If are unit vectors such that and the angle between then prove that

i) , ii) .

A problem in mathematics is given to 4 students A, B, C, D. Their chances of solving the problem, respectively, are

What is the probability that

(i) The problem will be solved?

(ii) At most one of them will solve the problem?

Find the intervals in which the following function is strictly increasing or strictly decreasing. Also, find the points of local maximum and local minimum, if any.

f(x)= (x–1)³(x+2)²

Using integration, find the area bounded by the tangent to the curve 4y = x² at the point (2, 1) and the lines whose equations are x =2y and x = 3y–3.

Find the distance of the point from the plane 3 x + y – z + 2 = 0 measured parallel to the line . Also, find the foot of the perpendicular from the given point upon the given plane.

OR

Find the equation of the plane passing through the line of intersection of the planes and and passing through the point (3, –2, –1). Also, find the angle between the two given planes.