Sets

M%3D%5Cbegin%7Bbmatrix%7Dsin%5E%7B4%7D%5C%2C%5Ctheta%20%26-1-sin%5E%7B2%7D%5C%2C%5Ctheta%5C%5C%201%2Bcos%5E%7B2%7D%5C%2C%5Ctheta%20%26cos%5E%7B4%7D%5C%2C%5Ctheta%5Cend%7Bbmatrix%7D%3D%5Calpha%20l%2B%5Cbeta%20M%5E%7B-1%7D

%5Ctherefore%20det%20%5Cleft(M%5Cright)%20%3D%20%7CM%7C%20%3D%20sin%5E%7B4%7D%5Ctheta%5Ccdot%20cos%5E%7B4%7D%5Ctheta%2Bsin%5E%7B2%7D%5Ctheta%20cos%5E%7B2%7D%5Ctheta%2B2

%3D%5Cleft%5C%7B%5Cleft(sin%5E%7B2%7D%5Ctheta%20cos%5E%7B2%7D%5Ctheta%2B%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B2%7D%2B%5Cfrac%7B7%7D%7B4%7D%5Cright%5C%7D

%5Cbegin%7Bbmatrix%7Dsin%5E%7B4%7D%5Ctheta%26-1-sin%5E%7B2%7D%5Ctheta%5C%5C%201%2Bcos%5E%7B2%7D%5Ctheta%26cos%5E%7B4%7D%5Ctheta%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D%5Calpha%260%5C%5C%200%26%5Calpha%5Cend%7Bbmatrix%7D%2B%5Cfrac%7B%5Cbeta%7D%7B%5Cleft%7CM%5Cright%7C%7D%5Cbegin%7Bbmatrix%7Dcos%5E%7B4%7D%5Ctheta%261%2Bsin%5E%7B2%7D%5Ctheta%5C%5C%20-1-cos%5E%7B2%7D%5Ctheta%26sin%5E%7B4%7D%5Ctheta%5Cend%7Bbmatrix%7D

%5Ctherefore%20%5Calpha%3Dcos%5E%7B4%7D%5Ctheta%2Bsin%5E%7B4%7D%5Ctheta%3D1-%5Cfrac%7B1%7D%7B2%7D%5Cleft(sin%5E%7B2%7D%5C%2C2%5Ctheta%5Cright)

and

%5Cbeta%3D-%5Cleft%7CM%5Cright%7C

%3D-%5Cleft%5C%7B%5Cleft(sin%5E%7B2%7D%5C%2C%5Ctheta%5Ccdot%20cos%5E%7B2%7D%5C%2C%5Ctheta%2B%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B2%7D%2B%5Cfrac%7B7%7D%7B4%7D%5Cright%5C%7D

%5Ctherefore%20%5Calpha_%7Bmin%7D%3D%5Cfrac%7B1%7D%7B2%7D

and

%5Cbeta_%7Bmin%7D%3D-%5Cfrac%7B37%7D%7B16%7D

%5Ctherefore%20%5Calpha%5E*%2B%5Cbeta%5E*%3D%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B37%7D%7B16%7D%3D-%5Cfrac%7B29%7D%7B16%7D

Step 1: Number of elements in S₁:
The set $S_1$ contains triples of the form $(i%2C%20j%2C%20k)$ , where $i%2C%20j%2C%20k%20%5Cin%20%5C%7B1%2C%202%2C%20%5Cdots%2C%2010%5C%7D$ .
The number of choices for $i$ , $j$ , and $k$ are 10 each.
Thus, the total number of elements in $S_1$ is:
$n_1%20%3D%2010%20%5Ctimes%2010%20%5Ctimes%2010%20%3D%201000$ .

Step 2: Number of elements in S₂:
The set $S_2$ contains pairs of the form $(i%2C%20j)$ where $1%20%5Cleq%20i%20%3C%20j%20%2B%202%20%5Cleq%2010$ and both $i$ and $j$ are in $%5C%7B1%2C%202%2C%20%5Cdots%2C%2010%5C%7D$ .
To find the total number of valid pairs $(i%2C%20j)$ , observe the following:
For each value of $i$ , the value of $j$ can range from $i%20%2B%201$ to $10$ , subject to the constraint $i%20%2B%20j%20%2B%202%20%5Cleq%2010$ .
Thus, for $i%20%3D%201$ , the possible values of $j$ are 2 through 9 (8 choices),
for $i%20%3D%202$ , the possible values of $j$ are 3 through 8 (7 choices), and so on.
Summing the choices gives the total number of elements in $S_2$ :
$9%20%2B%208%20%2B%207%20%2B%206%20%2B%205%20%2B%204%20%2B%203%20%2B%202%20%3D%2044$ .

Step 3: Number of elements in S₃:
The set $S_3$ contains quadruples of the form $(i%2C%20j%2C%20k%2C%20l)$ , where $1%20%5Cleq%20i%20%3C%20j%20%3C%20k%20%3C%20l%20%5Cleq%2010$ .
This is equivalent to choosing 4 distinct elements from a set of 10 elements, which can be done in $%5Cbinom%7B10%7D%7B4%7D$ ways.
Using the combination formula:
$%5Cbinom%7B10%7D%7B4%7D%20%3D%20%5Cfrac%7B10%20%5Ctimes%209%20%5Ctimes%208%20%5Ctimes%207%7D%7B4%20%5Ctimes%203%20%5Ctimes%202%20%5Ctimes%201%7D%20%3D%20210$ .

Step 4: Number of elements in S₄:
The set $S_4$ contains quadruples of the form $(i%2C%20j%2C%20k%2C%20l)$ , where $i%2C%20j%2C%20k%2C%20l$ are distinct elements in $%5C%7B1%2C%202%2C%20%5Cdots%2C%2010%5C%7D$ .
This is equivalent to choosing 4 distinct elements from a set of 10, and then arranging them in order (since the order matters).
The number of ways to do this is given by the permutation formula:
$P(10%2C%204)%20%3D%20%5Cfrac%7B10!%7D%7B(10-4)!%7D%20%3D%2010%20%5Ctimes%209%20%5Ctimes%208%20%5Ctimes%207%20%3D%205040$ .

Step 5: Final Answer:
The correct statements are:
A) $n_1%20%3D%201000$
B) $n_2%20%3D%2044$
D) $%5Cfrac%7Bn_4%7D%7B12%7D%20%3D%20420$ , since $%5Cfrac%7B5040%7D%7B12%7D%20%3D%20420$ .

About Sets - JEE-ADVANCED

Sets is a vital chapter for JEE-ADVANCED aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Sets PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Sets carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

About Sets - JEE-ADVANCED

Frequently Asked Questions

Why focus on Sets PYQs?

How to best use this analysis?

Sets

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

About Sets - JEE-ADVANCED

Frequently Asked Questions

Why focus on Sets PYQs?

How to best use this analysis?

Chapter Questions
3 MCQs