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Questions and Answers
What is the area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, and y = nx + 1?
What is the value of (log2 xy)² given that positive real numbers x and y satisfy log2 x = logy 16 and xy = 64?
What is the inverse of the function f(x) = 2x(x−1) with domain and range greater than or equal to 1?
Given A = (sin x)² + (cos x)⁴, what can be said about A for all real values of x?
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What is the area of the triangle determined by the points of tangency of circles with radii 1, 2, and 3?
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What are the equations of the diagonals for a parallelogram with opposite sides represented by x² − 5x + 6 = 0 and y² − 6y + 5 = 0?
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How many solutions of the equation tan x + sec x = 2 cos x lie in the interval [0, 2π]?
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What is the area of the region enclosed by the graph of the equation x² + y² = |x| + |y|?
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What is the number of terms with rational coefficients among the 1001 terms in the expansion of (x³/2 + y²/3)¹⁰?
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For how many different values of x does the term 2001 appear somewhere in the sequence defined by x, 2000, y...?
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What is the maximum possible value of x² − 6x + y² when |x + y| + |x − y| = 2?
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For how many integers n between 1 and 50, inclusive, is (n² − 1)!/((n!)ⁿ) an integer?
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What is D(96), the number of ways of writing the positive integer n as a product n = f₁ f₂ · · · fₖ where k ≥ 1?
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Let f₁(x) = 1 − x, and for integers n ≥ 2, let fₙ(x) = fₙ₋₁(n² − x). What is N + c where N is the largest value of n for which the domain of fₙ is nonempty?
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Study Notes
Parallelogram Area
- The area of the parallelogram formed by the lines ( y = mx ), ( y = mx + 1 ), ( y = nx ), and ( y = nx + 1 ) is given by ( |m - n| ).
Logarithmic Equations
- Positive real numbers ( x ), ( y ) (not equal to 1) satisfy ( \log_2 x = \log_y 16 ) and ( xy = 64 ); the value of ( ( \log_2 xy )^2 ) is 20.
Inverse Function
- The function ( f(x) = 2x(x-1) ) has an inverse in the domain and range of real values ( \geq 1 ).
Sine and Cosine Functions
- Given ( A = (\sin x)^2 + (\cos x)^4 ), the bounds for ( A ) are established as ( 0 \leq A \leq \frac{4}{3} ).
Triangle Area from Circles
- The area of the triangle formed by the points of tangency of circles with radii 1, 2, and 3 is ( \frac{5}{4} ).
Parallelogram Diagonals
- For the parallelogram defined by equations ( x^2 - 5x + 6 = 0 ) and ( y^2 - 6y + 5 = 0 ), the diagonals are represented by ( 4x + y = 13 ) and ( y = 4x - 7 ).
Mathematical Evaluation
- The mathematical expression given evaluates to 2.
Solutions to Trigonometric Equation
- The equation ( \tan x + \sec x = 2 \cos x ) has 2 solutions within the interval ( [0, 2\pi] ).
Area Enclosed by Graph
- The area enclosed by the graph of ( x^2 + y^2 = |x| + |y| ) is ( 2\pi + 2\sqrt{2} ).
Coefficient Counts in Expansion
- In the expansion of ( (x^{3/2} + y^{2/3})^{1000} ), there are 501 terms with rational coefficients.
Sequence Values
- The number of different values of ( x ) such that ( 2001 ) appears in the sequence formed by positive real numbers ( x, 2000, y ) is 2.
Maximum Expression Value
- For ( |x + y| + |x - y| = 2 ), the maximum value of the expression ( x^2 - 6x + y^2 ) is 9.
Integer Conditions
- For integers ( n ) between 1 and 50, inclusive, the condition ( \frac{(n^2 - 1)!}{(n!)^n} ) results in 34 valid integers.
Factorization Count
- The number of distinct ways to express ( n = 96 ) as a product of integers greater than 1 is 128.
Function Domain
- Establishing ( f_1(x) = 1 - x ) leads to ( N ) as the largest integer where the domain of ( f_N ) remains nonempty; ( N + c ) is defined.
Integral Calculation
- Integral calculations involving the fractional part of positive real numbers ( x ) are implied but not completed in the text.
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Description
Test your knowledge with this mock exam covering advanced mathematical concepts, including area calculations of geometrical shapes and logarithmic equations. You have 75 minutes to answer all 20 questions, presenting a comprehensive challenge for aspiring students. Good luck!
