IIT JAM Mathematical Statistics Past Year Papers - 2021 | IIT JAM Past Year Papers and Model Test Paper (All Branches) PDF Download

 Page 1


JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 1/21 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
 
 
 
  
Page 2


JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 1/21 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
 
 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 2/21 
 
 
Special Instructions/ Useful Data 
R The set of real numbers 
R
?? { ( ?? 1
, ?? 2
, … , ?? ?? ) : ?? ?? ? R, ?? = 1, 2, … , ?? } 
det( ?? ) Determinant of a matrix ?? 
?? ?? Identity matrix of order ?? × ?? , ?? = 2, 3, … 
?? ' First derivative of a real valued function ?? 
?? ''
 Second derivative of a real valued function ?? 
?? ?? Complement of an event ?? 
?? ( ?? ) Probability of an event ?? 
?? ( ?? |?? ) Conditional probability of an event ?? given the occurrence of event ?? 
?? ~ ?? The probability density/mass function of the random variable ?? is ??  
?? ( ?? ) Expectation of a random variable ?? 
Var ( ?? ) Variance of a random variable ?? 
?? ( ?? , ?? ) Continuous uniform distribution on the interval ( ?? , ?? ) , -8 < ?? < ?? < 8 
Poisson( ?? ) Poisson distribution with mean ?? , ?? ? ( 0, 8) 
?? ( ?? , ?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ? ( -8, 8) , ?? 2
? ( 0, 8) 
?? ?? 2
 Central chi-square distribution with ?? degrees of freedom, ?? = 1,2, …  
?? ?? ,?? ?? distribution with ( ?? , ?? ) degrees of freedom, ?? , ?? = 1,2, …   
F( ·) Distribution function of ?? ( 0, 1)  
|?? | Absolute value of ?? 
MLE Maximum Likelihood Estimator 
?? !  ?? · ( ?? - 1)? 3 · 2 · 1,   ?? = 1, 2, 3, … , and   0! = 1 
(
?? ?? ) 
?? !
?? ! ( ?? - ?? ) !
, ?? = 0,1,2, … , ??  and ?? = 1, 2, … ; (
0
0
) = 1 
max{ ?? 1
, ?? 2
, … , ?? ?? } Maximum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
min{ ?? 1
, ?? 2
, … , ?? ?? } Minimum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
ln?? Natural logarithm of ?? 
 
  
Page 3


JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 1/21 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
 
 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 2/21 
 
 
Special Instructions/ Useful Data 
R The set of real numbers 
R
?? { ( ?? 1
, ?? 2
, … , ?? ?? ) : ?? ?? ? R, ?? = 1, 2, … , ?? } 
det( ?? ) Determinant of a matrix ?? 
?? ?? Identity matrix of order ?? × ?? , ?? = 2, 3, … 
?? ' First derivative of a real valued function ?? 
?? ''
 Second derivative of a real valued function ?? 
?? ?? Complement of an event ?? 
?? ( ?? ) Probability of an event ?? 
?? ( ?? |?? ) Conditional probability of an event ?? given the occurrence of event ?? 
?? ~ ?? The probability density/mass function of the random variable ?? is ??  
?? ( ?? ) Expectation of a random variable ?? 
Var ( ?? ) Variance of a random variable ?? 
?? ( ?? , ?? ) Continuous uniform distribution on the interval ( ?? , ?? ) , -8 < ?? < ?? < 8 
Poisson( ?? ) Poisson distribution with mean ?? , ?? ? ( 0, 8) 
?? ( ?? , ?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ? ( -8, 8) , ?? 2
? ( 0, 8) 
?? ?? 2
 Central chi-square distribution with ?? degrees of freedom, ?? = 1,2, …  
?? ?? ,?? ?? distribution with ( ?? , ?? ) degrees of freedom, ?? , ?? = 1,2, …   
F( ·) Distribution function of ?? ( 0, 1)  
|?? | Absolute value of ?? 
MLE Maximum Likelihood Estimator 
?? !  ?? · ( ?? - 1)? 3 · 2 · 1,   ?? = 1, 2, 3, … , and   0! = 1 
(
?? ?? ) 
?? !
?? ! ( ?? - ?? ) !
, ?? = 0,1,2, … , ??  and ?? = 1, 2, … ; (
0
0
) = 1 
max{ ?? 1
, ?? 2
, … , ?? ?? } Maximum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
min{ ?? 1
, ?? 2
, … , ?? ?? } Minimum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
ln?? Natural logarithm of ?? 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 3/21 
 
SECTION – A 
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q. 1 – Q.10 carry one mark each. 
 
