SPMVVPGCET – 2013-14 APPLIED MATHEMATICS HALL TICKET NO: Time : 90 minutes Marks : 100 ______________________ Signature of the Candidate _______________________ Signature of the invigilator INSTRUCTIONS TO THE CANDIATES 1. The question paper contains three sections. Section-A contains 30 questions, Sectioncontains 30 questions and Section-C contains 40 questions. 2. Make sure that all the 100 questions are printed properly in the booklet. If the question paper booklet is defective, ask the Invigilator of your hall for replacement. 3. Do not write anything in the question paper booklet 4. Use blank pages marked ‘space for rough work’ for rough work. 5. Mark all your answers with HB pencil on the OMR sheet only. Answers marked in the question paper booklet will not be valued. 6. Read carefully the instructions on Side –I of the OMR sheet before answering. 7. At the time of leaving the examination hall, return both the question paper booklet and the OMR answer sheet to the Invigilator of the hall. SPMVVPGCET – 2013-14 APPLIED MATHEMATICS SECTION-A (DIFFERENTIAL EQUATIONS AND SOLID GEOMETRY) 1. The order of the differential equation dy dx (A) 3 (B) 4 (C) 2 + d2y dx2 = k d3y is dx3 (D) 1 2. The Degree of the differential equation a d2y = 1+dy2 3/2 is Dx2 dx (A) 3 (B) 4 (C) 2 (D) 1 3. The integrating factor of (1+y2) dx =(tan-1y__x) dy is Tan-1 (B) 1 (C) etan-1 y (D) ex X (A) tan-1 y (B) (C) etan-1 Y (D) ex 4. The general solution of 3ex tan y dx+(1_ex) sec2 y dy = 0 3ex tany dx+ (1-ex) sec2y dy = 0 (A) tan y = c(1-ex)3 (B) sec y = c(1-ex)3 (C ) tan y = c(1-ex)2 (D) sec y = c(1-ex)2 5. Which of the following is exact equation? (A) 2xy dx+(y2+x2) dy = 0 (B) (ysin 2x) dx-(y2+cos2x) dy = 0 (C ) (a2 – 2xy – y2) dx – (x-y)2 dy= 0 (D) (xy3 + y) dx+ 2(x2 y2 +x+y4) dy = 0 6. The Equation x2 – 2y2 + 2xy dy = 0 is Dx (A) Non-homogeneous (B) homogeneous (C ) exact (D) linear 7. The solution of x dx+ ydy + xdy-ydx = 0 is (A) X2+y2 + 2tan-1 x = 2c (B) x2+ y2+ 2tan -1 Y y = 2c x (C ) x2+ y2 + 2sin-1 x Y = 2c (D) x2+ y2 - tan -1 y 2 x = c 8.The orthogonal trajectories of the family of curves x2/3+y2/3 = a2/3 are (A) X4/3 - y4/3=c (B) y4/3 - x4/3 = c (D) x2/3 + y2/3 = c (C ) x4/3 + y4/3 =c 9.The general solution of P2 – 5P + 6 = 0, where P = dy is dx (A) (y-2x-c)(y+3x-c)=0 (B) (y+2x-c)(y-3x-c) = 0 (C ) (y-2x-c) (y-3x-c) =0 (D) (y+2x-c) (y+3x-c) = 0 10. The solution of (y-px) (p-1) = P is (A) Y = cx + c-1 C (C) y = cx + c c-1 11. The general solution of (D3 + 1) y = 0 is (B) y = cx – c c-1 (D) y= - cx – c-1 c (A) Y = c1 e-x +ex/2 c2cos √3 x + c3 sin √3 x 2 2 (B) y = c1 e1/2x + e-x c2cos √3 x+ c3 sin √3 x 2 2 (C ) y = c1 e-x e x/2 c2cos √3 x+c3 sin √3 x 2 2 (D ) y= c1 ex + e x/2 c2cos √3 x+ c3 sin √3 x 2 2 2 12. The particular integral of (D +2D+1)y = sin 3x is (A) 2cos3x+8sin3x (B) 3cos3x+4sin3x 100 50 (C ) 4cos3x+3sin3x 3cos3x+4sin3x -50 100 2 2 13. The complementary function of (D +a )y = tan