04-08-2025

2331CJA101031250005 JM

PHYSICS

SECTION-I

1) Density of ice is 0.9 g/ cc in the CGS system of units. The corresponding value in MKS units is

(A) 900
(B) 9
(C) 0.9
(D) 9000

2) Velocity of a particle depends on time t according to equation :- V = + bt +

The a, b, c and d represents the following quantities in order :-

(A) Distance, distance, acceleration, time

(B) Acceleration, distance, time, distance
(C) Acceleration, distance, distance, time
(D) Distance, acceleration, distance, time

3)

A physical quantity P is related to four observables a, b, c and d as follows : .

The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is
the percentage error in the quantity P ?

(A) 10 %
(B) 11 %
(C) 12 %
(D) 13 %

4)

(A)

(B)

(C)
(D)

5) Three coplanar vectors and have magnitudes 4, 3 and 2 respectively. If the angle between

any two vectors is 120° then which of the following vector may be equal to ?

(A)

(B)

(C)

(D)

6) If , then the value of C is:-

(A) 0
(B) 0.01
(C)
(D) Any value of C

7) Two balls are thrown with same speed v0 from the top of a cliff. The angles of their initial
velocities are θ above and below the horizontal as shown. How much farther along the ground does

the top ball hit than the bottom ball :-

(A)

(B)

(C)
(D)

8) The displacement x of a particle moving in one dimension is related to time t by the equation
where x is in meters and t in seconds. The displacement of the particle when its velocity is
zero is :-

(A) zero
(B) 4 m
(C) 1 m
(D) 0.5 m

9) Velocity of a particle moving along x-axis is related to co-ordinate as v = 2x2 – 6 m/s. Then
acceleration of particle at x = 1 is :-

(A) –4 m/s2
(B) –16 m/s2
(C) –8 m/s2
(D) 16 m/s2

10) In the arrangement shown in figure. If mA = mB = 2kg. String is massless and pulley is
frictionless. Block B is resting on a smooth horizontal surface and friction coefficient between blocks
A and B is µ = 0.5. What maximum horizontal force F can be applied so that block A does not slip

over the block B ? (g = 10 m/s2) :-

(A) 25 N
(B) 40 N
(C) 30 N
(D) 20 N

11) All surfaces shown in figure are smooth. For what ratio m1 : m2 : m3, system is in equilibrium. All

pulleys and strings are massless :-

(A) 1 : 2 : 1
(B) 2 : 2 : 1
(C) 2 : 1 : 2
(D) 1 : 2 : 2

12) A block lying on a long horizontal conveyor belt moving at a constant velocity receives a velocity
6 m/s relative to the ground in the direction opposite to the direction of motion of the conveyor.
After t = 2 sec, the velocity of the block becomes equal to the velocity of the belt. The coefficient of
friction between the block and the belt is 0.4. Then the speed of the conveyor belt is :-

(A) 1 m/s
(B) 2 m/s
(C) 3 m/s
(D) 6 m/s

13) Two blocks of equal mass 2 kg are placed on a rough horizontal surface as shown and a force
is applied on the upper block. The system is initially at rest. Find acceleration of the lower block.

(A) 2 m/s2
(B) 4 m/s2
(C) 5 m/s2
(D) 7 m/s2

14) A particle starts from point 'A' reaches to B in a uniform circular motion. Find angular
displacement of the particle (during which it reaches from A to B ) as seen from centre:- (if AC line

is diameter)

(A)
(B)

(C)

(D)

15) A pendulum of mass m and length ℓ is released from rest in a horizontal position. A nail at a
distance d below the pivot, causes the mass to move along the path indicated by the dotted line. The
minimum distance such that the mass will swing completely round in the circle shown in figure is d

then is :-

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

16) A particle is attached at one end of massless rod whose other end is fixed at O as shown in
figure. A particle is given minimum velocity at lower most point to complete vertical circular motion

about O. Find net force on the particle when it is at position P. Length of rod is ℓ. .

