REA GRE Practice Test 3

1. The generating function f(x) for the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, is given by

2. The number of solutions of \( p(x)=x^{2}+3x+2 \) in \( \mathbb{Z}_{6} \) is

3. Which of the following matrices is normal? \( (i=\sqrt{-1}) \)

4. Find the locus of all points (x, y), such that the sum of those distances from (0, 1) and (1,0) is 2.

5. Find \( \prod_{k=2}^{+\infty}(1-\dfrac{1}{k^{2}}) \)

6. The cross ratio of the following set of lines is

7. Find the characteristic of the ring \( \mathbb{Z}_{2}+\mathbb{Z}_{3} \).

8. Find \( \lim_{n\rightarrow+\infty}(\sqrt{n^{4}+i\,n^{2}}-n^{2}) \)

9. Which of the following is a solution of \( u(x)=x+\int_{0}^{x}(t-x)u(t)dt \) ?

10. Find \( \int_{0}^{1}(\ln\dfrac{1}{x})^{5}dx \)

11. Find the Laplace transform of f(x)=\( \{\begin{matrix}0&\text{if}&x\in&(-\infty,1)\\ 1&\text{if}&x\in(1,+\infty)\end{matrix} \)

12. Let \( T: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2} \) be defined by \( T(x,y)=\left[\begin{matrix}2x-y\\ x+3y\end{matrix}\right] \). Find the adjoint \( T^{*} \) of T.

13. The value of \( l=\int_{0}^{\dfrac{\pi}{2}}\dfrac{\cos\,x}{\cos\,x+\sin\,x}dx \); is

14. Find the discriminant of the ternary quadratic form \( x^{2}-y^{2}+z^{2}-2xy+4yz-6xz \)

15. The radius of curvature of \( f(x)=x+\dfrac{1}{x} \) at P(1,2) is

16. If \( \Gamma(p) \) represents the gamma function, then \( \int_{0}^{+\infty}e^{-x^{2}}dx \) is equal to \( \dfrac{1}{2}\Gamma(p) \) when p is equal to

17. The factor group \( \dfrac{(\mathbb{Z}_{2}\times \mathbb{Z}_{3})}{\langle 1,0 \rangle} \) has order

18. Let \( \mathcal{R}[0,1] \) denote the set of Riemann integrable functions defined on [0,1]. Which of the following is not satisfied by function d defined on \( \mathcal{R}[0,1] \) by \( d(f,g)=\int_{0}^{1}|f(x)-g(x)|dx \)?

19. Find the simple continued fraction for \( \dfrac{13}{42} \)

20. Which of the following polynomials satisfies an Eisenstein criterion for irreducibility over the rationals?

21. The number of degrees that the conic, defined by \( x^{2}-y^{2}+2\sqrt{3}xy=2 \), must be rotated in order to eliminate the xy term is

22. For the initial value problem \( y^{\prime\prime}+6y^{\prime}+9y=0 \), \( y(0)=3 \), \( y^{\prime}(0)=-11 \), find \( f(y) \), the Laplace transform of y.

23. Find the join of the subgroups \( \langle 4 \rangle \) and \( \langle 6 \rangle \) of \( \mathbb{Z}_{12} \)

24. For which n is the regular n-gon not constructible with a straightedge and compass?

25. Given \( T=\left[\begin{matrix}1&0\\ 1&1\end{matrix}\right] \), the sum of the elements in \( T^{n} \) is

26. Let \( b: \mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R} \) be the bilinear form defined by \( b(X;Y)=x_{1}y_{1}-2x_{1}y_{2}+x_{2}y_{1}+3x_{2}y_{2} \), where \( X=(x_{1},x_{2}) \) and \( Y=(y_{1},y_{2}) \). Find the \( 2\times 2 \) matrix B of b relative to the basis \( U=\{u_{1},u_{2}\} \) where \( u_{1}=(0,1) \) and \( u_{2}=(1,1) \).

27. Let \( X=\{a,b,c\} \). Which of the following classes of subsets of X does not form a topology on X?

28. The eigenvalues for the initial value-eigenvalue problem \( y^{\prime\prime}+\lambda y=0 \), \( y(0)=0 \), \( y(\pi)=0 \) are given by

29. For switching functions f, g, and h, the expression \( (f\vee g)\wedge(\overline{f}\vee h) \) is equivalent to

30. For the inner product \( \langle A, B \rangle=\text{trace}(B^{\prime}A) \) defined on the vector space of 2×2 matrices on \( \mathbb{R} \), find the square of the norm of \( T=\left[\begin{matrix}1&3\\ 2&-1\end{matrix}\right] \).

31. Find the limit of the series \( x^{4}+\dfrac{x^{4}}{1+x^{2}}+\dfrac{x^{4}}{(1+x^{2})^{2}}+\dfrac{x^{4}}{(1+x^{2})^{3}}+... \).

