02-04-2025

6501CJM80126025001 JM

PHYSICS

SECTION - I

1) An object is present on the principal axis of a concave mirror of focal length 15 cm. Object is at a
distance of 20 cm from mirror. If velocity of object is 5cms–1 towards mirror and velocity of mirror is

5 cms–1 towards object then velocity of image at the same instant is

(A) 35 cms–1 towards left

(B) 95 cms–1 towards left
(C) 30 cms–1 towards right
(D) 35 cms–1 towards right

2) An object is placed in front of a spherical mirror whose 2 times magnified image is formed on
screen. Then choose CORRECT option :-

(A) Mirror is concave m = +2

(B) Mirror is concave m = –2
(C) Mirror is convex m = +2
(D) Mirror is convex m = –2

3) A ray of light enters a rectangular glass slab of refractive index at angle of incidence 60°. It
ravels a distance of 5 cm inside the slab and emerges out of the slab. The perpendicular distance
between the incident and the emergent rays is

(A) cm

(B)
cm

(C)
cm
(D) 5 cm

4) An object is placed at 10 cm in front of a concave mirror of radius curvature 15 cm. Then

magnification of image is ______
(A) 3

(B)

(C) –3

(D)

5) There are two plane mirror inclined at 40º, as shown. A ray of light is incident on mirror M1. What
should be the value of angle of incidence ‘i’ so that the light ray retraces its path after striking the

mirror M2.

(A) 50º
(B) 40º
(C) 30º
(D) 20º

6) An object of height 4 cm kept above the principle axis of a spherical mirror form real image of
height 2 cm. Choose the correct option.

(A) Mirror is concave

(B) object is farther from the pole than the image.
(C) image formed is inverted.
(D) All of the above

7) A plane mirror is moving with velocity . A point object in front of the mirror moves
with a velocity . The mirror is parallel to x-y plane. The velocity of the image is:

(A)
(B)
(C)
(D)

8) A convex mirror of radius of curvature 20 cm is shown in figure. An object O is placed in front of

this mirror. Its ray diagram is shown. How many mistakes are there in the ray diagram (AB is
principal axis)

(A) 3
(B) 2
(C) 1
(D) 0

9) The image for the converging beam after refraction through the curved surface (in the given

figure) is formed at:

(A) x = 40 cm

(B)
x= cm

(C)
x=- cm

(D)
x= cm

10) A point object O is placed in front of a long glass rod having spherical end of radius of curvature
30 cm. The image would be formed at (measured from interface)

(A) 30 cm left
(B) Infinity
(C) 1 cm to the right
(D) 18 cm to the left

11) The radius of curvature of concave miror is 24 cm and the image is magnified by 1.5 times. The
real object distance is

(A) 20 cm or 4 cm
(B) 8 cm or 4 cm
(C) 16 cm or 20 cm
(D) 24 cm or 20 cm

12) Refractive index of a medium is µ. The incidence angle is twice that of refracting angle. The
angle of incidence is (Take: rarer medium to be air)

(A)

(B)

(C)

(D)

13) The distance of a real object from the pole of a concave mirror is equal to its radius of curvature.
The image must be:

(A) Real
(B) Inverted
(C) Same sized
(D) All the above

14) A plane mirror approaches a stationary person with acceleration 10 ms–2. The acceleration of his
image as seen by the person, will be

(A) 10 m/s2
(B) 20 m/s2
(C) 5 m/s2
(D) Can't determined

15) A plane mirror makes an angle of 30o with horizontal. If a vertical ray strikes the mirror, find the
angle between mirror and reflected ray

(A) 30o
(B) 45o
(C) 60o
(D) 90o

16) A transparent cube of 15 cm edge contains a small air bubble. Its apparent depth when viewed
through one face is 6 cm and when viewed through the opposite face is 4 cm. Then the refractive
index of the material of the cube is :

(A) 2.0
(B) 1.5
(C) 1.6
(D) 2.5

17) A point source of light B is placed at a distance L in front of the centre of a mirror of width d
hung vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror at a
distance 2L from it as shown. The greatest distance over which he can see the image of the light

source in the mirror is

(A)

(B) d
(C) 2d
(D) 3d

18) Short linear object of length lies along the axis of a concave mirror of focal length f at a
distance u from the pole of the mirror. The size of the image is approximately equal to

(A)

(B)

(C)

(D)

19) A ray of light is incident at the glass-water interface at an angle i, it emerges finally parallel to

the surface of water, then the value of would be

(A)
(B)

(C)

(D) 1

20)

Right face of the glass cube is silvered as shown. A ray of light incident on left face of the cube as
shown. Find the deviation of the incident ray when it comes out of the glass cube

(A)
(B)
(C)
(D)

SECTION-II

1) Two long plane mirrors M1 and M2 are kept inclined to each other. The angle between reflecting
surfaces is 40°. Find the maximum number of reflections , the ray will undergo.

2) A bird is flying 3m above the surface of water. If the bird is diving vertically down with speed =

6m/s, his apparent velocity (m/s) as seen by a stationary fish underwater is

3) The refractive index of glass and water with respect to air are 8/3 and 4/3 respectively. The
refractive index of glass with respect to water is

4) Figure shows a plane mirror on which a light ray is incident. If the incident light ray is turned by
10º and the mirror by 20º, as shown, find the angle (in degree) turned by the reflected ray.

5) In the figure shown find the total magnification after two successive reflections first on M1 and

then on M2 .

CHEMISTRY

SECTION - I

1) KMnO4 acts as an oxidising agent in alkaline medium. When alkaline KMnO4 is treated with KI,
iodide ion is oxidised to :

(A) I2
(B) IO–
–
(C) IO3
–
(D) IO4

2) Which of the following is likely to form white salts?

(A) Cu2+
(B) Sc3+
(C) Ti3+
(D) Fe2+

3) The 'spin only' magnetic moment [in units of Bohr magneton] µB of Ni+2 in aqueous solution would
be :-

(A) 0
(B) 1.73
(C) 2.84
(D) 4.90

4) Which of the following pairs of compound have nearly equal size :-

(a) Y and La (b) Zr and Hf
(c) Ti and Zr (d) Nb and Ta

(A) a and b
(B) b and c
(C) b and d
(D) a, b, c

5) The d-block element which is a liquid at room temperature, having high specific heat, less
reactivity than hydrogen and its chloride (MX2) is volatile on heating is:

(A) Cu
(B) Hg
(C) Ce
(D) Pm

6) The pink colour of permanganate ion is due to

(A) No unpaired electron

(B) Presence of unpaired electron.
(C) Charge transfer from metal to ligand orbital
(D) Charge transfer from ligand to metal orbital.

7) Among following which are amphoteric oxide?

Mn2O7, CrO3, Cr2O3, CrO, V2O5, V2O4

(A) V2O5, Cr2O3

(B) Mn2O7, CrO3
(C) CrO, V2O5
(D) V2O5, V2O4

8) Which of the following combinations in an aqueous medium will give a red colour or precipitate?

2+ 3-
(A) Fe + [Fe (CN)6]
(B) Co2+ + SCN–
(C) Cu2+ + OH–
(D) Fe3+ + SCN–
9) The electrode potential of M2+ / M of 3d-series elements shows positive value of :

(A) Zn
(B) Fe
(C) Co
(D) Cu

10) Which elements of lanthanoids exhibit oxidation sate of +4'.in MO2? (M = Lanthanide)

(A) Ce, Dy, Tb

(B) Yb, Eu, Sm
(C) Nd, Ce, Sm
(D) Ho, Tm, Yb

11) Inner orbital complex which is paramagnetic in nature ?

–3
(A) [Ni(CN)5]
–4
(B) [Fe(CN)6]
(C) [Fe(H2O)5(NO)]SO4
–3
(D) [Fe(CN)6]

12) [NiCl4]–2 and [Ni(CN)4]–2 show similarity in :-

(A) Geometry
(B) Magnetic nature
(C) Hybridisation of state of Ni
(D) Primary valency of Ni

4– –1 2–
13) The Δ0(CFSE) for [CoCl6] is 18000 cm . The splitting energy for [CoCl4] will be :-

(A) 18000 cm–1

(B) 16000 cm–1
(C) 8000 cm–1
(D) 2000 cm–1

14) EAN of metal carbonyl M(CO)x is 36 if atomic no. of metal M is 26, what is the value of x:

(A) 4
(B) 8
(C) 5
(D) 6

15) The oxidation state of Mo in its oxo-complex species [Mo2O4(C2H4)2(H2O)2]2– is:

(A) +2
(B) +3
(C) +4
(D) +5

16) The compound that is both paramagnetic and coloured is :-

(A) K2Cr2O7
(B) (NH4)2[TiCl6]
(C) VOSO4
(D) K3 [Cu(CN)4]

17) The correct order of intensity of colors of the compounds is :

2– 2– 2+
(A) [Ni(CN)4] > [NiCl4] > [Ni(H2O)6]
2+ 2– 2–
(B) [Ni(H2O)6] > [NiCl4] > [Ni(CN)4]
2– 2+ 2–
(C) [NiCl4] > [Ni(H2O)6] > [Ni(CN)4]
2– 2– 2+
(D) [NiCl4] > [Ni(CN)4] > [Ni(H2O)6]

18) Formation constant (kf) and dissociation constant (kd) data is given then select most stable
complex.

