B3.2-R3: BASIC MATHEMATICS 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) Find the square root of (3+4i).
b) Find the real eigenvalue and associated eigenvector of the following 6 0 0 0 0 ⅓ 0 ½ 0 A = c) Find 3 2 0
sin
x
ydy
x x∫
→ 0lim
d) Find the eccentricity of the hyperbola
5
x
2−
4
y
2−
10
x
−
16
y
−
31
=
0
e) Let Sn(
)
.
1
1
...
3
.
2
1
2
.
1
1
+
+
+
+
n
n
=
then find n. x→∞S
lim
f) Find the area of the region bounded by the curve and the lines
x = -2 and x = 1.
),
1
)(
1
)(
2
(
+
+
−
=
x
x
x
y
g) Find the derivative of
(
)(
)
3
2
1
2
6
4+
+
+
x
x
x
x
(7x4) 2.a) Find the value of k, for which the systems of equations x – ky +z = 0
kx + 3y – kz = 0 3x + y – z = 0,
has (i) trivial solution, (ii) non-trivial solutions.
x + 2 x (x + 1) (x + 2) (x + 1) (x + 2) b) If f (x) = 1 x + 1 2(x + 1) 2(x + 1) 3x (x + 1) (x – 1) x (x + 1) then find the value of f (100).
c) Determine the rank of the following matrix, for all values of .
x
5- x 2 1 2 1- x 0 1 0 1-x A = (6+6+6)3.
a) The function f (x) = x3 + 2ax + b,
0
≤ x
≤
1
bx2 + 3a ,x
>
0
is differentiable for all x > 0. Find the values of a and b. Draw the graph of the function .
( )
x
f
b) Evaluate∫
−−
π πdx
x
x
sin
1
sin
c) Find all functions
f
(x
)
such that 2)
1
(
)
(
=
−
−
x
dx
x
df
and draw the graph of in the x-y plane. Find in particular the function whose graph passes through the point =
(
.( )
x
f
(
x
0,
y
0)
1
,
1
)
(5+5+8) 4.
a) Find the equations of the common tangents to the circle (x – 2)2 + y2 = 4 and the parabola y2 = - 4x.
b) The lengths of two non-zero vectors a and b are equal. If the vectors a – 3b and 7a +5b are the right angles, then find the angle between the vectors a and b.
c) AOB is the positive quadrant of the ellipse 9x2 + y2 = 36. Find the area between the arc AB and the chord AB.
(6+6+6) 5. a) If
y
=
x
3e
2x, then find n ndx
y
d
andx
=
0
b) Find the extreme values of the functionf (x) =
x
1xc) Assume that x1 is a point of maxima and x2 is a point of minima of the function
f (x) = 2x3 – 9ax2 + 12a2x + 1
Find the value of a > 0 for which
x
12=
x
2(6+6+6) 6.
a) Find the derivative of
+
−
− 2 2 11
1
cos
x
x
with respect to
−
−
− 2 3 13
1
3
tan
x
x
x
. b) Evaluate∫
(
)
dx+
+
x
e
x
xcos
1
sin
1
e) State the mean-value theorem. For - in [a,b], determine all numbers
ξ
in the specified interval, such that3
)
(
x
x
f
=
x
( )
( ) ( )
a
b
a
f
b
f
f
−
−
=
ξ
'
Take a=0 and b=2.
7. a) If Zk=
,
1
,
2
,
...
3
sin
3
k+
i
kk
=
π
π
cos
then find the value of Z1 Z2 …….. ∞.b) From the top of a tower 15 feet high, the angles of depression at two points on opposite sides of it are 30º and 60º. Find the distance between the points.
c) If a , b , c are three vectors such that a + b + c =0, |a| = 3, |b|= 4, |c| = 5, then find the
value of a . b + b . c + c . a
d) Let P (x1, y1) and Q (x2, y2) be two distinct points on the parabola y = Ax2 + Bx + C, A≠0.Using Lagrange mean value theorem, find a point R on the arc PQ, where the tangent to the curve is parallel to the chord PQ.
B3.2-R3: BASIC MATHEMATICS NOTE:
Time: 3 Hours Total Marks: 100
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.
1.
a) Show that the points whose position vectors are and are collinear.
k
j
i
k
j
i
2
r
3
r
,
2
r
3
r
4
r
r
−
+
+
−
k
j
1
0
7
r
+
r
b) Show that the roots of the equation x2+x+1 =0 are of the form w and w2. Also, find the sum of the roots.
Ans
x
2+ x
+
1
=
0
( )
0
2
1
1
2
1
2
1
.
.
2
2 2 2=
−
+
+
+
⇒
x
x
1
4
1
2
1
2−
=
+
⇒ x
4
3
.
4
3
2
1
2 2i
x
=
−
=
+
⇒
i
i
x
2
3
4
3
.
2
1
=
2=
+
⇒
i
x
2
3
2
1 +
−
=
⇒
i
Let
2
3
2
1 +
−
=
ω
2 22
3
2
1
+
−
=
∴
ω
i
2 22
3
2
3
.
2
1
.
2
2
1
+
−
+
−
=
i
i
4
3
2
3
4
1
−
−
=
i
i
2
3
2
1 −
−
=
ω
∴ andω are the roots of the given equation
2x
2+ x
+
1
=
0
1
2
3
2
1
2
3
2
1
2
−
=
−
−
+
+
−
=
+
ω
i
i
ω
d) Evaluate
x
x
x 3 3 0sin
)
1
log(
+
→lt
Ans1
1
1
1
1
)
1
log(
)
1
log(
)
1
log(
)
1
log(
3 3 0 3 3 0 3 3 0 3 3 0 3 3 3 3 0 3 3 0 3=
=
=
+
=
+
=
+
=
+
→ → → → → →x
x
Sin
lt
x
x
lt
x
x
Sin
lt
x
x
lt
x
x
Sin
x
x
lt
x
Sin
x
lt
x x x x x xe) Show that the line y=x+2 is a tangent to the parabola y2=8x. Also, find the point of contact.
f) Test the Convergence of the series
∑
∞ =1+
.
)
1
(
2
3
nn
n
g) find the area included between the circle x2+y2=1 and the lines x=0 and x=1. h) Find points of maxima/minima of the function
f(x)=x+sin 2x; 0 ≤ x ≤ 2 Π
i) If a circle passes through the point (a,b) and touches both the co-ordinate axit, then show that a=b.
(3+3+3+3+3+3+3+4+3) 2.
a) State DeMoirre’s Theorem. Express the complex number
Z =
(
)
)
2
2
(
)
(
3θ
θ
θ
θ
θ
θ
iCos
Sin
iSin
Cos
iSin
Cos
+
−
+
In the form x-iy, where x and y are real numbers. Also find |z| and arg. z.
Ans
(
(
)
)(
θ
θ
)
θ
θ
θ
θ
cos
sin
2
sin
2
cos
sin
cos
3i
i
i
z
+
−
+
=
(
)
(
)
(
(
)
)
{
θ
θ
}(
θ
θ
)
θ
θ
cos
sin
2
sin
2
cos
sin
cos
3i
i
i
+
−
−
−
−
+
=
( )
( )
−
=
−
−
=
θ
θ
θ
θ
Cos
Cos
Sin
Sin
(
)
(
)
(
)
{
θ
θ
}(
θ
θ
)
θ
θ
cos
sin
2
sin
2
cos
sin
cos
3i
i
i
+
−
+
−
+
=
(
)
(
θ
θ
)
(
θ
θ
)
θ
θ
cos
sin
sin
cos
sin
cos
2 2 3i
i
i
i
+
−
+
+
=
−[
i
2=
−
1
⇒
−
i
2=
1
]
(
cos
θ
+
i
sin
θ
) (
5i
cos
θ
−
i
sin
θ
=
)
)
(
cos
θ
i
sin
θ
) (
4cos
θ
i
sin
θ
)(
cos
θ
i
sin
θ
i
+
+
−
=
(
θ
θ
)
4(
2θ
2 2θ
)
sin
cos
sin
cos
i
i
i
+
−
=
(
θ
θ
)
4(
2θ
2θ
)
sin
cos
sin
cos
+
+
=
i
i
(
)
4sin
cos
θ i
θ
i
+
=
( )
( )
(
cos
4
θ
i
sin
4
θ
)
i
+
=
( )
4
θ
sin
( )
4
θ
cos
i
2i
+
=
( )
4
θ
cos
( )
4
θ
sin
+
i
−
=
( )
4
θ
,
cos
sin
=
−
=
( )
4
θ
∴
x
y
( )
(
sin
4
)
2(
cos
( )
4
)
2sin
2( )
4
cos
2( )
4
1
1
2 2
+
=
−
+
=
+
=
=
=
x
y
θ
θ
θ
θ
z
( )
( )
(
( )
)
+
−
=
−
−
=
−
−
=
−π
−θ
π
−π
θ
θ
θ
π
4
2
tan
tan
4
cot
tan
4
sin
4
cos
tan
)
(
z
1 1 1Arg
θ
π
θ
π
π
θ
π
π
4
2
4
2
4
2
=
−
−
=
+
+
−
=
b) Find the value of λ so that the vectors and are co-planer.
k
j
i
k
j
i
r
r
r
r
r
r
−
+
+
−
2
,
λ
i
r
−
r
j
+
λ
k
r
c) Which of the following matrices is non-sungular? A =
, B =
6
3
2
1
0
2
0
2
Also, find its inverse.
Ans
=
6
3
2
1
A
0
6
6
6
3
2
1
=
−
=
=
A
∴A is a Singular Matrix
=
2
0
0
2
B
4
0
4
2
0
0
2
=
−
=
=
B
∴B is a Non-Singular Matrix
2
0
0
2
)
2
(
)
1
(
1 1 11=
−
⋅
=
+C
)
0
(
)
1
(
1 2 12=
−
⋅
=
+C
)
0
(
)
1
(
2 1 21=
−
⋅
=
+C
)
2
(
)
1
(
2 2 22=
−
⋅
=
+C
=
=
2
0
0
2
)
(
22 12 21 11C
C
C
C
B
Adj
=
=
=
−2
1
0
0
2
1
4
2
0
0
2
)
(
1B
B
Adj
B
d) Solve the following system of equations by Cramer’s rule or by Gauss-elimination method:
Ans
x
1−
2
x
2+
x
3=
0
L
L
L
(
1
)
)
2
(
2
3 2+
=
−
L
L
L
−
x
x
)
3
(
10
3
2
x
1− x
3=
L
L
L
By Cramer’s Rule
1
2
4
3
)
2
0
(
1
)
2
0
(
2
)
0
3
(
1
3
0
2
1
1
0
1
2
1
=
+
−
=
+
+
−
+
−
=
−
−
−
=
D
2
10
8
0
)
10
0
(
1
)
10
6
(
2
)
0
3
(
0
3
0
10
1
1
2
1
2
0
1=
−
+
−
+
+
=
−
+
=
−
−
−
−
=
D
0
4
0
4
)
4
0
(
1
)
2
0
(
0
)
10
6
(
1
3
10
2
1
2
0
1
0
1
3=
−
−
−
+
+
=
−
−
+
=
−
−
=
D
2
8
10
)
2
0
(
0
)
4
0
(
2
)
0
10
(
1
10
0
2
2
1
0
0
2
1
3−
−
=
−
−
+
+
+
+
=
−
+
=
−
−
=
D
2
1
2
0
1
0
2
1
2
3 3 2 2 1 1−
=
−
=
=
=
=
=
=
=
=
∴
D
D
x
D
D
x
D
D
x
By Gauss Elimination Method
)
3
(
10
3
2
)
2
(
2
)
1
(
0
2
3 1 3 2 3 2 1L
L
L
L
L
L
L
L
L
=
−
−
=
+
−
=
+
−
x
x
x
x
x
x
x
−
−
−
−
10
3
0
2
2
1
1
0
0
1
2
1
Step-1: Elimination of
x
1)
3
(
10
5
4
2
)
1
(
)
3
(
)
3
(
)
2
(
2
)
2
(
)
1
(
0
2
)
1
(
3 2 3 2 3 2 1L
L
L
L
L
L
L
L
L
=
−
⇒
×
−
=
−
=
+
−
⇒
=
+
−
⇒
x
x
x
x
x
x
x
−
−
−
−
10
5
4
0
2
1
1
0
0
1
2
1
Step-2: Elimination of
x
2)
3
(
2
4
)
2
(
)
3
(
)
3
(
)
2
(
2
)
2
(
)
1
(
0
2
)
1
(
3 3 2 3 2 1L
L
L
L
L
L
L
L
L
=
−
⇒
×
+
=
−
=
+
−
⇒
=
+
−
⇒
x
x
x
x
x
x
−
−
−
−
2
1
0
0
2
1
1
0
0
1
2
1
2
2
0
)
2
(
)
0
2
(
2
0
2
2
2
2
3 2 1 3 2 3=
+
=
−
−
×
=
−
=
=
+
−
=
+
=
−
=
∴
x
x
x
x
x
x
e) Find the multiplicative inverse of the complex number
i
i
2
3
3
2
−
+
in the form x+iy where x and y are real numbers.
