Sets

Step 1: Solve the first inequality $x%5E2%20-%207x%20%2B%206%20%5Cle%200$ .
Factor the quadratic expression:

x%5E2%20-%207x%20%2B%206%20%3D%20(x%20-%201)(x%20-%206)

.
The inequality becomes

(x%20-%201)(x%20-%206)%20%5Cle%200

.
The solution to this inequality is

1%20%5Cle%20x%20%5Cle%206

.
Step 2: Solve the second inequality $x%5E2%20-%203x%3E0$ .
Factor the quadratic expression:

x%5E2%20-%203x%20%3D%20x(x%20-%203)

.
The inequality becomes

x(x%20-%203)%3E0

.
The solution to this inequality is

x%3C0

x%3E3

.
Step 3: Find the integers $x$ that satisfy both inequalities.
We need integers

x

such that

(1%20%5Cle%20x%20%5Cle%206)

and

(x%3C0

x%3E3)

.
The integers satisfying both conditions are

x%20%3D%204%2C%205%2C%206

.
Step 4: Determine the number of elements in the set $S$ .
The set

S%20%3D%20%5C%7B4%2C%205%2C%206%5C%7D

.
The number of elements in the set

S

is 3.
Step 5: Conclusion.
The number of elements in the set

S

is 3.

To solve the given problem, we need to analyze the sets $A$ and $B$ defined as follows:

$A%20%3D%20%5C%7Bx%20%5Cin%20%5Cmathbb%7BR%7D%20%5Cmid%20%5Csin%5E%7B-1%7D(%5Csqrt%7Bx%5E2%2Bx%2B1%7D)%20%5Cin%20%5Cleft%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D%5C%7D$
$B%20%3D%20%5C%7By%20%5Cin%20%5Cmathbb%7BR%7D%20%5Cmid%20y%20%3D%20%5Csin%5E%7B-1%7D(%5Csqrt%7Bx%5E2%2Bx%2B1%7D)%2C%20x%20%5Cin%20A%5C%7D$

The function $%5Csin%5E%7B-1%7D(z)$ is defined in the domain $%5B-1%2C1%5D$ . Therefore, $%5Csqrt%7Bx%5E2%2Bx%2B1%7D$ must satisfy:
$-1%20%5Cleq%20%5Csqrt%7Bx%5E2%2Bx%2B1%7D%20%5Cleq%201$
Since $%5Csqrt%7Bx%5E2%2Bx%2B1%7D%20%5Cgeq%200$ , the inequality simplifies to:
$0%20%5Cleq%20%5Csqrt%7Bx%5E2%2Bx%2B1%7D%20%5Cleq%201$
Squaring both sides, we get:
$0%20%5Cleq%20x%5E2%2Bx%2B1%20%5Cleq%201$
Simplifying further:\
$0%20%5Cleq%20x%5E2%2Bx%20%5Cleq%200$
This gives:\
$x%5E2%2Bx%20%3D%200$
Solving for $x$ , we find:
$x(x%2B1)%20%3D%200$
Thus, $x%20%3D%200$ or $x%20%3D%20-1$ . Therefore, $A%20%3D%20%5C%7B-1%2C%200%5C%7D$ .

For set $B$ , since $y%20%3D%20%5Csin%5E%7B-1%7D(%5Csqrt%7Bx%5E2%2Bx%2B1%7D)$ and $x%20%5Cin%20A$ , the values are:\
1. For $x%20%3D%200$ , $%5Csqrt%7B0%5E2%2B0%2B1%7D%20%3D%201$ , so $y%20%3D%20%5Csin%5E%7B-1%7D(1)%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D$ .
2. For $x%20%3D%20-1$ , $%5Csqrt%7B(-1)%5E2%2B(-1)%2B1%7D%20%3D%200$ , so $y%20%3D%20%5Csin%5E%7B-1%7D(0)%20%3D%200$ .
Thus, $B%20%3D%20%5Cleft%5C%7B0%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5C%7D$ .

Now, calculate $A%5EC%20%5Ccap%20B$ , where $A%5EC$ is the complement of $A$ in $%5Cmathbb%7BR%7D$ , which means all real numbers except $-1$ and $0$ . Therefore:
$B%20%3D%20%5Cleft%5C%7B0%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5C%7D$ , and then $A%5EC%20%5Ccap%20B%20%3D%20%5Cleft%5C%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5C%7D$ . However, since only $%5C%7B%5Cfrac%7B%5Cpi%7D%7B3%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5C%7D$ fall within the complement in relation to $y$ , we have:

$A%5EC%20%5Ccap%20B%20%3D%20%5Cleft%5B%5Cfrac%7B%5Cpi%7D%7B3%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright%5D$

This corresponds to the correct answer: $A%5EC%20%5Ccap%20B%20%3D%20%5Cleft%5B%5Cfrac%7B%5Cpi%7D%7B3%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright%5D$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Sets

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs