Version 1 MATH 133 Final Exam - April 18, 2008 II.2 (15 points) Consider the matrix M = [1 1 | 1 2 1 2 3 5 | 1 3 4 5 5 14 6 -3 71 -5 8 -6 11 -13 23 -7 9 . (a) Find the reduced echelon form of M, showing your work. (You should find that M has rank 3.)

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3 Jan 2018

3. If det au 012 013 221 222 223 231 232 233) (A) 6 (B) (3a11 +231 3012 + 232 3013 + 233 = 4, then det 221 +211 222 +212 223 +013 has the value 231 + 2011 232 + 2a12 033 + 2013 - 6 (C) – 12 (D) 4 (E) None of the above.

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2 Jan 2018

(b) Show that one of the eigenspaces of A has dimension 2.

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31 Dec 2017

14. The function defined by (sin(81) if x < 0, f(x) = { 2x la cos(5:) if x > 0. is continuous if and only if a is (a) 5, (b) 8, (c)4, (d) -1, (e) 0.

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31 Dec 2017

(5) A function f(r) is defined for all positive real numbers and satisfies the equation $() = 23 for every 1 > 0. Find f'(4). Answer:

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31 Dec 2017

(b) Find the shortest distance between the line AB and the line CD.

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30 Dec 2017

2. (10 marks) ex-y = xy. Determine the (a) Consider the curve C implicitly defined by the equation equation of the tangent line to Cat (1,1). (b) Compute the derivative of y = (sin x)sin .

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30 Dec 2017

3.(bola) let n = ( ) (1 2 3 3. (8pts) Let A= 2 4 6 14 8 12 (a) Solve the homogeneous system AX = 0. (1)

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30 Dec 2017

3. Let u and v be vectors in R3 such that ||u|| = 2, |v|| = 5, and u.v=3. Then ||u - vll= (A) 35 (B) /43 (C) 43 (D) 23 (E) 29

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29 Dec 2017

[ ] , V3 = 12. Let V1 = -3 4 -3 -1 , V2 = [-11 -6 , V4 = [ 1] 0 L-1 Which one of the following statements is true ? (A) {V1, V2, V3} is a basis of R3, (9) (v.) bei of span {[:] 7:]} i] [- (B) {V1, V2} is a basis of Span ( 0] [ 0 (C) {V1, V2, V4} is a basis of Span{V1, V2, V3, V4}, (D) {V1, V2, V3} is a basis of Span{V1, V2, V3}, (E) {V1, V2} is a basis of Span{V3, V4}.

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29 Dec 2017

(4) Find the equations of both lines through the point (2, -1) that are tangent to the parabola given by y= 22 +2.. a tenere et bien

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27 Dec 2017

a. The plane has de estar contre : 1-6-6-li) mesur paper [7] 9 [ 2 3 6. The plane 7 has the vector equation 1 = + 2 + . A vector perpen- dicular to this plane is (A) (1, -1,27 (B) (1,-1,1]? (C) (7,9,117 (D) (1,0,–2,7 (E) (-1,7, -9]T

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27 Dec 2017

1. If A = ( ) has determinant -3, B =( le d ) lcd has determinant 2, 20 20 and C=1 ), then the determinant of 3adi (C-1) is la-2a 6 -- 26 ): the B) -12. C) - D) - (E) None of the above. 10 Wien

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26 Dec 2017

3. (10 marks) A camera is located 10 meters to the west from a straight north-south road along which a car is traveling to the south at a speed of 20 meters per second. The camera turns so that it is pointed at the car at all times. How fast, in radians per second, is the camera turning as the car passes closest to the camera?

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23 Dec 2017

2. (10pts) Suppose A is a 3 x 3 matrix such that det A=2. Find det B in each of the following cases: (a) B= -A(AT)A-1