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 you've sent. Question 12: Given the sets M = \x : -2 x 4\ and N = \x : 0 < x < 6\, we need to find their intersection M N. Step 1: Understand the sets as intervals. The set M includes all real numbers x such that x is greater than or equal to -2 and less than or equal to 4. In interval notation, this is [-2, 4]. The set N includes all real numbers x such that x is greater than 0 and less than 6. In interval notation, this is (0, 6). Step 2: Find the intersection of the intervals. The intersection M N consists of all values of x that are present in both sets. For x to be in both sets, it must satisfy both conditions: -2 x 4 AND 0 < x < 6 To find the combined range: • The lower bound for x must be greater than the largest of the two lower bounds. Comparing -2 and 0, the larger is 0. Since x > 0 in N, the intersection will have x > 0. • The upper bound for x must be less than the smallest of the two upper bounds. Comparing 4 and 6, the smaller is 4. Since x 4 in M, the intersection will have x 4. Combining these, we get 0 < x 4. The correct option is A. A. \x : 0 < x 4\ Question 13: If sqrt(3) - (2)/(sqrt(3)) = xsqrt(3), find the value of x. Step 1: Simplify the left side of the equation. To subtract the terms on the left side, we need a common denominator, which is sqrt(3). sqrt(3) - (2)/(sqrt(3)) = sqrt(3) × sqrt(3)sqrt(3) - (2)/(sqrt(3)) = (3)/(sqrt(3)) - (2)/(sqrt(3)) = (3 - 2)/(sqrt(3)) = (1)/(sqrt(3)) Step 2: Set the simplified left side equal to the right side and solve for x. (1)/(sqrt(3)) = xsqrt(3) To isolate x, divide both sides by sqrt(3): x = (1)/(sqrt(3) × 3) x = (1)/(3) The correct option is C. C. (1)/(3) Question 14: Given that the first term of an exponential sequence (G.P.) is (sqrt(3) - 2) and the second term is (7 - 4sqrt(3)), find the common ratio. Step 1: Recall the formula for the common ratio in a G.P. For a geometric progression, the common ratio r is found by dividing any term by its preceding term. r = second termfirst term Given: First term (a) = sqrt(3) - 2 Second term (ar) = 7 - 4sqrt(3) Step 2: Calculate the common ratio. r = 7 - 4sqrt(3)sqrt(3) - 2 To simplify this expression, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of sqrt(3) - 2 is sqrt(3) + 2. r = 7 - 4sqrt(3)sqrt(3) - 2 × sqrt(3) + 2sqrt(3) + 2 Step 3: Expand the denominator. Using the difference of squares formula (a-b)(a+b) = a^2 - b^2: (sqrt(3) - 2)(sqrt(3) + 2) = (sqrt(3))^2 - (2)^2 = 3 - 4 = -1 Step 4: Expand the numerator. (7 - 4sqrt(3))(sqrt(3) + 2) = 7(sqrt(3)) + 7(2) - 4sqrt(3)(sqrt(3)) - 4sqrt(3)(2) = 7sqrt(3) + 14 - 4(3) - 8sqrt(3) = 7sqrt(3) + 14 - 12 - 8sqrt(3) = (14 - 12) + (7sqrt(3) - 8sqrt(3)) = 2 - sqrt(3) Step 5: Combine the simplified numerator and denominator. r = 2 - sqrt(3)-1 r = -(2 - sqrt(3)) r = -2 + sqrt(3) r = sqrt(3) - 2 The correct option is D. D. sqrt(3) - 2 Question 15: If 5 + _2 y = 2_2 x, express y in terms of x. Step 1: Express the constant term as a logarithm with base 2. We know that k = _b (b^k). So, 5 = _2 (2^5). 5 = _2 32 Step 2: Substitute this into the given equation.