AMU - 2006 - MATHEMATICS

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AMU –Past papers: 

AMU –Past papers MATHEMATICS - UNSOLVED PAPER - 2006

SECTION – I: 

SECTION – I CRITICAL REASONING SKILLS

Problem: 

01 If 4a 2 + 9b 2 + 16c 2 = 2 (3ab + 6bc + 4ca), where a, b, c are non-zero numbers, then a, b, c are in : AP GP HP None of these Problem

Problem: 

Problem 02 It is given that is equal to : a. b. c. d. none of these

Problem: 

Problem 03 The polynomial (ax 2 + bx + c) (ax 2 – dx - c), , has : Four real roots At least two real roots At most two real roots No real roots

Problem: 

Problem 04 If (3 + i )z = (3 - i ) , then the complex number z is : a. b. c. d.

Problem: 

Problem 05 If then for the ∆ ABC e iA e iB e iC is : i 1 - 1 none of these

Problem: 

Problem 06 Numbers lying between 999 and 10000 that can be formed from the digits 0, 2, 3, 6, 7, 8 (repetition of digits not allowed) are : 100 200 300 400

Problem: 

Problem 07 In a club election the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote be 126, then the number of contestants is : 4 5 6 7

Problem: 

08 Problem ……. is equal to : for even value of n only for odd values of n only for all values of n none of these

Problem: 

Problem 09 + ………+ to is equal to : log ab log log none of these

Problem: 

Problem 10 The sum of infinite terms of the series + …..+ to , where a is a constant, is : a. b. c. d. none of these

Problem: 

Problem 11 The value of log 2 log 3 …log is equal to : 0 1 2 100!

Problem: 

12 The value of tan 20 0 + 2 tan 50 0 – tan 70 0 is : 1 0 tan 5 0 none of these Problem

Problem: 

Problem 13 A circular ring of radius 3 cm is suspended horizontally from a point 4 cm vertically above the centre by 4 string attached at equal intervals to its circumference. If the angles between two consecutive strings be θ , then cos θ is : a. b. c. d. none of these

Problem: 

Problem 14 The number of positive integral solutions of the equation is : One Two Zero None of these

Problem: 

Problem 15 If α is a repeated root of ax 2 + bx + c = 0, then is : 0 a b c

Problem: 

Problem 16 is equal to : 0 1 3 none of these

Problem: 

17 the domain of the function f(x) = log e (x – [x]) is : R R – Z (0 + ) Z Problem

Problem: 

Problem 18 If f(x + y, x - y) = xy , then the arithmetic mean of f(x, y) and f(y, x) is : x y 0 none of these

Problem: 

Problem 19 The equations of the three sides of a triangle are x = 2, y + 1 = 0 and x + 2y = 4. The coordinates of the circumecentre of the triangle are : (4, 0) (2, -1) (0, 4) (- 1, 2)

Problem: 

Problem 20 If the point (a, a) falls between the lines | x + y| = 4, then : | a | = 2 | a | = 3 | a | < 2 | a | < 3

Problem: 

Problem 21 The equation of the image of the pair of rays y = | x | by the line y = 1 is : y = | x | + 2 y = | x | - 2 y = | x | + 1 y = | x | - 1

Problem: 

Problem 22 Let P = (1, 1) and Q = (3, 2). The point R on the x-axis such that PR + RQ is minimum, is : a. b. c. (3, 0) d. (5, 0)

Problem: 

Problem 23 L is a variable line such that the algebraic sum of the distances of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass through : (1, 1) (2, 1) (1, 2) (2, 2)

Problem: 

Problem 24 C 1 is a circle of radius 2 touching the x-axis and the y-axis. C 2 is another circle of radius > 2 and touching the axes as well as the circle C 1 . Then the radius of C 2 is : 6 – 4 √2 6 + 4 √2 6 – 4 √3 6 + 4 √3

Problem: 

Problem 25 the locus of a point represented by is : an ellipse a circle a pair of straight lines none of these

Problem: 

Problem 26 the locus of the centre of the circle for which one end of a diameter is (1, 1) while the other end is on the line x + y = 3, is : x + y =1 2 (x - y) = 5 2x + 2y = 5 none of these

Problem: 

Problem 27 The locus of the middle points of chords of a parabola which subtend a right angle at the vertex of the parabola, is : A circle An ellipse A parabola A hyperbola

Problem: 

Problem 28 The point of intersection of the lines is : (0, 0, 0) (1, 1, 1) (-1, -1, -1) (1, 2, 3)

Problem: 

Problem 29 The equation of the plane which meets the axes in A, B, C such that the centroid of the triangle ABC is is given by : x + y + z = 1 x + y + z = 2 x + y + z =

Problem: 

Problem 30 The image of the point (5, 4, 6) in the plane x + y +2z – 15 = 0 is : (3, 2, 2) (2, 3, 2) (2, 2, 3) (-5, - 4,- 6 )

Problem: 

Problem 31 The radius of the circle x + 2y + 2z = 15, x 2 + y 2 + z 2 – 2y – 4z = 11 is : 2 √7 3 √5

Problem: 

Problem 32 A straight line which makes angle of 60 0 with each of y and z – axes, is inclined with x – axis at angle of : 30 0 45 0 60 0 75 0

Problem: 

Problem 33 The value of is : 0 30 x 30 -x 1

Problem: 

Problem 34 The rank of the matrix is : 4 3 2 1

Problem: 

Problem 35 The values of a for which the system of equations ax + y + z = 0, x – ay + z = 0, x + y + z = 0 possesses non-zero solutions, are given by : 1, 2 1, -1 0 none of these

Problem: 

Problem 36 If A is skew symmetric matrix of order n and C is a column matrix of order n x 1, then C T AC is : An identity matrix of order n An identity matrix of order 1 A zero matrix of order 1 None of these

Problem: 

Problem 37 If = kxyz , then the value of k is : 2 4 6 8

Problem: 

Problem 38 Let f(x) be a polynomial function of the second degree. If f(1) = f(1) and a 1 , a 2 , a 3 are in A.P., then f’(a 1 ), f’(a 2 ),f’(a 3 ) are in : A.P. G.P. H.P. None of these

Problem: 

Problem 39 The curve given by x + y = e xy has a tangent parallel to the y-axis at the point : (0, 1) (1, 0) (1, 1) (- 1, -1)

Problem: 

Problem 40 Let f(x) = 1 + 2x 2 + 2 2 x 4 + …+ 2 10 x 20 . Then f(x) has : More than one minimum Exactly one minimum At least one maximum None of these

Problem: 

Problem 41 The interval in which the function is decreasing, is : a. (-1, -1) b. (1, 1) c. (-1, 1) d. [- 1, 1]

Problem: 

Problem 42 A right circular cylinder which is open at the top and has a given surface area, will have the greatest volume if its height h and radius r are related by : 2 h = r h = 4 r h = 2r h = r

Problem: 

Problem 43 If f(x) = x2 – 2x + 4 on [1, 5], then the value of a constant c such that is : 0 1 2 3

Problem: 

Problem 44 The value of k for which the function is continuous at x = 0, is : k = 0 k = 1 k = - 1 none of these

Problem: 

Problem 45 If f(x) = cos x cos 2x cos 4x cos 8x cos 16x, then is : 0

Problem: 

Problem 46 Let and f(0) = 0. Then f(1) is : log (1 + ) none of these

Problem: 

Problem 47 The value of {g(x)}g’(x) dx , where g(1) = g (2), is equal to : 1 2 0 none of these

Problem: 

Problem 48 If then the value of f(1) is : 0 1 -

Problem: 

Problem 49 If f(x) = f(a + x) and is equal to : n k (n - 1) k (n + 1)k 0

Problem: 

Problem 50 If then the values of a and b are respectively : 1, 1 -1, -1 1, -1 - 1, 1

Problem: 

Problem 51 A vector has components 2a and 1 with respect to a rectangular Cartesian system. The axes are rotated through an angle about the origin in the anticlockwise direction. If the vector has components a + 1 and 1 with respect to the new system, then the values of a are : 1, - 1/3 0 - 1 , 1/3 1, -1

