If equation of the plane that contains the point $$(-2,3,5)$$ and is perpendicular to each of the planes $$2x+4y+5z=8$$ and $$3x-2y+3z=5$$ is $$\alpha x+\beta y+\gamma z+97=0$$ then $$\alpha +\beta +\gamma =$$

Let the image of the point $$\mathrm{P}(1,2,6)$$ in the plane passing through the points $$\mathrm{A}(1,2,0),\mathrm{B}(1,4,1)$$ and $$\mathrm{C}(0,5,1)$$ be $$\mathrm{Q}(\alpha ,\beta ,\gamma )$$. Then $$\left({\alpha }^2+{\beta }^2+{\gamma }^2\right)$$ is equal to :

Let the line $$\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$$ intersect the lines $$\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$$ and $$\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$$ at the points $$\mathrm{A}$$ and $$\mathrm{B}$$ respectively. Then the distance of the mid-point of the line segment $$\mathrm{AB}$$ from the plane $$2x-2y+z=14$$ is :

The shortest distance between the lines {{x + 2} \over 1} = {y \over { - 2}} = {{z - 5} \over 2} and {{x - 4} \over 1} = {{y - 1} \over 2} = {{z + 3} \over 0} is :

Let two vertices of a triangle ABC be (2, 4, 6) and (0, $$-$$2, $$-$$5), and its centroid be (2, 1, $$-$$1). If the image of the third vertex in the plane $$x+2y+4z=11$$ is $$(\alpha ,\beta ,\gamma )$$, then $$\alpha \beta +\beta \gamma +\gamma \alpha$$ is equal to :

Let P be the point of intersection of the line {{x + 3} \over 3} = {{y + 2} \over 1} = {{1 - z} \over 2} and the plane $$x+y+z=2$$. If the distance of the point P from the plane $$3x-4y+12z=32$$ is q, then q and 2q are the roots of the equation :

For $$\mathrm{a},\mathrm{b}\in \mathbb{Z}$$ and $$|\mathrm{a}-\mathrm{b}|\le 10$$, let the angle between the plane $$\mathrm{P}:\mathrm{ax}+y-\mathrm{z}=\mathrm{b}$$ and the line $$l:x-1=\mathrm{a}-y=z+1$$ be $${\cos }^{-1}\left(\frac{1}{3}\right)$$. If the distance of the point $$(6,-6,4)$$ from the plane P is $$3\sqrt{6}$$, then $$a^4+b^2$$ is equal to :

Let $$\mathrm{P}$$ be the plane passing through the line $$\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}$$ and the point $$(2,4,-3)$$. If the image of the point $$(-1,3,4)$$ in the plane P is $$(\alpha ,\beta ,\gamma )$$ then $$\alpha +\beta +\gamma$$ is equal to :

The shortest distance between the lines $$\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$$ and $$\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$$ is :

If the equation of the plane containing the line $$x+2y+3z-4=0=2x+y-z+5$$ and perpendicular to the plane $\vec{r}=(\hat{i}-\hat{j})+\lambda (\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k})$ is $a x+b y+c z=4$, then $$(a-b+c)$$ is equal to :

If the equation of the plane passing through the line of intersection of the planes $$2x-y+z=3,4x-3y+5z+9=0$$ and parallel to the line $$\frac{x+1}{-2}=\frac{y+3}{4}=\frac{z-2}{5}$$ is $$ax+by+cz+6=0$$, then $$a+b+c$$ is equal to :

One vertex of a rectangular parallelopiped is at the origin $$\mathrm{O}$$ and the lengths of its edges along $$x,y$$ and $$z$$ axes are $$3,4$$ and $$5$$ units respectively. Let $$\mathrm{P}$$ be the vertex $$(3,4,5)$$. Then the shortest distance between the diagonal OP and an edge parallel to $$\mathrm{z}$$ axis, not passing through $$\mathrm{O}$$ or $$\mathrm{P}$$ is :

A plane P contains the line of intersection of the plane $$\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})=6$$ and $$\vec{r}\cdot (2\hat{i}+3\hat{j}+4\hat{k})=-5$$. If $$\mathrm{P}$$ passes through the point $$(0,2,-2)$$, then the square of distance of the point $$(12,12,18)$$ from the plane $$\mathrm{P}$$ is :

Let the line $$\mathrm{L}$$ pass through the point $$(0,1,2)$$, intersect the line $$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$$ and be parallel to the plane $$2x+y-3z=4$$. Then the distance of the point $$\mathrm{P}(1,-9,2)$$ from the line $$\mathrm{L}$$ is :

Let $$9=x_{1}

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha (>$ 0 ), and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and 30 $-\alpha$. If the mean and variance of the marks of the combined class of $100+\mathrm{n}$ studants are respectively 50 and 350 , then the sum of variances of classes $A$ and $B$ is :

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is :

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of 100 consecutive positive integers $a_1,a_2,a_3,\dots .,a_{100}$ is 25 . Then $S$ is :

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $$\mu$$ and $${\sigma }^2$$ represent mean and variance of X, respectively, then $$10({\mu }^2+{\sigma }^2)$$ is equal to :

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :