All triangles in the set of triangles having a given side and a given angle opposite that side A are congruent B are similar C are equivalent D have the same inscribed circle E have the
Trang 137 In figure 37, what is the approximate area of triangle NJL?
(A) 11.79
(B) 13.85
(C) 15.31
(D) 17.10
(E) 17.82
38 A right circular cylinder is circumscribed about a sphere If S represents the surface area of the sphere
and T represents the total area of the cylinder, then
(A)
(B)
(C)
(D)
(E)
39 If |x – 2| < 5, what are the possible values of x?
(A) 0 < x < 5
(B) 0 < x < 2
(C) –3 < x ≤ 7
(D) –3 ≤ x < 7
(E) –3 < x < 7
40 How many even numbers greater than 40,000 may be formed using the digits 3, 4, 5, 6, and 9 if each
digit must be used exactly once in each number?
(A) 36
(B) 48
(C) 64
(D) 96
(E) 112
Fig 37
Trang 241 A regular octagon is formed by cutting off each corner of a square whose side is 6 Find the length of
one side of the octagon
(A) 2
(B)
(C)
(D)
(E)
42 Approximately, what is ?
(A) 1.42
(B) 3.18
(C) 5.38
(D) 7.00
(E) 8.67
43 All triangles in the set of triangles having a given side and a given angle opposite that side
(A) are congruent
(B) are similar
(C) are equivalent
(D) have the same inscribed circle
(E) have the same circumscribed circle
44 Find the value of
(A)
(B)
(C)
(D)
(E)
45 For what positive value of m will the line y = mx + 5 be tangent to the circle x2 + y2 = 9?
(A) 1
(B) 2
(C)
(D)
(E)
Trang 346 The graph of the curve whose parametric equations are x = a sin t and y = b cos t is a(n):
(A) ellipse
(B) circle
(C) parabola
(D) hyperbola
(E) straight line
47 The graph of y = |x – 2| + 2 consists of
(A) one straight line
(B) a pair of straight line rays
(C) the sides of square
(D) a circle
(E) a parabola
48 The converse of ~ p → q is equivalent to
(A) p → ~ q
(B) p → q
(C) ~ q → p
(D) q → p
(E) ~ p → ~ q
49 Write an equation of lowest degree, with real coefficients, if two of its roots are –1 and 1 + i.
(A) x3 + x2 + 2 = 0
(B) x3 – x2 – 2 = 0
(C) x3 – x + 2 = 0
(D) x3 – x2 + 2 = 0
(E) none of these
50 If logr 6 = S and log r 3 = T, then log r is equal to
(A) log2 r for any r
(B) 1 – S + T
(C) 1 – S – T
(D) logr 2 – 1
(E) zero, if r = 4
STOP
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY DO NOT WORK ON ANY OTHER TEST IN THIS BOOK.
Trang 4PRACTICE TEST 3
Answer Key
Math Level IIC
SOLUTIONS
1 The correct answer is (B) For y to be a function of x, there must be a unique value of y for any given
value of x This would be true only for y = x2 + 1 In A, C, D, and E, y may take on 2 or more values for
a given x.
2 The correct answer is (A) The set P ∩ Q includes as elements all rectangles that are also rhombi.
These elements make up the set of squares
3 The correct answer is (B).
4 The correct answer is (C) 82.5 = p and 25 = q
then (23)2.5 = p or p = 27.5 = (25)1.5
thus p = (25)3/2 = q3/2
5 The correct answer is (E) R is internally tangent to S and its diameter is half that of S.
Hence S has an area 4 times that of r, or 16 square inches.
1. B
2. A
3. B
4. C
5. E
6. D
7. C
8. D
9. D
10. A
11. E
12. D
13. D
14. B
15. C
16. D
17. E
18. B
19. C
20. A
21. E
22. C
23. D
24. B
25. A
26. B
27. C
28. B
29. D
30. D
31. B
32. A
33. E
34. A
35. B
36. C
37. B
38. A
39. E
40. A
41. D
42. B
43. E
44. B
45. E
46. A
47. B
48. A
49. D
50. B
Trang 56 The correct answer is (D) Consider a cube of edge 1.
Then the longest diagonal =
The diagonal of a base =
Hence the ratio =
7 The correct answer is (C) x2 – 6x + 9 + y2 + 4y + 4 = 12 + 13
(x – 3)2 + (y + 2)2 = 25
Center is (3, –2)
8 The correct answer is (D) sin 135º cos x + cos 135º sin x
+ sin 135º cos x – cos 135º sin x
9 The correct answer is (D).
10 The correct answer is (A).
11 The correct answer is (E) Consider y = sin x cos x
sin 2x has a period of 180º and reaches its maximum at x = 45 ° or
12 The correct answer is (D) There are 4! = 24 ways of lining up the 4 men Consider the certain two
as one unit and determine the number of ways of lining up three; 3! = 6 However the two men can
be next to each other in twice as many ways as indicated by merely switching places Hence, there are 12 ways of the four men lining up so that a certain two are always next to each other
Trang 613 The correct answer is (D).
