(Paper) AIEEE 2008 EXAMINATION PAPER : Code-A6 (Paper -4)

PAPER : AIEEE 2008 EXAMINATION PAPER (Code-A6)

Paper - 4

76. 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 a2 + b2 + c2 + 2abc is equal to
(1) –1 (2) 0
(3) 1 (4) 2
Ans. [3]


77. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not
two S are adjacent ?
(1) 6. 7. 8C4 (2) 6. 8. 7C4
(3) 7. 6C4 . 8C4 (4) 8. 6C4 . 7C4
Ans. [3]


78. The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is
48. If the terms of the geometric progression are alternately positive and negative, then the first term is
(1) – 12 (2) 12
(3) 4 (4) – 4
Ans. [1]


79. Let f(x) =



=



0 if x 1
if x 1
x 1
(x 1)sin 1
Then which one of the following is true ?
(1) f is differentiable at x = 0 and at x = 1 (2) f is differentiable at x = 0 but not at x = 1
(3) f is differentiable at x = 1 but not at x = 0 (4) f is neither differentiable at x = 0 nor at x = 1
Ans. [2]


80. How many real solution does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have ?
(1) 1 (2) 3
(3) 5 (4) 7
Ans. [1]


81. Suppose the cubic x3 – px + q has three distinct real roots where p > 0 and q > 0. Then which one of the
following holds ?
 (1) The cubic has minima at –
3
p and maxima at
3
p
(2) The cubic has manima at both
3
p and –
3
p
(3) The cubic has maxima at both
3
p and –
3
p
(4) The cubic has minima at
3
p and maxima at –
3
p
Ans. [4]


82. The value of 2 ∫



 π

4
sin x
sin x dx is -
(1) x – log | sin (x –
4
π
) | + c (2) x + log | sin (x –
4
π
) | + c
(3) x – log | cos (x –
4
π
) | + c (4) x + log | cos (x –
4
π
) | + c
Ans.[2]


83. The area of the plane region bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is equal to -
(1)
3
1 (2)
3
2
(3)
3
4 (4)
3
5
Ans.[3]


84. Let I = ∫
1
0 x
x sin dx and J = ∫
1
0 x
cos x dx. Then which one of the following is true ?
(1) I <
3
2 and J < 2 (2) I <
3
2 and J > 2
(3) I >
3
2 and J < 2 (4) I >
3
2 and J > 2
Ans.[1]


85. The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is -
(1) (y – 2) y′2 = 25 – (y – 2)2 (2) (y – 2)2 y′2 = 25 – (y – 2)2
(3) (x – 2)2 y′2 = 25 – (y – 2)2 (4) (x – 2) y′2 = 25 – (y – 2)2
Ans.[3]


86. The solution of the differential equation
dx
dy =
x
x + y
satisfying the condition y (1) = 1 is -
(1) y = x ln x + x2 (2) y = xe(x–1)
(3) y = x ln x + x (4) y = ln x + x
Ans.[3]


87. The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a
possible value of k is -
(1) 2 (2) –2
(3) –4 (4) 1
Ans.[3]


88. The point diametrically opposite to the point P(1, 0) on the circle x2 + y2 + 2x + 4y –3 = 0 is -
(1) (–3, 4) (2) (–3, –4)
(3) (3, 4) (4) (3, – 4)
Ans.[2]


89. 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) (1, 0) (1) (0, 1)
(3) (2, 0) (4) (0, 2)
Ans.[1]


90 A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is
2
1 . Then the
length of the semi-major axis is -
(1)
3
2 (2)
3
4
(3)
3
5 (4)
3
8
Ans.[4]


91. If the straight lines
k
x −1
=
2
y − 2
=
3
z − 3
and
3
x − 2
=
k
y − 3
=
2
z −1
intersect at a point, then the integer k is equal to
 (1) 5 (2) 2
(3) –2 (4) –5
Ans. [4]


92. The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point 



 −
2
, 13
2
0,17 .
Then
(1) a = 4, b = 6 (2) a = 6, b = 4
(3) a = 8, b = 2 (4) a = 2, b = 8
Ans. [2]


93. The non-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)
4
π (2)
2
π
(3) π (4) 0
Ans. [3]


94. The vector →a = αiˆ + 2 j ˆ + βkˆ lies in the plane of the vectors →b = iˆ + jˆ and →c = jˆ + kˆ and bisects the
angle between →b and →c . Then which one of the following gives possible values of α and β ?
(1) α = 1, β = 2 (2) α = 2, β = 1
(3) α = 1, β = 1 (4) α = 2, β = 2
Ans. [3]


95. 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 = 5, b = 2 (2) a = 1, b = 6
(3) a = 3, b = 4 (4) a = 0, b = 7
Ans. [3]


96. 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) 0 (2) 1
(3)
5
2 (4)
5
3
Ans. [2]


97. It is given that the events A and B are such that P(A) =
4
1 , P(A|B) =
2
1 and P(B|A) =
3
2 . Then P(B) is
(1)
3
1 (2)
3
2
(3)
2
1 (4)
6
1
Ans. [1]


98. 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 60º. 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 points A is 45º. Then the height of
the pole is
(1) ( 3 1)m
2
7 3 + (2) ( 3 1)m
2
7 3 −
(3) m
3 1
1
2
7 3
+
(4) m
3 1
1
2
7 3

Ans. [1]


99. The value of cot 



 + −
3
tan 2
3
cosec–1 5 1 is
(1)
17
3 (2)
17
4
(3)
17
5 (4)
17
6
Ans. [4]


100. The statement p → (q → p) is equivalent to
(1) p → (p ∨ q) (2) p → (p ∧ q)
(3) p → (p ↔ q) (4) p → (p → q)
Ans. [1]
Directions : Question number 101 to 105 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.

101. Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of
diagonal entries of A, Assume that A2 = 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 a correct explanation for Statement-1
(2) Statement-1 is true, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1
(3) Statement-1 is true, Statement -2 is false
(4) Statement-1 is false, Statement-2 is true
Ans. [4]


102. Statement-1:
For every natural number n ≥ 2.
1
1 +
2
1 + ........ +
n
1 > 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 a correct explanation for Statement-1
(2) Statement-1 is true, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1
(3) Statement-1 is true, Statement -2 is false
(4) Statement-1 is false, Statement-2 is true
Ans. [2]


103. Statement- 1:
Σ=
+
n
r 0
(r 1) nCr = (n +2) 2n–1
Statement -2:
Σ=
+
n
r 0
(r 1) nCr xr = (1 + x)n + nx (1 + x)n–1
(1) Statement-1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1
(2) Statement-1 is true, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1
(3) Statement-1 is true, Statement -2 is false
(4) Statement-1 is false, Statement-2 is true
Ans. [1]


104. In a 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 10C5
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 A's and 4 B's in a row.
(1) Statement-1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1
(2) Statement-1 is true, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1
(3) Statement-1 is true, Statement -2 is false
(4) Statement-1 is false, Statement-2 is true
Ans. [4]


105. 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 ot ~ (p ↔ ~q)
(1) Statement-1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1
(2) Statement-1 is true, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1
(3) Statement-1 is true, Statement -2 is false
(4) Statement-1 is false, Statement-2 is true
Ans. [4]