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ISI B.Math Entrance Exam
2010 ISI B.Math Entrance Exam
2010 ISI B.Math Entrance Exam
Part of
ISI B.Math Entrance Exam
Subcontests
(10)
2
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 2
In the accompanying figure ,
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
is the graph of a one-to-one continuous function
f
f
f
. At each point
P
P
P
on the graph of
y
=
2
x
2
y=2x^2
y
=
2
x
2
, assume that the areas
O
A
P
OAP
O
A
P
and
O
B
P
OBP
OBP
are equal . Here
P
A
,
P
B
PA,PB
P
A
,
PB
are the horizontal and vertical segments . Determine the function
f
f
f
. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("
y
=
x
2
y=x^2
y
=
x
2
",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("
y
=
2
x
2
y=2x^2
y
=
2
x
2
",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
",(24,576/10),W); label("
O
(
0
,
0
)
O(0,0)
O
(
0
,
0
)
",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("
P
P
P
",(20,400/25),E); label("
B
B
B
",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("
A
A
A
",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]
10
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 10
Consider a regular heptagon ( polygon of
7
7
7
equal sides and angles)
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
as in the figure below:-
(
a
)
.
(a).
(
a
)
.
Prove
1
sin
π
7
=
1
sin
2
π
7
+
1
sin
3
π
7
\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}
s
i
n
7
π
1
=
s
i
n
7
2
π
1
+
s
i
n
7
3
π
1
(
b
)
.
(b).
(
b
)
.
Using
(
a
)
(a)
(
a
)
or otherwise, show that
1
A
G
=
1
A
F
+
1
A
E
\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}
A
G
1
=
A
F
1
+
A
E
1
[asy] draw(dir(360/7)..dir(2*360/7),blue); draw(dir(2*360/7)..dir(3*360/7),blue); draw(dir(3*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(5*360/7),blue); draw(dir(5*360/7)..dir(6*360/7),blue); draw(dir(6*360/7)..dir(7*360/7),blue); draw(dir(7*360/7)..dir(360/7),blue); draw(dir(2*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(1*360/7),blue); label("
A
A
A
",dir(4*360/7),W); label("
B
B
B
",dir(5*360/7),S); label("
C
C
C
",dir(6*360/7),S); label("
D
D
D
",dir(7*360/7),E); label("
E
E
E
",dir(1*360/7),E); label("
F
F
F
",dir(2*360/7),N); label("
G
G
G
",dir(3*360/7),W); [/asy]
9
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 9
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with integer co-efficients. Assume that
3
3
3
divides the value
f
(
n
)
f(n)
f
(
n
)
for each integer
n
n
n
. Prove that when
f
(
x
)
f(x)
f
(
x
)
is divided by
x
3
−
x
x^3-x
x
3
−
x
, the remainder is of the form
3
r
(
x
)
3r(x)
3
r
(
x
)
where
r
(
x
)
r(x)
r
(
x
)
is a polynomial with integer coefficients.
8
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 8
Let
f
f
f
be a real-valued differentiable function on the real line
R
\mathbb{R}
R
such that
lim
x
→
0
f
(
x
)
x
2
\lim_{x\to 0} \frac{f(x)}{x^2}
lim
x
→
0
x
2
f
(
x
)
exists, and is finite . Prove that
f
′
(
0
)
=
0
f'(0)=0
f
′
(
0
)
=
0
.
6
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 6
Let each of the vertices of a regular
9
9
9
-gon (polygon of 9 equal sides and equal angles) be coloured black or white .
(
a
)
.
(a).
(
a
)
.
Show that there are two adjacent verices of same colour.
(
b
)
.
(b).
(
b
)
.
Show there are three vertices of the same colour forming an isosceles triangle.
5
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 5
Let
a
1
>
a
2
>
.
.
.
.
.
>
a
r
a_1>a_2>.....>a_r
a
1
>
a
2
>
.....
>
a
r
be positive real numbers . Compute
lim
n
→
∞
(
a
1
n
+
a
2
n
+
.
.
.
.
.
+
a
r
n
)
1
n
\lim_{n\to \infty} (a_1^n+a_2^n+.....+a_r^n)^{\frac{1}{n}}
lim
n
→
∞
(
a
1
n
+
a
2
n
+
.....
+
a
r
n
)
n
1
4
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 4
If
a
,
b
,
c
∈
(
0
,
1
)
a,b,c\in (0,1)
a
,
b
,
c
∈
(
0
,
1
)
satisfy
a
+
b
+
c
=
2
a+b+c=2
a
+
b
+
c
=
2
, prove that
a
b
c
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
≥
8
\frac{abc}{(1-a)(1-b)(1-c)}\ge 8
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
ab
c
≥
8
3
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 3
Show that , for any positive integer
n
n
n
, the sum of
8
n
+
4
8n+4
8
n
+
4
consecutive positive integers cannot be a perfect square .
1
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 1
Prove that in each year , the
1
3
t
h
13^{th}
1
3
t
h
day of some month occurs on a Friday .
7
1
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I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 7
We are given
a
,
b
,
c
∈
R
a,b,c \in \mathbb{R}
a
,
b
,
c
∈
R
and a polynomial
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
f(x)=x^3+ax^2+bx+c
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
such that all roots (real or complex) of
f
(
x
)
f(x)
f
(
x
)
have same absolute value. Show that
a
=
0
a=0
a
=
0
iff
b
=
0
b=0
b
=
0
.