List of top Mathematics Questions

If \( f(x) = \frac{e^x - 1}{e^x + 1} \), then \( \lim_{x \to \infty} f(x) \) is:
Evaluate: \[ \lim_{x \to 0} \frac{\sqrt{11 + |x|} - 6\sqrt{2 + |x|}}{6 - 2\sqrt{2 + |x|}} \]
If $[\cdot]$ denotes greatest integer function, then $\lim_{x \to 3} \dfrac{[-x]}{x} =$
Evaluate: $$ \lim_{x \to \infty} \log_e(\cosh x) + x $$
If $ a, b, c $ are distinct real numbers and: $$ \lim_{x \to \infty} \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} = \frac{1}{2} $$ then what is the value of $ a + 2c $?
Evaluate: $$ \lim_{x \to \infty} \frac{3|x| - x}{|x| - 2x} - \lim_{x \to 0} \frac{\log(1 + x^3)}{\sin^3 x} $$
If $\displaystyle \lim_{x \to \infty} x^n \log_e x = 0$, then $\log_e 12 =$
If \( \lim_{x \to 0} x^2 \left( \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \right) = k \), and \( \lim_{x \to 0} x^2 \left( \frac{e^{kx} - e^{-kx}}{e^{kx} + e^{-kx}} \right) = 1 \), then:
If $l, m$ $(l<m)$ are roots of $ax^2 + bx + c = 0$, then $\lim_{x \to a} \left| \dfrac{ax^2 + bx + c}{ax^2 + bx + c} \right|$ =
Let $f(x) = \begin{cases} \dfrac{1}{|x|}, & |x|>1 \\ ax^2 + b, & |x| \leq 1 \end{cases}$. If $\lim_{x \to 1} f(x)$ and $\lim_{x \to -1} f(x)$ exist, then the possible values for $a$ and $b$ are
\( \frac{x^3}{(x^2+1)(x^2+3)} = \)
If $\dfrac{13x + 43}{2x^2 + 17x + 30} = \dfrac{A}{2x + 5} + \dfrac{B}{x + 6}$, then $A^2 + B^2 = $
If \( y^2+z^2=3yz, z^2+x^2=8zx, x^2+y^2=4xy \), then the value of \( \frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} \) is
If \(\alpha, \beta\) are the roots of \(ax^2+bx+c=0\), then the quadratic equation whose roots are \( \sqrt{5}\alpha, \sqrt{5}\beta \) is
If \[ \frac{x^4 + 24x^2 + 28}{(x^2 + 1)^3} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2} + \frac{Ex + F}{(x^2 + 1)^3} \] then the value of $A + B + C + D + E + F =$ ?
The number of solutions of the equations \[ x + y + z = 1,\quad x^2 + y^2 + z^2 = 1,\quad x^3 + y^3 + z^3 = 1 \] is:
Given: $$ \frac{2x^2 + 1}{x^3 - 1} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + x + 1} $$ Find the value of $ 7A + 2B + C $
If the equivalent partial fraction of \[ \frac{x^3}{(2x - 1)(x + 2)(x - 3)} \] is of the form \[ A + \frac{B}{2x - 1} + \frac{C}{x + 2} + \frac{D}{x - 3} \] then the value of \( A + B + C = \) ?
If 2, 3, 6 are the roots of the polynomial \(f(x) = x^3+ax^2+bx+c\), where \(a,b,c \in C\). Then the value of a--c is
In a triangle \( ABC \), the identity holds: \[ \tan\frac{A}{2} \tan\frac{B}{2} + \tan\frac{B}{2} \tan\frac{C}{2} + \tan\frac{C}{2} \tan\frac{A}{2} = ? \]
In \( \triangle ABC \), let \( O \) be the circumcenter and \( G \) be the centroid. Then: \[ OG^2 = ? \]
Find the area of the triangle formed by the tangent to the curve \( xy = a^2 \) at \( (x_1, y_1) \) and the coordinate axes.
In a triangle, if \( b = 5 \), \( c = 6 \), and \( \tan\left(\frac{A}{2}\right) = \frac{1}{\sqrt{2}} \), then the side \( a = \) ?
In a triangle \( ABC \), the identity holds: \[ s \left( \frac{r_1 - r}{a} + \frac{r_2 - r}{b} + \frac{r_3 - r}{c} \right) = ? \]
Given $a_i^2 + b_i^2 + c_i^2 = 1$ and $a_i a_j + b_i b_j + c_i c_j = 0$ for $i \neq j$, and
\[ A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \] Then $\det(AA^T) = $ ?