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Let \(C\) be the curve given by the vector function \({\bf r}(t) =\langle 1- \frac{1}{2}t^2, t, t^2\rangle\). Find the equation of the osculating plane to \(C\) at the point \((1,0,0)\).

Use Green’s Theorem to compute \(\int_{C} {\bf F} \cdot {\bf dr}\) where \(C\) is the triangle with vertices \((0,2)\), \((-1,-1)\), \((3,-1)\) oriented counter-clockwise, and \({\bf F}= \langle y^{2}+\ln|1+x^{2}|, x(2y+1), e^{\sqrt{1+y^{2}}} \rangle.\)

Compute \(\oint_{C}{\bf F}\cdot d{\bf r}\) where \(C\) is the curve parametrized by \({\bf r}(t)= \langle \cos t,\sin t, \cos t+\sin t \rangle, \quad 0\leq t\leq 2\pi\) and where \({\bf F}=(xy+z){\bf i}+z^2{\bf j}+xyz{\bf k}\) [Hint: The curve lies on the surface \(z=x+y\)]

Evaluate the integral \(\int_{C} {\bf F} \cdot {\bf dr} \) where \({\bf F}=\left( \frac{1}{x}+y \right) {\bf i} + \left( \frac{1}{y} + x \right) {\bf j}\) and \(C\) is a smooth curve with starting point \((1,1)\) and end point \((2,3)\).  

TOTAL QUESTION 6: [10] TOTAL SECTION B: [20] GRAND TOTAL [50]

1.1 A group of elected people that make laws for their country. (1) A) Parliament   B) Electorate   C) President   D) Citizen  

5.1 Poverty can be described as (1)   A.     the state of not having enough material possessions or income for a person’s basic needs.     B.     having enough material possessions to live comfortably.     C.    a condition that results from eating a diet which does not supply a healthy amount of nutrients.     D.    the condition of not having adequate housing.  

3. Match column A with column B (5)   COLUMN A COLUMN B 3.1. To have supreme power or authority. A. Electorate 3.2. Example of a non-sovereign state. B. Sovereignty 3.3. It is a system of government in which power and authority are controlled by a single person or party. C.  Nation-state 3.4. A cultural group that is also a state. D. Hong Kong 3.5. All the people in a country who are allowed to vote. E. Totalitarian

6.4 Discuss the difference between a representative democracy and a direct democracy. (2)

5.7 Why do you think the Nubian life expectancy is so low? Provide one reason. (1)