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IIT-JAM Physics-2023
Solution-Mathematical Methods
Page 2
IIT-JAM Physics-2023
Solution-Mathematical Methods
2
Section A: Q.1-Q.10 Carry ONE mark each.
Q2. Which of the following fields has non-zero curl?
(a)
ˆ ˆˆ
xi yj zk ?? (b) ? ? ? ? ? ?
ˆ ˆˆ
y z i x z j x y k ? ? ? ? ?
(c)
? ?
22
ˆ ˆˆ
22 y i xy z j yzk ? ? ? (d)
ˆ ˆˆ
23 xyi yzj xzk ??
Ans: (d)
Solution.:
(a) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 0 0 0 0 0 0
i j k
i j k
x y z
x y z
? ? ?
? ? ? ? ? ? ?
? ? ?
(b) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
1 1 1 1 1 1 0
i j k
i j k
x y z
y z x z x y
? ? ?
? ? ? ? ? ? ?
? ? ?
???
(c) ? ? ? ? ? ?
22
ˆ ˆˆ
ˆ ˆˆ
2 2 0 0 2 2 0
22
i j k
i z z j k y y
x y z
y xy z yz
? ? ?
? ? ? ? ? ? ?
? ? ?
?
(d) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 2 3 0 0 0
23
i j k
i y j z k x
x y z
xy yz xz
? ? ?
? ? ? ? ? ? ?
? ? ?
Page 3
IIT-JAM Physics-2023
Solution-Mathematical Methods
2
Section A: Q.1-Q.10 Carry ONE mark each.
Q2. Which of the following fields has non-zero curl?
(a)
ˆ ˆˆ
xi yj zk ?? (b) ? ? ? ? ? ?
ˆ ˆˆ
y z i x z j x y k ? ? ? ? ?
(c)
? ?
22
ˆ ˆˆ
22 y i xy z j yzk ? ? ? (d)
ˆ ˆˆ
23 xyi yzj xzk ??
Ans: (d)
Solution.:
(a) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 0 0 0 0 0 0
i j k
i j k
x y z
x y z
? ? ?
? ? ? ? ? ? ?
? ? ?
(b) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
1 1 1 1 1 1 0
i j k
i j k
x y z
y z x z x y
? ? ?
? ? ? ? ? ? ?
? ? ?
???
(c) ? ? ? ? ? ?
22
ˆ ˆˆ
ˆ ˆˆ
2 2 0 0 2 2 0
22
i j k
i z z j k y y
x y z
y xy z yz
? ? ?
? ? ? ? ? ? ?
? ? ?
?
(d) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 2 3 0 0 0
23
i j k
i y j z k x
x y z
xy yz xz
? ? ?
? ? ? ? ? ? ?
? ? ?
3
Q4. The plot of the function ? ? 1 f x x ?? is
(a)
(b)
(c)
(d)
Ans: (b)
Solution. :
? ? 1 f x x ?? ? ? 0 0 1 1 f ? ? ? ?
? ? 1 1 1 0 f ? ? ? ; ? ? 1 1 1 1 1 0 f ? ? ? ? ? ? ?
? ? 1 2 1 1 f ? ? ? ; ? ? 2 2 1 2 1 1 f ? ? ? ? ? ? ?
Thus option (b) is correct.
x
1
? ? fx
0 1
1 ?
1 ?
x
1
? ? fx
1 0
1 ?
1 ?
1 ?
x
? ? fx
0 1
1
1 ?
x
1
1
1 ?
1 ?
0
? ? fx
Page 4
IIT-JAM Physics-2023
Solution-Mathematical Methods
2
Section A: Q.1-Q.10 Carry ONE mark each.
Q2. Which of the following fields has non-zero curl?
(a)
ˆ ˆˆ
xi yj zk ?? (b) ? ? ? ? ? ?
ˆ ˆˆ
y z i x z j x y k ? ? ? ? ?
(c)
? ?
22
ˆ ˆˆ
22 y i xy z j yzk ? ? ? (d)
ˆ ˆˆ
23 xyi yzj xzk ??
Ans: (d)
Solution.:
(a) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 0 0 0 0 0 0
i j k
i j k
x y z
x y z
? ? ?
? ? ? ? ? ? ?
? ? ?
(b) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
1 1 1 1 1 1 0
i j k
i j k
x y z
y z x z x y
? ? ?
? ? ? ? ? ? ?
? ? ?
???
(c) ? ? ? ? ? ?
22
ˆ ˆˆ
ˆ ˆˆ
2 2 0 0 2 2 0
22
i j k
i z z j k y y
x y z
y xy z yz
? ? ?
? ? ? ? ? ? ?
? ? ?
?
(d) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 2 3 0 0 0
23
i j k
i y j z k x
x y z
xy yz xz
? ? ?
? ? ? ? ? ? ?
? ? ?
3
Q4. The plot of the function ? ? 1 f x x ?? is
(a)
(b)
(c)
(d)
Ans: (b)
Solution. :
? ? 1 f x x ?? ? ? 0 0 1 1 f ? ? ? ?
? ? 1 1 1 0 f ? ? ? ; ? ? 1 1 1 1 1 0 f ? ? ? ? ? ? ?
? ? 1 2 1 1 f ? ? ? ; ? ? 2 2 1 2 1 1 f ? ? ? ? ? ? ?
Thus option (b) is correct.
x
1
? ? fx
0 1
1 ?
1 ?
x
1
? ? fx
1 0
1 ?
1 ?
1 ?
x
? ? fx
0 1
1
1 ?
x
1
1
1 ?
1 ?
0
? ? fx
Section A: Q.11-Q.30 Carry TWO marks each.
Q13. The Jacobian matrix for transforming from
? ? , xy to another orthogonal coordinates
system
? ? , uv as shown in the figure is
(a)
11
1
11
2
??
??
?
??
(b)
11
1
11
2
??
??
?
??
(c)
11
1
11
2
? ??
??
??
(d)
11
1
11
2
? ??
??
?
??
Ans: (c)
Solution.:
cos sin
sin cos
ux
vy
??
??
? ? ? ? ? ? ?
?
? ? ? ? ? ?
? ? ? ? ? ?
0
45 ? ?
00
00
11 cos 45 sin 45 1
11 sin 45 cos 45 2
? ?? ? ??
??
?? ??
?? ??
Q15. Inverse of the matrix
110
2 3 0
1 0 1
??
??
??
??
??
is
(a)
1 2 1
1 3 0
0 0 1
? ??
??
?
??
??
??
(b)
3 1 0
2 1 0
3 1 1
? ??
??
?
??
?? ?
??
(c)
1 1 0
2 3 0
1 0 1
?? ??
??
??
??
??
(d)
3 2 3
2 1 1
0 0 1
?? ??
??
?
??
??
??
Ans: (b)
Solution.:
110
3 0 2 0
det 2 3 0 1 1 0 3 2 1
0 1 1 1
1 0 1
A ? ? ? ? ? ? ?
Cofactor Matrix
3 2 3
1 1 1
0 0 1
ij
C
?? ??
??
????
??
??
??
??
3 1 0
2 1 0
3 1 1
T
ij
C
? ??
??
?? ? ? ?
??
??
?? ?
??
Thus
1
3 1 0
1
2 1 0
det
3 1 1
T
ij
AC
A
?
? ??
??
?? ? ? ?
??
??
?? ?
??
0
45
u
? ? 0,0
y
v
x
Page 5
IIT-JAM Physics-2023
Solution-Mathematical Methods
2
Section A: Q.1-Q.10 Carry ONE mark each.
Q2. Which of the following fields has non-zero curl?
(a)
ˆ ˆˆ
xi yj zk ?? (b) ? ? ? ? ? ?
ˆ ˆˆ
y z i x z j x y k ? ? ? ? ?
(c)
? ?
22
ˆ ˆˆ
22 y i xy z j yzk ? ? ? (d)
ˆ ˆˆ
23 xyi yzj xzk ??
Ans: (d)
Solution.:
(a) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 0 0 0 0 0 0
i j k
i j k
x y z
x y z
? ? ?
? ? ? ? ? ? ?
? ? ?
(b) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
1 1 1 1 1 1 0
i j k
i j k
x y z
y z x z x y
? ? ?
? ? ? ? ? ? ?
? ? ?
???
(c) ? ? ? ? ? ?
22
ˆ ˆˆ
ˆ ˆˆ
2 2 0 0 2 2 0
22
i j k
i z z j k y y
x y z
y xy z yz
? ? ?
? ? ? ? ? ? ?
? ? ?
?
(d) ? ? ? ? ? ?
ˆ ˆˆ
ˆ ˆˆ
0 2 3 0 0 0
23
i j k
i y j z k x
x y z
xy yz xz
? ? ?
? ? ? ? ? ? ?
? ? ?
3
Q4. The plot of the function ? ? 1 f x x ?? is
(a)
(b)
(c)
(d)
Ans: (b)
Solution. :
? ? 1 f x x ?? ? ? 0 0 1 1 f ? ? ? ?
? ? 1 1 1 0 f ? ? ? ; ? ? 1 1 1 1 1 0 f ? ? ? ? ? ? ?
? ? 1 2 1 1 f ? ? ? ; ? ? 2 2 1 2 1 1 f ? ? ? ? ? ? ?
Thus option (b) is correct.
x
1
? ? fx
0 1
1 ?
