IIT Mains 2014

Test Name: IIT Mains 2014

Question 1:

The pressure that has to be applied at the ends of a steel wire of length 10 cm to keep its length constant when its temperature is raised by 100°C is

(For steel, Young’s modulus is 2 × 1011 Nm–2 and coefficient of thermal expansion is 1.1 × 10–5 K–1.)

Option 1: 2.2 × 107 Pa
Option 2: 2.2 × 106 Pa
Option 3: 2.2 × 108 Pa
Option 4: 2.2 × 109 Pa
Correct Answer: 3

 

Solutions:Increase in length, ΔL, due to change in temperature = ΔT Here, L = Original length of the wire = 10 cm = 0.10 m α = Coefficient of thermal expansion = 1.1 × 10-5 K-1 ΔT = Change in temperature = 100°C ∴ ΔL = 0.10 × 1.1 × 10-5 × 100 Decrease in length, ΔL’, due to the applied pressure at the ends of the steel wire =

FAY×LHere, F = Force applied at both ends A = Area of cross-section of wire Y = Young’s modulus of elasticity = 2 × 1011 Nm-2 ∴ ΔL=

FA2×1011×0.10Given: ΔL = ΔL

⇒FA2×1011×0.10=0.10×1.1×10-5×100⇒FA=Pressure=1.1×10-5×100×2×1011                               =2.2×108 PaHence, the correct option is (3).

[[VIDEO:13836]]

Question 2:

A conductor lies along the z-axis at –1.5 ≤ z < 1.5 m and carries a fixed current of 10.0 A in

-a^zdirection (see figure). For a field

B→=3.0×10-4e-0.2x a^yT, what is the power required to move the conductor at a constant speed to x = 2.0 m and y = 0 m in 5 × 10–3 s? Assume parallel motion along the x-axis.

Option 1: 14.85 W
Option 2: 29.7 W
Option 3: 1.57 W
Option 4: 2.97 W
Correct Answer: 4

 

Solutions:Force acting on the current-carrying conductor due to magnetic field = ILB Here, I = Current through the conductor L = Length of the conductor B = Magnetic field strength Work done, W, in moving the conductor to x = 2.0 m and y = 0 from x = 0 and y = 0:

W=∫F.dx⇒W= ∫ILB.dx⇒W=∫021033×10-4e-0.2xdx⇒W=9×10-31-e-0.40.2

Power required, P = Work done/Time

⇒P=Wt=9×10-31-e-0.40.2×5×10-3=2.97 WHence, the correct option is (4).

[[VIDEO:13837]]

Question 3:

A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed ω rad/s about the vertical. About the point of suspension, the

Option 1: angular momentum changes in direction but not in magnitude
Option 2: angular momentum changes both in direction and magnitude
Option 3: angular momentum is conserved
Option 4: angular momentum changes in magnitude but not in direction
Correct Answer: 1

 

Solutions:  

Here, the angular momentum about the point of suspension is given as:

L→=mv→×l→From the above equation we can say that,

L→is always perpendicular to the plane containing

v→and

l→. During rotation direction of

v→is changing, therefore direction of

L→will also change. and magnitude of angular momentum remains constant.

Hence, the correct option is (A). [[VIDEO:13838]]

Question 4:

The current voltage relation of a diode is given by I = (e1000V/T – 1) mA, where the applied voltage V is in volts and the temperature T is in Kelvin. If a student makes an error measuring ±0.01 V while measuring the current of 5 mA at 300 K, what will be the error in the value of current in mA?

Option 1: 0.5 mA
Option 2: 0.05 mA
Option 3: 0.2 mA
Option 4: 0.02 mA
Correct Answer: 3

 

Solutions:Given:

 

I=e1000V/T-1 mA∵ I= 5 mA∴ 5=e1000V/T-1⇒e1000V/T=6

Again,I=e1000V/T-1⇒dIdV=e1000V/T×1000T⇒dI=6×1000T×dV⇒dI=6×1000300×0.01⇒dI=0.2 mAHence, the correct option is (C).

 

Question 5:

An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically by additional 46 cm. What will be length of the air column above mercury in the tube now?

(Atmospheric pressure = 76 cm of Hg)

Option 1: 38 cm
Option 2: 6 cm
Option 3: 16 cm
Option 4: 22 cm
Correct Answer: 3

 

Solutions:  Here, P + l = P0 = 76 cm of Hg ⇨ P = 76 − l

P08A=54-l76-lA⇒l=38 cm and l=92 cm

(l = 92 cm is not possible.)

∴ Length of the air column = 54

-38 = 16 cm

Hence, the correct option is (3).

[[VIDEO:13840]]

Question 6:

Match List-I (electromagnetic wave type) with List-II (its association/application) and select the correct option from the choices given below.

List-I List-II
(a) Infrared waves (i) To treat muscular strain
(b) Radio waves (ii) For broadcasting
(c) X-rays (iii) To detect fracture of bones
(d) Ultraviolet rays (iv) Absorbed by the ozone layer of the atmosphere
Option 1:
(a) (b) (c) (d)
(1) (iii) (ii) (i) (iv)
Option 2:
(a) (b) (c) (d)
(2) (i) (ii) (iii) (iv)
Option 3:
(a) (b) (c) (d)
(3) (iv) (iii) (ii) (i)
Option 4:
(a) (b) (c) (d)
(4) (i) (ii) (iv) (iii)
Correct Answer: 2

 

Solutions:(a) Infrared rays are used to treat muscular strain. (b) Radio waves are used for broadcasting. (c) X-rays are used to detect bone fractures. (d) Ultraviolet rays are absorbed by the ozone layer of the atmosphere.

Hence, the correct option is (2). [[VIDEO:13841]]

Question 7:

A parallel-plate capacitor is made of two circular plates separated by a distance of 5 mm and with a dielectric of dielectric constant 2.2 between them. When the electric field in the dielectric is 3 × 104 V/m, the charge density of the positive plate will be close to

Option 1: 3 × 104 C/m2
Option 2: 6 × 104 C/m2
Option 3: 6 × 10–7 C/m2
Option 4: 3 × 10–7 C/m2
Correct Answer: 3

 

Solutions:Electric field between the plates of the capacitor:

E=σKε0Here, σ = Charge density of the plate K = Dielectric constant of the dielectric

⇒σ=EKε0⇒σ=3×1042.28.85×10-12        =5.841×10-7        ≈6×10-7 C/m2Hence, the correct option is (3). [[VIDEO:13842]]

Question 8:

A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

Option 1: A screw gauge having 100 divisions in the circular scale and pitch as 1 mm
Option 2: A screw gauge having 50 divisions in the circular scale and pitch as 1 mm
Option 3: A metre scale
Option 4: A vernier calliper where 10 divisions in the vernier scale matches with 9 divisions in the main scale and the main scale has 10 divisions in 1 cm
Correct Answer: 4

 

Solutions:The value measured is 3.50 cm. So, the least count must be 0.01 cm or 0.1 mm. For the vernier calliper, 1 M.S.D. = 1/10 cm = 1 mm Also, 10 V.S.D. = 9 M.S.D. ⇨ 1 V.S.D. = 9/10 × M.S.D. = 0.9 mm Least count = 1 M.S.D. − 1 V.S.D. = 1 mm − 0.9 mm = 0.1 mm

Hence, the correct option is (4). [[VIDEO:13843]]

Question 9:

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is

Option 1: GMR1+22
Option 2: 12GMR1+22
Option 3: GMR
Option 4: 22GMR
Correct Answer: 2

 

Solutions:

The net force on any one particle will be along the x-direction (because components in the y-direction are equal and opposite and will cancel each other), and it is given as:

F=F2+2F1 cos45°⇒F=GM22R2+2GM2R22 cos45°       =GM24R2+2GM222R2       =GM2R214+12Because the particles are moving in a circle, this force will provide the necessary centripetal force.

⇒Mu2R=GMR214+12⇒u2 =GMR1+224⇒u=12GMR1+22Hence, the correct option is (2). [[VIDEO:13844]]

Question 10:

In a large building, there are 15 bulbs of 40 W, 5 bulbs of 100 W, 5 fans of 80 W and 1 heater of 1 kW. The voltage of the electric mains is 220 V. The minimum capacity of the main fuse of the building will be

Option 1: 12 A
Option 2: 14 A
Option 3: 8 A
Option 4: 10 A
Correct Answer: 1

 

Solutions:Power consumption of 15 bulbs of 40 W each = 600 W Power consumption of 5 bulbs of 100 W each = 500 W Power consumption of 5 fans of 80 W each = 400 W Power consumption of 1 heater of 1000 W = 1000 W ∴ Total power consumption, P, by all appliances = 2500 W Voltage of electric mains, V = 220 V Now, Current capacity of the main fuse:

i=PV=2500220=11.36 AThus, the minimum capacity of the fuse must be 12 A.

Hence, the correct option is (1). [[VIDEO:13846]]

Question 11:

A particle moves with simple harmonic motion in a straight line. In the first τ s, after starting from rest, it travels a distance a, and in the next τ s, it travels 2a in the same direction, then :

Option 1: amplitude of motion is 4a
Option 2: time period of oscillations is 6τ
Option 3: amplitude of motion is 3a
Option 4: time period of oscillations is 8τ
Correct Answer: 2

 

Solutions:Displacement equation of SHM: x = A(1 − cos ωt) Here, x = Displacement A = Amplitude of motion ω = Angular frequency t = Time

For the first τ s, we have: tτ  and x = aa = A(1 − cos ωτ)

Or, cos ωτ =

1-aA     …(i) For the next τ s, we have: t = 2τ  and x = (a + 2a) = 3a ∴ 3a = A(1 − cos 2ωτ)

Or, cos 2ωτ =

1-3aAOr,

2cos2ωτ-1=1-3aAOn substituting the value of cos ωτ from equation (i), we get:

21-aA2-1=1-3aAOn simplifying, we get:

aAaA-12=0Or,

aA=0,   aA=12 aA=0is not possible. Thus, the feasible solution is

aA=12.

