If `plambda^4+qlambda^3+rlambda^2+slambda+t=|[lambda^2+3lambda, lambda-1, lambda+3] , [lambda^2+1, 2-lambda, lambda-3] , [lambda^2-3, lambda+4, 3lambda]|` then `t=`

Let p lambda^4+qlambda^3+rlambda^2+slambda+t = |{:(lambda^2+3lambda,lambda-1,lambda-3),(lambda-1,-2lambda,lambda-4),(lambda-3,lambda+3," "3lambda):}| be an identity in lambda where p,q,r,s and t are constants. Then the value of t is

If veca satisfies vec axx( hat i+2 hat j+ hat k)= hat i- hat k , then vec a is equal to a. lambda hat i+(2lambda-1) hat j+lambda hat k ,lambda in R b. lambda hat i+(1-2lambda) hat j+lambda hat k ,lambda inR c. lambda hat i+(2lambda+1) hat j+lambda hat k ,lambda inR d. lambda hat i-(1+2lambda) hat j+lambda hat k ,lambda inR

If [(lambda^(2)-2lambda+1,lambda-2),(1-lambda^(2)+3lambda,1-lambda^(2))]=Alambda^(2)+Blambda+C , where A, B and C are matrices then find matrices B and C.

Find all values of lambda for which the (lambda-1)x+(3lambda+1)y+2lambdaz=0, (lambda-1)x+(4lambda-2)y+(lambda+3)z=0, 2x+(3lambda+1)y+3(lambda-1)z=0 possess non-trivial solution and find the ratios x:y:z, where lambda has the smallest of these value.

Let L_1-=ax+by+aroot(3)(b)=0 and L_2-=bx+ay+aroot(3)(a)=0 be two straight lines. The equations of the bisectors of the angle formed by the lines whose equations are lambda_1L_1-lambda_2L_2=0 and lambda_1L_1+lambda_2L_2=0 and lambda_1 and lambda_2 being non zero real numbers, are given by

If cos^-1 lambda + cos^-1 mu + cos^-1 gamma = 3 pi , then find the value of lambda mu + mu gamma + gamma lambda

If f(x)=lambda|sinx|+lambda^2|cosx|+g(lambda) has a period = pi/2 then find the value of lambda

Portion of asymptote of hyperbola x^2/a^2-y^2/b^2 = 1 (between centre and the tangent at vertex) in the first quadrant is cut by the line y + lambda(x-a)=0 (lambda is a parameter) then (A) lambda in R (B) lambda in (0,oo) (C) lambda in (-oo,0) (D) lambda in R-{0}

If the equation 4x^2-x-1=0 and 3x^2+(lambda+mu)x+lambda-mu=0 have a root common, then the irrational values of lambda and mu are a lambda=(-3)/4 b. lambda=0 c. mu=3/4 b. mu=0

If z=(lambda+3)+isqrt(5-lambda^2) then the locus of Z is