14 The correct answer is (B) The converse, inverse, and negative are not equivalent statements The
contrapositive, ~ Q → ~ P, is equivalent and this is the same as saying that Q is a necessary condition for P.
15 The correct answer is (C)
Hence,
16 The correct answer is (D) log x ≥ log 2 + log x1/2
log x – log x1/2 ≥ log 2
17 The correct answer is (E) 2 cos3 A sin A + 2 sin3 A cos A
= 2 sin A cos A (cos2 A + sin2 A)
= 2 sin A cos A = sin 2 A
18 The correct answer is (B).
Trang 719 The correct answer is (C) Let the roots be r, – r, and s
20 The correct answer is (A) Since log10 x = y, 10 y = x
Substituting, we get
21 The correct answer is (E)
22 The correct answer is (C)
Writing this to two significant figures, we get 8.6 ✕ 108
23 The correct answer is (D)
The exponent is the sum of an infinite geometric series
r
=
1 2 1 2
Thus 31 = 3
Trang 824 The correct answer is (B)
25 The correct answer is (A).
26 The correct answer is (B) He saves
Percent saved= − ⋅
≈ ⋅
≈
≈
50 2 1 414
50 586
29 3
29
x
Trang 9
27 The correct answer is (C) Longest dimension is along the diagonal
d2 = 242 + 82 + 62
d2 = 576 + 64 +35 = 676
28 The correct answer is (B) x2 – x – 6 < 0
(x – 3)(x + 2) < 0
Either x – 3 < 0 and x + 2 > 0 or x – 3 > 0 and x + 2 > 0
x < 3 and x > –2 or x > 3 and x < –2
29 The correct answer is (D).
30 The correct answer is (D)
or 1– K + 2 + 1 = 0
K = 4
31 The correct answer is (B).
Trang 1032 The correct answer is (A) By De Moivre’s Theorem
33 The correct answer is (E) Method 1: 17º ∆ 33º
Method 2:
34 The correct answer is (A).
35 The correct answer is (B)
36 The correct answer is (C)
Since the period of cos x is 360 °, the period of cos 2x is 180°.
37 The correct answer is (B)
Trang 1138 The correct answer is (A).
39 The correct answer is (E) If x – 2 > 0 or x > 2, then x – 2 < 5
x < 7
If x – 2 < 0 or x < 2, then – (x – 2) < 5
x – 2 > – 5
x > – 3 Thus, – 3 < x < 7
or |x – 2| < 5
– 5 < x – 2 < 5
Adding 2 to all three terms of inequality,
– 3 < x < 7
40 The correct answer is (A) The last digit may only be filled by 4 or 6, thus in 2 ways This leaves 3
remaining numbers for the first digit, 3 more for the second digit, 2 for the third digit and 1 for the fourth digit 3 · 3 · 2 · 1 · 2 = 36
Trang 1241 The correct answer is (D).
Reject the negative square root
x= −12 12 2− = −
42 The correct answer is (B)
43 The correct answer is (E) If the given side is p and the angle opposite is P, then the diameter, d, of
the circumscribed circle is given by
Hence, all circumscribed circles have the same diameter Thus all triangles have the same circum-scribed circle
Trang 1344 The correct answer is (B)
45 The correct answer is (E).
If the line is tangent to the circle, the discriminant of the quadratic must be zero
100m2 – 64(1 + m2) = 0
36m2 = 64
m =
4 3
46 The correct answer is (A).
Equation of an ellipse
47 The correct answer is (B) When x ≥ 2, y = x – 2 + 2; or the straight line y = x.
When x < 2, y = 2 – x + 2 = 4 – x, which graphs as another straight line.
Thus the graph is a pair of rays that form a “V.”
48 The correct answer is (A) The converse is q →~ p The contrapositive of this is p →~ q.
Trang 1449 The correct answer is (D) Conjugate of 1 + i, namely 1 – i, must also be a root of the equation.
Thus the roots are – 1, 1 + i, and 1 – i.
Sum of roots = 1
Product of roots = – 1 (1 + i)(1 – i) = –2
Product of roots two at a time
= – 1(1 + i) – 1 (1 – i) + (1 + i)(1 – i)
= – 1 – i – 1 + i + 1 + 1 = 0
Thus, the equation is x3 – x2 + 2 = 0
50 The correct answer is (B)
Substituting this value in second equation,