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Page 1
JAM 2023 Confidential Mathematical Statistics (MS)
MS 3/43
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1
Let ?? =(
1 -1 0
-1 2 -1
0 -1 1
). If a non-zero vector ?? =(?? ,?? ,?? )
?? ?R
3
satisfies
?? 6
?? =?? , then a subspace of R
3
that contains the vector ?? is
(A) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(B) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(C) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(D) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? -?? =0}
Page 2
JAM 2023 Confidential Mathematical Statistics (MS)
MS 3/43
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1
Let ?? =(
1 -1 0
-1 2 -1
0 -1 1
). If a non-zero vector ?? =(?? ,?? ,?? )
?? ?R
3
satisfies
?? 6
?? =?? , then a subspace of R
3
that contains the vector ?? is
(A) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(B) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(C) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(D) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? -?? =0}
JAM 2023 Confidential Mathematical Statistics (MS)
MS 4/43
Q.2 Let ?? =?? 1
?? 2
, where ?? 1
and ?? 2
are two 3×3 distinct matrices. Consider the
following two statements:
(I) The rows of ?? are linear combinations of rows of ?? 2
.
(II) The columns of ?? are linear combinations of columns of ?? 1
.
Then,
(A) only (I) is TRUE
(B) only (II) is TRUE
(C) both (I) and (II) are TRUE
(D) neither (I) nor (II) is TRUE
Q.3
Let ?? ~ ?? 6,2
and ?? ~ ?? 2,6
. If ?? (?? =2)=
216
343
and ?? (?? =
1
2
)=?? , then 686?? equals
(A) 246
(B) 254
(C) 260
(D) 264
Page 3
JAM 2023 Confidential Mathematical Statistics (MS)
MS 3/43
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1
Let ?? =(
1 -1 0
-1 2 -1
0 -1 1
). If a non-zero vector ?? =(?? ,?? ,?? )
?? ?R
3
satisfies
?? 6
?? =?? , then a subspace of R
3
that contains the vector ?? is
(A) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(B) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(C) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(D) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? -?? =0}
JAM 2023 Confidential Mathematical Statistics (MS)
MS 4/43
Q.2 Let ?? =?? 1
?? 2
, where ?? 1
and ?? 2
are two 3×3 distinct matrices. Consider the
following two statements:
(I) The rows of ?? are linear combinations of rows of ?? 2
.
(II) The columns of ?? are linear combinations of columns of ?? 1
.
Then,
(A) only (I) is TRUE
(B) only (II) is TRUE
(C) both (I) and (II) are TRUE
(D) neither (I) nor (II) is TRUE
Q.3
Let ?? ~ ?? 6,2
and ?? ~ ?? 2,6
. If ?? (?? =2)=
216
343
and ?? (?? =
1
2
)=?? , then 686?? equals
(A) 246
(B) 254
(C) 260
(D) 264
JAM 2023 Confidential Mathematical Statistics (MS)
MS 5/43
Q.4 Let ?? ~ ?? 4,2
. Then, ?? (?? =2) equals
(A) 0.60
(B) 0.62
(C) 0.64
(D) 0.66
Q.5
Let ?? 1
, ?? 2
,… be a sequence of ??.??.?? . random variables each having ?? (0,1) distribution.
Let ?? be a random variable having the distribution function ?? . Suppose that
lim
?? ?8
?? (
?? 1
+?? 2
+?? ?? 4
=?? )=?? (?? ), for all ?? ? R .
Then, ?????? (?? ) equals
(A)
1
12
(B)
1
32
(C)
1
48
(D)
1
64
Page 4
JAM 2023 Confidential Mathematical Statistics (MS)
MS 3/43
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1
Let ?? =(
1 -1 0
-1 2 -1
0 -1 1
). If a non-zero vector ?? =(?? ,?? ,?? )
?? ?R
3
satisfies
?? 6
?? =?? , then a subspace of R
3
that contains the vector ?? is
(A) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(B) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(C) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(D) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? -?? =0}
JAM 2023 Confidential Mathematical Statistics (MS)
MS 4/43
Q.2 Let ?? =?? 1
?? 2
, where ?? 1
and ?? 2
are two 3×3 distinct matrices. Consider the
following two statements:
(I) The rows of ?? are linear combinations of rows of ?? 2
.
(II) The columns of ?? are linear combinations of columns of ?? 1
.
Then,
(A) only (I) is TRUE
(B) only (II) is TRUE
(C) both (I) and (II) are TRUE
(D) neither (I) nor (II) is TRUE
Q.3
Let ?? ~ ?? 6,2
and ?? ~ ?? 2,6
. If ?? (?? =2)=
216
343
and ?? (?? =
1
2
)=?? , then 686?? equals
(A) 246
(B) 254
(C) 260
(D) 264
JAM 2023 Confidential Mathematical Statistics (MS)
MS 5/43
Q.4 Let ?? ~ ?? 4,2
. Then, ?? (?? =2) equals
(A) 0.60
(B) 0.62
(C) 0.64
(D) 0.66
Q.5
Let ?? 1
, ?? 2
,… be a sequence of ??.??.?? . random variables each having ?? (0,1) distribution.
