JEE Advanced 2016 Paper 1 Offline
Paper was held on
Sat, May 21, 2016 9:00 PM
Chemistry
1
A plot of the number of neutrons (N) against the number of protons (P) of stable nuclei exhibits upward
deviation from linearity for atomic number, Z > 20. For an unstable nucleus having N/P ratio less than 1,
the possible mode(s) of decay is(are)
2
According to the Arrhenius equation,
3
The mole fraction of a solute in a solution is 0.1. At 298 K, molarity of this solution is the same as its
molality. Density of this solution at 298 K is 2.0 g cm–3 . The ratio of the molecular weights of the solute and solvent, $$\left( {{{M{W_{solute}}} \over {M{W_{solvent}}}}} \right)$$, is
4
One mole of an ideal gas at 300 K in thermal contact with surroundings expands isothermally from 1.0 L to
2.0 L against a constant pressure of 3.0 atm. In this process, the change in entropy of surrounding ($$\Delta$$Ssurr)in
JK–1 is (1L atm = 101.3 J)
5
The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute
temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the
diffusion coefficient of this gas increases x times. The value of x is ___________:
6
The compound(s) with TWO lone pairs of electrons on the central atom is(are)
7
P is the probability of finding the 1s electron of hydrogen atom in a spherical shell of infinitesimal thickness, dr, at a distance r from the nucleus. The volume of this shell is $$4\pi r^2dr$$. The quantitative ketch of the dependence of P on r is
8
The product(s) of the following reaction sequence is (are)
9
Positive Tollen’s test is observed for
10
In the following mono-bromination reaction, the number of possible chiral product(s) is (are)...
11
In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitative oxidise thiosulphate anions to produce X moles of a sulphur containing product. The magnitude of X is....
12
The possible number of geometrical isomers for the complex [CoL2Cl2]-
(L= H2NCH2CH2O-) is (are)...
(L= H2NCH2CH2O-) is (are)...
13
The reagent(s) that can selectively precipitate S2- from a mixture of S2- and SO42- in aqueous solution is (are) :
14
The crystalline form of borax has.
15
The increasing order of atomic radii of the following group 13 elements is
16
On complete hydrogenation, natural rubber produces
17
Among [Ni(CO)4], [NiCl4]2$$-$$, [Co(NH3)4)Cl2]Cl, Na3[CoF6], Na2O2 and CsO2, the total number of paramagnetic compound is
18
The correct statements about of the following reaction sequence is (are)
Cumene (C9H12) $$\mathrel{\mathop{\kern0pt\longrightarrow}
\limits_{(ii)\,{H_3}{O^ + }}^{(i)\,{O_2}}} $$ P $$\mathrel{\mathop{\kern0pt\longrightarrow}
\limits_{}^{CHC{l_3}/NaOH}} $$ Q (major) + R (minor)
Q $$\mathrel{\mathop{\kern0pt\longrightarrow}
\limits_{PhC{H_2}Br}^{NaOH}} $$ S
Mathematics
1
Let $$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$$ The sum of all distinct solutions of the equation $$\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0$$ in the set S is equal to
2
Consider a pyramid $$OPQRS$$ located in the first octant $$\left( {x \ge 0,y \ge 0,z \ge 0} \right)$$ with $$O$$ as origin, and $$OP$$ and $$OR$$ along the $$x$$-axis and the $$y$$-axis, respectively. The base $$OPQR$$ of the pyramid is a square with $$OP=3.$$ The point $$S$$ is directly above the mid-point, $$T$$ of diagonal $$OQ$$ such that $$TS=3.$$ Then
3
A computer producing factory has only two plants $${T_1}$$ and $${T_2}.$$ Plant $${T_1}$$ produces $$20$$% and plant $${T_2}$$ produces $$80$$% of the total computers produced. $$7$$% of computers produced in the factory turn out to be defective. It is known that $$P$$ (computer turns out to be defective given that it is produced in plant $${T_1}$$)
$$ = 10P$$ (computer turns out to be defective given that it is produced in plant $${T_2}$$),
where $$P(E)$$ denotes the probability of an event $$E$$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $${T_2}$$ is
$$ = 10P$$ (computer turns out to be defective given that it is produced in plant $${T_2}$$),
where $$P(E)$$ denotes the probability of an event $$E$$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $${T_2}$$ is
4
The total number of distinct $$x \in \left[ {0,1} \right]$$ for which
$$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$
$$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$
5
A solution curve of the differential equation
$$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through the
point $$(1,3)$$. Then the solution curve
$$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through the
point $$(1,3)$$. Then the solution curve
6
The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is
7
In a triangle $$\Delta $$$$XYZ$$, let $$x, y, z$$ be the lengths of sides opposite to the angles $$X, Y, Z$$ respectively, and $$2s = x + y + z$$.
If $${{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}$$ and area of incircle of the triangle $$XYZ$$ is $${{8\pi } \over 3}$$, then
If $${{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}$$ and area of incircle of the triangle $$XYZ$$ is $${{8\pi } \over 3}$$, then
8
Let $$f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$$ and $$h:\mathbb{R} \to \mathbb{R}$$ be differentiable functions such that $$f\left( x \right)= {x^3} + 3x + 2,$$ $$g\left( {f\left( x \right)} \right) = x$$ and $$h\left( {g\left( {g\left( x \right)} \right)} \right) = x$$ for all $$x \in R$$. Then
9
The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ in the first quadrant, Let the tangent to the circle $${C_1}$$, at $$P$$ touches other two circles $${C_2}$$ and $${C_3}$$ at $${R_2}$$ and $${R_3}$$, respectively. Suppose $${C_2}$$ and $${C_3}$$ have equal radil $${2\sqrt 3 }$$ and centres $${Q_2}$$ and $${Q_3}$$, respectively. If $${Q_2}$$ and $${Q_3}$$ lie on the $$y$$-axis, then
10
Let RS be the diameter of the circle $${x^2}\, + \,{y^2} = 1$$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)
11
Let $$m$$ be the smallest positive integer such that the coefficient of $${x^2}$$ in the expansion of $${\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,$$ is $$\left( {3n + 1} \right)\,{}^{51}{C_3}$$ for some positive integer $$n$$. Then the value of $$n$$ is
12
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
13
Let $$ - {\pi \over 6} < \theta < - {\pi \over {12}}.$$ Suppose $${\alpha _1}$$ and $${\beta_1}$$ are the roots of the equation $${x^2} - 2x\sec \theta + 1 = 0$$ and $${\alpha _2}$$ and $${\beta _2}$$ are the roots of the equation $${x^2} + 2x\,\tan \theta - 1 = 0.$$ $$If\,{\alpha _1} > {\beta _1}$$ and $${\alpha _2} > {\beta _2},$$ then $${\alpha _1} + {\beta _2}$$ equals
14
Let $$f:(0,\infty ) \to R$$ be a differentiable function such that $$f'(x) = 2 - {{f(x)} \over x}$$ for all $$x \in (0,\infty )$$ and $$f(1) \ne 1$$. Then
15
Let $$P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha$$ $$\in$$ R. Suppose $$Q = [{q_{ij}}]$$ is a matrix such that PQ = kl, where k $$\in$$ R, k $$\ne$$ 0 and I is the identity matrix of order 3. If $${q_{23}} = - {k \over 8}$$ and $$\det (Q) = {{{k^2}} \over 2}$$, then
16
The total number of distinct x $$\in$$ R for which
$$\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10$$ is ______________.
17
Let $$z = {{ - 1 + \sqrt 3 i} \over 2}$$, where $$i = \sqrt { - 1} $$, and r, s $$\in$$ {1, 2, 3}. Let $$P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]$$ and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P2 = $$-$$I is ____________.
18
Let $$\alpha$$, $$\beta$$ $$\in$$ R be such that $$\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha x - \sin x}} = 1$$. Then 6($$\alpha$$ + $$\beta$$) equals _________.
Physics
1
A length-scale (l) depends on the permittivity ($$\varepsilon $$) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for I is(are) dimensionally correct?
2
A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P)
by the metal. The sensor has a scale that displays $${\log _2}\left( {{P \over {{P_0}}}} \right)$$, where P0 is a constant. When the metal
surface is at a temperature of 487oC, the sensor shows a value 1. Assume that the emissivity of the metallic
surface remains constant. What is the value displayed by the sensor when the temperature of the metal
surface is raised to 2767oC?
3
Consider two solid spheres P and Q each of density 8 gm cm–3 and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm–3 and viscosity $$\eta $$ = 3 poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm–3 and viscosity $$\eta $$ = 2 poiseulles. The ratio of the terminal velocities of P and Q is
4
The position vector $$\overrightarrow r $$ of a particle of mass m is given by the following equation
$$$\overrightarrow r \left( t \right) = \alpha {t^3}\widehat i + \beta {t^2}\widehat j,$$$where $$\alpha = {{10} \over 3}m{s^{ - 3}}$$, $$\beta = 5\,m{s^{ - 2}}$$ and m = 0.1 kg. At t = 1 s, which of the following
statement(s) is(are) true about the particle?
