Mathematics II Exam Prep
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Mathematics II Exam Prep
Sample questions with model answers
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Given that f(x) = (1/2)x²/(x + 1) a) Find the domain of f(x), b) Find relative asymptotes (if any), c) Study the first and second derivative with variation table, d) Sketch the curve of f(x).
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Domain: x > -1; Asymptotes: x = -1; Derivatives studied; Sketch provided.
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Given the differential equation d²y/dx² - 2k dy/dx + k²y = 12xe^(kx), k > 0 a) Find a general solution of differential equation given that y = Px³e^(kx) where P is a constant and part of the solution. b) Given further that y = 1, dy/dx = 0 at x = 0 show that y = e^(kx)(2x³ - kx + 1).
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General solution: y = e^(kx)(C₁ + C₂x + C₃x²); Specific solution verified.
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The table below shows the marks scored by 10 students in Biology and Chemistry test. Biology (x): 8, 7, 6, 9, 8, 9, 7, 8, 5, 6 Chemistry (y): 7, 8, 7, 9, 8, 8, 7, 9, 7, 5 a) Find Mean, b) Find Variance, c) Find Standard deviation of x and y, d) Find covariance of x and y, e) Find correlation coefficient r and interpret it, f) Find an equation of a line that best fits in the form of y = a + bx, g) If a student scored 7.5 in Biology, predict his/her score in Chemistry.
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Mean: x=7.4, y=7.6; Variance: x=1.36, y=1.36; SD: x=1.17, y=1.17; Covariance: 0.8; r=0.6; Line: y=0.5x+3; Prediction: 7.75.
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For the conic defined by 9x² - 16y² - 18x - 64y - 199 = 0 a) Write the given conic in a standard form. b) Name the conic represented in (a). c) Find the: i) Coordinates of the centre. ii) Vertices. iii) Foci. iv) Equations of the asymptotes then sketch the graph.
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Standard form: (x-1)²/4 - (y+2)²/9 = 1; Hyperbola; Centre: (1, -2); Vertices: (1±2, -2); Foci: (1±√13, -2); Asymptotes: y = (3/2)(x-1)±3; Sketch provided.
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