IIT JEE Advanced 2015 Paper 1 (Code 8)

Test Name: IIT JEE Advanced 2015 Paper 1 (Code 8)

Question 1:

Consider a hydrogen atom with its electron in the nth orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is (hc = 1242 eV nm)

Correct Answer: 2

 

Solutions:

Energy of electron in nth orbit, En=-13.6n2eVEnergy of incident photon, Ep=hcλ=1242 eV nm90 nm=13.8 eVThe kinetic energy of ejected electron, K = 10.4 eV By conservation of energy, we know that

Ep-En=K 

∴ 13.8 eV -13.6 eV n2=10.4 eV ⇒n2=13.63.4=4 ⇒n=2[[VIDEO:13854]]

Question 2:

A bullet is fired vertically upwards with velocity v from the surface of a spherical planet . When it reaches its maximum height, its acceleration due to the planet’s gravity is 1/4th of its value at the surface of the planet. If the escape velocity from the planet is

vesc=vN, then the value of N is (ignore energy loss due to atmosphere)

Correct Answer: 2

 

Solutions:Let R be the radius of planet and h be the maximum height to which the bullet reaches when thrown upwards from its surface.

Given that,

gh=14gswhere gh=acceleration due to gravity at height hgs=acceleration due to gravity on the surface of planet⇒GMh2=14GMR2⇒h=2RApplying conservation of mechanical energy at the two points, we get

-GMmR+12mv2=-GMmh+0 ∵velocity at maximum height will be zero⇒12mv2=GMm2R ∵h=2R⇒v2=GMR⇒v=GMR …..(1)The escape velocity from the planet is given by

vesc=2GMRFrom (1), we getvesc=v2⇒N=2[[VIDEO:13855]]

Question 3:

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speed v1 and v2, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed v1 = 3 m/s, then v2 in m/s is (g = 10 m/s2)

Correct Answer: 7

 

Solutions:Given that, both the discs reach their respective final points B and D with the same speed. So, their final kinetic energies are equal. Applying the conservation of energy:

3212mv12+mg×30=3212mv22+mg×27⇒v22=v12+4g=9+40=49⇒v2=7 m s-1[[VIDEO:13856]]

Question 4:

Two spherical stars A and B emit blackbody radiation. This radius of A is 400 times that of B and A emits 104 times the power emitted from B. The ratio

λAλBof their wavelengths λA and λB at which the peaks occur in their respective radiation curves is

Correct Answer: 2

 

Solutions:Let AA and TA be the area and temperature of star A and AB and TB be the area and temperature of star B.

Given that,

RA = 400 RB

where RA and RB are the radius of star A and B respectively.

Power emitted from star A (PA) is given by

PA=eσAATA4     …..1where σ = Stefan’s constant

Similarly, power emitted from star B (PB) is given by

PB=eσABTB4     …..2Dividing equation (1) by (2), we get

PAPB=AATA4ABTB4=4πRA24πRB2TA4TB4⇒TBTA=RARB·PBPA14From the Wein’s law, we haveλT=constant⇒λATA=λBTB ∴λAλB=TBTA=400 110414=2[[VIDEO:13857]]

Question 5:

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life T years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of nT years, then the value of n is

Correct Answer: 3

 

Solutions:We know that Power generated ∝ Activity

Let electric power available = A0

Total power requirement, A=12.5100×A0⇒A=18A0From the relation

AA0=12nWhere n = number of half life

⇒18=12n⇒n=3[[VIDEO:13858]]

Question 6:

A Young’s double slit interference arrangement with slits S1 and S2 is immersed in water (refractive index = 4/3) as shown in the figure. The positions of maxima on the surface of water are given by x2 = p2m2 λ2d2, where λ is the wavelength of light in air (refractive index = 1), 2d is the separation between the slits and m is an integer. The value of p is

Correct Answer: 3

 

Solutions:

Path length S2O = μx2+d2     where O is point of interference on the water surfacePath length S1O = x2+d2Path difference, ∆x=S2O-S1OFor constructive interference, Δx=mλ⇒x2+d2 μ-1=mλ   Given, μ=4/3     ⇒x2=9m2λ2-d2On comparing, p =3. [[VIDEO:13859]]

Question 7:

Consider a concave mirror and a convex lens (refractive index = 1.5) of focal length 10 cm each, separated by a distance of 50 cm in air (refractive index = 1) as shown in the figure. An object is placed at a distance of 15 cm from the mirror. Its erect image formed by this combination has magnification M1. When the set-up is kept in a medium of refractive index 7/6, the magnification becomes M2. Then magnitude

M2M1is

Correct Answer: 7

 

Solutions:For mirror in air,

1v-115=-110⇒1v=115-110=-130⇒v=-30Magnification, M=-vu=-2For lens in air, we have

1v-1-20=110⇒v=20 cmMagnification, M’=vu=-1Net magnification in air, M1=MM’=2Image formation by concave mirror is unaffected by presence of medium. So, magnification due to it, M will remain same. Focal length of lens will change when kept in other medium.

f’f=μ-1μμ’-1where μ is the refractive index of lens materialμ’ is the refractive index of medium⇒f’f=1.5-11.57/6-1⇒f’=74f=704 cmSo, calculating image distance again for the lens, we get

1v-1-20=470⇒v=140 cmMagnification, M’=vu=140-20=-7Net magnification, M2=MM’=14∴M2M1=142=7

Question 8:

An infinitely long uniform line charge distribution of charge per unit length λ lies parallel to the y-axis in the y-z plane at

z=32a(see figure). If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is

λLnε0(ε0 = permittivity of free space), then the value of n is

Correct Answer: 6

 

Solutions:Consider a cylindrical gaussian surface with line charge as axis passing through its centre, such that rectangular surface ABCD are at edges of the cylinder.

To calculate the flux through the cylindrical surface AB, calculate the angle subtended by the line charge to line AD or BC.

From the given figure,

tanθ=a223a⇒θ=300From gauss law, total flux through cylindrical surface =

Qε0Flux through surface ABCD will be ,

6003600×Qε0=16×Qε0∴ Total flux through ABCD will be

Q6ε0=λL6ε0Hence, the value of n is 6. [[VIDEO:13860]]

Question 9:

Two identical glass rods S1 and S2 (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light P is placed inside rod S1 on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside S2. The distance d is

Option 1: 60 cm
Option 2: 70 cm
Option 3: 80 cm
Option 4: 90 cm
Correct Answer: 2

 

Solutions:

On surface of glass  S2, refraction takes place from air to glass rod.μ2v-μ1u=μ2-μ1R⇒μ2∞-1u=μ2-110⇒u=-20 cm

as  distance between S1 and S2 is  d, then distance of I from S1 will be (d-20) now for refraction from S1 glass to air:

 

μ1v’-μ2u’=μ1-μ2Rv’=d-20 and u’=-50⇒1d-20-μ2-50⇒1-μ2-10⇒1d-20+μ250=μ2-110Substituting μ2=1.5, we get1d-20+3100=120⇒d=70 cmHence, the correct option is (B).

