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Category: IIT-JEE 2015 Physics Paper 2

Question 1:

A large spherical mass M is fixed at one position and two identical point masses m are kept on a line passing through the centre of M (see figure) .The point masses are connected by a rigid massless rod of length I and the assembly is free to move along the line connecting them.all three interact only through their mutual gravitational interaction.when the point mass nearer to M is at a distance r=3 l from M, the tension in the rod is zero for

m = k (\frac{M}{288}) .

The value of k is

Category: IIT-JEE 2015 Physics Paper 2

Question 2:

The energy of a system as a function of time t is given a E(t)=

A^{2} e x p (- α t),

where

α = 0.2 s^{- 1}

.The measurement of A has an error of 1.25%.If has error in the measurement of time is 1.50% ,the percentage error in the value of E(t) at t=5 s is

Category: IIT-JEE 2015 Physics Paper 2

Question 3:

The densities of two solid spheres A and B of the same radii R very with radial distance r as

p_{A} (r) = k (\frac{r}{R}) a n d p_{B} (r) = {(\frac{r}{R})}^{5}

,respectively ,where k is a constant.The moments of inertia of the individual spheres about axes passing through their centers are

I_{A} a n d I_{B}

,respectively.If

\frac{I_{B}}{I_{A}} = \frac{n}{10}

,the value of n is

Category: IIT-JEE 2015 Physics Paper 2

Question 4:

Four harmonic waves of equal frequencies and equal intensities

I_{0}

have phase angles 0,

π / 3, 2 π / 3 andπ .

when they are superposed,the intensity of the resulting wave is n

I_{0}

.The value of n is

Category: IIT-JEE 2015 Physics Paper 2

Question 5:

For a radioactive material,its activity A and rate of change of its activity R are defined as

A = \frac{d N}{d t} a n d R = - \frac{d A}{d t},

where N(t) is the number of nuclei at time t.Two radioactive sources P(mean lifeτ) and Q(mean life 2τ)have the same activity at t=0. their rate of change of activities at t =2τ are

R_{P} a n d R_{Q}

,respectively .If

\frac{R_{p}}{R_{Q}} = \frac{n}{e},

then the value of n is

Category: IIT-JEE 2015 Physics Paper 2

Question 6:

A monochromatic beam of light is incident at

60 °

on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle

θ

(n) with the normal (see the figure).For n=

\sqrt{3}

the value of

θ

is 60

°

and

\frac{d θ}{d n} = m

.The value of m is

Category: IIT-JEE 2015 Physics Paper 2

Question 7:

In the following circuit,the current through the resistor R

(= 2

Ω)is I Amperes. The value of I is

Category: IIT-JEE 2015 Physics Paper 2

Question 8:

An electron in an excited state of

L i^{2 +}

ion has angular momentum

3 h / 2 π .

The de Broglie wavelength of the electron in this state is

p {πα}_{0}

(where

α_{0}

is the Bohr radius).The value of p is

Category: IIT-JEE 2015 Physics Paper 2

Question 9:

Two spheres P and Q of equal radii have densities

p_{1} a n d p_{2}

,respectively. The spheres are connected by a massless string and placed in liquids

L_{1} a n d L_{2}

of densities

σ_{1} a n d σ_{2}

and viscosities

η_{1} a n d η_{2}

,respectively .They float in equilibrium with the sphere P in

L_{1} a n d s p h e r e Q i n L_{2}

and the string being taut (see figure),If sphere P alone in

L_{2}

has terminal velocity

V_{P}

and Q alone in

L_{1}

has terminal velocity

\vec{V_{Q}}

,then

A	$\frac{\|\vec{V_{P}}\|}{\|\vec{V_{Q}}\|} = \frac{η_{1}}{η_{2}}$	B	$\frac{\|\vec{V_{P}}\|}{\|\vec{V_{Q}}\|} = \frac{η_{2}}{η_{1}}$
C	$\vec{V_{P}} . \vec{V_{Q}} > 0$	D	$\vec{V_{P}} . \vec{V_{Q}} < 0$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2015 Physics Paper 2

Question 10:

In terms of potential difference V,electric current I,permittivity

ε_{0}

,permeability

μ_{0}

and speed of light c, the dimensionally correct equation (s) is (are)

A	$μ_{0} I^{2} = ε_{0} V^{2}$	B	$ε_{0} I = μ_{0} V$
C	$I = ε_{0} c V$	D	$μ_{0} c I = ε_{0} V$

A and B

A and C

A and D

None of these

Category: IIT-JEE 2015 Physics Paper 2

Question 11:

Consider a uniform spherical charge distribution of radius

R_{1}

central at the origin O.In this distribution ,a spherical cavity of radius

R_{2}

, centred at P with distance OP=

α = R_{1} - R_{2}

(see figure)is made.If the electric field inside the cavity at position

\vec{r} i s \vec{E}

(\vec{r}),

then the correct statement(s)is (are)

A	$\vec{E}$ is uniform,its magnitude is independent of $R_{2}$ but its direction on $\vec{r}$	B	$\vec{E}$ is uniform,its magnitude is depends on $R_{2}$ and its direction on $\vec{r}$
C	$\vec{E}$ is uniform,its magnitude is independent of $α$ but its direction on $\vec{α}$	D	$\vec{E}$ is uniform,its magnitude is depends on and its direction on $\vec{α}$

Category: IIT-JEE 2015 Physics Paper 2

Question 12:

In plotting stress versus strain curves for two materials P and Q, a students by mistake puts strain on the y-axis and stress on the x-axis as show in the figure .Then the correct statement(s) is (are)

A	P has more tensile strength than Q	B	P is more ductile than Q
C	P is more brittle than Q	D	The young's modulus of P is more than that of Q

A and B

B and C

C and D

Category: IIT-JEE 2015 Physics Paper 2

Question 13:

A spherical body of radius R consists of a fluid of constant of a fluid of constant density and is in equilibrium under its own gravity. If P(r) is the pressure at r(r

<

R),then the correct option (s) is (are)

A	$P (r = 0) = 0$	B	$\frac{P (r = 3 R / 4)}{p (r = 2 R / 3)} = \frac{63}{80}$
C	$\frac{P (r = 3 R / 5)}{P (r = 2 R / 5)} = \frac{16}{21}$	D	$\frac{P (r = 3 R / 4)}{P (r = R / 3)} = \frac{20}{27}$

A and B

B and C

C and D

Category: IIT-JEE 2015 Physics Paper 2

Question 14:

A parallel plate capacitor having plates of area S and plate separation d, has capacitance

C_{1}

in air. When two dielectrics of different relative permittivities

(ε_{1} = 2 a n d ε_{2} = 4)

are introduced between the two plates as shown in the figure,the capacitance becomes

C_{2}

.The ratio

\frac{C_{2}}{C_{1}} i s

A	$6 / 5$	B	$5 / 3$
C	$7 / 5$	D	$7 / 3$

Category: IIT-JEE 2015 Physics Paper 2

Question 15:

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially, the gas is at temperature

T_{1} p r e s s u r e p_{1} a n d v o l u m e V_{1}

and the spring is in its relaxed state. The gas is then heated very slowly to temperature

T_{2}

pressure

p_{2}

and volume

V_{2}

. During this process the piston moves out by a distance x.Ignoring the friction between the piston and the cylinder, the correct statement(s) is )(are)

A	If $V_{2} = 2 V_{1} a n d T_{2} = 3 T_{1},$ then the energy stored in the spring is $\frac{1}{4} p_{1} v_{1}$	B	If $V_{2} = 2 V_{1} a n d T_{2} = 3 T_{1},$ then the change in internal energy is $3 P_{1} V_{1}$
C	If $V_{2} = 2 V_{1} a n d T_{2} = 4 T_{1},$ then the work done by the gas is $\frac{7}{3} P_{1} V_{1}$	D	If $V_{2} = 2 V_{1} a n d T_{2} = 4 T_{1},$ then the heat supplied to the gas is $\frac{17}{6} P_{1} V_{1}$

A, B and C

B,C and D

C, D and A

None of these

Category: IIT-JEE 2015 Physics Paper 2

Question 16:

A fission reaction is given by

U_{92}^{236} \to X_{54}^{140} e + S_{38}^{94} r + x + y,

where x and y are two particles .considering

U_{92}^{236}

to be at rest , the kinetic energies of the products are denoted by

K_{X e} K_{s r}, K_{x} (2 M e V) a n d K_{y} (2 M e V),

respectively. let the binding energies per nucleon of

U_{92}^{236}, X_{54}^{140} e a n d S_{38}^{94} r b e 7.5 M e V, 8.5 M e V

,respectively .Considering different conservation laws, the correct option (s) is (are)

A	x=n, y=n, $K_{s r} = 129 M e V, K_{x e} = 86 M e V$	B	x=p, y= $e^{-}$ $K_{s r} = 129 M e V, K_{x e} = 86 M e V$
C	x=p, y=n, $K_{s r} = 129 M e V, K_{x e} = 86 M e V$	D	x=n, y=n $K_{s r} = 86 M e V, K_{x e} = 129 M e V$

Category: IIT-JEE 2015 Physics Paper 2

Question 17:

In a thin rectangular metallic strip, a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are I,

ω

and d, respectively. A uniform magnetic field

\vec{B}

is applied on the strip along the positive y-direction. Due to this, the charge carriers experience a net deflection along the z-direction . This result in accumulation of charge carriers on the surface PQRS and appearance of equal and opposite charges on the face opposite to PQRS.A potential difference along the z-direction is thus developed. charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross-section of the strip and carried by electrons.

Consider two different metallic strips (1 and 2) of the same material . Their lengths are the same, widths are

ω_{1} a n d ω_{2}

and thickness are

d_{1} a n d d_{2}

, respectively . Two points K and M are symmetrically located on the opposite face parallel to the x-y plane (see-figure).

V_{1} a n d V_{2}

are the potential differences between K and M in strips 1and 2, respectively.Then, for a given current I following through them in a given magnetic field strength B, the correct statements (s)is (are)

A	If $ω_{1} = ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = 2 V_{1}$	B	If $ω_{1} = ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = V_{1}$
C	If $ω_{1} = 2 ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = 2 V_{1}$	D	If $ω_{1} = 2 ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = V_{1}$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2015 Physics Paper 2

Question 18:

In a thin rectangular metallic strip, a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are I,

ω

and d, respectively. A uniform magnetic field

\vec{B}

Consider two different metallic strips (1 and 2) of same dimensions (length I, width w and thickness d) with carrier densities

n_{1} a n d n_{2}

, respectively .strip 1 is placed in magnetic field

B_{1}

and strip 2 is placed in magnetic field

B_{2}

,both along positive y- directions. Then

V_{1} a n d V_{2}

are the potential difference developed between K and M in strips 1 and 2, respectively assuming that the current I is the same for both the strips, the correct option (s) is (are)

A	If $B_{1} = B_{2} a n d n_{1} = 2 n_{2} t h e n V_{2} = 2 V_{1}$	C	If $B_{1} = B_{2} a n d n_{1} = 2 n_{2} t h e n V_{2} = V_{1}$
B	If $B_{1} = 2 B_{2} a n d n_{1} = 2 n_{2} t h e n V_{2} = 0.5 V_{1}$	D	If $B_{1} = 2 B_{2} a n d n_{1} = 2 n_{2}, t h e n V_{2} = V_{1}$

A and B

A and C

A and D

none of these

Category: IIT-JEE 2015 Physics Paper 2

Question 19:

Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index

η_{1}

surrounded by a medium of lower refractive index

η_{2} .

The light guidance in the structure takes place due to successive total internal reflection at the interface of the media

η_{1} a n d η_{2}

as shown in the figure . All rays with the angle of incidence I less than a particular value

i_{m}

are confined in the medium of refractive index

n_{1}

.The numerical aperture (NA) of the structure is defined as sin

i_{m} .

For two structures namly

S_{1} w i t h n_{1} = \sqrt{45 / 4} a n d n_{2} = \frac{3}{2}, a n d S_{2} w i t h n_{1} = 8 / 5

and

n_{1} = \frac{8}{5} a n d n_{2} = \frac{7}{5}

and taking the refractive index of water to be 4/3 and that of air to be 1, the correct option (s) is 9(are)

A	NA of $S_{1}$ immersed in water is the same as that of $S_{2}$ immersed in a liquid of refractive index $\frac{16}{3 \sqrt{15}}$	B	NA of $S_{1}$ immersed in liquid of refractive index $\frac{6}{\sqrt{15}}$ is the same as that of $S_{2}$ immersed in water
C	NA ofe $S_{1}$ placed in air is the same as that of $S_{2}$ immeresd in liquid of refractive index $\frac{4}{\sqrt{15}} .$	D	NA of $S_{1}$ placed in air is the same as that of $S_{2}$ placed in water

A and C

A and D

B and C

B and D

Category: IIT-JEE 2015 Physics Paper 2

Question 20:

Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index

η_{1}

surrounded by a medium of lower refractive index

η_{2} .

The light guidance in the structure takes place due to successive total internal reflection at the interface of the media

η_{1} a n d η_{2}

as shown in the figure . All rays with the angle of incidence I less than a particular value

i_{m}

are confined in the medium of refractive index

n_{1}

.The numerical aperture (NA) of the structure is defined as sin

i_{m} .

A	$\frac{N A_{1} N A_{2}}{N A_{1} + N A_{2}}$	B	$N A_{1} + N A_{2}$
C	$N A_{1}$	D	$N A_{2}$

Category: IIT-JEE 2015 Chemistry Paper 2

Question 21:

The number of hydroxyl group (s) in Q is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 22:

Among the following, the number of reaction (s) that produce (s) benzaldehyde is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 23:

In the complex acetybromidodicarbony lbis (triethyphosphine )iron (II),the number of Fe-C bond (s) is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 24:

Among the complex ions,

{[C o {(N H_{2} - C H_{2} - C H_{2} - N H_{2})}_{2} C I_{2}]}^{+}, {[C r C l_{2} {(C_{2} O_{4})}_{2}]}^{3 -}

{[F e {(H_{2} O)}_{4} {(O H)}_{2}]}^{+}, {[F e {(N H_{3})}_{2} {(C N)}_{4}]}^{-},

{[C o {(N H_{2} - C H_{2} - C H_{2} - N H_{2})}_{2} (N H_{3}) C I]}^{2 +}

and

{[C o {(N H_{3})}_{4} (H_{2} O) C I]}^{2 +}

,the number of complex ion(s)that show (s) cis-trans isomerism is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 25:

Three moles of

B_{2} H_{6}

are completely reacted with methanol. The number of moles of boron-containing product formed is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 26:

The molar conductivity of a solution of a weak acid HX(0.01M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.01M). If

λ_{x -}^{0} \approx λ_{y -}^{0},

the difference in their p

K_{a}

values ,

p K_{a} (H X) - p K_{a} (H Y)

, is (consider degree of ionization of both acids to be <<1)

Category: IIT-JEE 2015 Chemistry Paper 2

Question 27:

A closed vessel with rigid walls contains 1 mol of

U_{92}^{238}

and 1 mol of air at 298 k.considering complete decay of

U_{92}^{238} t o P_{82}^{206} b,

the ratio of the final pressure to the initial pressure of the system at 298 K is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 28:

In dilute aqueous

H_{2} S O_{4},

the complex diaquodioxalatoferrate (II) is oxidized by

M n O_{4}^{-}

For this reaction, the ratio of the rate of change of

[H^{+}]

to the rate of change of

[M n O_{4}^{-}]

Category: IIT-JEE 2015 Chemistry Paper 2

Question 29:

In the following reactions, the product S is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 30:

The major product U in the following reactions is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 31:

In the following reactions, the major product W is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 32:

The correct statement (s) regarding,(i)HCIO, (ii)

H C I O_{2,}

and (iii)

H C I O_{3} a n d (i v) H C I O_{4},

is (are)

A	The number of CI=O bonds in (ii) and (iii) together is two	B	The number of lone pairs of electrons on CI (ii) and (iii) together is three
C	The hybridization of CI in (iv) is $s p^{3}$	D	Amongst (i) to (iv), the strongest acid is (i)

A and B

B and C

C and D

Category: IIT-JEE 2015 Chemistry Paper 2

Question 33:

The pair(s) of ions where BOTH the ions are precipitated upon passing

H_{2} S

gas in presence of dilute HCI, is (are)

A	$B a^{2 +}, Z n^{2 +}$	B	$B i^{3 +}, F e^{3 +}$
C	$C u^{2 +}, P b^{2 +}$	D	$H g^{2 +}, B i^{3 +}$

A and B

B and C

C and D

Category: IIT-JEE 2015 Chemistry Paper 2

Question 34:

Under hydrolytic conditions, the compounds used for preparation of liners polymer and for chain termination respectively, are

A	$C H_{3} S i C I_{3} a n d S i {(C H_{3})}_{4}$	B	${(C H_{3})}_{2} S i C I_{2} a n d {(C H_{3})}_{3} S i C I$
C	${(C H_{3})}_{2} S i C I_{2} a n d C H_{3} S i C I_{3}$	D	$S i C I_{4} a n d {(C H_{3})}_{3} S i C I$

Category: IIT-JEE 2015 Chemistry Paper 2

Question 35:

When

O_{2}

is adsorbed on a metallic surface, electron transfer occurs from the metal to

O_{2}

. The TRUE statement (s) regarding this adsorption is (are)

A	$O_{2}$ is physisorbed	B	heat is released
C	occupancy of $π_{2 p} of O_{2}$ is increased	D	bond length of $O_{2}$ is increased

A, B and C

B, C and D

C, D and A

None of these

Category: IIT-JEE 2015 Chemistry Paper 2

Question 36:

One mole of a monoatomic real gas satisfies the equation p(V-b)=RT where b is a constant. The relationship of interatomic potential v(r) and interatomic distance r for the gas is given by

Category: IIT-JEE 2015 Chemistry Paper 2

Question 37:

In the following reactions

Compound X is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 38:

In the following reactions

The major compound Y is

Category: IIT-JEE 2015 Chemistry Paper 2

Question 39:

When 100mL of 1.0 M HCI was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure a temperature increase of

5.7 ° C

was measured for the beaker and its constant (Expt.1).because the enthalpy of a strong base is a constant

(- 57.0 k j m o l^{- 1})

,this experiment could be used to measure the calorimeter constant.In a second experiment (Expt.2),100mL of 2.0 M acetic acid (

k_{a} = 2.0 x 10^{- 5}

)was mixed with 100 mL of 1.0MNaOH(under identical conditions to Expt.1)where a temperature rise of 5.6

°

C was measured. (consider heat capacity of all solutions as 4.2

J g^{- 1} K^{- 1}

and density of all solutions as

1.0 g m L^{- 1})

Enthalpy of dissociation (in

k j m o l^{- 1})

of acetic obtained from the Expt.2 is

A	1.0	B	10.0
C	24.5	D	51.4

Category: IIT-JEE 2015 Chemistry Paper 2

Question 40:

When 100mL of 1.0 M HCI was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure a temperature increase of

5.7 ° C

was measured for the beaker and its constant (Expt.1).because the enthalpy of a strong base is a constant

(- 57.0 k j m o l^{- 1})

,this experiment could be used to measure the calorimeter constant.In a second experiment (Expt.2),100mL of 2.0 M acetic acid (

k_{a} = 2.0 x 10^{- 5}

)was mixed with 100 mL of 1.0MNaOH(under identical conditions to Expt.1)where a temperature rise of 5.6

°

C was measured. (consider heat capacity of all solutions as 4.2

J g^{- 1} K^{- 1}

and density of all solutions as

1.0 g m L^{- 1})

The pH of the solution after Expt.2 is

A	2.8	B	4.7
C	5.0	D	7.0

Category: IIT-JEE 2015 Math Paper 2

Question 41:

For any integer k, let

α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}), w h e r e i = \sqrt{- 1 .}

The value of the expression

\frac{\sum_{k = 1}^{12} |α_{k + 1} - α_{k}|}{\sum_{k = 1}^{3} |α_{4 k - 1^{-}} α_{4 k - 2}|} i s

Category: IIT-JEE 2015 Math Paper 2

Question 42:

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lines in between 130 and 140, then the common difference of this A.P. is

Category: IIT-JEE 2015 Math Paper 2

Question 43:

The coefficient of

x^{9}

in the expansion of

(1 + x) (1 + x^{2}) (1 + x^{3}) . . . . . . (1 + x^{100}) i s

Category: IIT-JEE 2015 Math Paper 2

Question 44:

Suppose that the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1 a r e (f_{1}, 0) a n d (f_{2}, 0) w h e r e f_{1} > 0 a n d f_{2} < 0 .

Let

p_{1} a n d p_{2}

be two parabolas with a common vertex at (0,0) and with foci at

(f_{1,} 0) a n d (2 f_{2,} 0),

respectively. Let

T_{1}

be a tangent to

p_{1}

which passes through

(2 f_{2}, 0) a n d T_{2}

be a tangent to

P_{2}

which passes through ((

f_{1}, 0

). If

m_{1}

is the slope of

T_{2}

, then the value of

(\frac{1}{m_{1}^{2}} + m_{2}^{2}) i s

Category: IIT-JEE 2015 Math Paper 2

Question 45:

Let m and n be two positive integers greater than 1.If

\lim_{α \to 0} (\frac{e^{\cos (α^{n})}}{α^{m}}) = - (\frac{e}{2})

then the value of

\frac{m}{n} i s

Category: IIT-JEE 2015 Math Paper 2

Question 46:

α = \int_{0}^{1} (e^{9 x + 3 \tan^{- 1}} x) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x

where

\tan^{- 1} x

takes only principal values,then the value of

(\log_{e} |1 + α| - \frac{3 π}{4}) i s

Category: IIT-JEE 2015 Math Paper 2

Question 47:

Let f :

ℝ \to ℝ

be a continuous odd function, which vanishes exactly at one point and

f (1) = \frac{1}{2} .

Suppose that F (x) =

\int_{- 1}^{x} f (t) d t f o r a l l x \in [- 1, 2] a n d G (x)

\int_{- 1}^{x} t |f (f (t))| d t

for all

x \in [- 1, 2] . I f \lim_{x \to 1} \frac{F (x)}{G (x)} = \frac{1}{14}, t h e n t h e v a l u e o f f (\frac{1}{2}) i s

Category: IIT-JEE 2015 Math Paper 2

Question 48:

Suppose that

\vec{p}, \vec{q} a n d \vec{r}

are three non-coplanar vectors in

ℝ^{3}

. Let the components of a vector

\vec{s}

along

\vec{p}, \vec{q} a n d \vec{r}

be 4,3 and 5, respectively . If the components of this vector

\vec{s}

along

(- \vec{p} + \vec{q} + \vec{r}), (\vec{p} - \vec{q} + \vec{r}) a n d (- \vec{p} - \vec{q} + \vec{r})

are x,y and z respectively , then the value of

2 x + y + z

Category: IIT-JEE 2015 Math Paper 2

Question 49:

Let S be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1} a n d x_{2}

satisfying the inquality

|x_{1} - x_{2}| < 1 .

which of the following intervals is (are) a subset (s) of S?

A	$(- \frac{1}{2}, - \frac{1}{\sqrt{5}})$	B	$(- \frac{1}{\sqrt{5}}, 0)$
C	$(0, \frac{1}{\sqrt{5}})$	D	$(\frac{1}{\sqrt{5}}, \frac{1}{2})$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2015 Math Paper 2

Question 50:

α = 3 \sin^{- 1} (\frac{6}{11}) a n d β = \cos^{- 1} (\frac{4}{9}),

where the inverse trigonometric function take only the principal values, then the correct option (s) is (are)

A	$\cos β > 0$	B	$\sin β < 0$
C	$\cos (α + β) > 0$	D	$\cos α < 0$

A, B and C

B, C and D

C, D and A

None of these

Category: IIT-JEE 2015 Math Paper 2

Question 51:

Let

E_{1} a n d E_{2}

be two ellipses whose centers are at the origin.The major axes of

E_{1} a n d E_{2}

lie along the x-axis and the y-axis, respectively. Let S be the circle

x^{2} + {(y - 1)}^{2} = 2 .

The straight line x+y=3 touches the curves S,

E_{1} a n d E_{2}

at P,Q and R, respectively .suppose that

P Q = P R = \frac{2 \sqrt{2}}{3} .

e_{1} a n d e_{2}

are the eccentricities of

E_{1} a n d E_{2},

respectively, than the correct expression(s) is (are)

A	$e_{1}^{2} + e_{2}^{2} = \frac{43}{40}$	B	$e_{1} e_{2} = \frac{\sqrt{7}}{2 \sqrt{10}}$
C	$\|e_{1}^{2} - e_{2}^{2}\| = \frac{5}{8}$	D	$e_{1} e_{2} = \frac{\sqrt{3}}{4}$

A and B

B and C

C and D

Category: IIT-JEE 2015 Math Paper 2

Question 52:

Consider the hyperbola

H : x^{2} - y^{2} = 1

and a circle S with center

N (x_{2}, 0) .

Suppose that H and S touch each other at a point P

(x_{1}, y_{1})

with

x_{1} > 1 a n d y_{1} > 0 .

The common tangent to H and S at P intersects the x-axis at point M.If (I,m) is centroid of the triangle

∆ P M N,

then the correct expression (s) is (are)

A	$\frac{d l}{d x_{1}} = 1 - \frac{1}{3 x_{1}^{2}} f o r x_{1} > 1$	B	$\frac{d m}{d x_{1}} = \frac{x_{1}}{3 (\sqrt{x_{1}^{2}} - 1)} f o r x_{1} > 1$
C	$\frac{d l}{d x_{1}} = 1 + \frac{1}{3 x_{1}^{2}} f o r x_{1} > 1$	D	$\frac{d m}{d y_{1}} = \frac{1}{3} f o r y_{1} > 0$

A, B and C

A, C and D

A, B and D

None of these

Category: IIT-JEE 2015 Math Paper 2

Question 53:

The option(s) with the values of

α

and L that satisfy the following equation is (are )

\frac{\int_{0}^{4 π} e^{t} (\sin^{6} α t + \cos^{4} α t) d t}{\int_{0}^{π} e^{t} (\sin^{6} α t + \cos^{4} α t) d t} = L ?

A	$α = 2, L = \frac{e^{4 π} - 1}{e^{π} - 1}$	B	$α = 2, L = \frac{e^{4 π} + 1}{e^{π} + 1}$
C	$α = 4, L = \frac{e^{4 π} - 1}{e^{π} - 1}$	D	$α = 4, L = \frac{e^{4 π} + 1}{e^{π} + 1}$

A and B

A and C

A and D

None of these

Category: IIT-JEE 2015 Math Paper 2

Question 54:

Let f,

g : [- 1, 2] \to ℝ

be continuous functions which are twice differential on the interval (-1,2). Let the values of f and g at the points-1,0 and 2 be as given in the following table:

	x=-1	x=0	x=2
f(x)	3	6	0
g(x)	0	1	-1

In each of intervals (-1,0) and (0,2) the function (f-3g)'' never vanishes.Then the correct statement (s) is 9(are)

A	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0)∪(0,2)	B	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0)
C	f'(x)-3g'9(x)=0 has exactly three solutions in(0,2)	D	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0) and exactly two solutions in (0,2)

A and B

B and C

C and D

Category: IIT-JEE 2015 Math Paper 2

Question 55:

Let f(x)=7

\tan^{8} x + 7 \tan^{6} x - 3 \tan^{4} x - 3 \tan^{2} x f o r a l l x \in (- \frac{π}{2}, \frac{π}{2})

.Then the correct expression (s) is (are)

A	$\int_{0}^{π / 4} x f (x) d x = \frac{1}{2}$	B	$\int_{0}^{π / 4} x f (x) d x = 0$
C	$\int_{0}^{π / 4} x f (x) d x = \frac{1}{6}$	D	$\int_{0}^{π / 4} x f (x) d x = 1$

A and B

B and C

C and D

Category: IIT-JEE 2015 Math Paper 2

Question 56:

Let f'(x)=

\frac{192 x^{3}}{2 + \sin^{4} πx}

for all x

\in ℝ w i t h f (\frac{1}{2}) = 0 .

If m

\leq \int_{1 / 2}^{1} (x) d x \leq M

,then the possible values of m and M are

A	m=13,M=24	B	$m = \frac{1}{4}, M = \frac{1}{2}$
C	m=-11,M=0	D	$m = 1, M = 12$

Category: IIT-JEE 2015 Math Paper 2

Question 57:

Let

n_{1} a n d n_{2}

be the number of red and black balls,respectively,in box I. Let

n_{3} a n d n_{4}

be the number of red and black balls,respectively ,in box II. One of two boxes, box I and Box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red.If the probability that this red ball was drawn from box II is

\frac{1}{3},

then the correct option (s) with the possible value of

n_{1}, n_{2}, n_{3} a n d n_{4}

is 9(are)

A	$n_{1} = 3, n_{2} = 3, n_{3} = 5, n_{4} = 15,$	B	$n_{1} = 3, n_{2} = 6, n_{3} = 10, n_{4} = 50,$
C	$n_{1} = 8, n_{2} = 6, n_{3} = 5, n_{4} = 20,$	D	$n_{1} = 6, n_{2} = 12, n_{3} = 5, n_{4} = 20$

A and B

B and C

C and D

Category: IIT-JEE 2015 Math Paper 2

Question 58:

Let

n_{1} a n d n_{2}

be the number of red and black balls,respectively,in box I. Let

n_{3} a n d n_{4}

be the number of red and black balls,respectively ,in box II. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is

\frac{1}{3},

then the correct option (s) with the possible values of

n_{1} a n d n_{2}

is a(are)

A	$n_{1} = 4 a n d n_{2} = 6$	B	$n_{1} = 2 a n d n_{2} = 3$
C	$n_{1} = 10 a n d n_{2} = 20$	D	$n_{1} = 3 a n d n_{2} = 6$

A and B

B and C

C and D

Category: IIT-JEE 2015 Math Paper 2

Question 59:

Let F :

ℝ \to ℝ

be a thrice differential function.suppose that F(1)=0, F(3)=-4 and F'(x)

< 0 f o r a l l x \in (1 / 2, 3) . L e t f (x) = x F (x) f o r a l l x \in ℝ

\int_{1}^{3} x^{2} F' (x) d x = - 12 a n d \int_{1}^{3} x^{3} F'' (x) d x = 40

,then the correct expression (s) is (are)

A	$9 f' (3) + f' (1) - 32 = 0$	B	$\int_{1}^{3} f (x) d x = 12$
C	$9 f' (3) + f' (1) + 32 = 0$	D	$\int_{1}^{3} f (x) d x = - 12$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2015 Math Paper 2

Question 60:

Let F :

ℝ \to ℝ

be a thrice differential function.suppose that F(1)=0, F(3)=-4 and F'(x

< 0 f o r a l l x \in (1 / 2, 3) . L e t f (x) = x F (x) f o r a l l x \in ℝ

The correct statements (s) is (are)

A	f'(1)<0	B	f(2) <0
C	$f (x) \neq 0 f o r a n y x \in (1, 3)$	D	$f' (x) = 0 f o r s o m e x \in (1, 3)$

A,B and C

B, C and D

C, D and A

None of these

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Question 1:

m = k (\frac{M}{288}) .

The value of k is

Correct Answer : 7

Question 2:

The energy of a system as a function of time t is given a E(t)=

A^{2} e x p (- α t),

where

α = 0.2 s^{- 1}

.The measurement of A has an error of 1.25%.If has error in the measurement of time is 1.50% ,the percentage error in the value of E(t) at t=5 s is

Correct Answer : 4

Question 3:

The densities of two solid spheres A and B of the same radii R very with radial distance r as

p_{A} (r) = k (\frac{r}{R}) a n d p_{B} (r) = {(\frac{r}{R})}^{5}

,respectively ,where k is a constant.The moments of inertia of the individual spheres about axes passing through their centers are

I_{A} a n d I_{B}

,respectively.If

\frac{I_{B}}{I_{A}} = \frac{n}{10}

,the value of n is

Correct Answer : 6

Question 4:

Four harmonic waves of equal frequencies and equal intensities

I_{0}

have phase angles 0,

π / 3, 2 π / 3 andπ .

when they are superposed,the intensity of the resulting wave is n

I_{0}

.The value of n is

Correct Answer : 3

Question 5:

For a radioactive material,its activity A and rate of change of its activity R are defined as

A = \frac{d N}{d t} a n d R = - \frac{d A}{d t},

where N(t) is the number of nuclei at time t.Two radioactive sources P(mean lifeτ) and Q(mean life 2τ)have the same activity at t=0. their rate of change of activities at t =2τ are

R_{P} a n d R_{Q}

,respectively .If

\frac{R_{p}}{R_{Q}} = \frac{n}{e},

then the value of n is

Correct Answer : 2

Question 6:

A monochromatic beam of light is incident at

60 °

on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle

θ

(n) with the normal (see the figure).For n=

\sqrt{3}

the value of

θ

is 60

°

and

\frac{d θ}{d n} = m

.The value of m is

Correct Answer : 2

Question 7:

In the following circuit,the current through the resistor R

(= 2

Ω)is I Amperes. The value of I is

Correct Answer : 1

Question 8:

An electron in an excited state of

L i^{2 +}

ion has angular momentum

3 h / 2 π .

The de Broglie wavelength of the electron in this state is

p {πα}_{0}

(where

α_{0}

is the Bohr radius).The value of p is

Correct Answer : 2

Question 9:

Two spheres P and Q of equal radii have densities

p_{1} a n d p_{2}

,respectively. The spheres are connected by a massless string and placed in liquids

L_{1} a n d L_{2}

of densities

σ_{1} a n d σ_{2}

and viscosities

η_{1} a n d η_{2}

,respectively .They float in equilibrium with the sphere P in

L_{1} a n d s p h e r e Q i n L_{2}

and the string being taut (see figure),If sphere P alone in

L_{2}

has terminal velocity

V_{P}

and Q alone in

L_{1}

has terminal velocity

\vec{V_{Q}}

,then

A	$\frac{\|\vec{V_{P}}\|}{\|\vec{V_{Q}}\|} = \frac{η_{1}}{η_{2}}$	B	$\frac{\|\vec{V_{P}}\|}{\|\vec{V_{Q}}\|} = \frac{η_{2}}{η_{1}}$
C	$\vec{V_{P}} . \vec{V_{Q}} > 0$	D	$\vec{V_{P}} . \vec{V_{Q}} < 0$

Correct Answer : D and A

Question 10:

In terms of potential difference V,electric current I,permittivity

ε_{0}

,permeability

μ_{0}

and speed of light c, the dimensionally correct equation (s) is (are)

A	$μ_{0} I^{2} = ε_{0} V^{2}$	B	$ε_{0} I = μ_{0} V$
C	$I = ε_{0} c V$	D	$μ_{0} c I = ε_{0} V$

Correct Answer : A and C

Question 11:

Consider a uniform spherical charge distribution of radius

R_{1}

central at the origin O.In this distribution ,a spherical cavity of radius

R_{2}

, centred at P with distance OP=

α = R_{1} - R_{2}

(see figure)is made.If the electric field inside the cavity at position

\vec{r} i s \vec{E}

(\vec{r}),

then the correct statement(s)is (are)

A	$\vec{E}$ is uniform,its magnitude is independent of $R_{2}$ but its direction on $\vec{r}$	B	$\vec{E}$ is uniform,its magnitude is depends on $R_{2}$ and its direction on $\vec{r}$
C	$\vec{E}$ is uniform,its magnitude is independent of $α$ but its direction on $\vec{α}$	D	$\vec{E}$ is uniform,its magnitude is depends on and its direction on $\vec{α}$

Correct Answer : D

Question 12:

A	P has more tensile strength than Q	B	P is more ductile than Q
C	P is more brittle than Q	D	The young's modulus of P is more than that of Q

Correct Answer : A and B

Question 13:

A spherical body of radius R consists of a fluid of constant of a fluid of constant density and is in equilibrium under its own gravity. If P(r) is the pressure at r(r

<

R),then the correct option (s) is (are)

A	$P (r = 0) = 0$	B	$\frac{P (r = 3 R / 4)}{p (r = 2 R / 3)} = \frac{63}{80}$
C	$\frac{P (r = 3 R / 5)}{P (r = 2 R / 5)} = \frac{16}{21}$	D	$\frac{P (r = 3 R / 4)}{P (r = R / 3)} = \frac{20}{27}$

Correct Answer : B and C

Question 14:

A parallel plate capacitor having plates of area S and plate separation d, has capacitance

C_{1}

in air. When two dielectrics of different relative permittivities

(ε_{1} = 2 a n d ε_{2} = 4)

are introduced between the two plates as shown in the figure,the capacitance becomes

C_{2}

.The ratio

\frac{C_{2}}{C_{1}} i s

A	$6 / 5$	B	$5 / 3$
C	$7 / 5$	D	$7 / 3$

Correct Answer : D

Question 15:

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially, the gas is at temperature

T_{1} p r e s s u r e p_{1} a n d v o l u m e V_{1}

and the spring is in its relaxed state. The gas is then heated very slowly to temperature

