List of top Mathematics Questions

Solve the differential equation \( \cos x \cos y \, dy - \sin x \sin y \, dx = 0 \).
Find the area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \).
If \( -1 \leq x \leq 1 \), then prove that \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \).
Find the mean of a number randomly selected from 1 to 15.
Evaluate: \( \int_0^{\pi/2} (1 - \cos 4x) \, dx \).
Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.
Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis, and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).
Solve the following inequations graphically and write the corner points of the feasible region: \( 2x + 3y \leq 6 \), \( x + y \geq 2 \), \( x \geq 0 \), \( y \geq 0 \).
Find the cartesian coordinates of the point whose polar coordinates are \( \left( 2, \frac{\pi}{4} \right) \).
If \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of lines and \( h^2 = ab \neq 0 \), then find the ratio of their slopes.
If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are the position vectors of the points A, B, C respectively and \( 5\mathbf{a} + 3\mathbf{b} - 8\mathbf{c} = 0 \), then find the ratio in which the point C divides the line segment AB.
Write inverse and contrapositive of the following statement: If \( x<y \), then \( x^2<y^2 \).
If \( A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \) is a non-singular matrix, then find \( A^{-1} \) by elementary row transformations. Hence write the inverse of \( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \).
Write the degree of the differential equation \( \frac{d}{dx} e^y + \frac{dy}{dx} = x \).
Write \( \int \cot x \, dx \).
Find the values of c which satisfy \( c \mathbf{u} = 3 \) where \( \mathbf{u} = \hat{i} + 2\hat{j} + 3\hat{k} \).
Let the probability mass function (p.m.f.) of a random variable X be \( P(X = x) = \binom{4{x} \left( \frac{5}{9} \right)^x \left( \frac{4}{9} \right)^{4-x} \), for \( x = 0, 1, 2, 3, 4 \), then E(X) is equal to _______.
}
Write the joint equation of co-ordinate axes.
The solution of the differential equation \( \frac{d}{dt} \log \frac{x}{t} = x \) is _______.
\( \int 3 \cos^3 x \, dx = \).
If y is a function of x and \( \log (x + y) = 2xy \), then the value of \( y'(0) = \).
If the corner points of the feasible solution are (0, 10), (2, 2), and (4, 0), then the point of minimum \( z = 3x + 2y \) is _______.
The area of the triangle with vertices (1, 2, 0), (1, 0, 2), and (0, 3, 1) in sq. unit is _______.
In \( \triangle ABC \), if \( c^2 + a^2 - b^2 = ac \), then \( \angle B = \).
If \( p \wedge q \) is F, \( p \to q \) is F, then the truth values of p and q are _______ respectively.