Question 10:

A conductor (shown in the figure) carrying constant current I is kept in the x-y plane in a uniform magnetic field

B→. If F is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is(are)

Option 1: If

B→is along

z^, F ∝ (L + R)

Option 2: If

B→is along

x^, F = 0

Option 3: If

B→is along

y^, F ∝ (L + R)

Option 4: If

B→is along

z^, F = 0

Correct Answer: 1, 2, 3

 

Solutions:The force on a current carrying wire is calculated by following formula:

∫dF→=i∫dl→×B→When

B→is along

z^the magnetic force on the wire is given by

F→=i[2 (L+R)i^×B→]=i[2 (L+R)B](-j^)⇒F ∝ (L + R)

When

B→is along

x^the magnetic force on the wire is given by

F→=i[2 (L+R)i^×B→]⇒F = 0

When

B→is along

y^the magnetic force on the wire is given by

F→=i[2 (L+R)i^×B→]=i[2 (L+R)B](k^)⇒F ∝ (L + R)

Hence, the correct options are (A), (B), (C).

Question 11:

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature T. Assuming the gases are ideal, the correct statement(s) is(are)

Option 1: The average energy per mole of the gas mixture is 2RT.
Option 2: The ratio of speed of sound in the gas mixture to that in helium gas is

6/5.

Option 3: The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/2.
Option 4: The ratio of the rms speed of helium atoms to that of hydrogen molecules is

1/2.

Correct Answer: 1, 2, 4

 

Solutions:Total energy of the mixture of hydrogen and helium is:

3RT2+5RT2=4RTAverage energy per moles will be 4RT2=2RTFor ratio of speed of sound in gas mixture to that in helium mixture:

Vsound =γRTMVsound, mixVsound, helium=γmixRTMmixγHeRTMHe⇒γmixMHeMmixγHeNow, γmix=1×52+1×721×32+1×52=32Mmix=1×2+1×42=3therefore, ⇒γmixMHeMmixγHe=65For the ratio of the rms speed of helium atoms to that of hydrogen molecules â€‹

vrms=3RTM         ∴vrms ∝1MLet the rms velocity for helium and hydrogen be vHe and vH2 respectvely.vHe vH2=1412=12Hence, the correct options are (A), (B) and (D).

Question 12:

In an aluminum (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure. The electrical resistivities of Al and Fe are 2.7 × 10–8 Ω m and 1.0 × 10–7 Ω m, respectively. The electrical resistance between the two faces P and Q of the composite bar is

Option 1: 247564μΩ
Option 2: 187564μΩ
Option 3: 187549μΩ
Option 4: 2475132μΩ
Correct Answer: 2

 

Solutions:The resistances due to aluminium bar and iron hole are calculated as under:

RAl=ρℓA=2.7×10-8×50×10-349-4×10-6=3×10-5 ΩRFe=1.0×10-7×50×10-34×10-6=125×10-5The two resistances will be in parallel. So,Req=RAl || RFe=3×10-5×125×10-53×10-5+125×10-5    =3×125128×10-5×106 μΩ = 187564μΩHence, the correct option is (B). [[VIDEO:13861]]

Question 13:

For photo-electric effect with incident photon wavelength λ, the stopping potential is V0. Identify the correct variation(s) of V0 with λ and 1/λ.

Option 1:  
Option 2:  
Option 3:
Option 4:
Correct Answer: 1, 3

 

Solutions:Stopping potential is given by

hcλ=eV0+φV0=hceλ-ϕeAs,  V0 ∝ 1λThe graphs given in (A) and (C) represent correct relations.

Hence, the correct options are (A) and (C). [[VIDEO:13862]]

Question 14:

Consider a vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

Option 1: If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm.
Option 2: If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.005 mm.
Option 3: If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm.
Option 4: If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.005 mm.
Correct Answer: 2, 3

 

Solutions:For vernier callipers,

1 division on main scale=18cm=0.125 cm5 divisions of the vernier scale=4 divisions of the main scale                                                       = 4×0.125=0.5 cm∴ 1 division of vernier scale=0.55=0.1 cmLeast count of vernier = 1 main scale division − 1 vernier scale division = 0.125 − 0.1 = 0.025 cm

Least count of screw gauge =

pitch100When the pitch of the screw gauge is twice the least count of vernier callipers Least count of the screw gauge =

0.05100 cm=0.5100 mm=0.005 mmWhen least count of linear scale of screw gauge is twice the least count of vernier callipers

Least count of linear scale = 2 ✕ 0.025 cm = 0.05 cm Then pitch of the screw gauge will be = 2 ✕ 0.05 cm = 0.1 cm = 1 mm ∴ Least Count of screw gauge =

1 mm100=0.01 mmHence, the correct options are (B) and (C).

Question 15:

Planck’s constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are)

Option 1: M∝c
Option 2: M∝G
Option 3: L∝h
Option 4: L∝G
Correct Answer: 1, 3, 4

 

Solutions:Let LKhx cy Gz

Here, x, y, z are the powers of h, c and G respectively. ∴ L ∝ (ML2T−1)x (LT−1)y (M−1L3T−2)z

L Mx−z L2x+y+3z T−x−y−2z

x − z = 0     …..(2) 2x + y + 3z = 1     …..(3) x + y + 2z = 0     …..(4)

On solving, we get

x=12, y=-32, z=12 Substituting in 1 we getL=Kh12c-32G12∴ Option (C) and (D) are correct.

Similarly Let Mha cb Gc

Here, a, b, and c are the powers of h, c and G respectively ∴ M ∝ (ML2T−1)x (LT−1)y (M−1L3T−2)zMMx−z L2x+y+3z Tx−y−2z

xz = 1    …..(5) 2x + y + 3z = 0     …..(6) x + y + 2z = 0     …..(7)

Solving the above equation, we get

x=12,  z=-12,   y=12∴ M∝h,       M∝c      ∴ Option (A) is correct.Hence, the correct options are (A), (C), and (D).

Question 16:

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ω1 and ω2 and have total energies E1 and E2, respectively. Then variation of their momenta p with position x are shown in the figures. If

ab=n2and

aR=n, then the correct equation(s) is(are)

Option 1: E1ω1 = E2ω2
Option 2: ω2ω1=n2
Option 3: ω1ω2 = n2
Option 4: E1ω1=E2ω2
Correct Answer: 2, 4

 

Solutions:Let m be the mass of each of the independent harmonic oscillators. From the figure, maximum momentum for first harmonic oscillator is b and maximum momentum of the second harmonic oscillator is R. Energy of first harmonic oscillator (E1) is given by

E1=12mω12a2=b22m         ⇒ab=1mω1=n2     …..1Energy of second harmonic oscillator E2 is given byE2=12mω22R2=R22m⇒1mω2=1     …..2          From 1 and 2 ω2ω1=n2On dividing the energies of two harmonic oscillator, we getE1E2=ω1ω22 aR2=1n2·ω1ω2·n2⇒E1ω1=E2ω2Hence, the correct options are (B) and (D).

