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 from the image. Problem 1: A die was rolled a number of times. The outcomes are: | Outcomes | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | Number | 30 | 26 | 20 | K | 23 | 24 | i) If the probability of obtaining a 1 is (1)/(5), find the value of K. Step 1: Calculate the total number of rolls in terms of K. Total rolls = 30 + 26 + 20 + K + 23 + 24 = 123 + K. Step 2: Use the given probability of obtaining a 1. The number of times 1 was obtained is 30. P(obtaining 1) = Number of times 1 appearedTotal number of rolls (30)/(123 + K) = (1)/(5) Step 3: Solve for K. 30 × 5 = 1 × (123 + K) 150 = 123 + K K = 150 - 123 K = 27 The value of K is 27. ii) The probability of obtaining an even number or a factor of 6. Using K=27, the total number of rolls is 123 + 27 = 150. Step 1: Identify the frequencies of even numbers. Even numbers are 2, 4, 6. Frequency of 2 = 26 Frequency of 4 = K = 27 Frequency of 6 = 24 Number of even outcomes = 26 + 27 + 24 = 77. Step 2: Identify the frequencies of factors of 6. Factors of 6 are 1, 2, 3, 6. Frequency of 1 = 30 Frequency of 2 = 26 Frequency of 3 = 20 Frequency of 6 = 24 Number of factor of 6 outcomes = 30 + 26 + 20 + 24 = 100. Step 3: Identify the frequencies of outcomes that are both even and a factor of 6. These are 2 and 6. Number of (even AND factor of 6) outcomes = 26 + 24 = 50. Step 4: Calculate the number of outcomes that are even OR a factor of 6. N(even or factor of 6) = N(even) + N(factor of 6) - N(even and factor of 6) N(even or factor of 6) = 77 + 100 - 50 = 127 Step 5: Calculate the probability. P(even or factor of 6) = (127)/(150) The probability is (127)/(150). Problem 2 (b): Solve: (5+x)/(6) + (4x-1)/(3) (3x+2)/(5) Step 1: Find the least common multiple (LCM) of the denominators (6, 3, 5), which is 30. Multiply the entire inequality by 30. 30 ((5+x)/(6)) + 30 ((4x-1)/(3)) 30 ((3x+2)/(5)) Step 2: Simplify each term. 5(5+x) + 10(4x-1) 6(3x+2) Step 3: Distribute and expand the terms. 25 + 5x + 40x - 10 18x + 12 Step 4: Combine like terms on each side. 45x + 15 18x + 12 Step 5: Move terms with x to one side and constants to the other. 45x - 18x 12 - 15 27x -3 Step 6: Divide by 27. x (-3)/(27) x -(1)/(9) The solution is x -(1)/(9). Problem 3: In the diagram, TU is a tangent to the circle. RVU = 100^, and ZUV = 36^. Find the value of ZSTU. (Assuming the points Z, S, V, U are on the circle, and R is also on the circle, forming a cyclic pentagon ZSVRU. The angle ZSTU is interpreted as ZSU.) Step 1: Identify angles subtended by the same arc. Given ZUV = 36^. This angle is subtended by arc ZV at point U on the circumference. Therefore, the angle subtended by the same arc ZV at point S on the circumference is also 36^. ZSV = 36^ Step 2: Apply the Alternate Segment Theorem. The angle between the tangent TU and the chord UV is TUV. This angle is equal to the angle subtended by the chord UV in the alternate segment, which is USV. So, TUV = USV. Step 3: Use the given angle RVU = 100^. Since Z, S, V, R, U are points on the circle, RVU is an angle subtended by arc RU at V. Therefore, RSU = RVU = 100^ (angles subtended by the same arc RU). Step 4: Find ZSU. We know RSU = 100^. We can write RSU = RSZ + ZSU. However, we need ZSU. We have ZSV = 36^. We also know RSU = RSV + VSU. From the diagram, it appears that Z, S, V, R are consecutive points on the circle. If RSU = 100^ and ZSV = 36^. We need ZSU. From the diagram, ZSU = RSU - RZ_1U (This is not correct). Let's use the tangent-chord theorem for TUZ. TUZ = USZ. We have TUV = USV. We know ZUV = 36^. We know RVU = 100^. From the diagram, TUV = TUZ + ZUV. This is incorrect. TUV is the angle between tangent TU and chord UV. TUZ is the angle between tangent TU and chord UZ. ZUV is an angle inside the circle. Let's re-evaluate the interpretation of ZSTU. It is likely ZSU. We have ZSV = 36^. We