vit - mathematics -2007 unsolved paper
TRANSCRIPT
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VIT – PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2007
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SECTION – I
Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them is
correct. Indicate you choice of the correct answer for each part in your answer-book by
writing the letter (a), (b), (c) or (d) whichever is appropriate
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01If the normal to the curve y = f(x) at (3, 4) makes an angle with the positive x-
axis, then f' (3) is equal to :
a. -1
b.
c. 1
d.
Problem
34
34
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02The function . Then the maximum value of f(x) is:
a.
b.
c.
d.
Problem
2 2xf x x e , x 0
1e
12e
2
1e
2
1e
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Problem03if is equal to :
a. sin u
b. cosec u
c. 2 tan u
d. 3 tan u
2 2 u ux y sin u x y ,then x y
x y
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Problem04The angle between the tangents at those points on the curve
where it meets x-axis is :
a.
b.
c.
d.
2 2x t 1 and y t t 6
1 4 tan
29
1 10 tan
49
1 5 tan
49
1 5 tan
49
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Problem05The value of is equal to:
a. 2
b.
c.
d.
4
1x 3 dx
52
12
32
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Problem06The area of the region bounded by the straight lines x = 0 and x = 2, and the
curves is equal to :
a.
b.
c.
d.
x 2y 2 and y 2x x
2 4log 2 3
3 4log 2 3
1 4log 2 3
4 3log 2 2
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Problem07The value of is equal to:
a.
b.
c.
d.
2 20
dx
a x
2
2a
a
12a
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Problem08The value of the integral is:
a.
b.
c.
d.
2X
2
1 xe dx
1 x
x2
1 xe c
1 x
x2
1 xe c
1 x
x
2
ec
1 x
xe 1 x c
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Problem09If , then the value of is equal
to:
a. x
b.
c. log x
d.
y yx sin dy y sin x dx and y 1
x x 2
ycos
x
1x
xe
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Problem10The differential equation of the system of all circles of radius r in the xy plane is :
a.
b.
c.
d.
2 23 22
2
dy d y1 r
dx dx
2 33 22
2
dy d y1 r
dx dx
3 22 22
2
dy d y1 r
dx dx
3 32 22
2
dy d y1 r
dx dx
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Problem11The general solution of the differential equation
is given by :
a.
b.
c.
d.
23x
2
d y dy 2 y 2e
dxdx
3x
x1 2
ey c c x e
8
-3x
-x1 2
ey c c x e
8
3x
-x1 2
ey c c x e
8
3x
-x1 2
ey c c x e
8
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Problem12The solution of the differential equation is :
a.
b.
c.
d.
3ydx x y dy 0
31xy y c
3
4xy y c
4 y 4xy c
34y y c
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Problem13The number of integral solutions of , is :
a.
b.
c.
d.
1 2 3 ix x x 0, with x 5
152C
162C
172C
182C
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Problem14Let A = {1, 2, 3, ... , n} and B = {a, b, c}, then the number of functions from A to
B that are onto is :
a.
b.
c.
d.
E
n n3 – 2
n n3 – 2 1
n3 2 1
n n3 3 2 1
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Problem15Everybody in a room shakes hands with everybody else. The total number of
hand shakes is 66. The total number of persons in the room is :
a. 9
b. 12
c. 10
d. 14
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Problem16In a group G = {1, 3,7, 9} under multiplication modulo 10, the inverse of 7 is :
a. 7
b. 3
c. 9
d. 1
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Problem17A box contains 9 tickets numbered 1 to 9 inclusive. If 3 tickets are drawn from
the box one at a time, the probability that they are alternatively either {odd,
even, odd} or {even, odd, even} is :
a.
b.
c.
d.
517
417
516
518
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Problem18 is equal to :
a.
b.
c.
d.
