Unclassified (maths)

The number of relations on the set $A = \{1, 2, 3\}$ containing at most 6 elements including $(1, 2)$, which are reflexive and transitive but not symmetric, is _____

If $\lim_{t \to 0} \left(\int_0^1 (3x + 5)^t dx\right)^{\frac{1}{t}} = \frac{2}{56} \left(\frac{8}{5}\right)^{\frac{2}{\alpha}}$, then $\alpha$ is equal to:

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to:

Let ⌊·⌋ denote the greatest integer function. If $\displaystyle\int_{0}^{e^3} \Bigl\lfloor\frac{1}{e^{x-1}}\Bigr\rfloor\,dx = \alpha - \ln 2,$ then $\alpha^3$ is equal to:

Let C be the circle x² + (y-1)² = 2, E_1 and E_2 be two ellipses whose centres lie at the origin and major axes lie on the x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E_1 and E_2 at P(x_1,y_1), Q(x_2,y_2) and R(x_3,y_3) respectively. Given that P is the midpoint of the line segment QR and PQ = \frac{2\sqrt{2}}{3}, the value of 9(x_1y_1 + x_2y_2 + x_3y_3) is equal to _____

Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is:

If the set of all $a \in \mathbb{R} \setminus \{1\}$, for which the roots of the equation $(1 - a)x^2 + 2(a - 3)x + 9 = 0$ are positive is $(-\infty, -\alpha] \cup [\beta, \gamma)$, then $2\alpha + \beta + \gamma$ is equal to:

Let the area of the triangle formed by the lines $x + 2 = y - 1 = z$, $\frac{x - 3}{5} = \frac{y}{-1} = \frac{z - 1}{1}$, and $\frac{x}{-3} = \frac{y - 3}{3} = \frac{z - 2}{1}$ be $A$. Then $A^2$ is equal to _____

Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ) = \frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to:

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is: