List of top Mathematics Questions

If \[ \int \sin(101x)\,\sin^{99}x\,dx = \frac{\sin(100x)\sin^{100}x}{k+5}+c, \] then \(\dfrac{k}{19}=\)

If \[ \Delta= \begin{vmatrix} f(x) & f\!\left(\dfrac{1}{x}\right)+f(x)\\ 1 & f\!\left(\dfrac{1}{x}\right) \end{vmatrix} =0 \] where \(f(x)\) is a polynomial and \(f(2)=17\), then \(f(5)=\) ?

The symbolic form of logic of the circuit given below is

Value of \[ \sum_{k=1}^{\infty}\sum_{r=0}^{k}\frac{1}{3^{k}}\binom{k}{r} \] is:

If \[ \sum_{r=1}^{n} a_r = \frac{n(n+1)(n+2)}{6}\quad \forall\, n \ge 1, \] then \[ \lim_{n\to\infty}\sum_{r=1}^{n}\frac{1}{a_r} = \]

If \(x,y,z\) are in A.P. with common difference \(d\) and the rank of the matrix \[ \begin{pmatrix} 4 & 5 & x \\ 5 & 6 & y\\ 6 & k & z \end{pmatrix} \] is \(2\), then the values of \(k,d\) are:

Let \( p = (1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}) \in {R^4 \) and \( f : {R}^4 \to {R} \) be a differentiable function such that \( f(p) = 6 \) and \( f(Ax) = A^3 f(x) \), for every \( A \in (0, \infty) \) and \( x \in {R}^4 \). The value of \[ 12 \frac{\partial f}{\partial x_1}(p) + 6 \frac{\partial f}{\partial x_2}(p) + 4 \frac{\partial f}{\partial x_3}(p) + 3 \frac{\partial f}{\partial x_4}(p) \] is equal to (answer in integer):}

(a) If orders of matrices A and B are \( p \times q \) and \( q \times r \) respectively, then order of AB is:

Evaluate the integral: \[ I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{\sin^4(4x)}{1 + e^{4x}} \, dx \]

Prove that: \[ \int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a - x) dx. \] Hence show that: \[ \int_0^\pi \sin x \,dx = 2 \int_0^{\pi/2} \sin x \,dx. \]

If the equation of the curve which passes through the point \( (1,1) \) satisfies the differential equation: \[ \frac{dy}{dx} = \frac{2x - 5y + 3}{5x + 2y - 3}, \] then the equation of the curve is: \vspace{0.5cm}