T_{2}

pressure

p_{2}

and volume

V_{2}

. During this process the piston moves out by a distance x.Ignoring the friction between the piston and the cylinder, the correct statement(s) is )(are)

A	If $V_{2} = 2 V_{1} a n d T_{2} = 3 T_{1},$ then the energy stored in the spring is $\frac{1}{4} p_{1} v_{1}$	B	If $V_{2} = 2 V_{1} a n d T_{2} = 3 T_{1},$ then the change in internal energy is $3 P_{1} V_{1}$
C	If $V_{2} = 2 V_{1} a n d T_{2} = 4 T_{1},$ then the work done by the gas is $\frac{7}{3} P_{1} V_{1}$	D	If $V_{2} = 2 V_{1} a n d T_{2} = 4 T_{1},$ then the heat supplied to the gas is $\frac{17}{6} P_{1} V_{1}$

Correct Answer : A, B and C

Question 16:

A fission reaction is given by

U_{92}^{236} \to X_{54}^{140} e + S_{38}^{94} r + x + y,

where x and y are two particles .considering

U_{92}^{236}

to be at rest , the kinetic energies of the products are denoted by

K_{X e} K_{s r}, K_{x} (2 M e V) a n d K_{y} (2 M e V),

respectively. let the binding energies per nucleon of

U_{92}^{236}, X_{54}^{140} e a n d S_{38}^{94} r b e 7.5 M e V, 8.5 M e V

,respectively .Considering different conservation laws, the correct option (s) is (are)

A	x=n, y=n, $K_{s r} = 129 M e V, K_{x e} = 86 M e V$	B	x=p, y= $e^{-}$ $K_{s r} = 129 M e V, K_{x e} = 86 M e V$
C	x=p, y=n, $K_{s r} = 129 M e V, K_{x e} = 86 M e V$	D	x=n, y=n $K_{s r} = 86 M e V, K_{x e} = 129 M e V$

Correct Answer : A

Question 17:

In a thin rectangular metallic strip, a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are I,

ω

and d, respectively. A uniform magnetic field

\vec{B}

Consider two different metallic strips (1 and 2) of the same material . Their lengths are the same, widths are

ω_{1} a n d ω_{2}

and thickness are

d_{1} a n d d_{2}

, respectively . Two points K and M are symmetrically located on the opposite face parallel to the x-y plane (see-figure).

V_{1} a n d V_{2}

A	If $ω_{1} = ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = 2 V_{1}$	B	If $ω_{1} = ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = V_{1}$
C	If $ω_{1} = 2 ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = 2 V_{1}$	D	If $ω_{1} = 2 ω_{2} a n d d_{1} = 2 d_{2}, t h e n V_{2} = V_{1}$

Correct Answer : D and A

Question 18:

In a thin rectangular metallic strip, a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are I,

ω

and d, respectively. A uniform magnetic field

\vec{B}

Consider two different metallic strips (1 and 2) of same dimensions (length I, width w and thickness d) with carrier densities

n_{1} a n d n_{2}

, respectively .strip 1 is placed in magnetic field

B_{1}

and strip 2 is placed in magnetic field

B_{2}

,both along positive y- directions. Then

V_{1} a n d V_{2}

are the potential difference developed between K and M in strips 1 and 2, respectively assuming that the current I is the same for both the strips, the correct option (s) is (are)

A	If $B_{1} = B_{2} a n d n_{1} = 2 n_{2} t h e n V_{2} = 2 V_{1}$	C	If $B_{1} = B_{2} a n d n_{1} = 2 n_{2} t h e n V_{2} = V_{1}$
B	If $B_{1} = 2 B_{2} a n d n_{1} = 2 n_{2} t h e n V_{2} = 0.5 V_{1}$	D	If $B_{1} = 2 B_{2} a n d n_{1} = 2 n_{2}, t h e n V_{2} = V_{1}$

Correct Answer : A and C

Question 19:

Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index

η_{1}

surrounded by a medium of lower refractive index

η_{2} .

The light guidance in the structure takes place due to successive total internal reflection at the interface of the media

η_{1} a n d η_{2}

as shown in the figure . All rays with the angle of incidence I less than a particular value

i_{m}

are confined in the medium of refractive index

n_{1}

.The numerical aperture (NA) of the structure is defined as sin

i_{m} .

For two structures namly

S_{1} w i t h n_{1} = \sqrt{45 / 4} a n d n_{2} = \frac{3}{2}, a n d S_{2} w i t h n_{1} = 8 / 5

and

n_{1} = \frac{8}{5} a n d n_{2} = \frac{7}{5}

and taking the refractive index of water to be 4/3 and that of air to be 1, the correct option (s) is 9(are)

A	NA of $S_{1}$ immersed in water is the same as that of $S_{2}$ immersed in a liquid of refractive index $\frac{16}{3 \sqrt{15}}$	B	NA of $S_{1}$ immersed in liquid of refractive index $\frac{6}{\sqrt{15}}$ is the same as that of $S_{2}$ immersed in water
C	NA ofe $S_{1}$ placed in air is the same as that of $S_{2}$ immeresd in liquid of refractive index $\frac{4}{\sqrt{15}} .$	D	NA of $S_{1}$ placed in air is the same as that of $S_{2}$ placed in water

Correct Answer : A and C

Question 20:

Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index

η_{1}

surrounded by a medium of lower refractive index

η_{2} .

The light guidance in the structure takes place due to successive total internal reflection at the interface of the media

η_{1} a n d η_{2}

as shown in the figure . All rays with the angle of incidence I less than a particular value

i_{m}

are confined in the medium of refractive index

n_{1}

.The numerical aperture (NA) of the structure is defined as sin

i_{m} .

A	$\frac{N A_{1} N A_{2}}{N A_{1} + N A_{2}}$	B	$N A_{1} + N A_{2}$
C	$N A_{1}$	D	$N A_{2}$

Correct Answer : D

Question 21:

The number of hydroxyl group (s) in Q is

Correct Answer : 4

Question 22:

Among the following, the number of reaction (s) that produce (s) benzaldehyde is

Correct Answer : 4

Question 23:

In the complex acetybromidodicarbony lbis (triethyphosphine )iron (II),the number of Fe-C bond (s) is

Correct Answer : 3

Question 24:

Among the complex ions,

{[C o {(N H_{2} - C H_{2} - C H_{2} - N H_{2})}_{2} C I_{2}]}^{+}, {[C r C l_{2} {(C_{2} O_{4})}_{2}]}^{3 -}

{[F e {(H_{2} O)}_{4} {(O H)}_{2}]}^{+}, {[F e {(N H_{3})}_{2} {(C N)}_{4}]}^{-},

{[C o {(N H_{2} - C H_{2} - C H_{2} - N H_{2})}_{2} (N H_{3}) C I]}^{2 +}

and

{[C o {(N H_{3})}_{4} (H_{2} O) C I]}^{2 +}

,the number of complex ion(s)that show (s) cis-trans isomerism is

Correct Answer : 6

Question 25:

Three moles of

B_{2} H_{6}

are completely reacted with methanol. The number of moles of boron-containing product formed is

Correct Answer : 6

Question 26:

The molar conductivity of a solution of a weak acid HX(0.01M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.01M). If

λ_{x -}^{0} \approx λ_{y -}^{0},

the difference in their p

K_{a}

values ,

p K_{a} (H X) - p K_{a} (H Y)

, is (consider degree of ionization of both acids to be <<1)

Correct Answer : 3

Question 27:

A closed vessel with rigid walls contains 1 mol of

U_{92}^{238}

and 1 mol of air at 298 k.considering complete decay of

U_{92}^{238} t o P_{82}^{206} b,

the ratio of the final pressure to the initial pressure of the system at 298 K is

