List of top Mathematics Questions

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:
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:
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:
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:
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 \(B^{100}\).
The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{12(3+[x])\,dx}{3+[\sin x]+[\cos x]} \] (where \([\,]\) denotes the greatest integer function) is:
Find the maximum distance between the two curves: \[ |z-2| = 4 \quad \text{and} \quad |z-2| + |z+2| = 5 \]
Let \[ f(x)=\lim_{\theta\to 0}\frac{\cos(\pi x)-x^{2/\theta}\sin(x-1)}{1-x^{2/\theta}(x-1)}. \] Statement 1: \(f(x)\) is discontinuous at \(x=1\).
Statement 2: \(f(x)\) is continuous at \(x=-1\).
Statement 1 : \(2^{513}+2^{013}+8^{13}+3^{13}\) is divisible by \(7\).
Statement 2 : The value of integral part of \((7+4\sqrt{3})^{25}\) is an odd number.