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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Hey Awesome, good to see you again. 1. If the set M = 0, 1, 2, 3, 4, find the number of proper subsets of M. Step 1: Count the number of elements in set M. The set M has 5 elements: \0, 1, 2, 3, 4\. So, n(M) = 5. Step 2: Calculate the total number of subsets. The total number of subsets for a set with n elements is 2^n. Total subsets = 2^5 = 32. Step 3: Calculate the number of proper subsets. A proper subset is any subset except the set itself. Number of proper subsets = Total subsets - 1 Number of proper subsets = 32 - 1 = 31. The correct option is C. The number of proper subsets of M is 31. 2. For what range of values of x is x^2 - 2x - 3 0 ? Step 1: Find the roots of the quadratic equation x^2 - 2x - 3 = 0. Factor the quadratic expression: (x-3)(x+1) = 0 The roots are x = 3 and x = -1. Step 2: Determine the range for which the inequality holds. Since the coefficient of x^2 is positive (1), the parabola opens upwards. The inequality x^2 - 2x - 3 0 is satisfied for values of x between and including the roots. -1 x 3 The correct option is D. The range of values for x is -1 x 3. 3. Given that M = 3 & 2 \\ -1 & 4 and N = 5 & 6 \\ -2 & -3 , calculate (3M - 2N). Step 1: Calculate 3M. 3M = 3 3 & 2 \\ -1 & 4 = 3 × 3 & 3 × 2 \\ 3 × (-1) & 3 × 4 = 9 & 6 \\ -3 & 12 Step 2: Calculate 2N. 2N = 2 5 & 6 \\ -2 & -3 = 2 × 5 & 2 × 6 \\ 2 × (-2) & 2 × (-3) = 10 & 12 \\ -4 & -6 Step 3: Subtract 2N from 3M. 3M - 2N = 9 & 6 \\ -3 & 12 - 10 & 12 \\ -4 & -6 3M - 2N = 9 - 10 & 6 - 12 \\ -3 - (-4) & 12 - (-6) 3M - 2N = -1 & -6 \\ -3 + 4 & 12 + 6 3M - 2N = -1 & -6 \\ 1 & 18 The correct option is C. The result of the calculation is -1 & -6 \\ 1 & 18 . 4. Simplify; (1)/(3) 8 + (1)/(3) 64 - 2 6. Step 1: Apply the logarithm property a b = b^a. 8^1/3 + 64^1/3 - 6^2 Step 2: Simplify the terms with exponents. 8^1/3 = [3]8 = 2 64^1/3 = [3]64 = 4 6^2 = 36 Substitute these values back into the expression: 2 + 4 - 36 Step 3: Apply the logarithm properties a + b = (ab) and a - b = ((a)/(b)). (2 × 4) - 36 8 - 36 ((8)/(36)) Step 4: Simplify the fraction inside the logarithm. ((8 ÷ 4)/(36 ÷ 4)) = ((2)/(9)) The correct option is B. The simplified expression is (2)/(9). 5. If 3^x+1 + 3^x-1 = 30, what is the value of x? Step 1: Rewrite the terms using exponent rules a^m+n = a^m a^n and a^m-n = (a^m)/(a^n). 3^x · 3^1 + (3^x)/(3^1) = 30 3 · 3^x + (1)/(3) · 3^x = 30 Step 2: Factor out 3^x. 3^x (3 + (1)/(3)) = 30 Step 3: Simplify the expression inside the parentheses. 3^x ((9)/(3) + (1)/(3)) = 30 3^x ((10)/(3)) = 30 Step 4: Isolate 3^x. 3^x = 30 × (3)/(10) 3^x = 3 × 3 3^x = 9 Step 5: Express 9 as a power of 3. 3^x = 3^2 Step 6: Equate the exponents since the bases are the same. x = 2 The correct option is B. The value of x is 2. 6. Find the equation of the tangent to the circle x^2 + y^2 + 8x - 12y = 15 at (2, 3). Step 1: Find the center of the circle. The general equation of a circle is x^2 + y^2 + 2gx + 2fy + c = 0. Rewrite the given equation: x^2 + y^2 + 8x - 12y - 15 = 0. Comparing coefficients, 2g = 8 g = 4 and 2f = -12 f = -6. The center of the circle is (-g, -f) = (-4, 6). Step 2: Find the slope of the radius connecting the center (-4, 6) to the point of tangency (2, 3). m_r = (y_2 - y_1)/(x_2 - x_1) = (3 - 6)/(2 - (-4)) = (-3)/(2 + 4) = (-3)/(6) = -(1)/(2) Step 3: Find the slope of the tangent line. The tangent line is perpendicular to the radius at the point of tangency. The product of the slopes of perpendicular lines is -1. m_t = -(1)/(m_r) = -(1)/((-1/2)) = 2 Step 4: Use the point-slope form of a linear equation $y - y_1