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Some people I know like numbers very much...

Ramanujan was lying on his deathbed in Chennai, India when his english chum G.H.Hardy took a plane from Cambridge to visit him in India. Well you see , he came to see him for the last time and complained to his indian friend:-

Hardy:- the taxi cab i took has the most un-interesting number..

Ramanujan:- which number?

Hardy:- 1729.

Ramanujan:- Hey there wait!!!! 1729... is NOT un interesting!

Hardy:- ...!

Ramanujan smiled weakly and said:-

- You see : 1729 is the smallest number which can be written as

1729 = 10^3 + 9^3

and

1729 = 12^3 + 1^3

that is, 1729 is the smallest number which can be broken down into two cubes in 2 different ways...

and the next such number is a very very large one in fact...

(internet source- life of Srinivasa Ramanujan)

question 2 is a favorite problem in high school vectors. i hope my small answer would be enough

**Qu:what is the vector equation of a Plane ( containing points P, say) which is parallel to xy plane and passes through point A (4,1,3) ??**

well

(1) as the plane is parallel to the xy plane, then... it is perpendicular to the z axis!!!

so, the normal vector is the k vector.

normal vector n = c k,

where c is a number and k is the vector along the z axis.

(2) if the plane passes through point A(4,1,3) then from anywhere in the plane, take a point P. We can contruct a vector lying on the plane itself

AP = OP - OA

which is at right angles to the normal vector of the plane.

(3) as vectors n and AP are at right angles to each other, their dot product is zero.

So, n. AP = 0

the vector equation is:

k. ( r - OA) = 0

Hi there!! I just wanted to write something not too foolish I hope..

A complex function is a function of complex number(s)

It can be assimple as f(z) = z^2 which is the square of the complex number z

A contour integral is obtained when you integrate f(z) w.r.t z

from one point on the complex plane ( the Argand diagram) to another point.

Usually when the path you follow starts from a point P & makes you travel along a path that brings u back again to the same point, this is probably the reason we use the french word Contour ( round about) integral... . {end of little comment.}

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