Let's go through each question step-by-step and solve them.

Question 1

Solve the inequality: $x^2 - 4 < 0$ To solve, we rewrite it as: $x^2 < 4$ Taking the square root on both sides, we get: $-2 < x < 2$ So the solution is $x \in (-2, 2)$, which corresponds to:

• Answer: B) $(-2, 2)$

Question 2

This question involves identifying the correct number of signs in the table for sine and cosine in each quadrant. Here's the general rule for trigonometric signs:

• 1st Quadrant: Sine (+), Cosine (+)
• 2nd Quadrant: Sine (+), Cosine (-)
• 3rd Quadrant: Sine (-), Cosine (-)
• 4th Quadrant: Sine (-), Cosine (+)

Counting the signs that are positive and negative across quadrants:

• There are 3 instances where signs are positive (one for each "+" symbol in each quadrant).

So the answer is:

• B) 3 ta

Question 3

For $h(x) = x + \frac{1}{x}$, we need to find $h\left(\frac{1}{2}\right)$.

Substitute $x = \frac{1}{2}$ into the function: $h\left(\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{\frac{1}{2}}$ $= \frac{1}{2} + 2 = 2.5$

So the answer is:

• D) 2.5

Question 4

Determine the truth or falsehood of statements about the function $y = x^2 + 2x + 4$.

Analysis:

1. Domain: The function $y = x^2 + 2x + 4$ is defined for all real values of $x$, so the statement "Aniqlanish sohasi $(-∞, ∞)$ bo'ladi" is true.

2. Minimum Value: This function is a parabola opening upwards, so it has a minimum value. Completing the square: $y = (x+1)^2 + 3$ The minimum value is $y = 3$ when $x = -1$, so the statement "Eng kichik qiymatga $x = -1$ da erishadi" is true.

3. No Zeros: The equation $x^2 + 2x + 4 = 0$ has no real roots (its discriminant is negative), so the function has no real zeros. Therefore, the statement "Funksiya nollarga ega emas" is true.

The answer is:

• A) to'g'ri, to'g'ri, to'g'ri

Let me know if you would like further explanations on any of the steps.

Additional Questions to Consider

1. How would the solution change in Question 1 if the inequality was $x^2 - 4 \leq 0$?
2. What are the general sign conventions for other trigonometric functions (like tangent or cotangent) in each quadrant?
3. How does the function $h(x) = x + \frac{1}{x}$ behave as $x \to 0$?
4. For Question 4, how would you verify the minimum of $y = x^2 + 2x + 4$ by taking derivatives?
5. How would the answer in Question 4 change if the function had a negative leading coefficient?

Tip

When solving inequalities involving squares, look for boundaries where the expression changes sign by identifying zero points.