Ans
( )
i
i
i
i
i
i
i
i
i
i
i
i
i
i
z
=
=
+
+
+
−
=
−
+
+
+
=
+
−
+
+
=
−
+
=
13
13
4
9
9
4
6
6
2
3
6
9
4
6
)
2
3
)(
2
3
(
)
2
3
)(
3
2
(
2
3
3
2
2 2 21
,
0
=
=
∴
x
y
(
)
i
i
i
y
x
y
i
y
x
x
iy
x
=
−
=
−
+
−
+
=
+
−
+
=
+
−1
0
0
1
1
1
0
0
.
2 2 2 2 1 (4+3+3+5+3) 3.a) Without expanding, show that the determinant
5
4
9
4
3
7
3
2
5
vanishes. Ans5
4
9
4
3
7
3
2
5
=
D
0
5
9
9
4
7
7
3
5
5
=
=
[
C
2=
C
2+
1 C
.
3]
b) If f(x) =25 x
−
2 , prove that4
3
3
)
3
(
)
(
3−
=
−
−
→x
f
x
f
lt
x . Ansf
(
x
)
=
25
−
x
24
16
9
25
3
25
)
3
(
=
−
2=
−
=
=
f
(
)(
)
(
3
)
(
25
4
)
4
25
4
25
3
4
25
3
)
3
(
)
(
2 2 2 3 2 3 3−
−
+
+
−
−
−
=
−
−
−
=
−
−
→ → →x
x
x
x
lt
x
x
lt
x
f
x
f
lt
x x x(
)
(
)
(
)
(
)
(
3
)
(
(
25
)
4
)
9
4
25
3
9
4
25
3
4
25
2 2 3 2 2 3 2 2 2 3−
−
+
−
−
=
+
−
−
−
=
+
−
−
−
−
=
→ → →x
x
x
lt
x
x
x
lt
x
x
x
lt
x x x(
)(
)
(
)
(
)
(
)
(
)
16
4
6
4
3
25
3
3
4
25
3
4
25
3
3
3
2 2 3 2 3−
+
=
−
+
+
−
=
+
−
+
−
=
+
−
−
+
−
−
=
→ →x
x
lt
x
x
x
x
lt
x x4
3
8
6
4
4
6
=
−
=
−
+
−
=
c) find the distance between the parallel lines 2x+4y = 7 and x+2y = 3.
d) Using the concept of ‘rank’ of a matrix, list for consistency the following system of equation 2x + 8y + 5z = 5, x + 2y – z = 2, x + y +z = -2.
e) If y = A cos mx + B Sin mx, then show that 2
0
2 2=
+
m
y
dx
y
d
Ans
y
=
A
Cos
mx
+
B
Sin
mx
)
(
A
Cos
mx
B
Sin
mx
dx
d
dx
dy
=
+
mx
Sin
B
dx
d
mx
Cos
A
dx
d
+
=
mx
Sin
dx
d
B
mx
Cos
dx
d
A
+
=
mx
Cos
Bm
mx
Sin
Am
+
−
=
mx
Sin
Am
mx
Cos
Bm
−
=
)
(
2 2mx
Sin
Am
mx
Cos
Bm
dx
d
dx
dy
dx
d
dx
y
d
−
=
=
mx
Sin
Am
dx
d
mx
Cos
Bm
dx
d
−
=
mx
Sin
dx
d
Am
mx
Cos
dx
d
Bm
−
=
mx
Cos
m
Am
mx
Sin
m
Bm
.
