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MATHEMATICS(HONOUS)PAPER-IV-2006
SECTION-I
1.Choose the correct answer of the following:
(a) Value of 𝜑 (360) is
(i)10 (ii)36 (iii)60 (iv)96
(b) The remainder , when 548 is divided by 12 is:
(i)11 (ii)9 (iii)1 (iv)7
(c) If 𝑛34 -9 is prime then n is equal to:
(i)6 (ii)8 (iii)9 (iv)none of these
(d) If S={1,2,3,4,5} then the number of transposition of S is:
(i)5 (ii)10 (iii)20 (iv)15
(e) the intersection of any two subgroups of a group is:
(i) Always a subgroup
(ii) not necessarily a subgroup
(iii) never a subgroup
(iv)none of these
(f) The equation y-2x=C represents the orthogonal trajectories of the family:
(i)y=C𝑒 −2𝑥 (ii) 𝑥 2 + 2𝑦 2 =C (iii)xy=C (iv)x+2y=C
(g) The singular solution of the equation 4𝑥𝑝2 = (3𝑥 − 𝑎)2 𝑖𝑠:
(i)x-a=0 (ii)x=0 (iii)3x-a=0 (iv)x+a=0
(h) Least force required to prevent a body of weight W from slipping down a rough plane
inclined at an angle ∝ with the horizontal is:
(i)W sin(∝ −𝜆) (ii)W sin(∝ +𝜆) (iii) W 𝑠𝑖𝑛(∝ +𝜆)/cos 𝜆 (iv) W 𝑠𝑖𝑛(∝ +𝜆)/sin 𝜆
(i) The differential of a particle executing S.H.M is:
𝑑2 𝑥
𝜇
𝑑2 𝑥
𝑑2 𝑥
𝑑2 𝑥
𝜇
(i) 𝑑𝑡 2 = 𝑥 2 (𝑖𝑖) 𝑑𝑡 2 = 𝜇𝑥(𝑖𝑖𝑖) 𝑑𝑡 2 = −𝜇𝑥 (𝑖𝑣) 𝑑𝑡 2 = − 𝑥 2
(j) The work done in stretching an elastic string of length / to double its length, when the
modulus of elasticity is X, is :
(i) 𝜆 l (ii) 𝜆 l/3 (iii) 𝜆 l/2 (iv) 𝜆 l/4
SECTION-II Group-A
12.(a) If a = b (mod n) and c = d (mod n) then show that:
(i)a + c≡b + d(modn) (ii) ac≡bd(modn)
(b) State and prove the fundamental theorem of Arithmetic.
13 (a) Show that538≡4(mod 11) .
(b) Prove that an integer p is a prime if and only if |p-1≡ −1 (mod p).
14. (a)Evaluate 𝜑(450)
1 𝑖𝑓 𝑛 = 1
(b)Prove the following for each integer n ≥1 :∑𝑑 𝜇(𝑑) {
0 𝑖𝑓 > 1
𝑛
15. (a) What will be the remainder when S4S is divided by 12 ?
(b) Solve the following system of linear congruence :
(i) x≡5(mod6)x = 4(mod 11)x = 3(mod 17)
Group – B
l6.(a) Prove that the set of four fourth roots of unity forms a group under ordinary multiplication,
(b) Prove that a non-empty subset H of a group G is a subgroup of G if and only if
a. b 𝜖 H => ab1 𝜖 H.
17. (a) Define homomorphism and isomorphism of groups.
(b) Let G and G' be any two groups and let e and e' be their identity elements. Let f.be a
homomorphism of G into G'. Then prove that the kernel of f is z normal subgroup of G.
18 (a) Define a cyclic permutation,
(b) State and prove Cayley's theorem.
19. (a) For any group G define left coset and right coset of a subgroup H of G determined by an
element a.