1 ?
x
1
? ? fx
1 0
1 ?
1 ?
1 ?
x
? ? fx
0 1
1
1 ?
x
1
1
1 ?
1 ?
0
? ? fx
Section A: Q.11-Q.30 Carry TWO marks each.
Q13. The Jacobian matrix for transforming from
? ? , xy to another orthogonal coordinates
system
? ? , uv as shown in the figure is
(a)
11
1
11
2
??
??
?
??
(b)
11
1
11
2
??
??
?
??
(c)
11
1
11
2
? ??
??
??
(d)
11
1
11
2
? ??
??
?
??
Ans: (c)
Solution.:
cos sin
sin cos
ux
vy
??
??
? ? ? ? ? ? ?
?
? ? ? ? ? ?
? ? ? ? ? ?
0
45 ? ?
00
00
11 cos 45 sin 45 1
11 sin 45 cos 45 2
? ?? ? ??
??
?? ??
?? ??
Q15. Inverse of the matrix
110
2 3 0
1 0 1
??
??
??
??
??
is
(a)
1 2 1
1 3 0
0 0 1
? ??
??
?
??
??
??
(b)
3 1 0
2 1 0
3 1 1
? ??
??
?
??
?? ?
??
(c)
1 1 0
2 3 0
1 0 1
?? ??
??
??
??
??
(d)
3 2 3
2 1 1
0 0 1
?? ??
??
?
??
??
??
Ans: (b)
Solution.:
110
3 0 2 0
det 2 3 0 1 1 0 3 2 1
0 1 1 1
1 0 1
A ? ? ? ? ? ? ?
Cofactor Matrix
3 2 3
1 1 1
0 0 1
ij
C
?? ??
??
????
??
??
??
??
3 1 0
2 1 0
3 1 1
T
ij
C
? ??
??
?? ? ? ?
??
??
?? ?
??
Thus
1
3 1 0
1
2 1 0
det
3 1 1
T
ij
AC
A
?
? ??
??
?? ? ? ?
??
??
?? ?
??
0
45
u
? ? 0,0
y
v
x
Q21. For a given vector
2
ˆ ˆˆ
F yi zj x k ? ? ? ? , the surface integral
? ?
ˆ
S
F rdS ? ? ?
?
over the
surface S of a hemisphere of radius R with the centre of the base at the origin is
(a)
2
R ? (b)
2
2
3
R ?
(b)
2
R ? ? (d)
2
2
3
R ?
?
Ans.: (a)
Solution.:
? ?
ˆ .
S
line
F rdS F dl ? ? ? ?
??
0 z ? , ˆˆ dl dxx dyy ?? . F dl ydx zdy ydx ? ? ? ? ? ?
Let cos xR ? ? , sin yR ? ? and sin dx R d ?? ?? .
Thus
? ? ? ? ? ?
22
22
00
. sin sin
line line
F dl ydx R R d R
??
? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
X
Y
Z
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FAQs on Physics - 2023 Solution Past Year Paper - IIT JAM Past Year Papers and Model Test Paper (All Branches)
| 1. What is the IIT JAM exam and what subjects does it cover? |  |
Ans.IIT JAM (Joint Admission Test for M.Sc.) is an entrance exam conducted for admission to various postgraduate programs in the field of science at IITs and other participating institutes. The exam primarily covers subjects such as Physics, Chemistry, Mathematics, Biological Sciences, and Mathematical Statistics.
| 2. How can I effectively prepare for the Physics section of the IIT JAM exam? | |
Ans.To effectively prepare for the Physics section of the IIT JAM exam, students should focus on understanding fundamental concepts, practicing numerical problems, and solving past year papers. Utilizing standard textbooks, joining coaching classes, and participating in study groups can also enhance preparation.
| 3. What types of questions are typically asked in the IIT JAM Physics exam? | |
Ans.The IIT JAM Physics exam typically includes multiple-choice questions (MCQs), multiple select questions (MSQs), and numerical answer type questions (NATs). These questions assess both theoretical understanding and problem-solving skills across various topics in Physics.
| 4. Is it necessary to refer to past year papers for IIT JAM Physics preparation? | |
Ans.Yes, referring to past year papers is crucial for IIT JAM Physics preparation as it helps students understand the exam pattern, types of questions asked, and frequently covered topics. It also aids in time management and boosts confidence by familiarizing students with the exam structure.
| 5. What are some common mistakes to avoid while preparing for the IIT JAM Physics exam? | |
Ans.Common mistakes to avoid while preparing for the IIT JAM Physics exam include neglecting conceptual clarity, not practicing enough numerical problems, procrastinating study schedules, and overlooking the importance of time management during mock tests. Regular review and consistent practice are essential for success.
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