List of top Mathematics Questions

The relation \( R = \{(x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is:
If \( f(x) = \frac{2^x}{2^x + \sqrt{2}} \), \(x \in \mathbb{R}\), then \( \sum_{k=1}^{81} f\left(\frac{k}{82}\right) \) is equal to:
The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is:
Let \(T_{n-1} = 28\), \(T_n = 56\), and \(T_{n+1} = 70\). Let A \((4\cos t, 4\sin t)\), B \((2\sin t, -2\cos t)\), and C \((3r_n - 1, r^2_n - n - 1)\) be the vertices of a triangle ABC, where \(t\) is a parameter. If \((3x - 1)^2 + (3y)^2 = a\), is the locus of the centroid of triangle ABC, then \(a\) equals:
The area (in sq. units) of the region \((x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3\) is:
Let O be the origin, the point A be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point B \( z_2 \) be such that \( \sqrt{3}|z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then:
Let the equation of the circle, which touches x-axis at the point \( (a, 0) \) and cuts off an intercept of length \( b \) on y-axis be \( x^2 + y^2 - cx + dy + e = 0 \). If the circle lies below x-axis, then the ordered pair \( (2a, b^2) \) is equal to:
Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \( x \) denotes the number of defective oranges, then the variance of \( x \) is:
The value of \( \cos \left( \sin^{-1} \left(-\frac{3}{5}\right) + \sin^{-1} \left(\frac{5}{13}\right) + \sin^{-1} \left(-\frac{33}{65}\right) \right) \) is:
If \( a = 1 + \sum_{r=1}^{6} (-3)^{r-1} \binom{12}{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is:
Let \(E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1\) be an ellipse. Ellipses \(E_i\) are constructed such that their centers and eccentricities are the same as that of \(E_1\), and the length of the minor axis of \(E_{i+1}\) is the length of the major axis of \(E_i\). If \(A_i\) is the area of the ellipse \(E_i\), then \(\frac{5}{\pi} \sum_{i=1}^{\infty} A_i\) is equal to:
Let \( f(x) \) be defined as follows:

\[ f(x) = \begin{cases} 3x, & \text{if } x < 0 \\ \min(1 + x + \lfloor x \rfloor, 2 + x \lfloor x \rfloor), & \text{if } 0 \leq x \leq 2 \\ 5, & \text{if } x > 2 \end{cases} \]

where \( \lfloor . \rfloor \) denotes the greatest integer function. If \( \alpha \) and \( \beta \) are the number of points where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals:
Let \(M\) denote the set of all real matrices of order 3 x 3 and let \(S = \{-3, -2, -1, 1, 2\}\). Let

\[ S_1 = \{A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j\}, \]

\[ S_2 = \{A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j\}, \]

\[ S_3 = \{A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j\}. \]

If \(n(S_1 \cup S_2 \cup S_3) = 125\), then \(\alpha\) equals:
Two numbers \(k_1\) and \(k_2\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \(i^{k_1} + i^{k_2}\) (where \(i = \sqrt{-1}\)) is non-zero equals:
If the image of the point \( (4, 4, 3) \) in the line \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} \) is \( (a, \beta, \gamma) \), then \( a + \beta + \gamma \) is equal to:
The sum of the squares of all the roots of the equation \( x^2 + [2x - 3] - 4 = 0 \) is:
Let for some function \( y = f(x) \), \(\int_0^x t f(t) \, dt = x^2 f(x), x>0\) and \( f(2) = 3 \). Then \( f(6) \) is equal to:
Find the variance of the numbers 8, 21, 34, ..., 320.
The total number of ways the word 'DAUGHTER' can be arranged so that all vowels don't occur together:
If \( A \) and \( B \) are binomial coefficients of the 30\(^\text{th}\) and 12\(^\text{th}\) terms of the binomial expansion \( (1 + x)^{2n-1} \), and \( 2A = 5B \), then the value of \( n \) is
There is a group A of 5 boys and 3 girls and another group B of 5 boys and 6 girls. How many ways can we invite 4 boys and 4 girls for a party with 5 from group A and 3 from group B?
The equation of chord of the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) with (3, 1) as mid-point is
Given that the arithmetic mean \( S_{12} = 57 \), \( S_{40} = 1030 \), find \( S_{30} - S_{10} \)?
Given the matrix \[ A = (a_{ij}), \] where \[ A_{11} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad A_{14} = \begin{bmatrix} 4 & 1 \\ 3 & 0 \end{bmatrix}, \quad A_{21} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad A_{23} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}, \quad \text{find} \, a_{23}. \]
By using relation \( \Delta G = \Delta H - T \Delta S \), which of the following is incorrect for a spontaneous reaction at a given temperature?