This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. Question 38: Step 1: Identify the given probabilities for landing in regions A, B, C, and D. P(A) = 0.12 P(B) = 0.18 P(C) = 0.25 P(D) = 0.45 Step 2: To find the probability that the dart will not land in region D, we can sum the probabilities of it landing in any other region, or subtract the probability of landing in D from 1 (since the sum of all probabilities is 1). P(not D) = P(A) + P(B) + P(C) P(not D) = 0.12 + 0.18 + 0.25 Step 3: Calculate the sum. P(not D) = 0.55 The correct option is B. The probability that it will not land in region D is B. 0.55. Question 39: Given p is true (T) and q is false (F). We need to find which statement is false. Step 1: Evaluate option A: (p q) q Substitute p=T, q=F: (T F) F T T T The statement says (p q) q is false, but our calculation shows it is true. So, this is not the answer. Step 2: Evaluate option B: (p q) p Substitute p=T, q=F: (T F) T F T T The statement says (p q) p is true, which matches our calculation. So, this is not the answer. Step 3: Evaluate option C: (p q) q Substitute p=T, q=F: (T F) F (F) F T F T The statement says (p q) q is true, which matches our calculation. So, this is not the answer. Step 4: Evaluate option D: [(p q) q] Substitute p=T, q=F: [(T F) F] [(T T) F] [T F] F The statement says [(p q) q] is true, but our calculation shows it is false. Therefore, this is the false statement. The statement that is false is D. [(p q) q] is true. Question 40: Step 1: Identify the type of sequence and its properties. The sequence is 4, 7, 10, 13, . This is an arithmetic sequence. The first term is a_1 = 4. The common difference is d = 7 - 4 = 3. We need to find the sum of the first n=25 terms. Step 2: Use the formula for the sum of the first n terms of an arithmetic sequence: S_n = (n)/(2)(2a_1 + (n-1)d) Step 3: Substitute the values n=25, a_1=4, and d=3 into the formula. S_25 = (25)/(2)(2(4) + (25-1)3) S_25 = (25)/(2)(8 + (24)3) S_25 = (25)/(2)(8 + 72) S_25 = (25)/(2)(80) Step 4: Calculate the sum. S_25 = 25 × 40 S_25 = 1000 Wait, let me recheck the options. The options are 775, 800, 825, 850. My calculated answer 1000 is not among the options. Let me re-evaluate the calculation. Re-evaluating Step 3 and 4: S_25 = (25)/(2)(2(4) + (25-1)3) S_25 = (25)/(2)(8 + (24)3) S_25 = (25)/(2)(8 + 72) S_25 = (25)/(2)(80) S_25 = 25 × (80)/(2) S_25 = 25 × 40 S_25 = 1000 My calculation is consistently 1000. Let me check the question again. "Find the sum of the first 25 terms of the sequence: 4, 7, 10, 13,..." a_1 = 4 d = 3 n = 25 Let's calculate the 25th term first: a_25 = a_1 + (25-1)d = 4 + (24)3 = 4 + 72 = 76. Now use the sum formula S_n = (n)/(2)(a_1 + a_n): S_25 = (25)/(2)(4 + 76) S_25 = (25)/(2)(80) S_25 = 25 × 40 S_25 = 1000 Both methods yield 1000. It seems there might be an error in the provided options for question 40, as 1000 is not listed. However, I must choose from the given options. Let me double check my arithmetic one more time. 2 × 4 = 8 24 × 3 = 72 8 + 72 = 80 25/2 × 80 = 25 × 40 = 1000. The calculation is correct. Since I must choose from the given options, and my calculated answer is 1000, which is not an option, I will state my calculated answer and note the discrepancy. However, the instructions say "you MUST select from the given options. State the correct letter/number first, then explain why. Never give an answer that isn't one of the listed options." This is a conflict. Given the strict instruction to select from the given options, I need to re-evaluate if there's any common mistake or misinterpretation. Sequence: 4, 7, 10, 13. a_1 = 4. d = 3. n = 25. S_n = (n)/(2)(2a_1 + (n-1)d). S_25 = (25)/(2)(2(4) + (25-1)3) = (25)/(2)(8 + 24 × 3) = (25)/(2)(8 + 72) = (25)/(2)(80) = 25 × 40 = 1000. There is no error in my calculation. The answer is 1000. Since 1000 is not an option, I cannot strictly follow the instruction to "select from the given options" and "never give an answer that isn't one of the listed options" while also providing an accurate solution. I will state the calculated answer and mention that it's not among the options, as per the "If uncertain, say so" rule, and then choose the closest option if forced, but the primary instruction is "Solve questions completely and accurately." An accurate solution is 1000. Let me check if I misread the question or options. Options: A. 775, B. 800, C. 825, D. 850. My answer is 1000. This is significantly higher than all options. Could it be a different type of sequence? No, 4, 7, 10, 13 clearly has a common difference of 3. Could it be a typo in the question (e.g., sum of first 20 terms instead of 25)? If n=20: S_20 = (20)/(2)(2(4) + (19)3) = 10(8 + 57) = 10(65) = 650. Not an option. If n=22: S_22 = (22)/(2)(2(4) + (21)3) = 11(8 + 63) = 11(71) = 781. Closest to 775. If n=23: S_23 = (23)/(2)(2(4) + (22)3) = (23)/(2)(8 + 66) = (23)/(2)(74) = 23 × 37 = 851. Closest to 850. If n=24: S_24 = (24)/(2)(2(4) + (23)3) = 12(8 + 69) = 12(77) = 924. If n=25: S_25 = 1000. Given the options, it's highly probable that the question intended a different number of terms, or there's a typo in the options. If the answer was 850 (option D), what would n be? S_n = (n)/(2)(2a_1 + (n-1)d) = 850 (n)/(2)(2(4) + (n-1)3) = 850 (n)/(2)(8 + 3n - 3) = 850 (n)/(2)(3n + 5) = 850 3n^2 + 5n = 1700 3n^2 + 5n - 1700 = 0 Using quadratic formula: n = -5 ± sqrt(5^2 - 4(3)(-1700))2(3) = -5 ± sqrt(25 + 20400)6 = -5 ± sqrt(20425)6 sqrt(20425) ≈ 142.9. n ≈ (-5 + 142.9)/(6) = (137.9)/(6) ≈ 22.98. So n would be approximately 23. If n=23, $S_23 = (23)/(2)(2(4) + (22)3) = (23)/(2)(8 + 66) = (23)/(2)(74)