Practice Test 3 (GR9367)

1. If \( f(g(x)) = 5 \) and \( f(x) = x + 3 \) for all real \( x \), then \( g(x) = \)

2. \( \displaystyle\lim_{x \to 0} \frac{\tan x}{\cos x} = \)

3. \( \displaystyle\int_0^{\log 4} e^{2x}\, dx = \)

4. Let \( A - B \) denote \( \{x \in A : x \notin B\} \). If \( (A - B) \cup B = A \), which of the following must be true?

5. If \( f(x) = |x| + 3x^2 \) for all real \( x \), then \( f'(-1) \) is

6. For what value of \( b \) is the value of \( \displaystyle\int_b^{b+1} (x^2 + x)\, dx \) a minimum?

7. In how many of the eight standard octants of \( xyz \)-space does the graph of \( z = e^{x+y} \) appear?

8. Suppose that the function \( f \) is defined on an interval by the formula \( f(x) = \sqrt{\tan^2 x - 1} \). If \( f \) is continuous, which of the following intervals could be its domain?

9. \( \displaystyle\int_0^1 \frac{x}{2 - x^2}\, dx = \)

10. If \( f''(x) = f'(x) \) for all real \( x \), and if \( f(0) = 0 \) and \( f'(0) = -1 \), then \( f(x) = \)

11. If \( \phi(x,y,z) = x^2 + 2xy + xz^{3/2} \), which of the following partial derivatives are identically zero?

I. \( \dfrac{\partial^2 \phi}{\partial y^2} \)   II. \( \dfrac{\partial^2 \phi}{\partial x \,\partial y} \)   III. \( \dfrac{\partial^2 \phi}{\partial z \,\partial y} \)

12. \( \displaystyle\lim_{x \to 0} \frac{\sin 2x}{(1+x)\log(1+x)} = \)

13. \( \displaystyle\lim_{n \to \infty} \int_1^n \frac{1}{x^n}\, dx = \)

14. At a 15 percent annual inflation rate, the value of the dollar would decrease by approximately one-half every 5 years. At this inflation rate, in approximately how many years would the dollar be worth \( \dfrac{1}{1{,}000{,}000} \) of its present value?

15. Let \( f(x) = \displaystyle\int_1^x \frac{1}{1+t^2}\, dt \) for all real \( x \). An equation of the line tangent to the graph of \( f \) at the point \( (2, f(2)) \) is

16. Let \( f(x) = e^{g(x)}h(x) \) and \( h'(x) = -g'(x)h(x) \) for all real \( x \). Which of the following must be true?

17. \( 1 - \sin^2\!\left(\arccos\dfrac{\pi}{12}\right) = \)

18. If \( f(x) = \displaystyle\sum_{n=0}^{\infty} (-1)^n x^{2n} \) for all \( x \in (0,1) \), then \( f'(x) = \)

19. Which of the following is the general solution of the differential equation \( \dfrac{d^3y}{dt^3} - 3\dfrac{d^2y}{dt^2} + 3\dfrac{dy}{dt} - y = 0 \)?

20. Which of the following double integrals represents the volume of the solid bounded above by \( z = 6 - x^2 - 2y^2 \) and below by \( z = -2 + x^2 + 2y^2 \)?

21. Let \( a \) be a number in the interval \( (0,1) \) and let \( f \) be a function defined on \( [0,1] \) by \[ f(x) = \begin{cases} a^2 & \text{if } 0 \le x < a, \\ \sqrt{x} & \text{otherwise.} \end{cases} \] Then the value of \( a \) for which \( \displaystyle\int_0^1 f(x)\,dx = 1 \) is

22. If \( b \) and \( c \) are elements in a group \( G \), and if \( b^5 = c^3 = e \), where \( e \) is the unit element of \( G \), then the inverse of \( b^2 c b^4 c^2 \) must be

23. Let \( f \) be a real-valued function continuous on \([0,1]\) and differentiable on \((0,1)\) with \( f(0)=1 \) and \( f(1)=0 \). Which of the following must be true?

