14-03-2023

1001CJM203322012 JM

PART-1 : PHYSICS

SECTION-I

1) A plano-convex lens of refractive index 1.5 and radius of curvature 30 cm is silvered at the curved
surface. Now, this lens has been used to form the image of an object. At what distance from this
lens, an object be placed in order to have a real image of the size of the object?

(A) 20 cm
(B) 30 cm
(C) 60 cm
(D) 80 cm

2) Young's double slit experiment is first performed in air and then in a medium other than air. It is
found that 8th bright fringe in the medium lies where 5th dark fringe lies in air. The refractive index of
the medium is nearly :-

(A) 1.59
(B) 1.69
(C) 1.78
(D) 1.25

3) A wave is given by

y = 3sin2π where y is in cm. Frequency of wave and maximum acceleration of the

particle will be :-

(A) 25 Hz, 4.7 × 103 cm/s2

(B) 50 Hz, 7.5 × 103 cm/s2
(C) 25 Hz, 4.7 × 104 cm/s2
(D) 25 Hz, 7.4 × 104 cm/s2

4) The driver of a car travelling with speed 30 m/sec towards a hill, sounds a horn of frequency 600
Hz. If the velocity of sound in air is 330 m/s, the frequency of reflected sound as heard by driver is :

(A) 500 Hz
(B) 550 Hz
(C) 555.5 Hz
(D) 720 Hz

5) A block of mass m lying on a rough horizontal plane is acted upon by a horizontal force P and
another force Q inclined at an angle θ to the vertical. The block will remain in equilibrium if the
coefficient of friction between it and the surface is :-

(A)

(B)

(C)

(D)

6) Three stones A, B and C are simultaneously projected from same point with same speed. A is
thrown upwards, B is thrown horizontally and C is thrown downwards from a building. When the
distance between stone A and C becomes 10m, then distance between A and B will be :-

(A) 10 m
(B) 5 m
(C) m
(D) m

7) A ball of mass m moving horizontally at a speed u collides with the bob of a simple pendulum at

rest. The mass of the bob is . If the collision is perfectly inelastic the height to which the composite
ball rise after the collision is :-

(A)

(B)

(C)

(D)

8) Two coaxial discs, having moments of inertia I1 and , are rotating with respective angular

velocities ω1 and , about their common axis. They are brought in contact with each other and
thereafter they rotate with a common angular velocity. If Ef and Ei are the final and initial total
energies, then (Ef – Ei) is :

(A)
(B)

(C)

(D)

9) A wheel having mass m has charges +q and –q on diametrically opposite points. It remains in
rotational equilibrium on a rough inclined plane in the presence of uniform vertical electric field E

(A)

(B)

(C)

(D) None

10) A transformer is used to light a 55 W and 110V lamp from a 220 V mains. If the main current is
0.5 amp, the efficiency of the transformer (in %) is :-

(A) 50
(B) 40
(C) 60
(D) 80

11) The magnetic field in a certain region of space is given by = 8.35 × 10–2 T. A proton is shot
into the field with velocity
= (2 × 105 + 4 × 105 ) m/s. The proton follows a helical path in the field. The distance moved by
proton in the x-direction during the period of one revolution in the yz-plane will be
(Mass of proton = 1.6 × 10–27 kg)

(A) 0.053 m
(B) 1.57 m
(C) 0.157 m
(D) 15.7 m

12) In the circuit shown in figure, the battery is an ideal one with emf V. The capacitor is initially

uncharged. Switch S is closed at time t = 0. The final charge Q on the

capacitor is :

(A)

(B)

(C) CV

(D)

13) A 110 V, 60 W lamp is run from a 220 V AC mains using a capacitor in series with the lamp,
instead of a resistor then the voltage across the capacitor is about:-

(A) 110 V
(B) 190 V
(C) 220 V
(D) 311 V

14) In an α-decay kinetic energy of α particle is 98 MeV and Q-value of the reaction is 100 MeV. The
mass number of the mother nucleus is. (Assume that daughter nucleus is in ground state):-

(A) 300
(B) 400
(C) 200
(D) 100

15) A proton and an α-particle are initially at a distance 'r' apart. Find the K.E. of α-particle at a
large separation from proton after being released

