1. find the centroid of a triangle mid points of whose sides are ( 1 , 2, -3 ) ( 3, 0, 1) and ( -1, 1, -4 )

2. find the coefficient of x⁵ y⁷ in ( x - 7y)¹²

3. the 3^rd 4^th and 5^thterms in the expansion of ( x + a)ⁿ are respectivle 84 ,280 ,and 560 find the values of x ,a ,and n

(1)

The Centroid of the triangle coincides with the Centroid of the triangle formed by joining the middle points of the sides of the triangle.

Hence, the co-ordinate of the Centroid of the given triangle is,

Hence, the co-ordinate of the Centroid of the given triangle is

(2)

Suppose  term contain  in the binomial expansion of

Now,

  …… (1)

This term will contain if,

So,  i.e.  term contain.

Putting  in equation (1), we get,

So,

Coefficient of

(3)

It is given that,

We know that,

Now,

   …… (1)

   …… (2)

   …… (3)

In order to eliminate x and a, we multiply (1) and (3) and then dividing the product by the square of (2),

So,

Now,

Putting  in above equation, we have,

  …… (4)

And,

Putting  and  in above equation, we have,

Putting  in equation (3),

Hence