Let ?? be a random variable having the distribution function ?? . Suppose that
lim
?? ?8
?? (
?? 1
+?? 2
+?? ?? 4
=?? )=?? (?? ), for all ?? ? R .
Then, ?????? (?? ) equals
(A)
1
12
(B)
1
32
(C)
1
48
(D)
1
64
JAM 2023 Confidential Mathematical Statistics (MS)
MS 6/43
Q.6 Let ?? 1
, ?? 2
, ?? 3
be a random sample from an ?? (?? ,1) distribution, where ?? ?R is an
unknown parameter. Then, which one of the following conditional expectations
does NOT depend on ?? ?
(A) ?? (?? 1
+?? 2
-?? 3
| ?? 1
+?? 2
)
(B) ?? (?? 1
+?? 2
-?? 3
| ?? 2
+?? 3
)
(C) ?? (?? 1
+?? 2
-?? 3
| ?? 1
-?? 3
)
(D) ?? (?? 1
+?? 2
-?? 3
| ?? 1
+?? 2
+?? 3
)
Q.7 For the function ?? : R×R? R defined by
?? (?? ,?? )=2?? 2
-???? -3?? 2
-3?? +7?? ,
the point (1,1) is
(A) a point of local maximum
(B) a point of local minimum
(C) a saddle point
(D) NOT a critical point
Page 5
JAM 2023 Confidential Mathematical Statistics (MS)
MS 3/43
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1
Let ?? =(
1 -1 0
-1 2 -1
0 -1 1
). If a non-zero vector ?? =(?? ,?? ,?? )
?? ?R
3
satisfies
?? 6
?? =?? , then a subspace of R
3
that contains the vector ?? is
(A) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(B) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(C) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? +?? =0}
(D) {(?? ,?? ,?? )
?? ?R
3
:?? =0, ?? -?? =0}
JAM 2023 Confidential Mathematical Statistics (MS)
MS 4/43
Q.2 Let ?? =?? 1
?? 2
, where ?? 1
and ?? 2
are two 3×3 distinct matrices. Consider the
following two statements:
(I) The rows of ?? are linear combinations of rows of ?? 2
.
(II) The columns of ?? are linear combinations of columns of ?? 1
.
Then,
(A) only (I) is TRUE
(B) only (II) is TRUE
(C) both (I) and (II) are TRUE
(D) neither (I) nor (II) is TRUE
Q.3
Let ?? ~ ?? 6,2
and ?? ~ ?? 2,6
. If ?? (?? =2)=
216
343
and ?? (?? =
1
2
)=?? , then 686?? equals
(A) 246
(B) 254
(C) 260
(D) 264
JAM 2023 Confidential Mathematical Statistics (MS)
MS 5/43
Q.4 Let ?? ~ ?? 4,2
. Then, ?? (?? =2) equals
(A) 0.60
(B) 0.62
(C) 0.64
(D) 0.66
Q.5
Let ?? 1
, ?? 2
,… be a sequence of ??.??.?? . random variables each having ?? (0,1) distribution.
Let ?? be a random variable having the distribution function ?? . Suppose that
lim
?? ?8
?? (
?? 1
+?? 2
+?? ?? 4
=?? )=?? (?? ), for all ?? ? R .
Then, ?????? (?? ) equals
(A)
1
12
(B)
1
32
(C)
1
48
(D)
1
64
JAM 2023 Confidential Mathematical Statistics (MS)
MS 6/43
Q.6 Let ?? 1
, ?? 2
, ?? 3
be a random sample from an ?? (?? ,1) distribution, where ?? ?R is an
unknown parameter. Then, which one of the following conditional expectations
does NOT depend on ?? ?
(A) ?? (?? 1
+?? 2
-?? 3
| ?? 1
+?? 2
)
(B) ?? (?? 1
+?? 2
-?? 3
| ?? 2
+?? 3
)
(C) ?? (?? 1
+?? 2
-?? 3
| ?? 1
-?? 3
)
(D) ?? (?? 1
+?? 2
-?? 3
| ?? 1
+?? 2
+?? 3
)
Q.7 For the function ?? : R×R? R defined by
?? (?? ,?? )=2?? 2
-???? -3?? 2
-3?? +7?? ,
the point (1,1) is
(A) a point of local maximum
(B) a point of local minimum
(C) a saddle point
(D) NOT a critical point
JAM 2023 Confidential Mathematical Statistics (MS)
MS 7/43
Q.8 Let ?? 1
, ?? 2
and ?? 3
be three events such that
?? (?? 1
n?? 2
)=
1
4
, ?? (?? 1
n?? 3
)=?? (?? 2
n?? 3
)=
1
5
and ?? (?? 1
n?? 2
n?? 3
)=
1
6
.
Then, among the events ?? 1
, ?? 2
and ?? 3
, the probability that at least two events occur,
equals
(A)
17
60
(B)
23
60
(C)
19
60
(D)
29
60
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Nov 22, 2024 Last updated |
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