NSCI0005 Mathematics for Natural Sciences A
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NSCI0005
Mathematics for Natural Sciences A
Answer all questions
1. (a) Find the determinant and inverse of the matrix: M = (3(7) 
1) .
(b) Solve the problem XM = Y to find X, where:
Y = ( 
) .
2. Find and classify all stationary points of the function:
f(x, y) = 2x2 + axy2 + y2 + 4x,
according to the value of the parameter a.
3. (a) Find the scalar equation of the plane P, which passes through the point (1, 1, 2) and is parallel to the two vectors:
u = ex + 3ey − ez
v = 2ex + 2ey + 2ez
(b) Find the shortest distance from the point (−1, 0, 4) to the plane P defined in part (a).
4. (a) Derive the Maclaurin series expansion for (x + 1)1/2, up to and including the term in x2 .
(b) Use your result from part (a) determine a rational approximation to ^0.9, giving your answer as a proper fraction.
(c) Find the percentage error in your answer in (b), giving your answer to three significant figures.
5. (a) Find the roots of:
z3 = − 1 − ^3i.
You may give your answers in either Cartesian or Polar form, but ensure they are given in principal range.
(b) Verify that the product of the roots found in part (a) is equal to − 1 − ^3i.
6. Consider the non-linear Ricatti-type ODE:
y\ = y2 − 
− 
(a) Verify that y = x−1 is a solution to the ODE.
(b) Show that the substitution y = 
+ 
allows the ODE to be rewritten as a linear ODE.
(c) Find the general solution w(x) of the linear ODE found in part (b) and so find the solution y(x) that satisfies the condition y(1) = 2.
7. For the second order ODE:
y\\ + 3y\ − 4y = ex ,
(a) Determine the complementary function yh .
(b) Use the method of undetermined coefficients to find a particular integral yp .
(c) Construct a solution that satisfies the conditions y(0) = 1, y\ (0) = 
2023-01-08