1 5 B 1If P A ,P B and P , then P A B
12 12 A 15
89180
90180
92180
91180
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Problem19If the probability density function of a random variable X is
,then P(X > 1.5 | X > 1) is equal to :
a.
b.
c.
d.
xf x in a 0 x 2
2
716
34
712
2164
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Problem20If X follows a binomial distribution with parameters n = 100 and , then P(X
=r) is maximum when r is equal to
a. 16
b. 32
c. 33
d. none of these
1p
3
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Problem21If and AB =1, then B is equal to:
a.
b.
c.
d.
1 tan A
-tan 1
2sec
A
A2
A
A2
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Problem22If x=-5 is a root of then the other roots are :
a. 3, 3.5
b. 1, 3.5
c. 1, 7
d. 2, 7
2x 1 4 8
2 2x 2 0
7 6 2x
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Problem23The simultaneous equations Kx + 2y - z = 1,(K -I)y - 2z = 2and (K + 2) z =3 have
only one solution when:
a. k = -2
b. k = -1
c. k = 0
d. k = 1
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Problem24If the rank of the matrix is 1,then the value of a is:
a. -1
b. 2
c. -6
d. 4
1 2 5
2 -4 a-2
1 -2 a+1
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Problem25If ,then all the roots of the equation
will be real if:
a. b>0,a<0,c>0
b. b<0,a>0,c>0
c. b>0,a>0,c>0
d. b>0,a>0,c<0
2 4 2b 4ac for the equation ax bx c 0
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Problem26If x>0 and ,then x
equals:
a. 9
b. 81
c. 1
d. 27
8 1643 3 3 3 3log x log x +log x +log x +log x . 4
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Problem27the number of real roots of equation is:
a. 0
b. 2
c. 4
d. 6
31 1
x x 0x x
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Problem28If H is the harmonic mean between P and Q, then the value of is:
a. 2
b.
c.
d.
H HP Q
PQP Q
12
P QPQ
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Problem29If is a unit vector, then is:
a.
b.
c.
d.
a . b b b x a x b
b
2
a b
a.b a
a
b
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Problem30If is the angle between the lines AB and AC where A, Band C are the three
points with coordinates (1, 2, -1), (2, 0, 3), (3, -1, 2) respectively, then
is equal to:
a. 20
b. 10
c. 30
d. 40
462 cos
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Problem31Let the pairs each determine a plane. Then the planes are
parallel, if:
a.
b.
c.
d.
a,b and c, d
a x c x b x d 0
a x c . b x d 0
a x b x c x d 0
a x b . c x d 0
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Problem32The area of a parallelogram with as diagonals
is :
a.
b.
c.
d.
ˆ ˆˆ ˆ ˆ ˆ3i j 2k and i 3j 4k
72
73
74
75
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Problem33 If
is equal to:
a. 2
b. (b)1
c. (c) -1
d. (d)0
2 12 10 8 6cos x cos x 1, then the value of sin x 3 sin x 3 sin x sin x 1
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Problem34The product of all values of is:
a. 1
b.
c.
d.
cos i sin
cos 3 i sin 3
cos 3 i sin 3
3/5cos i sin
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Problem35The imaginary part of is:
a.
b. 0
c.
d.
21 i
i 2i 1
45
25
45
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Problem36If , then is equal to :
a.
b.
c.
d.
1 1sin x sin y2
1 1cos x cos y
2
4
34
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Problem37The equation of a directrix of the ellipse is:
a. 3y = 5
b. y = 5
c. 3y = 25
d. y = 3
2 2x y + 1
16 25
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Problem38If the normal at on the parabola , meets the
parabola again at , then:
a.
b.
c.
d.
2ap , 2ap 2y 4ax
2aq , 2aq
2p pq 2 0
2p - pq 2 0
2q pq 2 0
2p pq 1 0
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Problem39The length of the straight line x - 3y = 1 intercepted by the hyperbola
is :
a.
b.
c.
d.
2 2x 4y 1
10
65
1
10
610
5
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Problem40The curve described parametrically by , y = 3t + 5
represents:
a. an ellipse
b. a hyperbola
c. a parabola
d. a circle
2x t 2t 1