The amplitude of motion, A = 2a Now, from equation (i), we have:

cos ωτ=1-12=12=cosπ3Or, ωτ=π3Using

ω=2πT, we get:

2πTτ=π3Or,

T=6τHence, the correct option is (2).

Question 12:

The coercivity of a small magnet where the ferromagnet gets demagnetised is 3 × 103 Am–1. The current required to be passed in a solenoid of length 10 cm and number of turns 100 so that the magnet gets demagnetised when inside the solenoid is

Option 1: 3 A
Option 2: 6 A
Option 3: 30 mA
Option 4: 60 mA
Correct Answer: 1

 

Solutions:We know:

Coercivity H=Bμ0And,B=μ0nI∴ H=nl    …iHere, n is the number of turns per unit length.

n=1000.1 m⇒n=103 m-1Given: H = 3 × 103 Am-1

On substituting the values of H and n in equation (i), we get: H=nI⇒I=Hn=3×103103=3 AHence, the correct option is (1).

Question 13:

The forward-biased diode connection is

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

 

Solutions:For the p–n junction to be forward bias, the p side should be at higher potential than the n side.

Hence, the correct option is (3).

Question 14:

During the propagation of electromagnetic waves in a medium,

Option 1: electric energy density is equal to the magnetic energy density
Option 2: both electric and magnetic energy densities are zero
Option 3: electric energy density is double of the magnetic energy density
Option 4: electric energy density is half of the magnetic energy density
Correct Answer: 1

 

Solutions:The average electric energy density of an electromagnetic wave propagating in a medium is given by

uE=12εE2=14εE02 ( using E=E02). The average magnetic energy density of an electromagnetic wave propagating in a medium is given by

uB=12B2μ=14B02 μ ( using B=B02). Using the relation

E0=B0με, it can be easily shown that

uE=uB.

Hence, the correct option is (1).

Question 15:

In the circuit shown here, point C is kept connected to point A till the current flowing through the circuit becomes constant. Afterwards, suddenly, point C is disconnected from point A and connected to point B at time t = 0. Ratio of the voltage across resistance and the inductor at t = L/R will be equal to

Option 1: –1
Option 2: 1-ee
Option 3: e1-e
Option 4: 1
Correct Answer: 1

 

Solutions:When point C is connected to point B, the resistor and inductor become parallel to each other. On applying Kirchhoff’s law, we get:

-VR-VL=0⇒VRVL=-1Hence, the correct option is (1).

Question 16:

A mass ‘m‘ is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

Option 1: 5g6
Option 2: g
Option 3: 2g3
Option 4: g2
Correct Answer: 4

 

Solutions:

If the string does not slip on the cylinder, it’s a case of pure rolling, i.e.

a=Rα Let the the mass ‘m’ moves downward with acceleration a. Then,

ma=mg-T…..(i) For the cylinder: Torque acting due to tension in the string is

TRand torque can be written as

Iα, where I is the moment of inertia. For a hollow cylinder:

I=mR2Therefore,

TR=mR2α

⇒T=mRα

⇒T=ma………(ii) Solving equations (i) and (ii),we get:

a=g2Hence, the correct option is (4).

Question 17:

One mole of a diatomic ideal gas undergoes a cyclic process ABC, as shown in the figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K, respectively. Choose the correct statement.

Option 1: The change in internal energy in process AB is −350 R.
Option 2: The change in internal energy in process BC is −500 R.
Option 3: The change in internal energy in the whole cyclic process is 250 R.
Option 4: The change in internal energy in process CA is 700 R.
Correct Answer: 2

 

Solutions:Process AB is an isochoric process, as shown. The change in internal energy in process AB is

∆UAB=nCVTB-TA            =1×5R2800-400=1000 RProcess CA is isobaric. The change in internal energy in process CA is

∆UCA=nCVTA-TC           =1×5R2400-600=-500 RSince, process ABC is cyclic,

∆UABC=0Process BC is an adiabatic process. So, the change in internal energy in process BC:

∆UAB+∆UBC+∆UCA=0⇒1000R+∆UBC-500R=0⇒∆UBC=-500R

Hence, the correct option is (2).

Question 18:

From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the particle to hit the ground is n times that taken by it to reach the highest point of its path.

The relation between H, u and n is

Option 1: 2gH = nu2(n − 2)
Option 2: gH = (n − 2)u2
Option 3: 2gH = n2u2
Option 4: gH = (n − 2)2u2
Correct Answer: 1

 

Solutions:At maximum height, velocity v will be zero. Therefore, time taken ‘t’ by the particle to reach the maximum height can be calculated using the first equation of motion as:

v=u-gtv=0⇒t=ugTime taken ‘t’ by the particle to hit the ground can be calculated using the second equation of motion as:

-H=ut’-12gt’2Given, t’ = nt

⇒-H=unt-12gn2t2Substituting the value of t from above, we get:

-H=unug-12gn2u2g2⇒2gH=nu2n-2Hence, the correct option is (1).

Question 19:

A thin convex lens made from crown glass

μ=32has focal length f. When it is measured in two different liquids of refractive indices

43and

53, it has focal lengths f1 and f2, respectively. The correct relation between the focal lengths is

Option 1: f2 > f and f1 becomes negative
Option 2: f1 and f2 become negative
Option 3: f1 = f2 < f
Option 4: f1 > f and f2 becomes negative
Correct Answer: 4

 

Solutions:The relation between the focal length of a lens and the refractive index of material of glass is expressed as:

1f=μ-11R1-1R2    …(i) When placed in a medium other than air, the focal length is expressed as:

1fm=μμm-11R1-1R2    …(ii) Here, μm is the refractive index of the medium. Dividing (i) by (ii), we get:

fmf=μ-1μμm-1For μm = 4/3 and fm = f1:

⇒f1f=32-13/24/3-1=4⇒f1=4fFor μm = 5/3 and fm = f2:

⇒f2f=32-13/25/3-1=-5⇒f2<0Hence, the correct option is (4).

Question 20:

Three rods of copper, brass and steel are welded together to form a Y – shaped structure. Area of cross – section of each rod is 4 cm2. The end of the copper rod is maintained at 100 °C, whereas the ends of brass and steel are kept at 0°C. Lengths of the copper, brass and steel rods are 46, 13 and 12 cm, respectively. The rods are thermally insulated from the surroundings, except at the ends. Thermal conductivities of copper, brass and steel are 0.92, 0.26 and 0.12 CGS units, respectively. Rate of heat flow through the copper rod is

Option 1: 4.8 cal/s
Option 2: 6.0 cal/s
Option 3: 1.2 cal/s
Option 4: 2.4 cal/s
Correct Answer: 1

 

Solutions:Rate of heat flow through a rod =

dQdt=-KA∆TlAs the rods form a Y-shape, Rate of heat flow through the copper rod(1) = Rate of heat flow through the brass rod(2) + Rate of heat flow through the steel rod(3)

dQ1dt=dQ2dt+dQ3dt    …iLet the temperature of the junction be

T °C. Now, substituting the values of K, A, l and ∆T in (i), we get:

0.92100-T46=0.26T-013+0.12T-012⇒T=40 °CSo, the rate of heat flow through the copper rod =

dQ1dt=0.92×4100-4046=4.8 cal/sHence, the correct option is (1).

Question 21:

A pipe of length 85 cm is closed at one end. Find the number of possible natural oscillations of the air column in the pipe whose frequencies lie below 1250 Hz. The velocity of sound in air is 340 m/s.

Option 1: 6
Option 2: 4
Option 3: 12
Option 4: 8
Correct Answer: 1

 

Solutions:We know that in fundamental mode,

λ4=lHere,

lis the length of the pipe.

λ=4×0.85The fundamental frequency of any wave:

f0=vλHere, v is the velocity of wave. Substituting the values of v and λ:

f0=vλ=3404×0.85=100 HzSo, frequency of the nth normal mode of vibration is given as

fn=(2n-1)f0Thus, the possible frequencies are, 100 Hz, 300 Hz, 500 Hz, 700 Hz, 900 Hz and 1100 Hz, as all these frequencies lie below 1250 Hz.

Hence, the correct option is (1). [[VIDEO:13847]]

Question 22:

There is a circular tube in a vertical plane. Two liquids that do not mix, and of densities d1 and d2, are filled in the tube. Each liquid subtends an angle of 90° at the centre. The radius joining their interface makes an angle α with vertical. The ratio

d1d2is

Option 1: 1+tan α1-tan α
Option 2: 1+sin α1-cos α
Option 3: 1+sin α1-sin α
Option 4: 1+cos α1-cos α
Correct Answer: 1

 

Solutions:Consider the following figure given below: In

∆OCD, OC=Rsinαand in

∆OBA, OB=Rcosα.

Equating pressure at P: Using

P=ρgh

Here, ρ=density, g=accerlation due to grravity and h is the height. d1gR(cosα-sinα)=d2gR(cosα+sinα)d1d2=cosα+sinαcosα-sinα=1+tanα1-tanαHence, the correct option is A.

Question 23:

A green light is incident from water on the air – water interface at critical angle (θ). Select the correct statement.

Option 1: The spectrum of visible light whose frequency is more than that of green light will come out to the air medium.
Option 2: The entire spectrum of visible light will come out of water at various angles to the normal.
Option 3: The entire spectrum of visible light will come out of water at an angle of 90° to the normal.
Option 4: The spectrum of visible light whose frequency is less than that of green light will come out to the air medium.
Correct Answer: 4

 

Solutions:The relation between refractive index and wavelength is expressed as:

μ∝1λ    …(i)For light of frequency (f‘) greater than green light i.e.

f’>fgreen, its wavelength (λ‘) will be smaller than that of the green light, i.e.

λ'<λgreen. (Using relation,

f=cλ) Now, from equation (i),

μ’>μgreenUsing Snell’s Law,

sinθ=1μ As frequency of visible light increases, refractive index increases. With the increase of refractive index, the critical angle decreases. So, the light with frequency greater than green light will undergo total internal reflection and the light with frequency less than green light will pass into the air.