(A)

(B)

(C)

(D)

16) In a given circuit diodes given are ideal, find current drawn from battery :

(A) 5
(B) 2
(C) 8
(D) 9

17) A silver ball of radius 4.8 cm is suspended by a thread in the vacuum chamber. UV light of
wavelength 200 nm is incident on the ball for some times during which a total energy of 1 × 10–7J
falls on the surface. Assuming on an average one out of 103 photons incident is able to eject electron.
The potential on sphere (in volt) will be

(A) 8
(B) 7
(C) 6
(D) 3

18) Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole
of nitrogen at temperature T0, while box B contains one mole of helium at temperature (7/3) T0. The
boxes are then put into thermal contact with each other, and heat flows between them until the
gases reach a common final temperature (Ignore the heat reach a capacity of boxes). Then, the final
temperature of the gases, Tf, in terms of T0 is :

(A)

(B)

(C)

(D)

19)

In the following figures heat is absorbed by the gas :-

(A)

(B)

(C)

(D)

20) The figure shows the P-V plot of an ideal gas taken through a cycle ABCDA. The part ABC is a
semicircle and CDA is half of an ellipse. Then,

(A) The process during the path A→B is isothermal

(B) Heat is absorbed by the gas during the path B→C→D
(C) Work done during the path A→B→C is zero
(D) Positive work is done by the gas in the cycle ABCDA

SECTION-II

1) A ray of light is incident at an angle of 60° on one face of a prism of angle 30°. The emergent ray
of light makes an angle of 30° with incident ray. The angle made by emergent ray with second face
of prism will be :-

2) Two beams, A and B, of plane polarized light with mutually perpendicular planes of polarization
are seen through a polaroid. From the position when the beam A has maximum intensity (and beam
B has zero intensity), a rotation of polaroid through 60° makes the two beams appear equally bright.

If the initial intensitites of the two beams are IA and IB respectively, then equals :

3) The maximum speed and acceleration of a particle executing SHM are 10 cm/s and 50 cm/s2. x is
the position of the particle from mean position where its speed is 8 cm/s is (in cm) then 10x is :-

4) The mass of planet is 1/9 of the mass of the earth and its radius is half that of the earth. If a body
weight 9 N on the earth. Its weight on the planet would be :-

5) The bulk modulus of rubber is 9.1 × 108N/m2. To what depth a rubber ball be taken in a lake so
that its volume is decreased by 0.1%?

6) A linear charge having linear charge density λ, penetrates a cube diagonally and then it penetrate

a sphere diametrically as shown. The ratio of flux coming cut of cube and sphere will be then x =

? :-
7) In given circuit charge on 1μF in steady state will be (in µC) :-

8) After five half lives percentage of original radioactive atoms left is , find x :

9) The maximum peak to peak voltage of an AM wave is 24 mV and the minimum peak to peak
voltage is 8 mV. The modulation factor is :–

10) 10 gm of ice at –5ºC is added to 10 gm of water at 60ºC. Specific heat of water = 1 cal/gm-ºC;
specific heat of ice = 0.5 cal/gm-ºC and latent heat of ice = 80 cal/gm. The resulting temperature of
the mixture is :-

PART-2 : CHEMISTRY

SECTION-I

1)

In which of the following molecule all bond length are equal :-

(A) SF4
(B) B2H6
(C) PCl5
(D) SiF4

2) The IUPAC name for the complex, [Co(NH3)2Cl(ONO)(Py)2]NO3 is ?

(A) Diamminechlorido dipyridinenitrito-N-cobaltate (III) nitrate

(B) Diammine chloridonitrito-O-bipyridine cobalt (III) nitrate
(C) Diamminechloridonitrito-O-dipyridine cobalt (III) nitrate
(D) Chloridodiamminenitrito-O-bis (pyridine) cobalt (III) nitrate

3) Which of the following statements is false ?

