Soru 1 (Functions & Transformations)
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The graph of y = f(x) is translated 3 units right and 2 units up. What is the equation of the transformed graph?
💡 Adım Adım Çözüm:
Horizontal shift 3 units right: replace x with (x - 3). Vertical shift 2 units up: add 2 => y = f(x - 3) + 2.
Soru 2 (Complex Numbers (Euler's Form))
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Express z = 2 (cos(π/3) + i sin(π/3)) in exponential form.
💡 Adım Adım Çözüm:
Using Euler's identity r (cos θ + i sin θ) = r e^(iθ) gives 2 e^(i π/3).
Soru 3 (Trigonometric Identities)
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Simplify (1 - cos(2θ)) / sin(2θ).
💡 Adım Adım Çözüm:
Using double angle formulas: 1 - cos(2θ) = 2 sin²(θ) and sin(2θ) = 2 sin(θ)cos(θ). Ratio = (2 sin²(θ)) / (2 sin(θ)cos(θ)) = sin(θ)/cos(θ) = tan(θ).
Soru 4 (Binomial Theorem)
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What is the coefficient of x³ in the expansion of (2x + 1)⁵?
💡 Adım Adım Çözüm:
Term formula: ⎛5╲3⎞ (2x)³ (1)² = 10 * 8x³ * 1 = 80x³. Coefficient = 80... Wait: ⎛5╲2⎞ = 10, (2)³ = 8 => 10 * 8 = 80.
Soru 5 (Vectors in 3D)
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Find the scalar (dot) product of vectors a = (2, -1, 4) and b = (3, 5, -2).
💡 Adım Adım Çözüm:
a • b = (2)(3) + (-1)(5) + (4)(-2) = 6 - 5 - 8 = -7.
Soru 6 (Logarithmic Functions)
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Solve for x: e^(2x) - 5 e^x + 6 = 0.
💡 Adım Adım Çözüm:
Let u = e^x => u² - 5u + 6 = 0 => (u - 2)(u - 3) = 0 => u = 2 or u = 3 => x = ln(2) or x = ln(3).
Soru 7 (Calculus (Optimization))
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A rectangle has a perimeter of 40 cm. What is the maximum possible area of the rectangle in cm²?
💡 Adım Adım Çözüm:
Perimeter = 2(x + y) = 40 => x + y = 20 => y = 20 - x. Area A(x) = x(20 - x) = 20x - x². Maximum occurs at x = 10 (square). Area = 10 * 10 = 100 cm².
Soru 8 (Normal Distribution)
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A continuous variable X is normally distributed with mean μ = 50 and standard deviation σ = 10. What is P(X > 50)?
💡 Adım Adım Çözüm:
The normal distribution is symmetric about its mean μ = 50. Therefore, P(X > μ) = 0.50 (50%).
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