Amity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -1 -AIEEE 2005 MATHEMATICS 1. The differential equation representing the family of curves ( ) 2 2 2 y c C = + , where C > 0, is a parameter, is of order and degree as follows : (1) order 1, degree 3 (2) order 2, degree 2 (3) order 1, degree 2 (4) order 1, degree 1 2. Area of the greatest rectangle that can be inscribed in the ellipse 2 2 2 2 1 x y a b + = is (1) ab (2) ab (3) 2ab (4) ab 3. 2 2 2 2 2 2 2 1 1 2 4 1 sec sec sec 1 nLimn n n n n ®¥é ù + + + ê ú ë û L equals (1) tan 1 (2) 1 tan1 2 (3) 1 sec1 2 (4) 1 cosec1 2 4. If the cube root of unity are 1, w, w2 then roots of equation (x – 1)3 + 8 = 0, are (1) –1, 1 – 2w, 1 – 2w2 (2) –1, 1 + 2w, 1 + 2w2 (3) –1, –1 + 2w, –1 – 2w2 (4) –1, –1, –1 5. If A2 – A + I = 0, then the inverse of A is (1) A – I (2) I – A (3) A + I (4) A 6. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is (1) an equivalence relation (2) reflexive and symmetric only (3) reflexive and transitive only (4) reflexive only 7. If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately (1) 25.5 (2) 24.0 (3) 22.0 (4) 20.5 8. Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid point of PQ is (1) x2 + 4y + 2 = 0 (2) x2 – 4y + 2 = 0 (3) y2 – 4x + 2 = 0 (4) y2 + 4x + 2 = 0 9. If C is the mid point of AB and P is any point outside AB, then (1) 2 0 PA PB PC + + = uuur uuur uuur r(2) 0 PA PB PC + + = uuur uuur uuur r (3) 2 PA PB PC + = uuur uuur uuur (4) PA PB PC + = uuur uuur uuurAmity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -2 -Mathematics AIEEE- 2005 10. ABC is a triangle. Forces P, Q, R acting along IA, IB and IC respectively are in equilibrium, where I is the incentre of DABC. Then P : Q : R is (1) cos :cos :cos 2 2 2 A B C (2) cos A : cos B : cos C (3) sin A : sin B : sin C (4) sin :sin :sin 2 2 2 A B C 11. In a triangle PQR, . 2 R p Ð = If tan 2P æ ö ç ÷ è ø and tan 2Q æ ö ç ÷ è ø are the roots of ax2 + bx + c = 0, a ¹ 0 then (1) b = c (2) b = a + c (3) a = b + c (4) c = a + b 12. If the coefficient of rth, (r + 1)th and (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation (1) m2 – m(4r + 1) + 4r2 – 2 = 0 (2) m2 – m(4r – 1) + 4r2 + 2 = 0 (3) m2 – m(4r – 1) + 4r2 – 2 = 0 (4) m2 – m(4r + 1) + 4r2 + 2 = 0 13. Let f : (–1, 1) ® B, be a function defined by 1 2 2 ( ) tan , 1 x f x x - = - then f is both one-one and onto when B is interval (1) , 2 2 p p é ù -ê ú ë û (2) , 2 2 p p æ ö -ç ÷ è ø (3) , 0 2p æ ö ç ÷ è ø (4) 0, 2p é ö÷ êë ø 14. If the coefficient of x7 in 11 2 1 ax bx é ù æ ö +ç ÷ ê ú è ø ë û equals the coefficient of x–7 in 11 2 1 ax bx é ù æ ö -ç ÷ ê ú è ø ë û , then a and b satisfy the relation (1) 1 ab = (2) ab = 1 (3) a – b = 1 (4) a + b = 1 15. If 13 z w z i = - and |w| = 1, then z lies on (1) a straight line (2) a parabola (3) an ellipse (4) a circle 16. If a2 + b2 + c2 = –2 and 2 2 2 2 2 2 2 2 2 1 (1 ) (1 ) ( ) (1 ) 1 (1 ) , (1 ) (1 ) 1 ax b x c x fx a x bx c x a x b x c x + + + = + + + + + + then f(x) is a polynomial of degree (1) 3 (2) 2 (3) 1 (4) 0 17. If z1 and z2 are two non-zero complex numbers such that |z1 + z2| = |z1| + |z2|, then arg z1 – arg z2 is equal toAmity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -3 -AIEEE-2005 Mathematics (1) 0 (2) 2 -p (3) 2p (4) –p 18. The value of a for which the sum of the squares of the roots of the equation x2 – (a – 2)x – a – 1 = 0 assume the least value is (1) 3 (2) 2 (3) 1 (4) 0 19. If the roots of the equation x2 – bx + c = 0 be two consecutive integers, then b2 – 4c equals (1) 2 (2) 1 (3) –2 (4) 3 20. The system of equations ax + y + z = a – 1 x + ay + z = a – 1 x + y + az = a – 1 has no solution, if a is (1) not –2 (2) 1 (3) –2 (4) either –2 or 1 21. The value of 6 50 56 4 3 1 r r C C - = +å is (1) 56C3 (2) 56C4 (3) 55C4 (4) 55C3 22. If 1 0 1 1 A é ù =ê ú ë û and 1 0 , 0 1 I é ù =ê ú ë û then which one of the following holds for all n ³ 1, by the principle of mathematical induction (1) An = nA + (n – 1)I (2) An = 2n–1A + (n – 1)I (3) An = nA – (n – 1)I (4) An = 2n–1A – (n – 1)I 23. If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number (1) 603 (2) 602 (3) 601 (4) 600 24. If 0 , n n x a ¥= =å 0 , n n y b ¥= =å 0 n n z c ¥= =å where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1 then x, y, z are in (1) Arithmetic – Geometric Progression (2) HP (3) GP (4) AP 25. If x is so small that x3 and higher powers of x may be neglected, then 3 32 12 1 (1 ) 1 2 (1 ) x x xæ ö + - + ç ÷ è ø - may be approximated asAmity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -4 -Mathematics AIEEE-2005 (1) 2 38 x - (2) 2 3 2 8 x x - (3) 2 3 1 8 x - (4) 2 3 3 8 x x + 26. If 1 1 cos cos , 2y x - - - =a then 4x2 – 4xy cosa + y2 is equal to (1) 4sin2a (2) –4sin2a (3) 2sin2a (4) 4 27. If in a DABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in (1) Arithmetic – Geometric Progression (2) H.P. (3) G.P. (4) A.P. 28. In a triangle ABC, let . 2 C p Ð = If r is the inradius and R is the circumradius of the triangle ABC, then 2(r + R) equals (1) a + b + c (2) c + a (3) b + c (4) a + b 29. A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? (1) 1 , 3 æ ù -¥ ç ú è û (2) ( ] , 4 -¥ - (3) ( , ) -¥ ¥ (4) [ ) 2, ¥ 30. Let a and b be the distinct roots of ax2 + bx + c = 0, then 2 2 1 cos( ) ( ) x ax bx c Lim x ®a- + + -a is equal to (1) 2 2 ( ) 2a - a - b (2) 2 1( ) 2 a-b (3) 2 2 ( ) 2 a a-b (4) 0 31. The normal to the curve x = a (cosq + qsinq), y = a (sinq + qcosq) at any point ‘q’ is such that (1) it passes through , 2 a a p æ ö - ç ÷ è ø (2) it is at a constant distance from the origin (3) it passes through the origin (4) it makes angle 2p + q with the x-axis 32. Let f be differentiable for all x. If f(1) = –2 and f ¢(x) ³ 2 for x Î [1, 6], then (1) f(6) < 5 (2) f(6) = 5 (3) f(6) ³ 8 (4) f(6) < 8 33. If f is a real-valued differentiable function satisfying |f(x) – f(y)| £ (x – y)2, x, y Î R and f(0) = 0, then f(1) equals (1) 2 (2) 1 (3) –1 (4) 0Amity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -5 -AIEEE-2005 Mathematics 34. Suppose f(x) is differentiable at x = 1 and 0 1 lim (1 ) 5, h f h h ® + = then f ¢(1) equals (1) 5 (2) 6 (3) 3 (4) 4 35. 2 2 (log 1) 1 (log ) x dx x ì ü - í ý + î þ ò is equal to (1) 2 1 x xe C x + + (2) 2 (log ) 1 x C x + + (3) 2 log (log ) 1 x C x + + (4) 2 1 x C x + + 36. A spherical ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is (1) 1 cm/min. 54p (2) 5 cm/min. 6p (3) 5 cm/min. 36p (4) 1 cm/min. 18p 37. Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), xaxxi and the ordinates 4 x p = and 4 x p = b > is sin cos 2 . 4p æ ö b b+ b+ b ç ÷ è ø Then 2 f p æ ö ç ÷ è ø is (1) 1 2 4p æ ö - - ç ÷ è ø (2) 1 2 4p æ ö - + ç ÷ è ø (3) 2 1 4pæ ö + - ç ÷ è ø (4) 2 1 4pæ ö - + ç ÷ è ø 38. Let f : R ® R be a differentiable function having f(2) = 6, 1 (2) . 