If the line, {{x - 3} \over 1} = {{y + 2} \over { - 1}} = {{z + \lambda } \over { - 2}} lies in the plane, 2x−4y+3z=2, then the shortest distance between this line and the line, {{x - 1} \over {12}} = {y \over 9} = {z \over 4} is :

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of $$\Delta$$ABC is :

If x = a, y = b, z = c is a solution of the system of linear equations x + 8y + 7z = 0 9x + 2y + 3z = 0 x + y + z = 0 such that the point (a, b, c) lies on the plane x + 2y + z = 6, then 2a + b + c equals :

The line of intersection of the planes $$\overrightarrow{r}.\left(3\hat{i}-\hat{j}+\hat{k}\right)=1$$ and $$\overrightarrow{r}.\left(\hat{i}+4\hat{j}-2\hat{k}\right)=2,$$ is :

The coordinates of the foot of the perpendicular from the point (1, $$-$$2, 1) on the plane containing the lines, {{x + 1} \over 6} = {{y - 1} \over 7} = {{z - 3} \over 8} and {{x - 1} \over 3} = {{y - 2} \over 5} = {{z - 3} \over 7}, is :

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line, {x \over 1} = {y \over 4} = {z \over 5} is Q, then PQ is equal to:

The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal perpendicular to both the lines {{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3} and {{x - 2} \over 2} = {{y + 1} \over { - 1}} = {{z + 7} \over { - 1}} is :

The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is :

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is :

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then {{a + b} \over {a - b}} is equal to :

If the sum of the first n terms of the series $$\sqrt{3}+\sqrt{75}+\sqrt{243}+\sqrt{507}+......$$ is $$435\sqrt{3},$$ then n equals :

If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

Let Sn = {1 \over {{1^3}}}+ {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n} \over {{1^3} + {2^3} + ...... + {n^3}}}. If 100 Sn = n, then n is equal to :

For any three positive real numbers a, b and c, 9(25$$a^2$$ + b2) + 25(c2 - 3$$a$$c) = 15b(3$$a$$ + c). Then

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :

The function f defined by f(x) = x3 $$-$$ 3x2 + 5x + 7 , is :

The tangent at the point (2, $$-$$2) to the curve, x2y2 $$-$$ 2x = 4(1 $$-$$ y) does not pass through the point :

The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point :

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

The area (in sq. units) of the region $$\left\{\left(x,y\right):x\ge 0,x+y\le 3,x^2\le 4yandy\le 1+\sqrt{x}\right\}$$ is