Maths Solutions 2004

AIEEE - 2004 (MATHEMATICS)



Important Instructions:

i) The test is of  hours duration.
ii) The test consists of 75 questions.
iii) The maximum marks are 225.
iv) For each correct answer you will get 3 marks and for a wrong answer you will get -1 mark.

1. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is
(1) a function (2) reflexive
(3) not symmetric (4) transitive

2. The range of the function   is
(1) {1, 2, 3} (2) {1, 2, 3, 4, 5}
(3) {1, 2, 3, 4} (4) {1, 2, 3, 4, 5, 6}

3. Let z, w be complex numbers such that  = 0 and arg zw = p. Then arg z equals
(1) (2)
(3) (4)

4. If z = x – i y and  , then   is equal to
(1) 1 (2) -2
(3) 2 (4) -1

5. If , then z lies on
(1) the real axis (2) an ellipse
(3) a circle (4) the imaginary axis.

6. Let   The only correct statement about the matrix A is
(1) A is a zero matrix (2)
(3)  does not exist   (4) , where I is a unit matrix

7. Let   . If B is the inverse of matrix A, then a is
(1) -2 (2) 5
(3) 2 (4) -1

8. If   are in G.P., then the value of the determinant  , is
(1) 0 (2) -2
(3) 2 (4) 1

9. Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
(1)   (2)  
(3)   (4)

10. If (1 – p) is a root of quadratic equation  , then its roots are
(1) 0, 1 (2) -1, 2
(3) 0, -1 (4) -1, 1

11. Let  . Then which of the following is true?
(1) S(1) is correct
(2) Principle of mathematical induction can be used to prove the formula
(3)
(4)

12. How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?
(1) 120 (2) 480
(3) 360 (4) 240

13. The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes  is empty is
(1) 5 (2)
(3) (4) 21

14. If one root of the equation   is 4, while the equation   has equal roots, then the value of ‘q’ is
(1) (2) 4
(3) 3 (4) 12

15. The coefficient of the middle term in the binomial expansion in powers of x of  and of   is the same if a equals
(1)   (2)
(3)   (4)

16. The coefficient of  in expansion of   is
(1) (n – 1) (2)
(3) (4)

17. If   and , then   is equal to
(1) (2)
(3) n – 1 (4)

18. Let  be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, m ¹ n,  , then a – d equals
(1) 0 (2) 1
(3) (4)

19. The sum of the first n terms of the series   is   when n is even. When n is odd the sum is
(1) (2)
(3) (4)

20. The sum of series   is
(1) (2)
(3) (4)

21. Let a, b be such that p < a - b < 3p. If sina + sinb =   and  cosa + cosb = , then the value of   is
(1) (2)
(3) (4)

22. If , then the difference between the maximum and minimum values of   is given by
(1) (2)
(3) (4)

23. The sides of a triangle are sina, cosa and   for some 0 < a < . Then the greatest angle of the triangle is
(1) (2)
(3) (4)

24. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is   and when he retires 40 meter away from the tree the angle of elevation becomes . The breadth of the river is
(1) 20 m (2) 30 m
(3) 40 m (4) 60 m

25. If , is onto, then the interval of S is
(1) [0, 3] (2) [-1, 1]
(3) [0, 1] (4) [-1, 3]

26. The graph of the function y = f(x) is symmetrical about the line x = 2, then
(1) f(x + 2)= f(x – 2) (2) f(2 + x) = f(2 – x)
(3) f(x) = f(-x) (4) f(x) = - f(-x)

27. The domain of the function   is
(1) [2, 3] (2) [2, 3)
(3) [1, 2] (4) [1, 2)

28. If , then the values of a and b, are
(1) (2) a = 1,
(3) (4) a = 1 and b = 2

29. Let . If f(x) is continuous in   is
(1) 1 (2)
(3) (4) -1

30. If , x > 0, then   is
(1) (2)
(3) (4)

31. A point on the parabola   at which the ordinate increases at twice the rate of the abscissa is
(1) (2, 4) (2) (2, -4)
(3) (4)

32. A function y = f(x) has a second order derivative f²(x) = 6(x – 1). If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x – 5, then the function is
(1) (2)
(3) (4)