−
.
−
=
mx
Cos
Am
mx
Sin
Bm
2−
2−
=
(
A
Cos
mx
B
Sin
mx
)
Am
Cos
mx
Bm
Sin
mx
m
y
m
2=
2+
=
2+
20
2 2 2 2 2 2 2=
+
+
−
−
=
+
my
Bm
Sin
mx
Am
Cos
mx
Am
Cos
mx
Bm
Sin
mx
dx
y
d
(Proved)
f) Find the value of k so that the function f(x) defined below is continuous at x =
2
Π
. f(x) =
=
≠
2
,
2
,
π
π
π
x
when
k
x
when
x
kCosx
− 2
Ans
The function
f
(x
)
is defined as:
2
,
2
,
2
cos
)
(
π
π
π
=
=
≠
−
=
x
when
k
x
when
x
x
k
x
f
At the point
x
=
π
2
k
f
=
2
π
x
x
k
lt
x
x
k
lt
x
f
lt
x x x→=
→−
=
→2
−
cos
2
1
2
cos
)
(
2 2 2 ππ
ππ
πPut
x
− 2
π
=
θ
2
0
π
θ
→
→
∴
as
x
(
)
(
)
2
1
.
2
sin
2
sin
2
1
2
cos
2
1
2
cos
2
1
0 0 0 2k
k
lt
k
k
lt
k
lt
x
x
k
lt
x−
=
=
=
−
=
−
+
=
−
∴
→ → → →θ
θ
θ
θ
θ
θ
π
π
θ θ θ πAs the function in continuous at
x
=
π
2
( )
2
(
)
2f
x
lt
f
x ππ
→=
∴
2
k
k
=
⇒
k
k
=
⇒ 2
0
2
−
=
⇒
k
k
0
=
⇒ k
(2+3+3+4+3+3) 4.a) Find the characteristics roots of the matrix
−
3
0
0
5
1
0
4
3
2
b) Find local maximum/minimum value (if any) for the function f(x) = x3 – 12x2 + 36x + 17, 1 ≤ x ≤ 10. c) Evaluate
∫
+ )
1
(
x
4x
dx
Ans
∫
∫
+
=
+
=
)
1
(
)
1
(
4x
2x
4dx
x
x
x
dx
I
u
dx
x
Put
2=
du
dx
x
du
dx
x
2
1
2
=
⇒
=
⇒
say
J
du
u
u
u
du
u
u
u
du
u
u
u
u
u
u
du
I
=
+
−
+
+
=
+
−
+
=
+
=
∴
∫
∫
∫
∫
)
1
(
2
1
)
1
(
)
1
(
2
1
)
1
(
)
1
(
2
1
)
1
(
2
1
2 2 2 2 2 2 2 2v
u
Put
2+1
=
dv
du
u
dv
du
u
2
1
2
=
⇒
=
⇒
)
1
(
log
4
1
log
4
1
4
1
2
1
2
1
1
2
1
2 2+
=
×
∫
=
∫
=
=
+
∫
v
u
v
dv
v
dv
u
du
u
( )
{
1
}
log
4
1
log
2
1
)
1
log(
4
1
log
2
1
1
2
1
log
2
1
2 2 2 2 2+
=
−
+
=
−
+
−
=
∫
u
u
x
x
u
du
u
u
J
C
x
x
x
x
+
−
+
=
+
−
=
log(
1
)
2
1
log
2
1
)
1
log(
4
1
log
2
1
2 4 2 4d) Find the asymptotes of the Curse x2y – xy2 + xy + y2 + x – y =0. Ans
x
2y
−
xy
2+
xy
+
y
2+
x
−
y
=
0
The given equation is of 3
rddegree where the term
and
are absent. So, it is
possible to exist an asymptotes parallel to y-axis and x-axis.
