Your FREE GRE Mathematics Subject Test Practice Questions 2026 – 150+ Q&A

GRE Math Practice Questions

1. What is the derivative of $f(x) = e^{2x} \sin(x)$?
   • $e^{2x} \cos(x)$
   • $2e^{2x} \sin(x) + e^{2x} \cos(x)$
   • $e^{2x} \sin(x) + e^{2x} \cos(x)$
   • $2e^{2x} \cos(x) - e^{2x} \sin(x)$

   Correct answer: $2e^{2x} \sin(x) + e^{2x} \cos(x)$

   Correct answer: $2e^{2x} \sin(x) + e^{2x} \cos(x)$. Explanation: Use the product rule for differentiation.

2. Find the integral of $\int \frac{1}{x \ln(x)} dx$.
   • $\ln | \ln(x) | + C$
   • $\ln |x| + C$
   • $\frac{1}{\ln(x)} + C$
   • $\frac{x}{\ln(x)} + C$

   Correct answer: $\ln | \ln(x) | + C$

   Correct answer: $\ln | \ln(x) | + C$. Explanation: Use substitution, setting $u = \ln(x)$.

3. What is the area enclosed by the curve $y = x^2$ and the lines $x = -1, x = 2,$ and $y = 0$?
   • 3 square units
   • 4.5 square units
   • 9 square units
   • 5.5 square units

   Correct answer: 3 square units

   Correct answer: 3 square units. Explanation: Calculate the definite integral of $x^2$ from -1 to 2.

4. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$.
   • Converges
   • Diverges
   • Converges conditionally
   • Cannot be determined

   Correct answer: Converges

   Correct answer: Converges. Explanation: Use the p-series test where $p = 2 > 1$, indicating convergence.

5. If $f(x) = \ln(x)$, what is $f''(e)$?
   • $\frac{1}{e}$
   • $-\frac{1}{e}$
   • $\frac{1}{e^2}$
   • $-\frac{1}{e^2}$

   Correct answer: $-\frac{1}{e^2}$

   Correct answer: $-\frac{1}{e^2}$. Explanation: Calculate the second derivative of $\ln(x)$ and evaluate at $x = e$.

6. Calculate the derivative of $f(x) = \sqrt{x}(x^2 + 1)$.
   • $\frac{1}{2\sqrt{x}}(x^2 + 1) + 2x\sqrt{x}$
   • $\frac{1}{2\sqrt{x}} + 2x\sqrt{x}$
   • $\frac{3}{2}x\sqrt{x}$
   • $\frac{1}{2}x^{-\frac{1}{2}} + 2x^\frac{3}{2}$

   Correct answer: $\frac{1}{2\sqrt{x}}(x^2 + 1) + 2x\sqrt{x}$

   Correct answer: $\frac{1}{2\sqrt{x}}(x^2 + 1) + 2x\sqrt{x}$. Explanation: Apply the product rule and simplify.

7. Evaluate the definite integral $\int_{0}^{2} (3x^2 - 2x) dx$.
   • 4
   • 6
   • 8
   • 10

   Correct answer: 4

   Correct answer: 4. Explanation: Find the antiderivative $x^3 - x^2$ and apply the Fundamental Theorem of Calculus: $(8 - 4) - 0 = 4$.

8. Find the derivative of $g(t) = t^4 \ln(t)$.
   • $4t^3 \ln(t)$
   • $4t^3 \ln(t) + t^3$
   • $4t^3 + t^3 \ln(t)$
   • $t^4 + 4t^3 \ln(t)$

   Correct answer: $4t^3 \ln(t) + t^3$

   Correct answer: $4t^3 \ln(t) + t^3$. Explanation: Use the product rule for differentiation.

9. Evaluate the integral: $\int (3x^2 - 2x + 1) dx$.
   • $x^3 - x^2 + x + C$
   • $x^3 - 2x^2 + x + C$
   • $x^3 - x^2 + 2x + C$
   • $x^3 + x^2 + x + C$

   Correct answer: $x^3 - x^2 + x + C$

   Correct answer: $x^3 - x^2 + x + C$. Explanation: Integrate each term separately.

10. Determine the limit: $\lim_{x \to \infty} \frac{4x^3 - x}{2x^3 + 5}$.
    • 0
    • 1
    • $\frac{4}{2}$
    • $\infty$

    Correct answer: $\frac{4}{2}$

    Correct answer: $\frac{4}{2}$. Explanation: Divide each term by $x^3$ and take the limit.

11. Find the area under the curve $y = e^{2x}$ from $x = 0$ to $x = 1$.
    • $e^2 - 1$
    • $\frac{e^2 - 1}{2}$
    • $e^2$
    • $\frac{e^2}{2}$

    Correct answer: $\frac{e^2 - 1}{2}$

    Correct answer: $\frac{e^2 - 1}{2}$. Explanation: Evaluate the definite integral of the function.

12. Evaluate: $\lim_{x \to 0} \frac{\sin(3x)}{x}$.
    • 0
    • 1
    • 3
    • Undefined

    Correct answer: 3

    Correct answer: 3. Explanation: Apply the standard limit $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ with a change of variable.

13. What is the derivative of $f(x) = x^5 \sin(x^2)$?
    • $5x^4 \sin(x^2) + 2x^6 \cos(x^2)$
    • $5x^4 \sin(x^2) + x^6 \cos(x^2)$
    • $5x^4 \cos(x^2)$
    • $x^5 \cos(x^2)$

    Correct answer: $5x^4 \sin(x^2) + 2x^6 \cos(x^2)$

    Correct answer: $5x^4 \sin(x^2) + 2x^6 \cos(x^2)$. Explanation: Apply the product rule and the chain rule.

14. Calculate the derivative of $f(x) = \frac{1}{x^2 + x + 1}$.
    • $\frac{-2x - 1}{(x^2 + x + 1)^2}$
    • $\frac{-2x + 1}{(x^2 + x + 1)^2}$
    • $\frac{2x - 1}{(x^2 + x + 1)^2}$
    • $\frac{2x + 1}{(x^2 + x + 1)^2}$

    Correct answer: $\frac{-2x - 1}{(x^2 + x + 1)^2}$

    Correct answer: $\frac{-2x - 1}{(x^2 + x + 1)^2}$. Explanation: Apply the quotient rule to find the derivative.

15. Evaluate the limit: $\lim_{x \to 0} \frac{e^{x} - 1}{x}$.
    • 0
    • 1
    • $e$
    • Undefined

    Correct answer: 1

    Correct answer: 1. Explanation: This is a standard limit for exponential functions.

16. Find the indefinite integral: $\int x \cos(x^2) dx$.
    • $\frac{1}{2}\sin(x^2) + C$
    • $\sin(x^2) + C$
    • $\cos(x^2) + C$
    • $\frac{1}{2}\cos(x^2) + C$

    Correct answer: $\frac{1}{2}\sin(x^2) + C$

    Correct answer: $\frac{1}{2}\sin(x^2) + C$. Explanation: Use substitution with $u = x^2$.

17. Compute the area between the curves $y = x^2$ and $y = x$ for $0 \le x \le 1$.
    • $\frac{1}{6}$
    • $\frac{1}{3}$
    • $\frac{1}{2}$
    • $\frac{2}{3}$

    Correct answer: $\frac{1}{6}$

    Correct answer: $\frac{1}{6}$. Explanation: Subtract the integrals of the two functions over the given interval.

18. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$.
    • Converges
    • Diverges
    • Converges conditionally
    • Cannot be determined

    Correct answer: Converges

    Correct answer: Converges. Explanation: The series converges absolutely: the terms $\frac{1}{n^3}$ form a convergent p-series with $p = 3 > 1$, so the alternating series converges. 'Converges conditionally' is incorrect because the absolute series already converges.

19. What is the maximum value of the function $f(x) = -x^2 + 4x - 3$ on the interval [0, 3]?
    • 1
    • 2
    • 3
    • 4

    Correct answer: 1

    Correct answer: 1. Explanation: Find the vertex of the parabola or use calculus to find the maximum.

20. Evaluate the integral: $\int \frac{dx}{x\sqrt{x^2 - 1}}$.
    • $\ln |x + \sqrt{x^2 - 1}| + C$
    • $\arcsin \frac{1}{x} + C$
    • $\operatorname{arcsec} x + C$
    • $\ln |x^2 - \sqrt{x^2 - 1}| + C$

    Correct answer: $\operatorname{arcsec} x + C$

    Correct answer: $\operatorname{arcsec} x + C$. Explanation: Use substitution and trigonometric identities.

21. Find the derivative of $g(y) = \arctan(\sqrt{y})$.
    • $\frac{1}{2\sqrt{y}(1+y)}$
    • $\frac{1}{2\sqrt{y}(1+y^2)}$
    • $\frac{1}{1+y}$
    • $\frac{1}{1+y^2}$

    Correct answer: $\frac{1}{2\sqrt{y}(1+y)}$

    Correct answer: $\frac{1}{2\sqrt{y}(1+y)}$. Explanation: Apply the chain rule and simplify.

22. Compute the limit: $\lim_{x \to \infty} \frac{\sqrt{x^2 + x} - x}{x}$.
    • 0
    • $\frac{1}{2}$
    • 1
    • Undefined

    Correct answer: 0

    Correct answer: 0. Explanation: Multiply by the conjugate and simplify.

23. If $x$ and $y$ are integers such that $x^2 + y^2 = 100$, what is the maximum possible value of $x + y$?
    • 10
    • 14
    • 17
    • 20

    Correct answer: 14

    Correct answer: 14. Explanation: The maximum value occurs when one variable is maximized while keeping the sum of squares equal to 100.

24. Find the roots of the equation $x^3 - 6x^2 + 11x - 6 = 0$.
    • 1, 2, 3
    • 1, 3, 6
    • 2, 4, 5
    • 3, 4, 5

    Correct answer: 1, 2, 3

    Correct answer: 1, 2, 3. Explanation: Use the Rational Root Theorem or factorization to find the roots.

25. If $3^{2x-1} = 27$, what is the value of $x$?
    • 1
    • 2
    • 3
    • 4

    Correct answer: 2

    Correct answer: 2. Explanation: Express 27 as a power of 3 and solve the resulting equation.

26. Given $f(x) = x^2 - 4x + 4$, for what value of $x$ is $f(x) = 0$?
    • 0
    • 2
    • 4
    • 6

    Correct answer: 2

    Correct answer: 2. Explanation: Solve the quadratic equation $x^2 - 4x + 4 = 0$.

27. If $(2x - 1)(x + 3) = 0$, what are the possible values of $x$?
    • $-3$ and $\frac{1}{2}$
    • $-3$ and $\frac{1}{3}$
    • $3$ and $\frac{1}{2}$
    • $3$ and $\frac{1}{3}$

    Correct answer: $-3$ and $\frac{1}{2}$

    Correct answer: $-3$ and $\frac{1}{2}$. Explanation: Set each factor equal to zero and solve for $x$.

28. Given a quadratic equation $x^2 + kx + 16 = 0$, where $k$ is a constant. If the equation has two distinct real roots, which of the following must be true?
    • $k > 4$
    • $k > 8$
    • $k^2 > 64$
    • $k^2 - 64 > 0$

    Correct answer: $k^2 - 64 > 0$

    Correct answer: $k^2 - 64 > 0$. Explanation: For two distinct real roots, the discriminant must be positive.

29. If $5^{2x-1} = 125$, what is the value of $x$?
    • 1
    • 2
    • 3
    • 4

    Correct answer: 2

    Correct answer: 2. Explanation: Express 125 as a power of 5 and solve the resulting equation.

30. What are the solutions to the equation $x^2 - 5x + 6 = 0$?
    • 2 and 3
    • 1 and 6
    • -2 and -3
    • -1 and -6

    Correct answer: 2 and 3

    Correct answer: 2 and 3. Explanation: Factorize the quadratic equation or use the quadratic formula.

31. If $x$ and $y$ satisfy the equation $3x - 2y = 6$, which of the following represents $y$ in terms of $x$?
    • $\frac{3x - 6}{2}$
    • $\frac{6 - 3x}{2}$
    • $2x - 3$
    • $3 - 2x$

    Correct answer: $\frac{3x - 6}{2}$

    Correct answer: $\frac{3x - 6}{2}$. Explanation: Solve the linear equation for $y$ in terms of $x$.

32. If $2x + 3y = 12$ and $3x + 2y = 12$, what is the value of $x - y$?
    • 0
    • 1
    • 2
    • 3

    Correct answer: 0

    Correct answer: 0. Explanation: Solve the system of linear equations and find $x - y$.

33. A set contains all integers $x$ such that $-2 < x < 3$. How many elements are in this set?
    • 3
    • 4
    • 5
    • 6

    Correct answer: 4

    Correct answer: 4. Explanation: The set includes the integers -1, 0, 1, and 2, which are 4 in total.

34. In a class of 30 students, 18 are studying French, 15 are studying German, and 7 are studying both. How many students are not studying either language?
    • 4
    • 6
    • 8
    • 10

    Correct answer: 4

    Correct answer: 4. Explanation: Use the principle of inclusion-exclusion. Total studying at least one language is $18 + 15 - 7 = 26$. So, $30 - 26 = 4$ are studying neither.

35. What is the area of a triangle with sides of length 13, 14, and 15?
    • 84
    • 88
    • 90
    • 92

    Correct answer: 84

    Correct answer: 84. Explanation: Use Heron's formula with semi-perimeter $s = \frac{13 + 14 + 15}{2} = 21$: Area $= \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84$.

36. If the sum of the first $n$ terms of an arithmetic sequence is given by $S_n = 3n^2 + 5n$, what is the first term of the sequence?
    • 8
    • 9
    • 10
    • 11

    Correct answer: 8

    Correct answer: 8. Explanation: The first term is $S_1 = 3(1)^2 + 5(1) = 8$.

37. In a certain sequence, the term $a_n = \frac{1}{n(n+1)}$ for $n \geq 1$. What is the sum of the first 10 terms of this sequence?
    • $\frac{10}{11}$
    • $\frac{1}{10}$
    • $\frac{1}{2}$
    • $\frac{9}{10}$

    Correct answer: $\frac{10}{11}$

    Correct answer: $\frac{10}{11}$. Explanation: Observe the telescoping nature of the series and compute the sum.

38. What is the remainder when $2^{100}$ is divided by 7?
    • 1
    • 2
    • 3
    • 4

    Correct answer: 2

    Correct answer: 2. Explanation: Use the properties of exponents and modular arithmetic to find the pattern of remainders.

39. A rectangle is inscribed in a circle of radius $r$. If the perimeter of the rectangle is $20$, what is the maximum possible area of the rectangle?
    • 25
    • 50
    • 75
    • 100

    Correct answer: 25

    Correct answer: 25. Explanation: Use the relationship between the radius of the circle and the dimensions of the rectangle.

40. If $x$ and $y$ are integers such that $x^2 + y^2 = 25$, what is the maximum possible value of $x$?
    • 3
    • 4
    • 5
    • 6

    Correct answer: 5

    Correct answer: 5. Explanation: Consider the possible integer pairs that satisfy the equation.

41. A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 9?
    • $\frac{5}{36}$
    • $\frac{1}{6}$
    • $\frac{1}{4}$
    • $\frac{4}{36}$

    Correct answer: $\frac{4}{36}$

    Correct answer: $\frac{4}{36}$. Explanation: Count the favorable outcomes and divide by the total possible outcomes.

42. What is the value of $n$ in the permutation equation $P(n, 3) = 60$?
    • 5
    • 6
    • 7
    • 8

    Correct answer: 5

    Correct answer: 5. Explanation: Solve the permutation equation $P(n, 3) = n(n-1)(n-2)$.

43. A circle has a circumference of $8\pi$. What is the area of the circle?
    • $16\pi$
    • $32\pi$
    • $64\pi$
    • $128\pi$

    Correct answer: $16\pi$

    Correct answer: $16\pi$. Explanation: Use the formula for circumference $C = 2\pi r$ to find the radius, then calculate the area.

44. What is the hundredth term of the arithmetic sequence 3, 6, 9, ... ?
    • 297
    • 300
    • 303
    • 306

    Correct answer: 300

    Correct answer: 300. Explanation: Use the formula for the $n$-th term of an arithmetic sequence.

45. What is the equation of the line perpendicular to $y = 2x + 3$ and passing through the point $(1, -2)$?
    • $y = -\frac{1}{2}x + \frac{1}{2}$
    • $y = -\frac{1}{2}x - \frac{3}{2}$
    • $y = -\frac{1}{2}x - 2$
    • $y = -\frac{1}{2}x - \frac{5}{2}$

    Correct answer: $y = -\frac{1}{2}x - \frac{3}{2}$

    Correct answer: $y = -\frac{1}{2}x - \frac{3}{2}$. Explanation: Find the slope of the perpendicular line and use the point to find the equation.

46. If a set contains all numbers $x$ such that $1 < x^2 < 9$, what is the range of $x$?
    • $-2 < x < 2$
    • $-3 < x < 3$
    • $1 < x < 3$
    • $-3 < x < -1$ and $1 < x < 3$

    Correct answer: $-3 < x < -1$ and $1 < x < 3$

    Correct answer: $-3 < x < -1$ and $1 < x < 3$. Explanation: Consider both positive and negative square roots of the given range.

47. The sum of the interior angles of a polygon is 1260 degrees. How many sides does the polygon have?
    • 7
    • 8
    • 9
    • 10

    Correct answer: 9

    Correct answer: 9. Explanation: Use the formula $180(n-2)$ where $n$ is the number of sides.

48. A cylinder has a radius of 3 and a height of 4. What is the volume of the cylinder?
    • $36\pi$
    • $48\pi$
    • $54\pi$
    • $72\pi$

    Correct answer: $36\pi$

    Correct answer: $36\pi$. Explanation: Use the formula for the volume of a cylinder $V = \pi r^2 h$.

49. If $\log_2 x + \log_2 y = 5$ and $\log_2 x - \log_2 y = 1$, what is the value of $x$?
    • 2
    • 4
    • 8
    • 16

    Correct answer: 8

    Correct answer: 8. Explanation: Solve the system of logarithmic equations.

50. Using integration by parts, what is the value of the integral of $x e^x$ with respect to $x$?
    • $x e^x + e^x + C$
    • $x e^x - e^x + C$
    • $\frac{x^2}{2} e^x + C$
    • $e^x(x + 1) + C$

    Correct answer: $x e^x - e^x + C$

    The antiderivative is $x e^x - e^x + C$. Integration by parts uses the formula $\int u \, dv = u v - \int v \, du$; choosing $u = x$ (so $du = dx$) and $dv = e^x \, dx$ (so $v = e^x$) gives $x e^x - \int e^x \, dx = x e^x - e^x + C$. The tempting $x e^x + e^x + C$ reverses the sign of the second term.

51. Apply L'Hopital's rule to evaluate $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$.
    • $\frac{1}{2}$
    • 1
    • Does not exist
    • 0

    Correct answer: $\frac{1}{2}$

    The limit equals $\frac{1}{2}$. The form is $\frac{0}{0}$, so differentiating top and bottom gives $\frac{e^x - 1}{2x}$, still $\frac{0}{0}$; a second application gives $\frac{e^x}{2}$, which approaches $\frac{1}{2}$ as $x \to 0$. Stopping after one application and concluding the limit is 0 misses the second indeterminate form.

52. The Mean Value Theorem guarantees a point $c$ in $(1, 4)$ for $f(x) = x^2$ where the instantaneous rate of change equals the average rate of change. What is $c$?
    • $\sqrt{5}$
    • 3
    • 2.5
    • 2

    Correct answer: 2.5

    The value is $c = 2.5$. The Mean Value Theorem requires $f'(c) = \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5$; since $f'(x) = 2x$, setting $2c = 5$ gives $c = 2.5$, which lies in $(1, 4)$. The midpoint 2.5 of the interval coincidentally equals the answer here because $f$ is quadratic, but the theorem is satisfied by the derivative condition.

53. By the Fundamental Theorem of Calculus, if $F(x)$ is the integral from 0 to $x$ of $\cos(t^2) \, dt$, what is $F'(x)$?
    • $-\sin(x^2) \cdot 2x$
    • $\cos(x^2)$
    • $\frac{1}{2} \sin(x^2)$
    • $\cos(x^2) \cdot 2x$

    Correct answer: $\cos(x^2)$

    $F'(x) = \cos(x^2)$. The Fundamental Theorem of Calculus (Part 1) states that the derivative of an integral with a constant lower limit and variable upper limit $x$ is simply the integrand evaluated at $x$. No chain-rule factor of $2x$ appears because the upper limit is $x$ itself, not $x^2$.

54. Using the chain rule, what is the derivative of $f(x) = (3x^2 + 1)^5$?
    • $30x(3x^2 + 1)^4$
    • $5(6x)^4$
    • $6x(3x^2 + 1)^4$
    • $5(3x^2 + 1)^4$

    Correct answer: $30x(3x^2 + 1)^4$

    The derivative is $30x(3x^2 + 1)^4$. The chain rule multiplies the derivative of the outer power, $5(3x^2 + 1)^4$, by the derivative of the inner function $3x^2 + 1$, which is $6x$, giving $5 \cdot 6x = 30x$ times $(3x^2 + 1)^4$. Forgetting the inner derivative leaves the incorrect $5(3x^2 + 1)^4$.

55. By the alternating series test, which condition on the terms $b_n$ (all positive) must hold for the series $\sum (-1)^n b_n$ to converge?
    • $b_n$ must equal $\frac{1}{n}$ exactly
    • $b_n$ must be increasing
    • $b_n$ must approach 0 and be eventually decreasing
    • The sum of $b_n$ must converge

    Correct answer: $b_n$ must approach 0 and be eventually decreasing

    The terms $b_n$ must approach 0 and be eventually monotonically decreasing. The alternating series test (Leibniz test) guarantees convergence when these two conditions hold, even if the series of absolute values diverges. Requiring the sum of $b_n$ to converge would describe absolute convergence, a strictly stronger condition than the test requires.

56. Green's theorem converts a line integral around a positively oriented simple closed curve $C$ into a double integral over the enclosed region $D$. For the field $F = (P, Q)$, the line integral of $P \, dx + Q \, dy$ equals the double integral over $D$ of which expression?
    • $\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}$
    • $\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial x}$
    • $\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$
    • $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$

    Correct answer: $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$

    The integrand is $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$. Green's theorem equates the counterclockwise circulation line integral of $P \, dx + Q \, dy$ to the double integral of the scalar curl, $Q_x - P_y$, over the enclosed region. Reversing this to $P_y - Q_x$ flips the sign and corresponds to clockwise orientation.

57. Using the limit comparison test with $b_n = \frac{1}{n^2}$, what can be concluded about the series $\sum_{n=1}^{\infty} \frac{n + 1}{n^3 + 5}$?
    • It converges
    • It diverges
    • It converges only conditionally
    • The test is inconclusive

    Correct answer: It converges

    The series converges. The limit comparison test compares $a_n = \frac{n+1}{n^3+5}$ with $b_n = \frac{1}{n^2}$; the limit of $\frac{a_n}{b_n}$ equals the limit of $\frac{n^2(n+1)}{n^3+5} = 1$, a finite positive number. Since $\sum \frac{1}{n^2}$ is a convergent p-series ($p = 2 > 1$), the given series also converges.

58. What is the Maclaurin series for $\cos(x)$?
    • $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$
    • $1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots$
    • $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
    • $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

    Correct answer: $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$

    The Maclaurin series for $\cos(x)$ is $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$, containing only even powers of $x$ with alternating signs. A Maclaurin series is a Taylor series centered at 0; the series with only odd powers ($x - \frac{x^3}{3!} + \cdots$) is the expansion of $\sin(x)$, not cosine.

59. Using Lagrange multipliers to extremize $f(x, y) = x + y$ subject to the constraint $x^2 + y^2 = 8$, what is the maximum value of $f$?
    • $2\sqrt{2}$
    • 2
    • 4
    • $\sqrt{8}$

    Correct answer: 4

    The maximum value is 4. Setting $\nabla f = \lambda \nabla g$ gives $1 = 2 \lambda x$ and $1 = 2 \lambda y$, so $x = y$; substituting into $x^2 + y^2 = 8$ yields $2x^2 = 8$, $x = 2$, so $x = y = 2$ and $f = 4$. The competing value $2\sqrt{2}$ is the distance $\sqrt{8}$, not the sum $x + y$.

60. For $f(x, y) = x^2 y + 3y^3$, what is the partial derivative of $f$ with respect to $y$?
    • $x^2 + 3y^2$
    • $2xy + 9y^2$
    • $2xy$
    • $x^2 + 9y^2$

    Correct answer: $x^2 + 9y^2$

    The partial derivative with respect to $y$ is $x^2 + 9y^2$. When differentiating partially with respect to $y$, $x$ is held constant, so $x^2 y$ differentiates to $x^2$ and $3y^3$ differentiates to $9y^2$. The expression $2xy$ is the partial derivative with respect to $x$ of the first term, not with respect to $y$.

61. The integral test can be applied to $\sum_{n=1}^{\infty} \frac{1}{n (\ln n)^2}$ starting at $n = 2$. What does it conclude?
    • The series converges
    • The series oscillates
    • The integral test does not apply
    • The series diverges

    Correct answer: The series converges

    The series converges. The integral test applies because $f(x) = \frac{1}{x (\ln x)^2}$ is positive, continuous, and decreasing for $x \ge 2$; the improper integral from 2 to infinity equals the limit of $-\frac{1}{\ln x}$, which converges to $\frac{1}{\ln 2}$. Since the integral converges, so does the series.

62. A spherical balloon is inflated so its volume increases at $36\pi$ cubic cm per second. Using related rates, how fast is the radius increasing when the radius is 3 cm? ($V = \frac{4}{3} \pi r^3$.)
    • 1 cm/s
    • $4\pi$ cm/s
    • 2 cm/s
    • 3 cm/s

    Correct answer: 1 cm/s

    The radius increases at 1 cm/s. Differentiating $V = \frac{4}{3} \pi r^3$ gives $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$; substituting $\frac{dV}{dt} = 36\pi$ and $r = 3$ gives $36\pi = 4 \pi (9) \frac{dr}{dt} = 36\pi \frac{dr}{dt}$, so $\frac{dr}{dt} = 1$. The related rates method links the rates through the chain rule applied to the volume formula.

63. Evaluate the line integral of the scalar field $f(x, y) = x$ along the curve $C$ parameterized by $x = t$, $y = t$ for $t$ from 0 to 1, with respect to arc length $ds$.
    • $\frac{\sqrt{2}}{2}$
    • 1
    • $\frac{1}{2}$
    • $\sqrt{2}$

    Correct answer: $\frac{\sqrt{2}}{2}$

    The line integral equals $\frac{\sqrt{2}}{2}$. With $x = t$ and $y = t$, $ds = \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt = \sqrt{2} \, dt$, and $f = t$, so the integral is $\int_0^1 t \sqrt{2} \, dt = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}$. Omitting the $\sqrt{2}$ arc-length factor gives the incorrect value $\frac{1}{2}$.

64. According to the epsilon-delta definition of a limit, the statement '$\lim_{x \to a} f(x) = L$' means: for every $\epsilon > 0$ there exists a $\delta > 0$ such that which condition holds?
    • If $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$
    • If $|f(x) - L| < \delta$ then $|x - a| < \epsilon$
    • If $x = a$ then $f(x) = L$
    • If $|x - a| < \epsilon$ then $|f(x) - L| < \delta$

    Correct answer: If $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$

    The correct condition is: if $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$. The epsilon-delta definition says that for any tolerance $\epsilon$ on the output, a positive $\delta$ on the input can be found so that all $x$ within $\delta$ of $a$ (but not equal to $a$) map to $f(x)$ within $\epsilon$ of $L$. Swapping the roles of $\epsilon$ and $\delta$ describes a different, incorrect statement.

65. What is the sum of the telescoping series $\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right)$?
    • 0
    • 1
    • $\frac{1}{2}$
    • Diverges

    Correct answer: 1

    The sum is 1. In a telescoping series, successive terms cancel: the partial sum $S_N = 1 - \frac{1}{N+1}$, and as $N$ approaches infinity, $\frac{1}{N+1}$ approaches 0, leaving 1. Recognizing the cancellation pattern is the key technique rather than applying a convergence test.

66. Evaluate the double integral over the rectangle $0 \le x \le 2$, $0 \le y \le 3$ of the function $f(x, y) = x y$.
    • 18
    • 6
    • 9
    • 12

    Correct answer: 9

    The double integral equals 9. Because the integrand separates as $x$ times $y$ over a rectangle, the value is $\left( \int_0^2 x \, dx \right) \left( \int_0^3 y \, dy \right) = 2 \cdot 4.5 = 9$. The value 18 would result from forgetting the factor of one-half in one of the single integrals.

67. Using the ratio test, what does the test conclude about the series $\sum_{n=1}^{\infty} \frac{n!}{10^n}$?
    • The ratio test is inconclusive
    • It converges
    • It converges conditionally
    • It diverges

    Correct answer: It diverges

    The series diverges. The ratio test computes the limit of $\frac{a_{n+1}}{a_n} = \frac{(n+1)! / 10^{n+1}}{n! / 10^n} = \frac{n+1}{10}$, which approaches infinity. Since this limit exceeds 1, the ratio test guarantees divergence; the factorial grows far faster than the exponential $10^n$.

68. For what values of the common ratio $r$ does the geometric series $\sum_{n=0}^{\infty} r^n$ converge, and to what value?
    • All $r$, converging to $\frac{1}{1 + r}$
    • $|r| \le 1$, converging to $\frac{1}{1 - r}$
    • $r > 0$, converging to $\frac{r}{1 - r}$
    • $|r| < 1$, converging to $\frac{1}{1 - r}$

    Correct answer: $|r| < 1$, converging to $\frac{1}{1 - r}$

    The geometric series converges if and only if $|r| < 1$, with sum $\frac{1}{1 - r}$. When the absolute value of the ratio is at least 1, the terms do not approach 0 and the series diverges. The endpoint $r = 1$ gives $1 + 1 + 1 + \cdots$, which diverges, so the inequality must be strict.

69. Apply the root test to the series $\sum_{n=1}^{\infty} \left( \frac{n}{2n+1} \right)^n$. What is the conclusion?
    • It converges to $\frac{1}{2}$
    • It converges
    • The root test is inconclusive
    • It diverges

    Correct answer: It converges

    The series converges. The root test evaluates the limit of the $n$th root of $|a_n|$, which is the limit of $\frac{n}{2n+1} = \frac{1}{2}$. Because this limit $\frac{1}{2}$ is less than 1, the root test guarantees convergence. The value $\frac{1}{2}$ is the limit used by the test, not the sum of the series.

70. How do you find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$?
    • The limit does not exist
    • Substitute directly to get 0
    • Factor and cancel to get 4
    • Apply the squeeze theorem to get 2

    Correct answer: Factor and cancel to get 4

    Factoring and canceling gives the limit 4. The expression $\frac{x^2 - 4}{x - 2}$ factors as $\frac{(x - 2)(x + 2)}{x - 2}$; canceling the common factor leaves $x + 2$, which evaluates to 4 at $x = 2$. Direct substitution initially yields the indeterminate form $\frac{0}{0}$, so algebraic simplification is the correct technique.

71. What is the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$?
    • $e$
    • 1
    • 0
    • Infinity

    Correct answer: Infinity

    The radius of convergence is infinity. Applying the ratio test to $\frac{x^n}{n!}$ gives the limit of $\frac{|x| \, n!}{(n+1)!} = \frac{|x|}{n+1}$, which approaches 0 for every $x$. Since the limit is less than 1 for all real $x$, the series (which equals $e^x$) converges everywhere.

72. What is the difference between absolute convergence and conditional convergence of a series?
    • Absolute applies only to positive series; conditional only to negative series
    • Absolute means $\sum |a_n|$ converges; conditional means the series converges but $\sum |a_n|$ diverges
    • They are two names for the same property
    • Absolute means the series diverges; conditional means it converges

    Correct answer: Absolute means $\sum |a_n|$ converges; conditional means the series converges but $\sum |a_n|$ diverges

    A series is absolutely convergent when the sum of the absolute values of its terms converges, whereas it is conditionally convergent when the series itself converges but the sum of absolute values diverges. The alternating harmonic series $\sum \frac{(-1)^n}{n}$ is the classic conditionally convergent example, since $\sum \frac{1}{n}$ diverges.

73. Using the direct comparison test, what is concluded about the series $\sum_{n=1}^{\infty} \frac{1}{n^2 + 3n}$?
    • It diverges because each term is at least $\frac{1}{n}$
    • It converges only conditionally
    • The comparison test does not apply
    • It converges because each term is at most $\frac{1}{n^2}$

    Correct answer: It converges because each term is at most $\frac{1}{n^2}$

    The series converges because $\frac{1}{n^2 + 3n} \le \frac{1}{n^2}$ for all $n \ge 1$, and $\sum \frac{1}{n^2}$ is a convergent p-series. The direct comparison test states that if terms of a positive series are dominated by the terms of a known convergent series, the given series also converges.

74. A function $f$ is differentiable at a point. Which statement about continuity and differentiability is true?
    • Differentiability at a point implies continuity there, but continuity does not imply differentiability
    • Continuity at a point implies differentiability there
    • Neither property implies the other
    • Differentiability implies the function is constant

    Correct answer: Differentiability at a point implies continuity there, but continuity does not imply differentiability

    Differentiability at a point implies continuity at that point, but the converse fails. The absolute value function $|x|$ is continuous at $x = 0$ yet not differentiable there because its left and right derivatives differ. Thus continuity is a necessary but not sufficient condition for differentiability.

75. Determine whether the sequence $a_n = \frac{3n + 1}{n + 2}$ converges, and if so, to what limit.
    • Converges to 3
    • Converges to $\frac{1}{2}$
    • Diverges to infinity
    • Converges to 0

    Correct answer: Converges to 3

    The sequence converges to 3. Dividing numerator and denominator by $n$ gives $\frac{3 + 1/n}{1 + 2/n}$; as $n$ approaches infinity, the $\frac{1}{n}$ and $\frac{2}{n}$ terms vanish, leaving $\frac{3}{1} = 3$. The ratio of the leading coefficients determines the limit when numerator and denominator have the same degree.

76. How can you tell whether the series $\sum_{n=1}^{\infty} \frac{1}{n}$ is convergent or divergent?
    • It converges because the terms approach $0$
    • It diverges to $-\infty$
    • It diverges; it is the harmonic series, a p-series with $p = 1$
    • It converges to $1$

    Correct answer: It diverges; it is the harmonic series, a p-series with $p = 1$

    The harmonic series $\sum \frac{1}{n}$ diverges. It is a p-series with $p = 1$, and p-series diverge whenever $p \le 1$. The fact that the individual terms $\frac{1}{n}$ approach $0$ is necessary but not sufficient for convergence, which is a common misconception.

77. How do you find the Taylor series of a function $f$ about a point $a$?
    • $\sum_{n=0}^{\infty} f(a)^n (x - a)^n$
    • $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$
    • Differentiate $f$ once and multiply by $(x - a)$
    • $\sum_{n=0}^{\infty} \frac{f^{(n)}(x)}{n!} a^n$

    Correct answer: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

    The Taylor series of $f$ about $a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$. Each coefficient is the $n$th derivative of $f$ evaluated at the center $a$, divided by $n!$. When the center $a$ is $0$, this reduces to the Maclaurin series.

78. For the scalar function $f(x, y) = x^2 + 3xy$, what is the gradient vector at the point $(1, 2)$?
    • $(8, 3)$
    • $(8, 6)$
    • $(2, 3)$
    • $(5, 6)$

    Correct answer: $(8, 3)$

    The gradient is $(8, 3)$. The gradient in multivariable calculus is the vector of partial derivatives: $f_x = 2x + 3y$ and $f_y = 3x$; at $(1, 2)$ these give $2(1) + 3(2) = 8$ and $3(1) = 3$, so $\nabla f = (8, 3)$. The gradient points in the direction of steepest increase of $f$.

79. What is the Taylor series (about $0$) of the common function $\frac{1}{1 - x}$ for $|x| < 1$?
    • $1 + x + x^2 + x^3 + \cdots$
    • $x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots$
    • $1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots$
    • $1 - x + x^2 - x^3 + \cdots$

    Correct answer: $1 + x + x^2 + x^3 + \cdots$

    The series for $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 + \cdots$, the geometric series with ratio $x$, valid for $|x| < 1$. This is one of the standard Taylor series of common functions. The alternating version $1 - x + x^2 - \cdots$ corresponds instead to $\frac{1}{1 + x}$.

80. Evaluate the improper integral $\int_{1}^{\infty} \frac{1}{x^2}\, dx$.
    • Diverges
    • 1
    • 2
    • $\frac{1}{2}$

    Correct answer: 1

    The improper integral equals $1$. The antiderivative of $\frac{1}{x^2}$ is $-\frac{1}{x}$; evaluating from $1$ to $b$ gives $-\frac{1}{b} + 1$, and as $b \to \infty$, $-\frac{1}{b}$ approaches $0$, leaving $1$. This contrasts with the integral of $\frac{1}{x}$, which diverges.

81. A particle's position is $s(t) = t^3 - 6t^2 + 9t$. At what time $t > 0$ does the particle change direction for the first time?
    • $t = 3$
    • $t = 1$
    • $t = 2$
    • $t = 0$

    Correct answer: $t = 1$

    The particle first changes direction at $t = 1$. Velocity is $s'(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3)$; the velocity changes sign at $t = 1$, going from positive to negative. The other root $t = 3$ is the second direction change, not the first.

82. What is the value of the integral $\int_{0}^{\pi/2} \sin(x)\cos(x)\, dx$?
    • $\frac{1}{2}$
    • 0
    • $\frac{\pi}{4}$
    • 1

    Correct answer: $\frac{1}{2}$

    The integral equals $\frac{1}{2}$. Using the substitution $u = \sin(x)$, $du = \cos(x)\, dx$, the integral becomes $\int_{0}^{1} u\, du = \frac{1}{2}$. Alternatively, $\sin(x)\cos(x) = \frac{1}{2}\sin(2x)$, whose integral over this interval also gives $\frac{1}{2}$.

83. Find the local minimum value of $f(x) = x^3 - 3x$ on its domain of all real numbers.
    • $-2$
    • $-1$
    • 0
    • 2

    Correct answer: $-2$

    The local minimum value is $-2$. Setting $f'(x) = 3x^2 - 3 = 0$ gives critical points $x = 1$ and $x = -1$; the second derivative $f''(x) = 6x$ is positive at $x = 1$, indicating a local minimum, and $f(1) = 1 - 3 = -2$. The point $x = -1$ yields a local maximum value of $2$.

84. What is the arc length of the curve $y = \frac{2}{3} x^{3/2}$ from $x = 0$ to $x = 3$?
    • $\frac{14}{3}$
    • 6
    • $\frac{13}{3}$
    • $\frac{16}{3}$

    Correct answer: $\frac{14}{3}$

    The arc length is $\frac{14}{3}$. The formula is $\int \sqrt{1 + (y')^2}\, dx$; here $y' = x^{1/2}$, so $1 + (y')^2 = 1 + x$, and $\int_{0}^{3} \sqrt{1 + x}\, dx = \frac{2}{3}(1 + x)^{3/2}$ evaluated from $0$ to $3$ equals $\frac{2}{3}(8 - 1) = \frac{14}{3}$.

85. Evaluate $\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^x$.
    • 3
    • $e^3$
    • $e$
    • $\infty$

    Correct answer: $e^3$

    The limit equals $e^3$. The standard limit $\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a$; here $a = 3$, so the value is $e^3$. This is the continuous-compounding form of the exponential limit definition.

86. What is the derivative of $f(x) = \arcsin(x)$?
    • $-\frac{1}{\sqrt{1 - x^2}}$
    • $\frac{1}{1 + x^2}$
    • $\frac{1}{\sqrt{x^2 - 1}}$
    • $\frac{1}{\sqrt{1 - x^2}}$

    Correct answer: $\frac{1}{\sqrt{1 - x^2}}$

    The derivative of $\arcsin(x)$ is $\frac{1}{\sqrt{1 - x^2}}$. This is a standard inverse-trigonometric derivative valid on $(-1, 1)$. The expression $\frac{1}{1 + x^2}$ is the derivative of $\arctan(x)$, a common point of confusion.

87. Using implicit differentiation, find $\frac{dy}{dx}$ for the curve $x^2 + xy + y^2 = 7$ at a general point.
    • $-\frac{2x + y}{2y}$
    • $-\frac{2x + y}{x + 2y}$
    • $-\frac{x + 2y}{2x + y}$
    • $\frac{2x + y}{x + 2y}$

    Correct answer: $-\frac{2x + y}{x + 2y}$

    The derivative is $-\frac{2x + y}{x + 2y}$. Differentiating implicitly gives $2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$; collecting $\frac{dy}{dx}$ terms yields $(x + 2y)\frac{dy}{dx} = -(2x + y)$, so $\frac{dy}{dx} = -\frac{2x + y}{x + 2y}$. The product rule on the $xy$ term contributes both $y$ and $x\frac{dy}{dx}$.

88. What is the value of the integral $\int_{0}^{1} \frac{1}{1 + x^2}\, dx$?
    • $\ln 2$
    • $\frac{\pi}{2}$
    • $\frac{\pi}{4}$
    • 1

    Correct answer: $\frac{\pi}{4}$

    The integral equals $\frac{\pi}{4}$. The antiderivative of $\frac{1}{1 + x^2}$ is $\arctan(x)$; evaluating from $0$ to $1$ gives $\arctan(1) - \arctan(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$. This is a frequently used definite integral connecting calculus to the value of $\pi$.

89. For the function $f(x, y) = x^2 - y^2$, classify the critical point at the origin using the second derivative test.
    • Saddle point
    • Inconclusive
    • Local maximum
    • Local minimum

    Correct answer: Saddle point

    The origin is a saddle point. The second-derivative (Hessian) test computes $D = f_{xx} f_{yy} - (f_{xy})^2 = (2)(-2) - 0 = -4$; a negative discriminant $D$ indicates a saddle point regardless of the signs of the individual second partials. The function increases along the $x$-axis and decreases along the $y$-axis.

90. Evaluate $\lim_{x \to 0^+} x \ln(x)$.
    • Does not exist
    • 0
    • 1
    • $-\infty$

    Correct answer: 0

    The limit is $0$. Rewriting $x \ln(x)$ as $\frac{\ln(x)}{1/x}$ gives the indeterminate form $\frac{-\infty}{\infty}$, and L'Hopital's rule differentiates to $\frac{1/x}{-1/x^2} = -x$, which approaches $0$. The product of a quantity approaching $0$ and one approaching $-\infty$ resolves to $0$ here.

91. What is the average value of $f(x) = x^2$ on the interval $[0, 3]$?
    • 4.5
    • 9
    • 3
    • 6

    Correct answer: 3

    The average value is $3$. The average value of a function on $[a, b]$ is $\frac{1}{b - a} \int_{a}^{b} f(x)\, dx$; here it is $\frac{1}{3} \int_{0}^{3} x^2\, dx = \frac{1}{3}(9) = 3$. The integral itself equals $9$, but the average requires dividing by the interval length.

92. Evaluate the integral over the region bounded by switching the order: $\int_{0}^{1} \int_{y}^{1} e^{x^2}\, dx\, dy$.
    • $e - 1$
    • $\frac{e - 1}{2}$
    • $\frac{e^2 - 1}{2}$
    • $\frac{1}{2}$

    Correct answer: $\frac{e - 1}{2}$

    The value is $\frac{e - 1}{2}$. Because $e^{x^2}$ has no elementary antiderivative, reversing the order of integration to $\int_{0}^{1} \int_{0}^{x} e^{x^2}\, dy\, dx$ gives $\int_{0}^{1} x e^{x^2}\, dx$, which by substitution $u = x^2$ equals $\frac{1}{2}(e - 1)$. Recognizing when to switch the order is essential here.

93. What is the curvature of a circle of radius $5$?
    • $\frac{1}{5}$
    • $\frac{1}{25}$
    • 25
    • 5

    Correct answer: $\frac{1}{5}$

    The curvature of a circle of radius $5$ is $\frac{1}{5}$. Curvature is defined as the reciprocal of the radius of the osculating circle, so for a circle of radius $r$ the curvature is constant and equals $\frac{1}{r}$. A larger radius corresponds to a gentler bend and thus smaller curvature.

94. Using partial fractions, evaluate $\int \frac{1}{x^2 - 1}\, dx$.
    • $\frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$
    • $\ln |x^2 - 1| + C$
    • $\arctan(x) + C$
    • $\frac{1}{2} \ln |x^2 - 1| + C$

    Correct answer: $\frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$

    The antiderivative is $\frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$. Decomposing $\frac{1}{x^2 - 1}$ into $\frac{1/2}{x - 1} - \frac{1/2}{x + 1}$ and integrating each term gives $\frac{1}{2} \ln|x - 1| - \frac{1}{2} \ln|x + 1|$, which combines into $\frac{1}{2} \ln$ of the ratio. Partial fraction decomposition is the standard technique for rational integrands with factorable denominators.

95. Using Green's theorem, evaluate the counterclockwise line integral $\oint -y\, dx + x\, dy$ around the circle $x^2 + y^2 = 4$.
    • $16\pi$
    • $4\pi$
    • $2\pi$
    • $8\pi$

    Correct answer: $8\pi$

    The line integral equals $8\pi$. Green's theorem converts the boundary integral of $P\, dx + Q\, dy$ into the double integral of $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ over the enclosed region; with $P = -y$ and $Q = x$, the integrand is $1 - (-1) = 2$, so the value is $2$ times the enclosed area. The disk of radius $2$ has area $4\pi$, giving $2 \times 4\pi = 8\pi$. Forgetting the factor of $2$ yields the tempting $4\pi$.

96. Apply the limit comparison test to determine whether the series $\sum_{n=1}^{\infty} \frac{2n + 1}{n^3 + n}$ converges.
    • The test is inconclusive
    • Converges, comparing with $\sum \frac{1}{n^2}$
    • Converges, comparing with $\sum \frac{1}{n}$
    • Diverges, comparing with $\sum \frac{1}{n}$

    Correct answer: Converges, comparing with $\sum \frac{1}{n^2}$

    The series converges by comparison with $\sum \frac{1}{n^2}$. The limit comparison test takes the ratio of the term $\frac{2n + 1}{n^3 + n}$ to $\frac{1}{n^2}$, which simplifies to $\frac{2n^3 + n^2}{n^3 + n}$ and approaches the finite positive limit $2$. Because the comparison series $\sum \frac{1}{n^2}$ converges as a p-series with $p = 2$, the original series converges as well.

97. Determine whether the geometric series $\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n$ converges, and if so to what value.
    • Converges to $\frac{2}{3}$
    • Diverges
    • Converges to $3$
    • Converges to $\frac{3}{2}$

    Correct answer: Converges to $3$

    The geometric series converges to $3$. A geometric series with first term $a$ and common ratio $r$ converges when $|r| < 1$, and its sum is $\frac{a}{1 - r}$. Here $a = 1$ (the $n = 0$ term) and $r = \frac{2}{3}$, so the sum is $\frac{1}{1 - 2/3} = \frac{1}{1/3} = 3$. The value $\frac{3}{2}$ mistakenly uses a ratio of $\frac{1}{3}$ instead of $\frac{2}{3}$.

98. Compute the determinant of the $3 \times 3$ matrix with rows $(2, 1, 3)$, $(0, 4, 1)$, and $(5, 2, 1)$.
    • $-51$
    • 51
    • $-39$
    • 9

    Correct answer: $-51$

    The determinant equals $-51$. Expanding along the first row by cofactors: $2(4 \cdot 1 - 1 \cdot 2) - 1(0 \cdot 1 - 1 \cdot 5) + 3(0 \cdot 2 - 4 \cdot 5) = 2(2) - 1(-5) + 3(-20) = 4 + 5 - 60 = -51$. Each $2 \times 2$ minor must keep the sign pattern $+\ -\ +$ along the top row, and sign errors on the middle and right terms produce the positive distractors.

99. A $2 \times 2$ matrix has trace $7$ and determinant $12$. What are its eigenvalues?
    • $-3$ and $-4$
    • $2$ and $6$
    • $1$ and $6$
    • $3$ and $4$

    Correct answer: $3$ and $4$

    The eigenvalues are $3$ and $4$. For a $2 \times 2$ matrix the eigenvalues are the roots of $\lambda^2 - (\text{trace})\lambda + (\text{determinant}) = \lambda^2 - 7\lambda + 12 = (\lambda - 3)(\lambda - 4)$. The pair must sum to the trace ($3 + 4 = 7$) and multiply to the determinant ($3 \cdot 4 = 12$); $2$ and $6$ multiply to $12$ but sum to $8$, so they fail the trace condition.

100. Find an eigenvector of the matrix with rows $(4, 1)$ and $(2, 3)$ corresponding to the eigenvalue $5$.
     • $(1, 0)$
     • $(2, 1)$
     • $(1, -2)$
     • $(1, 1)$

     Correct answer: $(1, 1)$

     The vector $(1, 1)$ is an eigenvector for eigenvalue $5$. Solving $(A - 5I)v = 0$ gives the equations $-x + y = 0$ and $2x - 2y = 0$, both requiring $x = y$, so any nonzero multiple of $(1, 1)$ works. Multiplying $A$ by $(1, 1)$ gives $(5, 5) = 5(1, 1)$, confirming it.

101. Which statement correctly defines an eigenvalue of a square matrix $A$?
     • A vector $v$ satisfying $Av = 0$
     • Any entry on the main diagonal of $A$
     • A scalar $\lambda$ such that $Av = \lambda v$ only when $v$ is the zero vector
     • A scalar $\lambda$ such that $Av = \lambda v$ for some nonzero vector $v$

     Correct answer: A scalar $\lambda$ such that $Av = \lambda v$ for some nonzero vector $v$

     An eigenvalue is a scalar $\lambda$ for which there exists a nonzero vector $v$ with $Av = \lambda v$. The nonzero requirement is essential, since $Av = \lambda v$ always holds trivially for $v = 0$ and would make every scalar an eigenvalue. Diagonal entries are eigenvalues only for triangular matrices, not in general.

102. Which statement best describes an eigenvector of a matrix $A$?
     • The column of $A$ with the largest entry
     • Any vector in the null space of $A$
     • A nonzero vector whose direction is unchanged (up to scaling) when multiplied by $A$
     • A vector orthogonal to every row of $A$

     Correct answer: A nonzero vector whose direction is unchanged (up to scaling) when multiplied by $A$

     An eigenvector is a nonzero vector $v$ such that $Av$ is a scalar multiple of $v$, meaning $A$ only stretches or compresses it without changing its line of direction. Null-space vectors are the special case of eigenvalue $0$, but a general eigenvector need not lie in the null space.

103. A linear map $T$ from $\mathbb{R}^5$ to $\mathbb{R}^4$ has a matrix of rank $3$. By the rank-nullity theorem, what is the dimension of the kernel of $T$?
     • 2
     • 5
     • 3
     • 1

     Correct answer: 2

     The kernel has dimension $2$. The rank-nullity theorem states $\text{rank} + \text{nullity} = \dim(\text{domain})$, so $\text{nullity} = 5 - 3 = 2$. The domain is $\mathbb{R}^5$ (the source space), not the codomain $\mathbb{R}^4$, so the relevant dimension to subtract from is $5$.

104. What is the rank of the matrix with rows $(1, 2, 3)$, $(2, 4, 6)$, and $(1, 1, 1)$?
     • 1
     • 0
     • 3
     • 2

     Correct answer: 2

     The rank is $2$. The second row is exactly twice the first, so it contributes no new information, leaving the first and third rows, which are linearly independent (not scalar multiples of each other). Row reduction yields exactly two nonzero rows, so the rank is $2$, not the full $3$.

105. Which condition guarantees that an $n \times n$ matrix $A$ is diagonalizable over the real numbers?
     • $A$ is invertible
     • $A$ has determinant equal to $1$
     • $A$ has trace equal to zero
     • $A$ has $n$ distinct real eigenvalues

     Correct answer: $A$ has $n$ distinct real eigenvalues

     Having $n$ distinct real eigenvalues guarantees diagonalizability, because eigenvectors corresponding to distinct eigenvalues are linearly independent, producing a full set of $n$ independent eigenvectors. Invertibility is unrelated to diagonalizability (a matrix can be invertible yet non-diagonalizable, like a Jordan block), and determinant or trace conditions say nothing about eigenvector independence.

106. Which set of vectors in $\mathbb{R}^3$ is linearly independent?
     • $(1, 1, 0), (0, 0, 0), (0, 1, 1)$
     • $(1, 2, 3), (2, 4, 6), (0, 0, 1)$
     • $(1, 2, 1), (2, 4, 2), (3, 6, 3)$
     • $(1, 0, 0), (0, 1, 0), (1, 1, 1)$

     Correct answer: $(1, 0, 0), (0, 1, 0), (1, 1, 1)$

     The set $(1, 0, 0), (0, 1, 0), (1, 1, 1)$ is linearly independent because no vector is a linear combination of the others, and the $3 \times 3$ matrix they form has nonzero determinant $1$. Any set containing the zero vector is automatically dependent, and the other rejected sets contain a vector that is a scalar multiple of another.

107. Which of the following is required for a set $V$ to be a vector space over a field $F$?
     • $V$ contains finitely many elements
     • Every nonzero element of $V$ has a multiplicative inverse in $V$
     • $V$ is closed under addition and under scalar multiplication by elements of $F$, with an additive identity and additive inverses
     • $V$ is closed under multiplication of its elements

     Correct answer: $V$ is closed under addition and under scalar multiplication by elements of $F$, with an additive identity and additive inverses

     A vector space must be closed under vector addition and scalar multiplication, contain a zero vector, and provide additive inverses, along with the usual associativity, commutativity, and distributivity axioms. Vector spaces are typically infinite, and there is no requirement that vectors multiply together or have multiplicative inverses, which distinguishes a vector space from a field.

108. For a linear transformation $T$ from a vector space $V$ to $W$, the image of $T$ is best described as which of the following?
     • The set of all vectors $w$ in $W$ such that $w = T(v)$ for some $v$ in $V$
     • The set of all $v$ in $V$ with $T(v) = 0$
     • The set of eigenvalues of $T$
     • A subspace of $V$

     Correct answer: The set of all vectors $w$ in $W$ such that $w = T(v)$ for some $v$ in $V$

     The image (range) of $T$ is the set of all outputs $w$ in $W$ achievable as $T(v)$ for some $v$ in $V$, and it is a subspace of the codomain $W$. The set of $v$ with $T(v) = 0$ is the kernel, which lives in the domain $V$, so confusing image with kernel is the trap here.

109. Let $T$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ send $(x, y)$ to $(x + y, x - y)$. What is the matrix representation of $T$ with respect to the standard basis?
     • Rows $(1, 1)$ and $(1, -1)$
     • Rows $(1, 1)$ and $(-1, 1)$
     • Rows $(1, -1)$ and $(1, 1)$
     • Rows $(1, 0)$ and $(0, -1)$

     Correct answer: Rows $(1, 1)$ and $(1, -1)$

     The matrix has rows $(1, 1)$ and $(1, -1)$. The columns of a standard-basis matrix are the images of the basis vectors: $T(1, 0) = (1, 1)$ is the first column and $T(0, 1) = (1, -1)$ is the second column, giving the matrix whose rows read $(1, 1)$ and $(1, -1)$. Swapping signs or transposing yields the distractors.

110. Which of the following is an example of an abelian group under the given operation?
     • The nonzero quaternions under multiplication
     • The set of $2 \times 2$ invertible real matrices under multiplication
     • The symmetric group $S_3$ under composition
     • The integers under addition

     Correct answer: The integers under addition

     The integers under addition form an abelian group, since addition is commutative: $a + b = b + a$ for all integers. The general linear group of $2 \times 2$ invertible matrices, the symmetric group $S_3$, and the quaternions are all groups but are non-commutative (non-abelian), so they fail the defining abelian property.

111. By Lagrange's theorem, which statement about a subgroup $H$ of a finite group $G$ must be true?
     • The order of $G$ divides the order of $H$
     • $G$ must be cyclic
     • $H$ must be a normal subgroup of $G$
     • The order of $H$ divides the order of $G$

     Correct answer: The order of $H$ divides the order of $G$

     Lagrange's theorem states that the order of any subgroup $H$ divides the order of the group $G$, because the left cosets of $H$ partition $G$ into equal-sized blocks. This forces $|H|$ to divide $|G|$, not the reverse, and Lagrange's theorem says nothing about normality or whether $G$ is cyclic.

112. A group $G$ of order $14$ acts on a finite set, and one element $x$ has an orbit of size $7$. By the orbit-stabilizer theorem, what is the order of the stabilizer of $x$?
     • 7
     • 1
     • 14
     • 2

     Correct answer: 2

     The stabilizer has order $2$. The orbit-stabilizer theorem says $|\text{orbit}| \times |\text{stabilizer}| = |G|$, so $|\text{stabilizer}| = 14 / 7 = 2$. Equivalently, the size of an orbit equals the index of the stabilizer, so a stabilizer of index $7$ in a group of order $14$ has order $2$.

113. A subgroup $N$ of $G$ is normal if and only if which condition holds for all $g$ in $G$?
     • $gN = Ng$ (the left and right cosets coincide)
     • $N$ has exactly two elements
     • $N$ is the trivial subgroup
     • $g$ is in $N$

     Correct answer: $gN = Ng$ (the left and right cosets coincide)

     $N$ is normal exactly when its left and right cosets agree, $gN = Ng$ for every $g$ in $G$, equivalently $gNg^{-1} = N$. This coset condition is what allows the quotient $G/N$ to inherit a well-defined group operation. The other choices describe special cases or unrelated properties, not the defining condition.

114. If $N$ is a normal subgroup of $G$ with $|G| = 24$ and $|N| = 8$, what is the order of the quotient group $G/N$?
     • 24
     • 3
     • 16
     • 8

     Correct answer: 3

     The quotient group $G/N$ has order 3. The order of a quotient is the index $[G:N] = |G| / |N| = 24 / 8 = 3$, which counts the distinct cosets of $N$ in $G$. Normality of $N$ is exactly what makes the set of cosets into a group of this size.

115. What is the order of the element $4$ in the additive group of integers modulo 6 ($\mathbb{Z}/6\mathbb{Z}$)?
     • 3
     • 2
     • 4
     • 6

     Correct answer: 3

     The order of $4$ in $\mathbb{Z}/6\mathbb{Z}$ is 3. The order is the smallest positive integer $k$ with $k \cdot 4 \equiv 0 \pmod{6}$: $4, 8 (\equiv 2), 12 (\equiv 0)$, so adding 4 three times gives 0. Equivalently, the order equals $6$ divided by $\gcd(4, 6) = 6/2 = 3$.

116. What is the kernel of a group homomorphism $\phi$ from $G$ to $H$?
     • The set of elements fixed by every automorphism of $G$
     • The center of $G$
     • The set of elements $g$ in $G$ with $\phi(g)$ equal to the identity of $H$
     • The image of $\phi$ inside $H$

     Correct answer: The set of elements $g$ in $G$ with $\phi(g)$ equal to the identity of $H$

     The kernel of $\phi$ is the set of all $g$ in $G$ that map to the identity element of $H$, and it is always a normal subgroup of $G$. By the first isomorphism theorem, $G$ modulo this kernel is isomorphic to the image of $\phi$. The image lives in $H$, not $G$, so confusing kernel with image is the trap.

117. How many groups of order 15 exist up to isomorphism?
     • 2
     • 1
     • 4
     • 3

     Correct answer: 1

     There is exactly one group of order 15 up to isomorphism, and it is cyclic. By the Sylow theorems, since $15 = 3 \cdot 5$, the number of Sylow 5-subgroups divides 3 and is congruent to 1 mod 5, forcing it to be 1, and likewise the Sylow 3-subgroup is unique; both normal subgroups give a direct product isomorphic to the cyclic group $\mathbb{Z}/15\mathbb{Z}$.

118. Which property must hold in an integral domain that need not hold in an arbitrary commutative ring with unity?
     • Addition is commutative
     • Every nonzero element has a multiplicative inverse
     • The ring contains a multiplicative identity
     • The product of two nonzero elements is nonzero

     Correct answer: The product of two nonzero elements is nonzero

     An integral domain is a commutative ring with unity that has no zero divisors, so a product of two nonzero elements is never zero. Requiring inverses for all nonzero elements would make it a field, a stronger condition; commutativity of addition and the existence of a unity already hold in any commutative ring with unity.

119. What distinguishes a field from a general commutative ring with unity in abstract algebra?
     • In a field, every nonzero element has a multiplicative inverse
     • A field cannot be finite
     • A field may have zero divisors
     • A field need not have a multiplicative identity

     Correct answer: In a field, every nonzero element has a multiplicative inverse

     A field is a commutative ring with unity in which every nonzero element has a multiplicative inverse, making the nonzero elements a group under multiplication. This forces a field to have no zero divisors, and finite fields such as the integers modulo a prime certainly exist, so the claim that fields cannot be finite is false.

120. Which condition must a nonempty subset $I$ of a commutative ring $R$ satisfy to be an ideal?
     • $I$ contains the multiplicative identity of $R$
     • $I$ is a field
     • $I$ is closed under multiplication only
     • $I$ is closed under addition and absorbs multiplication by every element of $R$

     Correct answer: $I$ is closed under addition and absorbs multiplication by every element of $R$

     An ideal $I$ must be an additive subgroup of $R$ and must absorb products: for every $r$ in $R$ and $a$ in $I$, the product $r \cdot a$ lies in $I$. This absorption property is precisely what lets the quotient ring $R/I$ be well defined. Containing the unity would force $I$ to equal all of $R$, so that is not a defining requirement.

121. Using Fermat's little theorem, what is the remainder when $3^{100}$ is divided by 7?
     • 2
     • 6
     • 1
     • 4

     Correct answer: 4

     The remainder is 4. Fermat's little theorem gives $3^6 \equiv 1 \pmod{7}$ (since 7 is prime and does not divide 3). Writing $100 = 6 \cdot 16 + 4$, we get $3^{100} \equiv 3^4 = 81$, and $81 = 7 \cdot 11 + 4$, so the remainder is 4.

122. Find the smallest positive integer $x$ satisfying $x \equiv 2 \pmod{3}$, $x \equiv 3 \pmod{5}$, and $x \equiv 2 \pmod{7}$.
     • 23
     • 53
     • 68
     • 8

     Correct answer: 23

     The smallest such $x$ is 23, found via the Chinese remainder theorem. The first two congruences combine to $x \equiv 8 \pmod{15}$; writing $x = 8 + 15k$ and imposing $x \equiv 2 \pmod{7}$ gives $1 + k \equiv 2 \pmod{7}$, so $k = 1$ and $x = 23$. Checking: $23 = 3 \cdot 7 + 2$, $23 = 5 \cdot 4 + 3$, $23 = 7 \cdot 3 + 2$.

123. What is the greatest common divisor of 252 and 198 computed via the Euclidean algorithm?
     • 54
     • 18
     • 6
     • 9

     Correct answer: 18

     The greatest common divisor is 18. The Euclidean algorithm gives $252 = 1 \cdot 198 + 54$, then $198 = 3 \cdot 54 + 36$, then $54 = 1 \cdot 36 + 18$, then $36 = 2 \cdot 18 + 0$; the last nonzero remainder, 18, is the gcd. Each step replaces the pair with the divisor and the remainder until the remainder is zero.

124. In modular arithmetic, what is $7 \times 8$ modulo 9?
     • 2
     • 6
     • 1
     • 5

     Correct answer: 2

     The result is 2. Compute $7 \cdot 8 = 56$, then reduce modulo 9: $56 = 9 \cdot 6 + 2$, so $56 \equiv 2 \pmod{9}$. Equivalently, reduce first ($7 \equiv -2$ and $8 \equiv -1$ modulo 9, giving $(-2)(-1) = 2$), which confirms the answer.

125. The binomial coefficient $C(n, k)$ counts the number of ways to choose $k$ items from $n$ without regard to order. What is the value of $C(10, 3)$?
     • 1024
     • 720
     • 30
     • 120

     Correct answer: 120

     The value of $C(10, 3)$ is 120. The binomial coefficient is computed as $n!$ divided by $k!(n-k)!$, which here gives $10! / (3! \cdot 7!) = (10 \cdot 9 \cdot 8) / (3 \cdot 2 \cdot 1) = 720 / 6 = 120$. The value 720 is the ordered count $P(10,3) = 10 \cdot 9 \cdot 8$, which fails to divide out the $3!$ orderings of the chosen items.

126. In the binomial expansion of $(x + y)^6$, what is the coefficient of the term $x^4 y^2$?
     • 15
     • 12
     • 6
     • 30

     Correct answer: 15

     The coefficient of $x^4 y^2$ is 15. By the binomial theorem, the coefficient of the term $x^{n-k} y^k$ in $(x+y)^n$ is the binomial coefficient $C(n, k)$. Here $n = 6$ and $k = 2$, so the coefficient is $C(6, 2) = 6! / (2! \cdot 4!) = 15$. The answer 30 comes from forgetting to divide by $2!$ when computing the combination.

127. Using the symmetry identity of the binomial coefficient, which expression is always equal to $C(n, k)$ for $0 \le k \le n$?
     • $C(n-1, k)$
     • $C(n, n-k)$
     • $C(n, k-1)$
     • $C(k, n)$

     Correct answer: $C(n, n-k)$

     The binomial coefficient satisfies $C(n, k) = C(n, n-k)$. Choosing $k$ items to include is equivalent to choosing the $n-k$ items to exclude, so the two counts must be identical. This symmetry is why Pascal's triangle rows read the same forwards and backwards. The form $C(k, n)$ equals zero for $k < n$ and is not equal in general.

128. A continuous random variable $X$ is uniformly distributed on the interval $[0, 4]$. What is the variance of $X$?
     • 2
     • $\frac{16}{3}$
     • $\frac{2}{3}$
     • $\frac{4}{3}$

     Correct answer: $\frac{4}{3}$

     The variance is $\frac{4}{3}$. For a uniform distribution on $[a, b]$, the variance is $(b - a)^2 / 12$. Here $(4 - 0)^2 / 12 = 16 / 12 = \frac{4}{3}$. The value 2 is the mean $(a+b)/2$, not the variance; confusing the two is a common error with the uniform probability distribution.

129. For a binomial probability distribution with $n = 10$ trials and success probability $p = 0.3$, what is the variance of the number of successes?
     • 2.1
     • 0.21
     • 7.0
     • 3.0

     Correct answer: 2.1

     The variance is 2.1. For a binomial distribution the variance equals $n$ times $p$ times $(1 - p)$, so $10 \cdot 0.3 \cdot 0.7 = 2.1$. The value 3.0 is the mean $np$, not the variance. Knowing that the binomial mean is $np$ while its variance is $npq$ is a standard probability-distribution fact.

130. A Poisson random variable models the number of events in a fixed interval with mean rate $\lambda = 2$. What is the probability of observing exactly zero events?
     • $2 e^{-2}$
     • $e^{-2}$
     • $1 - e^{-2}$
     • $e^{-1}$

     Correct answer: $e^{-2}$

     The probability of zero events is $e^{-2}$, approximately 0.135. The Poisson probability distribution gives $P(X = k) = (\lambda^k e^{-\lambda}) / k!$, so for $k = 0$ this is $(2^0 e^{-2}) / 0! = e^{-2}$. The expression $2 e^{-2}$ is $P(X = 1)$, not $P(X = 0)$.

131. How many nonnegative integer solutions are there to the equation $a + b + c + d = 10$?
     • 715
     • 286
     • 10000
     • 210

     Correct answer: 286

     There are 286 solutions. This is a stars-and-bars problem: distributing 10 identical units among 4 variables corresponds to $C(10 + 4 - 1, 4 - 1) = C(13, 3) = 286$. The stars-and-bars formula $C(n + r - 1, r - 1)$ counts the ways to place $r-1$ dividers among $n$ items. Treating the variables as distinct ordered objects gives the wrong count.

132. In how many distinguishable ways can the letters of the word MISSISSIPPI be arranged?
     • 1663200
     • 39916800
     • 34650
     • 11550

     Correct answer: 34650

     There are 34650 distinguishable arrangements. The word has 11 letters with repeats: 4 I's, 4 S's, 2 P's, and 1 M. The number of distinct permutations is $11!$ divided by the product $4! \cdot 4! \cdot 2! \cdot 1!$, which equals $39916800 / 1152 = 34650$. The value 39916800 is $11!$, which wrongly treats every letter as unique.

133. In how many distinct ways can 6 people be seated around a circular table, where seatings that differ only by rotation are considered the same?
     • 360
     • 720
     • 120
     • 60

     Correct answer: 120

     There are 120 ways. For circular permutations of $n$ distinct objects, fixing one person to remove rotational symmetry leaves $(n - 1)!$ arrangements of the rest, giving $5! = 120$. The value 720 is $6!$, the count for a straight line, which overcounts each circular arrangement 6 times by treating all rotations as different.

134. Three events $A$, $B$, and $C$ in a sample space are pairwise mutually exclusive. By the inclusion-exclusion principle, the probability of their union $P(A \cup B \cup C)$ equals which expression?
     • $P(A) P(B) P(C)$
     • $P(A) + P(B) + P(C)$
     • $P(A) + P(B) + P(C) - P(A)P(B)P(C)$
     • $1 - P(A) - P(B) - P(C)$

     Correct answer: $P(A) + P(B) + P(C)$

     For pairwise mutually exclusive events the union probability is simply $P(A) + P(B) + P(C)$. Inclusion-exclusion subtracts pairwise intersection probabilities, but mutual exclusivity makes every intersection empty with probability zero, so all correction terms vanish. The product form would apply to independence of a joint event, not to a union of disjoint events.

135. Two events $A$ and $B$ satisfy $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cap B) = 0.3$. Are $A$ and $B$ independent, and what is the basis for the conclusion?
     • Dependent, because $P(A \cap B)$ is not zero
     • Independent, because $P(A) + P(B) > 1$
     • Independent, because $P(A \cap B) = P(A)P(B)$
     • Dependent, because $P(A) > P(B)$

     Correct answer: Independent, because $P(A \cap B) = P(A)P(B)$

     The events are independent because $P(A \cap B) = P(A)P(B)$. Independence is defined by the multiplication rule: here $P(A)P(B) = 0.6 \cdot 0.5 = 0.3$, which matches the given joint probability exactly. Independence is unrelated to mutual exclusivity; disjoint events with positive probabilities are in fact dependent, so a nonzero intersection does not by itself imply dependence.

136. A factory has two machines producing identical parts: machine A makes 70% of parts with a 2% defect rate, and machine B makes 30% with a 5% defect rate. A randomly chosen part is defective. By Bayes' theorem, what is the probability it came from machine B?
     • About 0.30
     • About 0.52
     • About 0.62
     • About 0.39

     Correct answer: About 0.52

     The probability is about 0.52. By Bayes' theorem, $P(B \mid \text{defective}) = P(\text{defective} \mid B)P(B) / P(\text{defective})$. The total defect probability is $0.7(0.02) + 0.3(0.05) = 0.014 + 0.015 = 0.029$. Then $P(B \mid \text{defective}) = 0.015 / 0.029 = 0.517$. The prior 0.30 ignores that machine B's higher defect rate raises the posterior once a defect is observed.

137. A standard fair die is rolled once. What is the variance of the outcome?
     • 2.92
     • 3.50
     • 6.00
     • 1.71

     Correct answer: 2.92

     The variance is approximately 2.92, exactly $\frac{35}{12}$. The variance is $E[X^2]$ minus $(E[X])^2$. The mean is 3.5, and $E[X^2] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6 = 15.1667$, so the variance is $15.1667 - 12.25 = 2.9167$. The value 3.5 is the mean, not the variance.

138. A game pays 10 dollars if a fair coin lands heads and costs you 4 dollars if it lands tails. What is the expected net value of playing once?
     • 6 dollars
     • 3 dollars
     • 5 dollars
     • 7 dollars

     Correct answer: 3 dollars

     The expected value is 3 dollars. Expected value weights each outcome by its probability: $(1/2)(+10) + (1/2)(-4) = 5 - 2 = 3$. The answer 7 incorrectly averages the magnitudes 10 and 4 instead of accounting for the sign of the loss. A positive expected value means the game favors the player.

139. In a metric space, the closure of a set $A$ is defined as the smallest closed set containing $A$. What is the closure of the open interval $(0, 1)$ in the real line with the standard metric?
     • $[0, 1)$
     • $[0, 1]$
     • $(0, 1)$
     • $(0, 1]$

     Correct answer: $[0, 1]$

     The closure of $(0, 1)$ is the closed interval $[0, 1]$. The closure adds all limit points of the set; the endpoints 0 and 1 are limit points of $(0,1)$ even though they are not members, so they are included. The result $[0,1]$ is the smallest closed set containing the original interval. Leaving out either endpoint would not be closed.

140. In the real line with the standard topology, which of the following sets is closed but not open?
     • The whole real line $\mathbb{R}$
     • The empty set
     • The open interval $(1, 4)$
     • The closed interval $[1, 4]$

     Correct answer: The closed interval $[1, 4]$

     The closed interval $[1, 4]$ is closed but not open. It contains all its limit points (so it is closed), yet its endpoints have no surrounding open interval lying entirely inside the set (so it is not open). The empty set and all of $\mathbb{R}$ are both open and closed (clopen), and $(1,4)$ is open but not closed.

141. Which subset of the real line $\mathbb{R}$ with the usual topology is compact, by the Heine-Borel characterization?
     • The interval $[5, \infty)$
     • The set $\{2, 4, 6\}$ of three points
     • The interval $(3, 7]$
     • The set of rationals in $[0, 2]$

     Correct answer: The set $\{2, 4, 6\}$ of three points

     The finite set $\{2, 4, 6\}$ is compact. By the Heine-Borel theorem a subset of $\mathbb{R}$ is compact exactly when it is closed and bounded; a finite set of points is automatically both. The half-open interval $(3, 7]$ is not closed, the ray $[5, \infty)$ is not bounded, and the rationals in $[0, 2]$ are not closed because their limits can be irrational.

142. A key property of compactness in topology is that the continuous image of a compact set is compact. If $f$ is a continuous real-valued function and $K$ is a compact subset of its domain, what can be concluded about $f(K)$?
     • $f(K)$ is a single point
     • $f(K)$ is closed and bounded
     • $f(K)$ is connected but possibly unbounded
     • $f(K)$ is open

     Correct answer: $f(K)$ is closed and bounded

     The image $f(K)$ is closed and bounded. Continuity preserves compactness, so $f(K)$ is compact, and in the real line compactness is equivalent to being closed and bounded by Heine-Borel. This is precisely why a continuous function on a compact interval attains its maximum and minimum. Connectedness is preserved separately and does not guarantee boundedness on its own.

143. In topology, a space is connected if it cannot be written as the union of two disjoint nonempty open sets. Which property is preserved under a continuous function?
     • A continuous image of a connected set is always finite
     • A continuous image of a connected set is connected
     • A continuous image of a connected set is always open
     • A continuous image of a connected set is always closed

     Correct answer: A continuous image of a connected set is connected

     The continuous image of a connected set is connected. If the image could be split into two disjoint nonempty open pieces, their preimages would split the connected domain the same way, a contradiction. This is exactly the topological generalization of the Intermediate Value Theorem, where the connected interval maps to a connected (interval) image.

144. Which subset of the plane $\mathbb{R}^2$ with the standard topology is connected?
     • The set of points with integer coordinates
     • The complement of a closed disk's boundary circle
     • A single straight line through the origin
     • Two disjoint closed disks

     Correct answer: A single straight line through the origin

     A single straight line through the origin is connected; it is a continuous image of the connected real line and cannot be separated into two disjoint nonempty open pieces. Two disjoint disks form an obviously disconnected set, the integer lattice is a discrete set of isolated points, and removing a circle splits the plane into a disconnected inside-and-outside.

145. On the set of real numbers, define $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x$ is not equal to $y$. This is the discrete metric. Under this metric, which statement is true?
     • No single point is an open set
     • Every subset is open
     • The metric violates the triangle inequality
     • Distances can exceed 1

     Correct answer: Every subset is open

     Under the discrete metric every subset is open. Each point $x$ is contained in the open ball of radius $\frac{1}{2}$, which contains only $x$ itself, so every singleton is open; since arbitrary unions of open sets are open, every subset is open. The discrete metric does satisfy all metric axioms including the triangle inequality, and its distances never exceed 1.

146. The complex cube roots of 8 lie equally spaced on a circle in the complex plane. What is the radius of that circle (the modulus of each root)?
     • $\sqrt{8}$
     • 4
     • 8
     • 2

     Correct answer: 2

     The modulus of each $\sqrt[3]{8}$ is 2. The $n$-th roots of a complex number with modulus $r$ all have modulus $r^{1/n}$; here $r = 8$ and $n = 3$, so the modulus is $8^{1/3} = 2$. The three roots are 2, and 2 times the two primitive cube roots of unity, all sitting on a circle of radius 2 and spaced 120 degrees apart. The value 8 ignores taking the modulus's cube root $\sqrt[3]{8}$.

147. Newton's method is applied to find a root of $f(x) = x^2 - 2$ starting from $x_0 = 1$. Using the update $x_{n+1} = x_n - f(x_n)/f'(x_n)$, what is the first iterate $x_1$?
     • 1.4142
     • 1.5
     • 2.0
     • 1.25

     Correct answer: 1.5

     The first iterate is 1.5. Newton's method uses $x_1 = x_0 - f(x_0)/f'(x_0)$, where $f'(x) = 2x$. At $x_0 = 1$, $f(1) = -1$ and $f'(1) = 2$, so $x_1 = 1 - (-1)/2 = 1 + 0.5 = 1.5$. The value 1.4142 is the true root $\sqrt{2}$, which the iteration only approaches after further steps rather than reaching at $x_1$.

148. Find the volume of the solid generated by revolving the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$ about the x-axis.
     • $8\pi$
     • $4\pi$
     • $16\pi$
     • $\dfrac{16}{3}\pi$

     Correct answer: $8\pi$

     Correct answer: $8\pi$. Using the disk method, the volume is $V = \pi \int_{0}^{4} (\sqrt{x})^2\, dx = \pi \int_{0}^{4} x\, dx = \pi \left[ \tfrac{x^2}{2} \right]_0^4 = \pi \cdot 8 = 8\pi$.

149. Using the quotient rule, what is the derivative of $f(x) = \dfrac{x^2 + 1}{x - 1}$?
     • $\dfrac{2x}{1}$
     • $\dfrac{x^2 - 2x - 1}{(x-1)^2}$
     • $\dfrac{x^2 + 2x - 1}{(x-1)^2}$
     • $\dfrac{2x(x-1) + (x^2+1)}{(x-1)^2}$

     Correct answer: $\dfrac{x^2 - 2x - 1}{(x-1)^2}$

     Correct answer: $\dfrac{x^2 - 2x - 1}{(x-1)^2}$. The quotient rule gives $f'(x) = \dfrac{(2x)(x-1) - (x^2+1)(1)}{(x-1)^2} = \dfrac{2x^2 - 2x - x^2 - 1}{(x-1)^2} = \dfrac{x^2 - 2x - 1}{(x-1)^2}$.

150. At what x-value does the curve $y = x^3 - 6x^2 + 5$ have an inflection point?
     • $x = 4$
     • $x = 0$
     • $x = 2$
     • $x = 3$

     Correct answer: $x = 2$

     Correct answer: $x = 2$. An inflection point occurs where the second derivative changes sign. Here $y' = 3x^2 - 12x$ and $y'' = 6x - 12$; setting $y'' = 0$ gives $6x - 12 = 0$, so $x = 2$, and the concavity changes there.

151. For which values of $p$ does the p-series $\sum_{n=1}^{\infty} \dfrac{1}{n^p}$ converge?
     • $p \geq 1$
     • $p > 1$
     • $p < 1$
     • All real $p$

     Correct answer: $p > 1$

     Correct answer: $p > 1$. The p-series $\sum 1/n^p$ converges precisely when $p > 1$ and diverges when $p \leq 1$; for example $p = 1$ gives the divergent harmonic series.