List of top Mathematics Questions

The value of the integral \[ \int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}} \, dx \] is:
A function \( f : \mathbb{R} \to \mathbb{R} \) satisfies \[ f\!\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y) + f(0)}{3} \quad \text{for all } x,y \in \mathbb{R}. \] If \( f'' \) is differentiable at \( x = 0 \), then \( f \) is:
Question:
If \[ f(x) = \begin{cases} x^2 + 3x + a, & x \le 1 \\ bx + 2, & x > 1 \end{cases} \] for \( x \in \mathbb{R} \), and the function is everywhere differentiable, then:
The set of points of discontinuity of the function \[ f(x) = x - [x], \quad x \in \mathbb{R} \] is:
Let \[ A = \begin{bmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{bmatrix} \] If \[ |A|^2 = 25 \] then \( |\alpha| \) equals:
If \( a, b, c \) are positive real numbers each distinct from unity, then the value of the determinant \[ \begin{vmatrix} 1 & \log_a b & \log_a c \\ \log_b a & 1 & \log_b c \\ \log_c a & \log_c b & 1 \end{vmatrix} \] is:
If \( \cos^{-1}\alpha + \cos^{-1}\beta + \cos^{-1}\gamma = 3\pi \), then \( \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \) is equal to:
The number of reflexive relations on a set \( A \) of \( n \) elements is equal to:
If \( f(x)=\frac{3x-4{2x-3} \), then \( f(f(f(x))) \) will be:}
Let \( u+v+w=3 \), \( u,v,w \in \mathbb{R} \) and \( f(x)=ux^2+vx+w \) be such that \[ f(x+y)=f(x)+f(y)+xy,\quad \forall x,y \in \mathbb{R}. \] Then \( f(1) \) equals:
Let \( f(x)=\max\{x+[x],\, x-[x]\ \), where \( [x] \) is the greatest integer \( \le x \). Then} \[ \int_{-3}^{3} f(x)\,dx \] has the value:
The solution set of the equation \[ x \in \left(0,\frac{\pi}{2}\right), \quad \tan(\pi \tan x) = \cot(\pi \cot x) \] is:
If \( 0\le a,b \le 3 \) and the equation \[ x^2 + 4 + 3\cos(ax+b) = 2x \] has real solutions, then the value(s) of \( (a+b) \) is/are:
If the equation \[ \sin^2x - (p+2)\sin x - (p+3) = 0 \] has a solution, then \( p \) must lie in:
Three numbers are chosen at random without replacement from \( \{1,2,\dots,10\} \). The probability that the minimum of the chosen numbers is 3 or the maximum is 7 is:
The population \( p(t) \) of a certain mouse species follows \[ \frac{dp}{dt} = 0.5p - 450. \] If \( p(0)=850 \), then the time at which population becomes zero is:}

Let \[ f(\theta) = \begin{vmatrix} 1 & \cos\theta & -1 \\ -\sin\theta & 1 & -\cos\theta \\ -1 & \sin\theta & 1 \end{vmatrix} \] Suppose \( A \) and \( B \) are respectively maximum and minimum values of \( f(\theta) \). Then \( (A,B) \) is:

The number of common tangents to the circles \[ x^2+y^2-4x-6y-12=0,\quad x^2+y^2+6x+18y+26=0 \] is:
If \[ f(x)=\int_0^{\sin^2 x}\sin^{-1}\!\sqrt{t}\,dt, \quad g(x)=\int_0^{\cos^2 x}\cos^{-1}\!\sqrt{t}\,dt, \] then the value of \( f(x)+g(x) \) is:
The probability that a non-leap year selected at random will have 53 Sundays or 53 Saturdays is:
If \( \cos(\theta+\phi)=\frac{3}{5} \) and \( \sin(\theta-\phi)=\frac{5}{13} \), \( 0<\theta,\phi<\frac{\pi}{4} \), then \( \cot(2\theta) \) equals:
Let \( x-y=0 \) and \( x+y=1 \) be two perpendicular diameters of a circle of radius \( R \). The circle will pass through the origin if \( R \) equals:
Let \( f(x)=|x-\alpha|+|x-\beta| \), where \( \alpha,\beta \) are roots of \( x^2-3x+2=0 \). Then the number of points in \( [\alpha,\beta] \) at which \( f \) is not differentiable is:
If \( f(x) \) and \( g(x) \) are polynomials such that \[ \phi(x) = f(x^3) + xg(x^3) \] is divisible by \( x^2 + x + 1 \), then:
If \( a,b,c \) are in A.P. and the equations \[ (b-c)x^2 + (c-a)x + (a-b) = 0 \] \[ 2(c+a)x^2 + (b+c)x = 0 \] have a common root, then: