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Chennai Mathematical Institute B.Sc. Entrance Exam
2019 CMI B.Sc. Entrance Exam
2019 CMI B.Sc. Entrance Exam
Part of
Chennai Mathematical Institute B.Sc. Entrance Exam
Subcontests
(6)
6
1
Hide problems
CMI Entrance 19#6
(
a
)
(a)
(
a
)
Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*}
(
b
)
(b)
(
b
)
For
x
>
0
x > 0
x
>
0
define
F
(
x
)
=
∫
1
x
t
log
(
t
)
d
t
.
F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t .
F
(
x
)
=
∫
1
x
t
lo
g
(
t
)
d
t
.
\\ \\
1.
1.
1.
Determine the open interval(s) (if any) where
F
(
x
)
F ( x )
F
(
x
)
is decreasing and all the open interval(s) (if any) where
F
(
x
)
F ( x )
F
(
x
)
is increasing.\\ \\
2.
2.
2.
Determine all the local minima of
F
(
x
)
F ( x )
F
(
x
)
(if any) and all the local maxima of
F
(
x
)
F ( x )
F
(
x
)
(if any)
.
.
.
5
1
Hide problems
CMI Entrance #5
Three positive reals
x
,
y
,
z
x , y , z
x
,
y
,
z
satisfy \\
x
2
+
y
2
=
3
2
y
2
+
y
z
+
z
2
=
4
2
x
2
+
3
x
z
+
z
2
=
5
2
.
x^2 + y^2 = 3^2 \\ y^2 + yz + z^2 = 4^2 \\ x^2 + \sqrt{3}xz + z^2 = 5^2 .
x
2
+
y
2
=
3
2
y
2
+
yz
+
z
2
=
4
2
x
2
+
3
x
z
+
z
2
=
5
2
.
\\ Find the value of
2
x
y
+
x
z
+
3
y
z
2xy + xz + \sqrt{3}yz
2
x
y
+
x
z
+
3
yz
4
1
Hide problems
CMI Entrance 19#4
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram
.
.
.
Let
O
O
O
be a point in its interior such that
∠
A
O
B
+
∠
D
O
C
=
18
0
∘
.
\angle AOB + \angle DOC = 180^{\circ} .
∠
A
OB
+
∠
D
OC
=
18
0
∘
.
Show that
,
∠
O
D
C
=
∠
O
B
C
.
,\angle ODC = \angle OBC .
,
∠
O
D
C
=
∠
OBC
.
3
1
Hide problems
CMI Entrance 19#3
Evaluate
∫
0
∞
(
1
+
x
2
)
−
(
m
+
1
)
d
x
\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x
∫
0
∞
(
1
+
x
2
)
−
(
m
+
1
)
d
x
where
m
∈
N
m \in \mathbb{N}
m
∈
N
2
1
Hide problems
CMI entrance 19#2
(
a
)
(a)
(
a
)
Count the number of roots of
ω
\omega
ω
of the equation
z
2019
−
1
=
0
z^{2019} - 1 = 0
z
2019
−
1
=
0
over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*}
(
b
)
(b)
(
b
)
Find all real numbers
x
x
x
that satisfy following equation
:
:
:
\begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}
1
1
Hide problems
CMI entrance 19#1
For a natural number
n
n
n
denote by Map
(
n
)
( n )
(
n
)
the set of all functions
f
:
{
1
,
2
,
3
,
⋯
,
n
}
→
{
1
,
2
,
3
,
⋯
,
n
}
.
f : \{ 1 , 2 , 3 , \cdots , n \} \rightarrow \{ 1 , 2 , 3 , \cdots , n \} .
f
:
{
1
,
2
,
3
,
⋯
,
n
}
→
{
1
,
2
,
3
,
⋯
,
n
}
.
For
f
,
g
∈
f , g \in
f
,
g
∈
Map
(
n
)
,
f
∘
g
( n ) , f \circ g
(
n
)
,
f
∘
g
denotes the function in Map
(
n
)
( n )
(
n
)
that sends
x
→
f
(
g
(
x
)
)
.
x \rightarrow f ( g ( x ) ) .
x
→
f
(
g
(
x
))
.
\\ \\
(
a
)
(a)
(
a
)
Let
f
∈
f \in
f
∈
Map
(
n
)
.
( n ) .
(
n
)
.
If for all
x
∈
{
1
,
2
,
3
,
⋯
,
n
}
f
(
x
)
≠
x
,
x \in \{ 1 , 2 , 3 , \cdots , n \} f ( x ) \neq x ,
x
∈
{
1
,
2
,
3
,
⋯
,
n
}
f
(
x
)
=
x
,
show that
f
∘
f
≠
f
f \circ f \neq f
f
∘
f
=
f
\\
(
b
)
(b)
(
b
)
Count the number of functions
f
∈
f \in
f
∈
Map
(
n
)
( n )
(
n
)
such that
f
∘
f
=
f
f \circ f = f
f
∘
f
=
f