Hence, the correct option is (4). [[VIDEO:13848]]

Question 24:

Each of hydrogen (1H1), deuterium (1H2), singly-ionised helium (2He4)+ and doubly-ionised lithium (3Li6)++ have one electron around the nucleus. Consider an electron transition from n = 2 to n = 1. If the wavelengths of emitted radiations are λ1, λ2, λ3 and λ4, respectively, then approximately which one of the following is correct?

Option 1: λ1 = λ2 = 4λ3 = 9λ4
Option 2: λ1 = 2λ2 = 3λ3 = 4λ4
Option 3: 4λ1 = 2λ2 = 2λ3 = λ4
Option 4: λ1 = 2λ2 = 2λ3 = λ4
Correct Answer: 1

 

Solutions:We know that

1λ=RZ21n12-1n22Given, n1=1 and n2=21λ=RZ2112-122∴λ=43RZ2For hydrogen and deutrium, Z = 1,

⇒λ1=43R⇒λ2=43RFor helium, Z = 2,

⇒λ3=412RFor Lithium, Z = 3,

⇒λ4=427R⇒λ1=λ2=4λ3=9λ4Hence, the correct option is (1). [[VIDEO:13849]]

Question 25:

The radiation corresponding to 3 → 2 transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of 3 × 10−4 T. If the radius of the largest circular path followed by these electrons is 10.0 mm, the work function of the metal is close to

Option 1: 0.8 eV
Option 2: 1.6 eV
Option 3: 1.8 eV
Option 4: 1.1 eV
Correct Answer: 4

 

Solutions:The radius of the circular path followed by a particle is

r=mvqB…(i) And the maximum kinetic energy attained by the particle is

K(max)=p22m=(mv)22mFrom equation (i),

K(max)=(qrB)22mSubstituting q = 1.6 × 10-19 C, r = 10 mm = 10-2 m, B = 3 × 10-4 T and m = 9.1 × 10-31 kg, we get:

K(max)=(1.6×10-19×10-2×3×10-4)22×9.1×10-31×1.6×10-19=0.8 eVFor transition between 3 to 2:

E=hν=13.614-19⇒hν=13.6536

∴Work function =hν-K(max)                             =13.6536-0.8=1.1 eVHence, the correct option is (4). [[VIDEO:13850]]

Question 26:

A block of mass m is placed on a surface with a vertical cross-section given by

y=x36.If the coefficient of friction is 0.5, the maximum height above the ground, at which the block can be placed without slipping, is

Option 1: 13 m
Option 2: 12 m
Option 3: 16 m
Option 4: 23 m
Correct Answer: 3

 

Solutions:Slope:

tanθ=dydx⇒tanθ=dx36dx⇒tanθ=x22

At limiting equilibrium,

μ=tanθ⇒0.5=x22⇒x=±1 ∴y=16 mHence, the correct option is (3). [[VIDEO:13851]]

Question 27:

When a rubber-band is stretched by a distance x, it exerts a restoring force of magnitude F = ax + bx2 , where a and b are constants. The work done in stretching the unstretched rubber-band by L is

Option 1: aL22 +bL33
Option 2: 12aL22 +bL33
Option 3: aL2 + bL3
Option 4: 12 aL2 +bL3
Correct Answer: 1

 

Solutions:Given,

F=ax+bx2We know thatdW=F.dlOn integrating both sides of the above equation from 0 to L:

W=∫0L(ax+bx2)dx⇒W=aL22+bL33Hence, the correct option is (1). [[VIDEO:13852]]

Question 28:

On heating water, bubbles being formed at the bottom of the vessel detach and rise. Consider the bubbles to be spheres of radius R and making a circular contact of radius r with the bottom of the vessel. If r << R, and the surface tension of water is T, value of r just before the bubbles detach is (density of water is ρw)

Option 1: R2ρw gT
Option 2: R23ρw gT
Option 3: R2ρw g3T
Option 4: R2ρw g6T
Correct Answer: 3

 

Solutions:We know that on heating water, the water bubbles get detached from the surface of the vessel and rise.

So, the buoyant force = force due to surface tension

∫T×dlsinθ+∆P(πr2)=43πR3ρwgHere, dl is the small length element of the bubble touching the surface of the vessel. T is the tension of the water.

∆Pis the excess pressure inside the bubble.

ρwis the density of the water From

∆ABC, sinθ=rR

⇒T×2πr×rR+2TR(πr2)=43πR3ρwg⇒4πTr2R=43πR3ρwg⇒r2=R4ρwg3T∴r=R2ρwg3THence, the correct option is (3).

Question 29:

Two beams, A and B, of plane-polarised light with mutually perpendicular planes of polarisation are seen through a polaroid. From the position when the beam A has maximum intensity (and beam B has zero intensity), a rotation of the polaroid through 30° makes the two beams appear equally bright. If the initial intensities of the two beams are IA and IB respectively, then

IAIBequals

Option 1: 1
Option 2: 13
Option 3: 3
Option 4: 32
Correct Answer: 2

 

Solutions:Using Malus’ law,

I=I0cos2θ: For plane polarised beams of light A and B, after rotation of the polaroid, the intensities are

IA’=IAcos230°IB’=IBcos260°Given,

IA’=IB’

⇒IA×34=IB×14⇒IAIB=13Hence, the correct option is (2). [[VIDEO:13853]]

Question 30:

Assume that an electric field

E→=30 x2 i^exists in space. Then, the potential difference VA − VO (where VO is the potential at the origin and VA the potential at x = 2 m) is

Option 1: −80 J
Option 2: 80 J
Option 3: 120 J
Option 4: −120 J
Correct Answer: 1

 

Solutions:We know that

dV=-E.→dx→On integrating both sides of the above equation, according to the limits given in the question, we get:

∫V0VAdV=-∫0230x2dx⇒VA-V0=-30×3302=-80 VNone of the options given in the question correctly matches with the answer, as the unit of potential difference is volt or joule per coulomb and the options are given in joules.

Question 31:

The image of the line

x – 13 = y – 31 = z – 4-5in the plane

2x − y + z + 3 = 0 is the line :

Option 1: x + 33 = y – 51 = z – 2- 5
Option 2: x + 3- 3 = y – 5 – 1 = z + 25
Option 3: x – 33 = y + 5 1 = z – 2- 5
Option 4: x – 3- 3 = y + 5- 1 = z – 25
Correct Answer: 1

 

Solutions:The equations of the given line and the plane are:

x-13=y-31=z-4-5                 … (1)

2x

y + z + 3 = 0                   … (2)

The given line passes through the point (1, 3, 4) and is parallel to the vector, whose direction ratios are 3, 1 and

-5.

∵ al+bm+cn=2×3+-1×1+1×-5=6-1-5=0Thus, the given line and the plane are parallel to each other.

So, the direction ratios of the image of the given line are 3, 1 and

-5.

Let (x1, y1, z1) be the image of the point (1, 3, 4) in the given plane.

∴x1-12=y1-3-1=z1-41=-2ax2+by2+cz2+da2+b2+c2Here,

a=2, b=-1, c=1, d=3, x2,  y2,  z2=1, 3, 4 ∴x1-12=y1-3-1=z1-41=-22×1-3+4+322+-12+12=-2 ⇒x1=-3, y1=5,  z1=2Thus, the equation of the image of the given line is

x+33=y-51=z-2-5.

Hence, the correct option is (1).

Question 32:

If the coefficients of x3 and x4 in the expansion of (1 + ax + bx2) (1 − 2x)18 in powers of x are both zero, then (a, b) is equal to :

Option 1: 16, 2513
Option 2: 14, 2513
Option 3: 14, 2723
Option 4: 16, 2723
Correct Answer: 4

 

Solutions:Let us expand the given expression.

1+ax+bx21-2×18=1+ax+bx2C018+C118-2×1+C218-2×2+C318-2×3+C418-2×4+…Now, coefficient of x3 =

C318-23+a×C218-22+b×C118-21=0 ⇒-4×17×163×2+a×172×2-b=0 ⇒51a-3b-544=0                  … (1)

Coefficient of x4 =

C418-24+a×C318-23+b×C218-22=0 ⇒4×16×154×3×2-2a×163×2+b×12=0 ⇒32a-3b-240=0                  … (2)

Solving (1) and (2), we get:

a=16, b=2723Hence, the correct option is (4).

Question 33:

If a ϵ R and the equation

−3(x − [x])2 + 2 (x − [x]) + a2 = 0

(where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval :

Option 1: (−1,0) ∪ (0, 1)
Option 2: (1, 2)
Option 3: (−2, −1)
Option 4: (-∞, -2) ∪( 2, ∞)
Correct Answer: 1

 

Solutions:The given equation is as follows:

-3x-[x]2+2x-[x]+a2=0 ,  a∈R            … (1)

We know that,

{x}=x-[x].

∴-3×2+2x+a2=0⇒x2-23x-a23=0 ⇒x-132=a23+19 ∵x<1 ⇒ x-13<1-13⇒x-132<49and

a23+19>0 ∴0<a23+19<49 ⇒-13<a2<1⇒0≤a2<1           ∵a2≥0                  … (2)

Let a = 0.

From equation (1), we have:

-3x-[x]2+2x-[x]+0=0 ⇒x-[x]-3x+3[x]+2=0⇒x-[x]=0 or -3x+3[x]+2=0Take

x-[x]=0⇒x=[x], which is possible only when x is an integer.

But it is given that, the quadratic equation has no integral solution.

∴a≠0Thus, from equation (2), we have

a∈-1, 0∪0, 1.

Hence, the correct option is (1).

Question 34:

If a→×b→    b→×c→    c→×a→ = λa→  b→  c→2  then λis equal to :

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

 

Solutions:

a→×b→   b→×c→   c→×a→=a→×b→.  b→×c→×c→×a→=a→×b→.b→×c→ .a→c→-b→×c→.c→a→=a→×b→.b→×c→.a→c→-0=a→×b→.c→b→×c→.a→=a→   b→    c→b→     c→    a→=a→   b→    c→a→   b→    c→=a→   b→    c→2Comparing this with the given expression, we get:

λ=1Hence, the correct option is (4).

