Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The key points are given below: Unit Vector, Inverse, Matrix, Matrix Inversion, Cramers Rule, Angle, Perpendicular, Parallelogram, Moment, Parametric Equations
Typology: Exams
1 / 6
This page cannot be seen from the preview
Don't miss anything!
(NFQ — Level 7)
Answer any three questions. All questions carry equal marks [each 20 marks]. Maximum available marks is 60.
Examiners:
Dr. Brendan O’Regan Dr. Aine N´´ ı Sh´e
(a) For the matrices
(i) Verify that A(B + C) = AB + C;
(ii) Find the inverse of the matrix B;
(iii) Find a matrix K such that BK = C.
[6 marks]
(b) Determine the value of k for which the matrix
k k 3 4 4 − 2
is not invertible.
[5 marks]
(c) (i) Use the method of matrix inversion to solve the set of simulta- neous equations
x − 2 y + 2z = 7 2 x − 4 y + z = 8 − 2 x + 3y − z = − 7
(ii) Use Cramer’s rule to confirm your answer for the variable z.
[9 marks]
(a) An ellipse is defined by the equation
x^2 4
y^2 9
(i) Find the slopes of the tangents to this ellipse at x = 2.
(ii) For which points (x, y) are the tangents to this ellipse parallel to the y-axis?
[4 marks]
(b) A parabola has parametric equations
x = t^2 , y = 2t
(i) Evaluate
dy dx
at the point (1, −2);
(ii) Evaluate
d^2 y dx^2
at t = 3.
[5 marks]
(c) The surface area of a sphere is increasing at the rate of 16 cm^2. s−^1. Find the rate of change of the volume of this sphere when its radius is 12 cm.
[5 marks]
(d) The radius of a cylinder is increasing at 2 mm. s−^1 and its height is decreasing at 5 mm. s−^1. Use differentials to determine the rate at which the volume of the cylinder is changing when its radius is 8 cm and its height is 12 cm.
[6 marks]
(a) Determine each of the following integrals:
(i)
3
x^2 − 6 x + 25
dx
(ii)
1
te−^2 t^ dt
[9 marks]
(b) A force F (in N) acting on a body is described by
F = 200(1 − e−^1.^5 s)
where s is the displacement of the body in metres. Calculate the work done by the force in moving the body from s = 0 metres to s = 2 metres.
[5 marks]
(c) (i) Find the centroid of the area bounded by the curve y = sin x and the x-axis between x = 0 and x = π/2.
(ii) Find the volume generated when this area is revolved 360◦^ about the x-axis.
[6 marks]