Q.1  The value of the limit  
lim
?? ?8
((1 +
1
?? ) (1 +
2
?? )? (1 +
?? ?? ))
1
?? 
is equal to  
(A)    ??  
(B)     
1
??  (C)    
3
??  (D)    
4
??  
 
 
Q.2  Let ?? : R ? R be a function defined by 
?? ( ?? ) = ?? 7
+ 5 ?? 3
+ 11 ?? + 15,   ?? ? R. 
Then, which of the following statements is TRUE? 
 
(A) ?? is both one-one and onto  
(B) ?? is neither one-one nor onto  
(C) ?? is one-one but NOT onto  
(D) ?? is onto but NOT one-one   
 
 
Q.3  The value of the limit 
lim
?? ?0
?? -3?? - ?? ?? + 4?? 5( 1 - cos ?? )
 
is equal to  
(A)  1 (B)  0 
(C)  
2
5
 (D)  
8
5
 
 
 
Page 4


JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 1/21 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
 
 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 2/21 
 
 
Special Instructions/ Useful Data 
R The set of real numbers 
R
?? { ( ?? 1
, ?? 2
, … , ?? ?? ) : ?? ?? ? R, ?? = 1, 2, … , ?? } 
det( ?? ) Determinant of a matrix ?? 
?? ?? Identity matrix of order ?? × ?? , ?? = 2, 3, … 
?? ' First derivative of a real valued function ?? 
?? ''
 Second derivative of a real valued function ?? 
?? ?? Complement of an event ?? 
?? ( ?? ) Probability of an event ?? 
?? ( ?? |?? ) Conditional probability of an event ?? given the occurrence of event ?? 
?? ~ ?? The probability density/mass function of the random variable ?? is ??  
?? ( ?? ) Expectation of a random variable ?? 
Var ( ?? ) Variance of a random variable ?? 
?? ( ?? , ?? ) Continuous uniform distribution on the interval ( ?? , ?? ) , -8 < ?? < ?? < 8 
Poisson( ?? ) Poisson distribution with mean ?? , ?? ? ( 0, 8) 
?? ( ?? , ?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ? ( -8, 8) , ?? 2
? ( 0, 8) 
?? ?? 2
 Central chi-square distribution with ?? degrees of freedom, ?? = 1,2, …  
?? ?? ,?? ?? distribution with ( ?? , ?? ) degrees of freedom, ?? , ?? = 1,2, …   
F( ·) Distribution function of ?? ( 0, 1)  
|?? | Absolute value of ?? 
MLE Maximum Likelihood Estimator 
?? !  ?? · ( ?? - 1)? 3 · 2 · 1,   ?? = 1, 2, 3, … , and   0! = 1 
(
?? ?? ) 
?? !
?? ! ( ?? - ?? ) !
, ?? = 0,1,2, … , ??  and ?? = 1, 2, … ; (
0
0
) = 1 
max{ ?? 1
, ?? 2
, … , ?? ?? } Maximum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
min{ ?? 1
, ?? 2
, … , ?? ?? } Minimum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
ln?? Natural logarithm of ?? 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 3/21 
 
SECTION – A 
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q. 1 – Q.10 carry one mark each. 
 
Q.1  The value of the limit  
lim
?? ?8
((1 +
1
?? ) (1 +
2
?? )? (1 +
?? ?? ))
1
?? 
is equal to  
(A)    ??  
(B)     
1
??  (C)    
3
??  (D)    
4
??  
 
 
Q.2  Let ?? : R ? R be a function defined by 
?? ( ?? ) = ?? 7
+ 5 ?? 3
+ 11 ?? + 15,   ?? ? R. 
Then, which of the following statements is TRUE? 
 
