MATH 1210 - Section 002
Fall 2010
Nicolas Fourrier
Due on September 2, 2010
Homework 1 Name:
1. Evaluate the expression 1 1 a) [( ) 3 ]−2 64 a) 16
b)
5 4 −4 x y 2
c) (
b) 4
2. Simplify the expression a) a)
√ 3 6 2 c)
5x6 y 3 2x2 y 7
9−3 ∗ 95 − 1 ) 2 9−2
1 81
b)
p
81x6 y −4
b) 9x3 y −2
3. Rationalize the denominator of the expression r 3 2x a) √ b) xy y a)
√ 3 xy xy
√ b)
2xy y
4. Factor each expression completely a) 3x3 − x2 + 3x − 1 a) (3x − 1)(x2 + 1)
b) (x + y)2 − 1 b) (x + y − 1)(x + y + 1)
5. Perform the indicated operations and simplify each expression a) (x2 + 1)(4x3 − 3x2 + 2x) − (x4 − x3 + x2 )(2x) a) 2x5 − 5x4 + 8x3 − 3x2 + 2x 6. Find the real roots by factoring a) 3x2 − x − 4 = 0 a) (x + 1)(3x − 4)
b) (x2 − 4)(x2 + 4)(2x + 8) − (x2 + 8x − 4)(4x3 )
b) −2x5 + 40x4 − 16x3 − 32x − 128
b) −6x2 + x + 12 = 0 b) (−2x + 3)(3x + 4)
7. Solve the equation by using the quadratic formula a) x2 − 6x + 6 = 0 b) 2x2 + 7x − 15 = 0 √ √ a) b2 − 4ac = 12, x1 = 3 + 3, x2 = 3 − 3 2 b) b − 4ac = 169, x1 = −5, x2 = 1.5
8. Perform the indicated operations and simplify the expression 1 1 1 + y1 (x2 + 1) 2 − 2x2 (x2 + 1)− 2 x a) b) 2 1 1−x 1− xy
a)
x+y xy − 1
1
b) −(x2 + 1)− 2
√ √ 2 a+ b √ 9. Rationalize the denominator of the following expression √ 2 a− b √ 4a + 4 ab + b 4a − b 10. Find the values of x that satisfy x≥
2x − 3 ≥4 x+1
7 2
11. Evaluate the following expression | − 1| +
√
2| − 2|
√ 1 + 2 2 = 3.8284 12. The relationship between Celsius (C) and Fahrenheit (F) temperature is given by the formula: C = 59 (F − 32) a) If the temperature range for Montreal during the month of January is −15 < C < −5, find the range in degrees Fahrenheit in Montreal for the same period b) If the temperature range for New York City during the month of June is 59 < F < 77, find the range in degrees Celsius in New York City for the same period. a) 5 < F < 23 b) 15 < C < 25 13. A furniture store offers free setup and delivery services to all points within a 25 miles radius of its warehouse distribution center. If you live 16 miles east and 14 miles south of the warehouse, will you incur a delivery charge? Justify your answer √ 162 + 142 < 25 I will not be charged for delivery 14. Ship A leaves port sailing north at speed of 25 mph. A half hour later, ship B leaves the same port sailing east at a speed of 20 mph. Let t (in hours) denote the time ship B has been at sea. distance a) Find an expression in terms of t giving the distance between the two ships (hint: speed = e.g. time 5miles ) 5mph = 1hour b) Use the expression obtained in part (a) to find the distance between the two ships 2 hours after ship A left port p a) √ d = 252 (t + 0.5)2 + 202 t2 b) 3400 miles 15. Given the equation 2x + 3y = 4, answer the following questions and justify each answer a) Is the slope of the line described by this equation positive or negative? b) As x increases in value, does y increase or decrease? c) If x decreases by 2 units, what is the corresponding change in y?
a) Negative b) Decrease c) 34 16. If the line passing through the points (a,1) and (5,8) is parallel to the line passing through the points (4,9) and (a+2,1), what is the value of a? a = 26 17. Find an equation of the line that passes through the points: a) (2,1) and (2,5) b) (-1,-2) and (3,-4)
a) Vertical line, x = 2 b) Slope=− 12 and the equation is y = − 12 x −
5 2
18. Determine whether A(-1,7), B(2,-2) and C(5,-9) lie on a straight line Equation between A and B is y = −3x + 4 Point C does not satisfy this equation A, B and C do not lie on a straight line 19. Relationship between F and C is
F =
9 C + 32 5
a) Sketch the line with the given equation. b) What is the slope of the line? What does it represent? c) What is the F-intercept of the line? What does it represent? b) The slope is 59 . It represents the variation of F with respect to the variation of C. c) The F-intercept of the line is 32. It represents the value of F when the temperature is equal to 0C.
20. Show that two distinct lines with equations a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0, respectively, are parallel if and only if a1 b2 − b1 a2 = 0 The slope of the first equation has to be equal to the slope of the second equation. Means that Thus a1 b2 = b1 a2
a1 b1
=
a2 . b2
