Upsc Maths Optional Tutorial Sheets Paper I

Delhi Institute for Administrative Services
DIAS INDIA EDUTECH (PVT) LTD
CIVIL SERVICES EXAMINATION (MAINS)

MATHEMATICS PAPER I: Analytic Geometry
Unit 16: (Plane)
Q1. Find the equation of the plane parallel to the plane ax  by  cz  0 and passing through
the point  ,  ,   .
Q2. Find the equation of a plane- xy plane and passing through the points 1, 0,5  ,  0,3,1 .
Q3. Find the equation of the plane passing through the intersection of the planes x  y  z  6
and 2 x  3 y  4 z  5  0 and the point 1,1,1 .
Q4. Find the equation of the plane through the points  2, 2,1 and  9,3, 6  and perpendicular
to the plane 2 x  6 y  6 z  9 .
Q5. the plane lx  my  0 is rotated about its line of intersection with the plane z  0 through
an angle  . Prove that the equation of plane in its new position is
lx  my  z l 2  m 2 tan   0 .
Q6. A variable plane is at a constant distance p from the origin and meets the axes, which
are rectangular in A, B, C . Prove that the locus of the point of intersection of the planes
through A, B, C parallel to coordinate planes is
x 2  y 2  z 2  p 2
a b c
Q7. Prove that    0 represents a pair of planes.
yz zx x y
Q8. From a point P  x, y, z a plane is drawn at right angles to OP to meet the coordinate
r5
axes is A, B, C  . Prove that the area of the ABC is where r  0 P
2 xyz 
Q9. Two system of rectangular axes have the same origin. A plane cuts off intercepts
1 1 1 1 1 1
a, b, c, a, b, c from the axes respectively. Prove that  2 2  2 2 2.
2
a b c a  b  c
Q10. A variable plane is at a constant distance p from the origin and meets the axes in A, B, C .
Show that the locus of the centroid of tetrahedron O, A, B, C is x 2  y 2  z 2  16 p 2 .
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x y z
Q11. A point P moves on a fixed plane    1 . The plane through P perpendicular to
a b c
OP meets the axis in A, B, C . The plane through A, B, C parallel to coordinate axes
intersect in  . Show that the locus of 
1 1 1 1 1 1
2
 2 2   
x y z ax by cz
Q12. The plane x  2 y  3 z  0 is rotated through a right angle about its line of intersection
with the plane 2 x  3 y  4 z  5  0 . Find the equation of the plane in its new position.
(2008)

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Unit 17: Straight Line

Q1. A line with direction ratios 2, 7, -5 is drawn to intersect the lines
x y 1 z  2 x  11 y  5 z
  and  
3 2 4 3 1 1
Find the coordinates of the points of intersection and the length intercepted on it. (2007)
Q2. Find the locus of the point which moves so that its distance from the plane x  y  z  1 is
twice its distance from the line x   y  z . (2007)
x2 y 2
Q3. A line is drawn through a variable point on the ellipse   1, z  0 to meet two
a2 b2
fixed lines y  mx, z  c and y  mx, z  c . Find the locus of the line. (2008)
Q4. Find the equations of the straight line through the point  3,1, 2  to intersect the straight
line x  4  y  1  2  z  2  and parallel to the plane 4 x  5 z  y  0 . (2011)
Q5. Find the equation of the plane which passes through the points  0,1,1 and  2, 0, 1 and
parallel to the line joining the points  1,1, 2  ,  3, 2, 4  . Find also the distance between
the plane and line. (2013)

Q6. Find the image of the point 1, 2,3 in the plane 2 x  3 y  6 z  35  0 . (2007)
Q7. Find the distance of the point 1, 2,3 from the plane x  y  z  5 measured parallel to
x y z
the line   .
2 3 6
x 3 y 8 z 3
Q8. Find the S.D. between the lines and its equation   and
3 1 1
x3 y7 z6
 
3 2 4
Q9. Find the shortest distance between the lines and its equation 3x  9 y  5 z  0  x  y  z
and 6 x  8 y  3 z  13  0  x  2 y  z  3 .

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y z
Q10. Show that the equation to the plane containing the line   1; x  0 and parallel to the
6 c
x z x y z
line   1, y  0 is    1  0 and if 2d is the S.D. prove that
a c a b c
1 1 1 1
2
 2 2 2.
d a b c
Q11. Prove that the shortest distance between any two opposite edges of the tetrahedron
2a
formed by the planes y  z  0, z  x  0, x  y  0, x  y  z  a is and three lines of
6
S.D. intersect at the point x  y  z   a .
Q12. If the axes are rectangular the S.D. between the line
y  az  b, z  ax   ; y  az  b, z   x    is
    b  b       a  a

1/ 2
 2 2  a  a  2      2   a   a 2 
 
x  2 y 1 z  4
Q13. Prove that the lines   and 2 x  3 y  z  0  x  y  2 z  20 are
3 4 5
coplanar. Find also their point of intersection.
x y z x y z
Q14. Find the equation of the plane through the line   and   .
l m l n l m
Q15. Prove that all lines which intersect the lines y  mx, z  c; y   mx, z  c; and the
x -axis lie on the surface mxz  cy .
Q16. Prove that locus of a variable line which intersects the three gives lines
y  mx, z  c; y   mx, z  c, y  z , mx  c is the surface __________(Please Check)

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Unit 18: SPHERE

Q1. Find the equation of sphere passing through the points  0, 0, 0  0,1, , 1 1, 2, 0 1, 2,3 ,
coordinates of centre and its radius.

Q2. Find the equation to the sphere through the circle x 2  y 2  z 2  9, 2 x  3 y  4 z  5 and
the origin.
Q3. Find the equation of Tangent planes to the sphere x 2  y 2  z 2  zx  4 y  6 z  13  0
which are parallel to the plane x  y  z  0
Q4. Find the equation of the sphere which passes through the circle x 2  y 2  z 2  5,
x  2 y  3 z  3 and touch the plane 4 x  3 y  15
Q5. A sphere of radius k passes through the origin and meets the axes at A, B, C . Prove that
the centroid of the triangle ABC lies on the sphere a  x 2  y 2  z 2   4k 2
x y z
Q6. A variable plane is parallel to the given plane    0 and meets the axes in A, B, c
a b c
b c c a a b
yz     xz     xy     0
c b a c b a
Q7. A plane through a fixed point  a, b, c  cuts the axes in A, B, C . Show that the focus of the
a b c
centre of sphere OABC is    2 , 0 being the origin
x y z
Q8. Find out the equation of sphere passing through the origin and meeting the axis of x, y, z
respectively at A, B, C .
Q9. Prove that the circle x 2  y 2  z 2  2 x  3 y  4 z  5  0, 5 y  6 z  1  0 and
x 2  y 2  z 2  3x  4 y  5 z  6  0 , x  2 y  7 z  0 lie on the same sphere. Also find the

a
value of a for which x  y  z  touches the sphere.
3
Q10. Find the equation of a sphere which touches the sphere x 2  y 2  z 2  2 x  6 y  1  0 at
1, 2, 2  and passes through the origin.