3
y
x
3Equating the co-efficient of
x
2 with , we get:0
0
=
y
is a required asymptotesAgain the above equation can be written as:
(
1
)
20
2
y
−
x
−
y
+
xy
+
x
−
y
=
x
Equating the co-efficient of
y
2 with , we get:0
1
=
x
is another required asymptotes. Now, the given equation can be written as:0
)
(
x
−
y
+
xy
+
y
2+
x
−
y
=
xy
The asymptotes parallel to
x
− y
=
0
is:2
2
2 2 2 2 2 2+
−
=
+
−
=
−
+
+
+
−
=
−
+
+
+
−
∞ → ∞ → = ∞ →x
x
y
x
lt
y
x
x
x
x
x
x
lt
y
x
xy
y
x
y
xy
lt
y
x
x x x y x∴The required asymptotes is
x
− y
+
2
=
0
2
+
=
⇒
y
x
Thus,
y
=
0
,
x
=
1
,
y
=
x
+
2
are the required asymptotes.e) Verify the thpothesis and the conclusion of the Rolle’s theorem for the function f(x) = (x – 2)
x
on [0,2].(3+4+3+4+4) 5.
a) Write the equation of the ellipse 3x2 + 4y2 = 12 in standard form and sketch it. Clearly indicate its; center and vertices.
b) Evaluate:
)
2
1
4
4
(
2 2x
x
lt
x→−
+
−
(
)
−
−
−
=
−
+
−
=
−
−
−
=
−
+
−
→ → → →4
2
4
4
2
4
2
1
4
4
2
1
4
4
2 2 2 2 2 2 2 2x
x
lt
x
x
lt
x
x
lt
x
x
lt
x x x x(
)
(
)(
)
4
1
2
2
1
2
1
2
2
2
4
2
2 2 2 2
=
−
+
−
=
+
−
=
−
+
−
−
=
−
−
=
→ → →x
x
lt
x
x
lt
x
x
lt
x x xc) Show that the conic 9x2 – 24xy + 16y2 – 18x – 101y + 19 = 0 represents a parabola.
d) Evaluate
dx
x
x
+
1 0 23
∫
Say
I
dx
x
x
=
+
∫
1 0 23
u
x
Let
2+ 3
=
du
dx
x
du
u
dx
x
=
⇒
=
⇒
2
2
∫
=
∫
=
+
u
du
du
u
dx
x
x
3
2∫
]
2
3
3
2 3 2 3 1 0 2−
=
=
=
+
=
∫
dx
∫
du
z
x
x
I
3
0
2
z
1
x
e) Applying Leibnitz’s test to show that the series 1 -
...
4
1
3
1
2
1
+
−
+
is convergent (4+4+4+3+3) 6.a) Express the complex number
i
i
−
+
1
3
in polar form. Ans(
)
( )
( )( )
1
( )
1
3
1
3
1
3
3
1
1
1
3
1
3
2 2 2−
−
+
+
−
=
−
+
+
+
=
+
−
+
+
=
−
+
=
i
i
i
i
i
i
i
i
i
i
i
i
z
(
)
+
+
−
=
+
+
−
=
2
1
3
2
1
3
2
1
3
1
3
i
i
2
1
3
−
=
∴ x
and
2
1
3
+
=
y
( )
{
}
(
)
4
1
3
2
4
1
3
2
2
1
3
2
1
3
2 2 2 2 2 2=
+
+
=
+
+
−
=
+
=
x
y
r
2
4
8 =
=
−
+
=
−
+
=
−
+
=
=
− − − −3
1
1
3
1
1
tan
1
3
1
3
tan
2
1
3
2
1
3
tan
tan
1 1 1 1x
y
θ
2
5
4
10
4
10
tan
tan
6
4
tan
tan
6
tan
.
4
tan
1
6
tan
4
tan
tan
1 1π
π
1π
π
π
π
π
π
π
=
=
=
+
=
−
+
=
− − −
+
=
∴
12
5
sin
12
5
cos
2
π
i
π
z
b) Find the equation of the circle whose center is (1, 2) and which touches the line 3x+4y=1 c) If A =
and B = , then is it true that (AB)’ = -A’ B’?
