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1. Define/describe an isosceles spherical triangle. State its properties and use illustrations if necessary.
2. An isosceles spherical triangle has an angle A=B= 54° and side b = 82°. Find the measure of the third angle.
3. Determine the value of angle B of an isosceles spherical triangle ABC whose given parts are b=c= 54°28’ and a = 92°30’
4. Solve for the side b of a right spherical triangle ABC whose parts of an isosceles spherical triangle are a = 46°, b = 75° and C = 90°.
2. Use change of variables technique to integrate the following
Z
x(3x − 2) 1
2 dx
Determine the length of the arc (in radian measure) and the measure of the angle (in radian degree measures) generated by a point that starts (1,0)and terminates at the following:
1. Positive x-axis
2. Negative x-axis
3.Positive y-axis
4.Negative y-axis
Given that U is a function of x, y, and z
and A a vector field, prove that:
∇×(UA)=(∇U)×A+U(∇×A).
(a) Evaluate∫[
𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙.
(b) Use MATLAB to generate some typical integral curves of 𝑓(𝑥) =
𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙over the interval (−5,5).
Let E:=Exi^+Eyj^+Ezk^ and H:=Hxi^+Hyj^+Hzk^ be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations:∇⋅E=0,∇⋅H=0,∇×E=−1c∂H∂t,∇×H=1c∂E∂tprove that E and H satisfy the equation∇2Ei=1c2∂2Ei∂t2 and ∇2Hi=1c2∂2Hi∂t2Here, i=x,y or z.
Let r=xi^+yj^+zk^ and r=r .Show that:∇(lnr)=rr2. and∇×(rnr)=0.
If A= {1,3,4,6,7} B= {1,2,3,7,6,9,4} C={3,2,7,9,6,1,5,8,4} depict sets A,B and C in a Venn diagram
For each recurrence relation and initial conditions, find: (i) general solution;
(ii) unique solution with the given initial conditions:
(a) an = 3an−1 + 10an−2; a0 = 5, a1 = 11
The packaging on an electric light bulb states that the average length of life of bulbs is 1000 hours. A consumer association thinks that this is an overestimate and tests a random sample of 64 bulbs, recording the life x
hours, of each bulb. The results are summarised as follows:
∑x=63910.4,∑x2=63824061
A significance test, at the 10% level, is done to test where the statement on the packaging in overestimating the length of life of this type of light bulb. What is the null and alternative hypothesis for this test?