I. There exists \( x \in (0,1) \) such that \( f(x) = x \).
II. There exists \( x \in (0,1) \) such that \( f'(x) = -1 \).
III. \( f(x) > 0 \) for all \( x \in [0,1) \).

24. If \( A \) and \( B \) are events in a probability space such that \( 0 < P(A) = P(B) = P(A \cap B) < 1 \), which of the following CANNOT be true?

25. Let \( f \) be a real-valued function with domain \([0,1]\). If there is some \( K > 0 \) such that \( |f(x) - f(y)| \le K|x-y| \) for all \( x \) and \( y \) in \([0,1]\), which of the following must be true?

26. Let \( \mathbf{i}=(1,0,0) \), \( \mathbf{j}=(0,1,0) \), and \( \mathbf{k}=(0,0,1) \). The vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) are orthogonal if \( \mathbf{v}_1 = \mathbf{i}+\mathbf{j}-\mathbf{k} \) and \( \mathbf{v}_2 = \)

27. If the curve in the \( yz \)-plane with equation \( z = f(y) \) is rotated around the \( y \)-axis, an equation of the resulting surface of revolution is

28. Let \( A \) and \( B \) be subspaces of a vector space \( V \). Which of the following must be subspaces of \( V \)?

I. \( A + B = \{a+b : a \in A \text{ and } b \in B\} \)
II. \( A \cup B \)
III. \( A \cap B \)
IV. \( \{x \in V : x \notin A\} \)

29. \( \displaystyle\lim_{n\to\infty} \sum_{k=1}^{n}\left(\frac{1}{k} - \frac{1}{2^k}\right) = \)

30. If \( f(x_1,\dots,x_n) = \displaystyle\sum_{1 \le i < j \le n} x_i x_j \), then \( \dfrac{\partial f}{\partial x_n} = \)

31. If \( f(x) = \begin{cases} \sqrt{1-x^2} & \text{for } 0 \le x \le 1 \\ x-1 & \text{for } 1 < x \le 2, \end{cases} \) then \( \displaystyle\int_0^2 f(x)\,dx \) is

32. Let \( \mathbb{R} \) denote the field of real numbers, \( \mathbb{Q} \) the field of rational numbers, and \( \mathbb{Z} \) the ring of integers. Which of the following subsets \( F_i \) of \( \mathbb{R} \), \( 1 \le i \le 4 \), are subfields of \( \mathbb{R} \)?

\( F_1 = \{a/b : a,b \in \mathbb{Z} \text{ and } b \text{ is odd}\} \)
\( F_2 = \{a + b\sqrt{2} : a,b \in \mathbb{Z}\} \)
\( F_3 = \{a + b\sqrt{2} : a,b \in \mathbb{Q}\} \)
\( F_4 = \{a + b\sqrt[3]{2} : a,b \in \mathbb{Q}\} \)

33. If \( n \) apples, no two of the same weight, are lined up at random on a table, what is the probability that they are lined up in order of increasing weight from left to right?

34. \( \dfrac{d}{dx}\displaystyle\int_0^{x^2} e^{-t^2}\,dt = \)

35. Let \( f \) be a real-valued function defined on the set of integers and satisfying \( f(x) = \tfrac{1}{2}f(x-1) + \tfrac{1}{2}f(x+1) \). Which of the following must be true?

