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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 Physics Past Year Papers
| 1. What are the key topics covered in the Physics 2023 exam? |  |
Ans. The key topics covered in the Physics 2023 exam include mechanics, electricity and magnetism, thermodynamics, waves, optics, and modern physics.
| 2. How can I best prepare for the Physics 2023 exam? | |
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Ans. Some recommended study materials and resources for Physics 2023 exam preparation include textbooks like "Fundamentals of Physics" by Halliday, Resnick, and Walker, online lectures and tutorials, and practice questions from reputable sources.
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