32. The domain of \( f(x)=\int(x+2x^{2}+3x^{3}+...) dx \) is

33. Determine the number of homomorphisms from the group \( \mathbb{Z}_{8} \) onto the group \( \mathbb{Z}_{4} \).

34. The solution of \( x^{2}y^{\prime\prime}+6xy^{\prime}+6y=0 \) for \( x>0 \), is given by

35. The remainder of \( 5^{34} \) when divided by 17 is

36. Find the curl of \( \vec{u}=xyz\vec{i}+xy^{2}\vec{j}+yz\vec{k} \)

37. The inverse of the function \( f(x)=\dfrac{x}{x-1} \) is

38. Find the slope of the tangent line to the ellipse \( 2x^{2}+y^{2}+30=8y-12x \) at \( (x_{0},y_{0}) \), where \( x_{0}=-2 \) and \( y_{0}>4 \)

39. For matrices A=\( \left[\begin{matrix}1&1\\ 0&-1\end{matrix}\right] \), B=\( \left[\begin{matrix}0&1\\ 1&1\end{matrix}\right] \), C=\( \left[\begin{matrix}-1&0\\ 1&1\end{matrix}\right] \), and D=\( \left[\begin{matrix}3&3\\ 0&-2\end{matrix}\right] \), the matrix D is a linear combination (aA+bB+cC) of A, B, C for a,b,c, given by

40. Given \( p(x)=\sum_{k=1}^{+\infty}\dfrac{(x-2)^{k}}{k^{2}} \), find the interval in which \( p^{\prime}(x) \) converges

41. Find the Laplace transform of \( \int_{0}^{x}\sin\,2t\,dt \)

42. If \( f^{\prime}(x_{0})=\sqrt{3} \), then the tangent line to the graph of f at \( x_{0} \) makes an angle of \( \beta \) degrees with the positive x-axis. Find \( \beta \).

43. Let \( T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3} \) be defined by \( T(x,y,z)=(x+y,\,x-y+z,\,y+2z) \). Find the trace of \( T \).

44. Find the sum of \( \dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+... \).

45. Which of the following is not a proper ideal of the ring \( \mathbb{Z}_{12} \)?

46. Assuming that a person selects an answer to each of the first ten questions on this examination at random and that the selections are independent, what is the probability that he/she will guess exactly five answers correct?

47. Find the Jacobian of the transformation from the xy-plane to the uv-plane defined by \( u=f(x,y)=xe^{xy}, v=g(x,y)=ye^{xy} \)

48. On average, a baseball player gets a hit in one out of three attempts. Assuming that the attempts are independent, what is the probability that he gets exactly three hits in six attempts?

49. Find the number of units in the ring \( \mathbb{Z}_{8} \)

50. Define \( f(x)=x \) for \( x\in(0,1) \). Find the coefficient of the third term in the half range Fourier sine series.

51. Let V be the vector space of functions \( f:\mathbb{R}\rightarrow\mathbb{R} \). Let S be the subspace generated by \( \{e^{x},e^{2x},e^{-2x}\} \). Define \( D_{x} \) to be the derivative operator on S. Find the determinant of \( D_{x} \).

52. Find Green's function for \( y^{\prime\prime}+5y^{\prime}+6y=\sin\,x \)

53. The Maclaurin series for \( xe^{-x^{2}} \) is given by

54. The symmetric difference of the sets \( S=\{1,2,3,4,5\} \) and \( T=\{4,5,6,7,8\} \) is

55. Given \( x_{n+2}+6x_{n+1}+9x_{n}=0 \) \((n=0,1,2,\ldots)\), \( x_{0}=1 \), \( x_{1}=0 \), find \( x_{5} \).

56. Let V be the vector space of real polynomials with inner product \( (f,g)=\int_{0}^{1}f(x)g(x)dx \) where \( f,g \in V \). Find the cosine of the angle between \( f(x)=2 \) and \( g(x)=x \).

57. The integral \( \int_{-24}^{4}\dfrac{dx}{\sqrt[3]{{(x-3)}^{2}}} \)

58. Let n be a positive integer greater than 3. Then \( n^{3}+(n+1)^{3}+(n+2)^{3} \) is divisible by

59. If F is a finite field, then which of the following numbers can be the cardinality of F?

60. Consider the set \( S=\{2,3,4,6,8,9\} \) ordered by “s is a multiple of t”. How many minimal elements does S have?

61. Which of the following is a neighborhood of 0 relative to the usual topology τ for the real numbers?

62. Find the Cauchy number for the permutation σ=\( \left[\begin{matrix}1&2&3&4&5&6&7\\ 3&6&5&1&4&7&2\end{matrix}\right] \)\( \in S_{7} \).

63. Which of the following ordinary differential equations is exact?

64. Let G be a graph with vertices \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). If \( \text{val}(x_{1})=2 \), \( \text{val}(x_{2})=2 \), \( \text{val}(x_{3})=3 \), \( \text{val}(x_{4})=3 \), and \( \text{val}(x_{5})=4 \), where val(x) denotes the valence of vertex x, how many edges does G have?

65. If τ is the discrete topology on the real numbers \( \mathbb{R} \), find the closure of (a, b).

66. Define \( f: C^{3}\rightarrow C \) by \( f(c)=x-iy+(2+i)z \) where \( c=(x,y,z) \). Find \( \hat{c}\in C^{3} \) such that \( f(c)=(c,\hat{c}) \), where \( (c,\hat{c}) \) is the usual inner product on \( C^{3} \).