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


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25 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) - 2

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Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 1

Two points A and B move from rest along a straight line with constant acceleration f and f ' respectively. If A takes m sec. more than B and describes ‘n’units more than B in acquiring the same speed then [2005]  

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 1

As per question if point B moves s distance in t time then point A moves (s + n) distance in time (t + m) after which both have same velocity v.
Then using equation v = u + at we get

....(1)

Using equation v2 = u2+ 2 , as we get

v2 = 2 f (s + n) = 2 f's ....(2)

Also for point B using the eqn we get

Substituting values of t and s from equations (1) and (2) in the above relation, we get

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

 A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance                           [2005]

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

Let A and B be displaced by a distance x then Change in moment of (A + B) = applied moments
⇒ (A + B) x x = H ⇒ x 

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

Let x1 ,x2 , .............. xn be n observations such that = 400 and  = 80. Then the possible value of n among the following is [2005]

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

We know that for positive real numbers x1, x2, ...., xn, A.M. of  kth powers of x'is ≥ kth the power of A.M. of x'is

⇒ n ≥ 16 . So only possible value for n = 18

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 4

A particle is projected from a point O with velocity u at an angle of 60° with the horizontal. When  it is moving in a direction at right angles to its direction at O, its velocity then is given by [2005]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 4

u cos 60° = v cos 30° (as horizontal component of velocity remains the same)

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 5

The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to the smaller one is [2005]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 5

Let F be the larger force

Given R = 

Resolving F in horizontal and vertical direction

R =F cosθ ⇒ cosθ =

F' = Fsinθ =

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 6

ABC is a triangle, right angled at A. The resultant of the forces acting along with magnitudes  respectively is the force along , where D is the foot of the perpdicular from A onto BC. The magnitude of the resultant is [2006]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 6

If we consider unit vectors in the direction AB and AC respectively, then  as per quesiton, forces along AB and AC respectively are

∴ Their resultant along AD 

∴ Magnitude of resultant is

But from figure ΔABC ~ΔDBA

∴ The required magnitude of resultant becomes 

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 7

Let W denote the words in the English dictionary. Define the relation R by R = {(x, y) ∈ W × W| the words x and y have at least one letter in common.} Then R is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 7

Clearly ( x, x) ∈ R"x ∈ W . So R is relexive.
Let ( x, y) ∈ R ,  then ( y, x) ∈ R as x and y have at least one letter in common. So, R is symmetric.
But R is not transitive for example Let x = INDIA, y = BOMBAY and z = JOKER then ( x, y) ∈R (A is common) an d ( y, z) ∈ R (O is common) but ( x, z) ∉ R . (as no letter is common)

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 8

Suppose a population A has 100 observations 101, 102, ............., 200 and another population B has 100 obsevrations 151, 152, ................ 250. If VA and VB represent the variances of the two populations, respectively then is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 8

   (Here deviations are taken from the mean).
Since A and B both have 100 consecutive integers, therefore both have same standard deviation and hence the variance.

                  (As    is same in both the cases)

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 9

A particle has two velocities of equal magnitude  inclined to each other at an angle θ . If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then q is [2006]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 9

For two velocities u and u at an angle θ to each other the resultant is given by


Now in second case, the new resultant AE (i.e., R") bisects ∠CAB , therefore using angle bisector theorem in ΔABC , we get

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 10

A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4s prior to passing through P. If g = 10m/s 2 , then the height above the point P from where the body began to fall is [2006]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 10

Using   and 400 = 

Subtracting, we get 400 = 8g + 4gt  ⇒ t = 8 sec

 = 320

∴ Desired height = 320 + 400 = 720 m

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 11

The resultant of two forces Pn and 3n is a force of 7n. If the direction of 3n force were reversed, the resultant would be . The value of P is [2007]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 11

Given : Force P = Pn, Q = 3n, resultant R = 7n & P' = Pn,

Q' = (–3)n, R' =

We know that R2 = P2 + Q2 + 2PQ cosα
⇒ (7)2 = P2 + (3)2 + 2 × P × 3 cosα
⇒ 49 = P2 + 9 + 6P cosα
⇒ 40 = P2 + 6P cosα .....(i)

And= P2 + (–3)2 + 2P × –3 cosα
⇒ 19 = P2 + 9 – 6P cosα
⇒ 10 = P2 – 6P cosα.....(ii)
Adding (i) and (ii)  
50 = 2P2 ⇒ P2 = 25 ⇒ P = 5n.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 12

A particle just clears a wall of height b at a distance a and strikes the ground at a distance c from the point of projection.
The angle of projection is [2007]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 12

Let B be the top of the wall whose coordinates will be (a, b). Range (R) = c

B lies on the trajectory

⇒ 

The angle of projection,

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 13

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is [2007]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 13

Let the number of boys be x and that of girls be y.
⇒ 52x + 42y = 50(x + y) ⇒ 52x – 50x = 50y – 42y