Q11. Find the equation of a sphere inscribed in the tetrahedron whose faces are
x  0, y  0, z  0, zx  6 y  3 z  6  0
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Q12. Prove that the centres of spheres which touch the lines
y  mx z  c; y   mx, z  c lie upon the conicoid mxy  cz 1  m 2  0  
Q13. If any tangent plane to the sphere x 2  y 2  z 2  r 2 makes intercepts a, b, c on the
coordinate axes, prove that a 2  b 2  c 2  r 2

Q14. Shoe that the spheres x2  y 2  z 2  6 y  2z  8  0 and
x 2  y 2  z 2  6 x  8 y  4 z  20  0 intersect orthogonally. Find their planes of

intersection.
Q15. Find the equation of the sphere which touches the plane 3x  2 y  z  2  0 at the point
1, 2,1 and cuts orthogonally the sphere x 2  y 2  z 2  4 x  6 y  4  0
Q16. A sphere of constant radius r passes through the origin O and cuts the axes in A, B, C .
Prove that the focus of the foot of the perpendicular from O to the line ABC is given by
x  x 
2
2
 y2  z2 2
 y 2  z 2  4 r 2

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Unit 19: CONE & CYLINDER

Q1. Find the equation of right circular cylinder whose axis is x  2 y   z and radius 4 .
Prove that area of cross-section of this cylinder by the plane z  0 is 24
Q2. Find the equation of the right circular cylinder which passes through circle
x2  y 2  z 2  9 , x  y  z  3
x y z
Q3. Find the equation to the cylinder whose generators are parallel to the line   and
l m n
x2 y 2 z 2
which envelops the surface 2  2  2  1
a b c
x y z
Q4. Find the equation of the cylinder whose generator are parallel to the line   and
1 2 3
passes through the curve x 2  y 2  16, z  0
Q5. Find the equation of the cylinder which intersects the curve ax 2  by 2  cz 2  1 ,
lx  my  nz  p and whose generators are parallel to z - axis.
Q6. Show that the equation to the right circular cylinder described on the circle through three
points 1, 0, 0  0,1, 0  and  0, 0,1 as girding curve is x 2  y 2  z 2  yz  zx  xy  1
Q7. Show that ax 2  by 2  cz 2  2ux  2vy  2 wz  d  0 represents a cone if
u 2 v2  2
  d
a b c
Q8. Find the equation of a cone whose vertex is the point  ,  ,   and whos generating lines
x2 y 2
pass through the conic   1, z  0
a2 b2
Q9. Find out the vertex of the cone 2 y 2  8 yz  4 xz  8 xy  6 x  4 y  2 z  5  0
Q10. Find the equation of the cone whose vertex is 1, 2, 3 and griding curve is the circle
x2  y 2  z 2  4 , x  y  z  1
Q11. Find the equation of the cone with vertex at  0, 0, 0  and which passes through the curve
ax 2  by 2  cz 2  1  0   x 2   y 2  2 z

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x2 y 2
Q12. The section of a cone whose vertex is p and guiding curve the ellipse   1, z  0
a2 b2
x2 y 2  z 2
by the plane x  0 is a rectangular Hyperbola. Show that focus of p is  1
a2 b2
Q13. find the equations to the lines in which the plane 2 x  y  z  0 cuts the cone
4 x 2  y 2  3z 2  0
Q14. Prove that the plane ax  by  cz  0 cuts the cone xy  yz  xz  0 in perpendicular lines
1 1 1
if   0
a b c
x y z
Q15. If   represent one of a set of three mutually perpendicular generators of the
1 2 3
cone 5 yz  8 xz  3xy  0 find the equation of the other two
x2 y 2 z 2
Q16. Prove that cones ax 2  by 2  cz 2  0 and    0 are reciprocal
a b c

Q17. Prove that the angle between the lines given by x  y  z  0, ayz  bxz  cxy  0 is if
2
1 1 1
  0
a b c
Q18. Prove that the equation fx  gy  hz  0 represent a cone which touches the
coordinate planes and that the equation of its reciprocal cone is fyz  gzx  hxy  0
Q19. Find the laws of the vertices of enveloping cones by the plane z  0 are circles.

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Unit 20: Conicoids

Q1. Prove that the lines of intersection of pairs of tangent planes to ax 2  by 2  cz 2  0 which
touch along perpendicular generators lie on the cone
a2 b  c  x2  b2  c  a  y2  c2  a  b  z 2  0 . (2004)
x2 y 2 z 2
Q2. Tangent planes are drawn to the ellipsoid    1 to them through the origin
a 2 b2 c 2
generate the cone
 x   y   z 
2
 a2 x2  b2 y2  c2 z 2 (2004)
Q3. Obtain the equation of right circular cylinder on the circle through the points
 a, 0, 0  a, b, 0  and  0, 0, c  as the grinding curve. (2005)
Q4. Show that the plane 2 x  y  2 z  0 cuts the cone xy  yz  zx  0 in perpendicular lines.
(2007)
x y z
Q5. If   represent one of a set of three immutably + generators of the cone
1 2 3
5 yz  8 xz  3xy  0 find the equation of other two. (2008)
x2 y 2
Q6. Prove that the normals from the point  ,  ,   to the paraboloid   2 z lies on
a 2 b2
  a 2  b2
the cone    0. (2009)
x  y z 
Q7. Find the vertices of the show quadrilateral formed by the four generator of the
x2
Hyperboloid and 14, 2, 2   y 2  z 2  49 passing through 10, 5,1 . (2010)
4
x2 y 2 z 2
Q8. Three points P, Q, R are taken on the ellipsoid 2  2  2  1 so that the lines joining
a b c
P, Q, R to the origin are mutually perpendicular. Prove that the plane P, Q, R touches a
fixed sphere. (2011)
Q9. Show that the generators through any one of the ends of an equicojugate diameter of the
x2 y 2 z 2
principal elliptic section of the Hyperboloid    1 are inclined to each other at
a 2 b2 c 2

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an angle of 600 if a 2  b 2  6c 2 . Find also the condition for the generators to be

perpendicular to each other. (2011)
Q10. Show that the locus of a point from which the these mutually perpendicular tangent lines
can be drawn to paraboloid x 2  y 2  2 z  0 x 2  y 2  4 z  1 . (2012)

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Unit 21: 2nd Degree Equation, Reduction to Canonical Form

Q1. Reduce the equation:
11x 2  10 y 2  6 z 2  8 yz  4 zx  12 xy  7 zx  7 zy  36 z  150  0 to the standard form and
give the nature of the surface. Also find the equations of its axes.
Q2. Prove that the equation x 2  y 2  z 2  yz  zx  xy  3 x  y  4 z  4  0 represents an
1
ellipsoid the squares of whose semi axes are 2, 2, . Show that its principal axis is given
2
by x  1  y  1  z  2 .
Q3. Show that the equation 2 y 2  4 zx  2 x  4 y  6 z  5  0 represents a right circular cone.


