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JEE Advanced Level Test: Probability- 2 - JEE MCQ


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30 Questions MCQ Test - JEE Advanced Level Test: Probability- 2

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JEE Advanced Level Test: Probability- 2 - Question 1

The equation of line of intersection of the planes x + 2y + z = 3 and 6x + 8y + 3z = 13 can be written as

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 1

Let the d.r’s of a required line be a, b and c. Since, the normal to the given planes x + 2y + z = 3 and 6x + 8y + 3z = 13 are perpendicular to the line.
∴ a +2b+c = 0 and 6a + 8b + 3c = 0
⇒ 
⇒ 
or 
Also, line passes through (2, -1, 3).

JEE Advanced Level Test: Probability- 2 - Question 2

The direction cosines of two lines are such that l + m + n = 0, l2 + m2 – n2 = 0, then angle between them is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 2

l + m + n = 0
and  l2 + m2 - n2 = 0
l2 + m2 (- l - m)2 = 0
2lm = 0  (l = 0 or m = 0)
If l = 0 then n = - m
l : m : n = 0 : l : -1 
If m = 0 then n = - l 
l : m : n = l : 0 : -1 


⇒ θ = π/3

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JEE Advanced Level Test: Probability- 2 - Question 3

The point of intersection of the lines  and 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 3

Find general pt. of both line and equate them

⇒ (3λ - 1, 5λ - 3, 7λ - 5 ) .......(i)

⇒ (μ +2, 3μ + 4, 5μ + 6) .......(ii)
on solving λ = 1/2,  & μ = -3/2
on putting λ = 1/2, & μ = -3/2 we get 

JEE Advanced Level Test: Probability- 2 - Question 4

If the straight lines x = 1 + s, y = - 3 - λs, z = 1 + λs and x = t/2, y = 1 + t, z = -t + 2 with parameters s and t respectively are coplanar, then λ equals

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 4

Given line can be rewritten as

and 

since two line are coplanar, then

λ = -2

JEE Advanced Level Test: Probability- 2 - Question 5

Shortest distance between the line  and 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 5

Given, lines can be written as

and 
∴ Requires SD =

Where 

∴ Requires SD 

JEE Advanced Level Test: Probability- 2 - Question 6

If P(B) = 3/4,  = 1/3  = 1/3, then 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 6


JEE Advanced Level Test: Probability- 2 - Question 7

If events are independent and P(A) = 1/3, P(B) = 1/3, P(C) = 1/4 then   is equal to

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 7



 

JEE Advanced Level Test: Probability- 2 - Question 8

A fair die is tossed eight times. The probability that a third six is observed on the eight throw is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 8

Required probability 

JEE Advanced Level Test: Probability- 2 - Question 9

For k = 1, 2, 3, the box Bk contains k red balls and (k + 1) white balls. Let P(B1) = 1/2, P(B2) = 1/3, P(B3) = 1/6. A box is selected at random and a ball is drawn from it. If a red ball is drawn, then probability that it has come from box B2 is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 9

Use Baye’s Theorem formula


A : Red ball is drawn






= 14/39

JEE Advanced Level Test: Probability- 2 - Question 10

Let A and B be two event such that  stands for complement of event A. Then events A and B are

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 10





Independent event but not equally likely

JEE Advanced Level Test: Probability- 2 - Question 11

Three of the six vertices of a regular hexagon are chosen at random. The probability that triangle with three chosen vertices is an equilateral triangle is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 11

Total No. of ways = 6C3
No. of favorable ways = 2

JEE Advanced Level Test: Probability- 2 - Question 12

If A and B are two events such that P(A∪B) = 3/4, P(A∩B) 1/4,  = 2/3, then  

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 12


P(A∪B) = P(A) = P(A) + P(B) - P(A∩B)
P(B) = 2/3
 = P(B) - P(A∩B) = 

JEE Advanced Level Test: Probability- 2 - Question 13

Three critics review a book. Odds in favour of book are 5 : 2, 4 : 3 and 3 : 4, respectively for the three critics. The probability that majority are in favour of the book is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 13

Probability that first critic favour is P(ε1) = 5/7 
Similarly, P(ε2) = 4/7  P(ε3) = 3/7 
Majority are in favour if at least two favour 

JEE Advanced Level Test: Probability- 2 - Question 14

Out of a set of integers given by {1, 2, 3, …. 30}, three numbers are selected at random. Find the probability the sum of the number chosen is divisible by 3.

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 14

number of type 3k → 10
3k + 1 → 10
3k + 2 → 10
Either all the no. are of the same type or one No. from each type

JEE Advanced Level Test: Probability- 2 - Question 15

A is a set containing n element. A subset P of A is chosen at random. The set A is reconstructed by replacing the element of P. A set ‘Q’ is again chosen at random. Find probability such that P∩Q = φ.

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 15

For 1 element not to be present in P∩Q possibilities for n element ⇒ Total
No. of fovourable cases = 3n

JEE Advanced Level Test: Probability- 2 - Question 16

If the integers m and n are chosen at random from 1 to 100, then probability that a no. of form 7m + 7n is divisible by 5 equals 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 16

Consider last digit of power of 7 
last digit
74k → 1
74k+1 →​ 7
74K+2 →​ 9
74k+3 →​ 3
7m + 7n is divided by 5 if
(i) m → 4k + 1, n → 4k + 3
(ii) m→ 4k + 2 and n → 4k
(iii) m→ 4k+3 andn→ 4k+1
(iv) m → 4k and n → 4k + 2 

JEE Advanced Level Test: Probability- 2 - Question 17

If  are the probability of three mutually exclusive and exhaustive events, then the set of all value of p is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 17

Since events are mutually exclusive and exhaustive






Hence the set of value satisfying all the above inequalities are 

JEE Advanced Level Test: Probability- 2 - Question 18

A man alternately tosses a coin and throws a dice beginning with the coin. The probability that he gets a head in the coin before he gets a 5 or 6 in the dice is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 18

Probability of getting head = 1/2 and probability of throwing 5 or 6 with a dice = 2/6 1/3 . He starts with a coin and alternately tosses the coin and throws the dice and he will if he get a head before he get 5 or 6.