Question 35:

The variance of first 50 even natural numbers is:

Option 1: 8334
Option 2: 833
Option 3: 437
Option 4: 4374
Correct Answer: 2

 

Solutions:The first 50 even numbers are 2, 4, 6 … 100.

We know: Variance,

σ2=∑xi2N-x2∴ Variance of the first 50 even natural numbers

=22+42+…+100250-2+4+…+100502=412+22+…+50250-21+2+…+50502=4×505110150×6-2×50512×502=3434-2601=833Hence, the correct option is (2). [[VIDEO:13771]]

Question 36:

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45°. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to 30°. Then the speed (in m/s) of the bird is :

Option 1: 40 (2 – 1)
Option 2: 40(3 – 2)
Option 3: 202
Option 4: 20(3- 1)
Correct Answer: 4

 

Solutions:

Let AB be the height of the pole.

∴ AB = 20 m

Suppose the bird is at point C after 1 second.

∴ CD = 20 m

Let the speed of the bird be v m/s.

Distance covered by the bird in 1 sec = BD = v m/s × 1 s = v m

In ∆OAB:

tan45°=ABOB⇒1= 20 OB⇒OB=20 mIn ∆OCD:

tan30°=CDOD⇒13=20 20+v⇒20+v=203⇒v=203-1Thus, speed of the bird is

203-1m/s.

Hence, the correct option is (4). [[VIDEO:13772]]

Question 37:

The integral

∫0π1 + 4 sin2x2 – 4 sinx 2 dxequals :

Option 1: π − 4
Option 2: 2π3 – 4 -43
Option 3: 43 – 4
Option 4: 43 – 4 – π3
Correct Answer: 4

 

Solutions:Let

I = ∫0π1+4sin2x2-4sinx2 dx I = ∫0π1+4sin2x2-4sinx2 dx
 

=∫0π1-2sinx22 dx

=∫0π1-2sinx2 dx 

∴∫0π1-2sinx2 dx
  

= ∫0π31 – 2sinx2 dx + ∫π3π 2sinx2 – 1 dx                                                
 

=  x+ 4cosx20π3 +   -4cosx2- xπ3π
      

= π3 + 4cosπ6 -4cos0+ -4cosπ2 – π +4cosπ6+ π3

= π3 + 4·32 – 4 -π + 4·32 + π3

= 43 – 4 -π3

Hence, the correct option is (4). [[VIDEO:13773]]

Question 38:

The statement

~(p↔ ~q)is :

Option 1: equivalent to

p ↔ q

Option 2: equivalent to

~p ↔ q

Option 3: a tautology
Option 4: a fallacy
Correct Answer: 1

 

Solutions:

p q q p↔~q ~p↔~q p↔q
T T F F T T
T F T T F F
F T F T F F
F F T F T T

Since columns V and VI are same, the statement

~p↔~qis equivalent to

p↔q.

Hence, the correct option is (1). [[VIDEO:13774]]

Question 39:

If A is an 3 × 3 non − singular matrix such that AA’ = A’A and B = A−1 A’, the BB’ equals :

Option 1: I + B
Option 2: I
Option 3: B−1
Option 4: (B−1)’
Correct Answer: 2

 

Solutions:Given:

B = A−1 A’

AA’ = A’A

On pre-multiplying A with both the sides of the equation (B = A−1 A’), we get:

AB = A( A−1 A’) = (AA−1)A’ = IA’ = A’

∴ ABB’ = A’B’ = (BA)’ = (A−1 A’A)’ = (IA’)’ = A

  BB’ = I

Hence, the correct option is (2). [[VIDEO:13775]]

Question 40:

The integral

∫ 1 + x – 1x e x+1x dxis equal to :

Option 1: (x – 1) e x+1x + c
Option 2: x e x+1x +c
Option 3: (x + 1) e x+1x + c
Option 4: -x e x+1x + c
Correct Answer: 2

 

Solutions:

∫1+x-1xex+1xdx=∫ex+1x + x-1xex+1xdx=∫ex+1x+x1-1x2ex+1xdxLet f(x)=ex+1x⇒f'(x)=ex+1x(1-1×2)We know: ∫f(x)+xf'(x)dx=xf(x)+c∴ ∫ex+1x+x1-1x2ex+1xdx=xex+1x+cHence, the correct option is (2).

Question 41:

If z is a complex number such that |z|≥2, then the minimum value of

z + 12:

Option 1: is equal to

52

Option 2: lies in the interval (1, 2)
Option 3: is strictly greater than

52

Option 4: is strictly greater than

32but less then

52

Correct Answer: 2

 

Solutions:It is given that

z≥2.

Now, we need to evaluate the minimum value of

z+12.

We know:

x+y≥x-y ∴z +12⩾z-12         ⇒z +12⩾2-12⇒z +12⩾32Thus, the minimum value of

z+12is

32, which lies in the interval (1, 2).

Hence, the correct option is (2).

Question 42:

If g is the inverse of a function f and f’ (x) =

11 + x5, then g’ (x) is equal to :

Option 1: 1 + x5
Option 2: 5x4
Option 3: 11 + {g(x)}5
Option 4: 1 + {g(x)}5
Correct Answer: 4

 

Solutions: Given:

f’x=11+x5                            …1Also, g is the inverse of the function f.

⇒fogx=x⇒fgx=xDifferentiating both the sides with respect to x, we get:

f’gxg’x=1⇒g’x=1f’gx                                            …2 f’gx=11+gx5                                  Using 1⇒g’x=111+gx5  =1+gx5

⇒g’x = 1 + gx5Hence, the correct option is (4).

Question 43:

If α, β≠0, and f(n) = αn + βnand

31 + f(1)1 + f(2)1 + f(1)1 + f(2)1 + f(3)1 + f(2)1 + f(3)1 + f(4)= K

(1 – α)2 (1 – β)2 (α – β)2,then K is equal to :

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

 

Solutions:Given:

fn=αn+βn, α, β≠0      … (1)

Putting n = 1, 2, 3 and 4 in equation (1), we get:

f1=α+β,  f2=α2+β2,  f3=α3+β3,  f4=α4+β4∴

31+f11+f21+f11+f21+f31+f21+f31+f4=K1-α21-β2α-β2

⇒31+α+β1+α2+β21+α+β1+α2+β21+α3+β31+α2+β21+α3+β31+α4+β4=K1-α21-β2α-β2 ⇒1111αβ1α2β21111αβ1α2β2=K1-α21-β2α-β2 ⇒1111αβ1α2β22=K1-α21-β2α-β2 ⇒1-α21-β2α-β2=K1-α21-β2α-β2⇒K=1Hence, the correct option is (C). [[VIDEO:13778]]

Question 44:

Let

fk(x) = 1k (sink x + cosk x)where

x ε R and k≥ 1.Then f4(x) − f6(x) equals :

Option 1: 16
Option 2: 13
Option 3: 14
Option 4: 112
Correct Answer: 4

 

Solutions:

fkx=1ksinkx+coskx, x∈R, k≥1 ∴f4x-f6x=14sin4x+cos4x-16sin6x+cos6x ⇒f4x-f6x=1123sin4x+3cos4x-2sin6x+cos6xOn using the identity

x3+y3=x+yx2-xy+y2, we get:

⇒f4x-f6x=1123sin4x+3cos4x-2sin2x+cos2xsin4x+cos4x-sin2xcos2x ⇒f4x-f6x=1123sin4x+3cos4x-2sin4x-2cos4x+2sin2xcos2x ⇒f4x-f6x=112sin2x+cos2x+2sin2xcos2x ⇒f4x-f6x=112sin2x+cos2x2 ⇒f4x-f6x=112Hence, the correct option is (4).

Question 45:

Let

α and βbe the roots of equation px2 + qx + r = 0,

p≠0. If p, q, r are in A.P. and

1α+ 1β = 4, then the value of

α – βis :

Option 1: 619
Option 2: 2179
Option 3: 349
Option 4: 2139
Correct Answer: 4

 

Solutions:The given quadratic equation is

px2+qx+r=0.

αand

βare the roots of the given quadratic equation.

∴ α+β=-qp, αβ=rp

1α+1β=4⇒α+βαβ=4⇒-qprp=4

⇒ q=-4r                     …(1)

It is also given that p, q and r are in A.P.

∴ 2q=p+r                    …(2)

From (1) and (2), we get:

p=2q-r=-8r-r=-9r   …(3)

Now,

α-β=α+β2-4αβ=q2p2-4rp ∴ α-β=16r281r2+4r9r=2139            [Using (1) and (3)]

Hence, the correct option is (4).

Question 46:

Let A and B be two events such that

PA∪B=16, PA∩B=14 and PA=14,where

Astands for the complement of the event A. Then the events A and B are

Option 1: mutually exclusive and independent.
Option 2: equally likely but not independent.
Option 3: independent but not equally likely.
Option 4: independent and equally likely.
Correct Answer: 3

 

Solutions:Given:

PA∪B=16, PA∩B=14and

PA=14 PA∪B=16⇒PA∪B=1-16=56 PA=14⇒PA=1-14=34Now, we have:

PA∪B=PA+PB-PA∩B⇒PB=PA∪B-PA+PA∩B⇒PB=56-34+14⇒PB=13Now,

PA≠PBHence, the events A and B are not equally likely.

Also,

PAPB=34×13=14=PA∩BSo, A and B are independent events.

Thus, the events A and B are independent but not equally likely.

Hence, the correct option is (3). [[VIDEO:13779]]

Question 47:

If f and g are differentiable functions is [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 and f(1) = 6, then for some c∈[0, 1]:

Option 1: 2f‘ (c) = g‘(c)
Option 2: 2f‘ (c) = 3g‘ (c)
Option 3: f‘(c) = g‘(c)
Option 4: f‘(c) = 2g‘ (c)
Correct Answer: 4

 

Solutions:It is given that the functions f and g are differentiable functions in [0, 1]. So, f and g are continuos in [0, 1].