33. The normal to the curve x = a(1 + cosq), y = asinq at ‘q’ always passes through the fixed point
(1) (a, 0) (2) (0, a)
(3) (0, 0) (4) (a, a)

34. If 2a + 3b + 6c =0, then at least one root of the equation   lies in the interval
(1) (0, 1) (2) (1, 2)
(3) (2, 3) (4) (1, 3)

35. is
(1) e (2) e – 1
(3) 1 – e (4) e + 1

36. If , then value of (A, B) is
(1) (sina, cosa) (2) (cosa, sina)
(3) (- sina, cosa) (4) (- cosa, sina)

37. is equal to
(1) (2)
(3) (4)

38. The value of   is
(1) (2)
(3) (4)

39. The value of I =  is
(1) 0 (2) 1
(3) 2 (4) 3

40. If  dx, then A is
(1) 0 (2) p
(3) (4) 2p

41. If f(x) = , I1 =   and I2 =   then the value of   is
(1) 2 (2) –3
(3) –1 (4) 1

42. The area of the region bounded by the curves y = |x – 2|, x = 1, x = 3 and the x-axis is
(1) 1 (2) 2
(3) 3 (4) 4

43. The differential equation for the family of curves , where a is an arbitrary constant is
(1) (2)
(3) (4)

44. The solution of the differential equation y dx + (x + x2y) dy = 0 is
(1) (2)
(3) (4) log y = Cx

45. Let A (2, –3) and B(–2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line
(1) 2x + 3y = 9 (2) 2x – 3y = 7
(3) 3x + 2y = 5 (4) 3x – 2y = 3

46. The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is –1 is
(1)  and (2)  and
(3)  and (4)  and

47. If the sum of the slopes of the lines given by   is four times their product, then c has the value
(1) 1 (2) –1
(3) 2 (4) –2

48. If one of the lines given by   is 3x + 4y = 0, then c equals
(1) 1 (2) –1
(3) 3 (4) –3

49. If a circle passes through the point (a, b) and cuts the circle   orthogonally, then the locus of its centre is
(1) (2)
(3) (4)

50. A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is
(1) (2)
(3) (4)

51. If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of circumference 10p, then the equation of the circle is
(1) (2)
(3) (4)

52. The intercept on the line y = x by the circle   is AB. Equation of the circle on AB as a diameter is
(1) (2)
(3) (4)

53. If a ¹ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas   and , then
(1) (2)
(3) (4)

54. The eccentricity of an ellipse, with its centre at the origin, is . If one of the directrices is x = 4, then the equation of the ellipse is
(1) (2)
(3) (4)

55. A line makes the same angle q, with each of the x and z axis. If the angle b, which it makes with y-axis, is such that , then   equals
(1) (2)
(3) (4)

56. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
(1) (2)
(3) (4)

57. A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and
x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by
(1) (3a, 3a, 3a), (a, a, a) (2) (3a, 2a, 3a), (a, a, a)
(3) (3a, 2a, 3a), (a, a, 2a) (4) (2a, 3a, 3a), (2a, a, a)

58. If the straight lines x = 1 + s, y = –3 – ls, z = 1 + ls and x = , y = 1 + t, z = 2 – t with parameters s and t respectively, are co-planar then l equals
(1) –2 (2) –1
(3) – (4) 0

59. The intersection of the spheres   and   is the same as the intersection of one of the sphere and the plane
(1) x – y – z = 1 (2) x – 2y – z = 1
(3) x – y – 2z = 1 (4) 2x – y – z = 1

60. Let   and   be three non-zero vectors such that no two of these are collinear. If the vector   is collinear with   and   is collinear with   (l being some non-zero scalar) then   equals
(1) (2)
(3) (4) 0

61. A particle is acted upon by constant forces   and   which displace it from a point   to the point . The work done in standard units by the forces is given by
(1) 40 (2) 30
(3) 25 (4) 15

62. If   are non-coplanar vectors and l is a real number, then the vectors   and   are non-coplanar for
(1) all values of l (2) all except one value of l
(3) all except two values of l (4) no value of l

