[tabs title=”JEE MAINS Previous Year Paper” type=”centered”]

[tab title=”PHYSICS”]

PHYSICS

SECTION A

Q 1: A body weighs 49 N on a spring balance at the north pole. What will be its weight recorded on the same weighing machine, if it is shifted to the equator?

[Use g = GM/R2 = 9.8 ms-2 and radius of earth, R = 6400 km.]

(A) 49 N

(B) 49.83 N

(C) 49.17 N

(D) 48.83 N

Answer: (D)

At North Pole, weight

Mg = 49

Now, at equator

g’ = g – ω^{2}R

⇒ Mg’ = M(g – ω^{2}R)

⇒ weight will be less than Mg at equator.

Alter:

g is maximum at the poles.

Hence from options only (D) has lesser value than 49N.

Q 2: The logic circuit shown below is equivalent to :

Answer: (2)

Q 3: If one mole of an ideal gas at (P_{1}, V_{1}) is allowed to expand reversibly and isothermally (A to B) its pressure is reduced to one-half of the original pressure (see figure). This is followed by a constant volume cooling till its pressure is reduced to one-fourth of the initial value (B→C). Then it is restored to its initial state by a reversible adiabatic compression (C to A). The net work done by the gas is equal to:

(A) 0

(B) -RT/2( γ – 1)

(C) RT [ln2 -1/2( γ -1)]

(D) RT ln2

Answer: (C)

AB → Isothermal process

W_{AB} = nRT ln2 = RT ln2

BC → Isochoric process

W_{BC} = 0

CA → Adiabatic process

W_{CA} = P_{1}V_{1} – (P_{1}/4)X^{2}V_{1})/(1 – γ)

= P_{1}V_{1}/2(1 – γ)

= RT/2(1 – γ )

W_{ABCA} = RT ln2 + RT/2(1 – γ)

= RT [ln2 – 1/2(γ – 1)]

Q 4: Match List – I with List – II.

List – I | List – II |

(a) Source of microwave frequency | (i) Radioactive decay of nucleus |

(b) Source of infrared frequency | (ii) Magnetron |

(c) Source of Gamma Rays | (iii) Inner shell electrons |

(d) Source of X-rays | (iv) Vibration of atoms and molecules |

– | (v) LASER |

– | (vi) RC circuit |

Choose the correct answer from the options given below:

(A) (a)-(ii),(b)-(iv),(c)-(i),(d)-(iii)

(B) (a)-(vi),(b)-(iv),(c)-(i),(d)-(v)

(C) (a)-(ii),(b)-(iv),(c)-(vi),(d)-(iii)

(D) (a)-(vi),(b)-(v),(c)-(i),(d)-(iv)

Answer: (A)

(a) Source of microwave frequency – (ii) Magnetron

(b) Source of infrared frequency – (iv) Vibration of atom and molecules

(c) Source of gamma ray – (i) Radioactive decay of nucleus

(d) Source of X-ray – (iii) inner shell electron

Q 5: The period of oscillation of a simple pendulum is T = 2π√(L/g). Measured value of ‘L’ is 1.0 m from meter-scale having a minimum division of 1 mm and time of one complete oscillation is 1.95 s measured from stopwatch of 0.01 s resolution. The percentage error in the determination of ‘g’ will be:

(A) 1.33 %

(B) 1.30 %

(C) 1.13 %

(D) 1.03 %

Answer: (C)

T = 2π√(L/g)

T^{2} = 4𝜋^{2} [L/g]

g = 4𝜋^{2} [L/T2]

Δg/g = ΔL/L + 2ΔT/T

= [1mm/1m + 2(10×10-3)/1.95]×100

= 1.13 %

Q 6: When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is:

(A) elliptical

(B) parabolic

(C) straight line

(D) circular

Answer: (A)

We know that in SHM;

V = ω√(A^{2} – x^{2})

Elliptical

Alternate:

x = A sinωt ⇒ sin ωt = x/A

v = Aω cos ωt ⇒ cos ωt = v/Aω

Hence (x/A)^{2} + (v/Aω)^{2} = 1

Which is the equation of an ellipse.

Q 7: In the given figure, a body of mass M is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant k, the frequency of oscillation of given body is:

(A) (1/2π) √(2K/Mg sin α)

(B) (1/2π) √(K/Mg sin α)

(C) (1/2π) √(2K/M)

(D) (1/2π) √(K/2M)

Answer: (A)

Equivalent K = K + K = 2K

Now, T = 2π √(M/K_{eq})

⇒ T = 2π √(M/2K)

∴f = (1/2π) √(2K/M)

Q 8: Given below are two statements:

Statement I: PN junction diodes can be used to function as transistor, simply by connecting two diodes, back to back, which acts as the base terminal.

Statement II: In the study of transistor, the amplification factor β indicates ratio of the collector current to the base current.

In the light of the above statements, choose the correct answer from the options given below.

(A) Statement I is false but Statement II is true.

(B) Both Statement I and Statement II are true.

(C) Statement I is true but Statement II is false.

(D) Both Statement I and Statement II are false.

Answer: (A)

S-1

Statement 1 is false because in case of two discrete back to back connected diodes, there are four doped regions instead of three and there is nothing that resembles a thin base region between an emitter and a collector.

S-2

Statement-2 is true, as

β = I_{C}/I_{B}

Q 9: Figure shows a circuit that contains four identical resistors with resistance R = 2.0 Ω. Two identical inductors with inductance L = 2.0 mH and an ideal battery with emf E = 9.V. The current ‘i’ just after the switch ‘s’ is closed will be:

(A) 9A

(B) 3.0 A

(C) 2.25 A

(D) 3.37 A

Answer: (C)

Just when switch S is closed, inductor will behave like an infinite resistance. Hence, the circuit will be like

Given: V = 9 V

From V = IR

I = V/R

R_{eq}. = 2 + 2 = 4 Ω

i = 9/4 = 2.25 A

Q 10: A circular hole of radius (a/2) is cut out of a circular disc of radius ‘a’ shown in figure. The centroid of the remaining circular portion with respect to point ‘O’ will be:

(A) (10/11)a

(B) (⅔)a

(C) (⅙)a

(D) (⅚)a

Answer: (D)

Let σ be the surface mass density of disc.

Xcom= (m_{1}x_{1} – m_{2}x_{2})/(m_{1} – m_{2})

Where m = σπr^{2}

Xcom= (σ×πa^{2}×a) – (σπa^{2}/4) × 3a/2)/(σπa^{2} – σπa^{2}/4)

Xcom= (a – 3a/8)/(1 – ¼)

Xcom= (5a/8)/(¾)

Xcom= 5a/6

Q 11: The de Broglie wavelength of a proton and α-particle are equal. The ratio of their velocities is:

(A) 4:2

(B) 4:1

(C) 1:4

(D) 4:3

Answer: (B)

From De-Broglie’s wavelength:-

λ = h/mv

Given λP = λα

v ∝ 1/m

v_{p}/v_{α} = m_{α}/m_{p} = 4m_{p}/m_{p}

= 4/1

Q 12: On the basis of kinetic theory of gases, the gas exerts pressure because its molecules:

(A) suffer change in momentum when impinge on the walls of container.

(B) continuously stick to the walls of container.

(C) continuously lose their energy till it reaches wall.

(D) are attracted by the walls of container.

Answer: (A)

Based on kinetic theory of gases, molecules suffer change in momentum when impinge on the walls of container. Due to this they exert a force resulting in exerting pressure on the walls of the container.

Q 13: Two electrons each are fixed at a distance ‘2d’. A third charge proton placed at the midpoint is displaced slightly by a distance x (x<<d) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency:

(m = mass of charged particle)

(A) (q^{2}/2πε_{0}md^{3})^{1/2}

(B) (πε_{0}md^{3}/2q^{2})^{1/2}

(C) (2πε_{0}md^{3}/q^{2})^{1/2}

(D) (2q^{2}/πε_{0}md^{3})^{1/2}

Answer: (A)

Restoring force on proton:-

Fr = 2F_{1} sinθ where F_{1} = kq^{2}/(d^{2} + x^{2})

Fr = 2Kq^{2}x/[d^{2} + x^{2}]^{3/2}

x <<< d

Fr = 2kq^{2}x/d^{3}

= q^{2}x/2πε_{0}d^{3}

= kx

K = q^{2}/2πε_{0}d^{3}

Angular Frequency:-

ω = √(k/m)

ω = √(q^{2}/2πε_{0}md^{3})

Q 14: An X-ray tube is operated at 1.24 million volt. The shortest wavelength of the produced photon will be:

(A) 10^{-2} nm

(B) 10^{-3} nm

(C) 10^{-4} nm

(D) 10^{-1} nm

Answer: (B)

The minimum wavelength of photon will correspond to the maximum energy due to accelerating by V volts in the tube.

λmin = hc/eV

λmin = 1240nm-eV/1.24×10^{6}

λmin = 10^{-3} nm

Q 15: A soft ferromagnetic material is placed in an external magnetic fiel(D) The magnetic domains:

(A) decrease in size and changes orientation.

(B) may increase or decrease in size and change its orientation.

(C) increase in size but no change in orientation.

(D) have no relation with external magnetic field.

Answer: (B)

Atoms of ferromagnetic material in unmagnetized state form domains inside the ferromagnetic material. These domains have large magnetic moment of atoms. In the absence of magnetic field, these domains have magnetic moment in different directions. But when the magnetic field is applied, domains aligned in the direction of the field grow in size and those aligned in the direction opposite to the field reduce in size and also its orientation changes.

Q 16: According to Bohr atom model, in which of the following transitions will the frequency be maximum?

(A) n = 2 to n = 1

(B) n = 4 to n = 3

(C) n = 5 to n = 4

(D) n = 3 to n = 2

Answer: (A)

Since, ΔE is maximum for the transition from n = 2 to n = 1

f is more for transition from n = 2 to n = 1.

Q 17: Which of the following equations represents a travelling wave?

(A) y = Ae^{x^2} (*vt* + θ )

(B) y = A sin(15x – 2t)

(C) y = Aex cos(ωt – θ)

(D) y = A sin x cos ωt

Answer: (B)

Y = F(x, t)

For travelling wave y should be linear function of x and t and they must exist as (x ± vt)

Y = A sin (15x – 2t) which is a linear function in x and t.

Q 18: Zener breakdown occurs in a p-n junction having p and n both:

(A) lightly doped and have wide depletion layer.

(B) heavily doped and have narrow depletion layer.

(C) heavily doped and have wide depletion layer.

(D) lightly doped and have narrow depletion layer.

Answer: (B)

The zener breakdown occurs in the heavily doped p-n junction diode. Heavily doped p-n junction diodes have narrow depletion region. The narrow depletion layer width leads to a high electric field which causes the p-n junction breakdown.

Q 19: A particle is projected with velocity v0 along x-axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e. ma = -αx^{2}. The distance at which the particle stops:

(A) (2v_{0}/3α)^{1/3}

(B) (3v_{0}^{2}/2α)^{1/2}

(C) (3v_{0}^{2}/2α)^{1/3}

(D) (2v_{0}^{2}/3α)^{1/2}

Answer: Bonus

a = vdv/dx

Given:- v

_{i}= v

_{0}

V

_{f }= 0

X

_{i}= 0

X

_{f}= x

From Damping Force: a = -αx

^{2}/m

-v_{0}^{2}/2 = (-α/m) [x^{3}/3]

x = [3mv_{0}^{2}/2α]^{1/3}

Most suitable answer could be (3) as mass ‘m’ is not given in any options.

Q 20: If the source of light used in a Young’s double slit experiment is changed from red to violet:

(A) the fringes will become brighter.

(B) consecutive fringe lines will come closer.

(C) the central bright fringe will become a dark fringe.

(D) the intensity of minima will increase.

Answer: (B)

β = λD/d

As λv<λR

βv< βR

⇒ Consecutive fringe lines will come closer.

⇒ (2)

Section – B

Q 1: A uniform thin bar of mass 6 kg and length 2.4 meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the hexagon is ______×10^{-1} kg m^{2}.

Answer: 8

MOI of AB about P : IABP = (M/6)(l/6)^{2}/12

MOI of AB about O,

= (6/100) [(24×24/12×36) + (24×24/36)×3/4]

= 0.8 kg m^{2}

= 8×10^{-1} kg m^{2}

Q 2: Two solids A and B of mass 1 kg and 2 kg respectively are moving with equal linear momentum. The ratio of their kinetic energies (K.E.)A : (K.E.)B will be A/1. So the value of A will be ________.

Answer: 2

Given that, M_{1}/M_{2} = 1/2

we know that

K = p^{2}/2M

K_{1}/K_{2} = (p^{2}/2M_{1} )×2M_{2}/p^{2}

K_{1}/K_{2} = M_{2}/M_{1} = 2/1

A/1 = 2/1

⇒∴ A = 2

Q 3: An electromagnetic wave of frequency 3 GHz enters a dielectric medium of relative electric permittivity 2.25 from vacuum. The wavelength of this wave in that medium will be _______ ×10^{-2} cm.

Answer: 667

ε_{r} = 2.25

Assuming non-magnetic material ⇒ μ_{r} = 1

Hence refractive index of the medium

n = √(μ_{r}ε_{r })= √2.25 = 1.5

∴λ_{v}/λ_{m}=n

λ_{m} = C/f.n

= 3×10^{8}/3×10^{9}×1.5

= (⅔) 10^{-1} m

= 667×10^{-2} cm

Q 4: The root mean square speed of molecules of a given mass of a gas at 270C and 1 atmosphere pressure is 200 ms-1. The root mean square speed of molecules of the gas at 1270C and 2 atmosphere pressure is x/√3 ms-1. The value of x will be __________.

Answer: 400 m/s

Vrms = √(3RT_{1}/M_{0})

200 = √(3R×300/M_{0}) ….(1)

Also, x/√3 = √(3R×400/M_{0}) …(2)

(1)÷(2)

200/(x/√3) = √(300/400) = √(3/4)

⇒x = 400 m/s

Q 5: A signal of 0.1 kW is transmitted in a cable. The attenuation of cable is -5 dB per km and cable length is 20 km. The power received at the receiver is 10^{-x}W. The value of x is ______.

[Gain in dB = 10 log_{10}(P_{0}/P_{i})]

Answer: 8

Power of signal transmitted: P_{i} = 0.1 Kw = 100w

Rate of attenuation = -5 dB/Km

Total length of path = 20 km

Total loss suffered = -5×20 = -100dB

Gain in dB = 10 log_{10}(P_{0}/P_{i})

-100 = 10log_{10}(P_{0}/P_{i})

log_{10}(P_{i}/P_{0}) = 10

log_{10}(P_{i}/P_{0}) = log101010

100/P_{0} = 1010

⇒ P_{0} = 1/10^{8} = 10^{-8}

⇒ x = 8

Q 6: A series LCR circuit is designed to resonate at an angular frequency ω0 = 10^{5}rad/s. The circuit draws 16W power from 120 V source at resonance. The value of resistance ‘R’ in the circuit is _______ Ω.

Answer: 900

P = V^{2}/R

16 = 120^{2}/R

⇒ R = 14400/16

⇒ R = 900 Ω

Q 7: A uniform metallic wire is elongated by 0.04 m when subjected to a linear force F. The elongation, if its length and diameter is doubled and subjected to the same force, will be ________ cm.

Answer: 2

y = Fl/A∆l

F/A = y∆l/l

F/A = y×0.04/l …(1)

When length & diameter is doubled.

⇒ F/4A = y × ∆l/2l …(2)

(1)÷(2)

(F/A)/(F/4A) = (y×0.04/l)y×∆l/2l

4 = 0.04×2/∆l

∆l = 0.02

∆l = 2×10^{-2}

∴ x = 2

Q 8: A cylindrical wire of radius 0.5 mm and conductivity 5×107 S/m is subjected to an electric field of 10 mV/m. The expected value of current in the wire will be x3π mA. The value of x is ____.

Answer: 5

We know that

J = σE

⇒ J = 5×10^{7}×10×10^{-3}

⇒ J = 50×104 A/m^{2}

Current flowing;

I = J × πR^{2}

I = 50×10^{4} ×π(0.5×10^{-3})^{2}

I = 50×10^{4}×π×0.25×10^{-6}

I = 125×10^{-3}π

X = 5

Q 9: A point charge of +12 μC is at a distance 6 cm vertically above the centre of a square of side 12 cm as shown in figure. The magnitude of the electric flux through the square will be _____ ×10^{3} Nm^{2}/C.

Answer: 226

Using Gauss law, it is a part of cube of side 12 cm and charge at centre,

ϕ = Q/6ε_{0}= 12μc/6ε_{0}

12×10^{−6} /6×8.85×10^{−12}

= 226×103 Nm^{2}/C

Q 10: Two cars are approaching each other at an equal speed of 7.2 km/hr. When they see each other, both blow horns having frequency of 676 Hz. The beat frequency heard by each driver will be ________ Hz. [Velocity of sound in air is 340 m/s.]

Answer: 8

Speed = 7.2 km/h = 2 m/s

Frequency as heard by A

f’A= fB(v + v_{0})/(v – v_{s})

f’A = 676(340 + 2)/(340 – 2)

f’A = 684Hz

∴ fBeat = f’A- fB

= 684 – 676

= 8 Hz

[/tab]

[tab title=”CHEMISTRY”]

**Chemistry**

**
**

**SECTION A**

Q 1. Given below are two statements:

(A) Both Statement I and Statement II are false

(B) Statement I is false but Statement II is true

(C) Statement I is true but Statement II is false

(D) Both Statement I and Statement II are true

Answer: (C)

For survival of aquatic life dissolved oxygen is responsible for its optimum limit 6.5 ppm and optimum limit of BOD ranges from 10-20 ppm & BOD stands for biochemical oxygen demand.

Q 2. Which one of the following carbonyl compounds cannot be prepared by addition of water on an alkyne in the presence of HgSO_{4} and H_{2}SO_{4}?

Answer: (A)

Reaction of Alkyne with HgSO_{4} & H_{2}SO_{4} follow as

Hence, by this process preparation of CH_{3}CH_{2}CHO can’t be possible.

Q 3. Which one of the following compounds is non-aromatic?

Answer: (B)

(Not planar)

Hence, it is non-aromatic.

Q 4. The incorrect statement among the following is:

(A) VOSO_{4} is a reducing agent

(B) Red color of ruby is due to the presence of CO^{3+}

(C) Cr_{2}O_{3} is an amphoteric oxide

(D) RuO_{4} is an oxidizing agent

Answer: (B)

Red color of ruby is due to presence of CrO_{3} or Cr^{+6} not CO^{3+}

Q 5. According to Bohr’s atomic theory:

(a) Kinetic energy of electron is ∝ Z^{2} / n^{2}

(b) The product of velocity (v) of electron and principal quantum number (n). ‘vn’ ∝ z^{2}

(c) Frequency of revolution of electron in an orbit is ∝ Z^{3} / n^{3}

(d) Coulombic force of attraction on the electron is ∝ Z^{3} / n^{4}

Choose the most appropriate Answer from the options given below

(A) (C) only

(B) (A) and (D) only

(C) (A) only

(D) (A), (C) and (D) only

Answer: (B)

(a) KE = –TE = 13.6 × Z^{2} / n^{2} eV

KE α Z^{2} / n^{2}

(b) V = 2.188 × 10^{6} × z / n m/s

So, V_{n} ∝ Z

Frequency = V / 2πr

F α Z^{2} / n^{2} [∴ r α Z^{2} / n^{2} and v α Z / n ]

(d) Force ∝ Z^{2} / r^{2}

So, F ∝ Z^{3} / n^{4}

So, only statement (A) is correct.

Q 6. Match List-I with List-II

List- I | List-II |

(a) Valium | (iv) Tranquilizer |

(b) Morphine | (iii) Analgesic |

(c) Norethindrone | (i) Antifertility drug |

(d) Vitamin B_{12} |
(ii) Pernicious anemia |

(A) (A)-(iv), (B)-(iii), (C)-(ii), (D)-(i)

(B) (A)-(i), (B)-(iii), (C)-(iv), (D)-(ii)

(C) (A)-(ii), (B)-(iv), (C)-(iii), (D)-(i)

(D) (A)-(iv), (B)-(iii), (C)-(i), (D)-(ii)

Answer: (D)

(a) Valium (iv) Tranquilizer

(b) Morphine (iii) Analgesic

(c) Norethindrone (i) Antifertility drug

(d) Vitamin B_{12} (ii) Pernicious anemia

Q 7. Match List-I with List-II

List- I (Salt) | List-II(Flame colour wavelength) |

(a) LiCl | (i) 455.5 nm |

(b) NaCl | (ii) 970.8 nm |

(c) RbCl | (iii) 780.0 nm |

(d) CsCl | (iv) 589.2 nm |

Choose the correct Answer from the options given below:

(A) (A)-(ii), (B)-(i), (C)-(iv), (D)-(iii)

(B) (A)-(ii), (B)-(iv), (C)-(iii), (D)-(i)

(C) (A)-(iv), (B)-(ii), (C)-(iii), (D)-(i)

(D) (A)-(i), (B)-(iv), (C)-(ii), (D)-(iii)

Answer: (B)

Range of visible region: 390 nm – 760 nm

VIBGYOR

Violet – Red

LiCl Crimson Red

NaCl Golden yellow

RbCl Violet

CsCl Blue

So, LiCl which is crimson have wavelengths close to red in the spectrum of visible region which is as per given data.

Q 8. Match List-I and List-II.

List-I | List-II |

(A) | (i) Br_{2} / NaOH |

(B) | (ii) H_{2} / Pd-BaSO_{4} |

(C) | (iii) Zn (Hg) / Conc. HCl |

(D) | (iv) Cl_{2} / Red P, H_{2}O |

(A) (A)-(ii), (B)-(i), (C)-(iv), (D)-(iii)

(B) (A)-(iii), (B)-(iv), (C)-(i), (D)-(ii)

(C) (A)-(iii), (B)-(i), (C)-(iv), (D)-(ii)

(D) (A)-(ii), (B)-(iv), (C)-(i), (D)-(iii)

Answer: (D)

Q 9. In polymer Buna-S: ‘S’ stands for

(A) Styrene

(B) Sulphur

(C) Strength

(D) Sulphonation

Answer: (A)

Buna-S is the co-polymer of buta-1,3-diene & styrene

Q 10. Most suitable salt which can be used for efficient clotting of blood will be:

(A) Mg(HCO_{3})_{2}

(B) FeSO_{4}

(C) NaHCO_{3}

(D) FeCl_{3}

Answer: (D)

Blood is a negative sol, according to Hardy-Schulz’s rule, the cation with high charge has high coagulation power. Hence, FeCl_{3} can be used for clotting blood.

Q 11. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Hydrogen is the most abundant element in the Universe, but it is not the most abundant gas in the troposphere.

Reason R: Hydrogen is the lightest element.

In the light of the above statements, choose the correct Answer from the given below

(1) A is false but R is true

(2) Both A and R are true and R is the correct explanation of A

(3) A is true but R is false

(4) Both A and R are true but R is NOT the correct explanation of A

(A) A is false but R is true

(B) Both A and R are true and R is the correct explanation of A

(C) A is true but R is false

(D) Both A and R are true but R is NOT the correct explanation of A

Answer: (B)

Hydrogen is the most abundant element in the universe because all luminous bodies of the universe i.e. stars and nebulae are made up of hydrogen which acts as nuclear fuel and fusion reaction is responsible for their light.

Q 12. What is the correct sequence of reagents used for converting nitrobenzene into m- dibromobenzene?

Answer: (D)

Q 13. The correct shape and I-I-I bond angles respectively in I^{–}_{3} ion are:

(A) Trigonal planar; 120^{o}

(B) Distorted trigonal planar; 135^{o} and 90^{o}

(C) Linear; 180º

(D) T-shaped; 180º and 90º

Answer: (C)

I^{–}_{3} has sp^{3}d hybridization (2 BP + 3 LP) and linear geometry.

Q 14. What is the correct order of the following elements with respect to their density?

(A) Cr < Fe < Co < Cu < Zn

(B) Cr < Zn < Co < Cu < Fe

(C) Zn < Cu < Co < Fe < Cr

(D) Zn < Cr < Fe < Co < Cu

Answer: (D)

Fact Based

Density depends on many factors like atomic mass. atomic radius and packing efficiency.

Q 15. The Correct set from the following in which both pairs are in correct order of melting point is

(A) LiF > LiCl ; NaCl > MgO

(B) LiF > LiCl ; MgO > NaCl

(C) LiCl > LiF ; NaCl > MgO

(D) LiCl > LiF ; MgO > NaCl

Answer: (B)

Generally,

M.P. ∝ Lattice energy = KQ_{1}Q_{2} / r^{+} + r^{–}

∝ (packing efficiency)

Q 16. The calculated magnetic moments (spin only value) for species

[FeCl_{4}]^{2–}, [Co(C_{2}O_{4})^{3}]^{3–} and MnO_{2–}^{4} respectively are:

Answer: (C)

[FeCl_{4}]^{2–}

Fe^{2}+ 3d^{6} → 4 unpaired electrons. as Cl^{–} in a weak field liquid.

Q 17. Which of the following reagent is suitable for the preparation of the product in the following reaction?

(A) Red P + Cl_{2}

(B) NH_{2}-NH_{2}/ C_{2}H_{5}O^{–}Na^{+}

(C) Ni/H_{2}

(D) NaBH_{4}

Answer: (B)

It is a wolf-kishner reduction of carbonyl compounds.

Q 18. The diazonium salt of which of the following compounds will form a coloured dye on reaction with β-Naphthol in NaOH?

Answer: (C)

Orange bright dye.

Q 19. The correct order of the following compounds showing increasing tendency towards nucleophilic substitution reaction is:

(A) (iv) < (i) < (iii) < (ii)

(B) (iv) < (i) < (ii) < (iii)

(C) (i) < (ii) < (iii) < (iv)

(D) (iv) < (iii) < (ii) < (i)

Answer: (C)

Reactivity ∝ – M group present at o/p position.

Q 20. Match List-I with List-II

List- I(Metal) | List-II(Ores) |

(a) Aluminum | (i) Siderite |

(b) Iron | (ii) Calamine |

(c) Copper | (iii) Kaolinite |

(d) Zinc | (iv) Malachite |

(A) (A)-(iv), (B)-(iii), (C)-(ii), (D)-(i)

(B) (A)-(i), (B)-(ii), (C)-(iii), (D)-(iv)

(C) (A)-(iii), (B)-(i), (C)-(iv), (D)-(ii)

(D) (A)-(ii), (B)-(iv), (C)-(i), (D)-(iii)

Answer: (C)

Siderite FeCO_{3}

Calamine ZnCO_{3}

Kaolinite Si_{2}Al_{2}O_{5}(OH)_{4} or Al_{2}O_{3}.2SiO_{2}.2H_{2}O

Malachite CuCO_{3}.Cu(OH)_{2}

Section B

Q 1. The formula of a gaseous hydrocarbon which requires 6 times of its own volume of O_{2} for complete oxidation and produces 4 times its own volume of CO_{2} is C_{x}H_{y}. The value of y is

Answer: 8

C_{x}H_{y} + 6O_{2} 4CO_{2} + y/2 H_{2}O

Applying POAC on ‘O’ atoms

6 × 2 = 4 × 2 + y/2 × 1

y/2 = 4 ⇒ y = 8

Q 2. The volume occupied by 4.75 g of acetylene gas at 50°C and 740 mmHg pressure is _______L. (Rounded off to the nearest integer)

(Given R = 0.0826 L atm K^{–1} mol^{–1})

Answer: 5

T = 50C = 323.15 K

P = 740 mm of Hg = 740 / 760 atm

V = ?

moles (n) = 4.75 / 26 atm

V = 4.75 / 26 × 0.0821 × 323.15 / 740 × 760

V = 4.97 5 Lit

Q 3. Sucrose hydrolysis in acid solution into glucose and fructose following first order rate law with a half-life of 3.33 h at 25ºC. After 9h, the fraction of sucrose remaining is f. The value of is _________× 10^{–2} (Rounded off to the nearest integer)

[Assume: ln10 = 2.303, ln2 = 0.693]

Answer: 81

Sucrose ^{__}(Hydrolysis)→ Glucose + Fructoset

_{1/2}= 3.33h = 10 / 3h

C_{t} = C_{0 }/ {2 × T/t_{1/2}}

Fraction of sucrose remaining = f = Ct/Co = 1/ {2 × T/t_{1/2}}

1/f = 2^(t/t_{1/2})

Log(1/f) = *log*(2^(t/t_{1/2})) = t/t_{1/2} log(2)

9/ (10/3) × 0.3 – 8.1/10 – 0.81

= x × 10^{–2}

x = 81

Q 4. The total number of amines among the following which can be synthesized by Gabriel synthesis is _______

Answer: 3

Only 1^{o} amines can be prepared by Gabriel synthesis.

Q 5. 1.86 g of aniline completely reacts to form acetanilide. 10% of the product is lost during purification. Amount of acetanilide obtained after purification (in g) is ____× 10^{–2}.

Answer: 243

93 g Aniline produce 135 g acetanilide

1.86 g produce 135 × 1.86 / 93 = 2.70 g

At 10% loss, 90% product will be formed after purification.

Q 6. Among the following allotropic forms of sulphur, the number of allotropic forms, which will show paramagnetism is ______.

(a) α-sulphur (B) β-sulphur (C) S_{2}-form

Answer: (A)

S_{2} is like O_{2} i.e. paramagnetic as per molecular orbital theory.

Q 7. C_{6}H_{6} freezes at 5.5ºC. The temperature at which a solution of 10 g of C_{4}H_{10} in 200 g of C_{6}H_{6} freeze is ________ C. (The molal freezing point depression constant of C_{6}H_{6} is 5.12C/m).

Answer: 1

ΔT_{f} = i × K_{f }× m

= 1 × 5.12 × 10 / 58200 × 1000

∆T_{f} = 5.12 × 50 / 58 = 4.414

T_{f(solution)} = T_{k(solvent)} – ΔT = 5.5 – 4.414 = 1.086oC

≈ 1.09°C = 1 (nearest integer)

Q 8. Assuming ideal behavior, the magnitude of log K for the following reaction at 25ºC is x × 10^{–1}. The value of x is __________. (Integer Answer)

3HC ≡ CH(g) ⇌ C_{6}H_{6} (l)

[Given: Δ_{f}G° (HC = CH) = – 2.04 × 10^{5}] mol^{–1};

Δ_{f}G°(C_{6}H_{6}) = – 1.24 × 105 J mol^{–1;}

R = 8.314 J K^{–1} mol^{–1}]

Answer: 855

ΔG°_{r} = ΔG°_{f }[C_{6}H_{6} (l)] – 3 × ΔG°_{f} [HC = CH]

Q 9. The magnitude of the change in oxidising power of the MnO^{4-}/ Mn^{2+} couple is x × 10^{–4} V, if the H+ concentration is decreased from 1M to 10^{–4} M at 25°C. (Assume concentration of MnO^{4-} and Mn^{2+} to be same on change in H^{+} concentration). The value of x is _____. (Rounded off to the nearest integer).

[Given: 230RT / F = 0.059]

Answer: 3776

5e- + MnO^{–}_{4} + 8H → Mn^{+2} + 4H_{2}O

= 3776 × 10^{–4}

So, x = 3776

Q 10. The solubility product of PbI_{2} is 8.0 × 10^{–9}. The solubility of lead iodide in 0.1 molar solution of lead nitrate is x × 10^{–6} mol/L. The value of x is ________ (Rounded off to the nearest integer)

Given: √2 = 1.41

Answer: 141

PbI_{2}(s) ⇌ Pb^{2+} (aq) + 2I^{–} (aq)

S+0.1 2s

K_{SP} (PbI_{2}) = 8 × 109

K_{SP} = [Pb^{+2}][I^{–}]^{2}

8 × 10^{–9} = (S + 0.1) (2S)^{2} ⇒ (8 × 10^{–9} + 0.1) × 4S^{2}

⇒ S^{2} = 2 × 10^{–8}^{}S = 1.414 × 10^{–4} mol/Lit

= x × 10^{–6} mol/Lit

∴ x = 141.4 141

[/tab]

[tab title=”MATHS”]

Maths

SECTION A

Q 1: Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax^{2} + bx + 1 = 0, then the value of α^{2} + β^{2} – αβ is:

(A) 71/256

(B) -69/256

(C) 69/256

(D) -71/256

Answer: (D)

2b = a + c

(2a + 2)/3 = 10/3 and (2b + c)/3 = 7/3

⇒ a = 4

2b + c = 7, 2b – c = 4 },solving

b = 11/4 and c = 3/2

∴ Quadratic equation is 4x^{2}+ (11/4)x + 1 = 0

∴ The value of (α + β)^{2} – 3αβ = (121/256) – (¾)

= -71/256

Q 2: The value of the integral, where [x] denotes the greatest integer less than or equal to x , is :

(A) –4

(B) –5

(C) -√2 – √3 – 1

(D) -√2 – √3 + 1

Answer: (C)

I =

Put x – 1 = t ; dx = dt

I = -6 + (√2 – 1) + 2√3 – 2√2 + 6 – 3√3

I = -1 – √2 – √3

Q 3: Let f: R → R be defined as

Let A = {x ∈ R ∶ f is increasing}. Then A is equal to :

(A) (-5, -4) ∪ (4,∞)

(B) (-5, ∞)

(C) (-∞, -5) ∪ (4, ∞)

(D) (-∞, -5) ∪ (-4, ∞)

Answer: (A)

f’(x) = {-55 ; x< -5 6(x^{2} – x – 20) ; -5 < x < 4 6(x^{2}– x – 6) ; x > 4

⇒ f’(x) = {-55 ; x< -5 6(x – 5)(x + 4) ; –5 < x < 4 6(x – 3)(x + 2) ; x > 4

Hence, f(x) is monotonically increasing in (-5, -4) ∪ (4, ∞)

Q 4: If the curve y = ax^{2} + bx + c,x ∈ R passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are:

(A) a = 1, b = 1, c = 0

(B) a = -1, b = 1, c = 1

(C) a = 1, b = 0, c = 1

(D) a = 1/2, b = 1/2 ,c = 1

Answer: (A)

2 = a + b + c

dy/dx = 2ax + b, (dy/dx)(0,0) = 1

⇒ b = 1 and a + c = 1

Since (0, 0) lies on curve,

∴ c = 0, a = 1

TRICK: (0, 0) lies on the curve. Only option (1) has c = 0

Q 5: The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is:

(A) 65/2^{7}

(B) 135/2^{9}

(C) 65/2^{8}

(D) 35/2^{7}

Answer: (B)

Let A and B be two subsets.

For each x∈{1,2,3,4,5} , there are four possibilities:

x ∈ A∩B, x∈ A’∩B, x ∈ A∩B’, x ∈ A’∩B’

So, the number of elements in sample space =45

Required probability

= (^{5}C_{2} × 3^{3})/4^{5}

= (10×27)/2^{10}

= 135/2^{9}

Q 6: The vector equation of the plane passing through the intersection of the planes

and

and the point (1, 0, 2) is:

Answer: (B)

Family of planes passing through the intersection of planes is

The above curve passes through

(3 − 1) + λ(1 + 2) = 0

⇒ λ = -2/3

Hence, equation of plane is

⇒

TRICK: Only option (2) satisfies the point (1, 0, 2)

Q 7: If P is a point on the parabola y = x^{2} + 4 which is closest to the straight line y = 4x − 1, then the coordinates of P are :

(A) (–2, 8)

(B) (1, 5)

(C) (3, 13)

(D) (2, 8)

Answer: (D)

Tangent at P is parallel to the given line.

dy/dx|P = 4

⇒2x_{1} = 4

⇒x_{1} = 2

Required point is (2, 8)

Q 8: Let A and B be 3×3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 − B2A2)X = O, where X is a 3×1 column matrix of unknown variables and O is a 3×1 null matrix, has:

(A) a unique solution

(B) exactly two solutions

(C) infinitely many solutions

(D) no solution

Answer: (C)

AT = A, BT = –B

Let A^{2}B^{2} − B^{2}A^{2}= P

PT = (A^{2}B^{2} – B^{2}A^{2})T

= (A^{2}B^{2})T – (B^{2}A^{2})T

= (B^{2})T (A^{2})T– (A^{2})T (B^{2})T

= B^{2}A^{2} – A^{2}B

⇒ P is a skew-symmetric matrix.

∴ ay + bz = 0 …(1)

–ax + cz = 0 …(2)

–bx – cy =0 …(3)

From equation (1), (2), (3)

Δ = 0 and Δ_{1} = Δ_{2}= Δ_{3} = 0

∴ System of equations has infinite number of solutions.

Q 9: If n ≥ 2 is a positive integer, then the sum of the series n+^{1}C_{2} + 2(^{2}C_{2} + ^{3}C_{2} + ^{4}C_{2} + …. + ^{n}C_{2}) is

(A) n(n + 1)^{2} (n + 2)/12

(B) n(n – 1)(2n + 1)/6

(C) n(n + 1)(2n + 1)/6

(D) n(2n + 1)(3n + 1)/6

Answer: (C)^{2}C_{2} = ^{3}C_{3}

Let S = ^{3}C_{3} + ^{3}C_{2} + ……. + ^{n}C_{2} = n+^{1}C_{3} (∵ ^{n}C_{r} + ^{n}C_{r–1} = ^{n+1}C_{r})

∴ ^{n+1}C_{2} + ^{n+1}C_{3} + ^{n+1}C_{3}

= ^{n+2}C_{2} + ^{n+1}C_{3}

=(n + 2)!/3!(n – 1)! + (n + 1)!/3!(n – 2)!

=(n + 2)(n + 1)n/6 + (n + 1)(n)(n – 1)/6 = (n(n + 1)(2 + 1))/6

TRICK : Put n = 2 and verify the options.

Q 10: If a curve y = f(x) passes through the point (1, 2) and satisfies xdy/dx + y = bx4, then for what value of b,

(A) 5

(B) 62/5

(C) 31/5

(D) 10

Answer: (D)

dy/dx + y/x = bx^{3}

I.F. =

= x

∴ yx = ∫bx^{4} dx = bx^{5}/5 + c

The above curve passes through (1, 2).

2 = b/5 + c

Also,

⇒ (b/25) ×32 + c ln 2 – b/25 = 62/5

⇒ c = 0 and b = 10

Q 11: The area of the region: R{(x, y): 5x^{2} ≤ y ≤ 2x^{2} + 9} is:

(A) 9√3 square units

(B) 12√3 square units

(C) 11√3 square units

(D) 6√3 square units

Answer: (B)

Required area

= 12√3

Q 12: Let f(x) be a differentiable function defined on [0,2] such that f’(x) =f ‘(2 – x) for all x∈(0, 2), f(0) = 1 and f(2) = e^{2}. Then the value of is:

(A) 1 + e^{2}

(B) 1 – e^{2}

(C) 2(1 – e^{2})

(D) 2(1+e^{2})

Answer: (A)

f'(x) = f'(2 – x)

On integrating both sides, we get

f(x) = -f(2 – x)+c

Put x = 0

f(0) + f(2) = c

⇒ c = 1+ e^{2}

⇒ f(x) + f(2 – x) = 1 + e^{2}

I =

= 1+e^{2}

Q 13: The negation of the statement ~p∧(p∨q) is∶

(A) ~p∧q

(B) p∧~q

(C) ~p∨q

(D) p∨∼q

Answer: (D)

Negation of ~p∧(p∨q) is

∼[~p∧(p∨q)]

≡ p∨ ∼(p∨q)

≡ p∨(∼p∧∼q)

≡ (p∨∼p)∧(p∨∼q)

≡ T∧(p∨∼q), where T is tautology.