Using mean value theorem, we have:

f’c=f1-f01-0=6-2=4           …1Also,

g’c=g1-g01-0=2-0=2          …2From (1) and (2), we have:

f’(c) = 2 g’(c)

Hence, the correct option is (4). [[VIDEO:13780]]

Question 48:

Let the population of rabbits surviving at a time t be governed by the differential equation

dptdt=12pt-200. If p(0) = 100, then p(t) equals:

Option 1: 400 − 300 et/2
Option 2: 300 − 200 e−t/2
Option 3: 600 − 500 et/2
Option 4: 400 − 300 e−t/2
Correct Answer: 1

 

Solutions:It is given that the population of rabbits surviving at a time t be governed by the following differential equation:

dptdt= 12pt – 200

⇒ dpdt= p – 4002

⇒ dpp – 400= 12dt
Integrating both sides, we get: log p – 400 = 12t + c
Also, p(0) = 100

⇒ log 300 = c

logp – 400 = t2+ log 300

logp – 400300= t2

⇒p – 400 = 300et2 ⇒p-400=300et2  or  400-p=300et2
Now, since p(0) = 100, we neglect

p-400=300et2

⇒400 – p = 300et2

⇒p = 400 – 300 et2
Hence, the correct option is (1). [[VIDEO:13781]]

Question 49:

Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y), passing through origin and touching the circle C externally, then the radius of T is equal to:

Option 1: 32
Option 2: 32
Option 3: 12
Option 4: 14
Correct Answer: 4

 

Solutions: The given information can be represented diagrammatically as follows:

In the given figure, the smaller circle with centre (0, y) represents the circle T.

From figure, we have:

AD2 = BD2 + AB2   …(1)

The radius of the smaller circle is y.

∴ AD = 1+ y

From equation (1), we get:

1+y2=

1+1-y2 ⇒1+y2+2y = 1 + 1 + y2-2y⇒2y = 1 -2y⇒y = 14

Therefore, the radius of circle T is

14units.

Hence, the correct option is (4). [[VIDEO:13782]]

Question 50:

The area of the region described by A = {(x, y) : x2 + y2 ≤ 1 and y2 ≤ 1 − x} is :

Option 1: π2+43
Option 2: π2-43
Option 3: π2-23
Option 4: π2+23
Correct Answer: 1

 

Solutions:

A =

x,y: x2+y2≤1, y2≤1-xOn putting

y2=1-xin the equation

x2+y2=1, we get:

x2-x=0⇒x(x-1) = 0⇒x=0,1For x = 0, we get: y = 1, −1

For x = 1, we get: y = 0

Thus, the points of intersection are (0, 1), (1, 0), (0, −1).

∴ Required area = A =

2∫-101-x2dx+2∫011-xdx                                  =

2×21-x2 – 12sin-1x-10- 43 1-x3201 =

-sin-1-1-43-1=π2+43 sq unitsHence, the correct option is (1). [[VIDEO:13783]]

Question 51:

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lies in the fourth quadrant and is equidistant from the two axes then :

Option 1: 2bc − 3ad = 0
Option 2: 2bc + 3ad = 0
Option 3: 3bc − 2ad = 0
Option 4: 3bc + 2ad = 0
Correct Answer: 3

 

Solutions:The equations of the given lines are

4ax + 2ay + c = 0and

5bx + 2by + d =0.

The point of intersection of the lines lies in the fourth quadrant and is equidistant from the two axes. Let us assume the point of intersection to be (h, −h). Therefore, we have:

4ah – 2ah + c = 0          …15bh – 2bh + d =0          …2On solving the two equations, we have:

h2ad-2bc=-h5bc-4ad=18ab-10ab ⇒2ad-2bc = 4ad -5bc ⇒3bc-2ad =0Hence, the correct option is (3). [[VIDEO:13784]]

Question 52:

Let PS be the median of the triangle with vertices P(2, 2), Q(6, −1) and R(7, 3). The equation of the line passing through (1, −1) and parallel to PS is

Option 1: 4x − 7y − 11 = 0
Option 2: 2x + 9y + 7 = 0
Option 3: 4x + 7y + 3 = 0
Option 4: 2x − 9y − 11 = 0
Correct Answer: 2

 

Solutions:Since PS is the median of the triangle, S is the midpoint of QR.

∴ Coordinates of S:

6+72,-1+32=132,1The line is parallel to the median PS, therefore both the slopes will be equal.

Slope of PS=y2-y1x2-x1=2-12-132=-29Since the line passes through

1, -1and its slope is

-29, we can use the point slope form.

Equation of the line:

y+1=-29x-1 ⇒9y+9=-2x+2⇒2x+9y+7=0Hence, the correct option is (2).

Question 53:

limx→0sin π cos2 xx2is equal to:

Option 1: π2
Option 2: 1
Option 3: −π
Option 4: π
Correct Answer: 4

 

Solutions:

limx→0sinπcos2xx2=limx→0sinπ1-sin2xx2=limx→0sinπ-πsin2xx2We know:

sinπ-x=sin x ∴limx→0sinπ-πsin2xx2=limx→0sinπsin2xx2Multiplying the numerator and the denominator by

πsin2x, we get:

limx→0sinπsin2xx2 =limx→0sinπsin2xπsin2x×limx→0πsin2xx2                      =1 ×πlimx→0sinxx2=π                      ∵ limx→0sinxx=1Hence, the correct option is (4). [[VIDEO:13785]]

Question 54:

If X = {4n − 3n − 1 : n ∈ N} and Y = {9(n − 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to:

Option 1: N
Option 2: Y − X
Option 3: X
Option 4: Y
Correct Answer: 4

 

Solutions:Given: X = {4n − 3n − 1 : n ∈ N}= {(1+3)n − 3n − 1 : n ∈ N}

Expanding binomially, we get:

4n-3n-1=C0n+C1n3+C2n32……………..+Cnn3n-3n-1                  =1+3n+C2n32……………..+3n-3n-1                  =C2n32+C3n33…………….+3nTaking 9 as common, we get:9C2n+C3n3………………..+3n-2The set X has natural numbers that are multiples of 9.

Also, Y = {9(n − 1) : n ∈ N}

⇒ Y has all the multiples of 9

i.e.,

X⊆Y∴

X∪Y=YHence the correct option is (4). [[VIDEO:13786]]

Question 55:

The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangent to it is:

Option 1: (x2y2)2 = 6x2 + 2y2
Option 2: (x2y2)2 = 6x2 − 2y2
Option 3: (x2 + y2)2 = 6x2 + 2y2
Option 4: (x2 + y2)2 = 6x2 − 2y2
Correct Answer: 3

 

Solutions:x2 + 3y2 = 6

⇒x26+y22=1               … (1)

The equation of the tangent to the ellipse x2 + 3y2 = 6 is as follows:

y=mx±a2m2+b2 ∵a2 = 6, b2 = 2

∴y=mx±6m2+2         … (2)

Let P(h, k) be the foot of the perpendicular to the tangent.

The equation of the line that passes through the centre of the ellipse x2 + 3y2 = 6, i.e. (0, 0), and is perpendicular to

y=mx±6m2+2is as follows:

y=-1mx                      … (3)

Eliminating m from equations (2) and (3), we get the required locus as follows:

y=-xy×x±6×x2y2+2⇒x2+y22=6×2+2y2Hence, the correct option is (3).

Question 56:

Three positive numbers from an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is :

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

 

Solutions:Let the numbers in the G.P. be a, ar, ar2.

According to the question:

a, 2ar, ar2 are in A.P.

∴ 2(2ar) = a + ar2

⇒ 4r = 1 + r2

⇒ r2− 4r + 1 = 0

r=2±3Since three positive numbers form an increasing G.P.,

r=2 +3Hence, the correct option is (4).

Question 57:

If (10)9 + 2(11)1 (10)8 + 3(11)2 (10)7 + …..10 (11)9 = k(10)9, then k is equal to :

Option 1: 12110
Option 2: 441100
Option 3: 100
Option 4: 110
Correct Answer: 3

 

Solutions:Let

S = 109 + 2· 111·108 + 3· 112· 107+ …… + 10·119                 … (1) 1110·S = 111·108 +  2.112·107  +  ……..+ 9·119 + 1110             … (2)
Subtracting (2) from (1), we get: ⇒ -110S = 109 + 111·108 +112·107+ …….  + 119 – 1110 ⇒ -110S=1091+1110+11102+…+11109-1110

⇒ -110S = 109 111010 – 11110- 1 – 1110

⇒ -110S = 1110 – 1010 – 1110

S = 1011S = 100.109
Comparing with 

k109, we get:

k = 100

Hence, the correct option is (C). [[VIDEO:13787]]

Question 58:

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l2 = m2 + n2 is :

Option 1: π3
Option 2: π4
Option 3: π6
Option 4: π2
Correct Answer: 1

 

Solutions:The direction cosines of the two lines satisfy the relation

l+m+n=0          …1 l2=m2+n2          …2From (1) and (2), we have:

-m-n2=m2+n2⇒m2+n2+2mn=m2+n2⇒mn=0⇒m=0 or n=0Now, we know:

l2+m2+n2=1         …3

Here, m = 0, l = − n

Using (3), we have

-n2+0+n2=1⇒2n2=1⇒n2=12⇒n=±12∴

l1, m1, n1=-12, 0, 12 or 12, 0, -12Here, n = 0, l = − m

Using (3), we have:

-m2+m2+0=1⇒2m2=1⇒m2=12⇒m=±12∴

l2, m2, n2=-12, 12, 0 or 12, -12,0Let

θbe the angle between the two lines.

∴ cosθ=l1l2+m1m2+n1n2=-12×-12+0×12+12×0i.e. cosθ=12⇒θ=π3Hence, the correct option is (1). [[VIDEO:13788]]

Question 59:

The slope of the line touching both the parabolas y2 = 4x and x2 = −32y is :

Option 1: 12
Option 2: 32
Option 3: 18
Option 4: 23
Correct Answer: 1

 

Solutions:Let A (t2, 2t) be a point on the parabola y2 = 4x.

