QUESTION BANK
COMPLEX NUMBERS
Q1.
If w is a complex cube root of unity, prove that 1  w  w2  0 .
Q2.
Find the 7 roots of unity and show that the sum of their nth power always
vanishes unless n be a multiple of 7.
Q3.
Use De’ Moivre’s Theorem to solve the following equations:(i ) x 5  1  0
(ii ) x 7  1  0
(iii ) x 4  x 3  x 2  x  1  0
(iv) x 4  x 3  x 2  x  1  0
Q4.
Solve the equation x12  1  0 and find which of its roots satisfy the equation
x4  x2  1  0 .
Q5.
  
If u = log tan    prove , (i) tan h
 4 2
Q6.
Show that the roots of the equation (1  x)2 n  (1  x)2 n  0 are given by
i tan
Q7.
Q8.
Q9.
(ii) cosh  = sec 
(2r  1)
, where r=1, 2, 3, …., n.
4n
If sin  + sin  + sin  = 0 = cos  + cos  + cos , prove that
(i)
cos 3 = 3 cos ( +  + )
(ii)
 sin 3  = 3 sin ( +  + )
(iii)
 cos ( + ) =  sin ( + ) = 0
If x + iy = tan (A + iB); prove that
(i)
x2 + y2 + 2x cot 2A = 1
(ii)
x2 + y2 – 2y cot 2B + 1 = 0
Find the equation whose roots are
2 cos
Q10.

u
  = tan
2
2
2
4
6
, 2 cos
, 2 cos
7
7
7
If tan ( + i) = tan  + i sec ; show that
(i)
2 = n +

+
2
e2 = 
(ii)
Q11.
cot 
2
Show that multiplication of a complex no. by i corresponds to a counter clock
wise rotation of the corresponding vector through the angle
Q12.

.
2
Show that the equation whose roots are the nth powers of the roots of
x2 – 2x cos  + 1 = 0 is given by x2 – 2x cos n + 1 = 0
Q13.
If  and  are roots of x2 – 2x + 4 = 0, prove that n + n = 2n+1 cos
Q14.
Find the real and imaginary part of
sin-1 (cos  + i sin)
Q15.
 1  sin   i cos  
Show that 

 1  sin   i cos  
x




= cos x      i sin x   
2

2

Q16.
Prove that
128 sin3  cos5  = – sin 8 – 2sin 6 + 2sin 4 + 6sin 2
Q17.
Find the values of (–i)1/6
Q18.
Prove that i i  e
Q19.
Show that log(1  i tan  )  log sec   i
Q20.
Prove that

  4 n 1
2
(n = 0, 1, 2)
(1  i)1i
 sin(log 2)  i cos(log 2)
(1  i)1i
n
.
3
CONVERGENCE, DIVERGENCE OF AN INFINITE SERIES (RATIO, ROOT,
LOGARITHMIC TEST
x3 x5 x7
       is absolutely egt if –1 < x < 1.
3
5
7
Q1.
Prove that the series x 
Q2.
Examine the convergence of the series
Q3.
Test the convergence of
1
2 1
Q4.

x2
x4
x8
…..


3 2 4 3 5 4
Test the convergence of the series

1 
 1  n 
Q5.
2 1
3 1
4 1
 3
 3
 .......
3 1 4 1 5 1
3
 n3 2
Test the following series for absolute convergence
sin x sin 2 x sin 3 x



1
2
3
Q6.
Examine the series for convergence
a  x a  2 x  a  3x 


..........
L1
L2
L3
2
3
Q7.
Find the real and imaginary part of sin-1(cos  + isin)
Q8.
Find the interval of convergence of series
x
x 2 x3 x 4
……..