63. Let   be such that . If the projection   along   is equal to that of   along   and   are perpendicular to each other then   equals
(1) 2 (2)
(3) (4) 14

64. Let   and   be non-zero vectors such that . If q is the acute angle between the vectors   and , then sin q equals
(1) (2)
(3) (4)

65. Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.
Which of these is/are correct?
(1) only (a) (2) only (b)
(3) only (a) and (b) (4) (a), (b) and (c)

66. In a series of 2n observations, half of them equal a and remaining half equal –a. If the standard deviation of the observations is 2, then |a| equals
(1) (2)
(3) 2 (4)

67. The probability that A speaks truth is , while this probability for B is . The probability that they contradict each other when asked to speak on a fact is
(1) (2)
(3) (4)

68. A random variable X has the probability distribution:

X: 1 2 3 4 5 6 7 8  
p(X): 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05
For the events E = {X is a prime number} and F = {X < 4}, the probability P (E È F) is
(1) 0.87 (2) 0.77
(3) 0.35 (4) 0.50

69. The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is
(1) (2)
(3) (4)

70. With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then the forces are
(1) (2)
(3) (4)

71. In a right angle DABC, ÐA = 90° and sides a, b, c are respectively, 5 cm, 4 cm and 3 cm. If a force   has moments 0, 9 and 16 in N cm. units respectively about vertices A, B and C, then magnitude of   is
(1) 3 (2) 4
(3) 5 (4) 9

72. Three forces   acting along IA, IB and IC, where I is the incentre of a DABC, are in equilibrium. Then   is
(1) (2)
(3) (4)

73. A particle moves towards east from a point A to a point B at the rate of 4 km/h and then towards north from B to C at the rate of 5 km/h. If AB = 12 km and BC = 5 km, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively
(1)  km/h and   km/h (2)  km/h and   km/h
(3)  km/h and   km/h (4)  km/h and   km/h
74. A velocity   m/s is resolved into two components along OA and OB making angles 30° and 45° respectively with the given velocity. Then the component along OB is
(1)  m/s (2)  m/s
(3)  m/s (4)  m/s

75. If t1 and t2 are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then   is equal to
(1) (2)
(3) (4) 1


FIITJEE AIEEE - 2004 (MATHEMATICS)

ANSWERS


1. 3 16. 2 31. 4 46. 4 61. 1
2. 1 17. 1 32. 2 47. 3 62. 3
3. 3 18. 1 33. 1 48. 4 63. 3
4. 2 19. 2 34. 1 49. 2 64. 4
5. 4 20. 2 35. 2 50. 1 65. 3
6. 2 21. 1 36. 2 51. 1 66. 3
7. 2 22. 4 37. 4 52. 1 67. 3
8. 1 23. 3 38. 1 53. 1 68. 2
9. 4 24. 1 39. 3 54. 2 69. 4
10. 3 25. 4 40. 2 55. 3 70. 3
11. 4 26. 2 41. 1 56. 3 71. 3
12. 3 27. 2 42. 1 57. 2 72. 1
13. 4 28. 2 43. 3 58. 1 73. 1
14. 1 29. 3 44. 2 59. 4 74. 4
15. 3 30. 3 45. 1 60. 4 75. 2

FIITJEE AIEEE - 2004 (MATHEMATICS)

SOLUTIONS

1. (2, 3) Î R but (3, 2) Ï R.
Hence R is not symmetric.

2.

,
and
Þ  Þ x = 3, 4, 5 Þ Range is {1, 2, 3}.

3. Here w =   Þ arg  Þ 2 arg(z) – arg(i) = p  Þ arg(z) = .
4.
  .
5.

Þ R (z) = 0 Þ z lies on the imaginary axis.

6. A.A = .

7. AB = I   A(10 B) = 10 I
if .
8.
C3 ® C3 – C2, C2 ® C3 – C1
=   = 0 (where r is a common ratio).

9. Let numbers be a, b   , a and b are roots of the equation
.

10. (3)
     (since (1 – p) is a root of the equation x2 + px + (1 – p) = 0)

Þ (1 – p) = 0 Þ p = 1
sum of root is   and product   (where b = 1 – p = 0)
 Roots are 0, –1

11.
S(k + 1)=1 + 3 + 5 +............. + (2k – 1) + (2k + 1)
=    [from S(k) = ]
= 3 + (k2 + 2k + 1) = 3 + (k + 1)2 = S (k + 1).
Although S (k) in itself is not true but it considered true will always imply towards S (k + 1).

