Solve the irrational equation: sqrt(2x + 7) = x + 2.
Algebra Equations Inequalities
Question · Medium
What are all real solutions to the absolute value equation: |2x - 5| = 3x + 1?
Solve the system of equations: { x + y = 7, x² + y² = 25 }.
Question · Easy
Solve the logarithmic inequality: log3(x² - 2x) <= 1.
Question · Hard
For what values of the parameter 'a' does the quadratic equation x² - 2(a - 1)x + (a + 5) = 0 have two distinct real roots?
Solve the exponential equation: 4^x - 3 * 2^x - 4 = 0.
Simplify the algebraic expression: ((a^(3/2) - b^(3/2)) / (a^(1/2) - b^(1/2))) - (ab)^(1/2) for a > 0, b > 0, a != b.
Solve the rational inequality: (x² - 5x + 6) / (x - 1) >= 0.
Solve the equation: log_x(2x² - 3x - 4) = 2.
Find the sum of all integer solutions to the inequality: |3x - 2| <= 7.
If x1 and x2 are the roots of 2x² - 7x + 3 = 0, what is the value of 1/x1 + 1/x2?
Solve the equation: 2^(x + 2) + 2^(x - 1) = 36.
Solve the system: { 3^x * 2^y = 144, log_sqrt(2)(y - x) = 2 }.
Find all real roots of the biquadratic equation: x^4 - 13x² + 36 = 0.
Solve the irrational equation: sqrt(x + 5) + sqrt(x) = 5.
What is the domain of the function f(x) = ln((x - 3) / (x + 2))?
Functions Analysis Calculus
Find the derivative of the function f(x) = (x² - 1) / (x² + 1).
What is the equation of the normal line to the curve y = x³ - 3x at the point (2, 2)?
Find the absolute minimum value of the function f(x) = x^4 - 8x² + 3 on the closed interval [-1, 3].
Evaluate the definite integral: integral from 0 to pi/2 of (sin(x) * cos²(x)) dx.
What is the limit: lim as x -> infinity of ((2x + 1) / (2x - 3))^(3x)?
For which value of 'k' is the function f(x) = { (x² - 9)/(x - 3) for x != 3; k for x = 3 } continuous at x = 3?
Find the area of the figure bounded by the curves y = x² and y = 2x.
What is the inflection point (շրջման կետ) of the curve y = x³ - 6x² + 9x + 1?
Find the range of the function f(x) = 3*sin(x) - 4*cos(x) + 2.
What is the antiderivative F(x) of f(x) = 1 / (cos²(3x)) such that F(0) = 1?
Find the limit: lim as x -> 0 of (1 - cos(4x)) / (x²).
What is the interval of increase (աճման միջակայք) for the function f(x) = x * e^(-2x)?
Find the horizontal and vertical asymptotes of the rational function f(x) = (3x² + 5) / (x² - 4).
What is the value of the derivative of f(x) = ln(x² + 4) at x = 2?
Evaluate: cos(75°) * cos(15°).
Trigonometry Progressions Combinatorics
Solve the trigonometric equation: cos(2x) + 3*sin(x) - 2 = 0 on [0, 2*pi).
What is the exact value of arcsin(sin(5*pi / 6))?
In an arithmetic progression, the 4th term is a4 = 15 and the 9th term is a9 = 35. What is the sum of the first 12 terms (S12)?
In how many ways can a committee of 3 students and 2 teachers be chosen from a group of 8 students and 5 teachers?
Two standard six-sided dice are rolled simultaneously. What is the probability that the sum of the numbers rolled is equal to 8?
Simplify the expression: (1 - tan²(alpha)) / (1 + tan²(alpha)).
Solve the inequality: cos(x) > -1/2 on the interval [0, 2*pi).
What is the third term of a geometric progression where b1 = 5 and the common ratio q = -2?
How many distinct anagrams (arrangements) can be formed using all the letters of the word 'ARMENIA'?
What is the exact value of tan(arctan(2) + arctan(3))?
Find the sum of the series: 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^n) + ...
In a bag there are 4 red balls and 6 blue balls. If two balls are drawn at random without replacement, what is the probability that both balls are red?
What is the maximum value of the expression f(x) = sin²(x) + 2*cos(x) for all real x?
What is the common difference 'd' of an arithmetic progression where the sum of the first n terms is given by Sn = 3n² + 2n?
Find the length of the median to the hypotenuse in a right triangle whose legs measure 16 cm and 12 cm.
Geometry Planimetry Stereometry
In a rhombus ABCD, the diagonals measure AC = 24 cm and BD = 10 cm. What is the perimeter of the rhombus?
What is the area of a regular hexagon inscribed in a circle of radius R = 6 cm?
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