(A)

(B)

(C)

(D)

19) The number of water molecules not coordinated to copper ion directly in CuSO4. 5H2O is

(A) 4
(B) 1
(C) 2
(D) 3

20) For a d4 metal ion in an octahedral field, the correct electronic configuration is

(A)

(B)

(C)
(D)

SECTION-II

1) How many of the following statements are correct?

(i) Manganese exhibits +7 oxidation state in one of its oxide.
(ii) Ruthenium and Osmium exhibit +8 oxidation in their oxides.
(iii) Sc shows +4 oxidation state which is oxidizing in nature
(iv) Cr shows oxidising nature in +6 oxidation state.

2) How many of the following contains either half-filled or full filled "f-subshell" (inner/valency shell)
in their ground state electronic configuration ?
Pm, Eu, Gd, Ho, Yb, Lu, U, Cm, No and Lr

3) The volume (in mL) of 0.1 M AgNO3 for complete precipitation of chloride ions present in 30 ml.
of 0.01 M solution of as silver chloride is

4) Among Ni(CO)4, [NiCl4]2–, [Co(NH3)4Cl2]Cl, Na3[Co(F)6], Na2O2 and KO2, the total number of
paramagnetic compounds is:

5) Find the total number of complexes or compounds are coloured

[Ti(NH3)6]4⊕, CuCl2, [Cu(CH3CN)4]BF4, CuCl, VOCl2, MnO4⊝

MATHEMATICS

SECTION - I

1) If A is a skew symmetric matrix such that ATA = I, then A4n–1 is equal to -

(A) –AT
(B) I
(C) –I
(D) AT

2) If , then |A| |AdjA| is equal to -

(A) a25
(B) a27
(C) a81
(D) none of these

3) Let three matrices be A = ; B= and C = , then

tr(A) + tr + tr + tr + ....... + ∞ is equal to-

(A) 6
(B) 9
(C) 12
(D) none

4) Let P = and Q = [qij] be two 3×3 matrices such that Q–P5 = I3. Then is equal to
:

(A) 15
(B) 9
(C) 135
(D) 10

5) If A is a square matrix of order 50, then is equal to

(I is an identity matrix of order 50)

(A)

(B) 1

(C)

(D)

6) Let and and Q = PAPT, then PTQ2013P

(A)

(B)

(C)

(D)

7) If , then (λ +μ + δ) is equal to
(A) 1
(B) 2
(C) 3
(D) 4

8) If a, b and c are unequal natural numbers, then is equal to

(A) a + b + c
(B) a + b
(C) b + c
(D) 0

9) If the system of equations

x+y+z=6
2x + 5y + αz = β
x + 2y + 3z = 14
has infinitely many solutions, then α + β is equal to : [JEE Main 2022]

(A) 8
(B) 36
(C) 44
(D) 48

10) Let , where [t] denotes the greatest integer less than or equal to
t. If det(A) = 192, then the set of values of x is the interval:

(A) [68, 69)

(B) [62, 63)
(C) [65, 66)
(D) [60, 61)

11) Let the system of linear equations x + 2y + z = 2, αx + 3y – z = α, –αx + y + 2z = –α be

inconsistent. Then α is equal to :

(A)

(B)

(C)
(D)

12) |A3 × 3 | = 3, |B3 × 3 | = – 1 and |C2 × 2 | = +2 then |2ABC| =

(A) 23 (6)
(B) 23 (–6)
(C) 2 (–6)
(D) None of these

13) If , then f(100) is equal to -

(A) 0
(B) 1
(C) 100
(D) –100

14) Which of the following functions are identical function :

(A) ƒ (x) = ; g(x) =

(B) ƒ (x) = sec x – tan2x; g(x) = 1
2

(C) ƒ (x) = ; g(x) =

(D)

15) The system of linear equations

x + λy – z = 0
λx – y – z = 0
x + y – λz = 0
has a non-trivial solution for :

(A) Exactly one value of λ.