≡ p∨∼q

Q 14: For the system of linear equations: x − 2y = 1, x − y + kz = −2, ky + 4z = 6,k ∈ R

Consider the following statements:

(A) The system has unique solution if k ≠ 2,k ≠ −2.

(B) The system has unique solution if k = -2.

(C) The system has unique solution if k = 2.

(D) The system has no-solution if k = 2.

(E) The system has infinite number of solutions if k ≠ -2.

Which of the following statements are correct?

(A) (B) and (E) only

(B) (C) and (D) only

(C) (A) and (D) only

(D) (A) and (E) only

Answer: (C)

x -2y + 0.z = 1

x – y + kz = -2

0.x + ky + 4z = 6

= 4 – k2

For unique solution, 4 – k^{2} ≠ 0

k ≠ ±2

For k = 2,

x – 2y + 0.z =1

x – y + 2z = -2

0.x + 2y + 4z = 6

= -8 + 2 (-20)

⇒Δx= -48 ≠ 0

For k = 2,Δx ≠ 0

So, for k = 2, the system has no solution.

Q 15: For which of the following curves, the line x + √3y = 2√3 is the tangent at the point (3√3/2, 1/2)?

(A) x^{2} + 9y^{2} = 9

(B) 2x^{2} – 18y^{2} = 9

(C) y^{2} = x/(6√3)

(D) x^{2} + y^{2} = 7

Answer: (A)

Tangent to x^{2} + 9y^{2} = 9 at point (3√3/2, 1/2) is x(3√3)/2 + 9y(1/2) = 9

⇒3√3 x + 9y = 18

⇒x +√3 y = 2√3

⇒ Option (A) is true.

Q 16: The angle of elevation of a jet plane from a point A on the ground is 600. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 300. If the jet plane is flying at a constant height, then its height is:

(A) 1200√3 m

(B) 1800√3 m

(C) 3600√3 m

(D) 2400√3 m

Answer: (A)

v = 432 × 1000/(60×60) m/sec

= 120 m/sec

Distance PQ = v × 20 = 2400 m

In ΔPAC

tan 60^{0} = h/AC

⇒ AC = h/(√3)

In ΔAQD

tan 30^{0} = h/AD

⇒AD = √3h

AD = AC + CD

⇒√3 h = h/√3 + 2400

⇒2h/√3 = 2400

⇒ h = 1200√3 m

Q 17: For the statements p and q, consider the following compound statements:

(a) (~q∧(p→q)) → ~p

(b) ((p∨q))∧~p) → q

Then which of the following statements is correct?

(A) (A) is a tautology but not (B)

(B) (A) and (B) both are not tautologies

(C) (A) and (B) both are tautologies

(D) (B) is a tautology but not (A)

Answer: (C)

(A) is tautology.

(B) is tautology.

∴ (A) and (B) both are tautologies.

Q 18: Let a, b∈R. If the mirror image of the point P(a, 6, 9) with respect to the line (x – 3)/7 = (y – 2)/5 = (z – 1)/(-9) is (20, b, -a, -9), then |a + b| is equal to:

(A) 86

(B) 88

(C) 84

(D) 90

Answer: (B)

P(a, 6, 9), Q (20, b, -a-9)

Midpoint of PQ=((a+20)/2, (b+6)/2, -a/2) lie on the line.

=> (a + 20 – 6)/14 = (b + 6 – 4)/10 = (-a – 2)/(-18)

=> (a + 14)/14 = (a + 2)/18

=> 18a + 252 = 14a + 28

=> 4a = -224

a = -56

(b + 2)/10 = (a + 2)/18

=> (b + 2)/10 = (-54)/18

=>(b + 2)/10 = -3

=> b = -32

|a + b| = |-56 – 32|

= 88

Q 19: Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x∈R. If |f(x) f'(x) f'(x) f”(x)| = 0, for all x∈R, then the value of f(1) lies in the interval:

(A) (9, 12)

(B) (6, 9)

(C) (3, 6)

(D) (0, 3)

Answer: (B)

Given f(x) f”(x) – f'(x)^{2} =0

Let h(x) = f(x)/f'(x)

Then h'(x) = 0

⇒ h(x) = k

⇒ f(x)/f'(x) = k

⇒ f(x) = k f’(x)

⇒ f(0) = k f'(0)

⇒ k = 1/2

Now, f(x) = ½ f'(x)

⇒∫ 2dx = ∫ f'(x)/f(x) dx

⇒ 2x = ln |f(x)| + C

As f(0) = 1 ⇒ C = 0

⇒ 2x = ln |f(x)|

⇒ f(x) = ±e^{2x}

As f(0) = 1 ⇒ f(x) = e^{2x}

∴ f(1) = e^{2} ≈ 7.38

Q 20: A possible value of tan (¼ sin^{-1} √63/8) is:

(A) 1/(2√2)

(B) 1/√7

(C) √7 – 1

(D) 2√2 – 1

Answer: (B)

tan (¼ sin^{-1} √63/8)

Let sin^{-1}(√63/8) = θ

sin θ = √63/8

cos θ = 1/8

2 cos^{2}(θ/2) – 1 = 1/8

⇒ cos^{2} θ/2 = 9/16

cos θ/2 = 3/4

⇒(1- tan^{2} θ/4 )/(1 + tan^{2} θ/4) = 3/4

tan θ/4 = 1/√7

Section B

Q 1: If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x – 2)^{2}+ (y – 3)^{2} = 25 at the point (5, 7) is A, then 24A is equal to ___.

Answer: Q is wrong

Equation of normal at P is

(y – 7) = ((7 – 3)/(5 – 2))(x – 5)

⇒ 3y – 21 = 4x – 20

⇒ 4x – 3y + 1 = 0

⇒ M is (-1/4, 0)

Equation of tangent at P is

(y – 7) = (-¾) (x – 5)

⇒ 4y – 28 = -3x + 15

⇒ 3x + 4y = 43

⇒ N is (43/3, 0)

The question is wrong. The normal cuts at a point on the negative axis.

