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 $%5Cfrac%7B5040%7D%7B12%7D%20%3D%20420$ .

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Sets

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Chapter Questions
3 MCQs