Simplify the real numerical expression: E = (sqrt(3) + 1) / (sqrt(3) - 1) - (sqrt(3) - 1) / (sqrt(3) + 1).
Algebra And Number Sets
Question · Easy
Write the complex number z = (1 + 7i) / (3 - 4i) in standard algebraic form a + bi.
Compute the modulus of the complex number z = (3 - 4i) * (1 + i*sqrt(3)).
Evaluate the exact value of: E = log_2(24) - log_2(3) + log_3(root[4](27)).
In an arithmetic progression (a_n), the first term is a_1 = 4 and the common difference is d = 3. Find the sum of the first 20 terms, S_20.
Find the sum of the convergent infinite geometric series with first term b_1 = 12 and common ratio q = 1/3.
Determine all real values of the parameter m for which the quadratic equation x^2 - 2(m - 1)x + (m + 5) = 0 has two equal real roots.
Solve the system of linear inequalities in R: 2x - 5 <= 3x + 1 and (x + 4)/2 > 1 - x.
Compute the exact value of the complex power: W = ((1 + i*sqrt(3)) / (1 - i))^12.
Question · Medium
Solve the logarithmic equation: log_2(x) + log_4(x) + log_16(x) = 7.
Find the sum of all real solutions to the irrational equation: sqrt(2x + 3) - sqrt(x - 2) = 2.
Find the set of all real solutions to the exponential inequality: 4^x - 3 * 2^(x + 1) + 8 <= 0.
Find the product of all real solutions to the absolute value equation: |2x - 3| + |x + 1| = 5.
Solve the symmetric system for real numbers x and y with x < y: { x + y + x*y = 11, x^2 + y^2 + x*y = 19 }. What is the ordered pair (x, y)?
Three numbers a < b < c form an arithmetic progression. If a, b, c + 2 form a geometric progression and a + b + c = 12, find the value of c.
Determine the solution set in R of the rational inequality: (x^2 - 4x + 3) / (x^2 - 4) <= 0.
The cubic polynomial P(x) = x^3 - 6x^2 + 11x - 6 has roots x_1, x_2, x_3. Compute the sum of reciprocals: 1/x_1 + 1/x_2 + 1/x_3.
Find the value of the parameter m in R for which the linear system { m*x + y = 1, 4x + m*y = 2 } has infinitely many solutions.
Compute the sum of the telescoping series S = sum_{k=1}^{99} 1 / (k*(k + 1)).
Find the complex number z satisfying the equation |z| - z = 1 + 2i.
Question · Hard
Solve the inequality with variable base: log_x(3x - 2) >= 1.
A sequence is defined by a_1 = 1 and a_(n+1) = sqrt(6 + a_n) for all n >= 1. Find the limit L = lim_{n -> infinity} a_n.
How many real solutions exist for the equation x^2 - 5*floor(x) + 4 = 0, where floor(x) denotes the greatest integer less than or equal to x?
Simplify the real radical expression: X = root[3](20 + 14*sqrt(2)) + root[3](20 - 14*sqrt(2)).
Let matrix A = [[2, 1], [1, 2]] in M_2(R). Find the trace of the matrix power A^n for any positive integer n.
Determine the maximal real domain of the logarithmic function: f(x) = ln((x - 3) / (5 - x)).
Functions And Graphs
Let f: [1, +infinity) -> [-3, +infinity) be the bijective function defined by f(x) = x^2 - 2x - 2. Find the inverse function f^(-1)(x).
Which of the following functions defined on R is an odd function (satisfying f(-x) = -f(x))?
Find the fundamental period T of the real trigonometric function: f(x) = sin(4x) + cos(6x).
Let f(x) = x / (x - 1) for all x in R \ {1}. Compute the composite function (f o f)(x) on its domain.
Find the coordinates of the intersection points of the parabola y = x^2 - 3x + 2 and the line y = x - 1.
Find the range (image set) Im(f) of the function f: R -> R, f(x) = x / (x^2 + 1).
Find the equation of the oblique (slant) asymptote of the rational function f(x) = (2x^2 + 3x - 1) / (x - 1) as x -> +infinity.
Analyze the function f: R -> R, f(x) = x^3 - 3x. Which statement correctly characterizes f?
A function f: R -> R satisfies the relation 2*f(x) + f(1 - x) = 3*x^2 for all real numbers x. Find the exact value of f(2).
Determine the maximal real domain of the function: f(x) = arcsin(2x / (1 + x^2)).
Find the maximum value of the function f: R -> R defined by f(x) = 3*sin(x) - 4*cos(x) + 2.
Find the coordinates of the center of symmetry of the cubic function graph: f(x) = x^3 - 6x^2 + 9x + 2.
Determine the number of distinct real solutions to the equation: 2^x = 3 - x^2.
Simplify the expression f(x) = e^(2*ln(x)) - ln(e^(x^2)) for all x > 0.
Compute the exact value of: V = tan(2 * arctan(1/3)).
Find all values of the real parameter a for which the polynomial function f(x) = x^3 + a*x^2 + 3x - 5 is strictly increasing on R.
A continuous function f: R -> R satisfies f(x + y) = f(x) * f(y) for all x, y in R, and f(1) = 4. Find the value of f(-1/2).
Find the equation of the slant (oblique) asymptote of the function f(x) = sqrt(4x^2 + x + 1) as x -> -infinity.
Find the range of the function f: R -> R defined by f(x) = 4^x - 2^(x + 2) + 5.
Evaluate the limit of the rational sequence: L = lim_{x -> +infinity} (5x^3 - 2x + 7) / (2x^3 + 4x^2 - 1).
Mathematical Analysis Calculus
Compute the first derivative of the function f(x) = (x^2 + 1) * e^x.
Find the equation of the tangent line to the graph of y = x^3 - 2x + 4 at the point where x_0 = 1.
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