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AIEEE-2008 NARAYANA INSTITUTE : A-1/171 A, Janak Puri, N.D.58; FNS House, 63, Kalu Sarai, Sarvapriya Vihar, N.D.-16; D-11/141, Sec.-8, Rohini, N.D.-85,; F-38, South Extn.-I, New Delhi-49
[1]
AIEEE - 2008 Hints & Solutions(Code - D5) 27-4-2008
MATHS
CODE:
D5
HINTS & SOLUTIONS FOR AIEEE-2008
1.
The mean of the numbers a, b, 8, 5, 10 is 6 and the
variance is 6.80. Then which one of the following
gives possible values of a and b?
(1) a = 1, b = 6
(2) a = 3, b = 4
(3) a = 0, b = 7
(4) a = 5, b = 2
Key : (2)
2.
The vector
a
i 2 j 2 j
k
= +
+
+ lies in the plane
of the vectors

b i
j
= + and

c j k
= + and bisects
the angle between b and c . Then which one of
the following gives possible values of
and ?
(1)
= 2, = 1
(2)
= 1, = 1
(3)
= 2, = 2
(4)
= 1, = 2
Key : (2)
3.
The now-zero vectors a , b and c are related by
a = 8 b and c = 7 b . Then the angle between a
and c is
(1) 2
(2) (3) 0
(4) 4
Key : (2)
4.
The line passing through the points (5, 1, a) and (3,
2, 1) crosses the yz-plane at the point
17
13
0,
,
2
2
.
Then
(1) a = 6, b = 4
(2) a = 8, b = 2
(3) a = 2, b = 8
(4) a = 4, b = 6
Key : (1)
5.
If the straight lines x 1 y 2
z 3
k
2
3 =
=
and
x 2
y 3
z 1
3
k
2 =
=
intersect at a point then the
integer k is equal to
(1) 2
(2) 2
(3) 5
(4) 5
Key : (3)
6.
The differential equation of the family of circles with
fixed radius 5 units and centre on the line y = 2 is
(1) (y 2)
2
y
2
= 25 (y 2)
2
(2) (x 2)
2
y
2
= 25 (y 2)
2
(3) (x 2)
2
y
2
= 25 (y 2)
2
(4) (y 2) y
2
= 25 (y 2)
2
Key : (1)
7.
Let a. b. c be any real numbers. Suppose that there
are real numbers x, y, z not all zero such that
x = cy + bz, y = az + cx, and z = bx + ay. Then
a
2
+ b
2
+ c
2
+ abc is equal to
(1) 0
(2) 1
(3) 2
(4) 1
Key : (2)
8.
Let A be a square matrix all of whose entries are
integers. Then which one of the following is true?
(1) If det A = ± 1, then A
1
exists and all its entries
are integers
(2) If det A = ± 1, then A
1
need not exist NARAYANA INSTITUTE : A-1/171 A, Janak Puri, N.D.58; FNS House, 63, Kalu Sarai, Sarvapriya Vihar, N.D.-16; D-11/141, Sec.-8, Rohini, N.D.-85,; F-38, South Extn.-I, New Delhi-49
[2]
AIEEE - 2008 Hints & Solutions(Code - D5) 27-4-2008
(3) If det A = ± 1, then A
1
exists but all its entries
are not necessarily integers
(4) If det A
± 1, then A
1
exists and all its entries
are non-integers
Key : (1)
9.
The quadratic equations x
2
6x + a = 0 and x
2
cx
+ 6 = 0 have one root in common. The other roots
of the first and second equations are integers in the
ratio 4 : 3. Then the common root is
(1) 3
(2) 2
(3) 1
(4) 4
Key : (2)
10. How many different words can be formed by
jumbling the letters in the word MISSISSIPPI in
which no two S are adjacent?
(1) 6 . 8 .
7
C
4
(2) 7 .
6
C
4
.
8
C
4
(3) 8 .
6
C
4
.
8
C
4
(4) 6 . 7 .
8
C
4
Key : (2)
11. Let I =
1
0
sin x dx
x and J =
1
0
cos x dx
x . Then which
one of the following is true?
(1) I < 23 and J > 2
(2) I > 23 and J < 2
(3) I > 23 and J > 2
(4) I < 23 and J < 2
Key : (4)
12. The area of the plane region bounded by the curves
x + 2y
2
= 0 and x + 3y
2
= 1 is equal to
(1) 23
(2) 43
(3) 53
(4) 13
Key : (1)
13. The value of
sin x dx
2
sin x 4
is
(1) x log | sin(x
) | c + +
4
(2) x log | cos(x
) | c +
4
(3) x log | cos(x
) | c + +
4
(4) x log | sin(x
) | c +
4
Key : (1)
14. The statement p
(q p) is equivalent of
(1) p
(p q)

(2) p
(p
q)

(3)
p
(p
q)

(4)
p
(p q)

Key : (4)
15. The value of cot
1
1
5
2
cosec
tan
3
3
+ is
(1) 4
17
(2) 5
17
(3) 6
17
(4) 3
17
Key : (3)
Directions: Questions number 16 to 20 are
Assertion-Reason type questions. Each of these
questions contains two statements: Statement-1
(Assertion) and Statement-2 (Reason). Each of
these questions also has four alternative choices,
only one of which is the correct answer. You have
to select the correct choice.
16. Let A be a 2 × 2 matrix which real entries. Let I be
the 2 × 2 identity matrix. Denote by tr(A), the sum
of diagonal entries of A. Assume that A
2
= I.
Statement-1:
If A
I and A I, then det A = 1.
Statement-2:
If A
I and A I, then tr (A) 0.
(1) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for
Statement-1
(2) Statement-1 is true, Statement-2 is false
(3) Statement-1 is false, Statement-2 is true
(4) Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for
Statement-1 NARAYANA INSTITUTE : A-1/171 A, Janak Puri, N.D.58; FNS House, 63, Kalu Sarai, Sarvapriya Vihar, N.D.-16; D-11/141, Sec.-8, Rohini, N.D.-85,; F-38, South Extn.-I, New Delhi-49
[3]
AIEEE - 2008 Hints & Solutions(Code - D5) 27-4-2008
Key : (2)
17. Let p be the statement x is an irrational number, q
be the statement y is a transcendental number,
and r be the statement x is a rational number iff y
is a transcendental number.
Statement-1:
r is equivalent to either q or p.
Statement-2:
r is equivalent to ~ (p
~ q).
(1) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for
Statement-1
(2) Statement-1 is true, Statement-2 is false
(3) Statement-1 is false, Statement-2 is true
(4) Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for
Statement-1
Key : (3)
18. In the shop there are five types of ice-creams
available. A child buys six ice-creams.
Statement-1:
The number of different ways the child can buy the
six ice-creams is
10
C
5
.
Statement-2:
the number of different ways the child can buy the
six ice-creams is equal to the number of different
ways of arranging 6 As and 4 Bs in a row.
(1) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for
Statement-1
(2) Statement-1 is true, Statement-2 is false
(3) Statement-1 is false, Statement-2 is true
(4) Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for
Statement-1
Key : (3)
19. Statement-1:
n
n
n 1
r
r 0
(r 1) C
(n 2)2
. =
+
= + Statement-2:
n
n
r
n
n 1
r
r 0
(r 1) C x
(1 x)
nx(1 x)
. =
+
= +
+
+ (1) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for
Statement-1
(2) Statement-1 is true, Statement-2 is false
(3) Statement-1 is false, Statement-2 is true
(4) Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for
Statement-1
Key : (4)
20. Statement-1:
For every natural number n > 2,
1
1
1
...
n.
2
2
n
+
+ +
>
Statement-2:
For every natural number n > 2,
n(n 1) n 1.
+ < +
(1) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for
Statement-1
(2) Statement-1 is true, Statement-2 is false
(3) Statement-1 is false, Statement-2 is true
(4) Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for
Statement-1
Key : (4)
21. The conjugate of a complex number is 1
i 1
. Then
that complex number is
(1)
1
i 1 +
(2)
1
i 1 (3)
1
i 1
(4)
1
i 1
+
Key : (1)
22. Let R be the real line. Consider the following subsets
of the plane R × R:
S = {(x, y) : y x + 1 and 0 < x < 2}
T = {(x , y) : x y is an integer}
which one of the following is true?
(1) S is an equivalence relation on R but T is not
(2) T is an equivalence relation on R but S is not NARAYANA INSTITUTE : A-1/171 A, Janak Puri, N.D.58; FNS House, 63, Kalu Sarai, Sarvapriya Vihar, N.D.-16; D-11/141, Sec.-8, Rohini, N.D.-85,; F-38, South Extn.-I, New Delhi-49
[4]
AIEEE - 2008 Hints & Solutions(Code - D5) 27-4-2008
(3) Neither S nor T is an equivalence relation on R
(4) Both S and T are equivalence relations on R
Key : (2)
23. Let f : N
Y be a function defined as
f(x) 4x + 3 where
Y = {y
N : y = 4x + 3 for some x N}
Show that f is invertible and its inverse is
(1)
y 3
g(y)
4
+
=
(2)
y 3
g(y)
4 =
(3)
3y 4
g(y)
3
+
=
(4)
y 3
g(y) 4
4
+
= +
Key : (2)
24. AB is a vertical pole with B at the ground level and
A at the top. A man finds that the angle of elevation
of the point A from a certain point C on the ground
is 60°. He moves away from the pole along the line
BC to a point D such that CD = 7 m. From D the
angle of elevation of the point A is 45°. Then the
height of the pole is
(1) 7 3 ( 3 1)m
2 (2) 7 3
1
m
2
3 1
+
(3) 7 3
1
m
2
3 1 (4) 7 3 ( 3 1)m
2
+
Key : (4)
25. A die is thrown. Let A be the event that the number
obtained is greater than 3. Let B be the event that
the number obtained is less than 5. Then P(A
B)
is
(1) 1
(2) 25
(3) 35
(4) 0
Key : (1)
26. It is given that the events A and B are such that
P(A) = 14 , P(A | B) = 12 and (P(B | A) = 23 . Then
P(B) is
(1) 23
(2) 12
(3) 16
(4) 13
Key : (4)
27. A focus of an ellipse is at the origin. The directrix is
the line x = 4 and the eccentricity is 12 . Then the
length of the semi-major axis is
(1) 43
(2) 53
(3) 83
(4) 23
Key : (3)
28. A parabola has the origin as its focus and the line x
= 2 as the directrix. Then the vertex of the parabola
is at
(1) (0, 1)
(2) (2, 0)
(3) (0 , 2)
(4) (1 , 0)
Key : (4)
29. The point diametrically opposite to the point P(1, 0)
on the circle x
2