List of top Mathematics Questions

Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \cos 4x}{2 \sin^2 x + x \tan 7x} \]
Find the distance from the point \( A(2, 3, -1) \) to the given straight lines \( 2x - 2y + z + 3 = 0 \) and \( 3x - 2y + 2z + 17 = 0 \).
Evaluate the limit: \[ \lim_{x \to a} \frac{\log{(x^{a-1})}}{x-a} \]
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
In how many ways can 10 identical objects be put in 8 distinct boxes in such that no box is empty?
In how many ways can 3 blue, 4 white and 2 red balls be distributed into 4 distinct boxes?
Let \( T_n \) denote the number of triangles which can be formed using the vertices of a regular polygon of \( n \) sides. If \( T_{n+1} - T_n = 21 \), then \( n \) equals
If \( \sin \theta + \csc \theta = 2 \), then the value of \( \sin^{10} \theta + \csc^{10} \theta \) is
The standard deviation of a data is 6, when each observation is increased by 1, then the standard deviation of the new data is
If the arithmetic mean of the following data is 7, then \( a + b = \) \[ \begin{array}{|c|c|c|c|c|} \hline x_i & 4 & 6 & 7 & 9 \\ f_i & a & 4 & b & 5 \\ \hline \end{array} \]
Is equal to $$ \left| \begin{matrix} b^2 + c^2 & c^2 + b^2 \\ c^2 & c^2 + a^2 \\ b^2 & a^2 + b^2 \end{matrix} \right| $$ 
Compute the determinant of the \(n \times n\) matrix whose elements are identified by the condition \(a_{ij} = \min(i,j)\), where \(i\) is the row number and \(j\) is the column number.
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
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?
Negation of the statement \( (\rho \land r) \rightarrow (r \lor q) \) is
Determine the form of the conic section described by the equation \( x^2 + y^2 + 2xy - 8x + 8y = 0 \)
Let \( S \) be the set of all points with coordinates \( (x, y, z) \), where \( x, y, z \) are each chosen from the set \([0, 1, 2]\). How many equilateral triangles have all their vertices in \( S \)?
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?
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:
\[ \left| \begin{matrix} x+1 & x+2 & x+4 \\ x+3 & x+5 & x+8 \\ x+7 & x+10 & x+14 \end{matrix} \right| \]
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.