(A) ?? is both one-one and onto  
(B) ?? is neither one-one nor onto  
(C) ?? is one-one but NOT onto  
(D) ?? is onto but NOT one-one   
 
 
Q.3  The value of the limit 
lim
?? ?0
?? -3?? - ?? ?? + 4?? 5( 1 - cos ?? )
 
is equal to  
(A)  1 (B)  0 
(C)  
2
5
 (D)  
8
5
 
 
 
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 4/21 
Q.4  The value of the limit  
lim
?? ?8
? (
2?? ?? )
?? ?? =0
1
4
?? 
is equal to  
(A)  1 
(B)  
1
2
 
(C)  0 
(D)  
1
4
 
 
 
Q.5  Let { ?? ?? }
?? = 1
 be a sequence of independent and identically distributed random variables with  
probability density function 
?? ( ?? ) = {
1, if  0 < ?? < 1
0, otherwise 
. 
Then, the value of the limit  
lim
?? ?8
?? (- 
1
?? ? ln?? ?? = 1 +
1
v ?? ?? ?? =1
) 
is equal to  
(A)  
1
2
 
(B)  F( 1) (C)  0 (D)  F( 2) 
 
 
Q.6  Let ?? be a ?? ( 0, 1) random variable and let ?? = ?? 2
. If ?? is the correlation coefficient between the 
random variables ?? and ?? , then 48 ?? 2
 is equal to  
(A)  48 (B)  45 (C)  35 (D)  30 
 
 
Q.7  
Let ?? be a 3 × 3 real matrix. Let  (
1
2
3
) , (
1
1
1
) and (
0
-1
?? ) be the eigenvectors of ?? corresponding to 
three distinct eigenvalues of ?? , where ?? is a real number. Then, which of the following is NOT a 
possible value of ?? ? 
(A)  0 (B)  1 (C)  -2 (D)  2  
 
 
 
 
 
 
Page 5


JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 1/21 
Paper Specific Instructions 
 
1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire 
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are 
of different types. 
 
2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four 
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section 
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks 
each. 
 
3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar 
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the 
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no 
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20 
marks. 
 
4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type 
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. 
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and 
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.  
 
5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will 
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. 
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there 
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in 
Section – C (NAT) as well. 
 
6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other 
electronic gadgets are NOT allowed in the examination hall. 
 
 
7. The Scribble Pad will be provided for rough work. 
 
 
 
 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 2/21 
 
 
Special Instructions/ Useful Data 
R The set of real numbers 
R
?? { ( ?? 1
, ?? 2
, … , ?? ?? ) : ?? ?? ? R, ?? = 1, 2, … , ?? } 
det( ?? ) Determinant of a matrix ?? 
?? ?? Identity matrix of order ?? × ?? , ?? = 2, 3, … 
?? ' First derivative of a real valued function ?? 
?? ''
 Second derivative of a real valued function ?? 
?? ?? Complement of an event ?? 
?? ( ?? ) Probability of an event ?? 
?? ( ?? |?? ) Conditional probability of an event ?? given the occurrence of event ?? 
?? ~ ?? The probability density/mass function of the random variable ?? is ??  
?? ( ?? ) Expectation of a random variable ?? 
Var ( ?? ) Variance of a random variable ?? 
?? ( ?? , ?? ) Continuous uniform distribution on the interval ( ?? , ?? ) , -8 < ?? < ?? < 8 
Poisson( ?? ) Poisson distribution with mean ?? , ?? ? ( 0, 8) 
?? ( ?? , ?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ? ( -8, 8) , ?? 2
? ( 0, 8) 
?? ?? 2
 Central chi-square distribution with ?? degrees of freedom, ?? = 1,2, …  
?? ?? ,?? ?? distribution with ( ?? , ?? ) degrees of freedom, ?? , ?? = 1,2, …   
F( ·) Distribution function of ?? ( 0, 1)  
|?? | Absolute value of ?? 
MLE Maximum Likelihood Estimator 
?? !  ?? · ( ?? - 1)? 3 · 2 · 1,   ?? = 1, 2, 3, … , and   0! = 1 
(
?? ?? ) 
?? !
?? ! ( ?? - ?? ) !
, ?? = 0,1,2, … , ??  and ?? = 1, 2, … ; (
0
0
) = 1 
max{ ?? 1
, ?? 2
, … , ?? ?? } Maximum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
min{ ?? 1
, ?? 2
, … , ?? ?? } Minimum of real numbers ?? 1
, ?? 2
, … , ?? ?? ( ?? = 2)  
ln?? Natural logarithm of ?? 
 