Show also that the semi-vertical angle of this cone is and that its axis is given by
4
x  z  2  0, y  1 .
Q4. Show that 2 x 2  2 y 2  z 2  2 yz  2 xz  4 xy  x  y  0
Q5. Reduce 3z 2  6 yz  6 zx  7 x  5 y  6 z  3  0 to standard for and find the nature of the

surface represented by this equation.
Q6. Prove that 5 x 2  5 y 2  8 z 2  8 yz  8 zx  2 xy  12 x  12 y  6  0 represents a cylinder
1
whose cross-section is an ellipse of eccentricity and find the equations to its axis.
2
Q7. Reduce the surface 36 x 2  4 y 2  z 2  4 yz  12 zx  24 xy  4 x  16 y  26 z  3  0 to the
standard form and find the lotus rectum of a normal section.
Most general equation of 2nd degree in 3 coordinates:
F  x, y, z   ax 2  by 2  cz 2  2 fyz  2 gzx  2hxy  24 x  2vy  2 wz  d  0
It can be reduced to any one of the below mentioned forms by transformation of axes:
1 x 2  2 y 2  3 z 2   (1)
1 x 2  2 y 2  2 z (2)
By giving different values to 1 , 2 , 3 &  from (1) can be reduced to
(i) Ax 2  By 2  Cz 2  1 Ellipsoid
(ii) A  x 2  y 2   Cz 2  1 Ellipsoid of revolution

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 
(iii) A x 2  y 2  z 2  1 Sphere
(iv) Ax 2  By 2  Cz 2  1 Hyperboloid of one sheet
(v) Ax 2  By 2  Cz 2  1 Hyperboloid of two sheet
(vi) A  x 2  z 2   By 2  1 Hyperboloid of revolution
(vii) Ax 2  By 2  Cz 2  0 Cone
(viii) Ax 2  By 2  1 Elliptic cylinder
(ix) Ax 2  By 2  1 Hyperbolic cylinder
(x) Ax 2  By 2  0 Pair of intersecting planes
(xi) Ax 2  1 or By 2  1 or Cz 2  1 Pair of parallel planes
Homogenous part f  x, y, z   ax 2  by 2   z 2  2 fyz  2 gzx  2hxy  .

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MATHEMATICS PAPER I: O.D.E
Unit 22: Diffrential equations of First order and First Degree
dy
Q1. Solve x  y log y  xye x (2003)
dx
dy 1
Q2. Solve  y cos x  sin 2 x (2004)
dx 2
Q3. Solve y  xy  2 x 2 y 2  dx  x  xy  x 2 y 2  dy  0 (2004)
dy
Q4. Solve xy  x2  y 2  x2 y2  1 (2005)
dx
 1
 3 
Q5. Solve the D.E  xy 2  e x  dx  x y dy  0
2
(2006)

 
Q6. 
Solve 1  y 2   x  e  tan
1
y
 dydx  0 (2006)
Q7. Solve the D.E

dy 1 
cos 3 x  3 y sin 3 x  sin 6 x  sin 2 3x 0 x (2007)
dx 6 2
dy
Q8. Solve  xy 2 dx  4 xdx (2007)
y
Q9. Solve ydx   x  x 3 y 2  dy  0 (2008)
dy y2  x  y 
Q10. Solve  , y  0  1 (2009)
dx 3 xy 2  x 2 y  4 y 3
Q11. Show that D.E.
3 y 2
   
 x  2 y y 2  3 y  0 admits an integrating factor which is a function of x  y 2 .
Hence solve the equation. (2010)
Q12. Verify that
1 1 x
 Mx  Ny  d  ln xy    Mx  Ny  d ln    Mdx  Ndy
2 2  y
Hence show that:

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(i) If the D.E. Mdx  Ndy  0 is homogeneous, then Mx  Ny is an I.F. unless

Mx  Ny  0
(ii) If the D.E Mdx  Ndy  0 is not exact but is of the form
1
f1  x, y  ydx  f 2  x, y  xdy  0 , then is an I.F. unless Mx  Ny  0 . (2010)
Mx  Ny
dy
  4 x  y  1
2
Q13. Solve
dx
2
x
 
 y
dy 2 xy e
Q14. Solve  (2012)
dx   x 
2 2
x
   
y 2  1  e y    2 x 2 e  y 
 
 
Q15. Show that the D.E.
 2 xy log y  dx   x 2  y 2 
y 2  1 dy  0 is not exact. Find an integrating factor and hence
the solution of the equation. (2012)

dy
Q16. Solve:  cos  x  y   sin  x  y  (2013)
dx
Q17. Solve  5 x 3  12 x 2  6 x  dx  6 xy dy  0 (2013)

Ph: 011-40079000, 9350934622 14
Unit 23: D.E. of 1st order and Higher Degree
Q1. 
Solve the D.E px 2  y 2   px  y    p  1 2
by reducing to Clairut’s form using suitable
substitutions. (2003)
Q2. Show that the orthogonal trajectory of a system of confocal ellipses is self orthogonal
(2003)
Q3. Reduce the equation to Clairut’s equation and solve it:
dy
 px  y  py  x   2 p where p  (2004)
dx
Q4. Solve the D.E by reducing to it to Clairut’s from by using suitable substitution
x 2

 y 2 1  p   2  x  y 1  p   x  yp    x  yp   0
2 2
(2005)
Q5. Find the orthogonal trajectory of a system of coaxial circles x 2  y 2  2 gx  c  0 where

g is the parameter. (2005)
Q6. Solve x 2 p 2  yp  2 x  y   y 2  0 , using the substitution y  u and xy  v and find its
dy
singular solution where p  (2006)
dx

