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:
\[ 6 \times 6 = 36 \]
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:
\[ 6 \times 6 = 36 \]