The equation of tangent at A, yt = x + t2, is tangent to the parabola x2 + 32y = 0 at B.

x2+32xt+t=0 ⇒x2+32xt+32t=0Now, the discriminant of the above quadratic equation should be zero.

∴ 32t2-432t=0 ⇒3232t2-4t=0 ⇒32t2=4t⇒t3=8⇒t=2∴ Slope of tangent =

1t=12Hence, the correct option is (1). [[VIDEO:13789]]

Question 60:

If x = −1 and x = 2 are extreme points of f(x) = α log |x| = βx2 + x then :

Option 1: α=-6, β=12
Option 2: α=-6, β=-12
Option 3: α=2, β=-12
Option 4: α=2, β=12
Correct Answer: 3

 

Solutions:

fx=αlogx+βx2+x ⇒f’x=αx+2βx+1=2βx2+α+xxNow, x = −1 and x = 2 are extreme points of f (x).

∴ f’-1=0  and f’2=0 ⇒2β-12+α+-1=0   and   2β22+α+2=0 ⇒2β+α-1=0 and 8β+α+2=0On solving the two equations, we get:

α=2 and β=-12Hence, the correct option is (3).

Question 61:

Which one of the following properties is not shown by NO?

Option 1: It combines with oxygen to form nitrogen dioxide.
Option 2: It’s bond order is 2.5.
Option 3: It is diamagnetic in gaseous state.
Option 4: It is a neutral oxide.
Correct Answer: 3

 

Solutions:NO is a neutral oxide. It combines with oxygen to form NO2.

2NOg + O2g → 2NO2g

Number of electrons in an NO molecule = 7 + 8 = 15

The molecular electronic configuration of NO is

σ1s2 , σ*1s2, σ2s2, σ*2s2, σ2pz2 , π2px2=π2py2,  π*2px1 =π*2py0Bond order of NO =

Nb-Na2=10 – 52= 2.5It has one unpaired electron in the

πanti-bonding molecular orbital. As a result, it exhibits paramagnetic character in its gaseous state.

Hence, the correct option is (3).

Question 62:

If Z is a compressibility factor, van der Waals’ equation at low pressure can be written as

Option 1: Z=1-PbRT
Option 2: Z=1+PbRT
Option 3: Z=1+RTPb
Option 4: Z=1-aVRT
Correct Answer: 4

 

Solutions:Vander Waals’ equation for one mole of a gas is

P + aV2V-b = RTAt low pressure, the volume of a gas is extremely large. As a result, van der Waals’ constant b can be neglected in comparison to V. (V – b) = V

P + aV2V = RT⇒PV = RT – aV⇒PVRT = 1 – aRTV∵PVRT=Z∴Z=1-aRTVHere, Z is compressibility factor.

Hence, the correct option is (4). [[VIDEO:13886]]

Question 63:

The metal that cannot be obtained by electrolysis of an aqueous solution of its salts is

Option 1: Cu
Option 2: Cr
Option 3: Ag
Option 4: Ca
Correct Answer: 4

 

Solutions:Among the given metals, only Ca is an s-block element. It is known that during the electrolysis of s-block element, H2 gas is discharged at the cathode.

2H2O + 2e-→H2g + 2OH-aq.                   Eo = -0.83 VCa2+ + 2e- →Cas                                           Eo = -2.87 V

Since, the reaction for the higher value of Eo is preferred hence, during electrolysis at cathode the following reaction will occur.

2H2O + 2e- → H2 (g) + 2OH- (aq)Therefore, Ca cannot be obtained by the electrolysis of an aqueous solution of its salts.

Hence, the correct option is (4). [[VIDEO:13887]]

Question 64:

Resistance of 0.2 M solution of an electrolyte is 50 Ω. The specific conductance of the solution is 1.4 S m−1. The resistance of 0.5 M solution of the same electrolyte is 280 Ω. The molar conductivity of 0.5 M solution of the electrolyte in S m2 mol−1 is

Option 1: 5 × 103
Option 2: 5 × 102
Option 3: 5 × 10−4
Option 4: 5 × 10−3
Correct Answer: 3

 

Solutions:For 0.2 M solution of the electrolyte: Resistance, R = 50 ohm Specific conductance, σ = 1.4 Sm−1= 1.4

×10−2 S cm−1

Specific conductance can be expressed as: σ=la×1R la = R×σla= 50×1.4×10-2 cm-1(where l/a is called cell constant)

Now, for 0.5 M solution of the electrolyte: Resistance, = 280 ohm Molar conductivity,  λ is expressed as:

 

λ=σ×1000M σ=la×1R  =50×1.4×10-2280   = 2.5×10-3 S cm-1 λ = σ×1000M    = 2.5×10-3×10000.5    =5 S cm2 mol-1    = 5×10-4 S m2 mol-1Hence, the correct option is (3).

Question 65:

CsCI crystallises in body-centred cubic lattice. If ‘a‘ is its edge length, then which of the following expressions is correct?

Option 1: rCs++rCl-=32a
Option 2: rCs++rCl-=3a
Option 3: rCs++rCl-=3a
Option 4: rCs++rCl-=3a2
Correct Answer: 1

 

Solutions:CsCl crystallises in a bcc lattice, in which Cs+ ion is present at the centre and the Cl ions are present at the edges of the unit cell. For a bcc lattice, the relation between the nearest neighbour distance (d) and the edge length (a) is expressed as:

d = 32a

Now, the nearest neighbour distance is equal to the sum of the radii of the cation and the anion.

d = rCs+ + rCl-⇒rCs+ + rCl- = 32aHence, the correct option is (1).

Question 66:

Consider separate solutions of 0.500 M C2H5OH(aq), 0.100 M Mg3(PO4)2(aq), 0.250 M KBr(aq) and 0.125 M Na3PO4(aq) at 25° C. Which statement is true about these solutions, assuming all the salts to be strong electrolytes?

Option 1: 0.125 M Na3PO4(aq) has the highest osmotic pressure.
Option 2: 0.500 M C2H5OH(aq) has the highest osmotic pressure.
Option 3: They all have the same osmotic pressure.
Option 4: 0.100 M Mg3(PO4)2(aq) has the highest osmotic pressure.
Correct Answer: 3

 

Solutions:

Osmotic pressure is given by the formula mentioned below π = iCRT ​Where

π is osmotic pressure, i is van’t Hoff factor, C is concentration, R is universal gas constant and T is the temperature. We know,

α = i-1n-1Where

α is the dissociation constant and n is the number of particles released on dissociation. As

α= 1, Assuming all salts as strong electrolyte, therefore the relation between n and i can be written as: n = i 

Electrolyte n
C2H5OH 1 1
Mg3(PO4)2 5 5
KBr 2 2
 Na3PO4 4 4

(i) For 0.5 M C2H5OH:

 

π=1×0.5×R×T= 0.5 RT(ii) For 0.1 M Mg3(PO4)2:

 

π=5×0.1×R×T= 0.5 RT (iii) For 0.25 M KBr:

 

π=2×0.25×R×T= 0.5 RT(iv) For 0.125 M Na3PO4:

π=4×0.125×R×T= 0.5 RTAll the given solutions have the same osmotic pressure.

Hence, the correct option is (3).

Question 67:

In which of the following reactions, H2O2 acts as a reducing agent? (a) H2O2 + 2H+ + 2e → 2H2O (b) H2O2 − 2e → O2 + 2H+ (c) H2O2 + 2e → 2OH (d) H2O2 + 2OH − 2e → O2 + 2H2O

Option 1: (a), (c)
Option 2: (b), (d)
Option 3: (a), (b)
Option 4: (c), (d)
Correct Answer: 2

 

Solutions:

The substance which itself undergoes oxidation acts as a reducing agent. (a) (b) (c) (d)

In equations (b) and (d), H2O2 undergoes oxidation thus in both reactions it will acts as a reducing agent.

Hence, the correct option is (2).

Question 68:

In SN2 reactions, the correct order of reactivity for the compounds provided in brackets is

CH3Cl, CH3CH2Cl,CH32CHCl and CH33CCl

Option 1: CH3CH2Cl>CH3Cl>CH32CHCl>CH33CCl
Option 2: CH32CHCl>CH3CH2Cl>CH3Cl>CH33CCl
Option 3: CH3Cl>CH32CHCl>CH3CH2Cl>CH33CCl
Option 4: CH3Cl>CH3CH2Cl>CH32CHCl>CH33CCl
Correct Answer: 4

 

Solutions:The rate of bimolecular nucleophilic substitution reaction (SN2) depends on the steric-crowding in the alkyl halide. More the number of alkyl groups present around the carbon atom bearing the chloride group, lesser is the possibility of the attack of the nucleophile. So, the correct order of reactivity is CH3Cl > CH3CH2Cl > (CH3)2CHCl > (CH3)3CCl

Hence, the correct option is (4).

Question 69:

The octahedral complex of a metal iron M3+ with four monodentate ligands L1, L2, L3 and L4 absorbs wavelengths in the region of red, green, yellow and blue, respectively. The increasing order of ligand strength of the four ligands is

Option 1: L3 < L2 < L4 < L1
Option 2: L1 < L2 < L4 < L3
Option 3: L4 < L3 < L2 < L1
Option 4: L1 < L3 < L2 < L4
Correct Answer: 4

 

Solutions:

The energy order of light in the visible region is Violet > Indigo > Blue > Green > Yellow > Orange > Red.

Strong-field ligands absorb higher energy radiations and weak-field ligands absorb lower energy radiations. Therefore, the ligand that absorbs blue colour will be the strong-field ligand and the one that absorbs red colour will be the weak-field ligand. So, the increasing order of strength of ligands is L1<l3<l2<l4.</l</l</l

Hence, the correct option is (4).

