If `a , b , c` are non-zero, then the system of equations `(alpha+a)x+alphay+alphaz=0,alphax+(alpha+b)y+alphaz=0,alphax+alphay+(alpha+c)z=0' has a non-trivial solution if `alpha^(-1)=`(A) ` -(a^(-1)+b^(-1)+c^(-1))` (B) `a+b+c` (C) `alpha+a+b+c=1` (D) none of these

The value of |alpha| for which the system of equation alphax+y+z=alpha-1 x+alphay+z=alpha-1 x+y+alphaz=alpha-1 has no solution , is "____"

If a ,b ,c are non -zero real numbers and if the system of equations (a-1) x=y+z,(b-1)y=z+x,(c-1) z=x+y has a non -trivial solution, then prove that ab+bc+ca =abc .

If p q r!=0 and the system of equation (p+a)x+b y+c z=0 , a x+(q+b)y+c z=0 , a x+b y+(r+c)z=0 has nontrivial solution, then value of a/p+b/q+c/r is a. -1 b. 0 c. 1 d. 2

An ordered pair (alpha, beta) for which the system of linear equations (1+alpha)x+betay+z=2 , alphax+(1+beta)y+z=3 and alphax+betay+2z=2 has unique solution is: (a) (2,4) (b) (-3,1) (c) (-4,2) (d) (1,-3)

If int_0^1 e^(x^2)(x-alpha)dx=0 , then (a) alphalt2 (b) alphalt0 (c)"" 0ltalphalt1 (d) alpha="0

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

If cot^3alpha+cot^2alpha+cotalpha=1 then (a) cos2alpha.tanalpha=-1 (b) cos2alpha.tanalpha=1 (c) cos2alpha-tan2alpha=1 (d) cos2alpha-tan2alpha=-1

If alpha_1, ,alpha_2,.... ,alpha_n are the roots of equation x^n+n a x-b=0, show that (alpha_1-alpha_2)(alpha_1-alpha_3).......(alpha_1-alpha_n)=n(alpha_1^ (n-1)+a)

If alpha=pi/(14), then the value of (tanalphatan2alpha+tan2alphatan4alpha+tan4alphatanalpha) is 1 (b) 1//2 (c) 2 (d) 1//3

Write True or False If A alpha 1/C and C alpha 1/B , then A alpha B .