Download U0@Mac: PRECALCULUS MATHEMATICS DIAGNOSTIC TEST

U0@Mac: PRECALCULUS MATHEMATICS DIAGNOSTIC TEST - 2 hours
1. Simplify
(a) 5/2
(b) 5/6
(c) 5/12
(d) 16/15
2
3
2. Add 23 +
(a) 5/24
(b) 5/12
(c) 7/11
(d) 31/24
5
8
·
5
8
3. Express x13 +
(a) 0
2
2 y+2x
(b) y +x
x3 y 2
(c) x−2
3 y2
4
(d) x3 +xy+x
2 y2
4. Simplify
(a) 2x2 y 3
(b) 2xy 2
(c) 8x3 y 6
(d) 4x
y3
1
xy
2
x2 y 2
as a single fraction :
(2x2 y 3 )3
(xy)3
2
5. Simplify (a2 b) 3 ·
a1/3
+
¡ a ¢− 1
b
3
(a) b2/3
(b) ab
(c) ab
1/3
(d) ab 2/3
6. Solve for x in x2 + 7x + 12 = 0
(a) -4, -3
(b) 4, 3
(c) 7, 2
(d) -12, 2
1
7. Solve for x in
(a) x = 5/2
(b) x = 2
(c) x = 5
(d) x = 1
2
x−3
=
3
x−2
8. Solve for y in 4y 2 − 1 = 9y − 3
(a) y = 4 and 1/2
(b) y = 3 and 1/9
(c) y = 1 and 1/4
(d) y = 2 and 1/4
9. Factor x2 − 2x − 3
(a) (x − 3)(x + 1)
(b) (x + 2)(x − 1)
(c) (x − 2)(x + 1)
(d) (x − 3)(x − 1)
10. What is the equation of the line with slope
(a) y = 32 x + 3
1
3
passing through the point (x, y) = (3, 2)?
(b) y = 23 x − 2
(c) y = 13 x − 1
(d) y = 31 x + 1
11. At what point do the two lines x − y = 2 and y = 2x + 3 intersect?
(a) (1, 5)
(b) (−5, −7)
(c) (2, 9)
(d) (5, 7)
12. Consider the line y = x − 2 and the parabola y = x2 . Do they intersect? If yes, where?
(a) Yes, at (2, 1)
(b) No
(c) Yes, at (1, -1)
(d) Yes, at (-1, -3)
2
13. Convert the angles 60o and 210o to radians
(a) π/4 and 5π/4
(b) π/6 and 7π/6
(c) π/3 and 7π/6
(d) π/4 and 7π/4
14. An equivalent expression for cos θ is
(a) sin θ tan θ
(b) sin(tan θ)
sin θ
(c) tan
θ
θ
(d) tan
sin θ
15. What are√the values
√ of cos 0 , cos π/6 , cos π/4 , cos π/3 , cos π/2?
(a) 0 , 1/2
,
2/2
,
3/2 , 1
√
√
(b) 1 , 3/2 , 2/2, , 1/2 , 0
(c) 1 , − 3/2√, − 1, √
, − 1/2 , 0
(d) 0 , 1/4 , 2/2 , 3/4 , 1
16. What
√ values of sin π , sin 5π/6 , sin 3π/4 , sin 2π/3 , sin π/2?
√ are the
(a) 1 , 3/2√, 2/2, , 1/2 , 0√
(b) 1 , − 3/ √2 , − 1,
√ , − 1/ 2 , 0
(c) 0 , 1/4 , √2/2 , √3/4 , 1
(d) 0 , 1/2 , 2/2 , 3/2 , 1
17. Simplify e3x+2 · e2x−4
(a) e5x−2
(b) e3x+2 + e2x−4
2
(c) e6x −8
(d) ex−2
18. If y = f (x) = x2 , what is the inverse function x = f −1 (y)?
(a) x = y 2
(b) x = y −2
√
√
(c) x = y or − y
(d) x = y −1
19. If log a = 3 and log b = 4, what is log(a2 b)?
(a) log 36
(b) 13
(c) log 7
(d) 10
3
20. If y = f (x) = ex , what is the inverse function x = f −1 (y)?
(a) x = e−1
(b) x = ln y
(c) x = y −1
(d) x = 1/ey
21. If y = f (x) = sin x, what is the inverse function x = f −1 (y)?
(a) x = sin(y −1 )