Question 70:

For the estimation of nitrogen, 1.4 g of an organic compound was digested by Kjeldahl method and the evolved ammonia was absorbed in 60 mL of

M10sulphuric acid. The unreacted acid required 20 mL of

M10sodium hydroxide for complete neutralisation. The percentage of nitrogen in the compound is

Option 1: 3%
Option 2: 5%
Option 3: 6%
Option 4: 10%
Correct Answer: 4

 

Solutions:The percentage of nitrogen can be found by the following formula:

% N = 1.4 × Molarity of H2SO4 × 2(Volume of of H2SO4 required for the neutralization of ammonia)Mass of compound taken for estimation        …(i)

Mass of compound taken = 1.4 g Molarity of H2SO4 = M/10 Molarity of NaOH = M/10 Volume of NaOH used up for complete neutralisation = 20 mL Total volume of H2SO4 = 60 mL Volume of H2SO4 used for complete neutralisation with ammonia can be determined by the following steps: We know that na Ma Va = nb Mb Vb        …. (ii) Where na and nb are the basicity and acidity of acid and base, respectively. Ma and Mb are the molarities and Va and Vb are the volumes of acid and base, respectively. Ma = 1/ 10 M Mb = 1/10 M na = 2,  since H2SO4 is a dibasic acid nb = 1 as NaOH is monoacidic base Vb = 20 mL

On substituting the above values in formula (ii), we get:

2

× (1/10 )M

× Va = 1

× (1/10 )M

× 20 mL

Va = 10 mL

Hence, the volume of H2SO4 left unused is 10mL. Therefore, the volume of H2SO4 used up for neutralisation = (60 – 10) mL Volume of H2SO4 used for complete neutralisation of NH3 is 50 mL. ​ Now,

% N = 1.4 × (1/10) × 2(50)1.4           = 10This implies that the percentage of nitrogen present in 1.4 g of compound is 10.

Hence, the correct option is (4).

Question 71:

The equivalent conductance of NaCl at concentration C and at infinite dilution are λC and λ, respectively. The correct relationship between λC and λ is (where the constant B is positive)

Option 1: λC=λ∞-BC
Option 2: λC=λ∞+BC
Option 3: λC=λ∞+BC
Option 4: λC=λ∞-BC
Correct Answer: 1

 

Solutions:

The molar conductivity of a strong electrolyte varies with the concentration, according to the Debye-Huckel-Onsager equation mentioned below.

λc = λ∞- (B) CHere,

λc is the molar conductivity at any concentration,

λ∞ is the molar conductivity at infinite dilution, B is constant (whose nature depends upon electrolyte, temperature and nature of solvent) and C is the concentration of electrolyte.

Hence, the correct option is (1).

Question 72:

For the reaction

SO2g+12O2g⇌SO3g, if KP=KCRTx, where the symbols have usual meaning, then the value of x is (Assuming ideality)

Option 1: 12
Option 2: 1
Option 3: −1
Option 4: -12
Correct Answer: 4

 

Solutions:

Consider the following reaction: SO2(g) +

12O2(g) â‡Œ SO3 (g) For the above reaction, the relation between Kp and Kc is expressed as Kp = Kc (RT)x Here, Kp is the equilibrium constant in terms of pressure. Kc is the equilibrium constant in term of concentration. R is the gas constant T is the temperature. x is the change in the number of moles of gaseous substance.

 x = Number of moles of gaseous product − Number of moles of gaseous reactant        …………….(i) Number of moles of gaseous product = 1 Number of moles of gaseous reactant = ( 1 + 1/2) =  3/2 On substituting the above values in equation (i), we get: x = 1 − 3/2 x =  −1/2

Hence, the correct option is (4).

Question 73:

In the reaction,

CH3COOH→  LiAIH4   A→  PCI5  B→  Alc.KOH  C,the product C is

Option 1: ethylene
Option 2: acetyl chloride
Option 3: acetaldehyde
Option 4: acetylene
Correct Answer: 1

 

Solutions:LiAlH4 ( Lithium Aluminium hydride ) is a strong reducing agent and generally employed for the reduction of organic compounds. It reduces carboxylic acid, aldehyde and ketones to their respective alcohols.

CH3COOH →LiAlH4 CH3CH2OHPCl5, on reacting with alcohols, cleaves the carbon-oxygen bond to produce alkyl halide.

CH3CH2OH →PCl5 CH3CH2ClAlcoholic KOH is a well-known regent for the conversion of alkyl halide to alkenes.

CH3CH2Cl →Alc. KOH CH2=CH2When ethanoic acid is treated with LiAlH4, it yields ethanol (A). Ethanol, on treatment with PCl5, results in 1-Chloroethane (B). When 1-Chloroethane is treated with alcoholic KOH, it undergoes elimination reaction to produce ethene (C). The equation for the above reaction is as follows:

CH3COOH →   LiAlH4         CH3CH2OH →     PCl5         CH3CH2Cl →     Alc. KOH     CH2=CH2Ethanoic acid                  Ethanol                       Ethyl chloride                    EthyleneThe common name of ethene is ethylene.

Hence, the correct option is (1).

Question 74:

Sodium phenoxide, when heated with CO2 under pressure at 125°C, yields a product that on acetylation produces C.

The major product C would be

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

 

Solutions:Sodium phenoxide ion, when treated with carbon dioxide under high pressure (5 atm) and high temperature (125 oC), yields sodium salicylate (B). When sodium salicylate is treated with acetic anhydride, it results in the acetylation of OH group. The equation of the reaction is given below. The mechanism for the above reaction is as follows:

Hence, the correct option is (3). [[VIDEO:13889]]

Question 75:

On heating an aliphatic primary amine with chloroform and ethanolic potassium hydroxide, the organic compound formed is

Option 1: an alkyl cyanide
Option 2: an alkyl isocyanide
Option 3: an alkanol
Option 4: an alkanediol
Correct Answer: 2

 

Solutions:On heating an aliphatic primary amine with chloroform and ethanolic KOH, alkyl isocyanide is produced. This reaction is known as Carbylamine reaction or Isocyanide test. The alkyl isocyanide, thus formed, has very unpleasant or foul odour. This reaction is exhibited by only aliphatic and aromatic primary amines. Therefore, this test is used to distinguish primary amines from secondary and tertiary amines. The general equation for the reaction is as follows:

R-NH2 + CHCl3 + 3KOHethanolic →         ∆           R-NCalkyl isocyanide  + 3KCl + 3H2OHence, the correct option is (2). [[VIDEO:13890]]

Question 76:

Which is the correct statement for the molecule CsI3?

Option 1: It contains Cs3+ and I ions.
Option 2: It contains Cs+, I and lattice I2 molecule.
Option 3: It is a covalent molecule.
Option 4: It contains Cs+ and

I3-ions.

Correct Answer: 4

 

Solutions:Cesium triiodide is an ionic polyhalide compound that on dissociation produces Cs+ and

I3-ions. Therefore, it contains Cs+ and

I3-ions only. Moreover, Cs belongs to alkali metals. Therefore, it exhibits only +1 oxidation state.

Hence, the correct option is (4). [[VIDEO:13891]]

Question 77:

The equation which is balanced and represents the correct product(s) is

Option 1: [Mg (H2O)6]2+ + (EDTA)4−

→    excess NaOH      MgEDTA2++6H2O

Option 2: CuSO4 + 4KCN → K2[Cu(CN)4] + K2SO4
Option 3: Li2O + 2KCl → 2LiCl + K2O
Option 4: [CoCl(NH3)5]+ + 5H+ → Co2+ + 5NH4+ + Cl
Correct Answer: 4

 

Solutions:Among the given reactions, equation (4) is balanced and represents the correct product because [CoCl(NH3)5]decomposes under acidic medium. [CoCl(NH3)5]+ + 5H+ → Co2+ +

5NH4++ Cl

Hence, the correct option is (4). [[VIDEO:13892]]

Question 78:

For which of the following molecule. significant μ≠0 ? (a)                       (b) (c)                       (d)

Option 1: Only (c)
Option 2: (c) and (d)
Option 3: Only (a)
Option 4: (a) and (b)
Correct Answer: 2

 

Solutions:1, 4−benzenedithiol and 1, 4−dihydroxybenzene have non zero dipole moments. Because the two hydroxyl and two thiol groups are not symmetrical about C−O−H and C−S−H respectively, these do not lie in the same plane as the benzene ring. Hence, 1, 4−benzenedithiol and 1, 4−dihydroxybenzene show some value of dipole moment.

1, 4−dichlorobenzene and 1, 4−dicynobenzene are symmetrical and chlorine and cyanide groups lie in the same plane of benzene ring. So, the direction of the bond moment vectors cancel each other and show zero dipole moment.

                                

Hence, the correct option is (2). [[VIDEO:13893]]

Question 79:

For the non−stoichiometric reaction 2A + B → C + D, the following kinetic data were obtained in three separate experiments, all at 298 K.

Initial Concentration (A) Initial Concentration (B) Initial rate of formation of C (mol L−1S−1 )
0.1 M 0.1 M 0.2 M 0.1 M 0.2 M 0.1 M 1.2 × 10−3 1.2 × 10−3 2.4 × 10−3

The rate law for the formation of C is

Option 1: dcdt= k[A][B]2
Option 2: dcdt= k[A]
Option 3: dcdt= k[A][B]
Option 4: dcdt= k[A]2[B]
Correct Answer: 2

 

Solutions:The non-stoichiometric reaction is 2A + B → C + D. Now,

Rate of reaction =-12dAdt=-dBdt=dCdt=dDdtLet, the rate law for the formation of C be

dCdt= k [A]a [B]b where, a and b are order of reaction with respect to reactant A and B respectively. Data in the given table can be written as: k [0.1]a [0.1]b =

1.2×10-3    ……..(1) k [0.1]a [0.2]b =

1.2×10-3    ……..(2) k [0.2]a [0.1]b =

2.4×10-3    ………(3)

Dividing equation (1) by (2), we get:

k0.1a0.1bk0.1a0.2b=1.2×10-31.2×10-312b=1So,

12b=120b =0Dividing equation (1) by (3) we get,

k0.1a0.1bk0.2a0.1b=1.2×10-32.4×10-312a=121a=1Therefore,

dCdt=kA1B0dCdt=kAHence, the correct option is (2).

