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Semester B.Sc. Mathematics III UNIVERSITY OF CALICUT

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Semester B.Sc. Mathematics III UNIVERSITY OF CALICUT
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
(2014 Admn. onwards)
III Semester
B.Sc. Mathematics
CORE COURSE
CALCULUS AND ANALYTICAL GEOMETRY
Question Bank & Answer Key
1.
2.
3.
ln ln (1) = ........................
a) 0.2010
b) 0
ln x
dx = ..........................
x
(ln x) 2
1
a) 2  c
b)
c
2
x
 tan x dx
ln x
c
2
d)
1
 ln x  c
x
= ......................................
b) ln sec x  c
b)
c) sec2x+c
d) ln sin x  c
1
kx
1
x
d)
k
ln x
c) ax log a
d)
ax
log a
c)
ln a
ax
d)
a x 1
x 1
c)
ln 2
2
d)
2
ln 2
c)
d x
(a ) = .......................................
dn
a) xa x 1
6.
c)
d
(ln kx) = .................................
dn
a) kx
5.
d) 

a) ln cos x  c
4.
c) 1
a
x
b) ax
dx = ........................................
ax
a)
ln a
b) ax ln a
2
7.
 x2
x2
dx = .........................
1
a) 2
b)
1
ln 2
School of Distance Education
8.
Range of sin hx is ..........................
b) (, )
a) [1, 1]
9.
sin hx =
a)
11.
b) (, )
1
4
b)
5
4
c)
b) sec hx tan hx
1
 x
1
tanh x
b) tanh 1  
x 1
b) +1
x0
Lim
x0
b) 0
1
 x
c) coth 1  
d)
x
x 1
d)
1
1
tanh 1  
 x
c)
4
2x
x4 1
c) 0
d) 2
c) 
d) 
c) ln (ab)
d) 
c) (,)
d) (0, )
ax  bx
= ........................................
x
a
b
b
a
b) ln  
Range of tan hx is ...................................
a) [1, 1]
18.
c)  sec hx tan hx d) tan2 hx
sin x
= .........................
x2
Lim
a) ln  
17.
3
5
1 x
= ...............................
ln x
Lim
a) 1
16.
d)
 
d
cosh 1 x 2 = ...................................
dx
2x
a) 2 x sinh 1 x 2
b)
x2 1
a) 1
15.
4
5
cot h1x = .............................
 
14.
d) (, 1)
d
(sec hx)= .................................
dx
a)
13.
c) (, 1]
3
. Then cos hx is ............................
4
a) sec h2x
12.
d) (0, )
Range of cos hx is ...............................
a) [1, 1]
10.
c) (, 0]
b) (1, 1)
cos hx is ........................ function.
a) Odd
b) Even
c) Fluctuating
d) Neither even nor odd
CALCULUS AND ANALYTICAL GEOMETRY
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19.
20.
tan hx is .................... function.
a) Odd
b) Even
c) Neither even nor odd
d) None of these
Lim x cot x = ........................
x0
a) 0
21.
22.
Which of the following is not an indeterminate form
a) 1
b) 10
c) 0
lim
x 
c) 
b) 1
27.
b) 0
c) 1
d) None of these
The sequence {(1)n+1} is
a) Converges
b) Diverges
c) Has a convergent subsequence
d) None
an =
2n  1
. The Lim an = .............................
n 
3n  5
1
5
b)
2
5
c)
2
3
d)
3
2
What is wrong about the sequence {(1)n}
b) Converges
d) Absolutely convergent
1
n
The sequence{ } is
a) Diverges
30.
d) 7
 n 

