Home
Class 12
MATHS
" if " x(i) =a(i) b(i) C(i), i= 1,2,3 ar...

`" if " x_(i) =a_(i) b_(i) C_(i), i= 1,2,3` are three- digit positive integer such that each `x_(i)` is a mulptiple of 19 then prove that det`{{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}}` is divisible by 19.

Text Solution

Verified by Experts

`Delta = |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),((100a_(1)+10b_(1)+c_(1)),,(100a_(2)+10b_(2)+c_(2)),,(100a_(3)+10b_(3)+c)(3)):}|`
`|{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(x_(1),,x_(2),,x_(3)):}|= |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(19m_(1),,19m_(2),,19m_(3)):}|`
`19=|{:(a_(1) ,,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(m_(1),,m_(2),,m_(3)):}|=19n`
where `n= |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(m_(1),,m_(2),,m_(3)):}|` is certainly an integer.
Promotional Banner

Similar Questions

Explore conceptually related problems

If x_i=a_i b_i c+i,i=1,,23 are three-digit positive integer such that each x_1 is a multiple of 19, then for some integers n , prove that |a_1a_2a_3b_1b_2b_3c_1c_2c_3| is divisible by 19.

if Delta=det[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

if quad /_=[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

Given a_(i)^(2) + b_(i)^(2) + c_(i)^(2) = 1, i = 1, 2, 3 and a_(i) a_(j) + b_(i) b_(j) + c_(i) c_(j) = 0 (i !=j, i, j =1, 2, 3) , then the value of the determinant |(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))| , is

if a_(1)b_(1)c_(1), a_(2)b_(2)c_(2)" and " a_(3)b_(3)c_(3) are three-digit even natural numbers and Delta = |{:(c_(1),,a_(1),,b_(1)),(c_(2),,a_(2),,b_(2)),(c_(3),,a_(3),,b_(3)):}|" then " Delta is

Let veca=a_(1)hati+a_(2)hatj+a_(3)hatk,vecb=b_(2)hatj+b_(3)hatk and vecc=c_(1)hati+c_(2)hatj+c_(3)hatk gve three non-zero vectors such that vecc is a unit vector perpendicular to both veca and vecb . If the angle between veca and vecb is pi/6 , then prove that |{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|p=1/4 (a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))

Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(a) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(a) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(c ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(n) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}| is equal to