(A)

(B)

(C)

(D) None of these

17) A constant force F is pushing the block A of mass 5kg which is connected with block B of mass
10 kg by a light spring till the block B will slide. If 0.4 and 0.5 are the coefficient of friction between
A and ground and B and ground respectively. (where k = stiffness of the spring)
(A)
The maximum compression of the spring is equal to
(B) The minimum magnitude of F to move the block B is 45N
(C) The minimum magnitude of F to move the block B is 70N
(D) At maximum compression force F equals force of spring.

18) Two masses 'm' and '2m' are placed in fixed horizontal circular smooth hollow tube as shown.
The mass 'm' is moving with speed 'u' and the mass '2m' is stationary. After their first collision, the

time elapsed for next collision. (coefficient of restitution e = 1/2)

(A)

(B)

(C)

(D)

19) Mass 2m is kept on a smooth circular track of mass m which is kept on a smooth horizontal
surface. The circular track is given a horizontal velocity towards left and released. Find the

maximum height reached by 2m.

(A) R
(B) R/3
(C) R/4
(D) 2R/3

20) A block C of mass m is moving with velocity v0 and collides elestically with block A of mass m and
connected to another block B of mass 2m through a spring of spring constant k. What is k if x0 is the
compression of spring when velocity of A and B is same ?

(A)
(B)

(C)

(D)

SECTION-II

1) A body of mass 5kg rests on a rough horizontal surface of coefficient of friction 0.2. The body is
pulled through a distance of 10m by a horizontal force of 25 N. The kinetic energy (in J) acquired by
it is (g = 10 ms2)

2) All the surfaces are frictionless. Block of mass M is kept at the midpoint of AB. The system is
released from rest. The distance moved by the bigger block at the instant the smaller block reaches

the ground is x (cm). The value of x is :

3) A ball of mass 100 g is dropped from a height h = 10 cm on a massless platform fixed at the top
of vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed

by a distance . The spring constant is ________ Nm–1. (Use g = 10 ms–2)

4) A 20 kg car starts from rest at point 1 and moves without friction down the track as shown

The force exerted by the track on the car at point 2 where the radius
of curvature of the track is 20 m is given by 50x newton, then the value of x is ___.

5) One end of a spring of natural length ℓ and spring constant k is fixed at the ground and the other
is fitted with a smooth ring of mass m which is allowed to slide on a horizontal rod fixed at a height ℓ
(figure). Initially, the spring makes an angle of θ with the vertical when the system is released from

rest. If the speed of the ring when the spring becomes vertical is (2ℓ/3) m/s then find the value

of angle θ (in degree) :-

CHEMISTRY

SECTION-I

1) Molarity and Molality of a solution of an liquid (M. wt. = 50) in aqueous solution is 9 and 18
respectively. What is density of solution–

(A) 1g/cc
(B) 0.95 g/cc
(C) 1.05 g/cc
(D) 0.662 g/cc

2)

Monosodium glutamate (MSG) is salt of one of the most abundant naturally occuring non-essential
amino acid which is commonly used in food products like in "maggi" having structural formula as

Mass % of Na in MSG is-

(A) 14.8
(B) 15.1
(C) 13.6
(D) 16.5

3) Phosphoric acid (H3PO4) is prepared in a two step process.

(A) P4 + 5O2 → P4O10
(B) P4O10 + 6H2O → 4H3PO4
We allow 62 g of phosphorus to react with excess oxygen which form P4O10 with 85% yield. In the
step (B) 90% yield of H3PO4 is obtained. Produced mass of H3PO4 is :-

(A) 37.485 g
(B) 149.949 g
(C) 125.47 g
(D) 564.48 g

4)

For which of the following vant' Hoff factor is correctly matched -

Degree of
Salt dissociation i
(α)

(A) Na2SO4 80 % 2

(B) K3[Fe(CN)6] 75% 3.25

(C) [Ag(NH3)2]Cl 50 % 1.8

(D) [Cr(NH3)5Cl]SO4 90 % 2.8

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

5) For which of the following pair, the heat of mixing, ΔHmix, is approximately zero?