12. Since in half the arrangement A will be before E and other half E will be before A.
Hence total number of ways =   = 360.

13. Number of balls = 8
number of boxes = 3
Hence number of ways = 7C2 = 21.

14. Since 4 is one of the root of x2 + px + 12 = 0 Þ 16 + 4p + 12 = 0 Þ p = –7
and equation x2 + px + q = 0 has equal roots
Þ D = 49 – 4q = 0 Þ q = .
15. Coefficient of Middle term in
Coefficient of Middle term in

16. Coefficient of xn in (1 + x)(1 – x)n = (1 + x)(nC0 – nC1x + …….. + (–1)n –1 nCn – 1 xn – 1 + (–1)n  nCn  xn)
= (–1)n  nCn + (–1)n –1 nCn – 1 .

17.
 

18.          .....(1)
and             .....(2)
from (1) and (2) we get
Hence a – d = 0

19. If n is odd then (n – 1) is even Þ sum of odd terms .

20. ……..

put a = 1, we get
………..

21. sin a + sin b =   and cos a + cos b = .
Squaring and adding, we get
2 + 2 cos (a – b) =
Þ   Þ .

22.
=
Þ
min value of
max value of
Þ .

23. Greatest side is , by applying cos rule we get greatest angle = 120o.


24. tan30° =
Þ           …..(1)
tan60° = h/b  Þ h = ….(2)
Þ b = 20 m

25.   Þ
Þ range of f(x) is [–1, 3].
Hence S is [–1, 3].

26. If y = f (x) is symmetric about the line x = 2 then f(2 + x) = f(2 – x).

27. and   Þ

28.

29.

30. Þ x =
Þ lnx – x = y Þ .

31. Any point be ; differentiating y2 = 18x
Þ .
Þ Point is

32. f² (x) = 6(x – 1) Þ f¢ (x) = 3(x – 1)2 + c
and f¢ (2) = 3 Þ c = 0
Þ f (x) = (x – 1)3 + k  and f (2) = 1 Þ k = 0
Þ f (x) = (x – 1)3.

33. Eliminating q, we get (x – a)2 + y2 = a2.
Hence normal always pass through (a, 0).

34. Let f¢(x) =  Þ f(x) =
Þ , Now f(1) = f(0) = d, then according to Rolle’s theorem
Þ f¢(x) =   has at least one root in (0, 1)

35. =

36. Put x – a = t
Þ
=
A =

37.     =

38. =  = .

39. =   =   = 2.

40. Let I =   = (since f (2a – x) = f (x))
Þ I = p  Þ A = p.

41. f(-a) + f(a) = 1
I1 =   =
2I1 =  = I2  Þ  I2 / I1 = 2.


42. Area =   = 1.


43. 2x + 2yy¢ - 2ay¢ = 0
a =   (eliminating a)
Þ (x2 – y2)y¢ = 2xy.

45. y dx + x dy + x2y dy = 0.
Þ .

45. If C be (h, k) then centroid is (h/3, (k – 2)/3) it lies on 2x + 3y = 1.
Þ locus is 2x + 3y = 9.

46. where a + b = -1 and
Þ a = 2, b = -3 or a = -2, b = 1.
Hence .

47. m1 + m2 =   and m1 m2 =
m1 + m2 = 4m1m2 (given)
Þ c = 2.

48. m1 + m2 = , m1m2 =   and m1 = .
Hence c = -3.

49. Let the circle be x2 + y2 + 2gx + 2fy + c = 0 Þ c = 4 and it passes through (a, b)
Þ a2 + b2 + 2ga + 2fb + 4 = 0.
Hence locus of the centre is 2ax + 2by – (a2 + b2 + 4) = 0.

50. Let the other end of diameter is (h, k) then equation of circle is
(x – h)(x – p) + (y – k)(y – q) = 0
Put y = 0, since x-axis touches the circle
Þ x2 – (h + p)x + (hp + kq) = 0 Þ (h + p)2 = 4(hp + kq) (D = 0)
Þ (x – p)2 = 4qy.

51. Intersection of given lines is the centre of the circle i.e. (1,  - 1)
Circumference = 10p Þ radius r = 5
Þ equation of circle is x2 + y2 - 2x + 2y - 23 = 0.

52. Points of intersection of line y = x with x2 + y2 - 2x = 0 are (0, 0) and (1, 1)
hence equation of circle having end points of diameter (0, 0) and (1, 1) is
x2 + y2 - x - y = 0.

53. Points of intersection of given parabolas are (0, 0) and (4a, 4a)
Þ equation of line passing through these points is y = x
On comparing this line with the given line 2bx + 3cy + 4d = 0, we get
d = 0 and 2b + 3c = 0 Þ (2b + 3c)2 + d2 = 0.

54. Equation of directrix is x = a/e = 4 Þ a = 2
b2 = a2 (1 - e2) Þ b2 = 3
Hence equation of ellipse is 3x2 + 4y2 = 12.

55. l = cos q, m = cos q, n = cos b
cos2 q + cos2 q + cos2 b = 1 Þ 2 cos2 q = sin2 b = 3 sin2 q (given)
cos2 q = 3/5.

56. Given planes are
2x + y + 2z - 8 = 0, 4x + 2y + 4z + 5 = 0 Þ 2x + y + 2z + 5/2 = 0
Distance between planes =  = .

57. Any point on the line     is  (t1, t1 – a, t1) and any point on the line   is (2t2 – a, t2, t2).
Now direction cosine of the lines intersecting the above lines is proportional to
(2t2 – a – t1, t2 – t1 + a, t2 – t1).
Hence 2t2 – a – t1 = 2k ,  t2 – t1 + a  = k and t2 – t1 = 2k
On solving these, we get  t1 = 3a , t2 = a.
Hence points are (3a, 2a, 3a) and (a, a, a).

58. Given lines    are coplanar then plan passing through these lines has normal  perpendicular to these lines
Þ a - bl + cl = 0 and    (where a, b, c are direction ratios of the normal to the plan)
On solving, we get  l = -2.

59. Required plane is S1 – S2 = 0
where S1 = x2 + y2 + z2 + 7x – 2y – z – 13 = 0  and
S2 = x2 + y2 + z2 – 3x + 3y + 4z – 8 = 0
Þ 2x – y – z = 1.

60. ….(1)
and   ….(2)
(1) – 2´(2) Þ    Þ 1+ 2t2 = 0 Þ  t2 = -1/2 & t1 = -6.
Since   are non-collinear.
Putting the value of t1 and t2 in (1) and (2), we get .

61. Work done by the forces , where   is displacement
According to question  =
and . Hence   is 40.

63. Condition for given three vectors to be coplanar is  = 0 Þ l = 0, 1/2.
Hence given vectors will be non coplanar for all real values of l except 0, 1/2.

63. Projection of   along   and   along  is   and  respectively
According to question   Þ . and
= 14 Þ .

64. Þ
Þ   Þ   and
Þ   Þ cosq = –1/3 Þ sinq = .

65. Mode can be computed from histogram and median is dependent on the scale.
Hence statement (a) and (b) are correct.

66. and
S.D. =   Þ 2 =        Þ   Þ

67. event denoting that A speaks truth
event denoting that B speaks truth
Probability that both contradicts each other =   =

68. = 0.62 + 0.50 – 0.35 = 0.77

69. Given that n p = 4, n p q = 2 Þ q = 1/2 Þ p = 1/2 , n = 8 Þ p(x = 2) =

70. P + Q = 4, P2 + Q2 = 9 Þ P = .


71. F . 3 sin q = 9
F . 4 cos q = 16
Þ F = 5.
 
72. By Lami’s theorem
=
Þ .
 
73. Time T1 from A to B =   = 3 hrs.
T2 from B to C =   = 1 hrs.
Total time = 4 hrs.
Average speed =   km/ hr.
Resultant average velocity =   km/hr.

74. Component along OB =   m/s.

75. t1 = , t2 =   where a + b = 900
\ .