Q 2: If a + α = 1, b + β = 2 and af(x) + αf(1/x) = bx + β/2, x ≠ 0 then the value of the expression [f(x) + f(1/x) ]/(x + 1/x)

Answer: 2

af(x) + αf(1/x) = bx + β/x …(i)

Replace x by 1/x

af(1/x) + αf(x) = b/x + βx …(ii)

(i) + (ii)

(a + α)[f(x) + f(1/x) ]

= (x + 1/x)(b + β)

⇒ ( f(x) + f(1/x))/(x + 1/x)

= 2/1

= 2

Q 3: If the variance of 10 natural numbers 1,1,1,…,1,k is less than 10, then the maximum possible value of k is ________.

Answer: 11

⇒ 10(9 + k^{2}) – (81 + k^{2} + 18k) < 1000

⇒ 90 + 10k^{2} – k^{2} – 18k – 81 < 1000

⇒ 9k^{2}– 18k + 9 < 1000

⇒ (k – 1)^{2} < 1000/9

⇒ k – 1< (10√10)/3

⇒ k < (10√10)/3 + 1

Maximum possible integral value of k is 11.

Q 4: Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (-5, 0). If the locus of the point P is a circle of radius r, then 4r^{2} is equal to ___.

Answer: 56.25

Let P be (h, k), A(5, 0) and B(-5, 0)

Given PA = 3PB

⇒ PA^{2} = 9PB^{2}

⇒ (h-5)^{2} + k^{2} = 9[(h + 5)^{2} + k^{2}]

⇒ 8h^{2} + 8k^{2} + 100h + 200 = 0

∴ Locus of P is x^{2} + y^{2} + (25/2)x + 25 = 0

Centre = (-25/4, 0)

∴ r^{2} = (-25/4)2 – 25

= 625/16 – 25

= 225/16

∴4r^{2} = 4×225/16

= 225/4

= 56.25

Q 5: The number of the real roots of the equation (x + 1)^{2} + |x – 5| = 27/4 is ___.

Answer: 2

For x ≥ 5,

(x + 1)^{2}+ (x – 5) = 27/4

⇒ x^{2} + 3x – 4 = 27/4

⇒ x^{2} + 3x – 43/4 = 0

⇒ 4x^{2} + 12x – 43 = 0

x = (-12 ± √(144 + 688))/8

x = (-12 ± √832)/8

= (-12 ± 28.8)/8

= (-3 ± 7.2)/2

= (-3 + 7.2)/2, (-3 – 7.2)/2 (therefore, no solution)

For x < 5,

(x + 1)^{2}– (x – 5) = 27/4

⇒ x^{2} + x + 6 – 27/4=0

⇒ 4x^{2} + 4x – 3=0

x = (-4 ± √(16 + 48))/8

x = (-4 ± 8)/8

⇒ x = -12/8, 4/8

∴ 2 real roots.

Q 6: The students S_{1}, S_{2},… S_{10} are to be divided into 3 groups A, B, and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is ___.

Answer: 31650

Number of ways

= ^{10}C_{1} [2^{9} – 2] + ^{10}C_{2} [2^{8}– 2] + ^{10}C_{3} [2^{7} – 2]

= 27 [^{10}C_{1}×4 + ^{10}C_{2} ×2 + ^{10}C_{3}] – 20 – 90 – 240

= 128 [40 + 90 + 120 ] – 350

= (128 × 250) – 350

= 10 [3165]

= 31650

Q 7: For integers n and r, let

The maximum value of k for which the sum

exists, is equal to ____.

Answer: Bonus

(1+x)^{10} = ^{10}C_{0} + ^{10}C_{1}x + ^{10}C_{2}x^{2 }+ …… + ^{10}C_{10}x^{10}

(1+x)^{15}= ^{15}C_{0} + ^{15}C_{1}x +^{15}C_{2}x^{2} + …… + ^{15}C_{k-1}x^{k-1} + ^{15}C_{k+1}x^{k+1} + …^{15}C_{15}x^{15}

Coefficient of x^{k+1} in (1+x)^{25}

= ^{25}C_{k+1}^{25}C_{k} + ^{25}C_{k+1} = ^{26}C_{k+1}

For maximum value

as per given, k can be as large as possible.

Q 8: Let λ be an integer. If the shortest distance between the lines x – λ = 2y – 1 = -2z and x = y + 2λ = z – λ is √7/2√2, then the value of |λ| is __

Answer: 1

(x – λ)/1 = (y – 1/2)/(1/2) = z/(-1/2)

(x – λ)/2 = (y-1/2)/1 = z/(-1) …(1) Point on line = (λ, 1/2, 0)

x/1 = (y + 2λ)/1 = (z – λ)/1 …(2) Point on line = (0, -2λ, λ)

= |-5λ – 3/2|/√14

= √7/(2√2) (Given)

⇒ |10λ + 3| = 7

⇒ λ = -1 as λ is an integer.

⇒ |λ| = 1

Q 9: If i = √-1. If [(-1 + i√3)^{21}/(1 – i)^{24} + (1 + i√3)^{21}/(1 + i)^{24} ] = k, and n = [|k| ] be the greatest integral part of |k|. Then

is equal to ___

Answer: 310

= 2^{9} e^{i(20π)} +2^{9} e^{iπ}

= 2^{9} + 2^{9} (-1)

= 0 = k

∴ n = 0

∑(j=0)^{5} (j + 5)^{2}– ∑(j=0)^{5} (j + 5)

= [5^{2} + 6^{2} + 7^{2} + 8^{2} + 9^{2} + 10^{2} ] – [5 + 6 + 7 + 8 + 9 + 10]

= [(1^{2}+ 2^{2} +… + 10^{2}) – (1^{2} + 2^{2}+ 3^{2}+ 4^{2}) ] – [(1 + 2 + 3 + … + 10) – (1 + 2 + 3 + 4) ]

= (385 – 30) -[55 – 10]

= 355 – 45

= 310

Q 10: The sum of first four terms of a geometric progression (G.P.) is 65/12 and the sum of their respective reciprocals is 65/18. If the product of first three terms of the G.P. is 1, and the third term is α then 2α is____

Answer: 3

Let terms of GP are a, ar, ar^{2}, ar^{3}

a + ar + ar^{2} + ar^{3} = 65/12 …….(1)

1/a + 1/ar + 1/(ar^{2}) + 1/ar^{3}) = 65/18

⇒1/a (r^{3} + r^{2 }+ r + 1)/r^{3}) = 65/18 …….(2)

(1)/(2), we get

a^{2}r^{3} = 18/12

= 3/2

Also, a^{3} r^{3} = 1

⇒ a(3/2) = 1

⇒a = 2/3

(4/9) r^{3} = 3/2

⇒ r^{3} = 33/23

⇒ r = 3/2

α = ar^{2}

= 2/3.(3/2)^{2} = 3/2

∴2α = 3

[/tab]

[/tabs]