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 " 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.

The least positive integer n such that ((2i)/(1+i))^n is a positive integer is a. 16 b. 8 c. 4 d. 2

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat k and vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^2)

Let vec a=a_1 hat i+a_2 hat j+a_2 hat k , vec b=b_1 hat i+a_2 hat j+b_2 hat k , and vec c=c_1 hat i+c_2 hat j+c_2 hat k , be three non-zero vectors such that vec c is a unit vector perpendicular to both vectors vec a and vec b . If the angle between a and b is pi//6, then |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2 is equal to

The smallest positive integer n for which ((1+i)/(1-i))^n=1 is (a) 8 (b) 16 (c) 12 (d) None of these

If the roots of the cubic equation, x^3+a x^2+b x+c=0 are three consecutive positive integers, then the value of (a^2//b+1) is equal to?

If I is the incenter of A B Ca n dR_1, R_2,a n dR_3 re, respectively, the radii of the circumcircles of the triangles I B C ,I C Aa n dI A B , then prove that R_1R_2R_3=2r R^2dot

If the roots of the cubic, x^3+a x^2+b x+c=0 are three consecutive positive integers, then the value of a^2/(b+1) is equal to __________.

If A=d i ag\ (a\ \ b\ \ c) , show that A^n=d i ag\ (a^n\ \ b^n\ \ c^n) for all positive integer n .

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