(A) has a Cr – O – Cr bond

(B) is tetrahedral in shape
(C) K2Cr2O7 is less soluble in water than Na2Cr2O7
(D) In , there are 8 equivalent Cr – O bonds are present.
4) In reaction, Zn + NaOH → X, the product X is :-

(A) Na2ZnO2
(B) 2NaZnO2
(C) Zn(OH)2
(D) None of these

5) Which of the following is not correctly matched?

(A) ZnCO3 → Zn (Calcination followed by smelting)

(B) PbS → Pb (Partial roasting followed by self-reduction)
(C) Cu2S → Cu (Complete roasting followed by carbon reduction)
(D) Al(OH)3 → Al (Calcination followed by smelting)

6) When SbF5 reacts with XeF4 then determine hybridisation of central atom in cationic part of
product ?

(A) sp3
(B) sp3d
(C) sp3d2
(D) sp3d3

7) Which of the following oxide is different than other ?

(A) SO2
(B) Cl2O7
(C) Pb3O4
(D) Mn2O7

8)
Structure of B will be :

(A)

(B)
(C)

(D)

9) Which of the given following stability order is correct ?

(A)

(B)

(C)

(D)

10) Give the product of the following reaction sequence-

(A)

(B)

(C)

(D)

11) Which of the following molecules will not show optical activity ?

(A)
(B)

(C)

(D)

12) Which one of the following statements is not true regarding (+) Lactose?

(A) (+) Lactose, C12H22O11 contains 8-OH groups

(B) On hydrolysis (+) Lactose gives equal amount of D(+) glucose and D(+) galactose
(+) Lactose in a β,1:4-glycoside formed by the union of a molecule of D(+) glucose and a
(C)
molecule of D(+) galactose
(D) (+) Lactose is reducing sugar and does not exhibit mutarotation

13) The major product of the following reaction is

(A)

(B)

(C)

(D)

14) Identify C in the reactions :

(A)

(B)

(C)

(D) None of these

15) Arrange the following wavelengths (λ) of given emission lines of H atoms in increasing order
(a) (b)
(c) n = 5 n = 3 (d) n = 22 n = 20
Choose the correct option.

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

16) x ⇌ 2y KP
P ⇌ Q + R KP'

If degree of dissociation of x and P are same and KP = 2KP' then the ratio of total pressure =?

(A) 0.5
(B) 2
(C) 0.25
(D) 4

17)

If initially 100 gm of C14 is taken then. Find the amount of Radioactive isotope C14

( = 5730 years) after the storage of 22920 years.

(A) 3.12 gm
(B) 12.5 gm
(C) 6.25 gm
(D) None of these
18) The ionization constant of a weak acid is
1.6 × 10–5 and the molar conductivity at infinite dilution is 380 × 10–4 Sm2mol–1. If the cell constant is
0.01 m–1 then conductance of 0.01 M acid solution is :-

(A) 1.52 × 10–5 S

(B) 1.52 S
(C) 1.52 × 10–3 S
(D) 1.52 × 10–4 S

19) Pt | H2 | H+ || MnCu– | Mn+2, H+ | Pt

(g) aq. aq. aq. aq.
–3
0.1 atm 10 M 0.1 M 0.01M 0.01 M
Calculate the cell emf.

(A) 1.54
(B) 1.84
(C) 1.48
(D) 1.45

20) A buffer that is a mixture of acetic acid

(Ka = 2 × 10–5) and potassium acetate has pH = 5.18. The ratio in this buffer is approx
:-

(A) 1 : 1
(B) 3 : 1
(C) 5 : 1
(D) 1 : 3

SECTION-II

1) How many of the following pairs are correctly matched ?

(a) C-Reduction Process ⇒ Iron ore
(b) Self-Reduction Process ⇒ Lead ore
(c) Al-Reduction Process ⇒ Bauxite ore
(d) Electrolytic reduction Process ⇒ Sodium Salt

2) Write the no. of electrons possible with n = 5,

l = 3, s = –½.

3) One mole of Mg3N2 react with excess of water to form x mole of Ammonia and One mole of LiH

react with H2O to give y mole of H2 find =

4)
the value of x is :-

5) How many chiral carbon atoms are present in the following compound?

6) How many compounds gives chloroform with Cl2, NaOH ?

