08-12-2024

9610ZJM801335240013 JM

PART 1 : PHYSICS

SECTION-I

1) Two identical conducting rods are first connected independently to two vessels, one containing
water at 100°C and the other containing ice at 0°C. In the second case, the rods are joined end to
end and connected to the same vessels. Let q1 and q2 g/s be the rate of melting of ice in the two
cases respectively. The ratio q2/q1 is

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

2) The power radiated by a black body is P and it radiates maximum energy around the wavelength
λ0. If the temperature of the black body is now changed so that it radiates maximum energy around
wavelength 3/4λ0, the power radiated by it will increase by a factor of :-

(A) 4/3
(B) 16/9
(C) 64/27
(D) 256/81

3) An ideal gas mixture filled inside a balloon expands according to the relation PV2/3 = constant. The
temperature inside the balloon is

(A) increasing
(B) decreasing
(C) constant
(D) can’t be said

4) One mole of monoatomic ideal gas undergoes a cyclic process ABCA as shown in figure. Process
BC is adiabatic. The temperatures at A, B and C are 300, 600 and 450K respectively. Choose the
correct statement(s).

(A) In process CA change in internal energy is 225R.

(B) In process AB change in internal energy is –150R.
(C) In process BC change in internal energy is –225R.
(D) Change in internal energy during the whole cyclic process is +150R.

5) A liquid in a beaker has temperature θ(t) at time t and θ0 is temperature of surroundings, then
according to Newton's law of cooling the correct graph between loge (θ – θ0) and t is :

(A)

(B)

(C)

(D)

6) A refrigerator freezes 5 kg of water at 0°C into ice at 0°C in a time interval of 20 minutes. Assume
that room temperature is 20°C. Calculate the minimum power needed to accomplish it ?

(A) 24.4 Watt

(B) 0.1025 Watt
(C) 0.0244 Watt
(D) 102.5 Watt

7) Two rods of length L1 and L2 are made of material whose coefficient of linear expansion are α1 and
α2. If the difference between the two length is independent of temperature :

(A)

(B)

(C)
(D)

8) The pressure that has to be applied to the ends of a steel wire of length 10 cm to keep its length
constant when its temperature is raised by 100°C is :
(For steel Young's modulus is 2 × 1011 N m–2 and coefficient of thermal expansion is 1.1 × 10–5 K–1)

(A) 2.2 × 107 Pa

(B) 2.2 × 106 Pa
(C) 2.2 × 108 Pa
(D) 2.2 × 109 Pa

9) Find the pressure P, in the diagram as shown of monoatomic gas of one mole in the process ABC

if .

(A)

(B) 2P0

(C)

(D)

10) Which of the following is correct for the molecules of a gas in thermal equilibrium ?

(A) All have the same speed

(B) All have different speeds which remain constant
(C) They have a certain constant average speed
(D) They do not collide with one another.
11) A gas follows VT2 = const. Its volume expansion coefficient will be :-

(A)

–
(B)

(C)

–
(D)

12) When unit mass of water boils to become steam at 100°C. it absorbs Q amount of heat. The
densities of water and steam at 100°C are ρ1 and ρ2 respectively and the atmospheric pressure is p0.
The increase in internal energy of the water is

(A) Q

(B)

(C)

(D)

13) An ideal gas undergoes a cycle ABCA in which its pressure, P, and volume, V, are indicated in
the P-V diagram shown. It is known that the area under the P-V graph is the work done by the gas
during expansion. Find the net work done by the gas in the cycle ABCA.

(A) 6000 J
(B) 4000 J
(C) 3000 J
(D) 2000 J

14) I-V characteristic of a diode is shown in figure.

 If this diode is connected across a battery of
emf 10 V and a resistance of 12 kΩ in series, the current through diode in forward bias is nearly :

(A) 0.4 mA
(B) 0.83 mA
(C) 0.8 mA
(D) 1.2 mA

15) For the given logic, what will be Y when A and B both are high (equal to 1) and both are low

(equal to 0).

(A) 0, 0
(B) 1, 1
(C) 0, 1
(D) 1, 0

16) For the circuit shown below the current through zener diode is :-

(A) 14 mA
(B) zero
(C) 9mA
(D) 5mA

17) The combination of gates shown below yields :

(A) NAND gate

(B) NOT gate
(C) NOR gate
(D) OR gate

18) In the given figure, each diode has a forward bias resistance of 30Ω and infinite resistance in

reverse bias. The current I1 will be :

(A) 3.75 A
(B) 2.35 A
(C) 2 A
(D) 2.73 A

19) In the half-wave rectifier circuit shown. Which one of the following wave forms is true for VCD, if
the input is as shown?

(A)

(B)

(C)

(D)

20) Identify the semiconductor devices whose characteristics are given below, in the order (a), (b),
(c), (d) :

(A) Zener diode, Solar cell, Simple diode, Light dependent resistance
(B) Simple diode, Zener diode, Solar cell, Light dependent resistance
(C) Zener diode, Simple diode, Light dependent resistance, Solar cell
(D) Solar cell, Light dependent resistance, Zener diode, Simple diode

SECTION-II

1)

In a semiconductor, the number density of intrinsic charge carriers at 27°C is 1.5 × 1016/m3. If the
semiconductor is doped with impurity atom, the hole density increases to 4.5 × 1022/m3. The electron
density in the doped semiconductor is _______ × 109/m3.

2) A diode made forward biased by a two volt battery however there is a drop of 0.5 V across the
diode which is independent of current. Also a current greater then 10 mA produces large joule loss
and damages diode. If diode is to be operated at 5 mA, the series resistance to be put is .

Find the value of n.

3) An ideal gas is enclosed in a container as shown in figure. The spring constant is 100 N/m, the
area of the piston is 1 cm2 and the length of the gas column is 100 cm. The compression of the
spring is 10 cm. There is no friction between the walls and the piston. The whole system is heated to
make the temperature 2.4 times the original temperature. Find the distance moved by the piston (in
cm). Take atmospheric pressure = 105 Pa. If your answer is N fill the value of N/5.

4) Emissivity e is a property of surface. Suppose for a surface, emissivity e varies with kelvin temp. T
as e = CT (C is constant). If energy emission rate at temperature, 800 K from the surface is 64 W,
what will be the energy emission rate (in watt) at 400 K ?
5) The specific heat of a substance varies with temperature according to c = 0.2 + 0.16 T + 0.024 T2
with T in °c and c is cal/gK. Find the energy (in cal) required to raise the temp of 2g substance from
0° to 5°C

PART 2 : CHEMISTRY

SECTION-I

1) In the following conversion of sulphide of phosphorous P4S3 → P2O5 + SO2 Equivalent weight of
P4S3 (molecular weight = M) is :

(A)

(B)

(C)

(D)

2) Calculate the mass of anhydrous oxalic acid, which can be oxidised to CO2 (g) by 100 ml of an
MnO¯4 solution, 10 ml of which is capable of oxidising 50 ml of 1N I– to I2

(A) 45 gm
(B) 22.5 gm
(C) 30 gm
(D) 12.25 gm

3) An impure sample of KClO3 of 50% purity on decomposition produces 67.2 litre oxygen at 0ºC and
1 atm. The other product of decomposition is KCl. The initial mass of impure original sample (in gm)
taken is.

(A) 245
(B) 122.5
(C) 490
(D) None of these

4) Difference in wavelength of two extreme 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

5) The correct graph of ψ2(r)4πr2dr for 4s is

(A)

ψ2(r)4πr2dr
ψ2(r)4πr2dr

(B)

ψ2(r)4πr2dr

(C)

(D)

ψ2(r)4πr2dr

6) An azeotropic mixture of two liquid has a boiling point higher than either of them when it:-

(A) show positive deviation from Raoult's law

(B) shows negative deviation from Raoult's law
(C) show ideal behaviour
(D) is saturated

7) When 20 g of naphtholic acid (C11H8O2) is dissolved in 50 g of benzene (Kf = 1.72 K kg mol–1), a

freezing point depression of 2 K is observed. The Van't Hoff Factor (i) is

(A) 0.5
(B) 1
(C) 2
(D) 3

8) Find EMF of Cell

Pt | |H+(pH = 5) || H+ (pH = 7) | | Pt

given that

(A) 6
(B) –0.06 volt
(C) –0.12 volt
(D) 7
9) Given that

If Fe2+(aq), Fe+3(aq) and Fe solid are kept together then

(A) Conc. of Fe+3 increases

(B) Conc. of Fe+3 decreases
(C) Conc. of Fe+2, Fe+3 remains unchanged
(D) Conc. of Fe+2 decreases.

10) 75% of a first order reaction was completed in 32 minutes. When was 50% of the reaction
completed?

(A) 16 minutes
(B) 24 minutes
(C) 8 minutes
(D) 4 minutes

11)

For a zero order reaction

A → B ; K = 0.025 Ms–1
If 2M of A is taken initially the concentration of A after 40 sec is -

(A) 0.5M
(B) 1 M
(C) 0.25 M
(D) 2M

12) Select the correct option -

(A) Specific volume and molar heat capacity are extensive properteis
(B) Change in internal energy for an ideal gas for an isochoric process is zero
(C) Thermodynamics can predict rate at which process will takes place
(D) Free expansion is an irrereversible process.

13) Molar heat capacity of water in equilibrium with ice at constant pressure is

(A) zero
(B) Infinity
(C) 40.45 kJ K–1 mol–1
(D) 75.48 kJ K–1 mol–1

14) Which of the following thermodynamic properties must be associated with a reaction found to be
spontaneous at high temperatures but not spontaneous at low temperatures.

(A) ΔH < 0, ΔS < 0

(B) ΔH < 0 , ΔS > 0
(C) ΔH > 0 , ΔS > 0
(D) ΔH > 0 , ΔS < 0

15) (NH4)2CO3(s) 2NH3(g) + CO2(g) + H2O(g)

The value of Kp for above equilibrium is 64 atm4. The equilibrium pressure of the system is

(A) 4 atm
(B) 8 atm
(C) 2 atm
(D) 6 atm

16) For the equilibrium,

2H2O H3O⊕ + OH⊝, the value of ΔG° at 298 K is approximately:-

(A) –80 kJ mol–1

(B) –100 kJ mol–1
(C) 100 kJ mol–1
(D) 80 kJ mol–1

17) 2.0 mol of PCl5 were introduced in a vessel of 5.0 L capacity at a particular temperature. At
equilibrium, PCl5 was found to be 35% dissociated into PCl3 and Cl2. The value of Kc for the reaction
is:

(A) 1.89
(B) 0.377
(C) 0.75
(D) 0.075

18)

The equilibrium 2SO2(g) + O2(g) ⇌ 2SO3(g) shifts forward if :-

(A) A catalyst is used.

(B) An adsorbent is used to remove SO3 as soon as it is formed.
(C) Small amounts of reactants are removed.
(D) None of these

19) Solubility of Ca3(PO4)2 (Ksp = 10–13) in pure water is nearly-

(A) 10–3M
(B) 10–5M
(C) 10–6M
(D) 10–8M

20) At 25°C, Ka of weak monobasic acid, HA, is 5 × 10–6. Kb of A– is -

(A) 5 × 10–6
(B) 2 × 105
(C) 2 × 10–9
(D) 5 × 10–8

SECTION-II

1) Photon having energy equivalent to the binding energy of 4th state of He+ atom is used to eject an
electron from the metal surface of work function 1.4 eV. If electrons are further accelerated through
the potential difference of 4V then the minimum value of De–broglie wavelength associated with the
electron is (in Å)

2) If a compound AB dissociates to the extent of 75% in an aqueous solution, the molality of the
solution which shows a 2.5 K rise in the boiling point of the solution is ______ molal. (Rounded-off to
the nearest integer)
[Kb = 0.52K kg mol–1]

3) Calculate Ecell for following

M | M2⊕ (0.001 M) || M2⊕ (0.01 M) | M

[Given: = 0.06]
Fill your answer by multiplying it with 100.

4) For the reaction : 3A (g) → 2B(g), the rate of formation of 'B' at 298 K, is represented as

. The order of reaction is-

5) The valve on a cylinder containing initially 1 litres of an ideal gas at 7 atm and 25°C is opened to
the atmosphere, Whose pressure is 760 torr and the temperature is 25°C. Assuming that the process
is isothermal, how much work (in L. atm) is done (magnitude) on the atmosphere by the action of
expansion ?

PART 3 : MATHEMATICS

SECTION-I

1) Four common tangents of ellipse and hyperbola make a parallelogram

whose area is

(A)

(B)
(C) 17
(D) 34

2) A hyperbola passes through the foci of the ellipse and its transverse and conjugate
axes coincide with major and minor axes of the ellipse, respectively. If the product of their
eccentricities in one, then the equation of the hyperbola is:

(A)

(B)

(C)

(D)

3) The vertices of a hyperbola are at (0, 0) and (10, 0) and one of its foci is at (18, 0). The equation of
the hyperbola is

(A)
=1

(B)
=1

(C)
=1

(D)
=1

4) Locus of the point of intersection of the tangents at the points with eccentric angles ϕ and -

ϕ on the hyperbola = 1 is :

