Let S($$\alpha $$) = {(x, y) : y² $$ \le $$ x, 0 $$ \le $$ x $$ \le $$ $$\alpha $$} and A($$\alpha $$) is area of the region S($$\alpha $$). If for a $$\lambda $$, 0 < $$\lambda $$ < 4, A($$\lambda $$) : A(4) = 2 : 5, then $$\lambda $$ equals

If the system of linear equations

x – 2y + kz = 1
2x + y + z = 2
3x – y – kz = 3

has a solution (x,y,z), z $$ \ne $$ 0, then (x,y) lies on the straight line whose equation is :

Let ƒ(x) = a^x (a > 0) be written as
ƒ(x) = ƒ₁ (x) + ƒ₂ (x), where ƒ₁ (x) is an even function of ƒ₂ (x) is an odd function.
Then ƒ₁ (x + y) + ƒ₁ (x – y) equals

A student scores the following marks in five tests :

45, 54, 41, 57, 43.

His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is