(A) CH3COOCH3 + CHCl3

(B) CH3COOH + H2O
(C) C2H5OH + CH3OH
(D) CH3COCH3 + C6H6

6) A solution containing Na2CO3 and NaOH requires 300 ml of 0.1N HCl using phenolphthalein as an
indicator. Methyl orange is then added to the above titrated solution when a further 25 ml of 0.2N
HCl is required. The amount (in grams) of NaOH present in solution is (Molecular weights: NaOH =
40, Na2CO3 =106)

(A) 1g
(B) 1.5 g
(C) 2 g
(D) 4 g

7)

At low pressure, if RT = , then the volume occupied by a real gas is :

(A)

(B)
(C)

(D)

8) The total number of bimolecular collisions per unit volume per unit time is given by

where is molecule diameter, is the average speed and is the number of

molecules per unit volume. Which of the following facts is correct?

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

9) Difference in wavelength of two extreme spectral lines of H-atom in Balmer region is: (Where RH
is Rydberg constant)

(A) 7.2 / RH
(B) 0.25 / RH
(C) 4 / RH
(D) 3.2 / RH

10) The solubility of sparingly soluble salt increases in presence of complexating agents while it
decreases in presence of common ions. AgCl is a sparingly soluble salt (ksp = 10–10). The solubility of
AgCl decreases in AgNO3 and NaCl solution while it increases when taken in aqueous ammonia. The
formation constant kf for [Ag(NH3)2]+ is 108. (Given : MAgCl = 143.5 g/mole)
What is the solubility of AgCl (in mg) in 10 mL of 0.1 M AgNO3 solution.

(A) 9.25 × 10–9

(B) 1.43 × 10–6
(C) 6.96 × 10–9
(D) None of these

11) An acid-base indicator has Ka = 3.0 × 10–5. The acid form of the indicator is red and the basic
from is blue. The [H+] required to change the indicator from 75% blue to 75% red is

(A) 8 × 10–5 M
(B) 9 × 10–5 M
(C) 1 × 10–5 M
(D) 3.33 × 10–5 M

12)
Some reversible reactions (in List-I) are at equilibrium and the effect on system on increasing the
volume of system at 300K is given in List-II. Match them correctly.
 List-I List-II

(P) (1) Moles of reactant decreases

(Q) (2) Moles of reactant increases

(R) (3) Molar concentration of reactant increases

(S) (4) Molar concentration of reactant decreases

The correct option is

P Q R S

1 1 4 1 4

2 1 4 3 1

3 1 2 3 1

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

13) Dipole moment of a molecule is a vector sum of all the individual bond moments. Mathematically
it is μ = q × d, where q is magnitude of particle charges and d is the distance between centre of
opposite charges. Its main application includes, determination of polarity of bond, symmetry of
molecule and in calculation of % ionic character.
In HI molecule if observed dipole moment in 1.2 Debye and also the interionic distance between
atoms (dHI) is 1Å. The percentage covalent character in HI bond will be

(A) 25%
(B) 75%
(C) 50%
(D) 12.5%

14) Hydrogen bonding does NOT play an important role in which of the following :

(A) In vapour phase CH3COOH exist as dimer

(B) Boric acid is solid at room temperature
(C) slippery nature of graphite
(D) solubility of alcohol in water

15) For a process A + B → product, the rate of reaction is second order with respect to A and zero
order with respect to B. When 1 mole each of A & B are taken in 1 litre vessel, the initial rate of
disappearance of B is 1 × 10–2 mol/L-sec. The rate of reaction when 50% of the reactant have been
converted to product would be -
(A) 1 × 10–2 mol/L-sec
(B) 2.5 × 10–3 mol/L-sec
(C) 5 × 10–2 mol/L-sec
(D) 5 × 10–3 mol/L-sec

16) The reaction Cis-X trans-X is 1st order in both direction. At 25ºC the equlibrium
constant is 0.10 and the rate constant Kf = 3 × 10–4 sec–1. In an experiment starting with the pure cis
form, how long would it take for half of the equilibrium amount of the trans isomer to be formed -

(A) 150 sec.

(B) 200 sec.
(C) 240 sec.
(D) 210 sec.

