10: Mathematics Baseline Assessment 2024 Term I STION 1 Which one of the following numbers is rational? $$ 2,3 ; \sqrt{-25} ; \pi ; \sqrt[3]{17} $$ Determine the value (in scientific notation) of: $$ 3,7 \times 10^{-7} \times 2 \times 10^{4} $$ Simplify: 1.3.1 $$ \quad \frac{3^{x}+3^{x+2}}{3^{x+3}} $$ 1.3 .2 $$ \sqrt{0,06 y^{4}+0,1 y^{4}} $$ 1.3 .3 $$ 3(x-3)(x+3)-(x-1)^{2} $$ STION 2 Factorise the following: 2.1.1 $$ 2 x^{2}+6 x-20 $$ 2.1.2 $$ 9 x(5 a-b)+2(b-5 a) $$ Solve for $$ x $$ : 2.2.1 $$ x(2 x-1)=0 $$ $$ 2.2 .2 \quad 27 \cdot 3^{x}=1 $$

Question

UpStudy AI · Accepted Answer

Certainly! Below are the solutions to each of the questions you've provided in the **Mathematics Baseline Assessment 2024 Term I**.

---

## **QUESTION 1**

### **1.1. Identify the Rational Number**

**Options:**
- $$2.3$$
- $$\sqrt{-25}$$
- $$\pi$$
- $$\sqrt[3]{17}$$

**Solution:**
- **$$2.3$$:** This is a rational number since it can be expressed as $$\frac{23}{10}$$, where both numerator and denominator are integers.
- **$$\sqrt{-25}$$:** This is not a real number (it's imaginary) and hence not rational.
- **$$\pi$$:** This is an irrational number.
- **$$\sqrt[3]{17}$$:** This is also an irrational number.

**Answer:** **$$2.3$$ is the rational number.**

---

### **1.2. Value in Scientific Notation**

**Problem:**
$$
3.7 \times 10^{-7} \times 2 \times 10^{4}
$$

**Solution:**
Multiply the coefficients and add the exponents:
$$
3.7 \times 2 = 7.4
$$
$$
10^{-7} \times 10^{4} = 10^{-3}
$$
So,
$$
7.4 \times 10^{-3}
$$

**Answer:** $$7.4 \times 10^{-3}$$

---

### **1.3. Simplifications**

#### **1.3.1. Simplify $$\frac{3^{x} + 3^{x+2}}{3^{x+3}}$$**

**Solution:**
$$
\frac{3^{x} + 3^{x+2}}{3^{x+3}} = \frac{3^{x}(1 + 3^{2})}{3^{x} \cdot 3^{3}} = \frac{3^{x} \times 10}{3^{x} \times 27} = \frac{10}{27}
$$

**Answer:** $$\frac{10}{27}$$

#### **1.3.2. Simplify $$\sqrt{0.06 y^{4} + 0.1 y^{4}}$$**

**Solution:**
Combine like terms:
$$
0.06 y^{4} + 0.1 y^{4} = 0.16 y^{4}
$$
Take the square root:
$$
\sqrt{0.16 y^{4}} = \sqrt{0.16} \times \sqrt{y^{4}} = 0.4 \times y^{2} = 0.4y^{2}
$$

**Answer:** $$0.4y^{2}$$

#### **1.3.3. Simplify $$3(x-3)(x+3) - (x-1)^{2}$$**

**Solution:**
Expand each term:
$$
3(x-3)(x+3) = 3(x^{2} - 9) = 3x^{2} - 27
$$
$$
(x-1)^{2} = x^{2} - 2x + 1
$$
Subtract the second expression from the first:
$$
3x^{2} - 27 - (x^{2} - 2x + 1) = 3x^{2} - 27 - x^{2} + 2x - 1 = 2x^{2} + 2x - 28
$$
Factor out the common factor:
$$
2x^{2} + 2x - 28 = 2(x^{2} + x - 14)
$$
Since $$x^{2} + x - 14$$ doesn’t factor neatly, the simplified form is:
$$
2(x^{2} + x - 14)
$$