(B) Exactly two values of λ.
(C) Exactly three values of λ.
(D) Infinitely many values of λ.

16) Range of the function

, where is

(A) [0, 2]
(B) [2, 5]
(C) [–2, 0)
(D) [–1,5]

17) If , (where [.] denotes the greatest integer function) then value of

is

(A) -2
(B) -1
(C) 0
(D) 3

18) Let f : R – {3} → R – {1} be defined by . Let g : R → R be given as g(x) = 2x – 3. Then,

the sum of all the values of x for which f–1(x) + g–1(x) = is equal to

(A) 7
(B) 2
(C) 5
(D) 3

19) If then is

(A)

(B)

(C)

(D)

20) Let f : A → B be a function in which set A contains 5 distinct elements and set B contains 4
distinct elements. How many number of many one functions are possible:

(A) 5
(B) 1024
(C) 120
(D) None of these

SECTION-II

1) A is a square matrix of order n.

ℓ = maximum number of distinct entries if A is a triangular matrix
m = maximum number of distinct entries if A is a diagonal matrix
p = minimum number of zeroes if A is a triangular matrix
If ℓ + 5 = p + 2m, the value of n is

2) Let where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If
A = I3, then the value of x + y3 + z3 is_____.
2 3

3) If x, y & z are different even integers and minimum value of

is N, then is

4) The remainder when is divided by 5 is k then k equal to

5) Let ƒ : R → R, ƒ(x) = x3 + x + a – 9 is an odd function, then the value of 'a' is :

 ANSWER KEYS

PHYSICS

SECTION - I

Q. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A. B B B C B D B B A A A C D B C B D D B C

SECTION-II

Q. 21 22 23 24 25
A. 3 8 2 30 2

CHEMISTRY

SECTION - I

Q. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
A. C B C C B D A D D A D D C C B C C C B D

SECTION-II

Q. 46 47 48 49 50
A. 3 8 6 3 3

MATHEMATICS

SECTION - I

Q. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
A. D D A D D A C D C B D D A D C A C C B B

SECTION-II

Q. 71 72 73 74 75
A. 4 7 8 4 9
 SOLUTIONS

PHYSICS

1) ;

; (vI)x= –90 – 5 = –95 cm/s

9) v = 40 cm.

10) Use

R = +30 cm; u = –15 cm.

11) Use

and
f = –12 cm.

12) Use

16)
17)

According to the following ray diagram

Similarly IJ = d
so, GJ = GH + HI +IJ = d + d + d =3d

18) From mirror formula .......... (i)

Differentiating equation (1), we obtain

.......... (ii)

Also from equation (i) .......... (iii)

From equation (ii) and (IN) we get

Therefore size of image is

19) For glass-water interface ............ (i)

For water-air interface ............ (ii)

20)

Use Snell's law and laws of reflection, then add angle of deviations.

21)
 24) Angle turned by the reflected ray = 2 (20º) – (10º) = 30º clockwise.

25)

For M1: v1 = = – 60

∴ M=– = – 2.
For M2 : u = + 20. f = 10

∴ + = ⇒ v=20

m2 = – = – 1 m = m1 × m2 = + 2

CHEMISTRY

26)

KMnO4 + KI

0 0
27) Sc3+ = [Ar]4s 3d (It has zero unpaired e–)

28) Ni+2 4s2 3d8

n = 2,

29) Size of 4 d series 5 d series in group no 4 to 11

30)

Hg is liquid at room temperature.

31)
Momentarily transfer of e– from filled orbital of oxygen to vacant orbital of Mn+7.

32)

V2O5 and Cr2O3 are amphoteric oxides.

V2O4 and CrO are Basic oxides.
 Mn2O7 and CrO3 are Acidic oxides.

33) Fe(SCN)3 is red colour

34) Only copper shows positive value for electrode potential of M2+/M of 3d-series elements.

35)

36) In [Fe(CN)6]–3, hybridization of Fe is d2sp3 and it has 1 unpaired e–.