Question 17:

A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two point masses each of mass

M8at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is

89ωand one of the masses is at a distance of

35Rfrom O. At this instant the distance of the other mass from O is

Option 1: 23R
Option 2: 13R
Option 3: 35R
Option 4: 45R
Correct Answer: 4

 

Solutions:Let us consider that the other mass is at x distance from centre O. Applying conservation of angular momentum about the axis, we get

MR2ω=MR2+M835R2+M8x289ω⇒MR2ω=MR289ω+M935R2ω+M9x2ω⇒9MR2ω=8MR2ω+M35R2ω+Mx2ω⇒9R2=8R2+35R2+x2⇒x2=R2-35R2⇒x=45RHence, the correct option is (D). [[VIDEO:13863]]

Question 18:

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density λ are kept parallel to each other. In their resulting electric field, point charges q and −q are kept in equilibrium between them. The point charges are confined to move in the x direction only. If they are given a small displacement about their equilibrium positions, then the correct statement is

Option 1: Both charges execute simple harmonic motion.
Option 2: Both charges will continue moving in the direction of their displacement.
Option 3: Charge +q executes simple harmonic motion while charge −q continues moving in the direction of its displacement.
Option 4: Charge −q executes simple harmonic motion while charge +q continues moving in the direction of its displacement
Correct Answer: 3

 

Solutions:When +q is displaced towards the +ve x-direction (towards right), it experiences a repulsive force in opposite direction (towards left). Thus repulsive force dominates in this and resulting force is towards

x-direction (towards left)

The restoring force is given by:

F=q2kλd-x-2kλd+x=2k2xqλd2-x2F≅4kλqd2x ∵d2-x2≈d2 as x is very smallThis equation represents a simple harmonic motion as

F∝xNow, in case of charge

-q , when it is displaced towards +x direction (towards right), it gets attracted and â€‹continues moving in the direction of its displacement.

Hence, the correct answer is option C.

Question 19:

Match the nuclear processes given in column I with the appropriate option(s) in column (II).

Column I Column II
(A) Nuclear fusion (P) Absorption on thermal neutrons by

U92235

(B) Fission in a nuclear reactor (Q) Co2760nucleus
(C) β−decay (R) Energy production in stars via hydrogen conversion to helium
(D) γ-ray emission (S) Heavy water
(T) Neutrino emission
Correct Answer: 1#3|2#1,4|3#2,5|4#3

 

Solutions:(A) → (R); (B) → (P), (S); (C) → (Q), (T); (D) → (R) [[VIDEO:13865]]

Question 20:

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U0 are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

Column I Column II
(A) U1x=U021-xa22 (P) The force acting on the particle is zero at x = a.
(B) U2x=U02xa2 (Q) The force acting on the particle is zero at x = 0.
(C) U3x=U02xa2 exp-xa2 (R) The force acting on the particle is zero at x = −a
(D) U4x=U02xa-13xa3 (S) The particle experiences an attractive force towards x = 0 in the region |x| < a.
(T)  The particle with total energy

U04can oscillation about the point x = −a.

Correct Answer: 1#1,2,3,5|2#2,4|3#1,2,3,4|4#1,3,5

 

Solutions:

(A) U1=U021-xa22⇒Force, F=-dU1dx=-U02·21-xa2×-2xa2⇒F=U021-xa2×2xa2  Now, F=0 if x=0 and1-xa2=0⇒xa2=1⇒xa=±1⇒ x=±a∴A→P, Q, R, T (B) U2x=U02a2·x2F=dU2dx=-U02a2·2x⇒F=-U0a2·x    Now, F=0 if x=0∴ (B) →Q, SC U3x=U02xa2·e-xa2F=-dU3dx=U02xa2·e-xa2·-2xa2+e-xa2·U02·2xa2⇒F=-U02·2xa2-xa2·e-xa2+e-xa2=-U0·xa21-xa2·e-xa2=-U0a2·x·e-xa2 1-xa2Now, F=0 at x=0, x=±a. Also F is negative for x<a∴(C)→P, Q, R, S D U4x=U02xa-13xa3F=-dU4dx=-U021a-13·3x2a3⇒F=-U02a1-x2a2Now, F=0 at x=±a. Also F is negative for x<a.∴(D) →P, R, T[[VIDEO:13866]]

Question 21:

The number of resonance structures for N is

Correct Answer: 9

 

Solutions:

The resonance structures of N can be represented as

Thus, the species N has nine resonance structures.

Question 22:

The total number of lone pairs of electron in N2O3 is

Correct Answer: 8

 

Solutions:

The total number of lone pairs of electrons in N2O3 is 8.

Question 23:

For the octahedral complexes of Fe3+ in SCN (thiocyanato−S) and in CN ligand environments, the difference between the spin-only magnetic moments in Bohr magnetons (when approximated to the nearest integer) is [Atomic number of Fe = 26]

Correct Answer: 4

 

Solutions:Electronic configuration of Fe3+: [Ar] 3d5 CN is a strong field ligand which causes the pairing of electrons in the d-orbitals of metal, whereas, SCN is a weak field ligand.

Now, valence shell electronic configuration of Fe3+ in [Fe(CN)6]3− is given as:

Number of unpaired electrons, n = 1 Magnetic moment,

μ = nn+2 = 11+2 = 3B.M.

Now, valence shell electronic configuration of Fe3+ in [Fe(SCN)6]3− is given as

Number of unpaired electrons, n = 5 Magnetic moment,

μ = nn+2 = 55+2 = 35B.M.

Difference in magnetic moment values =

35 – 3 = 4 B.M.

Question 24:

Among the triatomic molecules/ions,

BeCl2, N3-, N2O, NO2+, O3, SCI2, ICI2-, I3-and XeF2 the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital (s) is [Atomic number : S = 16, Cl = 17, I = 53 and Xe = 54]

Correct Answer: 4

 

Solutions:Among the given triatomic molecules/ions, the species having linear geometry with hybridisation not involving the contribution from the d orbital(s), are BeCl2,

N3-, N2O and

NO2+.

They all involve sp hybridisation.

Thus, the total number of such molecules is 4.

Question 25:

Not considering the electronic spin, the degeneracy of the second excited state (n = 3) of H atom is 9, while the degeneracy of the second excited state of H is

Correct Answer: 3

 

Solutions:Order of energy for H is decided by the (n + l) rule.

Electronic configuration of H = 1s2 2s0 2p0

The second excited state of H is 2p, which has a degeneracy of 3.

Question 26:

All the energy released from the reaction X → Y, ∆rGo = −193 kJ mol−1 is used for oxidizing M+ as M+ → M3+ + 2e, Eo = −0.25 V. Under standard conditions, the number of moles of M+ oxidized when one mole of X is converted to Y is [F = 96500 C mol−1]

Correct Answer: 4

 

Solutions:Conversion of one mole of X to Y

X → Y     ΔrGo = -193 kJ mol-1Now, for the reaction

M+ → M3+ + 2 e-      Eo = -0.25 Vthe standard Gibbs free energy change is given as

ΔGo = -nFEo⇒ ΔGo = -2 × 96500 × -0.25             = 48.25 kJ mol-1Thus, 48.25 kJ mol−1 energy oxidise one mole of M+. So, 193 kJ mol−1 energy will oxidise ⇒

19348.25 = 4moles of M+.