 3 
a) Diverges
c) Increasing
29.
c) 6
Third term of the sequence (1)n+1 sin 
a)
28.
b) {a(n) = 1, nN}
d) {a(n) = 2n, nN}
b) 3
a) 1,
26.
d) None of these
a1= 2, an+1 =an+3. Then a2= .........................
a) 5
25.
d) 
Example of a constant sequence is ........................
a) {a(n) = n, nN}
c) {a(n) = 2+n, nN}
24.
d) 1
ln x
= ..........................
x
a) 0
23.
c) 
b) 1
b) Increasing
c) Decreasing
d) None of these
The sequence {(1)nn} is
a) Converges
CALCULUS AND ANALYTICAL GEOMETRY
b) Bounded above c) Bounded below d) Not bounded
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31.
32.
Which is true?
A) Every bounded sequence is converged
B) Every converged sequence is bounded
a) A and B are true
b) A is true b is false
c) A is false B is true
d) Both are false
Lim an = 2, Lim bn = 4, then Lim (an + bn) = ...................
n
n 
a) 6
33.
Lim
n
Lim
n
b) 1
c) 1
d) 
b)
2
3
c) 2
d) 
b)
3
2
c) 1
d) 
b) 1
c) 
d) None of these
b) 1
c) 
d) None of these
ln n
= ................
n
a) 0
37.
d) 16
3n
= ................
n2
a) 3
36.
c) 8
2n
Lim
= ................
n n  1
a) 0
35.
b) 10
 n 1
Lim 
 = ................
n
 n 
a) 0
34.
n 
Lim n
1
n
n
= ................
a) 0
1
38.
Lim x
c) 
d) None of these
39.
a) 0
b) 1
n
If |r |> 1, Lim r = ................
a) 0
c) 
d) None of these
c) 
d) None of these
c) e
d)
1
e
d)
1
24
40.
n
n
(x > 0) = ................
n
b) 1
If |r |<1, Lim rn = ................
n
a) 0
41.
Lim (1+
n
b) 1
1 n
) = ................
n
a) 1
42.
b) 0
 (1) n 1 
is
n 1 
 2

Fourth term of the sequence 
a)
1
25
CALCULUS AND ANALYTICAL GEOMETRY
b)
1
25
c)
1
24
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43.
Which of the following series diverges?

1
a)  n
n 0 2
1
b) (1)
n
n
1
c) 
n
2
d)   
3
n

44.
The series
a
n 1
n
converges, then
a) Lim an = c, a constant
b) Lim an = 0
c) Lim an = 
d) Lim an does not exist
n
n
n
n
n 1
The series 
n
n 1

45.
b) Converges to 1
d) None of these
a) Converges to 1
c) Diverges

46.
The series
n 1
47.
1
n
p
a) Converges for P  1
b) Diverges for P  1
c) Converges for P >1
d) Diverges for P > 1
Let {an} of {bn} is such that an  bn. Then
a) an converges if bn converges
c) an converges if bn diverges
48.
Let an> 0 and bn > 0 and Lim
n
a) l = 0
49.
an
= l. Then an and bn converges if
bn
c) l = 
b) l > 0
Let an> 0 and bn > 0 and Lim
n

n
n 1
n
is
1
b) Converges to 0
c) Diverges
d) None of these
a
an be a series such that Lim n 1 = l. Then
n  a
n
a) an converges if l > 1
c) an converges if l < 1
52.
b) bn converges if an converges
d) bn converges  an converges
3
a) Converges to 1
51.
d) None of these
an
= 0. Then
bn
a) an converges if bn converges
c) Both an and bn converges
50.
b) bn converges if an converges
d) bn diverges if an diverges
b) an diverges if l > 0
d) an diverges if l = 0
an be a series such with an  0 Lim (an) 1/n = l. Then
n 
a) an converges if l < 1
c) an converges if l =1
CALCULUS AND ANALYTICAL GEOMETRY
b) an diverges if l < 1
d) an diverges if l = 1
Page 5
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53.
54.
Let {an} be a sequence of positive terms such that an  an+1 and Lim an= 0, then
n
a) an converges
b) (1)nan converges
c) an converges but (1)nan diverges
d) None of these
If the series an(xc)n converges for x = k. Then it converges in
a) |r  c|< k
b) |r  c|  k
c) |r  c|> k
d) |r  c|  k
55.
If a series converges absolutely for all x. Then its radius of converges if
a) Finite
b) Infinite
c)Cannot be determined
d) None of these
56.
The radius of convergence of the power series

