

,

Important questions

,

Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

,

Previous Year Questions with Solutions

,

Extra Questions

,

Relations

,

Free

,

Previous year questions (2016-20): Miscellaneous (Sets

,

practice quizzes

,

Relations

,

study material

,

Objective type Questions

,

past year papers

,

Previous year questions (2016-20): Miscellaneous (Sets

,

ppt

,

MCQs

,

mock tests for examination

,

Relations

,

Sample Paper

,

Viva Questions

,

Exam

,

Summary

,

Semester Notes

,

shortcuts and tricks

,

Previous year questions (2016-20): Miscellaneous (Sets

,

video lectures

,

Statistics & Mathemati | JEE Main Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

,

pdf

;