12N / 223
CLASS: B.Sc. MATHEMATICS
St. JOSEPH’S COLLEGE (AUTONOMOUS) TIRUCHIRAPPALLI – 620 002
SEMESTER EXAMINATIONS – NOVEMBER 2012
TIME: 3 Hrs.
MAXIMUM MARKS: 100
SEM
SET
PAPER CODE
TITLE OF THE PAPER
I
2012
11UMA130201
BASIC MATHEMATICS
SECTION – A
Answer all the questions:
20 x 1 = 20
Choose the correct answer:
1. If y  e ax , then yn is ______.
a) eax
c) enax
b) a eax
d) an eax
2. The expansion of tan is _______.
2
4
3
5
a)
b)

2

2


 ...
 
 ...
2
15
3
15
3
5
2
4
c)
d)

2

2
 
 ...


 ...
3
15
2
15
3. The value of log 2 is _______.
a)
1 1 1
1     ...
2 3 4
c) 1  2 x  3x 2  ...
b)
1 1 1
1     ...
2 3 4
d) None of these
4. The expansion of ( 1  x ) 2 is _______.
a) 1  2 x  3x 2  4 x 3  ...
b) 1  2 x  3x 2  4 x 3  ...
2
3
c)
d) None of these
x x
x
1 
  ...
2! 3! 4!
5. The normal equation of a straight line is
a) p = r cos
b) p = r sin
c) p = r cos( - )
d) p = r sin( - )
Fill in the blanks:
6.
7.
The Cartesian formula for radius of curvature is ______.
8.
The expansion of ( 1  x )n is ______.
The general value of Log( x  iy ) is ______.
n
9.
 1
The value of lt 1   is _______.
n
n 
10. If the pole lies on the circumference of the circle c = a then the
equation of the circle is ______.
State True or False:
n
11. The nth derivative of sin(ax + b) is sin   ax  b 
 2

12. The expansion of cos4 is 8cos4 - 8cos2 + 1.
n
r nr
13. The general term of the expansion of ( x  a ) is u r 1  nc x a .
r
x
e e
14. The expansion
2
x
3
5
x
x
is x  
 ...
3! 5!
l
15. The polar equation of the conic is  1  e cos  .
r
Answer in one or two sentences:
16. Define envelop of a curve.
17. Prove that tanh 2 x 
2 tan hx
2
1  tanh x
.
18. Find the coefficient of xn in the expansion of
( 1  2 x  3x 2  4 x 3  ...)n
1
1  x 
log
.
2 1 x 
19. Write the expansion of
20. Define a conic.
SECTION – B
Answer all the questions:
5 x 4 = 20
21. a. Find the envelop of the family of straight line y  mx  a ,
m
where m is a parameter.
OR
b. Prove that the radius of curvature at any point of the cycloid

x = a( + sin) and y = a(1 – cos) is 4acos .
2
22. a. Expand sin 3  cos5  in a series of sines of multiples of .
OR
b. If sin( A  iB )  x  iy , prove that (i)
x
2
2

y
2
1
2
sin A cos A
(ii)
x
2
2
cosh B

y
2
2
 1.
sinh B
23. a. Find the coefficient of xn in the expansion of
OR
x 1
2
( x 1) ( x  2 )
.
13 2
when
b. Find the greatest term in the expansion of ( 1  x )
2
x .
3
24. a. Show that the coefficient of xr in the expansion of
r r 1
( 1 ) 2
2  3x
e
( 4  3r )
r!
2x
is
.
OR
b.
Show that loge 2 
1
1
1
2
3
(loge 2 )  (loge 2 )  ...  .
2!
3!
2
25. a. Show that in a conic the semi – latus rectum is the harmonic
mean between the segments of a focal chord.
OR
b.
Find the equation of the directrix of the conic
l
 1  e cos  .
r
SECTION – C
Answer any FOUR questions:
4 x 15 = 60
m
26. If y   x  1  x 2  ,prove that


(1  x 2 ) y
27. (i)
n 2
Express
 ( 2n  1 )xy
n 1
 ( n2  m2 ) y  0
sin 6
in terms of cos.
sin 
n
(10)
sin  5045
, show that  = 1°58’ approximately.


5046
sin 
 1 as   0 .  is measured in radians.
(5)

15 15.21 15.21.27
Sum the series to infinity


 ... (7+8)
16 16.24 16.24.32
(ii) If
28. (i)
 3x 
(ii) Find the first negative term in the expansion of 1  
4

23 4
.
2
29. (i)
1 3 1 3  3
Sum the series to infinity 1 

 ... .
2!
3!
(8)
(ii) If a,b,c denote three consecutive integers, show that
1
1
1
1
1
loge b  loge a  loge c 

 ... .
3
2
2
2ac  1 3 ( 2ac  1 )
30. (i)
Derive the polar equation of a conic.
(7)
(ii) A circle passing through the focus of a conic whose latus
rectum is 2l meets the conic in four points whose distances
from the focus are r1, r2, r3 & r4. Prove that
1 1 1 1 2
(8)
    .
r r r r
l
1
2
3
4
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