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4. a) Evaluate : integrate (x ^ 3 + 1/(x ^ 3)) dx
b) int 0 ^ pi 2 -4 sin xdx 3+4 cos x
2. a) Find the derivatives of the following functions with respect to x.
x ^ 3 + y ^ 3 = 3
y = (sin x) ^ tan x
b) Evaluate the 2 ^ (nd) order partial derivatives partial^ 2 u partial x^ 2 and partial^ 2 u partial y^ 2 if u=2x^ 3 +3x^ 2 y+xy^ +y^ .
5. a) Calculate the area under the curve y = x ^ 3 + 4x + 1 from x = - 3 to x = 3
b) Evaluate: integrate x * e ^ x dx from 0 to 4
A has four keys in his pocket, he tries to unlock a door and tries with each key until he finds the right one, let X be random variable number of keys tried including the right key to open the door,...
a) is x a discrete or continuous random variable? Explain
b) determine probability function for random variable x.
C) what is the probability that the man will try at most 2 keys before he finds the right one?
D) determine the mean and standard deviation of x.
Show that a tree with n vertices has exactly n-1 edges
Define spanning trees in a weighted graph, with an example
Show that every simple finite graph has two vertices of the same degree
Find the least square regression for the following.
X: 0 1 2 3 4
Y: 2 3 5 4 6
The City of South Lake Tahoe, CA has an Asian population of 1419 people, out of a total population of 23,609 (Source: U.S. Census Bureau, Census 2000). Suppose that a survey of 1419 self-reported Asians in the Manhattan, NY, area yielded the data in the table below.
Conduct a goodness of fit test to determine if the self-reported sub-groups of Asians in the Manhattan area fit that of the Lake Tahoe area.
Race Lake Tahoe Frequency Manhattan
Frequency
Asian Indian 131 174
Chinese 181 557
Filipino 1045 518
Japanese 80 54
Korean 12 29
Vietnamese 9 21
Other 24 26
Write short note on Caesar cipher with example