 (1)
n 1
n 1
a) 0
b) 1
xn
is .........
n
d) 
c) 2

57.
The radius of convergence of the series
x
n
is ..........
n 0
a) 1
b) 2
c)
2
d) 

58.
The series
x
n
is
n 0
a) Converges absolutely for | x | < 1
c) Has radius of converges ½
59.
Coefficient of (x 1)2 in the Taylor series expansion of f(x) =
a) 1
60.
b) ½
b)  ½
c) ½
d) 1
Coefficient of x in the Maclaurin’s series expansion of f(x) = cos x is
b) 0
c) 1
d) 2
c) (2, 0)
d) (1, 0)
Focus of the parabola y2 = 8x is
a) (0, 2)
63.
d) +1
3
a) 1
62.
c)  ½
1
at x = 2 is
x 1
Coefficient of x2 in the Maclaurin’s Series expansion of f(x) = ln (x +1) is
a) 1
61.
b) Converges for | x | > 1
d) None of these
b) (0, 1)
Equation of the directrix of the parabola y2 = 8x is
a) x = 2
b) y = 2
c) x =  2
d) y = 2
64.
Equation of the directrix of the parabola x2 = 16y is
65.
a) x = 4
b) y = 4
c) x =  4
d) y = 4
The vertex of the parabola whose focus is (1, 1) and whose directrix passes through
(3, 3) is
a) (2, 1)
CALCULUS AND ANALYTICAL GEOMETRY
b) (2, 1)
c) (1, 2)
d) (1, 2)
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66.
Focus of the ellipse
a) ( 3 , 0)
67.
69.
70.
5
3
9x2 9 y2

=1
20
16
b)
x2
y2

=1
4
2
b)
9x2 9 y 2

=1
16
20
b) y2  x2 = 8
5
4
2
x2
y
Foci of the hyperbola

= 1 is
2
8
d) x = 
2
y
3
d) x = 
4
y
3
c) y2  x2= 1
c)
d)
y2
x2

=1
2
4
3
2
16 x 2 20 y 2

=1
9
9
d) None
c)
x2
y2

=1
8
1
d)
x2
y2

=1
8
1
7
5
6
5
b)
c)
d)
7
4
b) (0,  10 )
c) ( 5 ,0)
d) (0,  5 )
The angle to be rotated so that the xy term in the equation 5x2  6xy + 5y2= 8 can
remove is
a)
76.
4
7
d)
Eccentricity of the hyperbola 9x2  16y2 = 144 is
a) ( 10 , 0)
75.
c)
Equation of hyperbola with one focus at (4, 0) and corresponding directrix x = 2 is
a)
74.
7
4
Equation of hyperbola with foci (2, 0) and eccentricity
a) x2  y2 = 8
73.
b)
Equation of the hyperbola with foci (0,  2 ) and asymptote y = x
a)
72.
d) (0, 1)
c) (1, 0)
4
3
2
x
y2
Equation of the asymptotes to the hyperbola

= 1 is
16 9
3
3
4
a) y =  x
b) x =  y
c) y =  x
4
4
3
2
2
x
y
Equation of the asymptote to the hyperbola

= 1 is
9 16
3
4
4
a) y =  x
b) x =  y
c) y =  x
4
3
3
a) x2  y2 = 1
71.
b) (0,  3 )
Electricity of the ellipse 16x2 + 9y2 = 144 is ............
a)
68.
x2
y2
+
= 1 is
3
2

6
b)

4
c)

2
d)
2
3
The equation Ax2 + Cy2 + Dx + Ey +F = 0 represent a ellipse if
a) (A > 0 and C > 0)
b) (A > 0 and C < 0)
c) (A < 0 and C > 0)
d) (A  0, C  0)
CALCULUS AND ANALYTICAL GEOMETRY
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77.
The equation Ax2 + Cy2 + Dx + Ey +F = 0 represent a circle if
a) (A = 0, C  0)
b) (A  0, C  0)
c) (A  0, C = 0)
78.
d) None
Ax2 + Cy2 + Dx + Ey + F = 0 represent
a) Parabola if AC  0
c) Ellipse if AC < 0
79.
x2 3xy + y2 x = 0 represent
a) Parabola
80.
b) Ellipse
b) Ellipse
c) Hyperbola
d) Circle
b) (x = a cos t, y = a sin t)
d) None
Tangent to the curve, x = 4 sin t, y = 2 cos t at t =
a) 
83.
d) Circle
Which of the following represent parametric equation of a circle
a) (x = a cos t, y = b sin t)
c) (x = a sec t, y = b tan t)
82.
c) Hyperbola
x2 +2xy + y2 + 2x  y + 2 = 0 represent
a) Parabola
81.
b) Ellipse if AC > 0
d) Hyperbola if AC > 0
1
2
b)
1
2