17) Which order is wrong :-

(A) Electronegativity – P < N < O < F

(B) Ist ionisation potential – B < Be < O < N
(C) Basic property – MgO > CaO > FeO > Fe2O3
(D) Reactivity – Be < Li < K < Cs

18)

List-I List-II

(Order) (Property)

(P) (1) Size of ion/atom

(Q) (2) Bond dissociation enthalpy

(R) F > O > N (3) First ionisation enthalpy

(S) F2 < O2 < N2 (4) Electronegativity

P Q R S

1 1 2 4 3

2 4 3 1 2

3 1 3 4 2

4 2 3 1 4
(A) 1
(B) 2
(C) 3
(D) 4
19) Find the Ka for 0.1M CH3COOH if it dissociates 1.30% at a certain temperature

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

20) 25g of an unknown hydrocarbon upon burning produces 88g of CO2 and 9g of H2O. This
unknown hydrocarbon contains.

(A) 20g of carbon and 5 g of hydrogen

(B) 24g of carbon and 1 g of hydrogen
(C) 18g of carbon and 7 g of hydrogen
(D) 22g of carbon and 3 g of hydrogen

SECTION-II

1) An aqueous solution of N2 gas obeys Henry's law at very low concentration of gas in water

. How many moles of N2 gas will dissolve in 18 kg. water at 27°C and 1.6 × 107 Pa ?
KH (Henry's law constant) for N2 gas in water at 27°C = 80 K bar.

2) Consider the following list of reagents :

Acidified K2Cr2O7, alkaline KMnO4, CuSO4, H2O2, Cl2, O3, FeCl3, HNO3 and Na2S2O3.
The total number of reagents that can oxidise aqueous iodide to iodine is -

3) KI(aq.) and KIO3(aq.) mixed together in presence of HCI to form I2 and KCI and liberated I2 required
100 mL of 0.1 M Na2S2O3 solution for complete reduction of I2.
If x = millimoles of KI used in the reaction.
and y = millimoles of HCI used in the reaction.
then the value of (3x - 2y) is :

4)
Andrew's isotherm of real gas A.
Identify sum of all "graph points" where gas A cannot be liquefied isothermally (among indicated
points only).

5) If 1023 radioactive atoms are added after each half life in a sample of 1024 radioactive atoms, then
number of atoms disintegrated between 2nd and 3rd half life is x × 1023

MATHEMATICS

SECTION-I

1)

If f(x) = sin2x + sin2(x + ) + cos x cos(x + ) and g( ) = 1 then graph of y = g(f(x)) is :-

(A) circle
(B) straight line
(C) Parabola
(D) None

2)

Which of the following statement is false?

2
(A) Range of f(x) = log10 (–x + 10x + 75) is (–∞,2]

(B)
Range of f(x) = is

(C)
and g(x) = are not identical

and
(D)
are not identical

3) The number of functions f from {1, 2, 3, ..., 20} onto {1, 2, 3, ....., 20} such that f(k) is a multiple
of 3, whenever k is a multiple of 4, is :-

(A) (15)! × 6!
(B) 56 × 15
(C) 5! × 6!
(D) 65 × (15)!

4) If f(x) is a polynomial such that :-

f(x) f = f(x) + f ; x ≠ 0 and f(4) = –63 then value of f

(A) –511
(B) – 26
(C) – 124
(D) –7

5) is equal to :-

(A)

(B)

(C)

(D)

6) (cot–1 3 + cot–1 7 + cot–1 13 + ....... + n terms) is :

(A)

(B) cot–1 2
(C) tan–1 2

(D)

7) If f(n + 1) =
and f(n) > 0 for all n ∈ N then is equal to

(A) 3
(B) –3

(C)

(D) 0

8) If sin–1x + sin–1 y = π and if x = λy, then the value of 392λ + 5λ must be :-

(A) 1526
(B) 1525
(C) 1524
(D) 1527
9)

(A) 0
(B) π – 2
(C) 2 – π
(D) 2

10) If cos4θ + α, sin4θ + α are the roots of the equation x2 + 2bx + b = 0 and cos2θ + β, sin2θ + β are
the roots of the equation x2 + 4x + 2 = 0 then find b?

