bitsat mathematics paper –iii

BITSAT MATHEMATICS PAPER –III
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1. For the following linear programming problem : minimize
constraints
2x + 3y ≥ 6
,
x + y ≤ 8 , y ≥ 1, x ≥ 0 ,
z = 4 x + 6y
subject to the
the solution is
(b)(0, 2) and (3 / 2, 1)
(a) (0, 2) and (1, 1)
(c) (0, 2) and (1, 6)
(d) (0, 2) and (1, 5)
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2. Section-wise expenditure of a State Govt. is shown in the given figure. The
(a) 25%
(b) 30%
(c) 32%
(d) 35%
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expenditure incurred on transport is
3. If the covariance between x and y is 10 and the variance of x and y are 16 and 9
respectively, then the coefficient of correlation between x and y is
(a) 0.61
(b) 0.79
(c) 0.83
(d)
0.93
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4. The mean and variance of a binomial distribution are 4 and 3 respectively, then
10
16
⎛1⎞ ⎛3⎞
C6 ⎜ ⎟ ⎜ ⎟
⎝4⎠ ⎝4⎠
6
(b)
6
16
⎛1⎞ ⎛3⎞
C6 ⎜ ⎟ ⎜ ⎟
⎝4⎠ ⎝4⎠
10
.s
a
(a)
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the probability of getting exactly six successes in this distribution is
(c)
10
12
⎛1⎞ ⎛3⎞
C6 ⎜ ⎟ ⎜ ⎟
⎝4⎠ ⎝4⎠
6
(d)
6
12
⎛1⎞ ⎛3⎞
C6 ⎜ ⎟ ⎜ ⎟
⎝4⎠ ⎝4⎠
6
5. In an entrance test there are multiple choice questions. There are four possible
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answers to each question of which one is correct. The probability that a student
w
w
knows the answer to a question is 90%. If he gets the correct answer to a
question, then the probability that he was guessing, is
(a)
37
40
(b)
1
37
(c)
36
37
(d)
1
9
6. If the product of three terms of G.P. is 512. If 8 added to first and 6 added to
second term, so that number may be in A.P., then the numbers are
(a) 2, 4, 8
(b) 4, 8, 16
(c) 3, 6, 12
(d) None of these
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7.
Let a relation R be defined by R = {(4, 5); (1, 4); (4, 6); (7, 6); (3, 7)} then
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is
(a) {(1, 1), (4, 4), (4, 7), (7, 4), (7, 7), (3, 3)}
(b) {(1, 1), (4, 4), (7, 7), (3, 3)}
(c) {(1, 5), (1, 6), (3, 6)}
(d) None of these
(a)
9.
If
be the universal set and
U
(b)
A∪B∪C
10. If
x
2 + (3 x − 2)
(a) 5
(b)
. Then
{( A − B ) ∪ (B − C ) ∪ (C − A )}′
(d)
A∩B∩C
( c) [2,+ ∞ )
(d) None of these
=
(c)
5
(d)
1/5
1/ 5
11. The value of p for which both the roots of the equation
less than 2, lies in
(b) (2, ∞)
(a) (4 / 5, 2)
is equal to
A ∩ (B ∪ C )
x belongs to the interval
(b) (− ∞, 2]
x = 3 − 5, then
(c)
A ∪ (B ∩ C )
log 0.04 (x − 1) ≥ log 0.2 (x − 1) then
(a) (1, 2]
A∪B∪C = U
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Let
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8.
R −1 oR
(c)
4 x 2 − 20 px + (25 p 2 + 15 p − 66 ) = 0 are
(d)
(−1, − 4 / 5)
(−∞, − 1)
12. Out of 10 white, 9 black and 7 red balls, the number of ways in which selection
of one or more balls can be made, is
3
5
7
+
+
+ .... ∞ =
1! 2! 3 !
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1+
(b) 2 e
(a) e
10C1.9C5
+
10C2.9C4
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14.
b) 84
(c) 879
+
10C3.9C3
c) 81
+
(d)
3e
10C4.9C2
+
(d) 892
4e
10C5.9C1
+
10C6
=
15. The values of θ lying between θ = 0 and θ = π/2 satisfying
0, are
a)
5π
24
,
7π
24
b)
7π
24
19C6
+ x then x =
d) – 81
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a) – 84
(c)
.s
a
13.
(b) 891
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(a) 881
,
11π
24
c)
5π
24
,
11π
24
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d)
5π
24
1 + sin 2 θ
cos 2 θ
4sin 4θ
sin 2 θ 1 + cos 2 θ
4sin 4θ
2
2
sin θ
cos θ 1 + 4sin 4θ
=
16. If A is any square matrix of order ‘n’.
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Observe the following list
List – II
A) |adj A|
1) |A|n – 2 A
B) adj (adj A)
2) |A|n(n – 1)
C) )adj A)-1
3) |A| (n −1)
D) |adj(adj A)|
4) |A|n – 1
1
|A|
a) A – 4; ;B – 5; C – 1; D – 3
b) A – 4; B – 5; C – 1; D – 2
c) A – 4; B – 1; C – 5; D – 2
d) A – 4; B – 1; C – 5; D – 3
17.
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5)
2
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List – I
The value of ‘c’ in Lagrange’s mean value theorem for f(x) = (x – a)m (x – b)n in
18. If
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A + B + C = 270 o ,
(b) 1
w
w
4
3
tan(cos
−1
(a) sin A
, if
2s = a + b + c ,
d) a + b
a+b
m+n
2
(c) 2
(d) 3
then K =
3
4
(c)
1
2
(d) 2
then x =
(b) ±
3
ΔABC
(b)
1⎞
⎛
x ) = sin ⎜ cot −1 ⎟ ,
2⎠
⎝
(a) ± 5
21. In
c)
ma + nb
m+n
cos 2 A + cos 2 B + cos 2C + 4 sin A sin B sin C =
cos 6 α + sin 6 α + K sin 2 2α = 1,
(a)
20. If
then
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(a) 0
19. If
b)
mb + na
m+n
.s
a
a)
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[a, b] is
5
3
(c)
±
5
3
then the value of
(b) cos A
(d) None of these
s(s − a) (s − b )(s − c)
−
=
bc
bc
(c)
tan A
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(d) None of these
22. A balloon is coming down at the rate of 4 m/min. and its angle of elevation is 45o
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from a point on the ground which has been reduced to 30o after 10 minutes.
Balloon will be on the ground at a distance of how many meters from the observer
(a)
(b) 20 (3 +
20 3 m
(c)
3 )m
(d) None of these
10 (3 + 3 ) m
23. The perpendicular distance from A(1, 4, –2) to BC where B = (2, 1, –2), C = (0, 5,
a)
3 26
7
b)
3
7
c)
26
7
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1) is
d) 2
b) IA + IB + IC
a) 0
25.
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24. If I is the centre of a circle inscribed in a ΔABC, then BC IA + CA IB + AB IC =
c)
IA + IB + IC
3
d) 2( IA + IB + IC )
A straight line passes through a fixed point
(h, k ) .
The locus of the foot of
perpendicular on it drawn from the origin is
(a)
(b) x 2 + y 2 + hx + ky = 0
c)
(d) None of these
3 x 2 + 3 y 2 + hx − ky = 0
The area (in square units) of the quadrilateral formed by the two pairs of lines
l 2 x 2 − m 2 y 2 + n(lx − my ) = 0
n2
2 | lm |
(b)
n2
| lm |
(c)
(d)
n
2 | lm |
is
n2
4 | lm |
.s
a
(a)
and
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l 2 x 2 − m 2 y 2 − n(lx + my ) = 0
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26.
x 2 + y 2 − hx − ky = 0
27. The locus of the middle points of chords of the circle
x 2 + y 2 − 2 x − 6 y − 10 = 0
which
(a)
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passes through the origin, is
x 2 + y 2 + x + 3y = 0
(b) x 2 + y 2 − x + 3 y = 0 (c)
(d)
x 2 + y 2 + x − 3y = 0
x 2 + y 2 − x − 3y = 0
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28. The locus of the centres of the circles which touch externally the circles
and
x 2 + y 2 = 4 ax
, will be
(a) 12 x 2 − 4 y 2 − 24 ax + 9 a 2 = 0
(b)
12 x 2 + 4 y 2 − 24 ax + 9 a 2 = 0
(c)
(d)
12 x 2 + 4 y 2 + 24 ax + 9 a 2 = 0
12 x 2 − 4 y 2 + 24 ax + 9 a 2 = 0
29. The equation of the common tangent of the parabolas
(a)
2 x + 3 y = 36
x 2 + y 2 = a2
(b) 2 x + 3 y + 36
=0
(c)
3 x + 2 y = 36
(d)
x 2 = 108 y
3 x + 2 y + 36 = 0
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and
y 2 = 32 x
, is
30. The equation of the tangent to the ellipse
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x-axis is
(a)
x 2 + 16 y 2 = 16
(b)
3x − y + 7 = 0
(c)
3x − y − 7 = 0
making an angle of
60 o
with
(d) None of these
3x − y ± 7 = 0
31. The reciprocal of the eccentricity of rectangular hyperbola, is
(a) 2
(b)
1
2
(c)
(d)
2
1
2
32. If three mutually perpendicular lines have direction cosines
then the line having direction cosines
l1 + l 2 + l 3 , m 1 + m 2 + m 3 and n1 + n 2 + n 3
co
m
(l 3 , m 3 , n 3 ) ,
make an angle of ..... with each other
(a) 0°
(b) 30 °
(c)
60 °
(d)
(l1 , m 1 , n1 ), (l 2 , m 2 , n 2 ) and
90 °
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33. A point moves in such a way that the sum of its distance from xy-plane and yzplane remains equal to its distance from zx-plane. The locus of the point is
(a)
34. If
(c)
(b) x + y − z = 0
x −y+z = 2
y cos x + x cos y = π
, then
y ′′(0 )
is
(b) π
(a) 1
(d)
x −y+z =0
(c) 0
x −y−z = 2
(d)–π
35. The slope of a curve at any point is the reciprocal of twice the ordinate at the
(a)
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point and it passes though the point (4, 3). The equation of the curve is
(b) y 2 = x − 5
x2 = y + 5
(c)
(d)
y2 = x + 5
x2 = y −5
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36. The differential equation of all circles in the first quadrant which touch the
.s
a
coordinate axes is of order
(a) 1
(c) 3
(d) None of these
The area enclosed between the curve
w
37.
(b) 2
w
w
(a)3
⎛π
1
38.
(b) 4
y = log e (x + e ) and
(c) 1
(d) 2
π
(d)
⎞
∫ log sin⎜⎝ 2 x ⎟⎠ dx =
0
(a) − log 2
⎡ 10
⎢
⎢⎣ n =1
⎤ ⎡ 10
sin 27 x dx ⎥ + ⎢
− 2 n −1
⎥⎦ ⎢⎣ n =1
∑∫
39.
(a)
27 2
(b) log 2
2n
∑∫
2 n +1
2n
(c)
⎤
sin 27 x dx ⎥
⎥⎦
(b) −54
2
log 2
−
π
2
log 2
equals
(c) 36
(d) 0
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the co-ordinate axes is
40.
∫
(x 4 − x )1 / 4
4 ⎛
1 ⎞
⎜1 − 3 ⎟
15 ⎝
x ⎠
(a)
41.
∫
is equal to
dx
x5
5/4
2
⎧ (log x − 1) ⎫
dx
⎨
2⎬
⎩ 1 + (log x ) ⎭
xe x
(a)
1+ x
2
(b)
+c
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4⎛
1 ⎞
⎜1 − 3 ⎟
5⎝
x ⎠
5/4
(c)
+c
4 ⎛
1 ⎞
⎜1 + 3 ⎟
15 ⎝
x ⎠
5/4
(d) None of these
+c
is equal to
(b)
+c
x
2
(log x ) + 1
(c)
+c
log x
2
(log x ) + 1
(d)
+c
x
2
x +1
+c
, is
(a) One-one but not onto
(c) One-one and onto both
43.
44.
(1 + x )1 / x − e
x →0
x
lim
equals
(b)
Onto but not one-one
(d)
Neither one-one nor onto
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⎧n − 1
⎪⎪ 2 , when n is odd
f (n) = ⎨
⎪− n , when n is even
⎪⎩ 2
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42. A function f from the set of natural numbers to integers defined by
(a) π / 2
(b) 0
log x n − [ x ]
, n ∈ N , ([ x ]
x →∞
[x ]
denotes greatest integer less than or equal to x)
lim
(c)
2/e
(d)– e / 2
(a) |
(c)
| x| >1
(d) None of these
c
b
b
b
a
c
c
d
11-20 d
c
c
a
b
c
a
b
b
b
21-30 b
b
a
a
a
a
d
a
b
c
31-40 c
a
c
b
c
a
c
a
d
a
41-45 b
c
d
a
b
w
b
w
w
b
is not differentiable for
(b) x = 1,−1
x| <1
KEY:
1-10
⎛ 2x ⎞
y = sin −1 ⎜
⎟
2
⎝1+ x ⎠
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Function
.s
a
45.
(d) Does not exist
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(a) Has value –1 (b) Has value 0 (c) Has value 1
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w
w
w
.s
a
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hi
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