ax is (A) yc = c1 cosax + c2sinax (B) yc = c1 sinax+c2 Cosax ax –ax (C ) yc = c1e +c2 e (D) yc = (c1 + c2x)eax 14 . If the particular integral of (D2 + a2) y = tan ax is (A) X (B) 1 (C) -1 (D) -x X x 15. The general solution of y = a √1 P2 is (B) y= a sin h x (A) Y = sin x + C a a (C ) y = a cos x + c (D) y = a cosh x a a + C + c 16. The equation of the plane through (-1,2,3) and whose normal has the direction ratios (3,4,5) is (A) 3x + 4y + 5z + 20 = 0 (B) x- 2y – 3z + 20 = 0 (C ) 3x + 4y + 5z – 20=0 (D) x -2y -3z -20 =0 17. The angle between the planes 2x -3y + 4z + 11 = 0 and 3x – 2y -3z + 27 = 0 is (A) (B) (C) (D) 2 3 4 6 18. The equation of the plane passing through the points (1,-2,4) and (3, -4,5) and Perpendicular to the xy plane is (A) Y-x+1 = 0 (B) x-y+1= 0 (C) x+y+1 = 0 (D) x+y-1 = 0 19. If the equation 6x2 +4y2-10z2 -11 xy + 3yz + 4zx = 0 (A) (B) (C) (D) 6 4 3 2 20. The equation of the line through (3,1,2) and equally inclined to the axes is (A) x-3 = y-1 = z-2 3 1 2 (C ) x-1 = y-1 = z-1 3 1 2 (B) x-3 = y-1 = z-2 1 1 1 (D) x-3 y-1 z-2 -1 1 1 21. Find the value of k so that the lines x+1 = y=2 = z-3 and x-1 = y+5 = z+6 -3 2k 2 3k 1 7 are perpendicular (A) -2 (B) (C) 2 (D) 5 22. The equation of the plane passing through the line (A) 4y -3z+1 = 0 (C ) 4y +3z-1= 0 = = (B) 3y-4z +1 = 0 (D) 4y – 3z -1 =0 23. The symmetric form of equation of the line x+y+z+1 = 0 = 4x+y -2z + 2 is (A) x-1 = y -2 (B) x+ 1 = y+1 3 3_ 3 3 _____ _______ = z _____ _____ = z 1 -2 1 2 -1 1 (D) x + 1 = y - 2 = z (C ) x + 1 = y + 2 3 3 1 2 1 ____ _____ = z 3 3 -1 2 1 24. The equation of the plane passing through the origin and the line x-3y+2z+3=0 = 3x-y+2z-5 (A) 9x -7y + 8z = 0 (B) 7x – 9y + 8z = 0 (C ) 7x – 9y – 8z =0 (D) 9x + 7y + 8z = 0 25. The centre of the sphere (x-x1) (x-x2) + (y-y1)(y-y2) + (z-z1) (z-z2) = 0 is (A) (x1, y1, z1) (B) (x1+x2, y1+y2, z1+z2) (C ) x1+x2, y1+y2 , z1+z2 2 2 2 (D) (x2, y2, z2) 26. The Circle x2+ y2 + z2 - 4x – 2y + 5z + 6 = 0, x + Y + 2z + 2 = 0 is a (A) small circle (B) great circle (C ) point circle (D) none 27. Find the plane of contact of (3, -1, 5) with reference to the sphere x2+ y2+ z2 – 2x + 4y + 6z – 11 = 0 (A) 2x – y + 8z + 1 = 0 (B) 2x+ y+ 8z +1 = 0 (C ) 2x – y – 8z – 1 = 0 (D) 2x +Y+ 8z – 1 =0 28. The relation between the spheres x2+ y2+ z2 +6y + 2z + 8 = 0 and (A) intersect each other (B) concentric (C ) orthogonal (D) touch externally 29. The reciprocal cone of ax2 + by2+ cz2 = 0 is (A) x2 + y2 + z2 = 0 (B) x2 - y2 + z2 = 0 -a b c a b c (C ) x2 + y2 + z2 = 0 (D) x2 + y2 – z2 = 0 A b c a b c 30. The equation of the right circular cylinder with radius r and axis x-a = y-b = z-y is 1 m n 2+ 2 2 2 2 2+ 2 2 (A) [(x-a) (y-b) + ( z-y) + r ] (l +m n ) = [l(x-a) –m (y-b) + n (z-y)] (B) [(x-a)2- (y-b)2- (z-y)2 r2] (l2+ m2+n2) = [l(x-a) + m(y-b) +n (z-y)]2 (C) [(x-a)2 + (y-b)2 + (z-y)2- r2 ] (l2-m2- n2) = [(l(x-a) + m (y-b)+ n(z-y)2] (D) [(x-a)2 + (y-b)2+ ( z-y)2 + r2 ] (l2+m2+n2) = [l(x-a) –m (y-b) + n (z-y)]2 SECTION – B (ABSTRACT ALGEBRA AND REAL ANALYSIS) 31. If ‘a’ is an element of a group such that 0(a) = n, then am = e (A) mn (B) n/ m (C) m/n (D) m+n 32. Find the order of the element ‘2’ of the group G={0,1,2,3,4,5}, the composition (A) 2 (B) 6 (C) 1 (D) 3 33. The improper normal subgroups of a group G are (A) {e},N (B) G,N (C) {e}, G (D) none 34. A group G is called simple if it has no (A) Proper subgroups (C ) improper subgroups (B) Psroper normal subgroups (D) improper normal subgroups 35. The statement of the fundamental theorem on homomorphism of groups, is (A) Every subgroup of prime order of cyclic group is cyclic (B) Every normal subgroup of a group is normal (C) Every homomorphic image of a group G is isomorphic to some quotient group of G (D) None 36. The number of generators of a cyclic group of order 8 is (A) 2 (B) 4 (C) 5 (D) 8 37. If the set G of rational numbers except 1 forms a group under binary operation (A) a (B) a-1 (C) a a a-1 38. The number of subgroups of a group of order 23 is (A) 2 (B) 22 (C) 23 1 a-1 (D) 1 39. The inverse element of ‘I’ in the multiplicative group {1, -1, I, -i} is (A) 1 (B) - 1 (C ) -i (D) i 40. (N, +) is a (A) group (B) monoid (C) semi group (D) abelian group 41. “The order of a subgroup of a finite group divides the order of the group” is the statement Of (A) Cauchy’s theorem (B) Fundamental theorem on homomorphism (C) Lagrange’s sdtheorem (D) Cayley’s theorem 42. If f = (2 3 4 1) is of degree 5, then f-1 = (A) (1 4 3 2 5) 43. If f = (A) (B) (1 3 4 2) 12345 53241 (C) (1 2 3 4 5) and g = 1 2 3 4 5 4 31 2 5 12345 42531 (D) (1 4 3 2) then gf = (B) 1 2 3 4 5 51324 (C ) 1 2 3 4 5 12534 (D) 1 2 3 4 5 25134 44. The number of elements of the alternating group A6 is (A) 6 (B) 720 (C) 360 (D) 36 45. The product of two even permutations is ______________ permutation. (A) Odd (C ) either even or odd (B) (D) even null 46. The null set ‘Ø’ is considered to be (A) Infinite (B) finite (C) aggregate (D) unbounded 47. If v is a lower bound and u is an upper bound of an aggregate, then (A) u< v (B) v<u (C) u ≠v (D) u=v 48. If every neighborhood of p has a point of S other than p itself, then p eR is said to be ___________ of a subset S of R (A) lower bound (B) upper bound (C) limit (D) limit point 49. If f(x) is strictly increasing at x = a, then (A) f’(a) > 0 (B) f’(a) <0 (C) f’(a)> 0 (D) f’(a) = 0 50. If |r|<1 or -1 <r<1, then the series ∑ (A) diverges (B) converges rn , (r R) is (C) oscillates (D) none 51. If Un = nn2 , then lim (n+1)n2 (A) e (B) 0 (C) 1 (D) 1/ 1 52. The alternating series 1 (A) Converges 1 1 + +.... 22 32 42 (B) diverges (C) absolutely converges (D) absolutely diverges 53. If f(x) = x – [x] at x=1, then f is (A) continuous (C ) Jump discontinuous (B) removable discontinuous (D) discontinuous 54. “If f : [a,b] -> R os continuous at c e (a,b) and f(c ) = 0, then exists σ > 0 such that x Є( c-σ, c+σ) => c(x) has the same sign as f(C )” is related to (A) Neighbourhood property (C ) Ollation property (B) Boundedness property D) Intermediate value property 55. If f(x) = x tan-1 1 for x‡ 0 and f(x) 0 for x=0, then R f’(0) = x (A) π (B) – π (C) 0 2 2 (D) 1 56. If f is bounded on [a,b] and p be a partition of [a,b], then L (P,f) is (A) < m(b-a) (B) > m(b-a) (C) < M(b-a) (D) > M(b-a) 57. 1∫ex dx = (A) 1 58. x + x3 + x5 3! 5! (A) Sinhx (B) e + (C) 2e (D) 0 x7 + … 00 is the expansion of 7! (B) coshx (C) sin x (D) cos x 59. f(x) = sin 1 is differentiable at (A) Zero only (C ) every point in R – {0} (B) every point in R (D) every point in Q 60. 3∫[x] dx= 0 (A) 0 (B) 3 (C) 9 2 (D) 1 SECTION - C (LINEAR ALGEBRA AND VECTOR CALCULUS) 61. If V (F) is a vector space and 0 Є V, then _______________ , a Є V (A) A.0=0 (B) 0.a = 0 (C) aa = 0 (D) 0.a = _ 0 62. If w = {(a1,a2,0) / a1, a2, ЄF}, then w is _______________ of vector space. (A) Subspace (B) basis (C) linearly independent subset (D) linearly dependent subset 63. If a ≠ 0, b ≠ 0 Є F and aa + bB = 0 for a, B Є V, then a, B are (A) Linearly dependent (C ) spans (B) linearly independent (D) none of these 64 If w1, w2, are two subsoaces if V (F), then w1! w2 is a subspace, if (A) W1 – W2 = {0} (B) W1 " W2 (C) W1 ∩ W2 = φ (D) W1 + W2 = {0} 65. The set S = {(1,0,0) (0,1,0) (0,0,1)} form a basis for Vn ® if n equals (A) 2 (B) 3 (C ) 4 (D) 1 66. Which of the following sets is linearly dependent? (A) {0} (B) {Ø} (C) {1} (D) {a} 67. Which of the following is not a vector space? (A) R(R) (B) C(C) (C) R(C) (D) C(R) 68. For what value of m1, the vector (m1, 3,1) is a linear combination of e1 = (3,2,1), e2 = (2,1,0)? (A) 3 (B) 2 (C ) 4 (D) 5 69. For what value of ‘α’ the vectors (α,1,0), (1,α,1) and (0, 1, α) in R3 are linearly dependent? (A) α = 0, 1 (B) α = #1, #2 (C) α = 0, #√2 (D) α = 0, # 2 70. If T ≠ 0, then T0 is (A) Identity transformation (C ) singular (B) null transformation (D) non-singular 71. Which of the following is a linear transformation? (A) T(x,y) = (x2,y) (C ) T(x,y) = (x+y, x-y) (B) T(x,y) = (sinx,y) (D) T(x,y) = |x-y| 72. If T: R2 R2 such that T(α, b) = (0, α) % (α, b) (A) (a, b) (B) 0 (C) (α2, b) R2 , then T2 (α, b) = (D) (α - b, b) 73. The nullity of a linear transformation T: U -> V is equal to the dimension of (A) Range space of T (C ) U (B) kernel of T (D) V 74. IF V* is a dual space of V, the dim V = (A) dim W (B) dim w* (C ) dim V* (D) dim v/w 75. If w1 and w2 are subspaces of a vector space V(F), then (w1+ w2)0 is equal to (A) w10 ∪ w20 (B) ) w10 ∩ w20 (C) w10 + w20 (D) w10 - w20 76. If a, β are orthogonal unit vectors, then ||α-β|| = (A) 0 (B) (C) √2 1 (D) 2 77. In the vector space V3 (R) with standard inner product, if θ is the angle between two nonzero vectors, then (A) |cosθ|>1 (B) |cosθ|<1 (C) | cosθ|< 1 (D) |cosθ| > 1 78. In the inner product space R2 , the unit vector are (A) 1 (B) (1, 0) (0,1) 79. If (a,β) Є V(R ) and ||a|| = ||β||, then a –β and a+β are (A) orthogonal (C ) 0 (B) orthonormal (D) linearly dependent 80. If u, v are two vectors in a complex inner product space with standard inner product, then ||u + v||2 ||u-v||2 + I ||u+iv||2 – I ||u-iv||2 is (A) 0 (B) 2<u, v> (C ) <u, v> (D) 1 81. if φ = 2xz4 – x2 y, then find grad φ at (2,-2,-1) (A) 10 i – 4j – 16 k (B) 6 i – 4 j – 16 k (C) 10 i – 2 j – 16 k (D) 10 i + 4 j – 16 K 82. Find the unit normal at (1,1,-2) to the surface φ = x3 + y3 + 3xyz (A) - i + j + k (B) - i – j – k √3 (C) - i – j + k √3 √3 (D) i + j + k √3 83. If F = 3xyi – y2j , then find & ' dr , where C is the curve y = 2x2 from (0, 0) to (1, 2) in the C XY – Plane ( ( ) ) (A) (B) (C) (D) ) ) ( ( 84. Using Green’s theorem, find ∫(xy + y2) dx + x2 dy, where C is the closed curve between y=x and y= x2 (A) - (B) * * (C) 20 85. Determine the constant ‘a’ so that the vector Solenoidal. (A) 2 (B) 0 (C ) -1 (D) -20 = (x+y)İ + (y-2z)j + (x+az) k is (D) -2 86. Using gauss divergence theorm, find ∫ r N dV (A) V (B) 2V (C) 4V (D) 3V 87. curl grad ø = (A) 0 (B) 1 (C ) -1 (D)ō 88. Find the angle between the surfaces x2+y2+z2 = 9 and z = x2+ y2- 3 at (2, -1, 2) 4 (B) θ= cos-1 2 (A) θ= cos-1 3√21 3√21 1 8 -1 -1 (C ) θ= cos 3√21 (D) θ= cos 3√21 89. The directional derivative of Ø(x,y,z) = x2yz+ 4xz2 at the point (1,-2,-1) in the direction of (A) 37 (B) 1 (C) 37 (D) 3 3 3 37 90. A particle moves along a curve whose parametic equations are x=e-1, y= 2 cos3t,z = 2 sin3t, where t is the time. Find its velocity (A) - e-t i – 6 sin 3t j + 6 cos 3t k (C ) - e-t i – 6 cos 3t j + 6 sin 3t k 91. div(curl F) = (A) 0 (B) 0 (B) (D) e-t i + 6 sin 3t j + 6 cos3t k e-t i + 6 sin 3t j - 6 cos3t k (C) 1 (D) 2 92. A vector function whose curl is the null vector is called ______________ function. (A) Solenoidal (B) Rotational (C ) Irrotational (D) Potential 93. If f = 3x2 i + 5xy2 j + xyz3K , then find div f at (1, 2, 3) (A) 80 (B) -80 (C) 40 (D) -40 94. Find the rate of increase of φ = x2yz3 at (2, 1, -1) (A) √160 (B) 14 (C) 44 (D) 43 95. Using gauss divergence theorem, find ∫∫F n dS, where = F = 4xzi – y2j + yzk and S is the s surface of the cube bounded by x = 0, x = 1,y = 0, Y = 1, z = 0, z = 1 (A) (B) (C) (D) 96. ∇ log|r| = r r (A) r2 (B) r2 (C) r (D) - r 97. If S is an open surface bounded by a closed curve C and F ia any continuously differentiable vector field, then & '. dr = c +' +' (A) & ,-. F × N dS (B) ∫∫ +/ +0 dx dy R (C ) ∫ curl F . N dS (D) ∫ grad F N dS 98. ∇ × (φ A) = (A) (∇φ) A + φ ( ∇A ) (C ) (∇φ) × A - φ ( ∇ × A ) 99. If ∇2 φ = 0, then ∇ φ is (A) Rotational (B) Solenoidal (B) (∇φ) A - φ ( ∇A ) (D) (∇φ) × A + φ ( ∇ × A ) vector (C) Irrotational (D) Null 100. If x = at2 , y = 2at are the equations of a curve C, 0 ≤ t ≤ 2 , then find ∫ (A) 2 log 4 (B) 2 log 6 (C) 0 1 2 C (D) 2 log 2

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