FIITJEE AIEEE  2004 (MATHEMATICS) - StudyGuideIndia

21 Pages·2007·0.2 MB·English

A IEEE - 2004 (MATHEMATICS)
Important Instructions:
1
i) The test is of 1 hours duration.
2
ii) The test consists of 75 questions.
iii) The maximum marks are 225. m
iv) For each correct answer you will get 3 marks and for a worong answer you will
c
get -1 mark.
.
a
i
d
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 n
(1) a function (2) rIeflexive
e
(3) not symmetric (4) transitive
d
2. The range of the function f(x)=7-xP i is
x-u3
(1) {1, 2, 3} (2) {1, 2, 3, 4, 5}
G
(3) {1, 2, 3, 4} (4) {1, 2, 3, 4, 5, 6}
y
3. Let z, w be complex numbersd such that z+iw= 0 and arg zw = p . Then arg z
equals
u
p 5p
(1) t (2)
4 S 4
3p p
(3) . (4)
4 w 2
w
æx yö
w 1 ç + ÷
4. If z = x – i y and , then èp qø is equal to
z3 =p+iq
(p2 +q2)
(1) 1 (2) -2
(3) 2 (4) -1
5. If z2 - 1=z2 +1, then z lies on
(1) the real axis (2) an ellipse
(3) a circle (4) the imaginary axis.
æ0 0 -1ö
ç ÷
6. Let A = 0 -1 0 . The only correct statement about the matrix A is
ç ÷
ç ÷
-1 0 0
è ø
(1) A is a zero matrix (2) A2 =I
(3) A-1does not exist (4)A =(-1)I, where I is a unit matrix
æ1 -1 1ö æ4 2 2ö
ç ÷ ç ÷
7. Let A = 2 1 -3 (10)B= -5 0 a . If B is the inverse of matrix A, then a is
ç ÷ ç ÷
ç ÷ ç ÷
1 1 1 1 -2 3
è ø è ø
(1) -2 (2) 5
(3) 2 (4) -1
8. If a,a ,a ,....,a ,.... are in G.P., then the value of the determinant
1 2 3 n
m
loga loga loga
n n+1 n+2
loga loga loga , is o
n+3 n+4 n+5
loga loga loga c
n+6 n+7 n+8
(1) 0 (2) -2 .
a
(3) 2 (4) 1
i
d
9. Let two numbers have arithmetic mean 9 and geometric mean 4. Then these
n
numbers are the roots of the quadratic equation
I
(1) x2 +18x+16=0 (2)e x2 - 18x- 16=0
(3) x2 +18x- 16=0 (d4) x2 - 18x+16=0
i
10. If (1 – p) is a root of quadratic equatiuon x2 +px+(1- p) =0, then its roots are
G
(1) 0, 1 (2) -1, 2
(3) 0, -1 (4) -1, 1
y
d
11. Let S(K)=1+3+5+...+(2K- 1) =3+K2. Then which of the following is true?
u
(1) S(1) is correct
t
(2) Principle of mathemaStical induction can be used to prove the formula
(3) S(K) Þ S(K +1)
.
(4) S(K)Þ S(K +1) w
w
12. How many ways are there to arrange the letters in the word GARDEN with the
vowels in alphawbetical 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) 8C
3
(3)38 (4) 21
14. If one root of the equation x2 +px+12=0 is 4, while the equation x2 +px+q=0
has equal roots, then the value of ‘q’ is
49
(1) (2) 4
4
(3) 3 (4) 12
2
AIEEE-PAPERS--3
15. The coefficient of the middle term in the binomial expansion in powers of x of
(1+ax)4and of (1- ax)6 is the same if a equals
5 3
(1) - (2)
3 5
-3 10
(3) (4)
10 3
16. The coefficient of xnin expansion of (1+x)(1- x)n is
(1) (n – 1) (2) (-1)n(1- n)
m
(3)(-1)n-1(n- 1)2 (4) (-1)n-1n
o
c
n 1 n r t
17. If S =å andt =å , then n is equal to .
n nC n nC S a
r=0 r r=0 r n
i
1 1 d
(1) n (2) n- 1
2 2
n
2n- 1
(3) n – 1 (4) I
e2
d
18. Let T be the rth term of an A.P. whose first term is a and common difference is d.
r i
u
1 1
If for some positive integers m, n, m „ n, T = and T = , then a – d equals
G m n n m
(1) 0 (2) 1
y
1 1 1
(3) d (4) +
mn m n
u
t
19. The sum of the first n teSrms of the series 12 +2×22 +32 +2×42 +52 +2×62 +... is
n(n+1)2
when n is ev.en. When n is odd the sum is
w
2
3n(n+1) w n2(n+1)
(1) (2)
2 2
w
n(n+1)2 én(n+1)ù2
(3) (4) ê ú
4 êë 2 úû
1 1 1
20. The sum of series + + +... is
2! 4! 6!