(A) 2
(B) 1
(C) – 2
(D) None of these

11) If α, β are the roots of the equation ax2 + bx +c = 0, then the equation ax2 – bx (x–1) + c (x–1)2 =
0 has roots :-

(A)

(B) α–1, β–1

(C)

(D)

12)

If α, β, γ are the roots of the equation 8x3 + 1001x + 2008 = 0 then the value of (α + β)3 + (β + γ)3 +
(γ + α)3 is :-

(A) 251
(B) 751
(C) 735
(D) 753

13) The equation 5x2 + 12x + 13 = 0 and ax2 + bx + c = 0 have a common root, where a, b, c are the
sides of ΔABC, then find ∠C?

(A) 45°
(B) 60°
(C) 90°
(D) 30°

14) If the roots of the equation

x2– 2ax + a2 + a – 3 = 0 are real and less than 3, then :-

(A) a < 2
(B) 2s ≤ a ≤ 3
(C) 3 < a ≤ 4
(D) a > 4

15) If p, q, r in G.P and tan–1 p, tan–1 q, tan–1 r are in A.P, then p, q, r satiesfy the relation :-

(A) p = q = r
(B) p ≠ q ≠ r
(C) p + q = r
(D) None of these

16)

If log52, log5(2x – 3) and log5 are in A.P. then the value of x is :-

(A) 0
(B) –1
(C) 3
(D) 4

17) If 1 + sin θ + sin2 θ + ... ∞ = 4 + 2 , 0 < θ < π, , then :-

(A)

(B)

(C)

(D)

18)

If in a ΔABC, ∠C = 90°, then the maximum value of sin A sin B is –

(A)

(B) 1
(C) 2
(D) None of these

19)

The number of roots of the equation x + 2 tan x = in the interval [0, 2π] is –
(A) 1
(B) 2
(C) 3
(D) infinite

20)

Which one of the following relations on R is an equivalence relation :

(A) a R1b ⇔ |a| = |b|

(B) aR2b ⇔ a ≥ b
(C) a R3b ⇔ a divides b
(D) a R4b ⇔ a < b

SECTION-II

1) If , then |a + b| is equal to :-

2) If has range [a,b], then value of b – a is equal to

3) If α2 = 5α – 3, and β2 = 5β – 3 also then p + q equals :

4) If f : R → R ; f(x) = 2x+1,g : R → R, g(x) = x3,

then (fog)–1 (55) equals

5) The number of solutions of the equation is equal to

 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. A D D A B C C A B D B B C C B C B B B D

SECTION-II

Q. 21 22 23 24 25
A. 150 20 120 20 53

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. B C B B C A C C D B A A B C B D C C A B

SECTION-II

Q. 46 47 48 49 50
A. 2 7 5 8 2

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. B D A D C D A A B A C D C A A C D A C A

SECTION-II

Q. 71 72 73 74 75
A. 7 5 22 3 4
 SOLUTIONS

PHYSICS

1)

2) [b] = ⇒ acceleration [c] = LT–1T = L ⇒ Distance

[d] = T ⇒ Time

[a] = ⇒ Distance

3)

4)

5)

6)

7)

Let t1 be time taken by top ball to strike ground & t2 be time taken by bottom ball to strike
ground.

⇒ (t1 – t2) =
Thus, R1 – R2 = (v0 cos θ) (t1 – t2)
=

8)
x = (t – 3)2
x = t2 – 6t + 9
v = 2t – 6
v = 0 at t = 3 sec
x = 0 at t = 3 sec

9)

v = 2x2 – 6

a=
a = (2x2 – 6)4x
a = (2 – 6) × 4
2
a = – 16 m/s

10)

a=

F – 10 =
F = 20 N

11)
T = m3g ......(i)
2T = m2g ......(ii)
T = m1g sin 30° ......(iii)

12)

a = – 4 m/s2; u = 6 m/s; t = 2 sec; v = 6 – 4 × 2 = –2 m/s

13)

N1 = 30, fmax1 = 18
N2 = 50, fmax2 =5

amax of lower block is = 6.5 m/s2

F – 5 = 4(6.5)
F = 31 N since we applied 25 N

∴ block move together; a = = 5 m/s2

14)

angle is measured from centre not C.

15) =
⇒ 2ℓ = 5ℓ – 5d

16)

mgl(1 – cos 37°) = – 4gℓ

∴ gl = – + 2gℓ

∴ v2 =

Fnet = =

17)

Fx – – µ1m1gx = 0
Kx = µ2 m2g
 18) Let the speeds of balls of mass m and 2m after collision be v1 and v2 as shown in figure.
Applying conservation of momentum

mv1 + 2mv2 = mu and –v1 + v2 =

solving we get v1 = 0 and v2 =

Hence the ball of mass m comes to rest and ball of mass 2m moves with speed . t =

19)

Let v be the final speed of block when it is at maximum height h. At that instant the speed of
circular track shall also be v.

From conservation of momentum

= (m + 2m) v ...(1)
From conservation of energy

m (2gR) = (m + 2m) v2 + 2mgh ...(2)

solving (1) and (2) we get

h= R

20) mv + 0 = (m+2m)v'

v' =

mv2 = + (m+2m)v'2

mv2 = + m

mv2 =

k= a

21) Kinetic energy acquired by body

= (Total work done on the body) – (work against friction)
= F × S – µmgS = 25 × 10 – 0.2 × 5 × 10 × 10
= 250 – 100 = 159 Joule

22) COM of the system remains at rest as no net external force is acting on the system.
 If bigger block moves towards right by a distance x, the smaller block moves towards left by a
distance (2.2 – x)
M (2.2 – x) = 10 Mx
⇒ x = 0.2 m
⇒ x = 20 m

23) By energy conservation

PE = KE

0.100 × 10 × (0.10) = k (0.05 × 0.05)

k= = 120 N/m

24)

25) kx2 = mv2

x=

CHEMISTRY

26) = molecular weight of solute in kg / mole

m = molarity
M = Molarity
d = 9 {0.05 + 1/18} = 0.95 g/cc

27)

28) Produced mass of

H3PO4 =

29) K3[Fe(CN)6] → 3K + [Fe(CN)6]–3

 1–α 3α α

i=
i = 1 + 3(0.75) ⇒ 3.25

30)

For ideal gas mixture.

31) SRP values of

32)

At low pressure,

i.e.,

33) Since and we have

34)

λ – λ' =

35)

AgCl → Ag+ + Cl–

(s + 0.1)
–10
10 = s(s + 0.1)
s = 10–9 mol/ L ⇒ s = 10–11 mol /10 mL
= 143.5 × 10–11 × 103 = 1.43 × 10–6 mg

36)

HIn⇋ H+ + In–
Initial [H+] =

Final [H+] =
[H+] required = 9 × 10-5 – 10-5 = 8 × 10-5

37)

Answer: 1
Answer: 1

38) % ionic = =
⇒ % covalent = 100 – 25 = 75%

39)

Explanation:
The question asks which of the given phenomena is NOT significantly influenced by hydrogen
bonding.

Concept:
Graphite's slippery nature is due to its layered structure. The layers are held together by weak
van der Waals forces, not hydrogen bonds. The layers can easily slide past each other, giving
graphite its slippery feel. The bonding within each layer is strong (covalent), but the forces
between layers are weak.

Answer : Slippery nature of graphite

Hence, option (3) is correct.

0
40) r = K [A]2 [B]
0
10–2 = K(1)2 (1) ⇒ K = 10–2
0
finally, r = 10–2(1/2)2 (1/2)
= 2.5 × 10–3

41)

Kf + Kb =

t= ln
t = 210 sec.

42)

Which order is wrong.

(C) Basic property – MgO > CaO > FeO > Fe2O3
↓
wrong order
Because correct order is,
⇒ CaO > MgO > FeO > Fe2O3
Basic nature of oxides :- when in peridic table the distance between the element and oxygen
increases,
Basis character increases.
(CaO > MgO : Basic property).

43)

Answer: 3
Answer: 3

44)

45)

H
C2yHy ≡ 24y gm C + y gm H
or
24 : 1 ratio by mass

46) P = KH.X ⇒ 1.6 × 107 = 80 × 103 × 105 ×

⇒
 47)

48)

49) At T > TC, real gas can't be liquefied by changing P & V. Point 1, 3 and 4 to be consider.

