The First-Class Testing Center (FCTC) Math Problems 2024

The First-Class Testing Center (FCTC) Math Problems
2024 include a variety of questions designed to test
different mathematical skills.

Below are a series of questions followed by their detailed solutions.



Question 1: Algebra

Problem: Solve for \(x\) in the equation \(2x + 3 = 15\).



Solution:

To isolate \(x\), first subtract 3 from both sides of the equation:

\[ 2x + 3 - 3 = 15 - 3 \]

\[ 2x = 12 \]

Next, divide both sides by 2:

\[ \frac{2x}{2} = \frac{12}{2} \]

\[ x = 6 \]



Question 2: Geometry

Problem: Find the area of a triangle with a base of 10 units and a height of 5 units.



Solution:

The area \(A\) of a triangle is given by:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Substitute the given values:

\[ A = \frac{1}{2} \times 10 \times 5 \]

\[ A = 25 \text{ square units} \]

, Question 3: Trigonometry

Problem: If \(\sin(\theta) = \frac{3}{5}\), find \(\cos(\theta)\) given that \(\theta\) is in the first quadrant.



Solution:

Using the Pythagorean identity:

\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]

Substitute \(\sin(\theta) = \frac{3}{5}\):

\[ \left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1 \]

\[ \frac{9}{25} + \cos^2(\theta) = 1 \]

\[ \cos^2(\theta) = 1 - \frac{9}{25} \]

\[ \cos^2(\theta) = \frac{25}{25} - \frac{9}{25} \]

\[ \cos^2(\theta) = \frac{16}{25} \]

\[ \cos(\theta) = \frac{4}{5} \]



Question 4: Calculus

Problem: Find the derivative of \(f(x) = 3x^2 + 2x + 1\).



Solution:

Using the power rule, where \(\frac{d}{dx}(x^n) = nx^{n-1}\):

\[ f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) + \frac{d}{dx}(1) \]

\[ f'(x) = 3 \cdot 2x^{2-1} + 2 \cdot 1x^{1-1} + 0 \]

\[ f'(x) = 6x + 2 \]



Question 5: Probability

Problem: What is the probability of rolling a sum of 7 with two six-sided dice?



Solution:

First, determine the total number of possible outcomes when rolling two dice. Each die has 6 faces, so:

\[ 6 \times 6 = 36 \]