2
3
4

Q9.
 n log x
n 1
Q10.
Examine the convergence of the series
Q11.
Test for the convergence of the series
n2  1
 n3  1
1 2 3
n
   .......... 
 ..........
2 3 4
n 1
Q12.
Test convergence

(i)
sin 2 n

2n
n 1
(ii)
 n


1
1/ n
n
n 1
Q13.
Test the convergence of the series:

(i)
sin 2 n

2n
n 1
(ii)
 n

1/ n

1
n
n 1
Q14.
Test for cgce
n 1
(a)

(b)
 (1)
n 2
n 1
3
n
xn
log n
n2
Q15. Test for cgce

x
n 1
n
1
( x  0)
 xn
( x  0)
SUCCESSIVE DIFFERENTIATION
LEIBNITZ THEOREM (WITHOUT PROOF)
x3
, (n  1)
( x  1)(2 x  3)
Q1.
Find the nth derivative of
Q2.
x3
Prove that the value of the nth derivative of 2
for x = 0 is zero when n is
x 1
even and {-Ln} when n is odd and > 1.
Q3.
If cos1 ( y / b)  log( x / n)n , show that x 2 yn  2  (2n  1) xyn 1  2n 2 yn  0
Q4.
If y 
sin 1 x
1 x
2
, prove that (1  x 2 ) yn 1  (2n  1) xyn  n 2 yn 1  0 and hence find
( yn ) 0
Q5.
If y = (x2 -1)n prove that (x2 – 1) yn+2 + 2xyn+1. Hence prove if Pn =
show that

d n ( x 2  1) n
dx n

d 
2 dPn 
 1 x
  n(n  1) Pn  0
dx 
dx 
Q6.
Determine yn(0) if y = emsin-1x
Q7.
if y1/ m  y 1/ m  2 x, prove that ( x 2  1) yn  2  (2n  1) xyn 1  (n 2  m 2 ) yn  0
Q8.
1 x 
If y = tan-1 
 find yn
1 x 
Q9.
If f(x) = m cos-1 x find fn(0) when n is even.
Q10.
If y = easin-1x, prove that
(1 – x2) yn+2 – (2n + 1) yn+1 x – (n2 + a2)yn = 0
Q11.
If y = sin-1 x prove that
(1 – x2) yn+2 – (2x + 1)x yn+1 – x2 yn = 0
Q12.


n
If y = x  x 2  1 , prove that
d2y
dy
(1  x ) 2  x  x 2 y  0
dx
dx
2
CURVATURE AND ASYMPTOTES
Q1.
Find the radius of curvature at the point(3a/2, 3a/2) on the curve x3 + y3 = 3axy.
Q2.
Find the radius of curvature at the point (x, y) on the curve xy=c2
Q3.
If  and ’ be the radii of curvature at the extremities of two conjugate diameters
of an ellipse, prove that (2/3 + `2/3) (ab)2/3 = a2 + b2.
Q4.
Prove that the radius of curvature at any point (x, y) of the curve x2/3 + y2/3 = a2/3
is three times the length of the perpendicular from the origin to the tangent at (x,
y).
Q5.
Prove that for the ellipse x=acos t, y = bsin t  
a 2b 2
, where p is the
p3
perpendicular from centre upon tangent at (acost, bsint).
Q6. Find all the asymptotes of curve
(i)
x2y2 (x2 – y2) = (x2 + y2)3
(ii)
y3 – xy2 – x2y + x3 + x2 – y2 = 1
Q7.
Find the asymptotes parallel to axes for the curve
Q8.
Find the asymptotes of curve
(i)
x3 + x2y – xy2 – y3 – 3x – y – 1 = 0
(ii)
y3 + x2y + 2xy2 + y + 1 = 0
a2 b2