I. The graph of \( f \) is a subset of a line.
II. \( f \) is strictly increasing.
III. \( f \) is a constant function.

36. If \( F \) is a function such that, for all positive integers \( x \) and \( y \), \( F(x,1)=x+1 \), \( F(1,y)=2y \), and \( F(x+1,y+1)=F(F(x,y+1),y) \), then \( F(2,2) = \)

37. If \( \det\begin{pmatrix}a&b&c\\d&e&f\\g&h&k\end{pmatrix}=9 \), then \( \det\begin{pmatrix}3a&3b&3c\\g-4a&h-4b&k-4c\\d&e&f\end{pmatrix}= \)

38. \( \displaystyle\lim_{n\to\infty}\frac{3}{n}\sum_{i=1}^{n}\left[\left(\frac{3i}{n}\right)^2 - \frac{3i}{n}\right] = \)

39. For a real number \( x \), \( \log(1 + \sin 2\pi x) \) is not a real number if and only if \( x \) is

40. If \( x \), \( y \), and \( z \) are selected independently and at random from the interval \([0,1]\), then the probability that \( x \ge yz \) is

41. If \( A = \begin{pmatrix}1&2\\0&-1\end{pmatrix} \), then the set of all vectors \( X \) for which \( AX = X \) is

42. What is the greatest value of \( b \) for which any real-valued function \( f \) that satisfies the following properties must also satisfy \( f(1) < 5 \)?

(i) \( f \) is infinitely differentiable on the real numbers;
(ii) \( f(0) = 1 \), \( f'(0) = 1 \), and \( f''(0) = 2 \); and
(iii) \( |f'''(x)| < b \) for all \( x \) in \((0,1)\).

43. Let \( n \) be an integer greater than 1. Which of the following conditions guarantee that the equation \( x^n = \displaystyle\sum_{i=0}^{n-1} a_i x^i \) has at least one root in the interval \((0,1)\)?

I. \( a_0 > 0 \) and \( \displaystyle\sum_{i=0}^{n-1} a_i < 1 \)
II. \( a_0 > 0 \) and \( \displaystyle\sum_{i=0}^{n-1} a_i > 1 \)
III. \( a_0 < 0 \) and \( \displaystyle\sum_{i=0}^{n-1} a_i > 1 \)

44. If \( x \) is a real number and \( P \) is a polynomial function, then \( \displaystyle\lim_{h\to 0}\frac{P(x+3h)+P(x-3h)-2P(x)}{h^2} = \)

45. Consider the system of equations \[ ax^2 + by^3 = c \\ dx^2 + ey^3 = f \] where \( a,b,c,d,e, \) and \( f \) are real constants and \( ae \neq bd \). The maximum possible number of real solutions \((x,y)\) of the system is

46. If \( x^3 - x + 1 = a_0 + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3 \) for all real numbers \( x \), then \( (a_0,a_1,a_2,a_3) \) is

47. Let \( C \) be the ellipse with center \((0,0)\), major axis of length \( 2a \), and minor axis of length \( 2b \). The value of \( \displaystyle\oint_C x\,dy - y\,dx \) is

48. Let \( G_n \) denote the cyclic group of order \( n \). Which of the following direct products is NOT cyclic?

49. Let \( X \) be a random variable with probability density function \( f(x) = \begin{cases}\tfrac{3}{4}(1-x^2) & \text{if } -1 \le x \le 1,\\ 0 & \text{otherwise.}\end{cases} \) What is the standard deviation of \( X \)?

50. The set of all points \((x,y,z)\) in Euclidean 3-space such that \( \begin{vmatrix}1&x&y&z\\1&1&0&0\\1&0&1&0\\1&0&0&1\end{vmatrix}=0 \) is

51. An automorphism \( \phi \) of a field \( F \) is a one-to-one mapping of \( F \) onto itself such that \( \phi(a+b)=\phi(a)+\phi(b) \) and \( \phi(ab)=\phi(a)\phi(b) \) for all \( a,b \in F \). If \( F \) is the field of rational numbers, then the number of distinct automorphisms of \( F \) is

52. Let \( T \) be the transformation of the \( xy \)-plane that reflects each vector through the \( x \)-axis and then doubles the vector's length. If \( A \) is the \( 2\times 2 \) matrix such that \( T\!\left(\begin{bmatrix}x\\y\end{bmatrix}\right) = A\begin{bmatrix}x\\y\end{bmatrix} \), then \( A = \)

53. Let \( r > 0 \) and let \( C \) be the circle \( |z|=r \) in the complex plane. If \( P \) is a polynomial function, then \( \displaystyle\int_C P(z)\,dz = \)

54. If \( f \) and \( g \) are real-valued differentiable functions and if \( f'(x) \ge g'(x) \) for all \( x \) in the closed interval \([0,1]\), which of the following must be true?

