If $$\lambda$$ $$\in$$ R is such that the sum of the cubes of the roots of the equation, x2 + (2 $$-$$ $$\lambda$$) x + (10 $$-$$ $$\lambda$$) = 0 is minimum, then the magnitude of the difference of the roots of this equation is :

Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of $$\Delta$$PTQ is :

If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of AB is :

A normal to the hyperbola, 4x2 $$-$$ 9y2 = 36 meets the co-ordinate axes $$x$$ and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the ocus of P is :

The locus of the point of intersection of the lines, $$\sqrt{2}x-y+4\sqrt{2}k=0$$ and $$\sqrt{2}kx+ky-4\sqrt{2}=0$$ (k is any non-zero real parameter), is :

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is :

If the angle between the lines, {x \over 2} = {y \over 2} = {z \over 1} and {{5 - x} \over { - 2}} = {{7y - 14} \over p} = {{z - 3} \over 4}\,\, is {\cos ^{ - 1}}\left( {{2 \over 3}} \right), then p is equal to :

The sum of the intercepts on the coordinate axes of the plane passing through the point ($$-$$2, $$-2,$$ 2) and containing the line joining the points (1, $$-$$1, 2) and (1, 1, 1) is :

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = 7 is :

A variable plane passes through a fixed point (3,2,1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz -plane through A, a second plane is drawn parallel zx-plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is :

An angle between the plane, x + y + z = 5 and the line of intersection of the planes, 3x + 4y + z $$-$$ 1 = 0 and 5x + 8y + 2z + 14 =0, is :

An angle between the lines whose direction cosines are gien by the equations, $$l$$ + 3m + 5n = 0 and 5$$l$$m $$-$$ 2mn + 6n$$l$$ = 0, is :

A plane bisects the line segment joining the points (1, 2, 3) and ($$-$$ 3, 4, 5) at rigt angles. Then this plane also passes through the point :

If L1 is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L2 is the line of intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z - 1 = 0, then the distance of the origin from the plane, containing the lines L1 and L2, is :

If $$\sum _{i=1}^9\left(x_i-5\right)=9$$ and $$\sum _{i=1}^9{\left(x_i-5\right)}^2=45$$, then the standard deviation of the 9 items $$x_1,x_2,.......,x_9$$ is

If the mean of the data : 7, 8, 9, 7, 8, 7, $$\lambda$$, 8 is 8, then the variance of this data is :

The mean of set of 30 observations is 75. If each observation is multiplied by a non-zero number $$\lambda$$ and then each of them is decreased by 25, their mean remains the same. Then $$\lambda$$ is equal to :

The mean and the standard deviation(s.d.) of five observations are9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is :

The sum of the first 20 terms of the series 1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ..., is :

Let An = \left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} $$-$$. . . . . + ($$-$$1)n-1 {\left( {{3 \over 4}} \right)^n} and Bn = 1 $$-$$ An. Then, the least dd natural numbr p, so that Bn > An , for all n$$\ge$$ p, is :