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Eccq 2202: Geometry Question Paper

Eccq 2202: Geometry

Course:Civil Engineering

Institution: Technical University Of Kenya question papers

Exam Year:2014

QUESTION ONE (30MARKS)
a) Use Gauss Elimination method to solve tge systems of equations.
i) - 4x +y +z =-200
x-4y +w =-200
x-4z+w =-100
y+z-4w=-100 (6marks)
ii) x-3y+2z= 2
5x-15y+7z=10 (6marks)
b) Sketch the surface whose equation is given by 8x²+6y²+6z²=12, giving all the relevant explanations. (6marks)
c) A plane passes through the points (4,0,0) (0,-2,0) and (0,0,5). Find its equation.
d) Reduce the following equations of the straight lines
i) 5x+2y-4z+3=0 to intercept form and
ii) 3x+2y-z+5=0 to normal form.
e) Show that the planes 4x-y+6z=0 and 8x-2y+12z-18=0 are parallel (6marks)

QUESTION TWO (20MARKS)
a) Given the equation of the surface x²+ 3y²+z²-wx+7y+4z-4=0 find tge equation of the trace on the xy and yz plane and identify the trace on the plane y=-1 (6marks)
b) Sketch the surface whose equation is given by 4x²+y²+4z²=16 (8marks)
c) Reduce the equation of the line 3x+2y+4z=5 and x-y+2z=4 ti a symmetric form. (6marks)

QUESTION THREE (20MARKS)
a) Describe Gauss elimination method (8marks)
b) Solve the following system using the inverse method
2x-y+3z=12
x+3y-z=11
2x-2y+5z=3 (6marks)
c) Solve the following system using the Gauss elimination method
x1+2x2+x3= 4
3x 1 + 4x 2 +2x 3=4
5x 1 + 3x 2 +5x 3=-1 (6marks)

QUESTION FOUR (20MARKS)
a) Prove that the intersection of two distinct planes in Sis either a line or a empty set. (8marks)
b) Identify the surfaces given by:-
i) 5x²+4y²-10x-10y-140y-3²z +69=0
ii) (x-³/2)²+( z+½)²=y+7/2
c) Discuss the locus of tge equation
x² - y² - z² = 1 (6marks)
9 4 16

QUESTION FIVE (20MARKS)
a) Give the definition of a three dimensional triple space (S, L, P) (5marks)
b) Prove that if L and M are distinct lines, assuming that LOM = O then there is a unique plane P such that LCP and MCP (5marks)
c) Prove that the intersection of two distinct pane in S is either a line or the empty set (10 marks)

Eccq 2202: Geometry Question Paper

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