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B3.2-R3: BASIC MATHEMATICS
: JANUARY 2006

NOTE:
1. Answer question 1 and any FOUR questions from 2 to 7.

2. Parts of the same question should be answered together and in the same sequence

Time: 3 Hours Total Marks: 100

1.a) If and , then find a vector of magnitude 4 perpendicular to both and .

b) Find

c) Find the value of the determinant

d) Evaluate

e) Find the area of region bounded by the parabola y 2=4x and its latus rectum.

f) Find the equation of tangent to the curve x=cos q and y=sin q at q = .

g) Test the convergence/divergence of the series:

(7x4)

2.a) Find the value of the constant p so that the vectors , and are coplanar.

b) Using DeMoivre’s theorem, find the value of .

c) Reduce the matrix to echelon form and hence find its rank.

(6+6+6)

3.a) Examine the continuity of the following function at x = 0

b) If y=log(1+cosx), and determine the value of c.

c) Solve the following system of equations by Cramer’s rule:

(6+6+6)

4.a) If and , find .

b) Examine the convergence and absolute convergence of the series: .

c) Find the points of local maxima or local minima, if any, of the function

.

d) Evaluate .

(3+6+6+3)

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5.a) Obtain the asymptotes of the curve: (x+y) 2=xy 2.

b) Find the vertex and focus of the parabola

y 2-4y-2x-8=0

c) Obtain the Taylor ’s series for the function at .

d) Draw the graph of the function

f(x) =

Is f(x) continuous at x = ?

(6+4+4+4)

6.a) i) Examine the validity of the Rolle’s theorem for the function

f(x) = cosx in .

ii) Verfiy Lagrange’s mean value theorem for the function

in [2, 4].

b) Exaluate .

c) Find the equation of the circle which touches y-axis and whose center at (1, 2).

d) If and , then find

(6+4+4+4)

7.a) Classify the following Conics in terms of parabola, ellipse or hyperbola:

i) x 2-3xy-y 2+10x-10y+21=0

ii) 22x 2-12xy+17y 2-112x+92y+178=0

b) Find the values of x and y if A = , B = and (A+B) 2=A 2+B 2.

c) Find when

d) Show that the following system of equations has infinite number of solutions:

x-2y+3z=0, 2x+4y+z=0, 3x+2y+4z=0

(6+4+4+4)

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