(A) x = a
(B) y = b
(C) x = ab
(D) y = ab

5) If the hyperbola cuts the circle at points

and , then the value of is equal to

(A) –8
(B) –9
(C) –10
(D) None of these
6) The normal to hyperbola xy = 4 at point P(t) i.e. meets the curve again at point

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

7) Let 9 = x1 < x2 < …< x7 be in an A.P. with common difference d. If the standard deviation of x1, x2
…, x7 is 4 and the mean is then is equal to :

(A)

(B) 34

(C)

(D) 25

8) Let observations such that and Then a possible value of n

amount the following is

(A) 9
(B) 12
(C) 15
(D) 18

9) Let μ be the mean and σ be the standard deviation of the distribution

where if [x] denotes the greatest integer ≤ x, then [μ2 + σ2] is equal

(A) 8
(B) 7
(C) 6
(D) 9

10) Let S be the set of all values of a1 for which the mean deviation about the mean of 100
consecutive positive integers a1, a2, a3, ….., a100 is 25. Then S is

(A) ϕ
(B) {99}
(C)
(D) {9}
11) Let the observations xi(1 ≤ i ≤ 10) satisfy the equations, and .
If µ and λ are the mean and the variance of the observations, x1 – 3, x2 – 3,....,x10 – 3, then the ordered
pair (µ, λ) is equal to:

(A) (6, 6)
(B) (3, 6)
(C) (6, 3)
(D) (3, 3)

12) The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of
these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new
mean and new s.d. become half of their original values, then q is equal to

(A) –20
(B) 10
(C) –10
(D) –5

13) Consider complex numbers z, w satisfying , if |z| = 1, then w will move -

(A) on a straight line parallel to real-axis

(B) on a straight line parallel to imaginary-axis

(C)
on a circle centred at
(D) on a pair of straight lines passing through origin

14) If |z – 1| = 2 and |w – i| = 3, where then the maximum value of |z – w| is then

the value of a is

(A) 4
(B) 6
(C) 5
(D) 7

15)

If z1 satisfies the equation |z – 3| = 4 and z2 satisfies the equation |z – 1| + |z + 1| = 3. If m = |z1 –

z2|min and M = |z1 – z2|max, then (m + 2M) is equal to

(A) 18
(B) 17
(C) 19
(D) 16
16) For which of the following holds good?

(A)

(B)
(C)
(D)

17) If , a = P + P2 + P4 and b = P3 + P5 +P6, then the value of

is-

(A)
(B)
(C)
(D) 2

18) Let A(z1), B(z2) and C(z3) be complex numbers, satisfying the equation |z| = 1 and also satisfying
the relation 3z1 = 2z2 + 2z3. Then |z2 – z3| is -

(A)

(B)

(C)

(D)

19)

If z lies on the curve arg(z – 1) = then maximum value of is-

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

20) If represent adjacent vertices of a regular polygon of n sides with centre at the origin & if

then the value of n is equal to:

(A) 8
(B) 12
(C) 16
(D) 24

SECTION-II

1) The mean square deviation of a set of n observations about a point c is defined as

The mean square deviation about -2 and 2 are 18 and 10 respectively, then standard
deviation of this set of observations is :

2) Let X1, X2, ..., X18 be eighteen observations such that and ,
where α and β are distinct real numbers. If the standard deviation of these observations is 1, then
the value of |α – β| is ______.

3) If are different complex number with then is equal to

4) Let z be a complex number having argument such that and satisfying equation

then is equal to
(Where |.| represent modulus)

5) The tangent at point P on the hyperbola = 1 passes through the point (0, –b) and the

normal at point P passes through the point . If e denote the eccentricity of hyperbola then
find the value of e2.
 ANSWER KEYS

PART 1 : 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. D D A C B D B C A C B B D C A B D C B B

SECTION-II

Q. 21 22 23 24 25
A. 5 3 4 2 8

PART 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 D D B A C B A B D B C B D D B A C

SECTION-II

Q. 46 47 48 49 50
A. 5 3 3 2 6

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

SECTION-II

Q. 71 72 73 74 75
A. 3 4 1 1 2
 SOLUTIONS

PART 1 : PHYSICS

1)

2) ⇒ …(i)
…(ii)
…(iii)

3)
V↑ ⇒ T↑

4)

dU = nCVdT dTAB = 300

= dTBC = – 150
TCA = –150
dUcycle = 0

5) Newtons law of cooling

⇒
Integrating
ln(θ – θ0) = – kt + C

6)
Q2 = mL = 5 × 103 × 80 × 4.2
W=
Q1 = Q2 + W

P1 =

7) ΔL = L2 – L1
L2 – L1 = L2(1 + α2Δt) – L1(1 +α1 Δt)
O = L2α2Δt – L1α1Δt
⇒ L2α2 = L1α1

