List of top Mathematics Questions

Let a circle \( C \) with radius \( r \) passes through four distinct points \( (0, 0), (k, 3k), (2, 3), (-1, 5) \), such that \( k \neq 0 \), then \( (10k + 2r^2) \) is equal to:
Evaluate the integral \( I = \int_0^\pi \frac{4 \cos^2 x + \sin^2 x}{8x} \, dx \).

If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and

and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to:

For the matrix [A] given below, the transpose is __________. 

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]

Integration of \(\ln(x)\) with \(x\), i.e. \(\int \ln(x)dx =\) __________.
 

If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
Total number of terms in an A.P. are even. Sum of odd terms is 24 and sum of even terms is 30. Last term exceeds the first term by \( \frac{21}{2} \). Find the total number of terms.
Let \( A = \{1,2,3\} \). The number of relations on \( A \), containing \( (1,2) \) and \( (2,3) \), which are reflexive and transitive but not symmetric, is ________.
Let \[ L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{-1} \quad {and} \quad L_2 : \frac{x+1}{1} = \frac{y-2}{2} = \frac{z-2}{2} \] be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \) where \( 5x - 11y - 8z = 1 \), then \( 5x - 11y - 8z \) equals:

The integral is given by:

\[ 80 \int_{0}^{\frac{\pi}{4}} \frac{\sin\theta + \cos\theta}{9 + 16 \sin 2\theta} d\theta \]

is equals to?

Find the value of \[ \sin 70^\circ \left( \cot 10^\circ \cot 70^\circ - 1 \right). \]
Let \( f(x) = \int_0^{x^2 \frac{t^2 - 8t + 15}{e^t}} dt, \, x \in \mathbb{R} \). Then the numbers of local maximum and local minimum points of \( f \), respectively, are:
For a \( 3 \times 3 \) matrix \( M \), let trace(M) denote the sum of all the diagonal elements of \( M \). Let \( A \) be a \( 3 \times 3 \) matrix such that \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). If \( B = \text{adj}(\text{adj}(2A)) \), then the value of \( |B| + \text{trace}(B) \) equals:
If the system of linear equations: \[ x + y + 2z = 6, \] \[ 2x + 3y + az = a + 1, \] \[ -x - 3y + bz = 2b, \] where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:
If \( y = y(x) \) is the solution of the differential equation, \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right), \] where \( -2 \leq x \leq 2 \), and \( y(2) = \frac{\pi^2 - 8}{4} \), then \( y^2(0) \) is equal to:
Let \[ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \quad \text{and} \quad H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1. \] Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to:
The area of the region enclosed by the curves \( y = x^2 - 4x + 4 \) and \( y^2 = 16 - 8x \) is:
If \[ \sum_{r=1}^{30} r^2 \left( \binom{30}{r} \right)^2 = \alpha \times 2^{29}, \] then \( \alpha \) is equal to _________.
Let the curve \((\)\( z(1 + i) + \overline{z(1 - i)} = 4\), \(\, z \in \mathbb{C}\) \()\), divide the region \(|z - 3| \leq 1\) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals:
The perpendicular distance of the line \( \frac{x - 1}{2} = \frac{y + 2}{-1} = \frac{z + 3}{2} \) from the point \( P(2, -10, 1) \) is:
In \( I(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} \, dx \), where \( m, n > 0 \), then \( I(9, 14) + I(10, 13) \) is:
For the function \( f(x) = \ln(x^2 + 1) \), what is the second derivative of \( f(x) \)?
Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).
Let \( D = \{a, b, c\} \). How many distinct ways can \( D \) be partitioned into non-empty subsets, representing equivalence relations?
What is the solution to the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) with the initial condition \( y(1) = 2 \)?