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 Section A: Algebra & Equations
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Solve for \( x \):
\[ 5^{2x-1} = 25^{x+3} \]

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Find all real solutions:
\[ \sqrt{x + 3} + \sqrt{x - 2} = 5 \]

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If \( f(x) = 2x^2 - 3x + 1 \), find \( f(f(2)) \)
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What is the length of
the altitude drawn to the hypotenuse?

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A circle is inscribed in an equilateral triangle with side length \( 6 \) cm
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What is the ratio
of the longer side to the shorter side?

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Evaluate:

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\[ \sin^2 15° + \sin^2 75° \] (Hint: Use complementary angles
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A cone has a base radius of \( 4 \) cm and a slant height of \( 5 \) cm
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What is the probability that the sum is divisible by 3 or 4?

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How many 4-digit numbers can be formed using the digits \( 1, 2, 3, 4, 5 \)
without repetition?

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A bag contains 6 red, 4 blue, and 5 green balls
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4 \), \( P(B) = 0
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7 \), find \( P(A \cap B) \)
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Prove that \( \sqrt{2} \) is irrational
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(Hint: Use modular arithmetic
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What is its average speed for the entire trip?

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 Answer Key & Detailed Solutions
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Section A: Algebra & Equations
\( x = -7 \)
a
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Since bases are equal, set exponents equal:
\[ 2x - 1 = 2x + 6 \Rightarrow -1 = 6 \]
a
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If the equation were \( 5^{2x-1} =
25^{x-3} \), then:
\[ 2x - 1 = 2x - 6 \Rightarrow -1 = -6 \] (No solution)
a
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\[ 5^{2x-1} = (5^3)^{x+3} \Rightarrow 2x - 1 = 3x + 9 \Rightarrow x = -10 \]

\( x = 6 \)
a
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Then \( a + b = 5 \)
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Square both sides:
\[ (a + b)^2 = 25 \Rightarrow a^2 + b^2 + 2ab = 25 \]
\[ (x+3) + (x-2) + 2ab = 25 \Rightarrow 2x + 1 + 2ab = 25 \]
a
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b
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Thus, \( \sqrt{x+3} = 3 \Rightarrow x = 6 \)
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First, \( f(2) = 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3 \)
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Then, \( f(f(2)) = f(3) = 2(3)^2 - 3(3) + 1 = 18 - 9 + 1 = 10 \)
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Correction: The correct answer is 10, not 21

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