Write the Parametric Equations - Mathematics Assignment Help

Assignment Task:

Task:

(1) (a) Find the equation of the line, L1, which passes through the points A : (x, y, z) = (0, −5, −3) and B : (x, y, z) = (3, 1, 0). (b) Find the equation of the line, L2, which passes through the points C : (x, y, z) = (−1, −3, 2) and D : (x, y, z) = (4, 3, 6).

(c) Show that L1 and L2 are not parallel lines.

(d) Write the parametric equations for L1 and L2, and then show that the lines L1 and L2 do not intersect.

(e) Find the vector, −−→BD, from the point B to the point D.

(f) Find a vector, n, which is perpendicular to both L1 and L2.

(g) Find the minimum distance between lines L1 and L2. Test Questions Continue Next Page Page: 3 of 14 Practice Test Trimester MCD4500 (2) Consider the system of linear equations x + 3y − 2z = −6 3x − y + 2z = 4 2x + 2y − z = −4

(a) Write the system of linear equations in matrix equation notation Ax = b.

(b) For the matrix A in part (a) find det(A), the determinant of the matrix A. Show all working, including 2 × 2-determinant working.

(c) Before we attempt to solve the system of linear equations, what does your answer in part

(b) tell us about the solution set corresponding to the system of linear equations? (

d) Write the system of linear equations in augmented matrix form [A| b].

(e) Use row operations to row reduce [A| b] to a form which supports your answer in part

(c). (f) Write the complete solution set to the system of linear equations. Test Questions Continue Next Page Page: 5 of 14 Practice Test Trimester MCD4500

(3) (a) Consider the definite integral Z 1 0 e √ x √ x dx. (i) Explain why this definite integral is an improper integral.

(ii) Determine if this improper integral converges or diverges.

(b) Consider the definite integral Z ∞ 0 arctan(2x) 1 + 4x 2 dx Recall that arctan(θ) is the inverse tangent function tan−1 (θ).

(i) Explain why this definite integral is an improper integral. (ii) Determine if this improper integral converges or diverges. Be sure to treat each improper integral with appropriate mathematical rigour. Simply treating the improper integral as if it was a proper integral will result in zero marks. Test Questions Continue Next Page Page: 7 of 14 Practice Test Trimester MCD4500

(4) (a) Use the definition of the hyperbolic functions to show the following identities

(i) cosh2 (t) − sinh2 (t) = 1.

(ii) sinh2 (t) = 1 2 cosh(2t) − 1 2 .

(iii) sinh(2t) = 2 sinh(t) cosh(t).

(b) Use an appropriate hyperbolic function substitution to evaluate the indefinite integral Z x 2 √ x 2 + 4 dx. Make sure your final answer is a function of x. The identities in part

(a) will be useful. Test Questions Continue Next Page Page: 9 of 14 Practice Test Trimester MCD4500

(5) (a) Use Integration By Parts to evaluate the definite integral Z e 1 x loge (x) dx.

(b) Use the comparison test to determine the convergence or divergence of the series X∞ n=2 1 n √ n2 − 1 .

(c) Evaluate the limit limx−→1 loge (x) sin(πx) . Make sure mathematical rigour is applied and key steps are clearly explained.

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