List of top Mathematics Questions

If the quadratic equation \( ax^2 + bx + c = 0 \) (\( a > 0 \)) has two roots \( \alpha \) and \( \beta \) such that \( \alpha < -2 \) and \( \beta > 2 \), then:
In \(\mathbb{R}\), a relation \(p\) is defined as follows: For \(a, b \in \mathbb{R}\), \(apb\) holds if \(a^2 - 4ab + 3b^2 = 0\).
Then:
Two smallest squares are chosen one by one on a chessboard. The probability that they have a side in common is:
If \(x y' + y - e^x = 0, \, y(a) = b\), then
\[ \lim_{x \to 1} y(x) \text{ is} \]
A unit vector in XY-plane making an angle \(45^\circ\) with \(\hat{i} + \hat{j}\) and an angle \(60^\circ\) with \(3\hat{i} - 4\hat{j}\) is:
In a plane, \(\vec{a}\) and \(\vec{b}\) are the position vectors of two points \(A\) and \(B\) respectively. A point \(P\) with position vector \(\vec{r}\) moves on that plane in such a way that
\[ |\vec{r} - \vec{a}| - |\vec{r} - \vec{b}| = c \quad (\text{real constant}). \]
The locus of \(P\) is a conic section whose eccentricity is:
The angle between two diagonals of a cube will be:
Let \(f : \mathbb{R} \to \mathbb{R}\) be a function defined by \(f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}\), then:
The equation \(2x^5 + 5x = 3x^3 + 4x^4\) has:
The expression \(\cos^2 \theta + \cos^2 (\theta + \phi) - 2 \cos \theta \cos (\theta + \phi)\) is:
If \(\alpha, \beta\) are the roots of the equation \(ax^2 + bx + c = 0\), then:
\[ \lim_{x \to \beta} \frac{1 - \cos(ax^2 + bx + c)}{(x - \beta)^2} \]
If \(U_n (n = 1, 2)\) denotes the \(n\)-th derivative (\(n = 1, 2\)) of \(U(x) = \frac{Lx + M}{x^2 - 2Bx + C}\) (\(L, M, B, C\) are constants), then \(PU_2 + QU_1 + RU = 0\) holds for:
If \(0 < \theta < \frac{\pi}{2}\) and \(\tan 30^\circ \neq 0\), then \(\tan \theta + \tan 2\theta + \tan 3\theta = 0\) if \(\tan \theta \cdot \tan 2\theta = k\), where \(k =\):
A biased coin with probability \(p\) (where \(0 < p < 1\)) of getting head is tossed until a head appears for the first time. If the probability that the number of tosses required is even is \(\frac{2}{5}\), then \(p =\):
If
\[ \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} \cdot A \cdot \begin{pmatrix} -3 & 2 \\ 5 & -3 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \]
then \(A\) is:
Consider the function f(x) = (x−2)logx. Then the equation xlogx = 2−x has:
The area bounded by the curves \(x = 4 - y^2\) and the Y-axis is:
For the real numbers \( x \) and \( y \), we write \( x \, P \, y \) iff \( x - y + \sqrt{2} \) is an irrational number.
Then the relation \( P \) is:
For every real number \(x \neq -1\), let \(f(x) = \frac{x}{x+1}\). Write \(f_1(x) = f(x)\) and for \(n \geq 2\), \(f_n(x) = f(f_{n-1}(x))\). Then \(f_1(-2), f_2(-2), \ldots, f_n(-2)\) must be:
Let N be the number of quadratic equations with coefficients from {0,1,2,...,9} such that 0 is a solution of each equation. Then the value of N is:
Let
\[ A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 1 \\ 7 \end{bmatrix}. \]
For the validity of the result \(AX = B\), \(X\) is:
The points of extremum of \[ \int_{0}^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} \, dt \] are:
If \(P(x) = ax^2 + bx + c\) and \(Q(x) = -ax^2 + dx + c\) where \(ac \neq 0\), then \(P(x) \cdot Q(x) = 0\) has:
Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function and \(f(1) = 4\). Then the value of
\[ \lim_{x \to 1} \int_{4}^{f(x)} \frac{2t}{x - 1} \, dt \]
\(\triangle OAB\) is an equilateral triangle inscribed in the parabola \(y^2 = 4ax, \, a>0\) with \(O\) as the vertex. Then the length of the side of \(\triangle OAB\) is: