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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:2010
QUESTION ONE (30 MARKS)
a) Find the angle between the following two planes: 2?? + ?? - 2?? = 1 ?????? ?? - 2?? - 2?? = 2. (3 Marks) b) Determine k so that the vectors ?? = 2,3??,-4,1,5 ?????? ?? = (6,-1,3,7,2??) are orthogonal. (3 Marks) c) Write the vector V=(1,-2,5) as linear combination of the vectors ??1 = 1,1,1 ,??2 = 1,2,3 ?????? ??3 = (2,-1,1). (4 Marks) d) Show that ?? = (??,??) ?? = 2?? is a subspace of R2. (4 Marks) e) Determine whether ??: R2 ? R2, defined by ??(??,??) = (?? - ??,?? + ??) is a linear transformation. (4 Marks) f) Define the following terms in relation to a linear transformation T: i. Nullity of T ii. Rank of T. (4 Marks) g) Let ??: R3 ? R2 be defined by ?? ??,??,?? = ?? - ??,?? + ?? + ?? . find kerf. (4 Marks) h) Find the dimension and a basis for ?? = (??,??,??,??) ?? + ?? + ?? + ?? = 0 . (4 Marks)
QUESTION TWO (20 MARKS)
a) Let ??: R3 ? R3 be defined by ?? ??,??,?? = ?? - ??,??,?? - ?? + ?? . Determine whether T is invertible and if so find the inverse of T. (6 Marks) b) Show that the transformation ??: R2 ? R2 defined ?? ??,?? = (?? + ??,??)is linear. (5 Marks) c) Let ??: R3 ? R3 be a linear mapping defined by ?? ??,??,?? = (?? + 2?? - ??,?? + ??,?? + ?? - 2??. Find a basis and the dimension of the kernel of T. (5 Marks)
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d) Show that the vectors ??1 = 6,2,3,4 ,??2 = 0,5,-3,1 ?????? ??3 = (0,0,7,-2) are linearly independent. (4 Marks)
QUESTION THREE (20 MARKS)
a) Find the equation of the plane through (4,1,-3) and (2,-1,-3) and perpendicular to the plane 2?? + 3?? - 4?? = 5. (5 Marks) b) Find the parametric equations of the line of intersection of the planes 3?? + 2?? - 4?? - 6 = 0 and ?? - 3?? - 2?? - 4 = 0. (5 Marks) c) Find the distance between the planes ?? + 2?? - 2?? = 3 and 2?? + 4?? - 4?? = 7. (5 Marks) d) Prove that ??.?? = ?? ?? , for any vectors u and v. (5 Marks)
QUESTION FOUR (20 MARKS)
a) Use the vectors ?? = ?? + ?? - 3??,?? = 2?? + ?? + 2?? ?????? ?? = 3?? - 2?? - ?? to prove that ?? × ?? + ?? = ?? × ?? + ?? × ??. (6 Marks) b) Show that the lines ?? = 3,5,7 + ?? 1,2,1 ?????? ?? = 1,2,3 + ??(2,3,5) do not meet. (5 Marks) c) i) Define a basis of a vector space. (2 Marks) ii) Determine whether 1,1,1 , 1,2,3 , 2,-1,1 form a basis for R3. (3 Marks) d) Show that 1,1,1 , 0,1,1 , 0,1,-1 spans R3. (4 Marks)
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