https://file.aiquickdraw.com/mathbot/file/1730352585023ginkuz55.jpg

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.

Solve the following mathematical problems in the image provided:
1. Solve the inequality x^2 - 4 < 0.
2. Determine the number of positive signs for sine and cosine in each quadrant based on the given table.
3. Calculate h(1/2) for the function h(x) = x + 1/x.
4. Identify the truth or falsehood of statements about the function y = x^2 + 2x + 4.

Explore solutions to Grade 10 math problems involving inequalities, trigonometric sign analysis in each quadrant, function evaluation, and determining the truth of statements about quadratic functions. Step-by-step explanations and solutions provided.

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