(b) Let G be a group and H be a subgroup of G. Let at a, b 𝜖 G. Then prove that;
(i)Ha=H↔a 𝜖 H (ii) Ha=Hb↔ 𝑎𝑏1 𝜖 H(iii) aH=bH↔ 𝑎−1 𝑏 𝜖 H
l0. (a) Prove that the relation of conjugacy on a group is an equivalence relation.
(b) Define Normalizer of an element of a group. Prove that for any element a
𝜖 G N (a) is a subgroup of G.
Group - C
11. (a) Solve sin px cosy = cos px . siny + p.
(b) Using substitution x + y = u and x2 + y2 = v transform the equation (x3 + y2) (𝑙 + 𝑝)2 −
2(𝑥 + 𝑦)(1 + 𝑝)(𝑥 + 𝑦𝑝) + (𝑥 + 𝑦𝑝)2 =0 to Clairau's form and hence find the complete
primitive.
12.(a)solve the following : (𝐷3 − 4𝑑2 + 5𝑑 − 2)𝑦 = 0
𝑑2𝑦
𝑑𝑦
(b) solve the following:𝑥 2 − 𝑑𝑥 2 + 4𝑥 𝑑𝑥 + 2𝑦 = 𝑥 + sin 𝑥
13.(a) Find the complementary function for the equation y2 + a2y = cosec ax
(b) Apply the method of variation of parameters to solve the equation
y2+xy = sec nx.
𝑑𝑥
14. (a) Solve the following: 𝑦 2 =
𝑑𝑦
𝑥2
𝑑𝑧
= 𝑥2𝑦2𝑥2
(b) Solve =the following:
𝑑𝑥
𝑥(𝑦 2 −𝑧 2 )
𝑑𝑦
𝑑𝑧
= 𝑦(𝑧 2 −𝑥 2 ) = 𝑧(𝑥 2 −𝑦 2 )
Group-D
5. (a) Prove that it is impossible for a particle to move from the rest so that its velocity varies
directly as the distance described from the commencement of motion.
(b) If in a simple harmonic motion u,v,w be the velocities at distances a,b,c from a fixed point
on the straight line which in not the centre of force show that the perodic time T is given by
the equation:
𝑢2 𝑣 2 𝑤 2
2
4𝜋(𝑏 − 𝑐)(𝑐 − 𝑎) = 𝑡 | 𝑎
𝑏
𝑐 |
1 1
1
16.(a) at a time t ,a moving point (x,y) has its co-ordinates given by the equation :
X=a(t+sin t)y=a(1-cos t) show that the particle moves with constant acceleration.
(b) Derive expression for the tangential and normal acceleration of a particle moving in the
plane.
17. (a)If x=a (cos𝜃+𝜃𝑠𝑖𝑛𝜃) and y=a(sin 𝜃 − 𝜃𝑠𝑖𝑛𝜃) and 𝜃 increases at a uniform rate W .then
show that the velocity of the point is a𝜃 w.
(b)prove that the work done in stretching an elastic string in equal to the extention produced
multiplied with the mean of initial and final tension.
18. (a) Define the angle of friction and the cone of friction.
(b) A body is placed on a rough plane inclined to the horizon at an greater than the angle of
friction and is supported by a force acting in a vertical plane through the line of greatest slope.
Then find the h between which the force must lie.
19. (a) Define Astatic Equilibrium.
(b)a square of side 2a is placed with its plane vertical between two smooth pegs which are in the
same horizontal line and at a distance C. Show that it will be in equilibrium when the inclination
𝑐2
of one of its edges to the horizon is either45°or 1/2 sin-1 𝑎2 − 𝑐 2 .
MATHEMATICS(HONOUS)PAPER-IV-2007
SECTION-I
1.