Hence, the 4 moles of M+ oxidised when one mole of X is converted to Y.

Question 27:

If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride−ammonia complex (which behaves as a strong electrolyte) is −0.0558 °C, the number of chloride(s) in the coordination sphere of the complex is [Kf of water = 1.86 K kg mol−1]

Correct Answer: 1

 

Solutions:Concentration of aqueous solution of cobalt (III) chloride-ammonia complex, m = 0.01 m Freezing point, Tf = −0.0558 oC = 272.9442 K ⇒ Depression in freezing point, ΔTf = 273 − (272.9442) = 0.0558 K Kf of water = 1.86 K Kg mol−1

Now, ΔTf = iKfm

⇒0.0558 = i × 1.86 × 0.01⇒ i = 3 

Thus, if the given complex behaves as a strong electrolyte, then the number of particles given by the complex upon dissociation in the aqueous solution is 3.

Thus, the complex has the molecular formula [Co(NH3)5Cl]Cl2.

Hence, the number of chloride in the coordination sphere of the complex is 1. [[VIDEO:13903]]

Question 28:

The total number of stereoisomers that can exist for M is

Correct Answer: 2

 

Solutions:Number of stereoisomers = 2n Here, n is the number of stereocentres. Thus, the total number of stereoisomers for the given compound is four as there are two stereocenters in the molecule as shown below:

These four stereoisomers will include geometrical as well as optical isomers, but this compound is unable to show geometrical isomerism. Therefore, only two optical isomers are possible for the given compound.

Question 29:

The correct statement (s) about Cr2+ and Mn3+ is (are) [Atomic numbers of Cr = 24 and Mn = 25]

Option 1: Cr2+ is a reducing agent
Option 2: Mn3+ is an oxidizing agent
Option 3: Both Cr2+ and Mn3+ exhibit d4 electronic configuration
Option 4: When Cr2+ is used as a reducing agent, the chromium ion attains d5 electronic configuration
Correct Answer: 1,2,3

 

Solutions:The electronic configurations of Cr2+ and Mn3+ ions are given below: Cr2+ = [Ar] 3d4 Mn3+ = [Ar] 3d4 So, both the ions show d4 electronic configuration.

Cr is reducing as its configuration changes from d4 to d3, the latter having a stable half-filled t2g level.

Mn3+ acts as a strong oxidising agent as it needs only one electron to attain the stable half-filled d5 electronic configuration.

Thus, the correct statements are A, B and C.

Question 30:

Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process is (are)

Option 1: Impure Cu strip is used as cathode
Option 2: Acidified aqueous CuSO4 is used as electrolyte
Option 3: Pure Cu deposits at cathode
Option 4: Impurities settle as anode−mud
Correct Answer: 2,3,4

 

Solutions:During refining of blister copper to obtain pure copper by the process of electrolytic refining, the impure copper is used as anode and acidified copper sulphate solution is used as an electrolyte. In this process, pure copper from anode is deposited on the strip of pure copper which is used as cathode, while the impurities settle down as anode mud.

The correct answer is B, C and D.

Question 31:

Fe3+ is reduced to Fe2+ by using

Option 1: H2O2 in presence of NaOH
Option 2: Na2O2 in water
Option 3: H2O2 in presence of H2SO4
Option 4: Na2O2 in presence of H2SO4
Correct Answer: 1,2

 

Solutions:Na2O2 in water furnishes H2O2 and NaOH which reduces Fe3+ to Fe2+ as shown in the reactions below: Na2O2 + 2 H2O → H2O2 + 2 NaOH 2 Fe3+ + H2O2 + 2 OH → 2 Fe2+ + 2 H2O + O2

Hence, the correct options are A and B.

Question 32:

The % yield of ammonia as a function of time in the reaction

N2 (g) + 3H2 (g) ⇌ 2NH3 (g), ΔH < 0

at (P, T1) is given below.

If this reaction is conducted at (P, T2), with T2 > T1, the % yield of ammonia as a function of time is represented by

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 2

 

Solutions:The given reaction is an exothermic reaction. When an exothermic reaction is carried out at high temperatures; then at equilibrium; according to Le Chatelier’s principle, the reaction will proceed in backward direction. Thus, on increasing the temperature, the percentage yield of ammonia will be given by the following graph:

Hence, the correct answer is option B.

Question 33:

If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are

Option 1: 12,18
Option 2: 1,14
Option 3: 12,12
Option 4: 14,18
Correct Answer: 1

 

Solutions:Oxygen atoms are arranged in ccp array. Number of oxygen atoms per unit cell = 4 Number of octahedral voids per unit cell = 4 Number of tetrahedral voids per unit cell = 8 Number of occupied octahedral voids per unit cell = 4m Number of occupied tetrahedral voids per unit cell = 8n

(4m

×3) + (8n

×2) + (4

×(−2)) = 0 12m + 16n − 8 = 0 3m + 4n − 2 = 0 3m + 4n = 2

This relation is satisfied by values

12,18. Hence, the correct answer is option A.

Question 34:

Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is(are)

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 2,4

 

Solutions:(A)

(B)

(C)

(D)

Hence, the correct options are B and D.

Question 35:

The major product of the following reaction is

Option 1:
Option 2:  
Option 3:
Option 4:
Correct Answer: 1

 

Solutions: Hence, the correct option is A.

Question 36:

In the following reaction, the major product is

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 4

 

Solutions: Hence, the correct option is D.

Question 37:

The structure of D-(+)-glucose is

The structure of L-(–)-glucose is

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 1

 

Solutions:      L-(–)-glucose

Hence, the correct option is A.

Question 38:

The major product of the reaction is

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 3

 

Solutions: Hence, the correct option is C.

Question 39:

Match the anionic species given in Column I that are present in the ore(s) given in Column II.

Column I Column II
(A) Carbonate (P) Siderite
(B) Sulphide (Q) Malachite
(C) Hydroxide (R) Bauxite
(D) Oxide (S) Calamine
(T) Argentite
Correct Answer: 1#1,2,4|2#5|3#2,3|4#3

 

Solutions:A → (P), (Q) and (S); (B) → (T); (C) → (Q) and (R); (D) → (R) (P) = Siderite = FeCO3 (Q) = Malachite = CuCO3.Cu(OH)2 (R) = Bauxite = AlOx(OH)3-2x (Here, 0 < x < 1) (S) = Calamine = ZnCO3 (T) = Argentite = Ag2S

Question 40:

Match the thermodynamic processes given under Column I with the expressions given under Column II.

