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Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category C
Problem 1
Problem 1
Part of
2019 LIMIT Category C
Problems
(2)
multi-choice matrix statements
Source: LIMIT 2019 CCS1 P1
4/28/2021
Which of the following are true?
<
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(
A
)
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∀
A
∈
M
n
(
R
)
,
A
t
=
X
−
1
A
X
for some
X
∈
M
n
(
R
)
<span class='latex-bold'>(A)</span>~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R)
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(
A
)
<
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>
∀
A
∈
M
n
(
R
)
,
A
t
=
X
−
1
A
X
for some
X
∈
M
n
(
R
)
<
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c
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a
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x
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>
(
B
)
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>
∀
A
∈
M
n
(
R
)
,
I
+
A
A
t
is invertible
<span class='latex-bold'>(B)</span>~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible}
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(
B
)
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>
∀
A
∈
M
n
(
R
)
,
I
+
A
A
t
is invertible
<
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c
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a
s
s
=
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a
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x
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>
(
C
)
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>
tr
(
A
B
)
=
tr
(
B
A
)
,
∀
A
,
B
∈
M
n
(
R
)
but
∃
A
,
B
,
C
such that
tr
(
A
B
C
)
≠
tr
(
B
A
C
)
<span class='latex-bold'>(C)</span>~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC)
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(
C
)
<
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>
tr
(
A
B
)
=
tr
(
B
A
)
,
∀
A
,
B
∈
M
n
(
R
)
but
∃
A
,
B
,
C
such that
tr
(
A
BC
)
=
tr
(
B
A
C
)
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>
(
D
)
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None of the above
<span class='latex-bold'>(D)</span>~\text{None of the above}
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(
D
)
<
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>
None of the above
linear algebra
matrix
differentiability of functions at 0
Source: LIMIT 2019 CCS2 P1
4/28/2021
Which of the following functions are differentiable at
x
=
0
x=0
x
=
0
?
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>
(
A
)
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>
f
(
x
)
=
{
tan
−
1
(
1
∣
x
∣
)
if
x
≠
0
π
2
if
x
=
0
<span class='latex-bold'>(A)</span>~f(x)=\begin{cases}\tan^{-1}\left(\frac1{|x|}\right)&\text{if }x\ne0\\\frac\pi2&\text{if }x=0\end{cases}
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>
(
A
)
<
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p
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>
f
(
x
)
=
{
tan
−
1
(
∣
x
∣
1
)
2
π
if
x
=
0
if
x
=
0
<
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′
>
(
B
)
<
/
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p
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>
f
(
x
)
=
∣
x
∣
1
/
2
x
<span class='latex-bold'>(B)</span>~f(x)=|x|^{1/2}x
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>
(
B
)
<
/
s
p
an
>
f
(
x
)
=
∣
x
∣
1/2
x
<
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=
′
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>
(
C
)
<
/
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p
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>
f
(
x
)
=
{
x
2
∣
cos
π
x
∣
if
x
≠
0
0
if
x
=
0
<span class='latex-bold'>(C)</span>~f(x)=\begin{cases}x^2\left|\cos\frac{\pi}x\right|&\text{if }x\ne0\\0&\text{if }x=0\end{cases}
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>
(
C
)
<
/
s
p
an
>
f
(
x
)
=
{
x
2
cos
x
π
0
if
x
=
0
if
x
=
0
<
s
p
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c
l
a
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s
=
′
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a
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x
−
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′
>
(
D
)
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/
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>
None of the above
<span class='latex-bold'>(D)</span>~\text{None of the above}
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(
D
)
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>
None of the above
function