Q7. Find the family of curves whose tangents form an angle with the hyperbolas
4
xy  c, c  0 . (2006)
dy
Q8. Solve the equation y  2 xp  yp 2  0 where p  . (2008)
dx
Q9. Determine the orthogonal trajectory of a finally of curves represented by the polar
equation r  a 1  cos   . (2011)
Q10. Obtain Chairut’s form of the D.E.

 dy  dy  2 dy
 x  y  y  x   a . Also find its general solution. (2011)
 dx  dx  dx
Q11. Find the orthogonal trajectories of the family of curves x 2  y 2  ax . (2012)

Q12. Obtain the equation of the orthogonal trajectory of the family of curves represented by
r n  a sin n (2013)

Ph: 011-40079000, 9350934622 15
Unit 24: D.E. of 2nd order with constant coefficients

d
Q1. Solve  D5  D  y  4  e x  cos x  x 3  where D  . (2003)
dx
Q2.  
Solve D 4  4 D 2  5 y  e x  x  cos x  (2004)
Q3. Solve  D 2

 2 D  2 y  e x tan x (2006)
 9 
Q4. Solve  D 3  6 D 2  12 D  8  y  12  e 2 x  e  x  (2007)
 4 
Q5. Obtain the general solution: y  2 y  2 y  x  e x cos x (2011)
Q6. Find the general solution of the equation y  y  12 x 2  6 x (2012)

Ph: 011-40079000, 9350934622 16
Unit 25: 2nd Order D.E. with Variable Coefficients
Solve 1  x  y  1  x  y  y  sin 2 log 1  x  .

2
Q1. (2003)
Q2. Solve the D.E. by Variation of parameters: x 2 y  4 xy  6 y  x 4 sec2 x (2003)
d2y dy
Q3. Solve  x  2  2
  2 x  5   2 y   x  1 e x (2004)
dx dx
d2y dy
Q4. 
Solve: 1  x 2
 dx 2
dx

 4x  1  x2 y  x (2004)
1
Solve:  x  1 D 3  2  x  1 D 2   x  1 D   x  1  y 
4 3 2
Q5. . (2005)
  x 1
Q6. Solve the D.E.:  sin x  x cos x  y  x sin xy  y sin x  0 , given that y  sin x is a
solution of this equation. (2005)

Q7. Solve the D.E. by variation of parameters: x 2 y  2 xy  2 y  x log x, x0 (2005)
d3y d2y y  1 
Q8. Solve: x 2 3
 2 x 2
 2  10 1  2  (2006)
dx dx x  x 
d2y dy
Q9. Solve: 2 x 2 2
 3x  3 y  x3 (2007)
dx dx
d2y dy
Q10. Solve by the method of variation of parameters: 2
 3  2 y  2e x (2007)
dx dx
Q11. Use the method of variation of parameters to find the general solution of
x 2 y  4 xy  6 y   x 4 sin x . (2008)
Q12. Solve the D.E.: x 3 y  3x 2 y  xy  sin  ln x   1 . (2008)
d2y
Q13. Solve by the method of Variation of parameters:  4 y  tan 2 x . (2011)
dx 2
Q14. Solve the D.E.: x  x  1 y   2 x  1 y  2 y  x 2  2 x  3 (2012)
d2y
Q15. Using the method of variation of parameters, solve 2
 a 2 y  sec ax . (2013)
dx
d2y dy
Q16. Find the general solution of x 2 2
 x  y  ln x sin  ln x  (2013)
dx dx

Ph: 011-40079000, 9350934622 17
d3 y 2
2 d y dy
Q17. Solve the D.E.: x3 3
 3 x 2
x  8 y  65cos  ln x  (2014)
dx dx dx
d2y dy
Q18. Solve the following D.E.: x 2
 2  x  1   x  2  y   x  2  e 2 x , when e x is a
dx dx
solution to its corresponding homogenous D.E. (2014)
dy
Q19. Solve by the method of variations of parameters  5 y  sin x . (2014)
dx

Ph: 011-40079000, 9350934622 18
Unit 26: Laplace Transforms
Q1. Using Laplace transform, solve the initial value problem y  3 y  2 y  4t  e3t with
y  0   1, y   0   1 (2008)
 s 1 
Q2. Find the inverse Laplace transform of F  s   ln  . (2009)
 s5
Q3. Find the D.E. of the family of circles in the xy - plane passing through  1,1 and 1,1 .
(2009)
d2y dx dx
Q4. Use Laplace transform to solve: 2
 2  x  et , x  0   2 and  1 .
dx dt dt t 0
(2011)
Q5. Using Laplace transforms, solve the initial value problem
y   2 y   y  et , y  0   1, y   0   1 (2012)
Q6. Using Laplace transform method, solve:  D 2  n 2  x  a sin  nt    with condition at
dx
x  0 and  0 at t  0 . (2013)
dt
Q7. Solve the initial value problem using Laplace transform:
d2y
 y  8e2t sin t , y  0   0, y  0   0 . (2014)
dt 2

Ph: 011-40079000, 9350934622 19
Mathematical Paper 1: Section B / Vector Analysis

Unit 27: Scalar & Vector Fields, Triple Products, Differentiation of vector
        
Q1. If a  r  b   a and a. r  3 where a  2iˆ  ˆj  kˆ and b  iˆ  2 ˆj  kˆ , then find r and


      b
Q2.   
If a , b , c be three unit vectors such that a  b  c  . Find the angles which a makes
2
   
with b and c , b and c being non parallel.
  
Q3. If a , b , c are the position vectors of the vertices A, B, C of a triangle, show that vector
1      
area of the  is
2

. b c  c a  a b 
     
Q4. Prove that  a  b , b  c , c  a   2  a b c 
Q5. Show that the four points whose position vectors are 3iˆ  2 ˆj  4kˆ, 6iˆ  3 ˆj  kˆ, 5iˆ  7 ˆj  3kˆ
and 2iˆ  2 ˆj  6kˆ are coplanar

Q6. Prove the identity
  
    a.c a.d
  
(i) a  b  c  d     
b .c b .d
       
   
(ii) a  b  c  d   abd  c   abc  a
 
du   dv   d     
Q7. If    u,    v , show that u  v     u  v 
dt dt dt
 
  dR 1  dr
Q8. If R be a unit vector in the direction of r , prove that R   2 r
dt r dt
   d 2r 2
Q9. If r  t   5t 2iˆ  tjˆ  t 3 kˆ , prove that
1  r  dt 2  dt  14iˆ  75 ˆj  15kˆ

   dR r dr
Q10. Let R be the unit vector along the vector r  t  . Show that R    (2002)
dt r 2 dt
     
Q11. Show that if a, b & c are the reciprocals to the non coplanar vector a , b , c , then any

vector r may be written as
         
 
r   r  a  a  r  b b   r  c  c (2003)

Ph: 011-40079000, 9350934622 20

Let the position vector of a particle moving on a plane curve be r  t  where t is the time.
Q12. Find the components of its acceleration along the radial and transverse directions. (2003)
1   
Q13. Show that the volume of tetrahedron ABCD is
6
 
AB  AC  AD . Hence find the volume
of the tetrahedron with vertices  2, 2, 2  ,  2,0, 0  ,  0, 2, 0  &  0, 0, 2  .