−
−
1
2
1
1
−1
4
2
1
Ans
and
−
−
=
1
2
1
1
A
−
=
1
4
2
1
B
−
−
=
+
−
+
−
=
−
−
−
=
5
2
3
3
1
4
4
2
1
2
4
1
1
4
2
1
1
2
1
1
AB
( )
−
−
=
5
3
2
3
/AB
−
−
=
1
1
2
1
/A
−
−
=
−
1
1
2
1
/A
−
=
1
2
4
1
/B
−
−
=
−
+
+
−
−
−
=
−
−
−
=
−
3
3
2
5
1
4
2
1
2
4
4
1
1
2
4
1
1
1
2
1
/ /B
A
( )
/ / /B
A
AB
≠
−
∴
d) Use DeMoivre’s Theorem to show that
Cos
3
θ
=
4
Cos
3θ
−
3
Cos
θ
Anscos
3
θ
=
4
cos
3θ
−
3
cos
θ
By Demoivre’s Theorem:
(
cos
3
θ
+
i
sin
3
θ
) (
=
cos
θ
+
i
sin
θ
)
3θ
θ
θ
θ
θ
θ
θ
sin
3
cos
33
cos
2sin
3
2sin
2cos
3sin
33
cos
+
i
=
+
i
+
i
+
i
⇒
θ
θ
θ
θ
θ
θ
θ
sin
3
cos
33
cos
2sin
3
sin
2cos
sin
33
cos
+
i
=
+
i
−
−
i
⇒
(
θ
θ
θ
) (
θ
θ
)
θ
θ
sin
3
cos
33
sin
2cos
3
cos
2sin
sin
33
cos
+
=
−
+
−
⇒
i
i
θ
θ
θ
θ
cos
3
sin
cos
3
cos
=
3−
2∴
[
If
,
x
+
iy
=
a
+
ib
,
then
x
=
a
]
(
θ
)
θ
θ
θ
cos
33
cos
1
cos
23
cos
=
−
−
⇒
θ
θ
θ
θ
cos
33
cos
3
cos
33
cos
=
−
+
⇒
)
(Pr
cos
3
cos
4
3
cos
θ
=
3θ
−
θ
oved
⇒
(4+4+5+5)7.
a) Sketch the graph of the function y = Sin3x in [0,Π].
b) Find the area enclosed between the parabola y = 4x2 , the x axis and the lines x=1 and x =2. c) Find
2
|
|
0x
x
x−
+ →lt
Ans0
2
0
2
0
0
2
0=
=
−
=
−
+ →x
x
lt
x
≤
−
=
≥
=
0
,
0
,
x
when
x
x
x
when
x
x
d) Assuming the validity of the Macularin’s series expansion, find the first four terms of the function f(x) = ex Cosx.
B3.2-R3: BASIC
MATHEMATICS
Question Papers
July, 2004
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 1,
ω
,ω
2 are the cube roots of unity, then find the value of(1+
ω
)(1+ω
2)(1+ω
4)(1+ω
5)b. Find the inverse of the matrix
2 2 2 2 2 1 2 1 2
using the Guass-Jordan method
c. Using the binomial theorem, find the coefficient of x4 in the expansion of 10 2 3 3 − x x d. Find 3 1 5 lim + + + ∞ → x x x x
e. Derive a reduction formula for I , =
∫
x (logx) dxm,n areintegersn m
n m
relating Im,n and Im,n-1.
f. Evaluate the definite integral
∫
+ π 0 2 cos 1 x xdxg. Test for convergence, the series
∑
∞[
]
= − − + 1 4 4 1 1 n n n
h. Find the equation of the tangent to the parabola y2=4(x+1) which is
i. Find the projection of the vector 2i+3j-k along the vector 4i+j+2k. (3+4+3+3+2+4+3+4+2) 2.
a. Find all the characteristic roots (elgen values) and the corresponding characteristic vector (elgen vectors) of the matrix
−1 2 2 1 2 0 2 2 1
b. Show that the length of the segment of the tangent line to the curve
x=acos3t, y=asin3t,cut off by the coordinate axis is constant.
c. Find the area of the region bounded by y=|x+5|, x=-1, x=-6 and the x-axis.
d. Obtain the first four terms of the Taylor series of f(x)= x about x=2. Estimate the error if this series is used in the interval[2,3].
