QuesSol_2003 - FACULTY OF ARTS AND SCIENCE University of...

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FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL/MAY 2003 MAT 133Y1Y Calculus and Linear Algebra for Commerce PART A. MULTIPLE CHOICE 1. [3 marks] A $100,000 mortgage is to be repaid over 10 years by equal monthly payments made at the end of each month; that is, the first payment is one month after the loan is made. If interest is 10% compounded semiannually, the amount of each payment is A $1,264.45 B $1,310.34 C $1,273.96 D $1,299.72 E $1,335.10 2. [3 marks] A $100 bond with 10 years until maturity has semiannual coupons at an annual coupon rate of 10%. If its annual yield is 9%, then its market price is A $108.48 B $107.76 C $105.97 D $109.03 E $106.50 3. [3 marks] If A is a 4 × 4 matrix and det( A ) = 5 , then det(3 A 2 ) = A 225 B 75 C 2025 D - 75 E 300 1
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4. [3 marks] f ( x ) = ( x 2 - 4) 1 / 3 has A an absolute minimum at x = 0 and no absolute maximum B absolute minimums at x = 2 and x = - 2 and a relative maximum at x = 0 C a relative minimum at x = 0 and absolute maximums at x = - 2 and x = 2 D a relative maximum at x = 0 and no relative minimums E no relative maximums or minimums 5. [3 marks] If the cost function is given by: c = 0 . 01 q 2 + 6 q + 100 Then average cost ¯ c is minimized when q = A 300 B 0 C 1 D 100 E 1000 6. [3 marks] lim x + ln(1 + e 2 x ) x A is 1 B is 4 C is 0 D is 2 E does not exist 2
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7. [3 marks] Z e 4 e 1 x (ln x ) 1 2 dx = A 1 e 2 B 1 2 e 4 C 1 2 e 4 - 1 2 e D 2( e - e ) E 2 8. [3 marks] If Z x 1 f ( t ) dt = e x ln x when x > 0 , then f (1) = A e B 2 C 1 D 0 E e x ln x 9. [3 marks] Z x 2 + 1 x 2 + x dx = A x + ln | x | - 2 ln | x + 1 | + C B ln | x | + ln | x + 1 | + C C x - 1 + 2( x + 1) - 1 + C D x - ln x + 2 ln( x + 1) + C E ln x x + 1 + C 3
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10. [3 marks] Using a subdivision of the interval [1 , 3] into 4 subintervals of equal length, the trapezoidal rule yields the following approximation for Z 3 1 1 ln( x + 1) dx : A 1 . 8722 B 2 . 7230 C 1 . 7342 D 0 . 9863 E 1 . 9409 11. [3 marks] If a > 0 , Z 0 axe - ax dx A diverges B equals a C equals 1 a D equals 1 E equals e - a 12. [3 marks] If dy dx = 2 xy and y = e when x = 0 then, when x = 1 , y = A 1 B e C e 2 D e 3 E e 4 4
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13. [3 marks] If f ( x, y ) = e xy , then when x = 2 and y = 3 , 3 f ∂x∂y 2 = A 24 e 6 B 16 e 6 C 30 e 6 D 12 e 6 E 18 e 6 14.
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