Column I Column II
(A) Freezing of water at 273 K and 1 atm (P) q = 0
(B) Expansion of 1 mol of an ideal gas into a vacuum under isolated conditions (Q) w = 0
(C) Mixing of equal volumes of two ideal gases at constant temperature and pressure in an isolated container (R) ΔSsys < 0
(D) Reversible heating of H2(g) at 1 atm from 300 K to 600 K, followed by reversible cooling to 300 K at 1 atm (S) ΔU = 0
(T) ΔG = 0
Correct Answer: 1#3,5|2#1,2,4|3#1,2,4|4#1,2,4,5

 

Solutions:

A → (R) and (T)

H2O (ℓ) ⇌ H2O(s)

ΔG = 0 and ΔSsys < 0

(B) → (P), (Q), (S)

Free expansion i.e. w = 0, ΔU = 0, q = 0

(C) → (P), (Q), (S)

q = 0, ΔU = 0, w = 0

(D) → (P), (Q), (S) and (T)

Question 41:

Let the curve C be the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y = –5, then the distance between A and B is

Correct Answer: 4

 

Solutions:Let P(t2, 2t) be a point on the curve y2 = 4x and Q(h, k) be its mirror image with respect to x + y + 4 = 0 on C.

Then,

h-t21=k-2t1=-2t2+2t+42⇒h-t2=k-2t=-t2-2t-4⇒h=-2t+4 and k=-t2+4If C intersects the line y = −5, then k = −5.

⇒-t2+4=-5⇒t=±1∴ h=-2, k=-5 and h=-6, k=-5So, the points A and B are (−2, −5) and (−6, −5).

⇒ AB = 4                    (Using Distance Formula)

Therefore, the distance between A and B is 4 units. [[VIDEO:13790]]

Question 42:

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

Correct Answer: 8

 

Solutions:Let the fair coin be tossed n number of times.

Given:

P(at least two heads)

≥0.96

⇒1

-P(no head)

-P(one head)

≥0.96

⇒1-nC012012n-nC11212n-1≥0.96 ⇒1-12n-n12n≥0.96 ⇒0.04≥n+112n ⇒125≥n+12n ⇒25≤2nn+1This inequality holds for

n≥8.

Therefore, the minimum number of times a fair coin needs to be tossed is 8.

Question 43:

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of

mnis

Correct Answer: 5

 

Solutions:Keeping all 5 five girls together and considering it as one entity, we have 6 entities (5 boys and a group of 5 girls) which can be arranged in 6! ways. But, the group of 5 girls can also arrange among themselves in the group in 5! ways. So, the number of ways in which 5 boys and 5 girls can stand in a queue such that all the girls stand consecutively in the queue is 6! × 5!. ∴ n = 6! × 5!

Now, 4 girls which stand in the queue consecutively can be selected out of 5 girls in

C45ways. Keeping all 4 five girls together and considering it as one entity, we have 6 entities (5 boys and a group of 4 girls) which can be arranged in 6! ways. But, the group of 4 girls can also arrange among themselves in the group in 4! ways. The remaining girl can be arranged in 5 ways. So, the number of ways in which 5 boys and 5 girls can stand in a queue such that exactly 4 girls stand consecutively in the queue is

C45× 6! × 4! × 5. ∴ m =

C45× 6! × 4! × 5

Thus,

mn=C45×6!×4!×56!×5!=5 

Therefore, the value of

mnis 5.

Question 44:

If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x – 3)2 + (y + 2)2 = r2, then the value of r2 is

Correct Answer: 2

 

Solutions:The end points of the latus rectum of the parabola are (1, ±2).

The equation of the normal to the parabola at the point (1, 2) is

y-2=-22x-1                  y-y1=-y12ax-x1⇒x+y-3=0Similarly, the equation of the normal to the parabola at the point(1, −2) is

x-y-3=0.

The centre of the given circle is (3, −2).

The normals to the parabola drawn at the end points of its latus rectum are tangents to the given circle.

So, the lengths of perpendicular drawn from the centre of the circle to the normals is equal to the radius of the circle.

∴3-2-312+12=r=3+2-312+12⇒r2=2Therefore, the value of r2 is 2.

Question 45:

Let f : ℝ → ℝ be a function defined by

fx=x,x≤20,x>2, where [x] is the greatest integer less than or equal to x. If

I=∫-12xf x22+fx+1dx,then the value of (4I – 1) is

Correct Answer: 0

 

Solutions:

I=∫-12xfx22+fx+1dx⇒I=∫-10xfx22+fx+1dx+∫01xfx22+fx+1dx+∫12xfx22+fx+1dx+∫22xfx22+fx+1dx    ….. 1We have

fx=x,x≤20,x>2So, fx2=x2=0,-1≤x<11,1≤x<2and fx2=0∀x>2 ∴I=∫-10x·02+0dx+∫01x·02+1dx+∫12x·12+0dx+∫22x·02+0dx           Using 1⇒I=14×212⇒I=142-1⇒I=14∴ 4I-1=0Therefore, the value of (4I − 1) is 0.

Question 46:

A cylindrical container is to be made from certain solid material with the following constraints. It has a fixed inner volume of V mm3, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is a radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of

V250πis

Correct Answer: 4

 

Solutions:Given:

Fixed inner volume,

V=πr2h   …..iLet the volume of the material used to make the container be v and the height of inner cylindrical part be h.

v=πr+222+πr+22h-πr2h⇒v=2πr+22+πh4r+4⇒v=2πr+22+4πhr+1⇒v=2πr+22+4r+1Vr2     Using i⇒v=2πr+22+4V1r+1r2⇒dvdr=4πr+2+4V-1r2-2r3For maxima or minima,

dvdr=0 ⇒4πr+2+4V-1r2-2r3=0⇒4π12+4V-10-2103=0    Given: r=10 mm⇒π12-12103V=0⇒V=103π⇒V250π=4Therefore, if the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of

V250πis 4. [[VIDEO:13791]]

Question 47:

Let F(x) =

∫xx2+π62cos2 t dt for all x ∈

ℝand f :

0, 12 →           [0, ∞)be a continuous function. For a

∈0, 12, if F'(a) + 2 is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a, then f(0) is

Correct Answer: 3

 

Solutions:We have

F(x) =

∫xx2+π61+cos2tdt =txx2+π6+sin2t2xx2+π6=x2-x+π6+12sin2x2+π3-sin2x∴ F’x=2x-1+12cos2x2+π34x-2cos2xNow,∫0a fxdx=F’a+2⇒∫0a fxdx=2a-1+12cos2a2+π34a-2cos2a+2Differentiating both sides w.r.t. a, we getfa=2+124cos2a2+π3-4asin2a2+π34a+4sin2aSubstituting a=0, we getf0=2+124×12-0+0=3Therefore, f(0) is 3. [[VIDEO:13792]]

Question 48:

The number of distinct solutions of the equation :

54cos2 2x + cos4 x + sin4 x + cos6 x + sin6 x = 2in the interval 10, 2π is

Correct Answer: 8

 

Solutions:

54cos22x+cos4x+sin4x+cos6x+sin6x=2 ⇒54cos22x+cos2x+sin2x2-2cos2xsin2x+cos3x+sin3x-3cos2xsin2x=2 ⇒54cos22x+1-12sin22x+1-34sin22x=2 ⇒54cos22x-54sin22x=0 ⇒54cos4x=0 ⇒cos4x=0Now,

x∈0,2π ⇒4x∈0,8π x=π8,3π8,5π8,7π8,9π8,11π8,13π8,15π8Therefore, the number of distinct solutions of the given equation is 8. [[VIDEO:13793]]

Question 49:

Let y(x) be a solution of the differential equation (1 + ex) y’ + yex = 1. If y(0) = 2, then which of the following statements is (are) true?