(e2 - 1) (e- 1)2
(1) (2)
2 2e
(e2 - 1) (e2 - 2)
(3) (4)
2e e
3
AIEEE-PAPERS--4
21 27
21. Let a , b be such that p < a - b < 3p . If sina + sinb = - and cosa + cosb =- ,
65 65
a- b
then the value of cos is
2
3 3
(1)- (2)
130 130
6 6
(3) (4) -
65 65
22. Ifu= a2cos2q+b2sin2q+ a2sin2q+b2cos2q, then the difference between the
m
maximum and minimum values of u2 is given by
(1)2(a2 +b2) (2) 2 a2 +b2 o
(3)(a+b)2 (4) (a- b)2 c
.
a
23. The sides of a triangle are sina , cosa and 1+sidniacosa for some 0 < a <p.
2
n
Then the greatest angle of the triangle is
(1)60o (2) I90o
e
(3)120o (4) 150o
d
i
24. A person standing on the bank of a ruiver observes that the angle of elevation of
the top of a tree on the opposite bGank of the river is 60o and when he retires 40
meter away from the tree the angle of elevation becomes30o. The breadth of the
y
river is
d
(1) 20 m (2) 30 m
(3) 40 m u (4) 60 m
t
S
25. Iff:R ®S, definedby f(x)=sinx- 3cosx+1, is onto, then the interval of S is
(1) [0, 3] . (2) [-1, 1]
w
(3) [0, 1] (4) [-1, 3]
w
26. The graph of the function y = f(x) is symmetrical about the line x = 2, then
(1) f(x + 2)= f(wx – 2) (2) f(2 + x) = f(2 – x)
(3) f(x) = f(-x) (4) f(x) = - f(-x)
sin-1(x- 3)
27. The domain of the function f(x)= is
9- x2
(1) [2, 3] (2) [2, 3)
(3) [1, 2] (4) [1, 2)
2x
æ a b ö
28. Iflim 1+ + =e2, then the values of a and b, are
ç ÷
x®¥è x x2ø
(1)aÎ R,bÎ R (2) a = 1, bÎ R
(3)aÎ R,b=2 (4) a = 1 and b = 2
4
AIEEE-PAPERS--5
1- tanx p é pù é pù æpö
29. Letf(x)= , x¹ , xÎ ê0, ú . If f(x) is continuous in ê0, ú,thenfç ÷ is
4x- p 4 ë 2û ë 2û è4ø
1
(1) 1 (2)
2
1
(3)- (4) -1
2
dy
30. Ifx=ey+ey+...to¥ , x > 0, then is
dx
x 1
(1) (2)
1+x x m
1- x 1+x
(3) (4) o
x x
c
31. A point on the parabola y2 =18x at which the ordina.te increases at twice the rate
a
of the abscissa is
i
(1) (2, 4) (2) (2, -4d)
æ-9 9ö æ9n9ö
(3)ç , ÷ (4) ç , ÷
è8 2ø Iè8 2ø
e
32. A function y = f(x) has a second order dderivative f† (x) = 6(x – 1). If its graph
passes through the point (2, 1) and ati that point the tangent to the graph is y =
u
3x – 5, then the function is
G
(1)(x- 1)2 (2) (x- 1)3
y
(3)(x+1)3 (4) (x+1)2
d
u
33. The normal to the curve x = a(1 + cosq ), y = asinq at ‘q ’ always passes through
t
the fixed point
S
(1) (a, 0) (2) (0, a)
.
(3) (0, 0) (4) (a, a)
w
34. If 2a + 3b + 6c =w0, then at least one root of the equation ax2 +bx+c =0 lies in
the interval
w
(1) (0, 1) (2) (1, 2)
(3) (2, 3) (4) (1, 3)
n 1 r
35. limå en is
n®¥ n
r=1
(1) e (2) e – 1
(3) 1 – e (4) e + 1
sinx
36. Ifò dx=Ax+Blogsin(x- a)+C, then value of (A, B) is
sin(x- a)
(1) (sina , cosa ) (2) (cosa , sina )
(3) (- sina , cosa ) (4) (- cosa , sina )
5
AIEEE-PAPERS--6
dx
37. ò is equal to
cosx- sinx
1 æx pö 1 æxö
(1) logtanç - ÷+C (2) logcotç ÷+C
2 è2 8ø 2 è2ø
1 æx 3pö 1 æx 3pö
(3) logtanç - ÷+C (4) logtanç + ÷+C
2 è2 8 ø 2 è2 8 ø
3
38. The value of ò|1- x2 |dx is
-2
28 14 m
(1) (2)
3 3
o
7 1
(3) (4)
c
3 3
.
a
p/2(sinx+cosx)2
39. The value of I = ò dxis di
1+sin2x
0 n
(1) 0 (2) 1
I
(3) 2 (4) 3
e
d
p p/2
40. If òxf(sinx)dx=A òf(sinx)dx, then A isi
u
0 0
(1) 0 G (2) p
(3)p y (4) 2p
4 d
u
f(a) f(a)
41. If f(x) =1+exex , I1 = òxSgt{x(1- x)}dx and I2 = òg{x(1- x)}dx then the value of II2
f(-a) f(-a) 1
.
is
w
(1) 2 (2) –3
(3) –1 w (4) 1
w
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 curvesx2 +y2 - 2ay=0, where a is an
arbitrary constant is
(1)2(x2 - y2)y¢=xy (2) 2(x2 +y2)y¢=xy
(3)(x2 - y2)y¢=2xy (4) (x2 +y2)y¢=2xy
44. The solution of the differential equation y dx + (x + x2y) dy = 0 is
1 1
(1)- =C (2) - +logy=C
xy xy
6
AIEEE-PAPERS--7
1
(3) +logy=C (4) log y = Cx
xy
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
x y x y x y x y
(1) + =-1and + =-1 (2) - =-1and + =-1
m
2 3 -2 1 2 3 -2 1
x y x y x y x y
(3) + =1and + =1 (4) - =1and o+ =1
2 3 2 1 2 3 -2 1
c
.
47. If the sum of the slopes of the lines given by x2 - 2acxy- 7y2 =0 is four times their
product, then c has the value i
d
(1) 1 (2) –1
(3) 2 (4) –2n
I
e
48. If one of the lines given by 6x2 - xy+4cy2 =0 is 3x + 4y = 0, then c equals
d
(1) 1 (2) –1
(3) 3 i(4) –3
u
G
49. If a circle passes through the point (a, b) and cuts the circle x2 +y2 =4
orthogonally, then the locus of iyts centre is
(1)2ax+2by+(a2 +b2 +4)=0 d (2) 2ax+2by- (a2 +b2 +4)=0
(3)2ax- 2by+(a2 +b2 +4)=0u (4) 2ax- 2by- (a2 +b2 +4)=0
t
S
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
w
(1)(x- p)2 =4qy (2) (x- q)2 =4py
w
(3)(y- p)2 =4qx (4) (y- q)2 =4px
w
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)x2 +y2 - 2x+2y- 23=0 (2) x2 +y2 - 2x- 2y- 23=0
(3)x2 +y2 +2x+2y- 23=0 (4) x2 +y2 +2x- 2y- 23=0
52. The intercept on the line y = x by the circle x2 +y2 - 2x=0 is AB. Equation of the
circle on AB as a diameter is
(1)x2 +y2 - x- y=0 (2) x2 +y2 - x+y=0
(3)x2 +y2 +x+y=0 (4) x2 +y2 +x- y=0
53. If a „ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection
of the parabolas y2 =4ax andx2 =4ay, then
7
AIEEE-PAPERS--8
(1)d2 +(2b+3c)2 =0 (2) d2 +(3b+2c)2 =0
(3)d2 +(2b- 3c)2 =0 (4) d2 +(3b- 2c)2 =0
1
54. The eccentricity of an ellipse, with its centre at the origin, is . If one of the
2
directrices is x = 4, then the equation of the ellipse is
(1)3x2 +4y2 =1 (2) 3x2 +4y2 =12
(3)4x2 +3y2 =12 (4) 4x2 +3y2 =1
55. A line makes the same angle q , with each of the x and z axis. If the angle b ,
m
which it makes with y-axis, is such thatsin2b=3sin2q, then cos2q equals
2 1 o
(1) (2)
3 5 c
3 2 .
(3) (4) a
5 5
i
d
56. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0
n
is
I
3 e5
(1) (2)
2 2
d
7 9
(3) i(4)
u
2 2
G
57. A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y
+ a y= z and
x + a = 2y = 2z. The co-ordidnates of each of the point of intersection are given
by u
(1) (3a, 3a, 3a), (a, a, a) (2) (3a, 2a, 3a), (a, a, a)
t
(3) (3a, 2a, 3a), (a, a, 2aS) (4) (2a, 3a, 3a), (2a, a, a)
.
w t
58. If the straight lines x = 1 + s, y = –3 – l s, z = 1 + l s and x = , y = 1 + t, z = 2 –
2
w
t with parameters s and t respectively, are co-planar then l equals
(1) –2 w (2) –1
1
(3) – (4) 0
2
59. The intersection of the spheres x2 +y2 +z2 +7x- 2y- z=13 and
x2 +y2 +z2 - 3x+3y+4z =8 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
r r r
60. Let a, b and c be three non-zero vectors such that no two of these are collinear.
r r r r r r
If the vector a+2b is collinear with c and b+3c is collinear with a (l being some
r r r
non-zero scalar) then a+2b+6c equals
8
AIEEE-PAPERS--9
r r
(1)l a (2) l b
r
(3)l c (4) 0
61. A particle is acted upon by constant forces 4ˆi+ˆj- 3kˆ and 3ˆi+ˆj- kˆ which displace
it from a point ˆi+2ˆj+3kˆ to the point5ˆi+4ˆj+kˆ. The work done in standard units
by the forces is given by
(1) 40 (2) 30
(3) 25 (4) 15
62. If a, b, c are non-coplanar vectors and l is a real number, then the vectors
a+2b+3c, l b+4c and (2l - 1)c are non-coplanar for m
(1) all values of l (2) all except one value of l
o
(3) all except two values of l (4) no value of l
c
.
63. Let u, v, w be such that u =1, v =2, w =3. If thea projection v along u is equal
to that of w along u and v, w are perpendicuilar to each other then u- v+w
d
equals
n
(1) 2 (2) 7
I
(3) 14 (4)e 14
d
64. Let a, b and c be non-zero vectors usiuch that(a´b)´c =1 b c a. If q is the acute
3
angle between the vectors b andcG, then sin q equals
1 y 2
(1) (2)
3 d 3
2 u 2 2
(3) (4)
3 t 3
S
65. Consider the following. statements:
w
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
w
(c) Variance is independent of change of origin and scale.
Which of thesew 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
(1) (2) 2
n
2
(3) 2 (4)
n
4 3
67. The probability that A speaks truth is , while this probability for B is . The
5 4
probability that they contradict each other when asked to speak on a fact is
9
AIEEE-PAPERS--10
3 1
(1) (2)
20 5
7 4
(3) (4)
20 5
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
¨ F) is
m
(1) 0.87 (2) 0.77
(3) 0.35 (4) 0.50
o
c
69. The mean and the variance of a binomial distribution are 4 and 2 respectively.
.
Then the probability of 2 successes is
a
37 219
(1) (2) i
d
256 256
128 28n
(3) (4)
256 I256
e
d
70. With two forces acting at a point, the maximum effect is obtained when their
resultant is 4N. If they act at right aingles, then their resultant is 3N. Then the
u
forces are
G
(1)(2+ 2)Nand(2- 2)N (2) (2+ 3)Nand(2- 3)N
y
æ 1 ö æ 1 ö æ 1 ö æ 1 ö
(3)ç2+ 2÷Nandç2- 2÷Nd (4) ç2+ 3÷Nandç2- 3÷N
è 2 ø è 2 ø è 2 ø è 2 ø
u
71. In a right angle D ABC, — tA = 90(cid:176) and sides a, b, c are respectively, 5 cm, 4 cm
rS
and 3 cm. If a force F has moments 0, 9 and 16 in N cm. units respectively
r
.
about vertices A, B awnd C, then magnitude of F is
(1) 3 (2) 4
w
(3) 5 (4) 9
rwr r
72. Three forces P,Q andR acting along IA, IB and IC, where I is the incentre of a D
r r r
ABC, are in equilibrium. Then P:Q:R is
A B C A B C
(1)cos :cos :cos (2) sin :sin :sin
2 2 2 2 2 2
A B C A B C
(3)sec :sec :sec (4) cosec :cosec :cosec
2 2 2 2 2 2
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
17 13 13 17
(1) km/h and km/h (2) km/h and km/h
4 4 4 4
10