  
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 3/21 
 
SECTION – A 
MULTIPLE CHOICE QUESTIONS (MCQ) 
Q. 1 – Q.10 carry one mark each. 
 
Q.1  The value of the limit  
lim
?? ?8
((1 +
1
?? ) (1 +
2
?? )? (1 +
?? ?? ))
1
?? 
is equal to  
(A)    ??  
(B)     
1
??  (C)    
3
??  (D)    
4
??  
 
 
Q.2  Let ?? : R ? R be a function defined by 
?? ( ?? ) = ?? 7
+ 5 ?? 3
+ 11 ?? + 15,   ?? ? R. 
Then, which of the following statements is TRUE? 
 
(A) ?? is both one-one and onto  
(B) ?? is neither one-one nor onto  
(C) ?? is one-one but NOT onto  
(D) ?? is onto but NOT one-one   
 
 
Q.3  The value of the limit 
lim
?? ?0
?? -3?? - ?? ?? + 4?? 5( 1 - cos ?? )
 
is equal to  
(A)  1 (B)  0 
(C)  
2
5
 (D)  
8
5
 
 
 
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 4/21 
Q.4  The value of the limit  
lim
?? ?8
? (
2?? ?? )
?? ?? =0
1
4
?? 
is equal to  
(A)  1 
(B)  
1
2
 
(C)  0 
(D)  
1
4
 
 
 
Q.5  Let { ?? ?? }
?? = 1
 be a sequence of independent and identically distributed random variables with  
probability density function 
?? ( ?? ) = {
1, if  0 < ?? < 1
0, otherwise 
. 
Then, the value of the limit  
lim
?? ?8
?? (- 
1
?? ? ln?? ?? = 1 +
1
v ?? ?? ?? =1
) 
is equal to  
(A)  
1
2
 
(B)  F( 1) (C)  0 (D)  F( 2) 
 
 
Q.6  Let ?? be a ?? ( 0, 1) random variable and let ?? = ?? 2
. If ?? is the correlation coefficient between the 
random variables ?? and ?? , then 48 ?? 2
 is equal to  
(A)  48 (B)  45 (C)  35 (D)  30 
 
 
Q.7  
Let ?? be a 3 × 3 real matrix. Let  (
1
2
3
) , (
1
1
1
) and (
0
-1
?? ) be the eigenvectors of ?? corresponding to 
three distinct eigenvalues of ?? , where ?? is a real number. Then, which of the following is NOT a 
possible value of ?? ? 
(A)  0 (B)  1 (C)  -2 (D)  2  
 
 
 
 
 
 
JAM 2021  MATHEMATICAL STATISTICS - MS 
MS 5/21 
Q.8  
If the series ? ?? ?? 8
?? =1
 converges absolutely, then which of the following series diverges? 
 
(A)    ? |?? 2?? |
8
?? =1
 
(B)    ?
?? ?? +?? ?? +1
2
8
?? =1
 
 
(C)    ? ( ?? ?? )
3 8
?? =1
 
(D)    ? (
1
( ln ?? )
2
+ ?? ?? )
8
?? =2
 
 
Q.9  There are three urns, labeled, Urn 1, Urn 2 and Urn 3. Urn 1 contains 2 white balls and 2 black 
balls, Urn 2 contains 1 white ball and 3 black balls and Urn 3 contains 3 white balls and 1 black 
ball. Consider two coins with probability of obtaining head in their single trials as 0.2 and 0.3. The 
two coins are tossed independently once, and an urn is selected according to the following scheme: 
Urn 1 is selected if 2 heads are obtained; Urn 3 is selected if 2 tails are obtained; otherwise Urn 2 is 
selected. A ball is then drawn at random from the selected urn. Then  
?? ( Urn 1 is selected | the ball drawn is white ) 
is equal to   
 
(A)    
6
109
 (B)   
12
109
 (C)    
1
18
 (D)    
1
9
 
 
 
Q.10  Let ?? be a random variable with probability density function 
?? ( ?? ) = 
1
2
?? -|?? |
,   - 8 < ?? < 8. 
Then, which of the following statements is FALSE? 
 
(A) ?? ( ?? |?? |) = 0 
(B) ?? ( ?? |?? |
2
) = 0 
(C) ?? (|?? | sin(
?? |?? |
)) = 0 
(D) ?? (|?? | sin
2
(
?? |?? |
)) = 0