Choose the corVect answer of the following :
(a) The value of ∅ (7) 𝜇 (8) is
(i) 42 (ii)0 (iii) 56 (iv) 4
(b) The number of solutions of the congruence 8x = 4 (mod 20) is
(i) one (ii) two (iii) four (iv) No solution
(c) If a, b 𝜖 G, then G is abelian if
(i)𝑎𝑏 2 = 𝑎2 𝑏 2 (𝑖𝑖)(𝑎𝑏)2 = 𝑏 2 𝑎2 (𝑖𝑖𝑖)(𝑎𝑏)2 = 𝑎𝑏 (𝑖𝑣)(𝑎𝑏)2 = 𝑏𝑎
(d) If H, K are two subgroups of a group G, then HK is a subgroup of G if
(i) HK-KH (ii) H=K(iii) HK' = ∅ (iv) 𝐻 −1KH =∅
(e) If G is a finite group, then for any a 𝜖 g
(i) a0(G) = e (ii) o(G) = e (iii) o(a) = e (iv)𝑎𝑒 = o(G)
(f) If x be an element of a group G such that x.x = x,then
(i)x = e (ii)x = x l(iii)x= {e}(iv)None
(g) What is the order of the following differential equation ?
𝑑2𝑦
𝑑2 𝑦
2
𝑑2 𝑦
2
+ (𝑑𝑥 3 ) + (𝑑𝑥 2 ) + 8𝑦 = 5𝑥
𝑑𝑥 4
(i) First order (ii) Eighth order (iii) Fourth order (iv) Sixth order
(h) if y = px +f/(p), what is tile complete primitive of the equation ?
(i)y = cx +f/(c) (ii)p = g (iii)x + f(p) = 0(iv)y = p(x)+ f (p)
(i) The value of 1/d-2 sin x is
.
•
-1
(i) cos X + 2 (/7) cosx + sinx (iii) 1/√5sin (x + tan ½) (iv) cosx
(j) Least force necessary to move a weight W along a rough horizontal plane is
(i) W tan 𝜆 (ii) W sin 𝜆 (iii) W (iv) w/ tan 𝜆
SECTION-II GROUP-A
2. (a) If n > 1 and a"-l is prime, then prove that a = 2 and n is a prime.
(b) Prove that ax = ay (modulo m) if and only if
x ≡ y (mod m/a,m)
3. (a) State and prove 'Division Algorithm'.
(b) Using Euclidean Algorithm, find the greatest common divisor 1819 and 3587 and hence
find integers x and y such that 1819+3587y=g.
4. (a) Define 'MObius function’.
(b) Prove that ∅(n) is multiplicative.
5. (a) Solve 42x=50 mod (76).
(b) State and prove 'Format's Theorem'.
GROUP-B
6. (a) The set Q1 of all rationals other than 1 with operation * define as
a*b = a + b-ab form a system, find the inverse of element a.
(b) prove that the set Z of integers is an infinite Abelian group for the binary composition *
defined by
a * b=a+b+ 1
7. (a)prove that a subgroup H of a group G is normal if glhg 𝜖 /H for every h 𝜖 H, g 𝜖 G.
(b)State and prove 'Lagrange Theorem'.
8. (a) Define an odd permutation and an even permutation.
(b) Prove that a permutation cannot be.both even and odd.
9. (a) Define Automorphism of groups. Give one example.
(b) Prove that the set of all automorphisms of a group forms a group respect to composite of
functions as composition.
10. (a) What do you mean by Factor group ?
(b)Prove that the set of all cosets of a normal subgroup is a group with respect to
multiplication of complexes as composition.
GROUP-C
11. (a) Solve p2 -9p +18 = 0.
(b) Solve y = 2 px + y2 p3.
12. (a) Find the orthogonal trajectories of the series of logarithmic spirals𝑟 = 𝑎𝜃 , where a varies.
(b) Prove that the system of confocal conics
𝑥2
𝑦2
𝑎2 +𝜆
+ 𝑏2 +𝜆 = 1 𝑖𝑠 𝑠𝑒𝑙𝑓 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙
13. (a) What is the general solution of the following equation ?
