Sma2113:Maths For Science Question Paper
Sma2113:Maths For Science
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE (30 MARKS)
a) Differentiate between a vector quantity and a scalar quantity. (2 Marks)
b) Given that ?? = 4 3 7
and ?? =
3 -1 0
Find; i. ?? - ?? ii. ?? - ?? . (3 Marks) c) Given the points ?? 2,-1,3 ?????? ??(1,1,4). Find the distance between P and Q. (3 Marks)
d) If ?? ? 0 show that 1 ?? ??
is the unique unit vector in the same direction as ?? . (4 Marks) e) Given the following lines in parametric form; ?? = 3 + ?? ?? = 1 - 2?? ?? = 3 + 3?? ?????? ?? = 4 + 2?? ?? = 6 + 3?? ?? = 1 + ?? Determine their point of intersection. (4 Marks) f) Find the two points trisecting the line segment ?? 2,3,5 ?????? ??(8,-6,2). (4 Marks) g) Determine whether the vectors
?? =
1 0 -2 5 , ?? =
2 1 0 -1 and ?? = 1 1 2 1
are linearly independent in R4. (4 Marks)
2
h) Prove that the following transformation ??: R2 ? R2 is linear ?? ??,?? = (2??,?? + ??). (6 Marks)
QUESTION TWO (20 MARKS)
a) If ?? ,?? ?????? ?? are vectors in a vector space V. simplify 2 ?? + 3?? - 3 2?? - ?? - 3 2 2?? + ?? - 4?? - 4( ?? - 2?? )]. (4 Marks) b) Consider the vectors ??1 = 1 + ?? + 4??2 and ??2 = 1 + 5?? + ??2 in ??2. Determine whether ??1 and ??2 lie in the span 1 + 2?? - ??2,3 - 5?? + 2??2 . (10 Marks) c) Prove that the triangle inequality ?? + ?? = ?? + ?? holds for all vectors.(6 Marks)
QUESTION THREE (20 MARKS)
a) i) Let ?? and ?? be position vectors of two points A and B. if M is the point two thirds the way from A to B, show that the position vector ?? of M is given by ?? = 1 3 ?? + 2 3 ?? . (6 Marks) ii) Conclude that if ??(??1,??2,??3) and ??(??1,??2,??3) then M has coordinates ?? 1 3 ??1 + 2 3 ??1, 1 3 ??2 + 2 3 ??2, 1 3 ??3 + 2 3 ??3 . (4 Marks) b) show that the line through ??0(??0,??0 with slope M has direction vector ?? = 1 ?? and equation ?? - ??0 = ??(?? - ??0). (10 Marks)
QUESTION FOUR (20 MARKS)
A transformation is represented by the matrix ?? =
1 2 3 0 -1 1 1 1 4 , determine
i. The Kernel and its basis of the transformation. (10 Marks) ii. The range and its basis of the transformation. (10 Marks)
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