Correct Answer : 9

Question 28:

In dilute aqueous

H_{2} S O_{4},

the complex diaquodioxalatoferrate (II) is oxidized by

M n O_{4}^{-}

For this reaction, the ratio of the rate of change of

[H^{+}]

to the rate of change of

[M n O_{4}^{-}]

Correct Answer : 8

Question 29:

In the following reactions, the product S is

Correct Answer : A

Question 30:

The major product U in the following reactions is

Correct Answer : B

Question 31:

In the following reactions, the major product W is

Correct Answer : A

Question 32:

The correct statement (s) regarding,(i)HCIO, (ii)

H C I O_{2,}

and (iii)

H C I O_{3} a n d (i v) H C I O_{4},

is (are)

A	The number of CI=O bonds in (ii) and (iii) together is two	B	The number of lone pairs of electrons on CI (ii) and (iii) together is three
C	The hybridization of CI in (iv) is $s p^{3}$	D	Amongst (i) to (iv), the strongest acid is (i)

Correct Answer : B and C

Question 33:

The pair(s) of ions where BOTH the ions are precipitated upon passing

H_{2} S

gas in presence of dilute HCI, is (are)

A	$B a^{2 +}, Z n^{2 +}$	B	$B i^{3 +}, F e^{3 +}$
C	$C u^{2 +}, P b^{2 +}$	D	$H g^{2 +}, B i^{3 +}$

Correct Answer : C and D

Question 34:

Under hydrolytic conditions, the compounds used for preparation of liners polymer and for chain termination respectively, are

A	$C H_{3} S i C I_{3} a n d S i {(C H_{3})}_{4}$	B	${(C H_{3})}_{2} S i C I_{2} a n d {(C H_{3})}_{3} S i C I$
C	${(C H_{3})}_{2} S i C I_{2} a n d C H_{3} S i C I_{3}$	D	$S i C I_{4} a n d {(C H_{3})}_{3} S i C I$

Correct Answer : B

Question 35:

When

O_{2}

is adsorbed on a metallic surface, electron transfer occurs from the metal to

O_{2}

. The TRUE statement (s) regarding this adsorption is (are)

A	$O_{2}$ is physisorbed	B	heat is released
C	occupancy of $π_{2 p} of O_{2}$ is increased	D	bond length of $O_{2}$ is increased

Correct Answer : B, C and D

Question 36:

One mole of a monoatomic real gas satisfies the equation p(V-b)=RT where b is a constant. The relationship of interatomic potential v(r) and interatomic distance r for the gas is given by

Correct Answer : C

Question 37:

In the following reactions

Compound X is

Correct Answer : C

Question 38:

In the following reactions

The major compound Y is

Correct Answer : D

Question 39:

When 100mL of 1.0 M HCI was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure a temperature increase of

5.7 ° C

was measured for the beaker and its constant (Expt.1).because the enthalpy of a strong base is a constant

(- 57.0 k j m o l^{- 1})

,this experiment could be used to measure the calorimeter constant.In a second experiment (Expt.2),100mL of 2.0 M acetic acid (

k_{a} = 2.0 x 10^{- 5}

)was mixed with 100 mL of 1.0MNaOH(under identical conditions to Expt.1)where a temperature rise of 5.6

°

C was measured. (consider heat capacity of all solutions as 4.2

J g^{- 1} K^{- 1}

and density of all solutions as

1.0 g m L^{- 1})

Enthalpy of dissociation (in

k j m o l^{- 1})

of acetic obtained from the Expt.2 is

A	1.0	B	10.0
C	24.5	D	51.4

Correct Answer : A

Question 40:

When 100mL of 1.0 M HCI was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure a temperature increase of

5.7 ° C

was measured for the beaker and its constant (Expt.1).because the enthalpy of a strong base is a constant

(- 57.0 k j m o l^{- 1})

,this experiment could be used to measure the calorimeter constant.In a second experiment (Expt.2),100mL of 2.0 M acetic acid (

k_{a} = 2.0 x 10^{- 5}

)was mixed with 100 mL of 1.0MNaOH(under identical conditions to Expt.1)where a temperature rise of 5.6

°

C was measured. (consider heat capacity of all solutions as 4.2

J g^{- 1} K^{- 1}

and density of all solutions as

1.0 g m L^{- 1})

The pH of the solution after Expt.2 is

A	2.8	B	4.7
C	5.0	D	7.0

Correct Answer : B

Question 41:

For any integer k, let

α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}), w h e r e i = \sqrt{- 1 .}

The value of the expression

\frac{\sum_{k = 1}^{12} |α_{k + 1} - α_{k}|}{\sum_{k = 1}^{3} |α_{4 k - 1^{-}} α_{4 k - 2}|} i s

Correct Answer : 4

Question 42:

Question 43:

The coefficient of

x^{9}

in the expansion of

(1 + x) (1 + x^{2}) (1 + x^{3}) . . . . . . (1 + x^{100}) i s

Correct Answer : 8

Question 44:

Suppose that the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1 a r e (f_{1}, 0) a n d (f_{2}, 0) w h e r e f_{1} > 0 a n d f_{2} < 0 .

Let

p_{1} a n d p_{2}

be two parabolas with a common vertex at (0,0) and with foci at

(f_{1,} 0) a n d (2 f_{2,} 0),

respectively. Let

T_{1}

be a tangent to

p_{1}

which passes through

(2 f_{2}, 0) a n d T_{2}

be a tangent to

P_{2}

which passes through ((

f_{1}, 0

). If

m_{1}

is the slope of

T_{2}

, then the value of

(\frac{1}{m_{1}^{2}} + m_{2}^{2}) i s

Correct Answer : 5

Question 45:

Let m and n be two positive integers greater than 1.If

\lim_{α \to 0} (\frac{e^{\cos (α^{n})}}{α^{m}}) = - (\frac{e}{2})

then the value of

\frac{m}{n} i s

Correct Answer : 2

Question 46:

α = \int_{0}^{1} (e^{9 x + 3 \tan^{- 1}} x) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x

where

\tan^{- 1} x

takes only principal values,then the value of

(\log_{e} |1 + α| - \frac{3 π}{4}) i s

Correct Answer : 9

Question 47:

Let f :

ℝ \to ℝ

be a continuous odd function, which vanishes exactly at one point and

f (1) = \frac{1}{2} .

Suppose that F (x) =

\int_{- 1}^{x} f (t) d t f o r a l l x \in [- 1, 2] a n d G (x)

\int_{- 1}^{x} t |f (f (t))| d t

for all

x \in [- 1, 2] . I f \lim_{x \to 1} \frac{F (x)}{G (x)} = \frac{1}{14}, t h e n t h e v a l u e o f f (\frac{1}{2}) i s

Correct Answer : 7

Question 48:

Suppose that

\vec{p}, \vec{q} a n d \vec{r}

are three non-coplanar vectors in

ℝ^{3}

. Let the components of a vector

\vec{s}

along

\vec{p}, \vec{q} a n d \vec{r}

be 4,3 and 5, respectively . If the components of this vector

\vec{s}

along

(- \vec{p} + \vec{q} + \vec{r}), (\vec{p} - \vec{q} + \vec{r}) a n d (- \vec{p} - \vec{q} + \vec{r})

are x,y and z respectively , then the value of

2 x + y + z

Correct Answer : 9

Question 49:

Let S be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1} a n d x_{2}

satisfying the inquality

|x_{1} - x_{2}| < 1 .

which of the following intervals is (are) a subset (s) of S?

