Math Problem Set: Algebra, Calculus, and Probability for High School Students

Math Problem Statement

A set of math problems including algebraic functions, calculus, probability, Bayes' theorem, and others

Solution

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Q5. If $f$ and $g$ are real-valued functions defined by $f(x) = x^2 + 7$ and $g(x) = 3x + 5$ , find the values of $f(-2) + g(-1)$ .

$f(x) = x^2 + 7$
$g(x) = 3x + 5$

Now let's evaluate $f(-2)$ and $g(-1)$ :

$f(-2) = (-2)^2 + 7 = 4 + 7 = 11$
$g(-1) = 3(-1) + 5 = -3 + 5 = 2$

So, $f(-2) + g(-1) = 11 + 2 = 13$ .

Q6. Evaluate $\int \frac{4x + 3}{x^2 - x - 2} \, dx$

This is an integral involving partial fraction decomposition.

First, factor the denominator:

$x^2 - x - 2 = (x - 2)(x + 1)$

Now, express the fraction as a sum of simpler fractions:

$\frac{4x + 3}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}$

Solve for $A$ and $B$ by equating:

$4x + 3 = A(x + 1) + B(x - 2)$

Expand and solve for $A$ and $B$ .

Q7. Define the modulus function and write its domain and range.

The modulus function, also known as the absolute value function, is defined as:

\begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$ - **Domain**: The set of all real numbers, $$(-\infty, \infty)$$. - **Range**: Non-negative real numbers, $$[0, \infty)$$. --- ### Q8. **A ball is drawn from a bag containing 5 white and 7 black balls. What is the probability of drawing one white and one black ball?** This question assumes two balls are drawn. The total number of balls is $$5 + 7 = 12$$. To find the probability of drawing one white and one black: $$P(\text{white then black}) = \frac{5}{12} \times \frac{7}{11} = \frac{35}{132}$$ Similarly, $$P(\text{black then white}) = \frac{7}{12} \times \frac{5}{11} = \frac{35}{132}$$. Thus, the total probability is: $$P(\text{one white, one black}) = \frac{35}{132} + \frac{35}{132} = \frac{70}{132} = \frac{35}{66}$$ --- ### Q9. **Define Bayes' theorem for mutually exclusive events.** Bayes' theorem is used for conditional probabilities, not mutually exclusive events. However, for conditional probability, it states: $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ For mutually exclusive events $$A$$ and $$B$$, $$P(A \cap B) = 0$$, so Bayes' theorem does not apply. --- Would you like more details or assistance on any specific question? Let me know if you'd like to focus on a particular problem. Here are 5 related questions to explore further: 1. How would you solve partial fraction decomposition for the integral in Q6? 2. What are some applications of Bayes' theorem in real-world scenarios? 3. How does the domain and range of other functions compare to the modulus function? 4. Can you find the probability of drawing 2 white or 2 black balls in Q8? 5. What happens to the probability if balls are drawn with replacement in Q8? **Tip**: Always double-check conditions (like replacement or mutual exclusivity) in probability problems to avoid common mistakes.

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Math Problem Analysis

Mathematical Concepts

Algebra
Calculus
Probability
Bayes' Theorem
Sets and Venn Diagrams
Sequences

Formulas

f(x) = x^2 + 7
g(x) = 3x + 5
Modulus function f(x) = |x|
Bayes' theorem P(A|B) = (P(B|A)P(A))/P(B)
Sum of sequences formula

Theorems

Bayes' theorem
Fundamental theorem of calculus

Suitable Grade Level

Grades 10-12

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