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Contests
National and Regional Contests
India Contests
ISI B.Stat Entrance Exam
2008 ISI B.Stat Entrance Exam
2008 ISI B.Stat Entrance Exam
Part of
ISI B.Stat Entrance Exam
Subcontests
(10)
3
1
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Sketch and find behaviour of y=cube root of (x^3-4x)
Study the derivatives of the function
y
=
x
3
−
4
x
y=\sqrt{x^3-4x}
y
=
x
3
−
4
x
and sketch its graph on the real line.
10
1
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Two lines and two circles are equivalent
Two subsets
A
A
A
and
B
B
B
of the
(
x
,
y
)
(x,y)
(
x
,
y
)
-plane are said to be equivalent if there exists a function
f
:
A
→
B
f: A\to B
f
:
A
→
B
which is both one-to-one and onto. (i) Show that any two line segments in the plane are equivalent. (ii) Show that any two circles in the plane are equivalent.
2
1
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High screen on a wall: Maximize the angle subtended
A
40
40
40
feet high screen is put on a vertical wall
10
10
10
feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?
4
1
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Tangents from two disjoint circles: show that AB=MN
Suppose
P
P
P
and
Q
Q
Q
are the centres of two disjoint circles
C
1
C_1
C
1
and
C
2
C_2
C
2
respectively, such that
P
P
P
lies outside
C
2
C_2
C
2
and
Q
Q
Q
lies outside
C
1
C_1
C
1
. Two tangents are drawn from the point
P
P
P
to the circle
C
2
C_2
C
2
, which intersect the circle
C
1
C_1
C
1
at point
A
A
A
and
B
B
B
. Similarly, two tangents are drawn from the point
Q
Q
Q
to the circle
C
1
C_1
C
1
, which intersect the circle
C
2
C_2
C
2
at points
M
M
M
and
N
N
N
. Show that
A
B
=
M
N
AB=MN
A
B
=
MN
8
1
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Divide 2,3,...9 into 4pairs
In how many ways can you divide the set of eight numbers
{
2
,
3
,
⋯
,
9
}
\{2,3,\cdots,9\}
{
2
,
3
,
⋯
,
9
}
into
4
4
4
pairs such that no pair of numbers has
gcd
\text{gcd}
gcd
equal to
2
2
2
?
9
1
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Define a*b=LCM[a,b]/GCD(a,b)
Suppose
S
S
S
is the set of all positive integers. For
a
,
b
∈
S
a,b \in S
a
,
b
∈
S
, define
a
∗
b
=
lcm
[
a
,
b
]
gcd
(
a
,
b
)
a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}
a
∗
b
=
gcd
(
a
,
b
)
lcm
[
a
,
b
]
For example
8
∗
12
=
6
8*12=6
8
∗
12
=
6
. Show that exactly two of the following three properties are satisfied: (i) If
a
,
b
∈
S
a,b \in S
a
,
b
∈
S
, then
a
∗
b
∈
S
a*b \in S
a
∗
b
∈
S
. (ii)
(
a
∗
b
)
∗
c
=
a
∗
(
b
∗
c
)
(a*b)*c=a*(b*c)
(
a
∗
b
)
∗
c
=
a
∗
(
b
∗
c
)
for all
a
,
b
,
c
∈
S
a,b,c \in S
a
,
b
,
c
∈
S
. (iii) There exists an element
i
∈
S
i \in S
i
∈
S
such that
a
∗
i
=
a
a *i =a
a
∗
i
=
a
for all
a
∈
S
a \in S
a
∈
S
.
1
1
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Maximize triangle area
Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer
5
1
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Easy geometry: r^2=xyz/(x+y+z)
Suppose
A
B
C
ABC
A
BC
is a triangle with inradius
r
r
r
. The incircle touches the sides
B
C
,
C
A
,
BC, CA,
BC
,
C
A
,
and
A
B
AB
A
B
at
D
,
E
D,E
D
,
E
and
F
F
F
respectively. If
B
D
=
x
,
C
E
=
y
BD=x, CE=y
B
D
=
x
,
CE
=
y
and
A
F
=
z
AF=z
A
F
=
z
, then show that
r
2
=
x
y
z
x
+
y
+
z
r^2=\frac{xyz}{x+y+z}
r
2
=
x
+
y
+
z
x
yz
7
1
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Real root of x^5+x=10 is irrational
Consider the equation
x
5
+
x
=
10
x^5+x=10
x
5
+
x
=
10
. Show that (a) the equation has only one real root; (b) this root lies between
1
1
1
and
2
2
2
; (c) this root must be irrational.
6
1
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evaluate this limit
Evaluate:
lim
n
→
∞
1
2
n
ln
(
2
n
n
)
\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}
lim
n
→
∞
2
n
1
ln
(
n
2
n
)