Question 80:

Which series of reactions correctly represents the chemical relations related to iron and its compounds ?

Option 1: Fe→      Cl2, heat         FeCl3→         heat, air          FeCl2→            Zn             Fe
Option 2: Fe→        O2, heat          Fe3O4→       CO, 600°C           FeO→         CO, 700°C          Fe
Option 3: Fe→       dil H2SO4          FeSO4→          H2SO4, O2            Fe2SO43→        heat          Fe
Option 4: Fe→         O2, heat         FeO→         dil H2SO4              FeSO4→            heat               Fe
Correct Answer: 1

 

Solutions:When dry chlorine gas is passed over heated iron fillings, anhydrous ferric chloride is formed. 2Fe + 3Cl2 → 2FeCl3 On heating FeCl3 in the presence of air, FeCl2 and Cl2 are produced.

2FeCl3 →∆2FeCl2 + Cl2On reduction of FeCl2 with Zn, Fe metal is obtained. FeCl2 + Zn → Fe + ZnCl2

In option (4), on heating Fe in the presence of oxygen and heat, Fe3O4 is obtained. So, this sequence is wrong. In option (3), on heating Fe2(SO4)3, oxide is formed. Thus, the given sequence of reactions is also wrong. In option (2), according to Ellingham diagram, the reduction of Fe3O4 with CO occurs in the range of 600-700 oC and reduction of FeO to Fe occurs at a higher temperature range. So, the given sequence is not correct as well.

Hence, the correct option is (1). [[VIDEO:13894]]

Question 81:

Considering the basic strength of amines in an aqueous solution, which one has the smallest pKb value ?

Option 1: (CH3)3N
Option 2: C6H5NH2
Option 3: (CH3)2NH
Option 4: CH3NH2
Correct Answer: 3

 

Solutions:The greater the basic strength of a compound, the smaller is its pKb value. In the case of aromatic amines, the lone pair of electrons on the N atom is involved in delocalisation with the aromatic

πelectron cloud of the benzene ring. Thus, aromatic amines are less basic than aliphatic amines. Among the alkyl amines, the secondary amines are most basic due to +I effect and formation of hydrogen bonding in aqueous solution. Thus, among the given compounds, (CH3)2NH is most basic and has the smallest pKb value.

Hence, the correct option is (3). [[VIDEO:13895]]

Question 82:

Which one of the following bases is not present in DNA?

Option 1: Cytosine
Option 2: Thymine
Option 3: Quinoline
Option 4: Adenine
Correct Answer: 3

 

Solutions:The following bases are present in DNA. A – Adenine G – Guanine T – Thymine C – Cytosine The compound, quinoline, is not present in DNA.

​Hence, the correct option is (3). [[VIDEO:13896]]

Question 83:

The correct set of four quantum numbers for the valence electrons of rubidium atom (Z = 37) is

Option 1: 5, 1, 1, +

12

Option 2: 5, 0, 1, +

12

Option 3: 5, 0, 0, +

12

Option 4: 5, 1, 0, +

12

Correct Answer: 3

 

Solutions:The electronic configuration of Rb is 1s22s22p63s23p64s23d104p65s1. ​So, the last electron enters the 5s atomic orbital. For this orbital, the values of quantum numbers are: n = 5 l = 0 ml = 0 ms =

+12Hence, the correct option is (3). [[VIDEO:13897]]

Question 84:

The major organic compound formed by the reaction of 1, 1, 1 − trichloroethane with silver powder is

Option 1: 2 − Butyne
Option 2: 2 − Butene
Option 3: Acetylene
Option 4: Ethene
Correct Answer: 1

 

Solutions:On reaction of 1, 1, 1− trichloroethane with silver powder, 2−butyne is obtained as the product.

Hence, the correct option is (1).\ [[VIDEO:13898]]

Question 85:

Given below are half – cell reactions. Mn2+ + 2e → Mn ; Eo = − 1.18 V 2(Mn3+ + e → Mn2+) ; Eo = + 1.51 V The Eo for 3Mn2+ → Mn + 2Mn3+ will be

Option 1: −0.33 V ; the reaction will not occur
Option 2: −0.33 V ; the reaction will occur
Option 3: −2.69 V ; the reaction will not occur
Option 4: −2.69 V ; the reaction will occur
Correct Answer: 3

 

Solutions:

Mn2++ 2e-→Mn;   Eo=-1.18V             ………………….. 1Mn3++ e-→Mn2+; Eo=+1.51 V           …………………….2For 3Mn2+ → Mn + 2Mn3+, we need to multiply equation 2 with 2 and  reverse the resultant equation.2Mn2+ →2Mn3+ + 2e- ; Eo=-1.51 V      …………………..3On adding equations 1 and 3, we get:3Mn2+ →Mn + 2Mn3+ ; Eo=-2.69 V∵ ∆G = -nFEo.Value of Eo is negative, the value of ∆G will be positive, which means that the reaction will be non-spontaneous.Hence, the correct option is (3). [[VIDEO:13899]]

Question 86:

The ratio of masses of oxygen and nitrogen in a particular gaseous mixture is 1 : 4. The ratio of number of their molecules is

Option 1: 1 : 8
Option 2: 3 : 16
Option 3: 1 : 4
Option 4: 7 : 32
Correct Answer: 4

 

Solutions:Given, mass of oxygen : mass of nitrogen = 1 : 4

Let, the mass of oxygen be x g. ∴ Mass of nitrogen = 4x g Number of moles =

Given massMolar massNumber of moles of

O2 = x32∴ Number of molecules of

O2 = x32×NAThe number of moles of

N2 = 4×28∴ Number of molecules of

N2 = 4×28×NASo, the ratio of number of molecules of oxygen to number of molecules of nitrogen is

=x32×NA : 4×28×NA=x32 : x7=7 : 32Hence, the correct option is (4).

Question 87:

Which one is classified as a condensation polymer?

Option 1: Teflon
Option 2: Acrylonitrile
Option 3: Dacron
Option 4: Neoprene
Correct Answer: 3

 

Solutions:Dacron is a polymer that is formed by the condensation of terephthalic acid and ethylene glycol. Neoprene and teflon are the addition polymers of isoprene and tetrafluoroethylene, respectively and acrylonitrile is a monomer of the polymer polyacrylonitrile.

Hence, the correct option is (3).

Question 88:

Among the following oxoacids, the correct decreasing order of acid strength is

Option 1: HClO4 > HClO3 > HClO2 > HOCl
Option 2: HClO2 > HClO4 > HClO3 > HOCl
Option 3: HOCl > HClO2 > HClO3 > HClO4
Option 4: HClO4 > HOCl > HClO2 > HClO3
Correct Answer: 1

 

Solutions:Acidic strength is defined by the resonance stability of the conjugated base obtained by the loss of hydrogen atom from the given acid.

HClO4 → ClO4- + H+                    ……. iHClO3 → ClO3- + H+                    ……. iiHClO2 → ClO2- + H+                    ……. iiiHOCl → ClO- + H+                      ……..ivFactors that affect the acidic strength:
1. More the resonating structures of the conjugated base more will be the acidic strength
2. More the number of surrounding oxygen atoms, more will be the oxidation number of the central Cl atom and hence, more will be the acidic strength
1. More the resonating structures of the conjugated base more will be the acidic strength

Conjugated base Number of resonating structures Resonance structures
4
3
2
0 Not possible

On observing the resonance structures of the conjugated bases, the order of the stability of the conjugated bases obtained can be written as:

ClO4- > ClO3- > ClO2- >ClO-2. More the number of surrounding oxygen atoms, more will be the oxidation number of the central Cl atom and hence, more will be the acidic strength

Conjugated base Number of surrounding oxygen atoms Oxidation number of central Cl atom
 

ClO4-

4 +7
 

ClO3-

3 +5
 

ClO2-

2 +3
 

ClO-

1 +1

So, on the basis of above two factors the order of the acidic strength is given as HClO4 > HClO3 > HClO2 > HClO

Hence, the correct option is (1). [[VIDEO:13900]]

Question 89:

For complete combustion of ethanol, C2H5OH(l) + 3O2(g) → 2CO2(g) + 3H2O(l), the amount of heat produced, as measured in a bomb calorimeter, is 1364.47 kJ mol−1 at 25°C. Assuming ideality of the enthalpy of combustion, ∆cH, for the reaction will be (R = 8.314 kJ mol−1)

Option 1: −1460.50 kJ mol−1
Option 2: −1350.50 kJ mol−1
Option 3: −1366.95 kJ mol−1
Option 4: −1361.95 kJ mol−1
Correct Answer: 3

 

Solutions:The complete combustion of ethanol is represented as:

C2H5OHl + 3O2g → 2CO2 g + 3H2OlGiven, ∆U = -1364.47 kJ mol-1∆ng = 2 – 3 = -1R = 8.314 J K-1 mol-1    = 8.314 × 10-3 kJ K-1 mol-1T= 298 K∆H = ∆U + ∆ngRT         = -1364.47 + (-1×8.314×2981000)           = – 1366.947 kJ mol-1Hence, the correct option is (3). [[VIDEO:13901]]

Disclaimer: Given value of R is wrong, standard value of R = 8.314 J K–1 mol–1

Question 90:

The most suitable reagent for the conversion of R−CH2−OH → R−CHO is

Option 1: CrO3
Option 2: PCC (Pyridinium Chlorochromate)
Option 3: KMnO4
Option 4: K2Cr2O7
Correct Answer: 2

 

Solutions:The most suitable reagent for the oxidation of primary alcohols to aldehydes in good amount is PCC. It is a mild oxidising agent.

R-CH2-OH →PCC R-CHO

Reagents like CrO3, KMnO4 and K2Cr2O7 oxidise the alcohol directly to carboxylic acid.

Hence, the correct option is (2). [[VIDEO:13902]]

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