(b) x = arcsin y = sin−1 y
(c) x = sin y −1
(d) x = 1/ sin y
22. What is the value of cos−1 (1/2)?
(a) π/6
(b) π/3
(c) 30o
(d) 45o
23. What is the value of arctan(1)?
(a) 0
(b) π/6
(c) π/4
(d) 30o
24. If x = 3 and y = 4, calculate yx − y x :
(a) -52
(b) 0
(c) 11
(d) -36
25. Simplify
(a) 1/2
(b) 2n−1
(c) 2
(d) 1/2n−1
2n
2(2n−2 )
4
26. Solve for x if log3 x = 2
(a) 2/ log 3
(b) 104/3
(c) 9
(d) 6/ log 3
27. Use polynomial long division to calculate
(a) x2 − 2x + 2
(b) x2 + x − 2
(c) x2 + 2x − 2
(d) x2 − x/2 + 2
x3 −2x2 −5x+6
x−3
28. If f (x) = x2 + 2x − 3, determine f (4):
(a) 0
(b) 9
(c) 1/21
(d) 21
29. Simplify 3 log4 2 + 2 log4 3
(a) log4 72
(b) log4 12
(c) log4 1
(d) 5 log4 5
30. Expand (a + b)3
(a) 3a + 3b
(b) a2 + abx + b2
(c) a3 + 3a2 b + 3ab2 + b3
(d) a3 + b3
31. What are the circumference and area of a circle of radius R?
(a) 4πR , 43 πR3
(b) 2πR , πR2
(c) 4πR2 , πR2
(d) 2πR , 43 πR3
32. If 3y = x, then y can be expressed as:
(a) y = x/3
(b) y = x/ log3
(c) y = logx 3
(d) y = log3 x
5
33. What is the area of a triangle with base 2 and height 3?
(a) 6
(b) 2
(c) 3
(d) 12
34. Solve the following pair of equations for x and y: x + 2y = 4 and 4x − 5y = 3
(a) x = 1 , y = 2
(b) x = −1 , y = 1
(c) x = 2 , y = 1
(d) x = −2 , y = −1
35. Find an equation for the line passing through the point (2, 1) and perpendicular to the
line 2x + y = 5 :
(a) 2x − 2y = 0
(b) x + 2y = 0
(c) x − 2y = 0
(d) 2x − y = 0
36. √
If f (x) =
(a) 1 − x2
√
(b) 1 − x
(c) 1 − x
(d) 1 + x2
√
x and g(x) = 1 − x2 , find f (g(x))
37. The lines 2x + 3y = 5 and 6x − 4y = 14 are:
(a) parallel
(b) perpendicular
(c) neither
(d) both
38. During the first and second year a business had earnings of $150,000 and $184,000 respectively. If the growth in sales follows a linear pattern, what will be the earnings during
the third year?
(a) $196,000
(b) $334,000
(c) $234,000
(d) $218,000
6
39. The curve described by the equation x2 y 4 + 2x4 y − 2 = 0 is symmetric about
(a) both the x-axis and the y-axis
(b) the y-axis
(c) the x-axis
(d) neither the x-axis nor the y-axis
40. Solve the equation ln(x − 3) − ln x = 1 for x
(a) x = e1/3
(b) ln 3 = 1
(c) x = ln 3−1
3
(d) x = 1−e
41. Find the values of x which satisfy |x + 4| = 2
(a) x = 2 or x = −4
(b) x = −2 or x = −6
(c) x = −2 or x = −4
(d) x = 2 or x = −6
42. Express as a single fraction and simplify
−3a
(a) x(x+a)
(b)
(c)
(d)
3h
x2 +a
−3
xa+x2
a
x(x+a)
43. Solve x2 + 3 ≤ 7
(a) x ≤ 2; x ≥ −2
(b) x ≤ −4; x ≥ 4
(c) x ≤ 4; x ≥ −4
(d) x ≥ 2; x ≤ −2
44. The graph of x + y − 3 = 0 is a
(a) plane
(b) circle
(c) parabola
(d) line
7
3
−3
x+a x
a
45. The graph of x2 + y 2 = 4 is a
(a) circle of radius 2
(b) circle of radius 4
(c) ellipse of eccentricity 1
(d) parabola
46. The graph of x2 + y = 4 is a
(a) circle centered at origin
(b) parabola opening upward
(c) ellipse
(d) parabola opening downward
47. The absolute value of -8 is
(a) 1/8
(b) 8
(c) -8
(d) -1/8
48. Simplify −3/4 ÷ 5/6
(a) -15/24
(b) 5/8
(c) -9/10
(d) 10/9
49. For an isosceles triangle of base 2b, height a and hypotenuse c, the Pythagorean Theorem states that
(a) c2 = a2 + b2
(b) a2 = c2 + b2
(c) b2 = a2 + c2
(d) c2 = −a2 − b2
50. If the interior angles in a triangle are labelled A, B, and C, then
(a) A + B + C ≤ 180o
(b) A + B + C = 180o
(c) A + B + C < 180o
(d) A + B + C = 90o
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