37)

[NiCl4]–2 sp3, Tetrahedral , paramagnetic, Primary valency = +2

–2 2
[Ni(CN)4] dsp , square planar, diamagnetic , primary valency = +2

38) » = 8000 cm–1

39)

[M(Co)x] EAN = 36
36 = 26 – 0 + 2x

x=5

40)

[MO2O4 (C2H4)2 (H2O)2]–2

2x + (–2)× 4 + 0 × 2 + 0 × 2 = – 2
2x = 6

x=+3

41) (Cr+6 does not have any unpaired e¯) is coloured but it is diamagnetic.
⇒ Ti and Cu+ do not have any unpaired e¯
+4

⇒ (V+4 has one unpaired e¯) is coloured and paramagnetic.

42)

43)
Stability of the complex

44)

45)

46) (i), (ii) and (iv) correct

Manganese exhibits +7 oxidation state in its oxide. (Mn2O7)
Ru & Os form RuO4 & OSO4 oxide in +8 oxidation state
Cr in +6 oxidation state acts as on oxidising agent
Sc does not show +4 oxidation state.

47)
 48)

49)

50) CuCl2 → Cu2⊕ = [Ar]3d9 (one unpaired electron)

VOCl2 → V4⊕ = [Ar]3d1 (one unpaired electron)
MnO4⊝ → Coloured due to change transfer spectrum.
All three are coloured complexes.

MATHEMATICS

51) AT = –A & ATA = I

⇒ A2 = –I ⇒ A4n = I
A4n–1 = A–1 ⇒ A4n–1 = AT (A is orthogonal)

52) We have |A| = a3

|A·adj(A)| = |A|n ; n = order
= |A|·|adj A| = (a3)3 = a9

53)

Now

54)
P" =

Q = P5 + I3

Aliter

P=I+X

X3 = 0
P5 = I + 5X + 10X2
Q = P2 + I = 2I + 5X + 10X2
55) ∵ xadj(x) = |x| I

and

⇒ x adj(x)
=

56) ⇒ PTP = I
⇒ PTQ2013P
⇒ PTQ2012PAPTP
⇒ PTQ2012PA
⇒ PTQ2011PAPTPA
⇒ PTQ2011PA2
.
.
.
⇒ PTQPA2012
⇒ PTPAPTPA2012

57) L.H.S =

58)

59)

x+y+z=6 …(1)
2x + 5y + αz = β …(2)
x + 2y + 3z = 14 …(3)
x+y=6–z
x + 2y = 14 – 3z
On solving
x = z – 2 ⇒ y = 8 – 2z in (2)
2(z – 2) + 5 (8 – 2z) + αz = β
(α – 8)z = β – 36 For having infinite solutions
α–8=0 & β – 36 = 0
α = 8, β = 36 (α + β = 44)

60)
R1 → R1 – R3 & R2 → R2 – R3

2[x] + 6 + [x] = 192 ⇒ [x] = 62

61)
= (6 + 1) – 2

= 14 + 2α

62)

2ABC is not defined

there is no solution

63)
C3 → C3 – C1 – C2
 ⇒ f(x) = 0
∴ f(100) = 0

64)

Option (A) :

both function has different domain

Not identical function.
Option (B) :

=1;
domain of Domain of g(x)
Not identical function.
Option (C) :

g(x) = but only if

Not identical function.
Option (D) :

and Both are identical function.

Option (D) is correct.

65)
1(λ + 1) – λ (–λ2 + 1) – 1 (λ + 1) = 0
λ + 1 + λ3 – λ – λ – 1 = 0
λ3 – λ = 0
λ(λ2 - 1) = 0
⇒ λ = 0, ±1
Three values

66)

67)
 N(x) is even function.

is odd function
is odd function

also,

68)

& g(x) = y = 2x – 3

∴
∴ sum of roots
x1 + x2 = 5

69) and verity options for this.

70) Since domain contains more elements than codomain hence all possible functions are
many one only.
∴ Total number of many one functions = 45 = 1024

71) ℓ = 1 + 2 + 3 + ...... + n ⇒ ℓ =

m = n + 1; p =
ℓ + 5 = p + 2m
n=4

72) Official Ans. by NTA (7)

A2 = I
⇒
⇒ A is orthogonal
So, x2 + y2 + z2 = 1 and xy + yz + zx = 0
⇒ (x + y + z)2 = 1 + 2 × 0
⇒x+y+z=1
Thus,
x3 + y3 + z3 = 3 × 2 + 1 × (1 – 0)
=7

73) Given determinant is

= (x – y)2 (y – z)2 (z – x)2
minimum value is 256
(take x = 2, y = 0, z = –2)

74)

75) ƒ(–x) = –ƒ(x)

–x3 – x + a – 9 = –x3 – x – a + 9
a=9