Problem: 

Problem 52 Let is a vector satisfying is : none of these

Problem: 

Problem 53 Let , where O, A and C are non-collinear points. Let p denote the area of the quadrilateral OABC and q denote the area of the parallelogram with OA and OC as adjacent sides. Then is equal to : 4 6

Problem: 

Problem 54 The value of x so that the four points A = {0, 2, 0}, B = (1, x, 0), C = (1, 2, 0) and D = (1, 2, 1) are coplanar, is : 0 1 2 3

Problem: 

Problem 55 Constant forces act on a particle at a point a. The work done when the particle is displaced from the point A to B where is : 3 9 20 none of these

Problem: 

Problem 56 The solution of the differential equation x given that y = 1, when x = , is : y = sin x – cos x y = cos x y = sin x y = sin x + cos x

Problem: 

Problem 57 From a point on the ground at a distance 70 feet from the foot of a vertical wall, a ball is thrown at an angle of 45 0 which just clears the top of the wall and afterwards strikes the ground at a distance 30 feet on the other side of the wall. The height of the wall is : 20 feet 21 feet 10 feet 105 feet

Problem: 

Problem 58 Three coplanar forces acting on a particle are in equilibrium. The angle between the first and the second is 60 0 and that between the second and the third is 150 0 . The ratio of the magnitude of the forces are : 1 : 1 : 1 : : 1 : 1 : 1 : : 1

Problem: 

Problem 59 A particle having simultaneous velocities 3 m/s, 5 m/s and 7 m/s, is at rest. The angle between the first two velocities is : 30 0 45 0 60 0 90 0

Problem: 

Problem 60 A cyclist is beginning to move with an acceleration of 1 m/s 2 and a boy, who is 40 m behind the cyclist, starts running at 9 m/s to meet him. The boy will be able to meet the cyclist after : 6 sec 8 sec 9 sec 10 sec

Problem: 

Problem 61 Two bodies slide from rest down two smooth inclined planes commencing at the same point and terminating in the same horizontal plane. The ratio of the velocities attained if inclinations to the horizontal of the planes are 30 0 and 60 0 respectively, is : : 1 2 : 1 : 1 1 : 2

Problem: 

Problem 62 A die is thrown 2n + 1 times. The probability that faces with even numbers show odd number of times, is : none of these

Problem: 

Problem 63 The probability that exactly one of the independent events A and B occurs, is equal to : P (A) + P (B) + 2P (A B) P (A) + P (B) – P (A B) P(A’) + P (B’) = 2P (A’ B’) None of these

Problem: 

Problem 64 A bag contains 30 tickets, numbered from 1 to 30. Five tickets are drawn at random and arranged in the ascending order. The probability that the third number is 20, is : none of these

Problem: 

Problem 65 The probability that at least one of the events A and B occur is 0.6 If A and B occur simultaneously with probability 0.2, then is : 0.4 0.8 1.2 1.4

Problem: 

Problem 66 The relation of “congruence modulo” is : Reflexive only Symmetric only Transitive only An equivalence relation

Problem: 

Problem 67 If flow values of switches x 1 , x 2 and x 3­ are respectively 0, 0 and 1, then the flow value of the circuit s = (x’ 1 .x’ 2 .x 3 ) + (x 1 .x’ 2 .x’ 3 ) + (x’ 1 .x 2 .x’ 3 ) is : 0 1 2 none of these

Problem: 

Problem 68 In a Boolean Algebra a v (a’ b) is equal to : a v b a b a’ b’

Problem: 

Problem 69 The range of the function is : [- 1, 1] (- 1, 1) (- 3, 3) (- 3, 1)

Problem: 

Problem 70 The coefficients of x in the quadratic equation x 2 + bx + c = 0 was taken as 17 in place of 13, its roots were found to be – 2 and –15. The correct roots of the original equation are : - 10, - 3 - 9, - 4 - 8, - 5 - 7, - 6

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