Math Answers

Find the values of a and b for which the following limit exists.
lim x→0 (ae^x +e^-x +bx)/(1-cosx)

true/false. justify your answer.
lim (x,y)→(0,0) (x^2-y^2)/(x^2+y^2) does not exist.

true/false.justify your answer.
The work done by the Force F(x, y) = (−y,x) in moving a particle along the
boundary of the ellipse 9x^2 + 4 − y^2 = 36 is 6.

true/false.justify your answer.
The set {(x, y, z) | x > 0, y > 0, z < 0} is a domain in R^3 .

true/false.justify your answer.
The level curves of z = y / x are hyperbolas.

true/false.justify your answer.
The function f (x, y, z) = e^(xyz) is integrable over [0, 1]×[0, 1]×[0,1]

Find the second Taylor polynomial of f (x, y) = 1 + 3 x^2 y − 4 y^3 at (1, 1)

The genetic features of a group of adult mice are such that the probability of an offspring
being albino is 0.2. If 50 offsprings are born to a group of such mice, find the probability that
15 or more of them are albinos.

Suppose the gene ‘A’ stands for tall and gene ‘a’ stands for short, and ‘A’ is dominant over
‘a’. Suppose the parents are of genotype aa and Aa.
i) In a family like this what is the probability of having a short child?
ii) In a family like this if there are four children, what is the probability that two children
are tall and two children are short?
iii) If you survey a large number of such families having four children, what is the average
number of tall children to be expected in a family?
iv) If you survey 256 families of this type, each having four children, how many families
are expected to have all tall children?

Near what points may the surface
x^3 + 3y^2 + 8xz^2 − 3z^3y =1
be represented as a graph of a differentiable function z = k (x, y)? Justify your
answer