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College Algebra Self-assessment C - answer Mathematical Relations Name: ___________________________________ Multiple choice 1. Every real number that can be written as a fraction is called a) digit b) rational c) irrational d) imaginary Answer b) rational e) complex 2. i 3 = a) 0 b) i c) -1 d) i e) -i 3. If the solutions of a quadratic equation are not real numbers, then the discriminant is a) positive b) negative c) zero d) undefined 4. The slope of a line perpendicular to the line 5x – 7y = 21 is a) 7 5 b) 7 5 c) 5 7 d) 5 7 c) -i b) negative b) e) none of these 7 5 5. The interval that represents all the non-negative real numbers is a) [0, ) 6. Which of the following points is symmetric to the point (-6, 7) with respect to the x – axis. a) (6, 7) b) (6, -7) c) (-6, -7) d) (7, -6) e) (-7, 6) 7) If one solution of a quadratic equation with rational coefficients is -5 + 3i, then the other solution is a) 5 + 3i b) 5 – 3i c) -5 – 3i d) 3 – 5i e) -3 +5i 8) The decimal 0.2222222….. is equivalent to which of the following fractions c) (-6, -7) a) [0, ) b) (0, ) 2 a) 10 2 b) 11 c) ( , 0) 22 c) 100 d ) ( , 0] 2 d) 9 c) -5 – 3i d) e) none of these 2 9 9) The two solutions of a quadratic equation are -3 and 7. Which of the following Hint: x 2 ( x1 x 2 )x x1x 2 0 is the quadratic equation? a) a) x 2 4x 21 0 b) x 2 4x 21 0 c) x 2 4x 21 0 d) x 2 4x 21 0 10) Which of the following intervals indicates all the real numbers which are greater a) (5, 10.6] than 5, but at most 10.6? a) (5, 10. 6] b ) [5, 10.6) c) (5, 10.6) d ) [5, 10.6] 11) Which of the following intervals indicates all the real numbers which are less than 5 or at least 7 ? a) (5, 7] b) [5, 7) c) ( , ) d ) ( . 5) [7, ) d) ( . 5) [7, ) 12) Which of the following is the solution of the inequality 5 – 3x > 2? a) x> 1 b) x < 1 c) x > -1 d) x < -1 e) none of these 13) Which of the following is equivalent to the expression |-7 +2| a) 9 b) -9 c) -5 d) 5 e) none of these -1- b) x < 1 d) 5 14) The center of the circle (x 3) 2 ( y 2) 2 16 is a) (3, -2) b) (-3, 2) c) (2, -3) d) (-2, 3) b) (-3, 2) 15) The domain of the circle (x 3) 2 ( y 2) 2 16 is a) (-7, 1) b) (-2, 6) c) [-7, 1] d) [-2, 6] e) [-3, 2] c) [-7, 1] Problem 1 . Find the solutions of the equation Solution: 15 – 2x = 7 or 15 – 2x = -7 -2x = -8 or -2x = -22 x = 4 or x =11 |15 – 2x| = 7 answer: __x = 4 or x =11 | Find the solution of the inequality |15 – 2x < 7 . Give answer in interval notation. (Solve by investigating the different interval on the number line determined by the solutions of the equation. ______________________________________________ 4 11 For x =0, the inequality is false, therefore the interval ( , 4) is not a solution. For x = 5, the inequality is true, therefore the interval (4, 11) is a solution. For x =12, the inequality is false, therefore the interval ( 4, ) is not a solution. The solution of the inequality is [4, 11] 4 11 2. Problem 2. Consider the linear relation 3x + 4y =8. a) Find the slope m of the relation. _A 3 m B 4 a) m _A 3 __________ B 4 b) Find the y-intercept of the relation When x = 0 , y = 2 b) __ 2 ____________ c) Use the slope and the fact that (0, 2) is a point of the relation to find another two points. (0 + 4, 2 – 3) = (4, -1) (4 + 4, -1 -3) = (8, -4) c) (4, -1) and (8, -4) or many other possible answers. -2- d) Use the points found to draw the line in the following Cartesian system. X = -2 y-axis (0, 2) Y=1 1 -1 1 x-axis (4, -1) e) After you draw the line, shade the region that represents the inequality 3x + 4y < 8. The point (0, 0) belongs to the inequality f) Draw the horizontal line y = 1 g) Draw the vertical line x = -2 Problem 3. Perform the operations required to simplify and write the answers in the forma a + bi a) (3 – 4i)(-5+3i) a) -3 + 29i Solution: 15 9i 20i 12i 2 15 29i 12( 1) 15 29i 12 3 29i b) (-2+3i) – (-5+6i) =-2 + 3i +5 -6i 3 – 3i b) 3 – 3i Problem 4. Find the solution of the equation 3x 2 x 2 4 Use the quadratic formula. a = _3____ b = __-1____ c = -2__ Careful! 2 b b 2 4ac 1 ( 1) 4(3)( 2) 1 25 1 5 6 4 2 x or 1 or 2a 2(3) 6 6 6 6 3 Problem 5. Find the equation of the circle with center (-1, 3) and radius r = 9. -3- Give the domain and range of the circle Equation: __ ( x 1) 2 ( y 3) 2 81 _____________________ Domain: ___[-1-9, -1+9] = [-10, 8]_________ Range: _____[3 – 9, 3 + 9] = [-6, 12]__________ Problem 6. Find an equation of the line that contains the points (-4, 1) and (3, -1) H int : y y1 m( x x1 ) y y1 1 1 2 m 2 x 2 x1 3 4 7 y y1 m( x x1 ) y 1 2 ( x 4) 7 7 y 7 2x 8 2x 7 y 1 Problem 7. Consider the linear relation: y = 5x – 7 a) Give the slope of the relation a) m = 5 b) Give the y-intercept of the relation b) b = -7 c) Give the equation of the transpose relation x = 5y -7 d) Give the equation of the relation that is symmetrical to y = 5x – 7 with respect to the y-axis. y = 5(-x) - 7 e) Give the equation of the relation that is symmetrical to y = 5x – 7 with respect to the x-axis. - y = 5x – 7 or y = -5x + 7 f) Give the equation of the relation that is symmetrical to y = 5x – 7 with respect to the origin -y = 5(-x) – 7 or y = 5x + 7 c) x = 6y -7 d) y = -5x - 7 e) y = 5x - 7 f) y = 5x + 7 -4-