8)

9) A → B B→C
W = P0V0 W = 0 (Isochoric)

ΔU = ΔU =
= 3V0(P – P0)

ΔUNet =

⇒P=

10) Vav = , as T = constant

∴ Vav = constant

11)

VT2 = constant
12) ...(1)

...(2)
ΔQ = Q [given] ...(3)
by 1,2 & 3

13)

W.D. area under curve

14) Cut in voltage of diode VC = 0.4 V

(Forward resistance is very small in comparison to R)

I = 0.8 mA

15)

16) here the maximum voltage across zener diode can be 80 V if whole current goes through
10 kΩ but zener diode conducts when its voltage is 90 V.
So current through it is zero.
17)

18)
As per diagram,
Diode D1 & D2 are in forward bias i.e. R = 30Ω
whereas diode D3 is in reverse bias i.e. R = infinite
⇒ Equivalent circuit will be
Applying KVL starting from point A

⇒ – 100 I1 + 200 = 0
I1 = 2
Option (3)

19) In reverse bias, current will be blocked by diode.

20) Zener diode works in breakdown region

So simple diode → (a), Solar cell → (c)
Zener diode → (b), Light dependent resistance → (d)

21) nenh = ni2

5 × 109 /m3

22)

23)

0
pi = p + = 2 × 105 P
pf = 2 × 105 + 106 x
vi = 1A
vf = (1 + x)A

= const.

= × (1 + x)A

2.4 =
2.4 = (1 + 5x) (1 + x)
2.4 = 1 + 5x2 + 6x
5x2 + 6x – 1.4 = 0

x= = 0.2 m = 20 cm

24) Using steafan's law

Energy radiate = σeT4
ΔET = σCT5 e = CT
 25)

dQ = mcdT

Q=

Q= dT
Q = 2 [0.2 × 5 + 0.08 × 25 + 0.008 × 125] = 2[1 + 2 + 1] = 8 cal

PART 2 : CHEMISTRY

26)

27) Let normality of KMnO4– solution is N

∴ N × 10 = 50 × 1 ⇒ N = 5
milli equivalant of MnO4– = milli equivalant of oxalic acid

5 × 100 = × 1000 ⇒ × 1000 = 5 × 100

⇒ w = 22.5 gram.

28) KClO3 → KCl + O2

Let, mass taken of KClO3 at initial = m gram

decomposed mass of KClO3 = gm

moles of KClO3 =

moles of O2 produced =

m = 490 gm

29)

λ – λ' =
30) ψ2(r)4πr2dr

31)

Solutions which shows large negative deviation from Raoult's Law form maximum boiling
azeotropes.

32) ΔTf = i × Kf × m

2 = i × 1.72 ×

2 = i × 1.72 ×

i=

33)

34) Fe and Fe+3 form galvanic cell

Fe|Fe+2||Fe+2, Fe+3|

= 0.77 – (–0.44) = 1.21

Fe, Fe+3 decreasing but Fe+2 increases = 1.21 V (spantanous)

35)

36) A → B
a 0
a–x x
X = Kt = 0.025 × 40 = 1M
∴ a – x = 2 – 1 = 1M

37) Free expansion is an irrereversible process. (correct statement)

38) By definition
For ,
temperature does not change if some heat is given to the system. Hence
39) ΔG = ΔH – TΔS

40) (NH4)2CO3(s) 2NH3(g) + CO2(g) + H2O(g)

2P P P
2
KP = (2P) . P.P = 64 atm
P = 2 atm
Ptotal = 2P + P + P = 8 atm

41)

ΔG = –RT ln K
ΔG° = –2.303 RT log K
ΔG° = –2.303 × 8.314 × 298 log (10–14)
ΔG° = 80000 J/mol = 80 kJ/mol

42)

43)

Product is removed , reaction moves forward

44) Ksp = 108 S5

S=

45)

Ka x Kb = Kw

46)
 Total energy = = = 3.4 eV
Now K.E. = 3.4 – 1.4 = 2 eV
Now, Total energy = 2 + 4 = 6 eV i.e. potential = 6 V

For electron, λ = so λ = 5 Å.

