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: Problem 48: Solve the equation 3x^2 + x - 2 = 0 Step 1: Factor the quadratic equation. We need two numbers that multiply to (3 × -2) = -6 and add up to 1. These numbers are 3 and -2. Rewrite the middle term: 3x^2 + 3x - 2x - 2 = 0 Step 2: Factor by grouping. 3x(x+1) - 2(x+1) = 0 (3x-2)(x+1) = 0 Step 3: Set each factor to zero and solve for x. 3x-2 = 0 3x = 2 x = (2)/(3) x+1 = 0 x = -1 The roots are x = -1 and x = (2)/(3). The correct option is D. D. x = -1, x = (2)/(3) Problem 49: Find the circumference of latitude 60^N to the nearest Km. (take the radius of the earth to be 6400km and = 3.142) Step 1: Calculate the radius of the circle of latitude. The radius of a circle of latitude (r) is given by r = R , where R is the radius of the Earth and is the latitude. r = 6400 km × (60^) r = 6400 km × 0.5 r = 3200 km Step 2: Calculate the circumference of this circle of latitude. The circumference (C) is given by C = 2 r. C = 2 × 3.142 × 3200 km C = 6.284 × 3200 km C = 20108.8 km Step 3: Round the circumference to the nearest Km. C ≈ 20109 km The correct option is A. A. 20,109km Problem 50: Find the quadratic equation whose roots are x = (1)/(2), x = (1)/(3) Step 1: Calculate the sum of the roots ( + ). Let = (1)/(2) and = (1)/(3). + = (1)/(2) + (1)/(3) = (3)/(6) + (2)/(6) = (5)/(6) Step 2: Calculate the product of the roots ( ). = (1)/(2) × (1)/(3) = (1)/(6) Step 3: Form the quadratic equation using the formula x^2 - ( + )x + = 0. x^2 - ((5)/(6))x + ((1)/(6)) = 0 Step 4: Multiply the entire equation by the least common multiple of the denominators (6) to clear the fractions. 6 (x^2 - (5)/(6)x + (1)/(6)) = 6 × 0 6x^2 - 5x + 1 = 0 The correct option is C. C. 6x^2 - 5x + 1 = 0 Drop the next question!