MathDB
Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2016 ISI Entrance Examination
2016 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
3
1
Hide problems
ISI B.STAT ENTRANCE 2016 #3
If
P
(
x
)
=
x
n
+
a
1
x
n
−
1
+
.
.
.
+
a
n
−
1
P(x)=x^n+a_1x^{n-1}+...+a_{n-1}
P
(
x
)
=
x
n
+
a
1
x
n
−
1
+
...
+
a
n
−
1
be a polynomial with real coefficients and
a
1
2
<
a
2
a_1^2<a_2
a
1
2
<
a
2
then prove that not all roots of
P
(
x
)
P(x)
P
(
x
)
are real.
4
1
Hide problems
ISI B.STAT 2016 #4
Given a square
A
B
C
D
ABCD
A
BC
D
with two consecutive vertices, say
A
A
A
and
B
B
B
on the positive
x
x
x
-axis and positive
y
y
y
-axis respectively. Suppose the other vertice
C
C
C
lying in the first quadrant has coordinates
(
u
,
v
)
(u , v)
(
u
,
v
)
. Then find the area of the square
A
B
C
D
ABCD
A
BC
D
in terms of
u
u
u
and
v
v
v
.
6
1
Hide problems
ISI B.STAT 2016 #6
Suppose in a triangle
△
A
B
C
\triangle ABC
△
A
BC
,
A
A
A
,
B
B
B
,
C
C
C
are the three angles and
a
a
a
,
b
b
b
,
c
c
c
are the lengths of the sides opposite to the angles respectively. Then prove that if
s
i
n
(
A
−
B
)
=
a
a
+
b
sin
A
cos
B
−
b
a
+
b
sin
B
cos
A
sin(A-B)= \frac{a}{a+b}\sin A \cos B - \frac{b}{a+b}\sin B \cos A
s
in
(
A
−
B
)
=
a
+
b
a
sin
A
cos
B
−
a
+
b
b
sin
B
cos
A
then the triangle
△
A
B
C
\triangle ABC
△
A
BC
is isoscelos.
8
1
Hide problems
ISI B.STAT 2016 #8
Suppose that
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
is a sequence of real numbers satisfying
a
n
+
1
=
3
a
n
2
+
a
n
a_{n+1} = \frac{3a_n}{2+a_n}
a
n
+
1
=
2
+
a
n
3
a
n
. (i) Suppose
0
<
a
1
<
1
0 < a_1 <1
0
<
a
1
<
1
, then prove that the sequence
a
n
a_n
a
n
is increasing and hence show that
lim
n
→
∞
a
n
=
1
\lim_{n \to \infty} a_n =1
lim
n
→
∞
a
n
=
1
.(ii) Suppose
a
1
>
1
a_1 >1
a
1
>
1
, then prove that the sequence
a
n
a_n
a
n
is decreasing and hence show that
lim
n
→
∞
a
n
=
1
\lim_{n \to \infty} a_n =1
lim
n
→
∞
a
n
=
1
.
5
1
Hide problems
Isi 2016 geometry
Prove that there exists a right angle triangle with rational sides and area
d
d
d
if and only if
x
2
,
y
2
x^2,y^2
x
2
,
y
2
and
z
2
z^2
z
2
are squares of rational numbers and are in Arithmetic ProgressionHere
d
d
d
is an integer.
2
1
Hide problems
ISI bstat 2016 #2
Consider the polynomial
a
x
3
+
b
x
2
+
c
x
+
d
ax^3+bx^2+cx+d
a
x
3
+
b
x
2
+
c
x
+
d
where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are integers such that
a
d
ad
a
d
is odd and
b
c
bc
b
c
is even.Prove that not all of its roots are rational..
1
1
Hide problems
ISI 2016 #1
In a sports tournament of
n
n
n
players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order
P
1
,
P
2
,
.
.
.
.
,
P
n
P_1,P_2, .... , P_n
P
1
,
P
2
,
....
,
P
n
such that
P
i
P_i
P
i
defeats
P
i
+
1
P_{i+1}
P
i
+
1
for all
1
≤
i
≤
n
−
1
1 \le i \le n-1
1
≤
i
≤
n
−
1
.
7
1
Hide problems
isi bstat 2016 #7
f
f
f
is a differentiable function such that
f
(
f
(
x
)
)
=
x
f(f(x))=x
f
(
f
(
x
))
=
x
where
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
.Also
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
.Find the value of
∫
0
1
(
x
−
f
(
x
)
)
2016
d
x
\int_0^1(x-f(x))^{2016}dx
∫
0
1
(
x
−
f
(
x
)
)
2016
d
x