JEE Advanced Level Test: Probability- 2 - Question 19

Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers are divisible by 2 and 3 is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 19

The numbers should be divisible by 6. Thus, the number of favourable ways is 16C3 (as there are 16 numbers in first 100 natural numbers, divisible by 6).
Required probability is 

JEE Advanced Level Test: Probability- 2 - Question 20

A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p, q and 1/2 respectively. If the probability that the student is successful is 1/2, then  

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 20

Let A, B and C be the events that the student is successful in test I, II and III respectively, then P(the student is successful)

P[A∩B∩C')∪(A∩B'∩C)∪(A∩B∩C)]
= P(A∩B∩C') + P(A∩B'∩C) + P(A∩B∩C) 
P(A).P(B).P(C') + P(A)P(B')P(C) + P(A)P(B)P(C)   
{∴ A, B, C are independent}



This equation has infinitely many values of p and q. 

JEE Advanced Level Test: Probability- 2 - Question 21

It is given that events A and B are such that P(A) = 1/4, P(A/B) = 1/2 and P(B/A) = 2/3, then P (B) is 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 21

P(A/B) = P(A⋂B)/P(B)
P(B/A) = P(B⋂A)/P(A)
2/3 = P(B⋂A)/(1/4)
P(A⋂B) = 2/3*1/4
= 1/6
Therefore, P(A/B) = P(A⋂B)/P(B)
1/2 = 1/6P(B)
P(B) = 1/3

JEE Advanced Level Test: Probability- 2 - Question 22

Out of a set of integers given by {1, 2, 3, …. 30}, three numbers are selected at random. Find the probability the sum of the number chosen is divisible by 3.

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 22

number of type 3k → 10
3k + 1 → 10
3k + 2 → 10
Either all the no. are of the same type or one No. from each type

JEE Advanced Level Test: Probability- 2 - Question 23

The equation of plane passing through the point (0, 7, -7) and containing the line 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 23

The equation of plane passing through (0, 7, -7) is a(x – 0) + b(y – 7) + c(z + 7) = 0
Plane contains line and passes through (-1, 3, -2)
∴ a(-1) + b(3 – 7) + c(-2 + 7) = 0
-3a + 2b + c = 0
∴ a : b : c = 1 : 1 : 1
x + y + z = 0 

JEE Advanced Level Test: Probability- 2 - Question 24

The angle between line  and plane 3x – 2y + 6z = 0 is (μ is scalar)

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 24

D. R’s of line are 2 : 1 : 2
D.R’s of Normal to plane is :3 : -2 : 6
Angle between line and plane

JEE Advanced Level Test: Probability- 2 - Question 25

Let P(3, 2, 6) be a point in space and Q be a point on line  Then value of m for which the vector  is parallel to the plane x-4y + 3z = 1 is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 25

 = i(- 2 - 3μ) + j(μ - 3) + k(5μ - 4)
 is parallel to x - 4y + 3z = 1
1(- 2 - 3μ) + 4(μ - 3) + 3(5μ - 4) = 0
μ = 1/4

JEE Advanced Level Test: Probability- 2 - Question 26

The equation of the plane through the point (-1, 2, 0) and parallel to the line  and 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 26


JEE Advanced Level Test: Probability- 2 - Question 27

Let A (1,1,1) , B (2, 3, 5) and C (-1, 0, 2) be three points, then equation of a plane parallel to the plane ABC which is at a distance 2 from origin is 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 27


A (1,1,1), B (2,3,5), C(-1,0,2) direction ratios of AB are < 1,2,4 >
Therefore, direction ratios of normal to plane ABC are < 2, -3,1 >
As a result, equation of the required plane is 2x – 3y +z = k then 

Hence, equation of the required plane is 

JEE Advanced Level Test: Probability- 2 - Question 28

The distance between the line  and the plane 

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 28


Therefore, the line and the plane are parallel. A point on the line is (2, -2, 3). Required distance = distance of (2, -2, 3) from the given plane x + 5y + z - 5 = 0

JEE Advanced Level Test: Probability- 2 - Question 29

If the distance between the plane x - 2y + z = d and the plane containing the line  and  then |d| is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 29

Equation of plane containing the given lines is 
⇒ (x - 1) (-1) - (y - 2) (-2) + (z - 3) (-1) = 0
⇒ -x  + 1 + 2y - 4 - z + 3
⇒ -x + 2y - z = 0
Given plane is
x - 2y + z = d .....(ii)
Eqs. (i) and (ii) are parallel
Now, distance between planes

⇒ |d| = 6

JEE Advanced Level Test: Probability- 2 - Question 30

A plane π passes through the point (1, 1, 1). If b, c, a are the direction ratios of a normal to the plane, where a, b, c (a < b < c) are the prime factors of 2001, then the equation of the plane π is

Detailed Solution for JEE Advanced Level Test: Probability- 2 - Question 30

2001 = 3 x 23 x 29 and (3+23+29)
= 55 ⇒ a = 3, b = 23, c = 29

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