A	$(- \frac{1}{2}, - \frac{1}{\sqrt{5}})$	B	$(- \frac{1}{\sqrt{5}}, 0)$
C	$(0, \frac{1}{\sqrt{5}})$	D	$(\frac{1}{\sqrt{5}}, \frac{1}{2})$

Correct Answer : D and A

Question 50:

α = 3 \sin^{- 1} (\frac{6}{11}) a n d β = \cos^{- 1} (\frac{4}{9}),

where the inverse trigonometric function take only the principal values, then the correct option (s) is (are)

A	$\cos β > 0$	B	$\sin β < 0$
C	$\cos (α + β) > 0$	D	$\cos α < 0$

Correct Answer : B, C and D

Question 51:

Let

E_{1} a n d E_{2}

be two ellipses whose centers are at the origin.The major axes of

E_{1} a n d E_{2}

lie along the x-axis and the y-axis, respectively. Let S be the circle

x^{2} + {(y - 1)}^{2} = 2 .

The straight line x+y=3 touches the curves S,

E_{1} a n d E_{2}

at P,Q and R, respectively .suppose that

P Q = P R = \frac{2 \sqrt{2}}{3} .

e_{1} a n d e_{2}

are the eccentricities of

E_{1} a n d E_{2},

respectively, than the correct expression(s) is (are)

A	$e_{1}^{2} + e_{2}^{2} = \frac{43}{40}$	B	$e_{1} e_{2} = \frac{\sqrt{7}}{2 \sqrt{10}}$
C	$\|e_{1}^{2} - e_{2}^{2}\| = \frac{5}{8}$	D	$e_{1} e_{2} = \frac{\sqrt{3}}{4}$

Correct Answer : A and B

Question 52:

Consider the hyperbola

H : x^{2} - y^{2} = 1

and a circle S with center

N (x_{2}, 0) .

Suppose that H and S touch each other at a point P

(x_{1}, y_{1})

with

x_{1} > 1 a n d y_{1} > 0 .

The common tangent to H and S at P intersects the x-axis at point M.If (I,m) is centroid of the triangle

∆ P M N,

then the correct expression (s) is (are)

A	$\frac{d l}{d x_{1}} = 1 - \frac{1}{3 x_{1}^{2}} f o r x_{1} > 1$	B	$\frac{d m}{d x_{1}} = \frac{x_{1}}{3 (\sqrt{x_{1}^{2}} - 1)} f o r x_{1} > 1$
C	$\frac{d l}{d x_{1}} = 1 + \frac{1}{3 x_{1}^{2}} f o r x_{1} > 1$	D	$\frac{d m}{d y_{1}} = \frac{1}{3} f o r y_{1} > 0$

Correct Answer : A, B and D

Question 53:

The option(s) with the values of

α

and L that satisfy the following equation is (are )

\frac{\int_{0}^{4 π} e^{t} (\sin^{6} α t + \cos^{4} α t) d t}{\int_{0}^{π} e^{t} (\sin^{6} α t + \cos^{4} α t) d t} = L ?

A	$α = 2, L = \frac{e^{4 π} - 1}{e^{π} - 1}$	B	$α = 2, L = \frac{e^{4 π} + 1}{e^{π} + 1}$
C	$α = 4, L = \frac{e^{4 π} - 1}{e^{π} - 1}$	D	$α = 4, L = \frac{e^{4 π} + 1}{e^{π} + 1}$

Correct Answer : A and C

Question 54:

Let f,

g : [- 1, 2] \to ℝ

be continuous functions which are twice differential on the interval (-1,2). Let the values of f and g at the points-1,0 and 2 be as given in the following table:

	x=-1	x=0	x=2
f(x)	3	6	0
g(x)	0	1	-1

In each of intervals (-1,0) and (0,2) the function (f-3g)'' never vanishes.Then the correct statement (s) is 9(are)

A	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0)∪(0,2)	B	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0)
C	f'(x)-3g'9(x)=0 has exactly three solutions in(0,2)	D	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0) and exactly two solutions in (0,2)

Correct Answer : B and C

Question 55:

Let f(x)=7

\tan^{8} x + 7 \tan^{6} x - 3 \tan^{4} x - 3 \tan^{2} x f o r a l l x \in (- \frac{π}{2}, \frac{π}{2})

.Then the correct expression (s) is (are)

A	$\int_{0}^{π / 4} x f (x) d x = \frac{1}{2}$	B	$\int_{0}^{π / 4} x f (x) d x = 0$
C	$\int_{0}^{π / 4} x f (x) d x = \frac{1}{6}$	D	$\int_{0}^{π / 4} x f (x) d x = 1$

Correct Answer : A and B

Question 56:

Let f'(x)=

\frac{192 x^{3}}{2 + \sin^{4} πx}

for all x

\in ℝ w i t h f (\frac{1}{2}) = 0 .

If m

\leq \int_{1 / 2}^{1} (x) d x \leq M

,then the possible values of m and M are

A	m=13,M=24	B	$m = \frac{1}{4}, M = \frac{1}{2}$
C	m=-11,M=0	D	$m = 1, M = 12$

Correct Answer : D

Question 57:

Let

n_{1} a n d n_{2}

be the number of red and black balls,respectively,in box I. Let

n_{3} a n d n_{4}

\frac{1}{3},

then the correct option (s) with the possible value of

n_{1}, n_{2}, n_{3} a n d n_{4}

is 9(are)

A	$n_{1} = 3, n_{2} = 3, n_{3} = 5, n_{4} = 15,$	B	$n_{1} = 3, n_{2} = 6, n_{3} = 10, n_{4} = 50,$
C	$n_{1} = 8, n_{2} = 6, n_{3} = 5, n_{4} = 20,$	D	$n_{1} = 6, n_{2} = 12, n_{3} = 5, n_{4} = 20$

Correct Answer : A and B

Question 58:

Let

n_{1} a n d n_{2}

be the number of red and black balls,respectively,in box I. Let

n_{3} a n d n_{4}

\frac{1}{3},

then the correct option (s) with the possible values of

n_{1} a n d n_{2}

is a(are)

A	$n_{1} = 4 a n d n_{2} = 6$	B	$n_{1} = 2 a n d n_{2} = 3$
C	$n_{1} = 10 a n d n_{2} = 20$	D	$n_{1} = 3 a n d n_{2} = 6$

Correct Answer : C and D

Question 59:

Let F :

ℝ \to ℝ

be a thrice differential function.suppose that F(1)=0, F(3)=-4 and F'(x)

< 0 f o r a l l x \in (1 / 2, 3) . L e t f (x) = x F (x) f o r a l l x \in ℝ

\int_{1}^{3} x^{2} F' (x) d x = - 12 a n d \int_{1}^{3} x^{3} F'' (x) d x = 40

,then the correct expression (s) is (are)

A	$9 f' (3) + f' (1) - 32 = 0$	B	$\int_{1}^{3} f (x) d x = 12$
C	$9 f' (3) + f' (1) + 32 = 0$	D	$\int_{1}^{3} f (x) d x = - 12$

Correct Answer : C and D

Question 60:

Let F :

ℝ \to ℝ

be a thrice differential function.suppose that F(1)=0, F(3)=-4 and F'(x

< 0 f o r a l l x \in (1 / 2, 3) . L e t f (x) = x F (x) f o r a l l x \in ℝ

The correct statements (s) is (are)

A	f'(1)<0	B	f(2) <0
C	$f (x) \neq 0 f o r a n y x \in (1, 3)$	D	$f' (x) = 0 f o r s o m e x \in (1, 3)$

Correct Answer : A,B and C

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