, CH3–CH2–OH , ,

, , ,

, , CH3–OH

7) The vapour pressure of CS2 at 50°C is 854 torr and a solution of 2.0 gm sulphur in 100 gm of CS2
has vapour pressure 848.9 torr. If the formula of sulphur molecule is Sn, then calculate the value of
"n" (approx)
(At. mass of S = 32) :-
[Assume very dilute solution]

8) 2.0 g sample contain mixture of SiO2 and Fe2O3, on very strong heating leave a residue weighing
1.96 g. The reaction responsible for loss of weight is Fe2O3(s) → Fe3O4(s) +O2(g), (unbalance
equation) What is the percentage by mass of SiO2 in original sample?
[Fe = 56, Si = 28]

9) Given the following bond enthalpies :

BE(N ≡ N) = 942 kJ/mol ;
BE(H – H) = 436 kJ/mol ;
BE(N – N) = 163 kJ/mol ;
BE(N – H) = 390 kJ/mol
Determine enthalpy change for the following polymerisation reaction per mole of N2(g) consumed -
nN2(g) + nH2(g) → –(– NH – NH –)n–
10) For the reaction at 300 K
A(g) + B (g) → C (g)
ΔU = –3.0 kcal ; ΔS = –10.0 cal/K
value of ΔG is ?

PART-3 : MATHEMATICS

SECTION-I

1) If ,
then the value of determinant

(A) 65
(B)

(C)

(D) 0

2) If
and det(A) = det(4I), where I is 3 × 3 identity matrix, then (a – b)3 + (b – c)3 + (c – a)3 can be equal to
-

(A) -24
(B) 26.5
(C) -27.25
(D) 22.5

3) Let and b denotes the

coefficient of x49 in (x – a1) (x – a2) .......

(x – a50), find

(A) 1200
(B) 1300
(C) 1400
(D) None
4) The number of 5-digit numbers with different digits such that the first as well as last digit are
divisible by 1 and 5, the second as well as fourth digits are divisible by 2 and 4 and the third digit is
divisible by 3 is

(A) 6
(B) 18
(C) 72
(D) 54

5) Let A(z1), B(z2) and C(z3) be the vertices of a triangle ABC such that z3 + iωz2 = (1 = + iω)z1 where
ω is the cube root of unity not equal to 1, then ΔABC is :

(A)
isosceles triangle and

(B)
isosceles triangle and
(C) right angle isosceles
(D) equilateral

6) 5 different balls are placed in 5 different boxes randamly. Find the probability that exactly two
boxes remin empty. Given each box can hold any number of balls :-

(A)

(B)

(C)

(D)

7) is equal to :-

(A) 0
(B) loge2

(C)

(D)

8) If y = y(x) is the solution curve of the differential equation + y tan x = x sec x, y(0) =

1, then y is equal to

(A)
(B)

(C)

(D)

9) The area enclosed by the closed curve C given by the differential equation , y(1) = 0
is 4π.Let P and Q be the points of intersection of the curve C and the y-axis. If normals at P and Q on
the curve C intersect x-axis at points R and S respectively, then the length of the line segment RS is

(A)

(B)

(C) 2

(D)

10) The distance of the point (1, 1, 9) from the point of intersection of the
line and the plane x + y + z = 17 is :

(A)
(B)
(C) 38
(D)

11) If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (–30, 0)
2
and is tangent to the parabola y = 30x, then the length of this chord is :

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

12) Let and and is a vector such that ,

and projection of on is 1, then the projection of on equals :

(A)

(B)

(C)
(D)

13) If the maximum distance of normal to the ellipse , b < 2, from the origin is 1, then
the eccentricity of the ellipse is:

(A)

(B)

(C)

(D)

14) A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x–axis and

y–axis respectively. If the perpendicular from origin O to this line makes an angle of with positive

direction of y–axis and the area of ΔOAB is , then a2 –b2 is equal to :

(A)

(B) 196

(C)

(D) 98

15) Let p and q be two statements, then ~(~ p q) (p q) is logically equivalent to

(A) q
(B) c
(C) p
(D) t

16) If a = cos 2 , b = sin 7, then :-

(A) a > 0, b > 0

(B) ab < 0
(C) a > b
(D) None of these

17) If f(x) be a polynomial function satisfying f(x).f = f(x) + f and f(4) = 65 then value of f(6)
is :

(A) 217
(B) 215
(C) 216
(D) 65

18)

(A) 100
(B) 1
(C) 5050
(D) –100

19) Let
Then which one of the following is true?

