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Math 114: Geometry And Linear Algebra Question Paper
Math 114: Geometry And Linear Algebra
Course:Bachelor Of Arts (Economic & Mathematics)
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
SECOND YEAR EXAMINATIONS FOR THE AWARD OF DEGREE OF
BACHELOR OF ARTS (ECONOMIC & MATHEMATICS)
MATH 114: GEOMETRY AND LINEAR ALGEBRA
STREAMS: BA (ECON&MATHS) Y2SI TIME: 2 HOURS
DAY/DATE: THURSDAY 25/4/2013 8.30 AM – 10.30 AM
INSTRUCTIONS:
Answer Question ONE (Compulsory) and any other TWO Questions.
Adhere to the instructions on the answer booklet
Do not write on the question paper.
QUESTION ONE (30 MARKS)
(a) Find the distance between the two lines
L_1=x+2y-2=0 and L_2=2x+4y+3=0 [4 Marks]
(b) Find the angle ? between the lines y=4x-7 and y=-3x+1 [3 Marks]
(c) Find the angle between the vector =i-2j+4k and =-4i+j-2k
[3 Marks]
(d) Determine the area A of the triangle PQR with vertices P (1,2,0) Q (3,0,-3)
and R (5,2,6). [4 Marks]
(e) Find an equation of the parabola with the point (1,1) as its focus F and the line
x+y+2=0 as its directrix L. [5 Marks]
(f) Write a polar equation for the hyperbola with Cartesian equation
x^2/144-y^2/25=1 [5 Marks]
(g) If A=[¦(1&2&2@2&1&2@2&2&1)] show that A^2-4A-5I=0 where I, 0 are the unit matrix and the null matrix of order 3 respectively. [3 Marks]
(h) Express (1+2i)/(1-3i) in the form r(cos???+sin??)? [3 Marks]
QUESTION 2 (20 MARKS)
(a) Determine the values of ?_1 ß & e when [¦(0&2ß&e@?&ß&-e@?&-ß&e)] is orthogonal [5 Marks]
(b) Solve the following equations using non-reduction (Gauss elimination)
x-y+2Z=3
x+2y+3Z=5
3x-4y-5Z=-13 [5 Marks]
(c) Express ((2+i)/(3-i))^2 into polar form. [5 Marks]
(d) Find the smallest positive integer n for which ((1+i)/(1-i))^n=1 [5 Marks]
QUESTION 3 (20 MARKS)
(a) (i) Let L_1 and L_2 be two lines which intersect in a point P, assumed to be above the
x – axis. Prove that the smallest angle ? between L_(1 ) and L_2 is given by
?=?tan?^(-1) ((m2-m1)/(1+m1m2)) where m1 and m2 are the gradients of line L_(1 ) and L_2
respectively. [5 Marks]
(ii) Find the angle between the lines v3.x-y+1 and v(3 ).x+y-1=0
[5 Marks]
(b) Analyse and identify the curves with the following equations
(i) x^2-2x+8y+9=0 [7 Marks]
(ii) (y-1)^2+(x-1)^2=1 [3 Marks]
QUESTION 4 (20 MARKS)
(a) Find the line of intersection l of the two planes
2x-3y+4Z-1=0 and
x+2y-2+3=0 [7 Marks]
(b) Write the equation of the plane through the point P_1 (-2,1,3) perpendicular to the vector =(4,5,-1 ) [3 Marks]
(c) Find the angle between the planes 6x+6y-3Z+5=0 and x-2y+2Z-4=0
[5 Marks]
(d) Analyse the graph of the equation x^2+?4y?^2+4x-8y+Z=0 [5 Marks]
QUESTION 5 (20 MARKS)
(a) Analyse and identify the curve with the equation x^2/16-y^2/9=1 [5 Marks]
(b) Prove that the distance d between the point p_1 (x_(1 ) ?,y?_1 ) and the line L with equation
AX+BX+C=0 is given by d=|?AX?_1+?BY?_1+C|/v(A^2+B^2 ) hence find the distance between the
point ( 3,1 ) and the line 3x + 4y-3=0 [10 Marks]
(c) Find an equation of the perpendicular bisector of the line segment joining the points (2,1) and (1,2) [5 Marks]
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