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# ptestSol - MAP 2302(0100 Fall-2011 1 Compute 1 3 1 2 2 What...

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MAP 2302 (0100) Fall-2011 Solutions to the PLACEMENT TEST 1. Compute 1 3 - 1 2 = - 1 6 2. What is larger 2010 2011 or 2011 2012 ? Note that 2010 2011 = 1 - 1 2011 and 2011 2012 = 1 - 1 2012 . Since 1 2012 < 1 2011 , it follows that 2010 2011 < 2011 2012 . 3. x - 4 x 2 - 8 x +16 + 1 x +4 - 2 x - 4 = x - 4 ( x - 4) 2 + 1 x +4 - 2 x - 4 = 1 x +4 - 1 x - 4 = - 8 ( x +4)( x - 4) = - 8 x 2 - 16 . 4. Solve the system 5 x + 2 y = 1 2 x + 5 y = 6 The answer: x = - 1 / 3 and y = 4 / 3 5. What is the derivative of π ? 0 6. R sin 2 tdt = - 1 2 cos 2 x . 7. Find the area of the rectangle spanned by the vertices: (0,0), (1,3), (5,2), and (6,5). Is this rectangle a parallelogram ? Let ¯ u = (1 , 3) and ¯ v = (5 , 2) be the corresponding vectors. Since ¯ u + ¯ v = (6 , 5), it is a parallelogram. The area is the absolute value of the determinant 1 5 3 2 which is 13. 1

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8. Solve the equation cos x sin x = 1 2 . Since 2 cos x sin x = sin 2 x , the equation turns into sin 2 x = 1. Therefore 2 x = π 2 + 2 πn , n = { 0 , ± 1 , ± 2 , ± 3 . . . } . Thus, x = π 4 + πn where n is any iteger. 9. Do the points (0 , 0 , 1), (1 , 0 , 0), ( 1 3 , 1 3 , 1 3 ), and (0 , 1 , 0) lie in one plane? Elementary solution: Yes, the point ( 1 3 , 1 3 , 1 3 ) is the center of mass of unit masses at (0 , 0 , 1), (1 , 0 , 0), and (0 , 1 , 0). Educated solution: Consider three vectors with issued from the 1st point
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ptestSol - MAP 2302(0100 Fall-2011 1 Compute 1 3 1 2 2 What...

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