Option 1: y(−4) = 0
Option 2: y(−2) = 0
Option 3: y(x) has a critical point in the interval (−1, 0)
Option 4: y(x) has no critical point in the interval (−1, 0)
Correct Answer: 1,3

 

Solutions:The given differential equation is

1+exy’+yex=1⇒y’+ex1+exy=11+exThis is a linear differential equation.

Integrating factor =

e∫ex1+exdx 

=eloge1+ex=1+exTherefore, the solution of the given equation is given by

y1+ex=∫11+ex.1+exdx+C 

⇒y1+ex=x+CGiven: y0=2∴21+1=C⇒C=4So,

y=4+x1+ex∴ y-4=0y-2=2x-2+1For critical points, dydx=0⇒1+ex-x+4ex1+ex2=0⇒1-3ex-xex=0⇒3+xex=1⇒3+x=e-x

From the graph we can see that the critical point lies in the interval (−1, 0).

Hence, the correct options are A and C.

Question 50:

Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented by the differential equation Py” + Qy’ + 1 = 0, where P, Q are functions of x, y and y’

here y’ = dydx, y” = d2ydx2, then which of the following statements is (are) true?

Option 1: P = y + x
Option 2: P = y − x
Option 3: P + Q = 1 − x + y + y’ + (y’)2
Option 4: P − Q = x + y − y’ − (y’)2
Correct Answer: 2,3

 

Solutions:Let the equation of the circle be

x-h2+y-h2=r2           …..1 

On differentiating (1) with respect to x, we get

x-h+y-hdydx=0      …..2⇒x-h+ydydx-hdydx=0 ⇒h=x+ydydx1+dydx                …..3Differentiating (2) with respect to x, we get

1+y-hd2ydx2+dydx2=0Substituting the value of h from (3), we get

1+y-x+ydydx1+dydxd2ydx2+dydx2=0⇒1+y-x1+dydxd2ydx2+dydx2=0⇒1+dydx+y-xd2ydx2+1+dydxdydx2=0⇒1+y’+y-xy”+1+y’y’2=0⇒y-xy”+1+y’+y’2y’+1=0∴P=y-x, Q=1+y’+y’2So, P + Q =

1-x+y+y’+y’2Hence, the correct options are B and C. [[VIDEO:13794]]

Question 51:

Let g:

ℝ →          ℝbe a differentiable function with g(0) = 0, g’ (0) = 0 and g'(1) ≠ 0. Let

fx = x x gx,x ≠00,x = 0and h(x) = e|x| for all x ∈

ℝ. Let (f ° h) (x) denote f(h(x)) and (h ° f) (x) denote h (f(x)). Then which of the following is (are) true ?

Option 1: f is differentiable at x = 0
Option 2: h is differentiable at x = 0
Option 3: f ° h is differentiable at x = 0
Option 4: h ° f is differentiable at x = 0
Correct Answer: 1,4

 

Solutions:We have

g(0) = 0, g‘(0) = 0 and g‘(1) ≠ 0

Consider option (A).

fx = gx,x>0-gx,x<00,x=0 f’0+=limh→0fh-f0h         =limh→0gh-g0h         =g’0         =0f’0-=limh→0f-h-f0-h         =limh→0-g-h+g0-h         =g’0         =0So, the function f(x) is differentiable at x = 0.

Consider option (B).

It is given that

hx=ex.

∴hx = ex,x>0e-x,x<01,x=0 h’0+=limx→0hx-h0x         =limx→0ex-1x         =1h’0-=limx→0h-x-h0-x         =limx→0ex-1-x         =-1So, h(x) is not differentiable at x = 0.

Consider option (C).

fohx=fhx=ghx                           ∵hx>0foh’0+=limx→0ghx-gh0x               =limx→0gex-g1x               =g’1foh’0-=limx→0gh-x-gh0-x               =limx→0gex-g1-x               =-g’1So, (foh)(x) is not differentiable at x = 0.

Consider option (D).

hofx=hfxhof’0+=limx→0hfx-hf0x               =limx→0hgx-h0x               = limx→0hgx-1x               =0            hof’0-=limx→0hf-x-hf0-x               =limx→0h-g-x-1-x               =0So, hof(x) is differentiable at x = 0.

Hence, the correct options are A and D. [[VIDEO:13795]]

Question 52:

Let

fx=sin π6sin π2sin xfor all x ∈ ℝ and

gx=π2sin xfor all x ∈ ℝ. Let (f ∘ g) (x) denote f(g(x)) and (g ∘ f) (x) denote g(f(x)). Then which of the following is(are) true?

Option 1: Range of f is -12, 12
Option 2: Range of f∘ g is -12, 12
Option 3: limx→0 fxgx=π6
Option 4: There is an x ∈ ℝ such that (g ∘ f) (x) = 1
Correct Answer: 1,2,3

 

Solutions:

fx=sinπ6sinπ2sinx 

Let π2sinx=θ            ⇒θ∈-π2, π2So, fx=sinπ6sinθLet π6sinθ=Φ⇒Φ∈-π6, π6∴ fx=sinΦ⇒fx∈-12, 12Now,fogx=sinπ6sinπ2sinπ2sinx Similarly,

fogx∈-12, 12 

limx→0f(x)g(x)=sinπ6sinπ2sinxπ6sinπ2sinx×π6sinπ2sinxπ2sinx=π6Now, gf(x)=π2sinsinπ6sinπ2sinx-π2sin12≤gf(x)≤π2sin12-0.75≤gf(x)≤0.75So, there is no x R such that (gof)(x) = 1.

Hence, the correct options are A, B and C. [[VIDEO:13796]]

Question 53:

Let ∆PQR be a triangle. Let

a→=QR→, b→=RP→ and c→=PQ→. If a→=12, b→=43 and b→·c→=24, then which of the following is(are) true?

Option 1: c→22-a→=12
Option 2: c→22+a→=30
Option 3: a→×b→+c→×a→=483
Option 4: a→·b→=-72
Correct Answer: 1,3,4

 

Solutions:In

∆PQR,

a→=QR→, b→=RP→ and c→=PQ→. So,

a→+b→+c→=0⇒b→+c→=-a→⇒b→+c→.b→+c→=-a→.-a→⇒b→2+c→2+2b→·c→=a→2⇒c→2=144-48-48⇒c→2=48⇒c→=43Thus,

∆PQRis an isosceles triangle.

