List of top Mathematics Questions

If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by \(l + m + n = 0\), \(2lm + 2mn - ln = 0\), then the angle between the lines is:
If
\[ \begin{vmatrix} x^k & x^{k+2} & x^{k+3} \\ y^k & y^{k+2} & y^{k+3} \\ z^k & z^{k+2} & z^{k+3} \end{vmatrix} = (x - y)(y - z)(z - x)\left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right), \] then the value of \(k\) is:
For the real numbers \( x \) and \( y \), we write \( x \, P \, y \) iff \( x - y + \sqrt{2} \) is an irrational number.
Then the relation \( P \) is:
For any integer \(n\),
\[ \int_{0}^{\pi} e^{\cos^2 x} \cdot \cos^3(2n + 1)x \, dx \text{ has the value.} \]
For every real number \(x \neq -1\), let \(f(x) = \frac{x}{x+1}\). Write \(f_1(x) = f(x)\) and for \(n \geq 2\), \(f_n(x) = f(f_{n-1}(x))\). Then \(f_1(-2), f_2(-2), \ldots, f_n(-2)\) must be:
If 1000! = 3n × m, where m is an integer not divisible by 3, then n =?
The numbers \(1, 2, 3, \ldots, m\) are arranged in random order. The number of ways this can be done, so that the numbers \(1, 2, \ldots, r \, (r < m)\) appear as neighbours is:
If $$ y = \tan^{-1} \left[ \frac{\log_e\left(\frac{e}{x}\right)}{\log_e(e x^2)} \right] + \tan^{-1} \left[ \frac{3 + 2\log_e x}{1 - 6\cdot\log_e x} \right], $$ then \( \frac{d^2y}{dx^2} = ? \) 
\(\triangle OAB\) is an equilateral triangle inscribed in the parabola \(y^2 = 4ax, \, a>0\) with \(O\) as the vertex. Then the length of the side of \(\triangle OAB\) is:
Let N be the number of quadratic equations with coefficients from {0,1,2,...,9} such that 0 is a solution of each equation. Then the value of N is:
The coefficient of \(a^{10}b^7c^3\) in the expansion of \((bc + ca + ab)^{10}\) is:
If \((1 + x + x^2 + x^3)^5 = \sum_{k=0}^{15} a_k x^k\), then \(\sum_{k=0}^{7} (-1)^k \cdot a_{2k}\) is equal to:
In \(\mathbb{R}\), a relation \(p\) is defined as follows: For \(a, b \in \mathbb{R}\), \(apb\) holds if \(a^2 - 4ab + 3b^2 = 0\).
Then:
Let \(f : \mathbb{R} \to \mathbb{R}\) be a function defined by \(f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}\), then:
If $a_i, b_i, c_i \in \mathbb{R}$ ($i = 1, 2, 3$) and $x \in \mathbb{R}$, and
$\begin{vmatrix} a_1 + b_1x & a_1x + b_1 & c_1 \\ a_2 + b_2x & a_2x + b_2 & c_2 \\ a_3 + b_3x & a_3x + b_3 & c_3 \end{vmatrix} = 0$,
then:
Let
\[ I(R) = \int_0^R e^{-R \sin x} \, dx, \quad R > 0. \]
Which of the following is correct?
The area bounded by the curves \(x = 4 - y^2\) and the Y-axis is:
If for the series \(a_1, a_2, a_3, \ldots\), etc., \(a_{n+1} - a_n\) bears a constant ratio with \(a_n + a_{n+1}\), then \(a_1, a_2, a_3, \ldots\) are in:
If \( a_1, a_2, \dots, a_n \) are in A.P. with common difference \( \theta \), then the sum of the series:
\[ \sec a_1 \sec a_2 + \sec a_2 \sec a_3 + \dots + \sec a_{n-1} \sec a_n = k (\tan a_n - \tan a_1), \]
where \( k = ? \)
Two smallest squares are chosen one by one on a chessboard. The probability that they have a side in common is:
If \( n \) is a positive integer, the value of:
\[ (2n + 1) \binom{n}{0} + (2n - 1) \binom{n}{1} + (2n - 3) \binom{n}{2} + \dots + 1 \cdot \binom{n}{n} \] is:
Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If \(a_n\) and \(b_n\) be the \(n\)-th term of A.P. and G.P. respectively, then:
The equation \(r \cos \theta = 2a \sin^2 \theta\) represents the curve:
If the foot is perpendicular from (1, 2, 3) to the line \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z-1}{1}\) is \(( a, \beta, \gamma)\), then find \(a + \beta + \gamma\)
Two vertices of the parallelogram \( ABCD \) are given as \( A(-1, 2, 1) \) and \( B(1, -2, 5) \). If the equation of the line passing through \( C \) and \( D \) is: \[ \frac{x - 4}{1} = \frac{y + 7}{-2} = \frac{z - 8}{2}, \] find the distance between sides \( AB \) and \( CD \). Hence, find the area of the parallelogram \( ABCD \).