(A) f is neither differentiable at x = 0 nor at x = 1

(B) f is differentiable at x = 0 and at x = 1
(C) f is differentiable at x = 0 but not at x = 1
(D) f is differentiable at x = 1 but not at x = 0

20) Function f(x) = 2x + cot–1x – is increasing in

(A) (–∞, 0)
(B) (0, ∞)
(C) (–∞, ∞)
(D) No where

SECTION-II

1) If a,b,c are non-zero real numbers then the minimum value of the expression

equals-

2) The integral value of a for which the equation (x2 + x + 2)2 – (a – 3)(x2 + x + 2)(x2 + x + 1) + (a –
4)(x2 + x + 1)2 = 0 has atleast one real roots

3) If where l, m, n ∈ , m and n are coprime

then l + m + n is equal to ____.
4) Let f : be a differentiable function such that . If f(0) = e–2, then 2f(0) –
f(2) is equal to _____.

5) If the distance of the point (1, –2, 3) from the plane x + 2y – 3z + 10 = 0 measured parallel to the

line, is , then the value of |m| is equal to ______.

6) Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2 + y2 = 8 and y2
= 16x. If one
of these tangents touches the two curves at Q and R, then (QR)2 is equal to

7) If and , then the standard deviation of the 9 items x1, x2, ....., x9 is-

8) A chimney of 20 m height standing on the top of a building subtends an angle whose tangent is
at a distance of 70 m from the foot of the building, then the height of building is :

9) Number of positive solutions satisfying the equation

10) is equal to
 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. A C D D A C C D B A C A B C D B D C C D

SECTION-II

Q. 21 22 23 24 25 26 27 28 29 30
A. 90 3 12 4 500 2 8 25 50 0

PART-2 : CHEMISTRY

SECTION-I

Q. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
A. D C D A D B C A A B C D A A C A C B C B

SECTION-II

Q. 51 52 53 54 55 56 57 58 59 60
A. 3 7 2 3 4 5 8 40 272 -600

PART-3 : MATHEMATICS

SECTION-I

Q. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
A. A A A A A B B A D D D A B A C B A C A C

SECTION-II

Q. 81 82 83 84 85 86 87 88 89 90
A. 6 6 63 1 2 72 2 50 1 0
 SOLUTIONS

PART-1 : PHYSICS

1) f = = 10 cm
∴ R = 2F = 20 cm

2) (y8)Bright, medium = (y5)Dark, air

⇒m=

4)

n' =

n" =

= 720 Hz.

6) Let the stones be projected at t = 0 sec with a speed u from point O. Then an observer, at
rest at t = 0 and having constant acceleration equal to acceleration due to gravity, shall
observe the three stones move with constant velocity as shown.

In the given time each ball shall travel a distance 5 metre as seen by this observer. Hence the
required distance between A and B will be = metre

7)

mu = ⇒v=
 8)

Ei =

I1ω1 +

ω=

Ef = =

⇒ Ef – Ei = =

10)

11) p = V11T

12)

VC = VR = iR =

q = CVC =

13)
14)
⇒ 98 A = 100 A – 400
∴ A = 200

15)
Ki + Ui = Kb + Ub

16) D1 diode is in R.B and D2 diode is in FB

i= = 2 amp.

17)

Number of electrons ejected

20)

WABCDA = +ve
21)
∵ δ = i1 + i2 – A
30° = 60° + i2 – 30°
i2 = 0°
Means, emergent ray goes along the normal & it will make angle with the second face of the
prism is θ = 90° – 0° = 90°

22) IA cos2 60° = IB cos2 30°

23) vmax. = Aω So

10 = Aω ⇒ ω2 =
amax. = ω2A

50 = ω2A ⇒ ω2 = ∴ =
⇒ A = 2cm

Hence ω = = 5 s–1
Now v2 = ω2(A2 – x2)
64 = 25(4 – x2)

4 – x2 = = 2.56
2
x = 4 – 2.56 = 1.44
x= = ±1.2 cm

24)

gp = ⇒ gp =

gp = , wp = mgp = mg = 9N

wp = = 4N

25) K =
9.1 × 108 = ⇒ h = 91 m

26) Flux coming out of the cube ....(i)

and from sphere ϕ2 = ...(ii)

27) Ceq =
q = CeqV
= 2μF × 12V
= 24 μc

∴ q1 = = 8μc

28) N = N0 ⇒

29) Here Vmax = = 12 mV

and Vmin = = 4 mV

30)

(a) Let us first decide whether whole of the ice melts or not. The question is solved in steps as
follows :
(i) Heat supplied by 10gm of water in cooling
from 60ºC to 0ºC = mS.Δt
10 × 1 × (60 – 0) = 600 cal
(ii) Heat required to raise the temperature of
10 gm ice from –5ºC to 0ºC.
= 10 × 0.5 × (0 – (–5)) = 25 cal
(iii) Heat required to melt 10 gm ice at 0ºC into
water at 0ºC
= mL = 10 × 80 = 800 cal
Now the heat required to melt the ice completely,
= (a) + (b) = 25 + 800 = 825 cal
Since heat supplied by the water to lower its
temperature to 0ºC is only 600 cal, therefore
whole of the ice does not melt. The resulting temperature of the mixture is thus 0ºC.

PART-2 : CHEMISTRY

31)
all bond lengths are equal

32) Correct IUPAC name is :

Diamminechloridonitrito-O-dipyridinecobalt (III) nitrate.

33) In , all 8 bonds are not equivalent.

34)

35) Al(OH)3 Al2O3 + H2O↑(Calcination)

Al2O3 → Al (Electrolytic reduction)

36)

37) SO2, Cl2O7 and Mn2O7 (all are acidic)

Pb3O4(2PbO + PbO2) (Amphotericoxide)

38) By using NaBH4 reduction of only aldehyde group takes place.

39)

40)

41) Due to presence of POS in option (3), it is meso compound, rest are optically active
because no POS/COS.

42)

43)
It is a case of intramolecular aldol condensation which occurs as follows
44)

46) x ⇌ 2y
1 2y
1-α 2-α

P⇌ Q+R
1 Q R
1-α α α

KP = 2KP'

4Pa = 2P12 ⇒

47) = 4 i.e. after (4) half lifes [A]t = = = = = 6.25gm

49) Anode : H2 → 2H+ + 2e–

Cathode :
Ecell = Ecathode – EAnode

= [1.51 – log

=
= 1.48

50) pH = pKa + log

5.18 = (5 – log 2) + log

5.18 – 4.7 = 0.48 = log 3 = log

∴ =3:1

51)

Al-reduction process is used to reduce Cr2O3, Mn3O4 etc. (not for bauxite ore)

52) 5f s = –½

53)

Mg3N2 + 6H2O → 2NH3 + 3Mg(OH)2

(x)
LiH + H2O → H2 + LiOH
(y)
54)

55)

56) Chloroform given by

CH3–CH2–OH, ,

, ,

57)
(for very dilute solution )

(mol. wt)solute = 254.5

mol. wt of Sn = 32 × n
∴ 32 × n = 254.5
n = 7.95
=8

58)

3Fe2O3(s) → 2Fe3O4 + O2
480 g Fe2O3 provide 16 g O2.
 For loss of 0.04 g O2 → 0.04 × = 1.2 g
Fe2O3% by mass of

SiO2 = × 100 = 40%

59) nN2(g) + nH2(g) → –(– NH – NH –)n–

ΔH = [n × BE (N º N) + n × BE (H – H)
– BE(N – N) × 2n + 2n × BE (N – H)]

⇒ ΔH = [BE(N ≡ N) + BE(H – H) –
[BE (N – N) + BE (N – H)] × 2

PART-3 : MATHEMATICS

61) ∴
⇒ (a + b)2 + (b + c)2 + (c + a)2< 0

62) A =

⇒ det(A) = (a – b)2 (b – c)2 (c – a)2

& det (4I) = 64
⇒ (a – b)(b – c)(c – a) = ±8
∵ (a – b) + (b – c) + (c – a) = 0
∴ (a – b)3 + (b – c)3 + (c – a)3
= 3(a – b)(b – c)(c – a) = ±24

63) ar =
b = Coeff of x49 = – (a1 + a2 + ...... a50)

64) Total number of diagonals

= 10C2 – 10 = 35
Number of pair of diagonals intersecting inside
= 10C4 = 210
Number of ways of selecting any two diagonals

= 6/17

65) Given Z3 + iwZ2 = (1 + iw)Z1

Z3 – Z1 = iw(Z1 – Z2)

= with

66) Each ball can be placed in 5 ways

∴ Total no of ways = 55
2 empty boxes can be selected in ways and 5 balls can be placed in the remaining 3 boxes
in group of 2,2,1 or 3,1,1 in

= 150 ways
∴ favourable cases = .150

∴ P=

67)

68) Here I.F. = sec x

Then solution of D.E :
y(sec x) = x tan x – ln(sec x) + c
Given y(0) = 1 ⇒ c = 1
∴ y(sec x) = x tan x – ln(sec x) + 1

At x = ,y=
69)

(2 – y) dy = (x + a) dx

+ ax + c

a+c = as y (1) = 0
2 2
X + y + 2ax – 4y – 1 – 2a = 0
πr2 = 4π
r2 = 4
4=
(a + 1)2 = 0

P, Q =
Equation of normal at P, Q are y – 2 = (x – 1)
y–2=– (x – 1)

R=

S=

RS =

70) y = =t
⇒ x = 3 + t, y = 2t + 4, z = 2t + 5
for point of intersection with x + y + z = 17
3 + t + 2t + 4 + 2t + 5 = 17
⇒ 5t = 5 ⇒ t = 1
⇒ point of intersection is (4, 6, 7)
distance between (1, 1, 9) and (4, 6, 7)
is

71) Equation of tangent to y2 = 30 x

y = mx +

Pass thru (–30, 0) : 0 = –30m + ⇒ m2 = 1/4

⇒m= or m =

At = , ⇒ x – 2y + 30 = 0
72)
Let
⇒ 15x – 20y – 25z + 25 = 0
⇒ 3x – 4y – 5z = –5
Also x + y + z = 4

and ⇒ 4x + 3y = 5
⇒

Projection of and =

73)

Equation of normal is
2x secθ – by cosecq = 4 – b2

Distance from (0, 0) =

Distance is maximum if
4sec2θ + b2 cosec2θ is minimum

74)
Equation of straight line :

or

Comparing both :

Now area of

75)

~ (~ pq) (p q) ≡ [~(~ p)(~ q)] (p q)

≡ [p (~ q) )(p q]

≡ p [(~q) q]

≡p F≡p

76) a = cos 2 cos 114º = – sin 24º < 0

b = sin 7 sin 399º – sin 39º > 0
so ab < 0

77) f(x) = (1 ± xn)

81)

82) Let x2 + x + 1 = y
(y + 1)2 – (a – 3)y (y + 1) + (a – 4)y2 = 0

i.e.
D≥0
 a∈I

83)
2x21 + 3x14 + 6x7 = t
42(x20 + x13 + x6) dx = dt

=
l = 48, m = 8, n = 7
l + m + n = 63

84)
y·ex = k · ex + c
f(0) = e–2
⇒ c = e–2 – k
∴ y = k + (e–2 – k)e–x

now

⇒ k = e–2 – 1
∴ y = (e–2 – 1) + e–x
f(2) = 2e–2 – 1, f(0) = e–2
2f(0) – f(2) = 1

85)

DC of line ≡

Q lies on x + 2y – 3z + 10 = 0

⇒
⇒
r2m2 = m2 + 10

⇒ ⇒ m2 = 4
|m| = 2

86)

y = ± x ± 4. Point of contact on parabola

Let m = 1,
R (4, 8)
Point of contact on circle Q (–2, 2)

87)

= 2.00

88)

Given, tan a =

⇒ h = 50
Hence, 50 m