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

About Sets - AP-EAPCET

Sets is a vital chapter for AP-EAPCET 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
2 MCQs

Official Solution

Official Solution

About Sets - AP-EAPCET

Frequently Asked Questions

Why focus on Sets PYQs?

How to best use this analysis?

Sets

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Sets - AP-EAPCET

Frequently Asked Questions

Why focus on Sets PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs