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Chennai Mathematical Institute B.Sc. Entrance Exam
2017 CMI B.Sc. Entrance Exam
2017 CMI B.Sc. Entrance Exam
Part of
Chennai Mathematical Institute B.Sc. Entrance Exam
Subcontests
(6)
3
1
Hide problems
CMI 2017 #3
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial of degree strictly less than
100
100
100
and such that it does not have
(
x
3
−
x
)
(x^3-x)
(
x
3
−
x
)
as a factor. If
d
100
d
x
100
(
p
(
x
)
x
3
−
x
)
=
f
(
x
)
g
(
x
)
\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}
d
x
100
d
100
(
x
3
−
x
p
(
x
)
)
=
g
(
x
)
f
(
x
)
for some polynomials
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
then find the smallest possible degree of
f
(
x
)
f(x)
f
(
x
)
.
2
1
Hide problems
CMI 2017 #2
Let
L
L
L
be the line of intersection of the planes
x
+
y
=
0
~x+y=0~
x
+
y
=
0
and
y
+
z
=
0
~y+z=0
y
+
z
=
0
.(a) Write the vector equation of
L
L
L
, i.e. find
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
and
(
p
,
q
,
r
)
(p,q,r)
(
p
,
q
,
r
)
such that
L
=
{
(
a
,
b
,
c
)
+
λ
(
p
,
q
,
r
)
∣
λ
∈
R
}
L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}
L
=
{(
a
,
b
,
c
)
+
λ
(
p
,
q
,
r
)
∣
λ
∈
R
}
(b) Find the equation of a plane obtained by
x
+
y
=
0
x+y=0
x
+
y
=
0
about
L
L
L
by
4
5
∘
45^{\circ}
4
5
∘
.
6
1
Hide problems
CMI 2017 #6
You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.(a) A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio.(b) Suppose, a square
A
B
C
D
ABCD
A
BC
D
is inscribed in the hexagon such that
A
A
A
and
C
C
C
are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon.(c) Suppose, the side of the hexagon is of length
1
1
1
. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.(d) Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.
5
1
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CMI 2017 #5
Each integer is colored with exactly one of
3
3
3
possible colors -- black, red or white -- satisfying the following two rules : the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black.(a) Show that, the negative of a white number must be colored black and the sum of two black numbers must be colored white.(b) Determine all possible colorings of the integers that satisfy these rules.
4
1
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CMI 2017 #4
The domain of a function
f
f
f
is
N
\mathbb{N}
N
(The set of natural numbers). The function is defined as follows :
f
(
n
)
=
n
+
⌊
n
⌋
f(n)=n+\lfloor\sqrt{n}\rfloor
f
(
n
)
=
n
+
⌊
n
⌋
where
⌊
k
⌋
\lfloor k\rfloor
⌊
k
⌋
denotes the nearest integer smaller than or equal to
k
k
k
. Prove that, for every natural number
m
m
m
, the following sequence contains at least one perfect square
m
,
f
(
m
)
,
f
2
(
m
)
,
f
3
(
m
)
,
⋯
m,~f(m),~f^2(m),~f^3(m),\cdots
m
,
f
(
m
)
,
f
2
(
m
)
,
f
3
(
m
)
,
⋯
The notation
f
k
f^k
f
k
denotes the function obtained by composing
f
f
f
with itself
k
k
k
times.
1
1
Hide problems
CMI 2017 #1
Answer the following questions :(a) Evaluate
lim
x
→
0
+
(
x
x
x
−
x
x
)
~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)
lim
x
→
0
+
(
x
x
x
−
x
x
)
(b) Let
A
=
2
π
9
A=\frac{2\pi}{9}
A
=
9
2
π
, i.e.
40
40
40
degrees. Calculate the following
1
+
cos
A
+
cos
2
A
+
cos
4
A
+
cos
5
A
+
cos
7
A
+
cos
8
A
1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A
1
+
cos
A
+
cos
2
A
+
cos
4
A
+
cos
5
A
+
cos
7
A
+
cos
8
A
(c) Find the number of solutions to
e
x
=
x
2017
+
1
e^x=\frac{x}{2017}+1
e
x
=
2017
x
+
1