D C
A
B
(2005)
   
Q14. If A  2iˆ  kˆ, B  iˆ  ˆj  kˆ, C  4iˆ  3 ˆj  7 kˆ determine a vector R satisfying
     
R  B  C  B and R. A  0 (2006)

Q15. Show that F   2 xy  z 3  iˆ  x 2 ˆj  3 xz 2 kˆ is a conservative force field. Find the scalar

potential F and work done in moving an object in this field from 1, 2,1 to  3,1, 4  .
(2008)
x2 y2
Q16. Find the work done in moving the particle once round the ellipse   1, z  0
25 16
under the field of force given by

F   2 x  y  z  iˆ   x  y  z  ˆj   3 x  2 y  4 z  kˆ . (2009)
 

Q17. Show that the vector field defined by the vector function V  xyz yziˆ  xzjˆ  xykˆ is
conservative. (2010)
   
Q18. For any vectors a & b given respectively by a  5t 2iˆ  tjˆ  t 3kˆ and b  sin tiˆ  cos tjˆ
d  d  
determine (i)
dt
 
ab and (ii)
dt
ab .   (2011)
  
Q19. Examine within the vectors u, v & w are coplanar, where u, v, w are the scalar
function whether defined by
u  x yz
v  x2  y 2  z 2

Ph: 011-40079000, 9350934622 21
w  yz  xz  xy (2011)

Q20. if A  x 2 yziˆ  2 xz 3 ˆj  xz 2 kˆ

B  2 ziˆ  yjˆ  x 2 kˆ
2  
Find the value of
xy
 
A  B at 1, 0, 2  . (2012)

Ph: 011-40079000, 9350934622 22

Unit 28: Gradient, Divergence and Curl
 
Q1. If a & b are constant vectors, then show that:
     
(i)    x   a  x   2 x  a
   2a   b  x   2b   a  x 
    
 

(ii)    a  x   b  x (1992)
Q2. Prove that the angular velocity of rotation at any point is equal to one half of the curl of

the velocity vector V . (1993)

Q3. Show that r n r is an irrotational vector for any value of n , but is solenoidal only if
n  3 . (1994, 2006)
 
Q4. If r  xiˆ  yjˆ  zkˆ and r  r , show that:

(i) r  grad f  r   0

(ii)    r n r    n  3 r n

(1996)
 
Q5. If r1 and r2 are the vectors joining the fixed points A  x1 , y1 , z1  and B  x2 , y2 , z2 
 
respectively to a variable point P  x, y , z  , then find the values of Grad  r1  r2  and
 
r1  r2 . (1998)
   
Q6. 
Evaluate   F for F   x3  y 3  z 3  3 xyz .  (1999)
Q7. In what direction from the point  1,1,1 is the directional derivative of f  x 2 yz 3 is
maximum? Compute the magnitude. (2000)


Q8. Show that the vector field defined by F  2 xyz 3iˆ  x 2 z 3 ˆj  3 x 2 yz 2 kˆ is irrotational. Find

also the scalar U such that F  Grad U . (2001)
Q9. Find the directional derivative of f  x 2 yz 3 along x  e t , y  1  2 sin t , z  t  cos t at
t  0. (2001)
 
Q10.
 a b
Show that  



 
ar   
a 3r   
  3  5  a  r  , where a is any constant vector.
3 3
r r r r
(2001)

Ph: 011-40079000, 9350934622 23
  
Q11.   
Show that Curl curl V  Grad divV   2V  (2002)
Q12. Prove that the divergence of a vector field is invariant with respect to coordinate
transformations. (2003)
          
Q13.    
Prove the identity:   A2   2 A  A  2 A    A , where   iˆ  ˆj  kˆ .
x y z
(2003)
              
Q14.        
Prove the identity:  A  B  B   A  A   B  B    A  A    B   (2004)
   
Q15. Show that if A & B are irrotational, then A  B is solenoidal. (2004)
Q16. Prove that the curl of a vector field is independent of the choice of coordinates. (2005)
  1  1
Q17. Show that    kˆ  grad   grad  kˆ  grad   0 , where r is the distance from the
 r  r

origin and k is the unit vector in the direction 0 Z . (2005)
Q18. Find the values of constants a, b and c so that the directional derivative of the function
f  axy 2  byz  cz 2 x at the point 1, 2, 1 has maximum magnitude 64 in the direction

parallel to z - axis. (2006)

Q19. Prove that r n r is an irrotational vector for any value of n , but is solenoidal only if
n3 0.
(2006)

Q20. If r denotes the position vector of a point and if r̂ be the unit vector in the direction of
   1
r , r  r determine    in terms of r̂ and r . (2007)
r
   
Q21. For any constant vector a , show that the vector represented by    a  r  is always
 
parallel to the vector a , r being the position vector of a point  x, y, z  measured from
the origin. (2007)

 
Q22. If r  xiˆ  yjˆ  zkˆ , find the value(s) of n in order that r n r may be (i) Solenoidal (ii)
Irrotational. (2007, 2011)

Ph: 011-40079000, 9350934622 24
d 2 f 2 df
Prove that  2 f  r   where r   x 2  y 2  z 2  . Hence find f  r  such that
1/ 2
Q23. 2

dr r dr
2 f  r   0 . (2008)
 
Q24.  
Show that   r n  n  n  1 r n 2 , where r  x 2  y 2  z 2 . (2009)
Q25. Find the directional derivative of

(i) 4 xz 3  3x 2 y 2 z 2 at  2, 1, 2  along Z - axis.
(ii) x 2 yz  4 xz 2 at 1, 2,1 in the direction of 2iˆ  ˆj  2kˆ . (2009)
Q26. Find the directional derivative of f  x, y   x 2 y 3  xy at the point  2,1 in the direction

of a unit vector which makes an angle of with the x - axis. (2010)
3
     
Q27.      
Prove that   f   f  V  f V , where f is a scalar function. (2010)
  