(4𝐷2 + 4𝑑 + 1 )y = 0
(b) Solve:
𝑑2 𝑦
+ 4𝑦 = sin 2𝑥
𝑑𝑥 2
14. (a) Solve (y + z) dx + (z + x) dy + (x + y) dz = 0.
(b)State and prove necessary and sufficient condition for the integrability of the differential
equation Pdx + Qdy + Rdz = 0
GROUP-D
15. (a) What are the conditions of equilibrium of a rigid body ?
(b)A solid cone, of height h and semi-vertical angle a, is placed with I base against a smooth
vertical wall and is supported by a string attaché to the vertex and to a point in the wall,
Show that the greatest possible length of the string is
ℎ√1 +
16
9
tan2 ∝
16. (a) Define limiting equilibrium and coefficient of friction.
(b) A hemispherical shell rests on a rough inclined plane, whose angle c friction is 𝜃. Show
that the inclination of the plane base of the rim to the horizontal cannot be greater than sin-1
(2 sin 𝜃).
17.(a) A particle executing SHM in a straight line about a point O has two velocities V, and V2 at
distances x1 and 𝑥2 respectively from O. Find it periodic time.
(b) A particle is projected with a velocity V directed away from a point at distance b from the
point of projection. If the acceleration be 𝜇x(x=0 distance from the fixed point) and always
towards the fixed point, find the amplitude of Simple Harmonic Motion.
18. (a) Find the work done in extending an elastic light string to double its length.
(b) A light elastic string of natural length a and modulus of elasticity X b suspended by one
end, to the other end is tied a particle of weight mg the particle is slightly pulled down and
released. Discuss the motion
19. (a) Write down the expressions for the radial and transverse acceleration
(b) A point moves in a plane curve so that its tangential and normal accelerations are equal
and the tangent rotates with Constant angular velocity. Find the path.
MATHEMATICS(HONOUS)PAPER-IV-2008
1. Choose the Correct answer of the following :
(a) 11 [28 ] ! = p (mod9) then value of p is =
(i) 29 (ii) 28 (iii) 1 (iv) -1
(b) The value of ∅ (900) is :
(i) 240 (ii) 360 (iii) 120 (iv) 425
(c)
A binary operation * defined on the set Q+ of all positive
rationals by a * b = ab/3, a,b 𝜖 Q+. the identity element is
(i)1 (ii)3 (iii)1/3 (iv)9
(d) If a, b s G, a group, then G is abelian if:
(i) (ab)2 =𝑎2 𝑏 2 (ii) (ab)2 =𝑏 2 𝑎2 (iii) (ab)2 = ab (iv) (ab)2=ba
(e). G is a group of prime order p. If a a a 𝜖 G and a≠e then the order of a is :
(i) 1 (ii) p-1 (iii) p (iv) 0
(f). The orthogonal trajectories of the rectangular hyperbola xy = 𝑎2 is :
(i) x2 + y2 =c2 (ii) x2 - y2 =c2 (iii) 2x2 + 3y2 =c2 (iv) 4x2 + y2 =c2
(g) Singular solution of the equation y = xp +a/p is :
(i) x2 = 4ay (ii) y2=4ax (iii) (x-a)2=4y (iv) x2=(y-a)
(h) The solution of the differential equation (y dx + xdy) (a-z) + xy dz = 0 is given by :
(i) xy = c(a-z) (ii) yx2=c(z-a) (iii) xy2=c(a-z) (iv) x2y2=c(a-z) (i)
(i)
If 𝜇 is the coefficient of friction and 𝜆 is the angle of friction then :
(i) 𝜇= cosλ (ii) 𝜇= sinλ (iii) 𝜇= tanλ (iv) 𝜇= cotλ.
(j) If the velocity vary as (distance)", then:
(i) n >1/2 (ii) n > 1/2 (iii) n =1/2 (iv) n =0
Section-II Group - A
2. (a) Bind the general solution of 70x +112y= 168.
(b) Prpve that (a,b) [a,b] = ab, wheie a and b are two positive integers
3. (a) Prove that 41 divides 220 -1
(b) Prove that p is prime if anv only if (p-1) != -1 (mod p).