47) α = 0.75, n = 2
i = 1 – a + na = 1 – 0.75 + 2 × 0.75 = 1.75
ΔTb = ikbm
or, 2.5 = 1.75 × 0.52 × m

or, m =
∴ nearest integer answer will be 3

48) For concentration cell, =0

Ecell = 0 –
Ecell = 0.03 V
= 100 × Ecell
=3V

49)

or, = ln k + n ln [A] ⇒

50)

T = Constant
P1V1 = P2V2

7×1= × V2 ⇒ V2 = 7 litre
W = –Pext (ΔV)
= –1 × (7–1)
= – 6 L.atm
W = – 6 L. atm
Magnitude of work = 6 atm. L

PART 3 : MATHEMATICS

51) Let m is the slope of common tangent

52) For ellipse

for hyperbola
Let hyperbola be

∵ it passes through (3,0) ⇒

⇒

∴ Hyperbola is

53) Centre of hyperbola ≡ (5, 0)

∴ 2a = 10
∴ ae =13
b2 = a2 e2 – a2
b2 = 169 – 25
∴ b2 = 144

– =1

54)
Tangent at ϕ ,

at
∴ (bsecϕ)h – (a tanϕ)k = ab
(bcosecϕ)h – (a cotϕ)k = ab

K=– = =– =b

55)

We know that if a circle cuts a rectangular hyperbola, then arithmetic mean of points of

intersections is the mid-point of centre of hyperbola and circle. So,

56)

Slope of normal

Passes through .

Thus,

57) 9 = x1 < x2 < …….< x7

9, 9 + d, 9 + 2d, ………9 + 6d
0, d, 2d, …….6d

16 = 4d2
d2 = 4
d=2
= 6 + 9 + 10 + 9

58) Since, root mean square arithmetic mean

Hence, possible value of n = 18.

59)
⇒ 3k2 + 16k – 12k – 64 = 0

⇒ k = or (rejected)

60) let a1 be any natural number

are values of

Mean deviation about mean

= 25
So, it is true for every natural no.

61)
 ⇒ Mean of observation
⇒ µ = mean of observation (xi – 3)
= (mean of observation (xi – 5)) + 2
=1+2=3
Variance of observation

– (Mean of (xi – 5))2 = 3

⇒ λ = variance of observation (xi – 3)
= variance of observation (xi – 5) = 3
∴ (µ, λ) = (3, 3)

62) 20p – q = 10 ...(i)

and 2|p| = 1 ⇒ p = ± ...(ii)

So, p = – and q = –20

63) , |z| = 1

⇒ |w| = |w – 1|
⇒ w lies on perpendicular
bisector of line joining

(0,0) and (1,0) i.e.

64)

Max. |z – w| = c1c2 + r1 + r2

where c1,c2, r1,r2 are the centres & radius of the given two circles.

65)

|z1 – z2|min = 0

(∴ curves are intersecting)

⇒m=0
& |z1 – z2|max = AC

∴ m + 2M = 17.

66)
Similarly

67)

i.e.

ab = (P + P2 + P4) (P3 + P5 + P6)

= P4 + P6 + P7 + P5 + P7 + P8 + P7 + P9 + P10
= P4 + P6 + P5 + P8 + P9 + P10 + 3
= P4 + P6 + P5 + P + P2 + P3 + 3 = 2

68)

A, B, C lies on unit circle centered at origin. Also

Let DE = 3k & EA = K

4K = 1 ⇒ BE = ⇒ |z2 – z3| = BC =
69)
Take the image of B in line y = x – 1 say Q.

∴ Max|PA – PB| = |PA – PQ| = 2

70)
Let

Let ' ' be angle subtended by side at centre (origin)

71)
and
and
and
and

and
72)

Hence

Given
⇒ 90 – 18β2 + 36β(α + 2) – 18(α + 2)2 = 18
⇒ 5 – β2 + 2αβ + 4β – α2 – 4α – 4 = 1
⇒ (α – β)2 + 4(α – β) = 0 ⇒ |α – β| = 0 or 4
As α and β are distinct |α – β| = 4

73)

74)

⇒ |z – 5i| = 5
OP = OQ sinθ
⇒ |z| = 10 sinθ
z = 10sinθ (cosθ + i sinθ)

75) The equation of tangent at P(x, y) is =1

It passes through (0, –b), so 0 + = 1 ⇒ y1 = b

Also, equation of normal at P(x1, y) is = a2e2

It passes through , so

= a2e2

⇒ x1 = So, P
 As P(x1, y1) lies on the hyperbola = 1, so =1

⇒ = 2 ⇒ e4 = 4 ⇒ e2 = 2 Ans.