1
x2 y2
Q9.
Find asymptotes parallel to the axes of curve y2 x – a2 (x+a) = 0.
Q10.
Find the curvature of x=4 cos t, y = 3 sin t. at what points on the ellipse does the
curvature have greatest and least values. What are magnitudes.
Q11.
If ρ is the radius of curvature at any point P on the parabola y 2  4ax and S its
focus then show that  2 varies as SP 3 . Also, show that the radius of curvature at
the vertex is equal to the length of the semi-latus rectum.
Q12.
The
tangents
at
two
points
P
and
Q
on
the
cycloid
x  a(  sin  ), y  a(1  cos  ) are at right angles. If 1 and 2 are the radii of
curvature at these points then show that 12   2 2  16a 2 .
Q13.
Find the point on the curve y  e x at which the curvature is maximum and show
that the tangent to the curve at that point forms with the coordinate axes, a triangle
whose sides are in the ratio 1: 2 : 3 .
MACLAURIN’S & TAYLOR’S SERIES
ERROR AND APPROXIMATION
CURVE TRACING
Q1.
Calculate the approximate value of
10 to four places of decimal by taking the
first four terms of an appropriate expansion.
Q2.
Find the change in total surface area of a right circular cone when the altitude is
constant and the radius changes by r.
Q3.
If A is the area of a  having sides equal to a, b, c and s is the semi-perimeter,
prove that the error in A resulting from a small error in measurement of c is given
by A 
Q4.
1 1
1
1
1 
A 


C
4 s s  a s  b s  c
A soap bubble of radius 2cm shrinks to radius 1.9cm. Estimate the decrease in
(i) Volume
Q5.
(ii) Surface area
Apply Maclaurin’s theorem to prove that
e ax sin bx  bx  abx 2 
b(3a 2  b 2 ) 3
x  .......
6
Q6.
 11 
Apply Taylor’s theorem to find f   is f(x) = x3 + 3x2 + 15x - 10
 10 
Q7.
Show that sin 1 x  x 
12 3 12.3 2 5
x 
x  ....... and hence find approximate value
6
120
of .
Q8
Prove that e a sin
1
x
 1  ax 
show that e  1  sin  
(ax) 2 a(12  a 2 ) 3
(2 2  a 2 ) 4

x  a2
x  ...... and
2
6
24
sin 2  2 sin 3 

 ......
2
6
Q9.
Using Maclaurin’s series, give the expansion of sin-1x and sin x.
Q10.
which trigonometric function cannot be expanded by Maclaurin’s Theorem?
Q11.
Trace the curve x2/3 + y2/3 = a2/3
Q12.
Trace the curve x 
Q13.
Trace the curve r=a+bcosθ.
3at
3at 2
,
y

1 t3
1 t3
Q14.
Trace the Folium of Descartes x3  y 3  3axy .
Q15.
Trace the curve y 2 ( x  a)  x 2 (3a  x) .
Q16.
Find the asymptotes of the following curves:(i )
x3  2 x 2 y  xy 2  2 y 3  4 y 2  2 xy  y  1  0
(ii ) x 3  4 y 3  3 x 2 y  y  x  3  0
(iii ) x 3  5 x 2 y  8 xy 2  4 y 3  x 2  3xy  2 y 2  1  0
(iv) 4 x 3  3 xy 2  y 3  2 x 2  xy  y 2  1  0.
REDUCTION FORMULA
Q1.
Derive the reduction formula for  sec n xdx , Use to find  sec 6 xdx

2
Q2.
If I n   tan n x dx , show that In + In-2 =
0
1
and deduce I5.
n 1
 /2
Q3.
 sin
If Im,n =
m
x cos n x dx ; prove that Im,n =
0
n 1
Im, n-2; where m, n I.
m 1
 /2
 sin
Evaluate
6
x cos 6 x dx
n
 d , prove that u n 
0
 /2
Q4.
If u n 
  .sin
0
n 1
1
u n  2  2 . Hence find u3.
n
n
 /2
Q5.
Using properties of definite integral show that
 log sin x
dx 
0
Q6.
Evaluate
2
m 1 n 1
 x y dxdy over the positive quadrant of the ellipse
 /4
Q7.


 log 1  tan   d  8
log e 2 prove.
0
 /2
Q8.
 
