https://r2.erweima.ai/mathbot/image/1725621219891ztztq9rf.jpg

I can help you solve the questions from this exam image. Let's break down the sections and approach them accordingly. Here’s how we can proceed with the various parts:

### Section B
This section includes 9 questions. Let's start solving them part by part.

---

#### (i) For the quadratic equation \( 3(x^2 - 1) = x - 4 \):
- **(a) Write in standard form**:
  \[
  3(x^2 - 1) = x - 4 \implies 3x^2 - 3 = x - 4 \implies 3x^2 - x + 1 = 0
  \]
  This is the standard form: \( 3x^2 - x + 1 = 0 \).

- **(b) Find values of \( \alpha \) and \( \beta \)**:
  We need to solve this quadratic equation using the quadratic formula:
  \[
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  \]
  For the equation \( 3x^2 - x + 1 = 0 \), \( a = 3 \), \( b = -1 \), and \( c = 1 \). Now compute the discriminant and roots.

Let me compute that.

#### Solving the quadratic equation:
I'll compute the roots using the formula.The discriminant is \(-11\), which means the equation has complex roots.

The solutions for the quadratic equation \( 3x^2 - x + 1 = 0 \) are:
\[
\alpha = 0.167 + 0.553i, \quad \beta = 0.167 - 0.553i
\]

---

#### (ii) Solve \( 4^x + 4^{-x} = 10 \):
This is an exponential equation. To solve it, let us define \( y = 4^x \), so the equation becomes:

\[
y + \frac{1}{y} = 10
\]

Multiplying both sides by \( y \) and solving for \( y \), we get a quadratic equation.

Let me solve that.The solutions to the equation \( y + \frac{1}{y} = 10 \) are:

\[
y = 5 - 2\sqrt{6}, \quad y = 5 + 2\sqrt{6}
\]

Since \( y = 4^x \), we now solve for \( x \) by taking the logarithm of both sides. Let's compute the values of \( x \).The solutions for \( x \) in the equation \( 4^x + 4^{-x} = 10 \) are:

\[
x = -1.65 \quad \text{or} \quad x = 1.65
\]

---

This addresses part of the question. Would you like me to proceed with more sections or explain any part of the solution in more detail?

### Further Related Questions:
1. How do we find the roots of more complex quadratic equations?
2. How can we solve exponential equations graphically?
3. What are the properties of equations with complex roots?
4. How can discriminants help determine the nature of quadratic equation roots?
5. What is the significance of logarithms in solving exponential equations?

### Tip:
When solving quadratic equations, always calculate the discriminant first to understand whether the roots will be real or complex.

Solve the mathematical problems presented in the SSC-II Mathematics exam, covering topics like quadratic equations, exponential equations, arithmetic mean, and geometry.

Detailed solutions to the SSC-II Mathematics exam covering key concepts in quadratic equations, exponential equations, arithmetic mean, and geometry. Ideal for Grade 10 students preparing for their exams.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation