The equation of the plane passing through the intersection of planes $2x - y + z = 3$, $4x - 3y + 5z = -9$ and parallel to the line $\dfrac{x+1}{2} = \dfrac{y+3}{4} = \dfrac{z-3}{5}$ is...

If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then the sum of values of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ and $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is ...

The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ of a linear programming problem subject to constraints $5x + 10y \leq 50$, $x + y \geq 1$, $y \leq 4$ and $x \geq 0, y \geq 0$ is $3\lambda$. Then the value of $\lambda$ is

In a multiple-choice examination, there are $10$ questions with one correct option out of $4$ options for each question. A student gets $4$ marks for each correct answer and $1$ mark is deducted for each incorrect answer. The probability that a student, guessing randomly on every question, scores $30$ marks in this exam is...