If un   x sin x dx , n > 1, prove that un + n(n-1) un-2 = n 
2
0
n

Q9.
Show that
x
f (sin x) dx 
0
 /2
Q10.
Evaluate

0
dx
4  5 sin x


2 0
f (sin x) dx
n 1
log
1
2
x2 y2

 1.
a2 b2
AREA UNDER CURVE
VOLUME & SURFACE AREA OF SOLID OF REVOLUTION
LENGTH OF CURVE
Q1.
Find the volume of the solid formed by the revolution of the cissoid y2(a-x) = a2x
about its asymptote.
Q2.
Find the length of cardioid r = a(1 – cos ) lying outside the circle r = a cos .
Q3.
Find the area of the surface generated by revolving an arc of the cycloid x = a( +
sin ); y = a(1 – cos ) about the tangent at the vertex.
Q4.
Find the whole area of the curve a2 x2 = y3 (2a-y)
Q5.
Find the length of the arc of the cycloid x = a(t-sin t), y = a(1 – cos t)
Q6
Find the volume of the solid generates by revolving one loop catenary y = c cos
h(x/c) about the axis of x.
Q7.
Find the perimeter fo the curve r = a(1 – cos )
Q8.
The part of parabola y2 = 4ax cut off by the latus rectum is revolvrd about the
tangent at the vertex. Find the volume of the reel thus generated.

Q10.
For the cycloid x = a( + sin ), y = a (1 – cos ), prove that   4a cos
Q11.
Show that the area of the loop of the curve a2y2 = x2 (2a – x) (x – a) is
Q12.
Find the perimeter of the cardiode r = a(1 – cos ) & show that arc of the upper
half of the curve is bisected by the line  =
2
3a 2
8
2
3
Q13.
Find the area of the loop of curve xy2 + (x + a)2 (x + 2a) = 0
Q14.
Find the volume of solid generated by revolving about the x-axis , the area
enclosed by the arch of the cycloid x = a( + sin ), y = a(1 + cos ) about the xaxis
Q15.
Find the area bounded by the parabola y2 =4ax and its latus rectum.
Q16.
Find the volume of the solid generated by rotating 4x2 + y2 = 4 about x-axes
Q17.
Find the surface area of a sphere of radius “a”.
Q18.
Find the area of the portion of the cylinder x2 + z2 inside the cylinder x2 + y2 = 16.
Q19.
Find the volume bounded by paraboloid z = 2x2 and the cylinder z = 4 – y2.
x  y  a and the coordinate axes
Q20.
Find the area bounded by the parabolic arc
Q21.
Three sides of a trapezium are equal, each being 6 inches long, find the area of the
trapezium when it is maximum.
Q22.
Find the volume bounded by the parabola z=2x2 – y2 and the cylinder z=4 – y2.
Q23.
Find the volume of the solid generated by the revolution of the curve y =
about its asymptote.
Q24.
Fine the length of the curve y2 = x3 from origin to the point (1,1).
a3
a2  x2
MATRICES
Q1.
State Cayley Hamilton’s Theorem. Write down the eigen values of A2 if
 1 0 0
A   2  3 0
 1
4 2
Q2.
 1 2 0
Verify Cayley Hamilton theorem for A   1 1 2 . Hence find A-1.


 1 2 1
Q3.
1 0 0 
Find eigen values and eigen vectors of A  0 2 1 .
2 0 3
Q4.
Suppose An =(0) and B is an invertible matrixof the same size as A, show that
(BAB-1)n =(0).
Q5.
2 1 1 
Find the Characteristic equation of matrix A  0 1 0 and hence find the


1 1 2
matrix represented by A8  5 A 7  7 A 6  3 A5  A 4  5 A3  8 A 2  2 A  I .
Q6.
0 1 2 
Find the inverse of the matrix A  1 2 3 by E-row operations.
3 1 1
Q7.
2 3 4  1 
Find the rank of the matrix A   5 2 0  1  .
 4 5 12  1
Q8.
  1 2  2
Reduce the matrix A   1
2
1  to the diagonal form.
 1  1 0 
Q9.
For what values of d and µ the system of equations x + y+ z = 6, x + 2y + 3z =10,
x + 2y + d z = µ have (i) No solution (ii) Unique solution (iii) more than one
solution.
Q10.
1 0  1
Find the eigen values and eigen vectors of A  1 2 1  and diagonalise it.


