Download PART A - MATHEMATICS (Solutions)

Solution To AIEEE-2008
Part A - Mathematics
1.
Let f : N ® Y be a function defined as f(x) = 4x + 3 where
Y = {y Î N : y = 4x + 3 for some x Î N}. Show that f is invertible and its inverse is
1.
(1)
(2)
(3)
(4)
(3)
f(x) = 4x + 3
f¢(x) = 4 > 0
\
f(x) is strictly increasing function
Þ
f(x) is one-one and onto
Þ
f-1(x) exist
Putting f(x) = y
y = 4x + 3
Þ
2.
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) Both S and T are equivalence relations on R
(2) S is an equivalence relation of R but T is not
(3) T is an equivalence relation on R but S is not
(4) Neither S nor T is an equivalence relation on R
2.
(3)
3.
The conjugate of a complex number is
3.
(1)
(2)
(3)
(4)
(2)
.
4.
. Then that complex numbers is
z=
The quadratic equations
x2 = 6x + a = 0
and x2 - 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 roots is
4.
(1) 4
(2) 3
(3) 2
(4) 1
(3)
x2 - 6x + a = 0
(1)
x2 - cx + 6 = 0
(2)
Given second root equation(1)/second root of equation(2) = 4/3 = k
Let one root is a and other root is 4k of first equation
(x - a) (x - 4k) = 0,
Þ
x2 - (a + 4k)x + 4k a = 0
(3)
(x - a) (x - 3k) = 0
Þ
x2 - (a + 3k)x + 3ka = 0
(4)
By comparison (3) with (1) and (4) with (2)
a + 4k = 6,
(5)
a + 3k = c, 3ka = 6 Þ a = 2/k
(5) Þ 2/k + 4k = 6 Þ k = 1, k = ½
k = ½ is not allowed as it gives
second root of second equation as 3 ´ ½ = 3/2 which is not an integer.
\
5.
k = 1, a = 2.
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 ± 1, then A-1 exists and all its entries are non-integers
(2) If det A = ±1, then A-1 exists and all its entries are integers
(3) If det A = ±1, then A-1 need not exist
(4) If det A = ±1, then A-1 exists but all its entries are not necessarily integers
5.
(2)
6.
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 + b2 + c2 + 2abc is equal
to
6.
(1) -1
(2) 0
(3) 1
(4) 2
(3)
(for non trivial solution)
1 - a2 + c (-c - ab) + b (-ac - b) = 0
a2 + b2 + c2 + 2abc = 1
7.
7.
How many different words can be formed by jumbling the letters in the word
MISSISSIPPI in which no two S are adjacent?
(1) 6 . 7 . 8C4
(2) 6 . 8 . 7C4
(3) 7 . 6C4 . 8C4
(4) 8 . 6C4 . 7C4
(3)
First we took M, I, I, I, P, P, I and arranged then arrangement =
ways.
There are 8 spaces available for 4 S.
So, total number of arrangement according to requirement =
=
8.
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
8.
(4) -4
(1)
Given a + ar = 12 (where a first term and r common ratio)
ar2 + ar3 = 48
r2(a + ar) = 48
r2 = 4
r = ±2 Þ r = -2
(is valid)
\
9.
first term is -12 and second term is -12 ´ (-2) = 24
Let
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
9.
(2)
LHD =
=
= a finite value which lies between -1 and 1.
RHD = sin(¥) = a finite value which lies between -1 and 1.
RHD ¹ LHD.
\
10.
10.
f(x) is differentiable at x = 0 but not at x = 1.
How many real solutions does the equation x 7 + 14x5 + 16x3 + 30x - 560 = 0 have?
(1) 1
(2) 3
(3) 5
(4) 7
(1)
f(x) = x7 + 14x5 + 16x3 + 30x - 560
f¢(x) = 7x6 + 70x4 + 48x2 + 30 > 0
Þ
f(x) is strictly increasing function
Þ
hence f(x) cuts x axis only at one point
11.
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
and maxima at
(2) The cubic has minima at both
and -
(3) The cubic has maxima at both
and -
(4) The cubic has minima at
11.
and maxima at -
(4)
f¢(x) = 0
4x2 - p = 0
x=±
12.
The value of
is
(1)
(2)
(3)
(4)
12.
(2)
I=
=
=
= x + ln(sinx - cosx) + c
= x + logsin(x - p/4) + k where k = log
13.
13.
+ c.
The area of the plane region bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is equal
to
(1)
(2)
(3)
(4)
(3)
x = -2y2
x = 1 - 3y2
x1 = -2y2
x2 = 1 - 3y2
x1 = x2
Þ
(for cutting points)
-2y2 = 1 - 3y2
y2 = 1
y = ±1
Area =
Area =
=
=
=
.
14.
Let
and
.
Then which one of the following is true?
14.
(1)
and J < 2
(2)
and J > 2
(3)
and J < 2
(4)
and J > 2
(1)
\
I < 2/3
J < 2.
15.
15.
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)2y¢2 = 25 - (y - 2)2
(3) (x - 2)2y¢2 = 25 - (y - 2)2
(4) (x - 2)y¢2 = 25 - (y - 2)2
(2)
Equation of circle
(x - h)2 + (y - 2)2 = 25
Differentiating with respect to x
2(x - h) + 2(y - 2)y¢ = 0
(x - h) = -(y - 2) y¢
Þ
16.
(y - 2)2 y¢2 + (y - 2)2 = 25.
The solution of the differential equation
satisfying the condition y(1) = 1 is
16.
(1) y = xlnx + x2
(2) y = xe(x - 1)
(3) y = xlnx + x
(4) y = ln x + x
(3)
I.F. =
y.
y = xlogx + x as y(1) = 1 Þ c = 1.
17.
17.
The perpendicular bisector of the line segment joining P(1, 4) and Q(k , 3) has yintercept -4. Then a possible value of k is
(1) 2
(2) -2
(3) -4
(4) 1
(3)
Slope of PQ =
Since the perpendicular bisector passing
through R and S
Slope of RS =
We know slope of RS ´ slope of PQ = -1
Þ
18.
k = ± 4 Þ k = -4.
The point diametrically opposite to the point P(1, 0) on the circle x 2 + y2 + 2x + 4y - 3 =
0 is
(1) (-3, 4)
(2) (-3, -4)
(3) (3, 4)
18.
(4) (3, -4)
(2)
Equation of circle
x2 + y2 + 2x + 4y - 3 = 0
Centre (-1, -2)
Since P and Q are diametrically opposite.
19.
19.
\
O is the mid point of P and Q.
\
(x, y) = (-3, -4)
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)
(2) (0, 1)
(3) (2, 0)
(4) (0, 2)
(1)
x-coordinate of vertex is
=1
and y-coordinate of vertex is
20.
A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity
is
. Then the length of the semi-major axis is
20.
(1)
(2)
(3)
(4)
(4)
(a - ae) = e ´
Using above formula
(4 - x )
=x
2 - x/2 = x
2 = 3x/2
x = 4/3 = OP
PQ = 4 - 4/3 = 8/3
2a - a = 8/31
a = 8/3
21.
If the straight lines
and
intersect at a point, then the integer k is equal to
21.
(1) 5
(2) 2
(3) -2
(4) -5
(4)
Two lines
and
if
are intersecting
= 0.
Þ
22.
Þ
2k2 + 5k - 25 = 0
Þ
k = -5.
The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the
point
22.
. Then
(1) a = 4, b = 6
(2) a = 6, b = 4
(3) a = 8, b = 2
(4) a = 2, b = 8
(2)
(1)
(2)
(3)
Putting x = 0 in (1) taking with (3)
=
Þ
a = 6.
Taking (1) and (3)
Þ (1 - b) = -3 Þ b = 4.
23.
23.
The non-zero vectors
,
angle between
is.
(2)
(3) p
(4) 0
= 8
and
= -7
. Then the
(3)
are antiparallel vector so angle between
The vector
=a
+2
+b
and
= p.
lies in the plane of the vectors
+
and bisects the angle between
possible values of a and b?
24.
are related by
(1)
and
24.
and
and
and
=
+
and
=
. Then which one of the following gives
(1) a = 1, b = 2
(2) a = 2, b = 1
(3) a = 1, b = 1
(4) a = 2, b = 2
(3)
a-2+b=0
Þ
25.
25.
a+b=2
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
(3)
= arithmetical mean
=6
a+b=7
(6 - a)2 + (6 - b)2 + 22 + 12 = 5 ´ 6.8
(6 - a)2 + (6 - b)2 = 13
Þ
26.
26.
a = 3, b = 4.
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 È B) is
(1) 0
(2) 1
(3)
(4)
(2)
A = {4, 5, 6}
B = {1, 2, 3, 4}
P(A È B) = P(A) + P(B) - P(A Ç B)
=
27.
27.
=
= 1.
It is given that the event A and B are such that P(A) =
Then P(B) is
(1)
(2)
(3)
(4)
(1)
=
, P(A|B) =
and P(B|A) =
.
.
28.
28.
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 point A is 45o. Then the height of the pole is
(1)
(2)
(3)
(4)
(1)
tan60o =
=
29.
29.
.
The value of
is
(1)
(2)
(3)
(4)
(4)
=
=
=
=
=
30.
The statement p ® (p ® q) is equivalent to
(1) p ® (p Ú q)
(3) p ® (p « q)
30.
(2) p ® (p Ù q)
(4) p ® (p ® q)
(2)
Directions: Questions number 31 to 35 are Assertion-Reason type questions. Each of these
question 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.
31.
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 ¹ I and A ¹ -I, then detA = -1.
Statement - 2 :
If A ¹ I and A ¹ -I, then tr(A) ¹ 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
31.
(3)
32.
Statement - 1:
For every natural number n ³ 2,
Statement - 2 :
For every natural number n ³ 2,
(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
32.
(2)
33.
Statement - 1:
Statement - 2 :
(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
33.
(1)
34.
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
34.
(4)
35.
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).
35.
(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
(3)