Q28. If u and v are two scalar fields and f is the vector field such that uf   V  , find the
  

value of f    f . 
(2011)
Q29. Calculate  2  r n  and find its expression in terms of vector, r being the distance of any
point  x, y, z  from the origin, n being constant and  2 being Laplace operator. (2013)

Ph: 011-40079000, 9350934622 25

Unit 29: Curvature and Torsion
Find the length of the arc of the twisted curve r   3t ,3t 2 , 2t 3  from the point t  0 to the

Q1.
point t  1 . Find also the unit tangent t , unit normal n and the unit binormal b at t  1 .
(2001)
Q2. Find the curvature k for the space curve:
x  a cos  , y  a sin  , z  a tan  (2002)
Q3. Find the radii of curvature and torsion at a point of intersection of the surfaces
x
x 2  y 2  c 2 , y  x tanh . (2003)
c
Q4. Show that the Frenet-Serret Formula can be written in the form
  
dT   dN   dB     
  T , N ,    B where    T  kB (2004)
dS dS dS
Q5. Find the curvature and the torsion of the space curve
  
x  a 3u  u 3 , y  3au 2 , z  a 3u  u 2 .  (2005)

Q6. The parametric equation of a circular helix is r  a cos uiˆ  a sin ujˆ  cukˆ , where c is a
constant and u is a parameter. Find the unit tangent vector tˆ at the point u and the arc
dtˆ
length measured from u  0 . Also find where S is the arc length. (2005)
dS
 
Q7. If the unit tangent vector t and binomial b makes angles  &  respectively with a
 sin  d k
constant unit vector a , prove that  . (2006)
sin  d 
Q8. Find the curvature and torsion at any point of the curve:
x  a cos 2t y  a sin 2t z  2a sin t . (2007)
Q9. Show that for the space curve
2
x  t, y  t 2 z  t 3
3
The curvature and torsion are same at every point. (2008)

Ph: 011-40079000, 9350934622 26
k 
Q10. Find for the curve r  t   a cos t iˆ  a sin t ˆj  b t kˆ (2010)

Q11. Derive the Frenet-Serret Formulae. Define the curvature and torsion for a space curve.
Compute them for the space curve
2 3
x  t, y  t2 z  t .
3
Show that the curvature and torsion are equal for the curve. (2012)

Q12. A curve in space is defined by the vector equation r  t 2iˆ  2t ˆj  tkˆ . Determine the angle
between the tangents to this curve at the points t  1 and t  1 . (2013)
 1 t ˆ 1 t2 ˆ
Q13. Show that the curve x  t   tiˆ  j k lies in a plane. (2013)
t t

Q14. Find the curvature vector at any point of the curve r  t   t cos tiˆ  t sin tjˆ, 0  t  2 .
Give its magnitude. (2014)

Ph: 011-40079000, 9350934622 27

Unit 30: Line, Surface and Volume Integrals
 
Q1. Evaluate    F  nˆ dS where S is the upper half surface of the unit sphere
S

x 2  y 2  z 2  1 and F  ziˆ  xjˆ  ykˆ (1993)
  
Q2. If F  yiˆ   x  2 xz  ˆj  xykˆ evaluate    F  ndS
ˆ where S is the surface of the sphere
x 2  y 2  z 2  a 2 above the xy - plane. (1994)

Q3. Let the region V be bounded by the smooth surface S and let n denote outward drawn

unit normal vector at a point on S , if  is harmonic in V , then show that S n
 dS is 0.
(1995)

Q4. Verify Gauss’s divergence theorem for F  xyiˆ  z 2 ˆj  2 yzkˆ on the tetrahedron
x  y  z  0, x  y  z  1 . (1996)

Q5. Verify Gauss’s theorem for F  4 xiˆ  2 y 2 ˆj  z 2 kˆ taken over the region bounded by
x 2  y 2  4, z  0 & z  3 . (1997)
Q6. Evaluate by Green’s theorem: C

e  x sin y dx  e  x cos y dy , where C is rectangle whose
   
vertices are  0, 0  ,  , 0  ,   ,  and  0,  . (1999)
 2  2
  
Q7. Evaluate  F  n ds where F  2 xyiˆ  yz 2 ˆj  xzkˆ and S is the surface of the parallel
S
piped bounded by x  0, y  0, z  0 and x  2, y  1 & z  3 . (2000)


Q8. Verify Gauss’s divergence theorem for A   4 x, 2 y 2 , z 2  taken over the region bounded
by x 2  y 2  4, z  0 & z  3 . (2001)
Q9. Let D be a closed and bounded region having boundary S . Further, let f be a scalar
function having second order partial derivatives defined on it. Show that
  f grad f   nˆ dS    grad  f  2 f  dV

2
f

S V

Ph: 011-40079000, 9350934622 28
hence or otherwise evaluate:   f grad f   nˆ dS

S
for f  2x  y  2z over
x2  y 2  z 2  4 . (2002)
 
Q10. Evaluate 
S
curl A  dS , where S is the open surface x 2  y 2  4 x  4 z  0, z  0 and

    
A  y 2  z 2  x 2 iˆ  2 z 2  x 2  y 2 ˆj  x 2  y 2  3 z 2 kˆ .  (2003)
      dV       nˆ dS

2 2
Q11. Derive the identity: , where V is the
V S
volume bounded by the closed surface S . (2004)


Q12. Verify Stoke’s theorem for f   2 x  y  iˆ  yz 2 ˆj  y 2 zkˆ where S is the upper half
surface of the sphere x 2  y 2  z 2  1 and C is its boundary. (2004)
 x dydz  x ydzdx  x 2 zdxdy by Gauss’s divergence theorem, where S is the

3 2
Q13. Evaluate
S
surface of the cylinder x 2  y 2  a 2 bounded by z  0 & z  b . (2005)


Q14. Verify Stoke’s theorem for the function: F  x 2iˆ  xyjˆ , integrated round the square in the
plane z  0 and bounded by the lines x  0, y  0, x  a & y  a, a  0 . (2006)
Q15. Determine 
C
ydx  zdy  xdz by using Stoke’s theorem where C is the curve defined by
 x  a   y  a  z 2  2a 2 , x  y  2a that starts from the point  2a, 0, 0  and goes at

2 2
first below the z - plane. (2007)

 
Q16. Evaluate  A  dr along the curve x 2  y 2  1, z  1 from
C
 0,1,1 to 1, 0,1 if

A   yz  2 x  iˆ  xzjˆ   xy  2 z  kˆ . (2008)
 