2 2 3 
Q11.
1 1 2 
Find the rank of the matrix A by reducing it to the normal form A  1 2 2 .


2 2 3
Q12.
Find the values of a and b & t
such that the rank of the matrix
1  2 3 1 
A  2 1  1 2  is 2.
6  2 a b
Q13.
3 0 0
Find the sum of roots of A where A  8 4 0 .
6 2 5
Q14.
1
1
Find the rank of matrix A  
2

3
Q15.
Use the method of E-row transformations to compute the inverse of
2
1 1 1
3  2 1 
.
0  3 2

3 0 3
 1 2 5
A   2 3 1 .
 1 1 1
Q16.
2  1 1 
Find the rank of A  1 0 2 by reducing it to echelon form.
3  1 3
Q17.
1 3 6
Find the Ch. Roots of A-1 where A  0 2 5 .
0 0 3
Q18.
For what values of  the equations x + y + z = 1, x + 2y +4z =  , x + 4y +10z
= 2 have a solution and solve them completely in each case.
Q19.
Find the non-singular matrices P and Q such that PAQ is in the normal form
2
1 1

where A  1 2 3  .


0  1  1
Q20.
Prove that diagonal elements of a Hermitian matrix are all real.
Q21.
If A and B are Hermitian, Show that AB-BA is skew Hermitian.
Q22.
Show that every square matrix is expressible as the sum of a hermitian matrix
and a skew hermitian matrix.
DIFFERENTIAL EQUATIONS
Q1.
Find the values of λ for which the diff. eqn. ( xy 2  x 2 y)dx  ( x  y) x 2 dy  0 is
exact.
Q2.
Solve the initial value problem
e x (cos ydx  sin ydy)  0, y(0)  0 .
Q3.
Obtain the complete solution of the diff. eqn.
d2y
dy
 7  6 y  e 2 x and
2
dx
dx
determine constants so that y = 0 when x =0.
Q4.
Solve ( D 2  4 D  4) y  8(e 2 x  x 2  sin 2 x),
Q5.
Solve (a)
dy
 y( x  y)
dx
Q6.
Use method of variation of parameters, solve
Q7.
Solve
(i) y sin 2 xdx  ( y 2  cos 2 x)dy  0
(ii) (e y  1) cos 2 xdx  e y sin ydy  0.
(iii) ( D 3  3D  2) y  x 2 e  x ,
D
Solve the simultaneous equations
dx
dx
 x  y  et ,
 y  x  et .
dt
dt
Q9.
Apply the variation of parameters
(i) x 2 y 2  xy1  y  x 2 e x
(ii) y 2  a 2 y  cos ax
Q10.
Solve
d
dx
dy
 e x y  x 2 e  y
dx
(b) x( x  y )
Q8.
D
d
dx
d2y
 4 y  tan 2 x
dx 2
d
dx
(i)
( D 4  4 D 3  8 D 2  8 D  4) y  0, D 
(ii)
y(axy  e x )dx  e x dy  0
(iii)
( D 4  2 D 2  1) y  x 2 cos 2 x,
(iv)

x
(1  e x / y )dx  e x / y 1  dy  0
y

(v)
( y 3  2 x 2 y)dx  (2 xy 2  x 3 )dy  0
(vi)
(1  x 2 ) y' '(1  x) y' y  4 cos log( 1  x)
(vii)
y' 'a 2 y  sec ax using method of variation of parameters.
(viii)
dx
dy
 4x  3 y  t,
 2x  5 y  et
dt
dt
D
d
dx