Now,

c→22-a→=482-12=12Let

θbe the angle between

b→and

c→. Now,

b→·c→=24⇒b→c→cosθ=24⇒cosθ=12⇒θ=120°                   θ≠60°⇒∠PQR=∠QRP=30°               ∵PQ→=RP→ 

a→+b→+c→=0⇒a→×b→+a→×c→=0⇒a→×b→=c→×a→∴a→×b→+c→×a→=2a→×b→=212×43×sin30°=483Also,

a→+b→+c→=0⇒a→+b→=-c→⇒a→+b→.a→+b→=-c→.-c→⇒a→2+b→2+2a→·b→=c→⇒144+48+2a→·b→=48⇒a→·b→=-72Hence, the correct options are A, C and D.

Question 54:

Let X and Y be two arbitrary, 3 ✕ 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 ✕ 3, non-zero, symmetric matrix. Then which of the following matrices is(are) skew symmetric?

Option 1: Y3Z4 − Z4Y3
Option 2: X44 + Y44
Option 3: X4Z3 − Z3X4
Option 4: X23 + Y23
Correct Answer: 3,4

 

Solutions:If X and Y are skew-symmetric matrices and Z is symmetric matrix, then

XT=-X,  YT=-Y and ZT=Z A Y3Z4-Z4Y3T     =Y3Z4T-Z4Y3T     =Z4T·Y3T-Z4T·Y3T     =Y3Z4-Z4Y3Therefore,

Y3Z4-Z4Y3is a symmetric matrix.

B X44+Y44T     =X44T+Y44T     =X44+Y44Therefore,

X44+Y44is a symmetric matrix.

C X4Z3-Z3X4T     =X4Z3T-Z3X4T     =Z3T·X4T-X4T·Z3T     =-X4Z3-Z3X4Therefore,

X4Z3-Z3X4is a skew-symmetric matrix.

D X23+Y23T     =X23T+Y23T     =-X23-Y23     =X23+Y23Therefore,

X23+Y23is a skew-symmetric matrix.

Hence, the correct options are C and D.

Question 55:

Which of the following values of α satisfy the equation

1+α21+2α21+3α22+α22+2α22+3α23+α23+2α23+3α2=-648α?

Option 1: −4
Option 2: 9
Option 3: −9
Option 4: 4
Correct Answer: 2,3

 

Solutions:

1+α21+2α21+3α22+α22+2α22+3α23+α23+2α23+3α2=-648α R2→R2-R1, R3→R3-R21+α21+2α21+3α23+2α3+4α3+6α5+2α5+4α5+6α=-648αR2→R2-R31+α21+2α21+3α2-2-2-25+2α5+4α5+6α=-648αC2→C2-C1, C3→C3-C11+α2α2+3α4α1+2α-2005+2α2α4α=-648α ⇒24α·α2+3α-2α·4α1+2α=-648α⇒-8α3=-648α⇒-8α3+648α=0⇒α=0, ±9Hence, the correct options are B and C. [[VIDEO:13797]]

Question 56:

In ℝ3, consider the planes P1 : y = 0 and P2 : x + z = 1. Let P3 be a plane, different from P1 and P2, which passes through the intersection of P1 and P2. If the distance of the point (0, 1, 0) from P3 is 1 and the distance of a point (α, β, γ) from P3 is 2, then which of the following relations is (are) true?

Option 1: 2α + β + 2γ + 2 = 0
Option 2: 2α − β + 2γ + 4 = 0
Option 3: 2α + β − 2γ − 10 = 0
Option 4: 2α − β + 2γ − 8 = 0
Correct Answer: 2,4

 

Solutions:

Given:P1: y=0P2: x+z=1Let P3 be the plane which passes through the intersection of planes P1 and P2. ⇒P3: P2+λP1=0⇒x+z-1+λy=0Now, the distance of the point (0, 1, 0) from P3 is 1.

⇒0+λ+0-11+1+λ2=1⇒λ2+1-2λ=2+λ2⇒-2λ=1⇒λ=-12So, the equation of plane is x+z-1+-12y=0i.e. P3: 2x-y+2z-2=0Also, the distance of the point

α, β, γfrom P3 is 1.

⇒2α-β+2γ-222+1+22=2⇒2α-β+2γ-2=±6⇒2α-β+2γ=-4 or 8⇒2α-β+2γ+4=0 or 2α-β+2γ-8=0Hence, the correct options are B and D. [[VIDEO:13798]]

Question 57:

In

ℝ3, Let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P1 : x + 2y – z + 1 = 0 and P2 : 2x – y + z – 1 = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1. Which of the following points lie(s) on M?

Option 1: 0,-56,- 23
Option 2: -16,-13,16
Option 3: -56, 0,16
Option 4: -13, 0,23
Correct Answer: 1,2

 

Solutions:

The given line is parallel to P1 and P2 that means its direction cosines will be perpendicular to normal of both the planes.L1⊥n1→ and L1⊥n2→ , where n1→  and n2→ are normal vectors perpendicular to the two planes P1 and P2 respectively.Thus, the vector parallel to the given line is n1→× n2→.n1→× n2→=i^j^k^12-12-11=i ^-3j^-5k^So, the equation of the line is x-01=y-0-3=z-0-5Hence, the general point on the line is λ, -3λ, -5λ.The coordinates of the foot of perpendicular drawn from the points on line L to the plane P1 are given by  x-λ1=y+3λ2=z+5λ-1=-1λ-6λ+5λ+11+4+1x-λ=y+3λ2=z+5λ-1=-16x=λ-16y=-3λ-13z=-5λ+16These equations are satisfied by the points given in options A and B.Hence, the correct options are (A) and (B). [[VIDEO:13799]]

Question 58:

Let P and Q be distinct points on the parabola y2 = 2x such that a circle with PQ as diameter passes through the vertex 0 of the parabola. If P lies in the first quadrant and the area of the triangle ∆OPQ is

32, then which of the following is(are) the coordinates of P?

Option 1: 4, 22
Option 2: 9, 32
Option 3: 14, 12
Option 4: 1, 2
Correct Answer: 1,4

 

Solutions:

The equation of the given parabola is y2 = 2x.

∴ 4a = 2

a =

12The parabola can be written parametrically as

x = at2

y = 2at

So, the coordinates of any distinct points P and Q on the parabola are

t122, t1 and t222, t2, respectively.

PQ is the diameter of the circle.

We know that the angle in the semicircle is

90°.

