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FIITJEE AIEEE - 2004 (MATHEMATICS)
Important Instructions:
i) The test is of
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
(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
(1)
(3)
4. If z = x – i y and
(1) 1 (2) -2
(3) 2 (4) -1
5. If
(1) the real axis (2) an ellipse
(3) a circle (4) the imaginary axis.
6. Let
(1) A is a zero matrix (2)
(3)
7. Let
(1) -2 (2) 5
(3) 2 (4) -1
8. If
(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)
(3)
10. If (1 – p) is a root of quadratic equation
(1) 0, 1 (2) -1, 2
(3) 0, -1 (4) -1, 1
11. Let
(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)
14. If one root of the equation
(1)
(3) 3 (4) 12
15. The coefficient of the middle term in the binomial expansion in powers of x of
(1)
(3)
16. The coefficient of
(1) (n – 1) (2)
(3)
17. If
(1)
(3) n – 1 (4)
18. Let
(1) 0 (2) 1
(3)
19. The sum of the first n terms of the series
(1)
(3)
20. The sum of series
(1)
(3)
21. Let a, b be such that p < a - b < 3p. If sina + sinb =
(1)
(3)
22. If
(1)
(3)
23. The sides of a triangle are sina, cosa and
(1)
(3)
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
(1) 20 m (2) 30 m
(3) 40 m (4) 60 m
25. If
(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
(1) [2, 3] (2) [2, 3)
(3) [1, 2] (4) [1, 2)
28. If
(1)
(3)
29. Let
(1) 1 (2)
(3)
30. If
(1)
(3)
31. A point on the parabola
(1) (2, 4) (2) (2, -4)
(3)
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)
(3)
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
(1) (0, 1) (2) (1, 2)
(3) (2, 3) (4) (1, 3)
35.
(1) e (2) e – 1
(3) 1 – e (4) e + 1
36. If
(1) (sina, cosa) (2) (cosa, sina)
(3) (- sina, cosa) (4) (- cosa, sina)
37.
(1)
(3)
38. The value of
(1)
(3)
39. The value of I =
(1) 0 (2) 1
(3) 2 (4) 3
40. If
(1) 0 (2) p
(3)
41. If f(x) =
(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
(1)
(3)
44. The solution of the differential equation y dx + (x + x2y) dy = 0 is
(1)
(3)
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)
(3)
47. If the sum of the slopes of the lines given by
(1) 1 (2) –1
(3) 2 (4) –2
48. If one of the lines given by
(1) 1 (2) –1
(3) 3 (4) –3
49. If a circle passes through the point (a, b) and cuts the circle
(1)
(3)
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)
(3)
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)
(3)
52. The intercept on the line y = x by the circle
(1)
(3)
53. If a ¹ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas
(1)
(3)
54. The eccentricity of an ellipse, with its centre at the origin, is
(1)
(3)
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
(1)
(3)
56. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
(1)
(3)
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 =
(1) –2 (2) –1
(3) –
59. The intersection of the spheres
(1) x – y – z = 1 (2) x – 2y – z = 1
(3) x – y – 2z = 1 (4) 2x – y – z = 1
60. Let
(1)
(3)
61. A particle is acted upon by constant forces
(1) 40 (2) 30
(3) 25 (4) 15
62. If
(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
(1) 2 (2)
(3)
64. Let
(1)
(3)
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)
(3) 2 (4)
67. The probability that A speaks truth is
(1)
(3)
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)
(3)
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)
(3)
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
(1) 3 (2) 4
(3) 5 (4) 9
72. Three forces
(1)
(3)
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)
(3)
74. A velocity
(1)
(3)
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
(1)
(3)
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
3. Here w =
4.
5.
Þ R (z) = 0 Þ z lies on the imaginary axis.
6. A.A =
7. AB = I
8.
C3 ® C3 – C2, C2 ® C3 – C1
=
9. Let numbers be a, b
10. (3)
sum of root is
11.
S(k + 1)=1 + 3 + 5 +............. + (2k – 1) + (2k + 1)
=
= 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 =
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.
and
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 =
Squaring and adding, we get
2 + 2 cos (a – b) =
Þ
22.
=
Þ
min value of
max value of
Þ
23. Greatest side is
| 24. tan30° = Þ tan60° = h/b Þ h = Þ b = 20 m | |
25.
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.
28.
29.
30.
Þ lnx – x = y Þ
31. Any point be
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
Þ f¢(x) =
35.
36. Put x – a = t
Þ
=
A =
37.
38.
39.
40. Let I =
Þ I = p
41. f(-a) + f(a) = 1
I1 =
2I1 =
| 42. Area = | |
43. 2x + 2yy¢ - 2ay¢ = 0
a =
Þ (x2 – y2)y¢ = 2xy.
45. y dx + x dy + x2y dy = 0.
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.
Þ a = 2, b = -3 or a = -2, b = 1.
Hence
47. m1 + m2 =
m1 + m2 = 4m1m2 (given)
Þ c = 2.
48. m1 + m2 =
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
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
Þ a - bl + cl = 0 and
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.
and
(1) – 2´(2) Þ
Since
Putting the value of t1 and t2 in (1) and (2), we get
61. Work done by the forces
According to question
and
63. Condition for given three vectors to be coplanar is
Hence given vectors will be non coplanar for all real values of l except 0, 1/2.
63. Projection of
According to question
64.
Þ
Þ
65. Mode can be computed from histogram and median is dependent on the scale.
Hence statement (a) and (b) are correct.
66.
S.D. =
67.
Probability that both contradicts each other =
68.
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 = T2 from B to C = Total time = 4 hrs. Average speed = Resultant average velocity = | |
74. Component along
75. t1 =
\
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