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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
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))
If the mean deviation about the median of the numbers a, 2a,.......,50a is 50, then  a  equals [2011]
Median is the mean of 25th and 26th observation
The negation of the statement [2012] "If I become a teacher, then I will open a school", is :
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.
Let x_{1} , x_{2},...., x_{n} be n observation s, an d let be their arithmetic mean and σ^{2} be the variance. [2012]
Statement1 : Variance of 2x_{1}, 2x_{2}, ..., 2x_{n} is 4σ^{2}.
Statement2 : Arithmetic mean 2x_{1}, 2x_{2}, ..., 2x_{n }is 4 .
KEY CONCEPT : If each observation is multiplied by k, mean gets multiplied by k and variance gets multiplied by k^{2}. Hence the new mean should be and new variance should be k^{2}σ^{2}.
So statement1 is true and statement2 is false.
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]
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
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]
Given n(A) = 2, n(B) = 4, n(A × B) = 8
Required number of subsets =
Consider
Statement1 : (p ^ ~ q) ^ (~ p ^ q) is a fallacy.
Statement2 : (p → q) ⇿ (~ q → ~ p) is a tautology. [JEE M 2013]
Statement2 : (p → q) ⇿ (~q → ~p) ≡ (p → q) ⇿ (p → q)
which is always true.
So statement 2 is true
Statement1 : (p ^ ~q) ^ (~p ^ q) = p ^ ~q ^ ~p ^ q = p ^ ~p ^ ~q ^ q = f ^ f = f
So statement1 is true
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]
If initially all marks were x_{i} then
Now each is increased by 10
Hence, variance will not change even after the grace marks were given.
If X = {4^{n}  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]
∴ X = {x : x is a multiple of 9}
Also, {All multiples of 9}
Clearly X ⊂ Y. ∴ X ∪ Y = Y
The variance of first 50 even natural numbers is [JEE M 2014]
First 50 even natural numbers are 2, 4 , 6 ....., 100
Variance =
⇒
= 3434 – 2601
⇒ σ^{2} = 833
The statement : ~ (p ⇿ ~ q) is : [JEE M 2014]
Clearly equivalent to p ⇿q
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]
n(A) = 4, n (B) = 2 ⇒ n(A × B) = 8
The number of subsets of A × B having atleast 3 = elements = ^{8}C_{3} +^{ 8}C_{4 }+ ... + ^{8}C_{8 }
= 28 – ^{8}C_{0} – ^{8}C_{1 }– ^{8}C_{2}
= 256 – 1 – 8 – 28 = 219
The negation of ~ s ν (~ r ∧ s) is equivalent to : [JEE M 2015]
~[~sν(~r ∧ s)]
= s∧~(~r ∧ s)
= s∧(r ν~s)
= (s ∧ r) ν (s ∧~ s)
= (s ∧ r) ν 0
= s ∧ r
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]
Sum of 16 observations = 16 × 16 = 256
Sum of resultant 18 observations = 256 – 16 + (3 + 4+5) = 252
Mean of observations =
If f(x) + and S = {x ∈ R : f(x) = f(–x)}; then S: [JEE M 2016]
.......(1)
.......(2)
Adding (1) and (2) ⇒
Substracting (1) from (2)
On adding the above equations
x^{2} = 2 or x =
The Boolean Expression (p ∧ ~ q ) ν q ν (~ p∧ q) is equivalent to: [JEE M 2016]
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]
⇒ 3a^{2}  32a + 84= 0
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]
...(1)
tan 60^{0}
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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