Q17. Evaluate  F  ndS ˆ where F  4 xiˆ  2 y 2 ˆj  z 2 kˆ and S is the surface of the cylinder
V
boundary by x 2  y 2  4 , Z  0 and Z  3 . (2008)

 
Q18. Using divergence theorem, evaluate  A  ndS
ˆ where A  x 3iˆ  y 3 ˆj  z 3 kˆ and S is the
surface of the sphere x 2  y 2  z 2  a 2 . (2009)

Ph: 011-40079000, 9350934622 29
  
Q19. Find the value of     F   dS
S
taken over the upper portion of the surface
x 2  y 2  2ax  az  0 and the bounding curve lies in the plane z  0 when


F   y 2  z 2  x 2  iˆ   z 2  x 2  y 2  ˆj   x 2  y 2  z 2  kˆ . (2009)
 
Q20. Use the divergence theorem to evaluate  V  ndA
ˆ where V  x ziˆ  yjˆ  xz kˆ2 2
and S is
S
the boundary of the region bounded by the paraboloid z  x 2  y 2 and the plane z  4 y .
(2010)
Q21. Verify the Green’s theorem for e  x sin ydx  e  x cos ydy , the path of integration being the
    
boundary of the square whose vertices are  0, 0   ,  and  0,  . (2010)
2 2  2
  
Q22.

 
If u  4 yiˆ  xjˆ  2 zkˆ calculate the    u  dS over the hemisphere given by
S
x2  y 2  z 2  a2 , z  0 . (2011)

Q23. Verify the Gauss’s divergence theorem for the vector u  x 2iˆ  y 2 ˆj  z 2 kˆ taken over the
cube x, y ,  0, z2  1. (2011)
  xy  y  dx  x dy , where C
2 2
Q24. Verify Green’s theorem in the plane for is the closed
C
curve of the region bounded by y  x and y  x 2 . (2012)

   
Q25. 
If F  yiˆ   x  2 xz  ˆj  xykˆ , evaluate    F  ndS 
ˆ , where S is the surface of the
S
sphere x 2  y 2  z 2  a 2 above the xy plane. (2012)
  a 
1 / 2
Q26. By using divergence theorem of Gauss, evaluate the 2
x 2  b2 y 2  c2 z 2  dS ,
S
where S is the surface of the ellipsoid ax 2  by 2  cz 2  1 and a, b and c being all

positive constants. (2013)
Q27. Using Stoke’s theorem to evaluate the  C
 y 3dx  x 2 dy  z 3dz , where C is the
intersection of the cylinder x 2  y 2  1 and the plane x  y  z  1 . (2013)

Ph: 011-40079000, 9350934622 30
Q28. Evaluate by Stoke’s theorem: 

ydx  zdy  xdz , where  is the curve given by
x 2  y 2  z 2  2ax  2ay  0, x  y  2a , starting from  2a, 0, 0  and then going below the
z - plane. (2014)

Ph: 011-40079000, 9350934622 31

Unit 31: Simple Harmonic Motion (Motion in a Plane)
Q1. A particle moving with uniform acceleration describes distances S1 and S 2 metres in
successive intervals of time t1 and t2 seconds. Express the acceleration in terms of
S1 , S 2 , t1 and t2 . (2004)
 a4 
Q2. A particle whose mass is m , is acted upon by a force m  x  3  towards the origin. If it
 x 

starts from rest at a distance a , show that it will arrive at origin in time . (2006, 2012)
4
Q3. A particle is performing simple harmonic motion of period T about a centre O . It passes
through a point p  op  p  with velocity v in the direction op . Show that the time which
T VT
elapses before it returns to P is tan 1 . (2007)
 2 p
Q4. One end of a light elastic string of natural length l and modulus of elasticity 2 mg is
attached to a fixed point O and the other end to a particle of mass m . The particle
initially held at rest at O is let fall. Find the greatest extension of the string during the
2l

motion and show that the particle will reach O again after a time   2  tan 1 2  g
.
(2009)
Q5. (i) After a ball has been falling under gravity for 5 seconds it passes through a pane of glass
and loses half of its velocity. If it now reaches the ground in 1 second, find the height of
glass above the ground. (2011)
(ii) A particle of mass m moves on straight line under an attractive force mn 2 x towards a
dx
point O on the line, where x is the distance from O . If x  a and  u when t  0 ,
dt
find x  t  for any time t  0 . (2011)
Q6. The velocity of a train increases from 0 to v at a constant acceleration f1 , then remains
constant for an interval and again decreases to 0 at a constant retardation f 2 . If the total
distance described is x , find the total time taken. (2011)
Ph: 011-40079000, 9350934622 32
Q7. A particle is performing a simple harmonic motion (SHM) of a period T about a centre
O with amplitude a and it passes through a point P , where OP  b in the direction of
T b
OP . Prove that the time which elapses before it returns to P is cos 1 . (2014)
 a
Q8. A particle is acted on by a force parallel to the axis of y whose acceleration (always
towards the axis of x ) is  y 2 and when y  a , it is projected parallel to the axis of x
2m
with velocity . Find the parametric equation of the path of the particle. Here  is a
a
constant. (2014)

Ph: 011-40079000, 9350934622 33

Unit 32: Projectile Motion
Q1. Prove that the velocity required to project a particle from a height h to fall at a horizontal
distance a from a point of projection is at least equal to g a 2  h2  h . (2004)
Q2. If V1 , V2 ,V3 are the velocities at three points A, B, C of the path of a projectile, where the
inclinations to the horizon are  ,    ,   2 and if t1 , t2 are the times of describing
1 1 2 cos 
the arcs AB, BC respectively, prove that V3t1  V1t2 and   . (2010)
V1 V3 V2
Q3. A projectile aimed at a mark which is in the horizontal plane through the point of
projection falls a meter short of it when the angle of projection is  and goes y meter
beyond when the angle of projection is  . If the velocity of projection is assumed same
in all cases, find the correct angle of projection. (2011)

Ph: 011-40079000, 9350934622 34

Unit 33: Constrained Motion
Q1. If a particle slides down a smooth cycloid, starting from a point whose actual distance
2 xb  2 xT 
from the vertex is b , prove that its speed at any time t is sin   , where T is the
T  T 
time of complete oscillation of the particle. (2003)
Q2. A particle is projected along the inner side of a smooth vertical circle of radius a so that
velocity at the lowest point is u . Show that 2ag  u 2  5ag . The particle will the highest
 