50)

After 1st half life number of atoms = 5 x 1023 + 1023

After 2nd half life number of atoms = 3 × 1023 + 1023d
Number of atoms disintegrated between 2nd and 3rd half life = 2 ×1023

MATHEMATICS

51)

y = g(f(x)) = ⇒y=1
∴ straight line.

52)

Answer: and
 are not identical

53) ƒ(k) = 3m (3,6,9,12,15,18)

for k = 4,8,12,16,20 6.5.4.3.2 ways
For rest numbers 15! ways
Total ways = 6!(15!)

54) f(x) = 1 – x3
And

∴ f(2) = 1 –8 = – 7

55)

Put x = cosθ

56) (tan–1(1/3) + tan–1(1/7) + tan–1 (1/13) + ....... + n terms).

57)

As n → ∞ f(n) = f(n + 1) = k say

We have f(n + 1) =

⇒ f(n + 1) =

⇒k=
⇒ k2 = 9 or k = 3
⇒

58) ∵ – cos–1 π + – cos–1 y = π

or cos–1 x + cos–1 y = 0

⇒ xy – =1
⇒ (xy – 1) = (1 – x )(1 – y2)
2 2

⇒ x2y2 + 1 – 2xy = 1 – x2 – y2 + x2y2

or x2 + y2 – 2xy = 0
⇒ (x – y)2 = 0
∴ x=y
∴ λ=1
Then, 392λ + 5λ = 392 + 5 = 1521 + 5 = 1526

59) = sin–1(sin(2))
=π–2

60)

Difference of roots for both equations are equal

⇒ (cos4θ + α) – (sin4θ + α)
= (cos2θ + β) – (sin2θ + β)
⇒
⇒ b2 – b – 2 = 0 Þ b = 2 or –1

61) ....(1)

If t = α, then t = ⇒α=

⇒x= so roots are

62) 8x3 + 1001x + 2008 = 0 → α, β, γ

S.O.R ⇒ α + β + γ = 0 ⇒ α3 + β3 + γ3 = 3αβγ …(1)
Now, (α + β)3 + (β + γ)3 + (γ + α)3 = – γ3 – α3 – β3

⇒
⇒ 753

63) Roots of 5x2 + 12x + 13 = 0 are imaginary so

= k (let)

So cos C =

= =0

cos C = 0 ⇒ ∠C =

64) Let f(x) = x2 –2ax + a2 + a–3

acc. to given condition f(3) > 0, and D > 0 solving these gives a ∈ (–∞,2)U (3, ∞) and a
< 3 and a ≤ 3, Hence Common of these three results a < 2

65) p,q,r → G.P

tan–1p, tan–1q, tan–1r → A.P
2 tan–1q = tan–1p + tan–1r

2tan–1q = tan–1
∴p=q=r

66)

2b = a + c

⇒ 2log5(2x – 3) = log52 + log5

⇒ (2x – 3)2 = 2 ⇒ 22x – 7.2x – 8 = 0

⇒x=3

67)

1 – sinθ =

sinθ =

θ= or
68) sin A sin B = × 2 sin A sin B

= [cos (A – B) – cos(A + B)]

= [cos(A – B) – cos 90°]

= cos(A – B) ≤
∴ Maximum value of sin A sin B =
Hence, (1) is the correct answer.

69) We have, x + 2 tan x = ⇒ tan x = –

Now, the graph of the curve y = tan x and

y= , in the interval [0, 2π] intersect at three points. The abscissa of these three
points. The abscissa of these three points are the roots of the equation.
Hence, (3) is the correct answer.

70)

a R1b ⇔ |a| = |b| is reflexive, symmetric and transitive so this relation is equivalence relation.

71)
...(i)
...(ii)

72)

ƒmax when cos–1x = π

ƒmin when cos–1x = 0

∴
∴ b–a=5

73)

Clearly have Roots

74) y = f(g(x)) = 2x3 + 1

x3 =

x=

fog–1(x) =

75) Case-1 Power = 0

x2 – 6 = 0
⇒x=
Case-2 Base = 1
|x – 2| = 1
⇒ x – 2 = ±1
⇒ x = 3,1