48 f æ ö ¢ =ç ÷ è ø Then ( ) 3 2 6 4 lim 2 f x x t dt x ® - ò equals (1) 12 (2) 18 (3) 24 (4) 36 39. The area enclosed between the curve y = loge (x + e) and the coordinate axes is (1) 3 (2) 4 (3) 1 (4) 2 40. The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom; then S1 : S2 : S3 is (1) 2 : 1 : 2 (2) 1 : 1 : 1 (3) 1 : 2 : 1 (4) 1 : 2 : 3 41. If 2 1 1 02 , x I dx = ò 3 1 2 02 , x I dx = ò 2 2 3 12 , x I dx = ò and 3 2 4 1 2x I dx = ò then (1) I3 = I4 (2) I3 > I4 (3) I2 > I1 (4) I1 > I2 42. The line parallel to the x-axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx – 2ay – 3a = 0, where (a, b) ¹ (0, 0) is (1) above the x-axis at a distance of 32 from it (2) above the x-axis at a distance of 23 from itAmity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -6 -Mathematics AIEEE-2005 (3) below the x-axis at a distance of 32 from it (4) below the x-axis at a distance of 23 from it 43. If (log log 1), dy x y y x dx= - + then the solution of the equation is (1) log y cx x æ ö= ç ÷ è ø (2) log x cy y æ ö= ç ÷ è ø (3) log x y cy y æ ö= ç ÷ è ø (4) log y x cy x æ ö= ç ÷ è ø 44. If a vertex of a triangle is (1, 1) and the mid points of two sides through the vertex are (–1, 2) and (3, 2), then the centriod of the triangle is (1) 7 1, 3 æ ö ç ÷ è ø (2) 1 7 , 3 3 æ ö ç ÷ è ø (3) 7 1, 3 æ ö -ç ÷ è ø (4) 1 7 , 3 3 -æ ö ç ÷ è ø 45. If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct points P and Q then the line 5x + by – a = 0 passes through P and Q for (1) infinitely many values of a (2) exactly two values of a (3) exactly one value of a (4) no value of a 46. If non-zero numbers a, b, c are in H.P., then the straight line 1 0 x y a b c + + = always passes through a fixed point. That point is (1) (1, –2) (2) 1 1, 2 æ ö - ç ÷ è ø (3) (–1, 2) (4) (–1, –2) 47. If a circle passes through the point (a, b) and cuts the circle x2 + y2 = p2 orthogonally, then the equation of the locus of its centre is (1) x2 + y2 – 2ax – 3by + (a2 – b2 p2) = 0 (2) 2ax + 2by – (a2 + b2 + p2) = 0 (3) x2 + y2 – 3ax – 4by + (a2 + b2 – p2) = 0 (4) 2ax + 2by – (a2 – b2 + p2) = 0 48. An ellipse has OB as semi minor axis, F and F ¢ its focii and the angle FBF ¢ is a right angle. Then the eccentricity of the ellipse is (1) 14 (2) 13 (3) 12 (4) 12 49. A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is (1) a hyperbola (2) a parabola (3) an ellipse (4) a circle 50. The angle between the lines 2x = 3y = –z and 6x = –y = – 4z is (1) 45º (2) 30º (3) 0º (4) 90ºAmity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -7 -AIEEE-2005 Mathematics 51. If the angle q between the line 1 1 2 1 2 2 x y z + - - = = and the plane 2 4 0 x y z - + l + = is such that 1 sin 3 q= the value of l is (1) 34 (2) 4 3 - (3) 53 (4) 3 5 - 52. The locus of a point P(a, b) moving under the condition that the line y = ax + b is a tangent to the hyperbola 2 2 2 2 1 x y a b - = is (1) a parabola (2) a hyperbola (3) an ellipse (4) a circle 53. The distance between the line r = 2i – 2j + 3k + l(i – j + 4k) and the plane r.(i + 5j + k) = 5 is (1) 3 10 (2) 10 3 (3) 10 9 (4) 10 3 3 54. For any vector a, the value of (a × i)2 + (a × j)2 + (a × k)2 is equal to (1) 2a2 (2) 4a2 (3) 3a2 (4) a2 55. If the plane 2ax – 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2 + y2 + z2 + 6x – 8y – 2z = 13 and x2 + y2 + z2 – 10x + 4y – 2z = 8 then a equals (1) –2 (2) 2 (3) –1 (4) 1 56. Let a, b and c be distinct non-negative numbers. If the vectors ai + aj + ck, i + k and ci + cj + bk lie in a plane, then c is (1) equal to zero (2) the Harmonic Mean of a and b (3) the Geometric Mean a and b (4) the Arithmetic Mean of a and b 57. If a, b, c are non-coplanar vectors and l is a real number then [ ] 2 ( ) é ù l + l l = + ë û a b b c ab c b for (1) exactly three values of l (2) exactly two values of l (3) exactly one value of l (4) no value of l 58. Let a = i – k, b = xi + j + (1 – x)k and c = yi + xj + (1 + x – y)k. Then [a, b, c] depends on (1) both x and y (2) neither x nor y (3) only y (4) only x 59. Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is (1) 89 (2) 79 (3) 29 (4) 19 60. A random variable X has Poisson distribution with mean 2. Then P(X > 1.5) equals (1) 2 3 1 e - (2) 2 3 e (3) 2 2 e (4) 0Amity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -8 -Mathematics AIEEE-2005 61. Let A and B be two events such that ( ) 1 , 6 P A B È = ( ) 14 P A B Ç = and ( ) 1 , 4 P A = whree A stands for component of event A. Then events A and B are (1) independent but not equally likely (2) mutually exclusive and independent (3) equally likely and mutually exclusive (4) equally likely but not independent 62. A lizard, at initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard catch the insect after (1) 21 s (2) 24 s (3) 20 s (4) 1 s 63. The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is (1) 3 : 2 (2) 3:2 2 (3) 2 : 1 (4) 3: 2 64. Two points A and B move from rest along a straight line with constant acceleration f and f ¢ respectively. If A takes m sec. more than B and describes ‘n’ units more than B in acquiring the same speed then (1) 2 1( ) 2 f f m f f n ¢ ¢ + = (2) 2 1 ( ) 2 f fn ffm ¢ ¢ - = (3) 2 ( ) f f m f f n ¢ ¢ - = (4) 2 ( ) f f m f f n ¢ ¢ + = 65. A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with the. The resultant of A and B after combining is displaced through a distance (1) 2( ) H A B + (2) H A B - (3) 2H A B - (4) H A B + 66. The sum of the series : 1 1 1 1 4.2! 16.4! 64.4! + + + +K ad inf. is (1) 1 2e e - (2) 1 2e e + (3) 1 e e - (4) 1 e e + 67. If a1, a2, a3, ...., an are in G.P., then the determinant 1 2 3 4 5 6 7 8 log log log log log log log log log n n n n n n n n n a a a a a a a a a + + + + + + + + D = is equal to (1) 4 (2) 2 (3) 1 (4) 0 68. If both the roots of the quadratic equation x2 – 2kx + k2 + k – 5 = 0 are less than 5, then k lies in the interval (1) (–¥, 4) (2) [4, 5] (3) (5, 6] (4) (6, ¥) 69. If the equation anxn + an – 1xn – 1 + ... + a1x = 0, a1 ¹ 0, n ³ 2, has a positive root x = a, then the equation nanxn–1 + (n – 1)an–1xn–2 + ... + a1 = 0 has a positive root, which isAmity Institute for Competitive Examinations : Phones: 26963838, 26850005/6, 25573111/2/3/4, 95120-2431839/42 -9 -AIEEE-2005 Mathematics (1) greater than or equal to a (2) equal to a (3) greater than a (4) smaller than a 70. A real valued function f(x) satisfies the functional equation f(x – y) = f(x)f(y) – f(a – x)f(a + y), where a is a given constant and f(0) = 1, f(2a – x) is equal to (1) f(a) + f(a – x) (2) f(–x) (3) –f(x) (4) f(x) 71. The plane x + 2y – z = 4 cuts the sphere x2 + y2 + z2 – x + z – 2 = 0 in a circle of radius (1) 2 (2) 2 (3) 3 (4) 1 72. If the pair of lines ax2 + 2(a + b)xy + by2 = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then (1) 3a2 + 10ab + 3b2 = 0 (2) 3a2 + 2ab + 3b2 = 0 (3) 3a2 – 10ab + 3b2 = 0 (4) 3a2 – 2ab + 3b2 = 0 73. The value of 2 cos , 0, 1 xx dx a a p -p > + ò is (1) ap (2) 2p (3) ap (4) 2p 74. A particle is projected from a point O with velocity u at an angle of 60º with the horizontal. When it is moving in a direction at right angles to its direction at O, its velocity then is given by (1) 23u (2) 3 u (3) 3u (4) 2u 75. Let x1, x2, ..., xn be n observations such that 2 400 i x = å and 80 i x = å . Then a possible value of n among the following is (1) 9 (2) 12 (3) 15 (4) 18