is
4
c)  1
d) 1
Length of the circle x = cos t, y = sin t in the first quadrant is
a) 3.14
b) 1.57
c) 6.25
d) none
2
84.
Find
d y
if x = cos t, y = sin t
dx 2
a) cosec2 t
85.
86.
87.
d) cosec3 t

) is
4


a) (3, )
b) (3, 9 )
4
4
5
Cartesian equation of r =
is
sin   2 cos 
c) (3, 9
a) x = 2y +5
c) y = 5x+2
d) x = 5x+2
c) Origin
d) The line  =
b) y = 2x+5

)
4
d) (3, 5

)
4
The equation r2 = sin 2 is symmetric about
b) y – axis

4
Which of the following represent equation of a circle?

4

Cartesian equation of r sin ( + ) = 2 is
6
a) 2 x + y = 4
b) 3 x + y = 4
a) r = 2
89.
c) cosec3 t
The polar co-ordinate equal to (3,
a) x – axis
88.
b)  cosec2 t
CALCULUS AND ANALYTICAL GEOMETRY
b)  =
c) r2 = 1
d) r sin  = 1
c)
d)
3y + x = 4
3y + x = 2
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90.

Which of the following point lie on the curve r =  sin  
3
1 
)
2 2
1 3
)
2 2
a) ( ,
91.

)
2
b) (1,
d) None

)
4
d) (1, 0)
c) r = 2 cos 
b) r = 4 sin 
d) r = 2 cos 

) and passing through origin is
2
c) r = 4 cos 
d) r = 4 cos 
Polar equation for the hyperbola with eccentricity 5 at direction t =  6 is
6
1  5 cos
b) r =
6
1  5 sin 
The polar equation r =
12
represent
3  3 sin 
a) Parabola
b) Ellipse

96.
c) (1,
Polar equation of the circle with centre at (2,
a) r =
95.

)
2
b) r = 4 cos 
a) r = 4 sin 
94.
3
)
2
Polar equation of a circle passing through origin, radius 2 and centre on +ve x-axis is
a) r = 4 sin 
93.
c) (1,
One of the point of intersection of the Cardiods r = 1 + cos , at r =1- cos , is
a) (0,
92.
b) ( ,
c) r =
30
1  5 sin 
d) r =
30
1  5 sin 
c) Hyperbola
d) None of these
c) 2 ln 2
d)
6
 tan 2 x dx = ............
0
1
ln 2
a)
97.
b)
b) e
c)
1
e
d) 
b) x
c)
1
ex
d)
ln (ex) = .....................
a) ex

99.
2
ln 2
ln1 1 = ..............
a) 1
98.
1
ln 2
2
1
x
2
e
sin x
cos x dx = ...............
0
 1
2
a) e
b) e 1
c) e ½ 1
d) 1
b) a
c) xa
d) x
c) 
d) x
x
100. loga (a ) = ..................
a) ax
1
101. sin x + cos
1
x = .................
a) 1
CALCULUS AND ANALYTICAL GEOMETRY
b)

2
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School of Distance Education
1
102.
1
e tan x
0 1  x 2 dx = ..................