55. Let \( p \) and \( q \) be distinct primes. There is a proper subgroup \( J \) of the additive group of integers which contains exactly three elements of the set \( \{p,\,p+q,\,pq,\,p^q,\,q^p\} \). Which three elements are in \( J \)?

56. For a subset \( S \) of a topological space \( X \), let \( \text{cl}(S) \) denote the closure of \( S \) in \( X \), and let \( S' = \{x : x \in \text{cl}(S-\{x\})\} \) denote the derived set of \( S \). If \( A \) and \( B \) are subsets of \( X \), which of the following statements are true?

I. \( (A \cup B)' = A' \cup B' \)
II. \( (A \cap B)' = A' \cap B' \)
III. If \( A' \) is empty, then \( A \) is closed in \( X \).
IV. If \( A \) is open in \( X \), then \( A' \) is not empty.

57. Consider a procedure for determining whether a given name appears in an alphabetized list of \( n \) names (binary search). If \( n \) is very large, the maximum number of steps required by this procedure is close to

58. Which of the following is an eigenvalue of the matrix \( \begin{pmatrix}2&1-i\\1+i&-2\end{pmatrix} \) over the complex numbers?

59. Two subgroups \( H \) and \( K \) of a group \( G \) have orders 12 and 30, respectively. Which of the following could NOT be the order of the subgroup of \( G \) generated by \( H \) and \( K \)?

60. Let \( A \) and \( B \) be subsets of a set \( M \) and let \( S_0 = \{A,B\} \). For \( i \ge 0 \), define \( S_{i+1} \) inductively to be the collection of subsets \( Y \) of \( M \) that are of the form \( C \cup D \), \( C \cap D \), or \( M - C \) (the complement of \( C \) in \( M \)), where \( C,D \in S_i \). Let \( S = \displaystyle\bigcup_{i=0}^{\infty} S_i \). What is the largest possible number of elements of \( S \)?

61. A city has square city blocks formed by a grid of north-south and east-west streets. One automobile route from City Hall to the main firehouse is to go exactly 5 blocks east and 7 blocks north. How many different routes from City Hall to the main firehouse traverse exactly 12 city blocks?

62. Let \( R \) be the set of real numbers with the topology generated by the basis \( \{[a,b) : a < b,\; a,b \in R\} \). If \( X \) is the subset \([0,1]\) of \( R \), which of the following must be true?

I. \( X \) is compact.
II. \( X \) is Hausdorff.
III. \( X \) is connected.

63. Let \( R \) be the circular region of the \( xy \)-plane with center at the origin and radius 2. Then \( \displaystyle\iint_R e^{-(x^2+y^2)}\,dx\,dy = \)

64. Let \( V \) be the real vector space of real-valued functions defined on the real numbers and having derivatives of all orders. If \( D \) is the mapping from \( V \) into \( V \) that maps every function in \( V \) to its derivative, what are all the eigenvectors of \( D \)?

65. If \( f \) is a function defined by a complex power series expansion in \( z - a \) which converges for \( |z-a| < 1 \) and diverges for \( |z-a| > 1 \), which of the following must be true?

66. Let \( n \) be any positive integer and \( 1 \le x_1 < x_2 < \cdots < x_{n+1} \le 2n \), where each \( x_i \) is an integer. Which of the following must be true?

I. There is an \( x_i \) that is the square of an integer.
II. There is an \( i \) such that \( x_{i+1} = x_i + 1 \).
III. There is an \( x_i \) that is prime.