∴∠POQ=90°So, OP ⊥ OQ

Slope of the line OP × Slope of the line OQ = −1

t212t22×t112t12=-1⇒t2t1=-t22t124⇒ t2t1=-4Area of

△OPQ=32                   (Given)

∴12t144+t12×t244+t22=32⇒12×4×t12+4t22+416=32                    ∵t1t2=-4⇒4×16+4t12+t22+1616=18⇒8+t12+t22=18⇒t12+t22-10=0⇒t14-10t12+16=0         ∵t2=-4t1⇒t12=2, 8⇒t1=±2, ±22P lies in first quadrant. So, t1 can be 2 or 22.∴Coordinates of P are 12×2, 2 or 12×8, 22 i.e. 1, 2 or 4, 22.Hence, the correct options are A and D. [[VIDEO:13800]]

Question 59:

Column I Column II
(A) In

ℝ2, if the magnitude of the projection vector of the vector

αi^+βj^ on 3i^+j^ is 3 and if α=2+3β,then possible value(s) of

αis (are)

(P) 1
(B) Let a and b real numbers such that the function

fx=-3ax2-2,x<1bx+a2,    x≥1is differentiable for all

x ∈ ℝ.Then possible value(s) of a is (are)

(Q) 2
(C) Let

ω≠1be a complex cube root of unity. If

3-3ω+2ω24n+3 +2+3ω-3ω24n+3+-3+2ω+3ω24n+3=0,the possible value(s) of n is (are)

(R) 3
(D) Let the harmonic mean of two positive real numbers a and b be 4. If q is a positive real number such that a, 5, q, b is an arithmetic progression, then the value(s) of lq − al is (are) (S) 4
(T) 5
Correct Answer: 1#1|2#1,2|3#1,2,4,5|4#2,5

 

Solutions:(A)

Projection of the vector αi^ +βj^ on 3i^ +j^ =αi^ +βj^.3 i^ +j^32+1=3α+β2Projection of the vector αi^ +βj^ on 3i^ +j^=3    Given∴3α+β2=3⇒3α+β=23      …..1Now, α=2+3β             Given⇒α-3β=2            …..2Solving (1) and (2), we get

α=2and

β=0Thus, the possible value of

αis 2.

(A) → (P)

(B)

fx=-3ax2-2,    x <1bx+a2,        x≥1 is differentiable for all x∈R. f’1+=limh→0 f1+h-f1h          =limh→0 b1+h+a2-b-a2h          = bf’1-=limh→0 f1-h-f1-h          =lim     h→0 -3a1-h2-2-b-a2-h         =lim                  h→0 -3a1+h2-2h-2-b-a2-h         =lim     h→0 -3a-3ah2+6ah-2-b-a2-h         =lim                h→0  -3ah2+6ah-3a-2-b-a2-h    For f to be differntiable -3a-2-b-a2 has to be 0, otherwise it will be not defined.So, lim     h→0 -3ah2+6ah-h=-6aAs f is differentiable,

f‘(1+) = f‘(1)

∴b=-6a          …..(1) Also, -3a-2-b-a2=0⇒-3a-2+6a-a2=0               Using 1⇒a2-3a+2=0⇒a=1, 2Thus, the possible values of

αare 1 and 2.

(B) → (P), (Q)

(C)

Let a=3-3ω+2ω2∴aω=3ω-3ω2+2And aω2=3ω2-3+2ωSo,3-3ω+2ω24n+3+2+3ω-3ω24n+3+-3+2ω+3ω24n+3=0⇒a4n+3+aω4n+3+aω24n+3=0⇒a4n+31+ω4n+3+ω24n+3=0⇒1+ω4n+3+ω24n+3=0Therefore, 4n+3 should not be multiple of 3 or n should not be multiple of 3.Thus, the values of n are 1, 2, 4 and 5.

(C) → (P), (Q), (S), (T)

(D)

Harmonic mean of a and b=2aba+bHarmonic mean of a and b=4               Given⇒2aba+b=4⇒aba+b=2      …..1 It is given that a, 5, q, b are in AP.∴10=a+q       …..22q=5+b    ⇒q=5+b2Substituting the value of q in 2, we get10=a+5+b2⇒2a+b=15      ⇒b=15-2a             …..3From 1 and 3, we geta15-2aa+15-2a=2⇒15a-2a2=-2a+30⇒2a2-17a+30=0⇒2a2-12a-5a+30=0⇒2aa-6-5a-6=0⇒a-62a-5=0⇒a=6, 52∴q=4, 152⇒q-a=2, 5, 32Thus, the values of

q-aare 2 and 5.

(D) → (Q), (T)

Question 60:

Column I Column II
(A) In a triangle ∆XYZ, let a, b and c be the lengths of the sides opposite to the angles X, Y and Z, respectively. If 2(a2 − b2) = c2 and

λ=sin X-Ysin Zthen possible values of n sin Z for which cos(nπλ) = 0 is (are)

(P) 1
(B) In a triangle ∆XYZ, let a, b and c be the lengths of the sides opposite to the angles X, Y and Z, respectively. If 1 + cos 2X − 2cos 2Y = 2 sin X sin y, then possible value(s) of

abis (are)

(Q) 2
(C) If ℝ2, let

3 i^+j^, i^+3 j^ and βi^+1-βj^be the position vectors of X, Y and Z with respect to the origin O, respectively. If the distance of Z from the bisector of the acute angle of

OX→ with OY→ is 32, then possible value(s) of |β| is (are)

(R) 3
(D) Suppose that F(α) denotes the area of the region bounded by x = 0, x = 2, y2 = 4x and y = |αx − 1| + |αx − 2| + ax, where α ∈{0, 1}. Then the value(s) of F(α) +

832, when α = 0 and α = 1, is (are)

(S) 5
(T) 6
Correct Answer: 1#1,3,4|2#1|3#1,2|4#4,5

 

Solutions:(A) In ∆XYZ, a, b and c be the length of the sides opposite to the angles X, Y and Z, respectively. It is given that 2(a2b2) = c2 ∴ 2(sin2X − sin2Y) = sin2Z          (Uisng Sine Rule) ⇒ 2 sin(X + Y) sin(X − Y) = sin2Z        …..(1) As, X + Y + Z =

180° 

⇒X+Y=180°-Z 

⇒sinX+Y=sin180°-Z⇒sinX+Y=sinZ∴ 2sinZ sin(X − Y)  sin2Z

⇒sinX-YsinZ=12Given: λ=sinX-YsinZ⇒λ=12∴cosnπλ=cosnπ2=0   ⇒nπ2=2k+1π2⇒n=2k+1Thus, the possible values of n are 1, 3 and 5.

(A) → (P), (R), (S)

(B) In ∆XYZ, a, b and c be the length of the sides opposite to the angles X, Y and Z, respectively.

1+cos2X-2cos2Y=2sinX sinY⇒1+1-2sin2X-21-2sin2Y=2 sinXsinY⇒2sin2Y-sin2X=sinXsinY⇒2-sin2Xsin2Y=sinXsinYNow,

sinXsinY=abLet sinXsinY=ab=m∴ 2-m2=m⇒m2+m-2=0⇒m=1, -2Thus, the possible values of

abare 1 and −2.

(B) → (P)

(C)

The angle made by the bisector of acute angle of OX→ and OY→ is 45°.The equation of the acute angle of the bisector of OX→ and OY→ is x=y or x-y=0.Distance of βi ^+1-βj^ from x-y = β-1-β2 =322β-1=32β=±3+1⇒2β=4, -2⇒β=2, -1⇒β=2, 1Thus, the possible values of

βare 1 and 2.

(C) → (P), (Q)

(D)

For α=0, y=3.Now, F0 denotes the area of the region bounded by x=0, x=2, y2=4x and y=3.∴f0=∫023-2xdx          =6-823∴ f0+823=6

For α=1,y=x-1+x-2+xF1=∫013-x-2xdx+∫12x+1-2xdx     =5-823⇒F1+823=5Thus, the possible values of

Fα+832are 5 and 6. (D) → (S), (T)

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