List of top Mathematics Questions

Consider sequences of positive real numbers of the form \( x, 2000, y, \dots \), in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of \( x \) does the term 2001 appear somewhere in the sequence?
Six ants simultaneously stand on the six vertices of a regular octahedron with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
In how many ways can 5 men and 3 women be seated in a row such that no two women sit adjacent?
If \( \tan \frac{\pi}{18}, x \) and \( \tan \frac{\pi}{18} \) are in AP and \( \tan \frac{5\pi}{18} \) are in AP, then the value of \( \frac{x}{y} \) will be
For real \( x \), let \( f(x) = x^3 + 5x + 1 \), then:
Given the vertices of a triangle are \( A(1, -1, -3) \), \( B(2, 1, -2) \), and \( C(-5, 2, -6) \). Compute the length of the bisector of the interior angle at vertex A.
Let \( n \) be a 5-digit number and let \( q \) and \( r \) be the quotient and remainder respectively, when \( n \) is divided by 100. For how many values of \( n \) is \( q + r \) divisible by 11?
\[ \left| \begin{matrix} x+1 & x+2 & x+4 \\ x+3 & x+5 & x+8 \\ x+7 & x+10 & x+14 \end{matrix} \right| \]
Negation of the statement \( (\rho \land r) \rightarrow (r \lor q) \) is
Solve for $x \ (a \neq 0)$ $$ \sqrt{(a + x)^2 + 4\sqrt{(a - x)^2}} = 5\sqrt{a^2 - x^2} $$
\( 8 \cos^4 x - 8 \cos^2 x + 1 \) is equal to
Choose the most appropriate options. If \(a, b, c\) are positive real numbers, then \[ \frac{1}{\log_{abc}} + \frac{1}{\log_{abc}} + \frac{1}{\log_{abc}} = \]
The line \( y = mx + C \) will be tangent to the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) if \( C \) is equal to
The eccentricity of the hyperbola \( \frac{\sqrt{1999}}{3} \left( x^2 - y^2 \right) = 1 \) is
The value of \( \int \frac{\sin^2 x \cos^2 x}{(\sin^3 x + \cos^3 x)^2} \, dx \) is
The integral \( \int e^{\sec x} \tan x \sec x \, dx \) is equal to
The integral \( \int 34x^4 dx \) is equal to
The value of \( \int_0^{\frac{\pi}{2}} e^x \cos x \, dx \) is equal to:
The series \( 1 + 1 + \frac{3}{2^2} + \frac{4}{2^3} + \frac{5}{2^4} + \cdots \) is equal to
The value of the integral \( \int_{-3}^{5} |x - 3| \, dx \) is
In a class, there are 10 boys and 8 girls. When 3 students are selected at random, the probability that 2 girls and 1 boy are selected, is
If A and B are independent events such that \( P(B) = \frac{2}{7} \), \( P(A \cup B) = 0.8 \), then \( P(A) = \)?
It is known that \( AB = 2a - 6b \) and \( AC = 3a + b \), where \( a \) and \( b \) are mutually perpendicular unit vectors. Determine the angles of the AABC.
Let \( f(x) = x^2 + 6x + 1 \) and let \( R \) denote the set of points \( (x, y) \) in the coordinate plane such that \( f(x) + f(y) \) so and \( f(x) - f(y) \leq 0 \). Which of the following is closest to the area of \( R \)?
The determinant of the matrix \[ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \] is equal to