List of top Mathematics Questions

In a triangle ABC, \( (r_2 + r_3)\sec^2\left(\frac{A}{2}\right) = \)
(Note: Based on the answer key and standard identities, the function is interpreted as \( \sec^2 \)).
The number of common tangents that can be drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is
The value of the greatest positive integer \( k \), such that \( 49^k + 1 \) is a factor of \( 48(49^{125} + 49^{124} + \dots + 49^2 + 49 + 1) \) is
Find the value of the determinant \( \begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} \).
If A = $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ and B = $\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$ then the matrix AB is equal to
If \[ \lim_{x \to \infty} \left( \frac{e}{1 - e} \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha, \] then the value of \[ \frac{\log_e \alpha}{1 + \log_e \alpha} \] equals:
If \(\alpha, \beta\) are the roots of the equation \(x^2 + 3x + k = 0\) and \(\alpha + 1/\beta\), \(\beta + 1/\alpha\) are the roots of the equation \(4x^2 + px + 18 = 0\) then k satisfies the equation
In a triangle ABC, if \( r_1=4 \), \( r_2=8 \) and \( r_3=24 \), then \( a:b:c = \)
If \( f: \mathbb{R}-\{0\} \to \mathbb{R} \) is defined by \( 3f(x) + 4f\left(\frac{1}{x}\right) = \frac{2-x}{x} \), then \( f(3) = \)
If \( f(x) = x^{2}+bx+c \) and \( f(1+k) = f(1-k) \) \( \forall K \in \mathbb{R} \), for two real numbers b and c, then
If \(x = 2 \sin 60^\circ \cos 60^\circ\) and \(y = \sin 230^\circ - \cos 230^\circ\), and \(x^2 = ky^2\), the value of \(k\) is
If \( A = \begin{bmatrix} 4 & \lambda & -3 0 & 2 & 5 1 & 1 & 3 \end{bmatrix} \), then \( A^{-1} \) exists if:
1+(1+3)+(1+3+5)+(1+3+5+7)+... to 10 terms =
AB and CD are diameters of a circle with centre O and radius 7 cm. If \(\angle BOD = 30^\circ\), then find the area and perimeter of the shaded region.
Solution of the differential equation \( \frac{dy}{dx} + x = 0 \) represents a family of
The expression for K is given as a sum of terms up to infinity. Find the value of K.
One of the roots of the equation \((x+1)^4 + 81 = 0\) is
An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ball drawn is not red is
If \( x = \alpha, y = \beta, z = \gamma \) is the solution of the system of equations \( 2x+3y+z=-1 \), \( 3x+y+z=4 \), \( x-3y-2z=1 \), then the value of \( \beta \) is
The tangents drawn from a point (2,-1) touch the circle \(x^2+y^2+4x-2y+1=0\) at the points A and B. If C is the centre of the circle, then the area (in sq. units) of the triangle ABC is
The particular integral of \[ \frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=e^{-2x} \] is
The solution of the differential equation \[ \frac{d^3y}{dx^3}+3\frac{d^2y}{dx^2}+2\frac{dy}{dx}=0 \] is
The solution of the differential equation \[ \frac{dy}{dx}+\frac{y}{x}=x^2 \] under the condition \(y(1)=1\) is
The solution of the differential equation \[ \frac{dy}{dx}=1+y^2 \] is
The differential equation \[ \frac{dy}{dx}=-\left(\frac{x+y}{1+x^2}\right) \] is