DOEACC Society 2006 DOEACC B Level B3.2 Basic Mathematics ( ) - Question Paper

B3.2-R3 BASIC MATHEMATICS
NOTE:
Time: three Hours Total Marks: 100
1.
a) If ( ) ( ) two (cos sin ),
1
1 one three q i q
i
i i = +
-
+ + then obtain the value of q .
b) If A and B are symmetric matrices, then show that AB – BA is skew-symmetric matrix.
c) If w (#1) is a cube root of unity, then show that
A = w + w(1/ 2+ three / 8+ nine / 32+ ..........¥ )
is real. Hence, obtain the value of A.
d) Test the convergence of the series å¥
= 1
3
n 2
n
n
.
e) Evaluate the integral ò ( + ) ( + )
= 2
0 two 2
2
1 4
3 dx
x x
I x
f) obtain the coefficient of x2 in the binomial expansion of
8
2
1
÷ø
ö ç
è
æ -
x
x
g) Let a = 7, b = 2, a ´ b = 3i - 2j + 6k. obtain the acute angle ranging from the vectors a and
b.
(7x4)
2.
a) obtain the inverse of the matrix
- one two 0
A = - one 1 1
0 one 0
using Gauss-Jordan elimination method.
b) obtain all the eigen values of the matrix B = 2I + 3A – A2, where I is an identity matrix of
order three and
3 one - 1
A = - two one 2
0 one 2
c) Using the concept of rank, obtain the values of a and b for which the system of equations
3x – y + 2z = 3
2x + y + 3z = 5
x – 2y + az = b
has no solution.
(6+6+6)
B3.2-R3 Page one of three July, 2006
1. ans ques. one and any 4 ques. from two to 7.
2. Parts of the identical ques. should be answered together and in the identical
sequence.
3.
a) Show that
2 2
tan log 2
a b
ab
a ib
i a ib e -
= ú
û
ù
êë
é
÷ø
ö ç
è
æ
+
-
.
b) obtain the value of x, when
2
........
2 4
........ cos
2 4
sin
4 6
1 2
2 3
1 p = ÷ ÷
ø
ö
ç çè
æ
¥ - + - + ÷ ÷
ø
ö
ç çè
æ
- x - x + x - ¥ - x x x .
c) obtain all the asymptotes to the curve
y x2 - four = x2 .
(6+6+6)
4.
a) obtain the limit
( )
sin (2 )
lim cos two one 6
3
0 x
x
x
-
®
.
b) obtain the domain of the function
f ( x) = sin- one (2x) + p / six .
c) The function y = a cos x + b tan x + x has extreme values at x = 0 and x = p / six . obtain the
values of a and b.
d) obtain
dx
dy
when y = ( x) x sin cos one - .
(4+4+4+6)
5.
a) Show that ò ( ) = ò ( ) / 2
0
0
sin two sin p p f x dx f x dx .
b) Evaluate the integral
ú ú ú
û
ù
ê ê ê
ë
é
= ò
® 4
0
3
0
sin
lim
x
tdt
I
x
x
c) The area of the region bounded by the curves y = x – x2 and y = mx equals 9/2. obtain the
value of m.
(6+6+6)
6.
a) obtain the value of p, for which the formula
px2 + xy + y2 – 5x – y + p = 0
represents a pair of straight lines.
B3.2-R3 Page two of three July, 2006
b) Write the formula
4x2 + 9y2 – 32x + 54y + 109 = 0
in standard form of the formula of the ellipse. Hence, determine the eccentricity and the
coordinates of foci.
c) obtain the product of the perpendicular distances from the foci to a tangent to the
hyperbola
1 2
2
2
2
- =
b
y
a
x
(6+6+6)
7.
a) obtain the sum of the series
+ ¥
+ +
+
+
+ .................. upto
1 two 3
7
1 2
5
1
3
2 two 2 two 2 2
b) obtain a unit vector a which is horizontal and perpendicular to the vector b = 4i – 3j + 7k.
c) 2 vectors a = 2i – 2j + k and b = 2i + j – k are provided. Write the vector b as sum of
vectors b1 and b2 such that b1 is parallel to a and b2 is perpendicular to a. obtain the
vectors b1 and b2.
(5+5+8)
B3.2-R3 Page three of three July, 2006

Earning:   Approval pending.