a) e 4 1
103.
dx
 7  3x
a)
2
c) 1 e

d) e
4
4
= .....................
1
tan 1
2
104. (1)n

b) e 2 1
7x
3
1
tan 1
2
b)
3x
7
c)
1
tan 1
7
2x
3
d)
None
1
is ...............
n
a) Converges
b) Diverges
c) Absolutely convergent
d) None
dy
= ..............
dx
105. Let y = x sin hx cos hx then
a) x cos hx
106. ln [x +
c) x cos hx + x
d) x cos hx  sin hx
c)tan h 1 x
d)sec h 1 x
b) 0
c) 1
d) sec h2x tan hx
b) 1
c) 
d) None
b) x sin hx
x 2  1 ] = .............
a) cos h 1 x
b) sin h 1 x
107. y = sec h2 x + tan h2 x then
a) 2 sec hx tan hx
dy
= ..............
dx
1
108.
 1 n
Lim  2  = ..............
n
n 
a) 0
109. Length of Latus Rectum of the ellipse
a)1
b)2
c)
2
x2 y2
+
= 1 is
4
2
d)4 2
110. Which of the following function is odd?
a) f(x) = x
111. If f(x) =
112.
a2  1
ax 1
5x  4
b) sin x + cos x
, if x  2
2( x  1) , if x 2
2
x2
b) 6
Lim
1
is equal to ............
1 x
a) 0
CALCULUS AND ANALYTICAL GEOMETRY
d)None
Lim f(x) = .............
a) 0
x 1
c) log(x + 1  x 2
b) 
c) 14
d) Does not exist
c) 
d) Does not exist
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School of Distance Education
113. y = sin (sin x). Then
dy
 cos (sin x) cos x = .............
dx
a) 0
b) cos x
c) sin x
d) 1
114. For differentiability of a function continuity is ............
a) Sufficient
b) Necessary
c) Sufficient and necessary
d) None of these
115. The function f(x) = | x | is ..............
a) Continuous at x = 0
b) Discontinuous at x = 0
c) Differentiable at x = 0
d) Not differentiable at x =0
116. The series 1 
1 1 1
+  + .... is ..............
2 4 8
a) Convergent
b) Absolutely convergent
c) Divergent
d) All of the above
117. In a series of +ve terms Un if Lim Un  0 then the series
n
a) Convergent
b) Divergent
c) Not converged d) Oscillatory
118. The series nm xn is converged if ..............
a) x > 1 and x = 1 when m <  1
b) x > 1 and x = 1 when m >  1
c) x < 1 and x = 1 when m <  1
d) x < 1 and x = 1 when m >  1
119. The series 1 +
1
1
1
+
+
+ .... is
2
3
4
a) Convergent but not absolutely
b) Oscillatory
c) Divergent
d) Absolutely convergent
120. x= asect y=btant is the parametric representation of
a) Parabola
CALCULUS AND ANALYTICAL GEOMETRY
b) Ellipse
c)Hyperbola
d)Circle.
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School of Distance Education
Answer Key
1.b,
2.b,
3.b,
4.c, 5.c,6.a, 7.b, 8.b, 9.c, 10.b, 11.c, 12,b, 13.d, 14.a, 15.c, 16.a, 17.b, 18.b,
19.a, 20.b, 21.b, 22.a, 23.b, 24,a, 25.c,26.b,27.c,28.c,29.c,30.d,31.c,32.b,33.c,34,c , 35.d,
36.a,37.b,38,b,39.c,40.a,41.c,42.b,43.c,44.c,45.c,46.c,47,a,48.a,49.a,50.c,51.c,52.a,53.b
54.b,55.b,56.b,57.a,58.a,59.b,60.b,61.b,62.c,63.d,64.b,65.b,66.c,67.b,68.a,69.c,70.c,
71.a,72.a,73.b,74.a,75.b,76.a,77.b,78.b,79.c,80.b,81.b,82.a,83.b,84.d,85.c,86.b,87.b,88.
a,89.c,90.a,91.b,92.b,93.a,94.c,95.a,96b,97.b,98.b,99.b,100.d,101.b,102.a,103.a,104.c,
105.d,106.a,107.d,108.b,109.d,110.d,111.b,112.d,113.a,114.b,115.d,116.b,117.b,118.c,
119.c,120.c
Prepared by:
Aboobacker P
Assistant Professor,
Department of Mathematics,
WMO Arts & Science College, Muttil
Wayanad 673122
Scrutinised by :
Dr.D.Jayaprasad,
Principal, Sreekrishna College, Guruvayur
Chairman, Board of Studies in Mathematics (UG)
CALCULUS AND ANALYTICAL GEOMETRY
Page 12
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