**Answer:** $$2(x^{2} + x - 14)$$

---

## **QUESTION 2**

### **2.1. Factorisation**

#### **2.1.1. Factorise $$2x^{2} + 6x - 20$$**

**Solution:**
Factor out the greatest common factor (GCF) first:
$$
2x^{2} + 6x - 20 = 2(x^{2} + 3x - 10)
$$
Now, factor $$x^{2} + 3x - 10$$:
Find two numbers that multiply to $$-10$$ and add to $$3$$: $$5$$ and $$-2$$.

$$
x^{2} + 3x - 10 = (x + 5)(x - 2)
$$

Combine with the GCF:
$$
2(x + 5)(x - 2)
$$

**Answer:** $$2(x - 2)(x + 5)$$

#### **2.1.2. Factorise $$9x(5a - b) + 2(b - 5a)$$**

**Solution:**
Notice that $$b - 5a = -(5a - b)$$. So,

$$
9x(5a - b) + 2(b - 5a) = 9x(5a - b) - 2(5a - b) = (5a - b)(9x - 2)
$$

**Answer:** $$(5a - b)(9x - 2)$$

---

### **2.2. Solve for $$x$$**

#### **2.2.1. Solve $$x(2x - 1) = 0$$**

**Solution:**
Set each factor equal to zero:
$$
x = 0 \quad \text{or} \quad 2x - 1 = 0
$$
Solve $$2x - 1 = 0$$:
$$
2x = 1 \Rightarrow x = \frac{1}{2}
$$

**Answer:** $$x = 0$$ or $$x = \frac{1}{2}$$

#### **2.2.2. Solve $$27 \cdot 3^{x} = 1$$**

**Solution:**
Express $$27$$ as a power of $$3$$:
$$
27 = 3^{3}
$$
So,
$$
3^{3} \cdot 3^{x} = 3^{x + 3} = 1
$$
Recall that $$3^{0} = 1$$:
$$
x + 3 = 0 \Rightarrow x = -3
$$

**Answer:** $$x = -3$$

---

I hope these solutions help you understand each problem thoroughly! If you have any further questions or need additional explanations, feel free to ask.
**Mathematics Baseline Assessment 2024 Term I Solutions** --- **Question 1:** 1. **Identify the Rational Number:** - **Options:** - $$2.3$$ - $$\sqrt{-25}$$ - $$\pi$$ - $$\sqrt[3]{17}$$ - **Answer:** $$2.3$$ is the rational number. 2. **Value in Scientific Notation:** $$ 3.7 \times 10^{-7} \times 2 \times 10^{4} = 7.4 \times 10^{-3} $$ - **Answer:** $$7.4 \times 10^{-3}$$ 3. **Simplifications:** - **1.3.1:** $$ \frac{3^{x} + 3^{x+2}}{3^{x+3}} = \frac{10}{27} $$ - **Answer:** $$\frac{10}{27}$$ - **1.3.2:** $$ \sqrt{0.06 y^{4} + 0.1 y^{4}} = 0.4 y^{2} $$ - **Answer:** $$0.4 y^{2}$$ - **1.3.3:** $$ 3(x-3)(x+3) - (x-1)^{2} = 2(x^{2} + x - 14) $$ - **Answer:** $$2(x^{2} + x - 14)$$ --- **Question 2:** 1. **Factorisation:** - **2.1.1:** $$ 2x^{2} + 6x - 20 = 2(x - 2)(x + 5) $$ - **Answer:** $$2(x - 2)(x + 5)$$ - **2.1.2:** $$ 9x(5a - b) + 2(b - 5a) = (5a - b)(9x - 2) $$ - **Answer:** $$(5a - b)(9x - 2)$$ 2. **Solve for $$x$$:** - **2.2.1:** $$ x(2x - 1) = 0 \Rightarrow x = 0 \text{ or } x = \frac{1}{2} $$ - **Answer:** $$x = 0$$ or $$x = \frac{1}{2}$$ - **2.2.2:** $$ 27 \cdot 3^{x} = 1 \Rightarrow x = -3 $$ - **Answer:** $$x = -3$$ --- I hope these solutions help you understand each problem thoroughly! If you have any further questions or need additional explanations, feel free to ask.

Solución de tutoría real

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions