problem stringlengths 14 176 | thinking stringlengths 89 387 | answer stringlengths 1 36 | difficulty stringclasses 4
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What is 15% of 240? | To find 15% of 240, I can convert 15% to a decimal, which is 0.15. Then, I multiply 240 by 0.15. 240 * 0.15 = 36. Alternatively, 10% is 24, and 5% is 12. 24 + 12 = 36. | 36 | easy |
A car travels 120 miles in 2 hours. What is its average speed? | Average speed is total distance divided by total time. Distance = 120 miles. Time = 2 hours. Speed = 120 / 2 = 60 miles per hour. | 60 mph | easy |
Solve for x: 3x + 5 = 20 | I need to isolate x. First, subtract 5 from both sides: 3x = 15. Then, divide both sides by 3: x = 5. | 5 | easy |
What is the perimeter of a rectangle with length 5 and width 3? | The perimeter of a rectangle is calculated as 2 * (length + width). Length = 5, width = 3. Perimeter = 2 * (5 + 3) = 2 * 8 = 16. | 16 | easy |
Calculate 2^5. | 2^5 means multiplying 2 by itself 5 times. 2 * 2 = 4. 4 * 2 = 8. 8 * 2 = 16. 16 * 2 = 32. | 32 | easy |
What is the sum of the interior angles of a pentagon? | The formula for the sum of interior angles of an n-sided polygon is (n - 2) * 180 degrees. For a pentagon, n = 5. Sum = (5 - 2) * 180 = 3 * 180 = 540 degrees. | 540 degrees | easy |
Solve the quadratic equation: x^2 - 5x + 6 = 0 | I need to factor the quadratic. I am looking for two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3. So, the factored form is (x - 2)(x - 3) = 0. Setting each factor to zero gives x = 2 and x = 3. | x = 2, x = 3 | medium |
What is the probability of rolling a sum of 7 with two standard 6-sided dice? | Total possible outcomes when rolling two dice is 6 * 6 = 36. The combinations that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 favorable outcomes. Probability = 6 / 36 = 1/6. | 1/6 | medium |
An object starts from rest and accelerates at 2 m/s^2 for 5 seconds. What is the distance traveled? | I will use the kinematic equation: d = v0*t + 0.5*a*t^2. Initial velocity v0 = 0. Acceleration a = 2. Time t = 5. d = 0 + 0.5 * 2 * (5^2) = 1 * 25 = 25 meters. | 25 meters | medium |
What is the output of the Python code: `len([x for x in range(10) if x % 2 == 0])`? | The code creates a list comprehension. `range(10)` generates numbers 0 through 9. The condition `x % 2 == 0` filters for even numbers. The even numbers in this range are 0, 2, 4, 6, 8. There are 5 such numbers. The `len()` function returns the count, which is 5. | 5 | medium |
Calculate the amount after 2 years if $1000 is invested at 5% annual compound interest. | The formula for compound interest is A = P(1 + r)^t. Principal P = 1000. Rate r = 0.05. Time t = 2. A = 1000 * (1 + 0.05)^2 = 1000 * (1.05)^2 = 1000 * 1.1025 = 1102.5. | $1102.50 | medium |
Find the determinant of the 2x2 matrix: [[1, 2], [3, 4]] | For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here, a=1, b=2, c=3, d=4. Det = (1 * 4) - (2 * 3) = 4 - 6 = -2. | -2 | medium |
What is the area of a circle with a diameter of 10? | The area of a circle is pi * r^2. The diameter is 10, so the radius r is 5. Area = pi * 5^2 = 25pi. | 25\pi | medium |
Evaluate: log2(32) + log2(8) | Using logarithm properties, log2(32) is the power to which 2 must be raised to get 32, which is 5. log2(8) is the power to which 2 must be raised to get 8, which is 3. Adding them together: 5 + 3 = 8. | 8 | medium |
Find the next number in the sequence: 2, 6, 18, 54, ... | I need to find the pattern. 6 / 2 = 3. 18 / 6 = 3. 54 / 18 = 3. This is a geometric sequence with a common ratio of 3. The next number is 54 * 3 = 162. | 162 | medium |
Evaluate the indefinite integral: \int x e^x dx | I will use integration by parts: \int u dv = uv - \int v du. Let u = x, so du = dx. Let dv = e^x dx, so v = e^x. Applying the formula: x e^x - \int e^x dx = x e^x - e^x + C. | x e^x - e^x + C | hard |
An ideal gas expands isobarically at 100 kPa, increasing its volume by 2 m^3. What is the work done by the gas? | Work done by a gas during isobaric (constant pressure) expansion is W = P * \Delta V. Pressure P = 100 kPa = 100,000 Pa. Change in volume \Delta V = 2 m^3. W = 100,000 * 2 = 200,000 Joules, or 200 kJ. | 200 kJ | hard |
What is the time complexity of Dijkstra's algorithm when implemented with a binary min-heap? | Dijkstra's algorithm extracts the minimum vertex and relaxes its edges. With a binary min-heap, extracting the minimum takes O(log V) and happens V times, giving O(V log V). Decreasing the key (relaxing edges) takes O(log E) and happens E times, giving O(E log E). Since E <= V^2, log E is O(log V). Total time complexit... | O((V + E) \log V) | hard |
What is the pH of a 0.01 M aqueous solution of HCl? | HCl is a strong acid, meaning it completely dissociates in water. Therefore, the concentration of H+ ions [H+] is equal to the concentration of HCl, which is 0.01 M. pH is calculated as -log10([H+]). pH = -log10(0.01) = -log10(10^-2) = 2. | 2 | hard |
Find the sum of the infinite geometric series where the first term a = 1 and the common ratio r = 1/3. | The sum of an infinite geometric series is S = a / (1 - r), provided |r| < 1. Here, a = 1 and r = 1/3. Since |1/3| < 1, the series converges. S = 1 / (1 - 1/3) = 1 / (2/3) = 3/2. | 3/2 | hard |
Calculate the de Broglie wavelength of an electron with a momentum of 2 x 10^-24 kg m/s. (Planck's constant h = 6.626 x 10^-34 J s) | The de Broglie wavelength is given by \lambda = h / p. Planck's constant h = 6.626e-34. Momentum p = 2e-24. \lambda = 6.626e-34 / 2e-24 = 3.313e-10 meters. | 3.313 x 10^-10 meters | hard |
How many edges are in a complete graph with 5 vertices (K_5)? | A complete graph K_n has an edge between every pair of distinct vertices. The number of edges is given by the combination formula nC2, or n(n-1)/2. For n = 5, Edges = 5 * 4 / 2 = 10. | 10 | hard |
Find the derivative of f(x) = \arctan(x^2). | I need to use the chain rule. The derivative of \arctan(u) is 1 / (1 + u^2) * du/dx. Here, u = x^2, so du/dx = 2x. Substituting u into the derivative formula gives: f'(x) = 1 / (1 + (x^2)^2) * 2x = 2x / (1 + x^4). | \frac{2x}{1 + x^4} | hard |
Calculate the capacitance of a parallel-plate capacitor with plate area 0.01 m^2 and separation 0.001 m in a vacuum. (Permittivity of free space \epsilon_0 = 8.85 x 10^-12 F/m) | The capacitance of a parallel-plate capacitor is C = \epsilon_0 * A / d. Area A = 0.01. Distance d = 0.001. \epsilon_0 = 8.85e-12. C = 8.85e-12 * (0.01 / 0.001) = 8.85e-12 * 10 = 8.85e-11 Farads. | 8.85 x 10^-11 F | hard |
Find the remainder when 2^100 is divided by 7. | I need to find 2^100 mod 7. Let's look at the pattern of powers of 2 mod 7. 2^1 = 2. 2^2 = 4. 2^3 = 8 = 1 mod 7. The cycle length is 3. I divide the exponent 100 by the cycle length 3. 100 = 3 * 33 + 1. The remainder of the exponent division is 1. Therefore, 2^100 mod 7 = 2^1 mod 7 = 2. | 2 | very_hard |
What is the time complexity of Strassen's algorithm for matrix multiplication? | Standard matrix multiplication is O(n^3). Strassen's algorithm uses a divide-and-conquer approach, dividing the matrices into 7 subproblems of half the size instead of 8. The recurrence relation is T(n) = 7T(n/2) + O(n^2). By the Master Theorem, since log2(7) is approx 2.807, which is greater than 2, the time complexit... | O(n^{\log_2 7}) \approx O(n^{2.807}) | very_hard |
Calculate the relativistic momentum of a proton moving at 0.8c. (Rest mass of proton = m) | Relativistic momentum is p = \gamma m v, where \gamma = 1 / \sqrt{1 - v^2/c^2}. Here, v = 0.8c. v^2/c^2 = 0.64. 1 - 0.64 = 0.36. \sqrt{0.36} = 0.6. So, \gamma = 1 / 0.6 = 5/3. Now, p = (5/3) * m * (0.8c) = (5/3) * (4/5) * mc = (4/3) mc. | \frac{4}{3} mc | very_hard |
Evaluate the closed-form expression for the sum: \sum_{k=1}^n k^3 | The sum of the first n cubes is a known identity. It is equal to the square of the sum of the first n integers. The sum of the first n integers is n(n+1)/2. Therefore, the sum of the cubes is [n(n+1)/2]^2, which expands to n^2(n+1)^2 / 4. | \frac{n^2(n+1)^2}{4} | very_hard |
Calculate the degree of unsaturation (Index of Hydrogen Deficiency) for a molecule with the formula C8H10O. | The formula for the degree of unsaturation (DoU) for a molecule C_c H_h N_n O_o X_x is DoU = c - (h/2) + (n/2) + 1. Oxygen and sulfur do not affect the DoU. Here, c = 8, h = 10. DoU = 8 - (10/2) + 1 = 8 - 5 + 1 = 4. | 4 | very_hard |
How many onto (surjective) functions exist from a set A with 4 elements to a set B with 2 elements? | The total number of functions from A to B is |B|^|A| = 2^4 = 16. An onto function means every element in B is mapped to by at least one element in A. The only way a function is NOT onto is if it maps all elements of A to a single element in B. Since B has 2 elements, there are exactly 2 such constant functions. Therefo... | 14 | very_hard |
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