have RSU = 100^. From the diagram, it looks like Z, S, V, R are in order around the circle. Then ZSU = ZSV + VSU. We need VSU. By the alternate segment theorem, VSU = TUV. We know ZUV = 36^. We know RVU = 100^. The angle TUV is the angle between the tangent TU and the chord UV. The angle TUZ is the angle between the tangent TU and the chord UZ. The angle TUV is not directly given. Let's assume the angle 100^ is ZRV. If Z, R, V, U are points on the circle, then ZRV + ZUV = 180^. So 100^ + 36^ = 136^ 180^. So ZRVU is not a cyclic quadrilateral. Let's assume the angle 100^ is VUR. This is the angle between the tangent TU and the chord UV. If VUR = 100^, then by the alternate segment theorem, USV = 100^. We are given ZUV = 36^. Since Z, S, V, U are on the circle, ZSV = ZUV = 36^. We need to find ZSU. ZSU = ZSV + VSU. ZSU = 36^ + 100^ = 136^ This interpretation seems plausible. Let's assume ZRVU = 100^ means VUR = 100^. Step 1: Identify the angle in the alternate segment. Given that TU is a tangent at U, and UV is a chord. The angle between the tangent and the chord, TUV, is equal to the angle subtended by the chord in the alternate segment, USV. From the problem statement, ZRVU = 100^. Given the diagram, this is most likely referring to TUV = 100^ (the angle between the tangent and the chord UV). So, USV = 100^. Step 2: Identify angles subtended by the same arc. Given ZUV = 36^. This angle is subtended by arc ZV at point U on the circumference. The angle subtended by the same arc ZV at point S on the circumference is ZSV. Therefore, ZSV = ZUV = 36^. Step 3: Calculate ZSU. From the diagram, ZSU = ZSV + VSU. ZSU = 36^ + 100^ ZSU = 136^ The value of ZSTU (interpreted as ZSU) is 136^. Problem 4: Given that (7-2x), 9, (5x+17) are consecutive terms of a Geometric Progression with common ratio r>0. Find the values of x. Step 1: Use the property of a Geometric Progression (GP). In a GP, the ratio of consecutive terms is constant. So, (9)/(7-2x) = (5x+17)/(9). Step 2: Cross-multiply to form an equation. 9 × 9 = (7-2x)(5x+17) 81 = 35x + 119 - 10x^2 - 34x Step 3: Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 81 = -10x^2 + x + 119 10x^2 - x + 81 - 119 = 0 10x^2 - x - 38 = 0 Step 4: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=10, b=-1, c=-38. x = -(-1) ± sqrt((-1)^2 - 4(10)(-38))2(10) x = 1 ± sqrt(1 - (-1520))20 x = 1 ± sqrt(1 + 1520)20 x = 1 ± sqrt(1521)20 x = (1 ± 39)/(20) Step 5: Find the two possible values for x. x_1 = (1 + 39)/(20) = (40)/(20) = 2 x_2 = (1 - 39)/(20) = (-38)/(20) = -(19)/(10) = -1.9 Step 6: Check the common ratio r for each value of x, ensuring r>0. For x=2: The terms are (7-2(2)), 9, (5(2)+17). Terms are (7-4), 9, (10+17). Terms are 3, 9, 27. The common ratio r = (9)/(3) = 3. Since r=3 > 0, x=2 is a valid solution. For x=-1.9: The terms are (7-2(-1.9)), 9, (5(-1.9)+17). Terms are (7+3.8), 9, (-9.5+17). Terms are 10.8, 9, 7.5. The common ratio r = (9)/(10.8) = (90)/(108) = (5)/(6). Since r=(5)/(6) > 0, x=-1.9 is also a valid solution. The problem asks for "the values of x", implying both are valid. The values of x are 2 and -1.9. Problem 5: Two positive numbers are in the ratio 3:4. The sum of thrice the first number and twice the second is 68. Find the smaller number. Step 1: Represent the two numbers using the given ratio. Let the two positive numbers be 3k and 4k, where k is a positive constant. The first number is 3k. The second number is 4k. Step 2: Form an equation based on the sum of thrice the first and twice the second number. Thrice the first number = 3 × (3k) = 9k. Twice the second number = 2 × (4k) = 8k. The sum is 68: 9k + 8k = 68 Step 3: Solve for k. 17k = 68 k = (68)/(17) k = 4 Step 4: Find the two numbers. First number = 3k = 3 × 4 = 12. Second number = 4k = 4 × 4 = 16. Step 5: Identify the smaller number. The smaller number is 12. The smaller number is 12. Drop the next question.