4. (a) Evaluate T (3000) and 𝜎- (3000).
(b) State and prove fundamental theorem of Airthmetic
5. (a) Show that 503 is prime
(b) Solve the system of linear congruence :
(i)x =2 (mod 3) (ii) X = 3 (mod 5) (iv) x= 2 (mod 7)
6. (a)Prove that identity element in a group is unique.
(b) Prove that every subgroup of a cyclic group is cyclic.
7. (a) If G be a group and let A, and B be normal subgroup of G. Show that A ∩ B is also a
normal subgroup of G.
(b) prove that every homomorphic image of a group G is isomorphic to some quotient group of
G.
8. (a) Define a normal subgroup of a group G with an example.
(b) State and prove Csyley's theorem on permutation group.
9. (a) Define a cyclic permutation.
(b) Prove that a non-empty subset H of a group G is a subgroup of G if an only if a, b 𝜖 H =>
ab 𝜖 H
10. (a)Define homomoiphism and isomorphism or group
(b) Show that anv quotient group G/m of a gioup G with respect to a noimal subgioup M is a
homomoiphic image of G.
Group – C
𝑦𝑑𝑦
ll. (a) Solve x dy + y dx + 𝑥𝑑𝑦 − 𝑥 2 +𝑦 2 = 0
(b) Using the substitutions y = u , xy = v, solve the equation
x2p2 + yp (2x+y) + 𝑦 2 =0 where p = and also find its singular solution.
12. (a) Find the orthogonal trajectories of r = a𝑒 𝜃𝑐𝑜𝑡𝑢 .
(b) Find the orthogonal trajectories of the family of co-axial trajectories x2 + y2 + 2gx + c = 0
where g is the parameter.
13. (a)Solve the following :
𝑑
(𝐷2 + 𝐷 + 1)𝑦 = 𝑥 2 ; 𝑤ℎ𝑒𝑟𝑒 𝐷 = 𝑑𝑥
(b)Solve (D-l)2y = x sinx.
14. (a)Solve the following :
𝑑𝑥
𝑑𝑦
𝑑𝑧
=
=
(𝑥 2 − 𝑦 2 ) 2𝑥𝑦 (𝑥 + 𝑦)𝑧
(b) Solve : yz (y +z) dx+zx (z+x) dy+xy (x+y)dz=0
Group – D
15. (a)Define static equilibrium.
(b)Forces P, Q, R act along the lines x = 0, y=0 and x cos𝛼t + y sin𝛼 = p. Find the magnitude
of the resultant and the equation of its line of action.
16. (a)Define angle of friction and core of friction. A solid hemisphere of weight w rests in
limiting equilibrium with its curved surface on a rough inclined plance and its plane face is kept
horizontally by a weight p attached to a point on its rim. Prove that the co-efficient
𝑝
of friction is
√𝑤(2𝑝−𝑤)
17. (a)Define Simple Harmonic Motion and with its differential equation.
(b)A praticle starts from the origin and the. components of its velocity parallel to the axis of
co-ordinates at time t are 2t + 3 and 4t. Find the path.
18. (a) Define impulsive Force and its impulse.
(b)A particle starts from rest and moves along a straight line with an acceleration which is
always directed towards a fixed point and varies as the distance from the fixed point. Discuss
the motion.
19. (a) Define conservative and non-conservative forces.
(b) A-particle whose mass is m, is acted upon by a force m𝜇.
𝑎4
(𝑥 + 𝑥 3 ) towards the origin, if it starts from rest distance a, show that it will arrive at the
origin in time 𝑎
𝜆
√𝜇
.
20. (a) Prove that the energy of a stretched elastic string is equal to half the product of the
tension and the extension.
(b) Find the extension of a heavy elastic string of weight W and natural length I hanging
from on end and supporting a weight W at the other, where X is the modulus of
elasticity of the string.