List of top Mathematics Questions

Let \(x=x(t)\) be the solution curve of the differential equation \[ \frac{dx}{dt}=-kx, \] with \[ x(0)=100,\quad x\!\left(\frac{1}{2}\right)=80. \] If \(x(t_\alpha)=5\), then \(t_\alpha\) is equal to:
Let \(\alpha,\beta,\gamma\ (0<\alpha,\beta,\gamma<\tfrac{\pi}{2})\) be the angles between non–zero vectors \(\vec a\) and \(\vec b\), \(\vec b\) and \(\vec c\), \(\vec c\) and \(\vec a\) respectively. If \(\theta\) is the angle that the vector \(\vec a\) makes with the plane containing \(\vec b\) and \(\vec c\), then:
If \[ a_n=(2n^2-n+2)(n!) , \] then \[ \sum_{n=1}^{20} a_n \] is equal to:
Let \(\alpha\) and \(\beta\) be the roots of the equation \[ 2x^2-5x-1=0. \] For \(n\in\mathbb{N}\), let \[ P_n=\alpha^n+\beta^n. \] Then the value of \[ \frac{2P_{11}\,(2P_{10}-5P_9)}{P_8\,(5P_{10}+P_9)} \] is equal to:
Let \(f\) be a differentiable function satisfying \[ f(x+y)=f(x)+f(y)-xy \quad \text{for all } x,y\in\mathbb{R}. \] If \[ \lim_{h\to 0}\frac{f(h)}{h}=3, \] then the value of \[ \sum_{n=1}^{10} f(n) \] is equal to:
Let \[ f(t)=\int_{0}^{t} e^{x^2}\Big((1+2x^2)\sin x+x\cos x\Big)\,dx. \] Then the value of \(f(\pi)-f\!\left(\frac{\pi}{2}\right)\) is equal to:
If both the roots of the equation \[ x^2-2ax+a^2-1=0 \quad (a\in\mathbb{R}) \] lie in the interval \((-2,2)\), then the equation \[ x^2-(a^2+1)x-(a^2+2)=0 \] has:
All the words (with or without meaning) formed using all the five letters of the word GOING are arranged as in a dictionary. Then the word OGGIN occurs at the place which is:
Let \(f:[-2a,2a]\to\mathbb{R}\) be a thrice differentiable function and define \[ g(x)=f(a+x)+f(a-x). \] If \(m\) is the minimum number of roots of \(g'(x)=0\) in the interval \((-a,a)\) and \(n\) is the minimum number of roots of \(g''(x)=0\) in the interval \((-a,a)\), then \(m+n\) is equal to:
Let \(A=[a_{ij}]\), \(\det(A)\neq 0\), and \(B=[b_{ij}]\) be two \(3\times 3\) matrices. If \[ b_{ij}=3^{\,i-j}\,a_{ij}\quad \text{for all } i,j=1,2,3, \] then:
Let \(m\) and \(n\) be non–negative integers such that for \[ x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),\qquad \tan x+\sin x=m,\quad \tan x-\sin x=n. \] Then the possible ordered pair \((m,n)\) is:
If \( f(3) = 18, f'(3) = 0 \) and \( f''(3) = 4 \), then the value of \[ \lim_{x \to 3} \ln \left( \frac{f(x+2)}{f(3)} \right)^{\frac{18}{(x-3)^3}} \] is equal to
If \[ I(x) = 3\int \frac{dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}, \quad I(0) = \frac{\sqrt{3}}{4}, \] then find \( I(1) \):
\[ x - ny + z = 6 \\ x - (n - 2)y + (n + 1)z = 8 \\ (n - 1)y + z = 1 \] Let \( n \) be the number on the die when rolled randomly. Then \( P \) (that system equation has a unique solution) = \( \frac{k}{6} \). Then sum of value of \( k \) and all possible values of \( n \) is:
If \[ \lim_{x \to 0} \frac{e^{a(x-1)} + 2\cos(bx) + e^{-x}(c - 1)}{x \cos x - \ln(1 + x)} = 2, \] Then the value of \( a^2 + b^2 + c^2 \) is:
If \( P(10, 2\sqrt{15}) \) lies on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and the length of the latus rectum is 8, then the square of the area of \( \Delta PS_1S_2 \) is [where \( S_1 \) and \( S_2 \) are the foci of the hyperbola].
If \( a, b, c \) are in A.P. where \( a + b + c = 1 \) and \( a, 2b, c \) are in G.P., then the value of \( 9(a^2 + b^2 + c^2) \) is equal to:
Let \( \mathbf{a} = \sqrt{2} \hat{i} \) and \( \mathbf{b} = 5\hat{j} + \hat{k} \). If \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) and \( \mathbf{c} \) lies in the \( y \)-\( z \) plane such that \( |\mathbf{c}| = 2 \), then the maximum value of \( |\mathbf{c} \cdot \mathbf{d}| \) is equal to:
If \[ \int_0^6 \left( x^3 + \lfloor x^{1/3} \rfloor \right) \, dx = \alpha \] and \[ \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{\sin^6 x + \cos^6 x} \, dx = \beta, \] then the value of \( ab^2 \) is equal to:
If \( \alpha, \beta \) are roots of the equation \( 12x^2 - 20x + 3 = 0 \), \( \lambda \in \mathbb{R} \). If \( \frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2} \), then the sum of all possible values of \( \lambda \) is:
Let the domain of the function \[ f(x) = \log_3 \log_5 \left( 7 - \log_2 \left( x^2 - 10x + 15 \right) \right) + \sin^{-1} \left( \frac{3x - 7}{17 - x} \right) \] be \( (\alpha, \beta) \), then \( \alpha + \beta \) is equal to:
If \( \cos(\alpha + \beta) = -\frac{1}{10} \) and \( \sin(\alpha - \beta) = \frac{3}{8} \), where \[ 0<\alpha<\frac{\pi}{3} \quad \text{and} \quad 0<\beta<\frac{\pi}{4}, \] and \[ \tan(2\alpha) = \frac{3\left(1 - \sqrt{55}\right)}{\sqrt{11} \left(s + \sqrt{5}\right)}, \] and \( r, s \in \mathbb{N} \), then \( r^2 + s \) is:
Find the value of \[ \frac{6}{3^{26}} + 10\cdot\frac{1}{3^{25}} + 10\cdot\frac{2}{3^{24}} + \cdots + 10\cdot\frac{2^{24}}{3^{1}} : \]
The sum of coefficients of \(x^{499}\) and \(x^{500}\) in the expression: \[ (1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \cdots + x^{1000} \] is:
Let matrix \[ A=\begin{pmatrix} 3 & -4\\ 1 & -1 \end{pmatrix} \] and \(A^{100} = 100B + I\). Find the sum of all the elements in \(B^{100}\).