⇒ 2x = 8y ⇒  and 

Required % of boys = 

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 14

A body weighing 13 kg is suspended by two strings 5m and 12m long, their other ends being fastened to the extremities of a rod 13m long. If the rod be so held that the body hangs immediately below the middle point, then tensions in the strings are [2007]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 14

 

∵ 132  = 5 2 + 122 ⇒ AB2 = AC2 + BC2
⇒ ∠ACB = 90°
Q m is mid point of the hypotenuse AB, therefore MA = MB = MC ⇒ ∠A = ∠ACM =θ
Applying  Lami’s theorem at C, we get

⇒ T1 = 13 sinθ and T2 = 13 cosθ

⇒ T1 = 5 kg and T2 = 12 kg

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 15

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a  and b ? [2008]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 15

Mean  of a, b, 8, 5, 10 is 6

= ⇒ a + b = 7 ...(i)

Variance of  a, b, 8, 5, 10 is 6.80

⇒ a2 – 12a + 36 + (1 –a )2 + 21= 34   [using eq. (i)]
⇒ 2a2 –14a + 24 = 0 ⇒ a2 – 7a + 12 = 0
⇒ a = 3  or   4      ⇒ b = 4   or   3
∴ The possible values of a and b are a = 3 and b =  4 or, a = 4 and b = 3

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 16

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “ x is a rational number if f y is a transcendental number”. [2008] Statement-1 : r is equivalent to either q or p Statement-2 : r is equivalent to  ~(p⇿~q).

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 16

(None) p :  x is an irrational number
q : y is a transcendental number
r : x is a rational number iff y is a transcendental number. clearly r :~p ⇿q
Let us use truth table to check the equivalence of ‘r’ and ‘q or p’; ‘r’ and ~ (p ⇿~ q)

From columns (1), (2) and (3), we observe, none of the these statements are equivalent to each other.
∴ Statement 1as well as statement 2 both are false.
∴ None of the options is correct.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 17

The statement p → (q→p) is equivalent to [2008]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 17

Let us make the truth table for the given statements, as follows :

From table we observe p → (q→p) is equivalent to p→(pνq)

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 18

Statement-1 : ~ ( p ⇿ ~q) is equivalent to p ⇿ q .
Statement-2 : ~ ( p ⇿ ~q) is a tautology [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 18

The truth table for the logical  statements, involved in statement 1, is as follows :

We observe the columns for ~ (p ⇿ ~q) and p ⇿ q are identical, therefore ~(p ⇿ ~q) is equivalent to p ⇿ q
But ~ (p ⇿ ~q) is not a tautology as all entries in its column are not T.
∴ Statement-1 is true but statement-2 is false.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 19

Statement-1 : The variance of first n even natural numbers 
Statement-2 : The sum of first n natural numbers isand the sum of squares of first n natural numbers is      [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 19

For the numbers 2, 4, 6, 8, ......., 2n

And Var = 

∴ Statement-1 is false.  Clearly, statement - 2 is true.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 20

If A, B and C are three sets such that A ∩ B = A∩C and A ∪ B= A∪C , then [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 20

Let x∈ A and x ∈ B ⇔ x ∈A∪B
⇔ x ∈ A∪C (∵ A ∪ B = A∪C) ⇔ x∈C
∴ B = C.
Let x ∈A and x ∈ B ⇔ x ∈ A∩B
⇔ x ∈ A∩C (∵A ∩ B = A∩C)
⇔ x∈C ∴ B=C

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 21

If the mean deviation of the numbers 1, 1 + d, 1 + 2d, .... 1 + 100d from their mean is 255, then d is equal to : [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 21

Mean = 

∵ Mean deviation from the mean = 255

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 22

Let S be a non-empty subset of R. Consider the following statement : P : There is a rational number x ∈ S such that x > 0.Which of the following statements is the negation of the statement P ? [2010]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 22

P : there is a rational number x ∈S such that x > 0
~ P : Every rational number  x ∈S satisfies x ≤ 0

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 23

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w};   and q are integers such that n, q ≠ 0 and qm = pn}.Then                                   [2010]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 23

x Ry need not implies yRx ∴ R is not symmetric and hence not an equivalence relation

Given qm = pn  

 qm = pn, ps= rq

⇒ ms = rn (transitive).

S is an equivalence relation.

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 24

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is
[2010]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 24

Variance of combined data

Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 25

 Let R be the set of real numbers.Statement-1: A = {(x, y) ∈ R × R : y – x is an integer} is an equivalence relation on R.
Statement-2: B = {(x, y) ∈ R ×  R : x = ay for some rational number a} is an equivalence relation on R. [2011]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) - 2 - Question 25

x – y is an integer.
∵  x – x = 0 is an integer ⇒ A is reflexive.
Let x – y is an integer ⇒  y – x is an integer
⇒  A is symmetric Let x – y, y – z are integers
⇒  x – y + y – z is also an integer
⇒  x – z is an integer ⇒  A is transitive
∴  A is an equivalence relation.
Hence statement 1 is true.
Also B can be considered as

xBy if a rational number

is a rational number

But    a rational number need not imply   a rational number because

is rational ⇒   is not rational

∴  B is not an equivalence relation.

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