A speaks truth in 60% of the cases, while B is 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ?
From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find :
(a) The probability distribution of X
(b) Mean of X
(c) Variance of X
The Cartesian equation of line is : 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, – 5, 0) and parallel to the given line.
Water is dripping out from a conial funnel of semi-verticle angle at the uniform rate of 2 cm2/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Using matrices, solve the following system of equations :
2x - 3y + 5z = 11
3x + 2y - 4z = - 5
x + y - 2z = - 3
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height.
Let VAB be a cone of greatest volume inscribed in a sphere of radius 12. It is obvious that for maximum volume the axis of the cone must be along a diameter of the sphere.
Let VC be the axis of the cone and O be the centre of the sphere such that OC = x.
Then,
VC = VO + OC = R + x = (12 + x) = height of cone
Applying Pythagoras theorem to triangle OAC,
OA2 = AC2 + OC2
AC2 = (12)2 - x2 = 144 - x2
Let V be the volume of the cone, then
V =
=
= ...(i)
=
=
Now,
i.e., 144 - 24x - 3x2 = 0
i.e., x2 + 8x - 48 = 0
i.e., (x + 12)(x - 4) = 0
i.e., x = - 12 or x = 4
Thus, V is maximum when x = 4
Putting x = 4 in (1), we obtain
Height of the cone = x + R = 4 + 12 = 16 cm