The area of a parallelogram whose adjacent sides are represented by the vectors `vec a = 2 hat i + hat j +3 hat k` and `vec b = hat i- hat j ` is

Answer

A, B, C, D

Explanation

`atoq,btos,ctop,dtor`.
a. (x+1)(x+3)..(x+19)
Here, number of factors=10
`therefore` The coefficent of `x^(19)=1+3+5+..+19`
`=10/2(1+19)=100`
b. Coefficent of `x^(8)`= Sum of product of numbers 1,3,5,7,
….19 taken two at a time
Let the coeffiecnt be S. Then
`(1+3+5+...+19)^(2)=1^(2)+3^(2)+5^(2)+7^(2)+.....+19^(2)+2xxS`
`thereforeS=((10/2(1+19))^(2)-sum_(r=1)^(10)(2r-1)^(2))/2`
`=(10000-1330)/2=4335`
c. Coefficent of `x^(9)` in (x+1)(x+2)`(x+2^(2))...(x+2^(9))`
`=1+2+2^(2)+..+2^(9)`
`=1(2^(10-1)/(2-1))`
`=2^(10)-1=1023`
d. Cofficient of `x^(8)`=Sum of product of numbers `1,2,2^(2),2^(3),....,2^(9)`
taken two at a time
Let the coefficient be S. Then
`(1+2+2^(2)+..+2^(9))^(2)`
`=(1)^(2)+(2)^(2)+(2^(2))^(2)+...+(2^(9))^(2)+2xxS`
`thereforeS=(((2^(10)-1)/(2-1))^(2)-((2^(2))^(10)-1)/(2^(2)-1))/2`
`=(2^(20)-3cdot2^(10)+2)/3`
`=(1021xx2^(10)+2)/3`