Let the plane passing through the point ($$-$$1, 0, $$-$$2) and perpendicular to each of the planes 2x + y $$-$$ z = 2 and x $$-$$ y $$-$$ z = 3 be ax + by + cz + 8 = 0. Then the value of a + b + c is equal to :

A plane P contains the line $$x+2y+3z+1=0=x-y-z-6$$, and is perpendicular to the plane $$-2x+y+z+8=0$$. Then which of the following points lies on P?

Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $$\overrightarrow{r}.\left(\hat{i}+\hat{j}+4\hat{k}\right)=16$$ and $$\overrightarrow{r}.\left(-\hat{i}+\hat{j}+\hat{k}\right)=6$$. Then which of the following points does NOT lie on P?

The distance of the point (1, $$-$$2, 3) from the plane x $$-$$ y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, $$-$$6 is :

Equation of a plane at a distance \sqrt {{2 \over {21}}} from the origin, which contains the line of intersection of the planes x $$-$$ y $$-$$ z $$-$$ 1 = 0 and 2x + y $$-$$ 3z + 4 = 0, is :

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl + lm = 0, is :

The equation of the plane passing through the line of intersection of the planes $$\overrightarrow{r}.\left(\hat{i}+\hat{j}+\hat{k}\right)=1$$ and $$\overrightarrow{r}.\left(2\hat{i}+3\hat{j}-\hat{k}\right)+4=0$$ and parallel to the x-axis is :

Let the equation of the plane, that passes through the point (1, 4, $$-$$3) and contains the line of intersection of the planes 3x $$-$$ 2y + 4z $$-$$ 7 = 0 and x + 5y $$-$$ 2z + 9 = 0, be $$\alpha$$x + $$\beta$$y + $$\gamma$$z + 3 = 0, then $$\alpha$$ + $$\beta$$ + $$\gamma$$ is equal to :

The distance of the point ($$-$$1, 2, $$-$$2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x $$-$$ 2y + z = 0 is :

Let the acute angle bisector of the two planes x $$-$$ 2y $$-$$ 2z + 1 = 0 and 2x $$-$$ 3y $$-$$ 6z + 1 = 0 be the plane P. Then which of the following points lies on P?

The distance of line $$3y-2z-1=0=3x-z+4$$ from the point (2, $$-$$1, 6) is :

Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

Let in a series of 2n observations, half of them are equal to a and remaining half are equal to $$-$$a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a2 + b2 is equal to :

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are :

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and {{20} \over 3}, respectively, then the value of | a $$-$$ b | is equal to :

Let the mean and variance of the frequency distribution \matrix{ {x:} & {{x_1} = 2} & {{x_2} = 6} & {{x_3} = 8} & {{x_4} = 9} \cr {f:} & 4 & 4 & \alpha & \beta \cr } be 6 and 6.8 respectively. If x3 is changed from 8 to 7, then the mean for the new data will be :

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $$\sqrt{13.44}$$, then the standard deviation of the second sample is :

If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and {{37} \over 4} respectively, then (a $$-$$ b)2 is equal to :

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. if $$\alpha$$ and $$\sqrt{\beta }$$ are the mean and standard deviation respectively for correct data, then ($$\alpha$$, $$\beta$$) is :

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :