Evaluate `|^x C_1^x C_2^x C_3^y C_1^y C_2^y C_3^z C_1^z C_2^z C_3|`

Show that |^x C_r^x C_(r+1)^x C_(r+2)^y C_r^y C_(r+1)^y C_(r+2)^z C_r^z C_(r+1)^z C_(r+1)|=|^x C_r^(x+1)C_(r+1)^(x+2)C_(r+2)^y C_r^(y+1)C_(r+1)^(y+2)C_(r+2)^z C_r^(z+1)C_(r+1)^(z+2)C_(r+1)| .

Evaluate |{:(.^(x)C_(1),,.^(x)C_(2),,.^(x)C_(3)),(.^(y)C_(1),,.^(y)C_(2),,.^(y)C_(3)),(.^(x)C_(1),,.^(z)C_(2),,.^(z)C_(3)):}|

The general solution of the differential equation (d^2y)/dx^2=e^(-3x) is (A) y=9e^(-3x)+C_1x+C_2 (B) y=-3e^(-3x)+C_1x+C_2 (C) y=3e^(-3x)+C_1x+C_2 (D) y=e^(-3x)/9+C_1x+C_2

The determinant |[ C(x,1) ,C(x,2), C(x,3)] , [C(y,1) ,C(y,2), C(y,3)] , [C(z,1) ,C(z,2), C(z,3)]|= (i) 1/3xyz(x+y)(y+z)(z+x) (ii) 1/4xyz(x+y-z)(y+z-x) (iii) 1/12xyz(x-y)(y-z)(z-x) (iv) none

Consider the system of equations a_(1) x + b_(1) y + c_(1) z = 0 a_(2) x + b_(2) y + c_(2) z = 0 a_(3) x + b_(3) y + c_(3) z = 0 If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0 , then the system has

Show that |a b c a+2x b+2y c+2z x y z|=0

|{:(.^(x)C_(r),,.^(x)C_(r+1),,.^(x)C_(r+2)),(.^(y)C_(r),,.^(y)C_(r+1),,.^(y)C_(r+2)),(.^(z)C_(r),,.^(z)C_(r+1),,.^(z)C_(r+2)):}| is equal to

Let a=xhati+12hatj-hatk,b=2hati+2xhatjj+hatk and c=hati+hatk . If b,c,a in that order form a left handed system, then find the value of x. [x_(1)a+y_(1)b+z_(1)c,x_(2)a+y_(2)b+z_(2)c,x_(3)a+y_(3)b+z_(3)c] =|(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3))|[abc] .