3
2 u 2  2ag
point and will describe a parabola whose latus rectum is .
27a 2 g 3
(2005)
Q3. Two particles connected by a fine string are constrained to move in a fine cylendrial tube
in a vertical plane. The axis of the cycloid is vertical with vertex upwards. Prove that the
tension in the string is constant throughout motion. (2005)
Q4. A particle is free to move on a smooth vertical circular wire of radius a . It is projected
horizontally from the lowest point with velocity 2 ag . Show that the reaction between
the particle and the wire is zero after a time

a
g
log  5 6 .  (2006)
Q5. A particle is projected with velocity v from the cusp of a smooth inverted cycloid down
a v
the arc. Show that the time of reaching the vertex is 2 cot 1 . (2009)
g 2 ag

Ph: 011-40079000, 9350934622 35
Unit 34: Central Orbits
Q1.   1
A particle of mass m moves under a force m 3au 4  2  a 2  b 2  u 5 , u  , a  b and
r
  0 being given constants. It is projected from as apse at a distance a  b with velocity

. Show that its orbit is given by r  a  b cos  , where  r ,   are the plane polar
ab
coordinates of a point. (2008)
Q2. A body is describing an ellipse of eccentricity e under the action of a central force
directed towards a focus and when at the nearer apse, the centre of force is transferred to
other focus. Find the eccentricity of the new orbit in terms of the original orbit. (2009)
Q3. A particle moves with a central acceleration   r 5  9r  being projected from an apse at a
distance 3 with velocity 3 2u . Show that its path is x 4  y 4  9 . (2010)

Ph: 011-40079000, 9350934622 36
Unit 35: Work, Energy and Impulse

Q1. A shot of mass m is projected from a gun of mass M by an explosion which generates a
2mE
kinetic energy E . Show that the gun recoils with a velocity  and the
M  M  m
2ME
initial velocity of the shot is .
M  M  m
Q2. A shell of mass M is moving with velocity v . An internal explosion generates an

amount of energy E and breaks the shell into two portions whose masses are in the ratio
m1 : m2 . The fragments continue to move in the original line of motion of the shell. Show
2m2 E 2m1 E
that their velocities are v  and v  .
m1M m2 M
Q3. A bullet of mass m moving with velocity v , strikes a block of mass M , which is free to
move in the direction of the motion of the bullet and is embedded in it. Show that a
M
portion of the K.E. is lost. If the block is afterwards struck by an equal bullet
M m
moving in the same direction with the same velocity. Show that there is a further loss of
mM 2v 2
K.E. equal to .
2  m  M  M  2m 
Q4. A gun of mass M fires a shell of mass m horizontally and the energy of explosion is
such as would be sufficient to project the shell vertically to a height h . Prove that the
1/ 2
 2m 2 gh 
velocity of the recoil is   .
 M  M  m  
Q5. A train of mass M lb is ascending a smooth incline of 1 in n and when the velocity of
the train is v ft/sec, its acceleration is f ft/sec2. Prove that the effective HP of the engine
Mv  nf  g 
is .
550 ng

Ph: 011-40079000, 9350934622 37

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Delhi Institute for Administrative Services

DIAS INDIA EDUTECH (PVT) LTD

CIVIL SERVICES EXAMINATION (MAINS)

and 2 x  3 y  4 z  5  0 and the point 1,1,1 .

Show that the locus of the centroid of tetrahedron O, A, B, C is x 2  y 2  z 2  16 p 2 .

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

Unit 17: Straight Line

line x  4  y  1  2  z  2  and parallel to the plane 4 x  5 z  y  0 . (2011)

the plane and line. (2013)

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

    b  b       a  a

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

Unit 18: SPHERE

coordinates of centre and its radius.

the centroid of the triangle ABC lies on the sphere a  x 2  y 2  z 2   4k 2

Q9. Prove that the circle x 2  y 2  z 2  2 x  3 y  4 z  5  0, 5 y  6 z  1  0 and

x 2  y 2  z 2  3x  4 y  5 z  6  0 , x  2 y  7 z  0 lie on the same sphere. Also find the

1, 2, 2  and passes through the origin.

coordinate axes, prove that a 2  b 2  c 2  r 2

x 2  y 2  z 2  6 x  8 y  4 z  20  0 intersect orthogonally. Find their planes of

1, 2,1 and cuts orthogonally the sphere x 2  y 2  z 2  4 x  6 y  4  0

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

Unit 19: CONE & CYLINDER

Q7. Show that ax 2  by 2  cz 2  2ux  2vy  2 wz  d  0 represents a cone if

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

Unit 20: Conicoids

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

an angle of 600 if a 2  b 2  6c 2 . Find also the condition for the generators to be

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

Unit 21: 2nd Degree Equation, Reduction to Canonical Form

Q3. Show that the equation 2 y 2  4 zx  2 x  4 y  6 z  5  0 represents a right circular cone.

Q4. Show that 2 x 2  2 y 2  z 2  2 yz  2 xz  4 xy  x  y  0

Q5. Reduce 3z 2  6 yz  6 zx  7 x  5 y  6 z  3  0 to standard for and find the nature of the

By giving different values to 1 , 2 , 3 &  from (1) can be reduced to

(ii) A  x 2  y 2   Cz 2  1 Ellipsoid of revolution

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

(iv) Ax 2  By 2  Cz 2  1 Hyperboloid of one sheet

(v) Ax 2  By 2  Cz 2  1 Hyperboloid of two sheet

(vi) A  x 2  z 2   By 2  1 Hyperboloid of revolution

(viii) Ax 2  By 2  1 Elliptic cylinder

(ix) Ax 2  By 2  1 Hyperbolic cylinder

(x) Ax 2  By 2  0 Pair of intersecting planes

(xi) Ax 2  1 or By 2  1 or Cz 2  1 Pair of parallel planes

Homogenous part f  x, y, z   ax 2  by 2   z 2  2 fyz  2 gzx  2hxy  .

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

CIVIL SERVICES EXAMINATION (MAINS)

Q7. Solve the D.E

Q9. Solve ydx   x  x 3 y 2  dy  0 (2008)

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

(i) If the D.E. Mdx  Ndy  0 is homogeneous, then Mx  Ny is an I.F. unless

the solution of the equation. (2012)

18/1, 2nd Floor, Old Rajender Nagar, New Delhi

Unit 23: D.E. of 1st order and Higher Degree

Q5. Find the orthogonal trajectory of a system of coaxial circles x 2  y 2  2 gx  c  0 where

Q6. Solve x 2 p 2  yp  2 x  y   y 2  0 , using the substitution y  u and xy  v and find its

Q10. Obtain Chairut’s form of the D.E.