(5+5+4+4) 3.
a. Find the complex numbers, which satisfy both the equations 1 4 2 and 9 5 4 6 = − − = − − z z i z z b. If xyz = 1 and y x z x z y z y x
= 1, then find the value of x3+y3+z3.
c. Find the sides of a rectangle of greatest area that can be inscribed in the ellipse
4x2+9y2=36
d. Find the area of the region bounded by{(x,y):x2+y2≤16andx+y≥4}. e. Find a unit vector perpendicular to both the vectors 2i−3j+6k,i+ j+k
(4+2+5+4+3) 4.
a. If z=x+iy where x and y are variables, then find the locus represented
by the equation 1 1 1 = + − z z
b. Find the values of the parameters ka and a such that the system of equations a kx x x x x x x x x = + + − = + − = + − 3 2 1 3 2 1 3 2 1 2 2 3 2 3 2
has (i) unique solution, (ii) infinite number of solutions, (iii) no solution. c. Evaluate the integral
∫
[ ]
x dx2
0
2 where [x2] denotes the greatest integer function
at
x
2.d. Test for convergence, the series
∑
+ − xn n n 2 1 2 2e. A stone is dropped in quiet water. The water moves in circles. The radii of the circles are increasing at the rate of 0.2 cm/sec. Find the rate at which the area of a circle is increasing when radius is 5 cm.
(2+5+4+5+2) 5. a. 9 8 7 6 5 4 3 2 1 i log log log log log log log log log t determinan the of value the find then n, progressio geometric a form they and 9 ,..., 2 , 1 , 0 a If a a a a a a a a a i= >
b. Find the equations of the tangents to the ellipse 16x2+3y2=1, which are
perpendicular to the line 3x=4y+1
c. Using the DeMoivre's theorem, find the values of
(
1− 3i)
1/4.d. Examine whether the vectors i+2j+3k, 3i+4j+5k, 6i+7j+8k are linearly dependent or linearly independent.
e. Find a point on the curve
y
=
x
which is nearest to the point(2,0).(4+4+3+3+4) 6.
a. Find the limit
x e x x 1 0 lim sin3 − → .
b. Find the conic, which is represented by the equation
9x2-4y2+36x+8y-4=0
Hence, find its (i) centre, (ii) vertices, (ii) eccentricity c. Find the rank of the matrix
− − − − − − 13 11 8 11 1 7 2 1 5 1 4 3 2 3 1 2
d. Using vectors, find the unit normal to the plane containing the points
A(1,-2,3), B(2,1,0), C(3,2,1).
e. Evaluate the integral
∫
log[ 1−x+ 1+x]dx(2+4+4+3+5) 7.
a. Find the values of a and b such that the function
f(x)=x-3, for x ≥ 2 =ax+b, for 0 x ≤ 2
=-2x-1, for x < 0
is continuous for all x.
b. Prove that the feet of perpendiculars from the foci of the ellipse 1 2 2 2 2 = + b y a x
upon any tangent to this ellipse lie on the auxiliary circle. c. The following vectors are given:
. 2 2 3 c and 2 ,b i j k i j k k j i a= + + = − + = + +
Determing a vector dsuch that d•a=0 ,andd×b=c×b.
d. Find the intervals in which f(x)=sinx+sinx,0<x≤2
π
, is increasing or decreasing or neither increasing nor decreasing.e. Evaluate the integral
∫
2 / 0 ) log(tan ) 2 sin( π dx x x (3+4+4+4+3)
B3.2-R3: BASIC MATHEMATICS NOTE:
Time: 3 Hours Total Marks: 100
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.
1.
a) Using DeMoivre's theorem, find all the values of
z
=
1
+
3
i
. Ansz
=
1
+
3
i
=
(
1
+
3
i
)
12 We have,
+
=
+
=
+
3
sin
3
cos
2
2
3
2
1
2
3
i
i
π
i
π
1
2 13
2
sin
3
2
cos
2
+
+
+
=
k
π
π
i
k
π
π
Hence,(
)
2 1 2 1 2 13
2
sin
3
2
cos
2
3
1
+
+
+
=
+
i
k
π
π
i
k
π
π
+
+
+
=
3
2
2
1
sin
3
2
2
1
cos
2
12k
π
π
i
k
π
π
, wherek
=
0
,
1
,
2
b) Write the matrix A=
as the sum of a symmetric and a skew-symmetric matrix.
4
1
3
2
Ans Let =