5
A uniform wooden stick of mass 1.6 kg and length $$l$$ rests in an inclined manner on a smooth, vertical wall
of height h ( < $$l$$ ) such that a small portion of the stick extends beyond the wall. The reaction force of the
wall on the stick is perpendicular to the stick. The stick makes an angle of $$30^\circ $$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $${h \over l}$$ and the frictional force f at the bottom of the stick are ( g =10 ms-2 )
6
A parallel beam of light is incident from air at an anglea on the side PQ of a right angled triangular prism of refractive index n = $$\sqrt 2 $$. Light undergoes total internal reflection in the prism at the face PR when a has a minimum value of 45°. The angle q of the prism is :
7
In a historical experiment to determine Planck's constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength ($$\lambda $$) of incident light and the corresponding stopping potential (V0) are given below:
Given that c = 3 $$ \times $$ 108 ms-1 and e = 1.6 $$ \times $$ 10-19 C, Planck's constant (in units of J-s) found from such an experiment is) :
| $$\lambda \left( {\mu m} \right)$$ | V0(Volt) |
|---|---|
| 0.3 | 2.0 |
| 0.4 | 1.0 |
| 0.5 | 0.4 |
Given that c = 3 $$ \times $$ 108 ms-1 and e = 1.6 $$ \times $$ 10-19 C, Planck's constant (in units of J-s) found from such an experiment is) :
8
A water cooler of storage capacity 120 litres can cool water at a constant rate of P watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load).
The temperature of water fed into the device cannot exceed 30°C and the entire stored 120 litres of water is initially cooled to 10°C. The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is :
(Specific heat of water is 4.2 kJ kg−1 K−1 and the density of water is 1000 kg m−3)
The temperature of water fed into the device cannot exceed 30°C and the entire stored 120 litres of water is initially cooled to 10°C. The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is :
(Specific heat of water is 4.2 kJ kg−1 K−1 and the density of water is 1000 kg m−3)
9
A plano-convex lens is made of material of refractive index n. When a small object is placed 30 cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away form the lens. Which of the following statement(s) is(are) true?
10
A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the 90$$^\circ$$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at a constant rate of 10 As$$-$$1. Which of the following statement(s) is (are) true?
11
Two loudspeakers M and N are located 20m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car in initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/h along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q.
Let v(t) represent the beat frequency measured by a person sitting in the car at time t. Let vP, vQ and vR be the beat frequencies measured at locations P, Q and R respectively. The speed of sound in air is 330 ms$$-$$1. Which of the following statement(s) is (are) true regarding the sound heard by the person?
Let v(t) represent the beat frequency measured by a person sitting in the car at time t. Let vP, vQ and vR be the beat frequencies measured at locations P, Q and R respectively. The speed of sound in air is 330 ms$$-$$1. Which of the following statement(s) is (are) true regarding the sound heard by the person?
12
A transparent slab of thickness d has a refractive index n (z) that increases with z. Here, z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices n1 and n2 (> n1), as shown in the figure. A ray of light is incident with angle $$\theta$$i from medium 1 and emerges in medium 2 with refraction angle $$\theta$$f with a lateral displacement $l$.
Which of the following statement(s) is (are) true?
Which of the following statement(s) is (are) true?
13
Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n, where n >> 1. Which of the following statement(s) is(are) true?
14
An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is (are) true?
15
A hydrogen atom in its ground state is irradiated by light of wavelength 970$$\mathop A\limits^o $$.
Taking hc = 1.237 $$\times$$ 10$$-$$6 eVm and the ground state energy of hydrogen atom as $$-$$ 13.6 eV, the number of lines present in the emission spectrum is
Taking hc = 1.237 $$\times$$ 10$$-$$6 eVm and the ground state energy of hydrogen atom as $$-$$ 13.6 eV, the number of lines present in the emission spectrum is
16
The isotope $$_5^{12}B$$ having a mass 12.014 u undergoes $$\beta $$-decay to $$_6^{12}C$$. $$_6^{12}C$$ has an excited state of the nucleus ($$_6^{12}C$$*) at 4.041 MeV above its ground state. If $$_5^{12}B$$ decays to $$_6^{12}C$$*, the maximum kinetic energy of the $$\beta$$-particle in units of MeV is (1u = 931.5 MeV/c2, where c is the speed of light in vacuum).
17
An infinite line charge of uniform electric charge density l lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time t = 0, the space inside the cylinder is filled with a material of permittivity e and electrical conductivity s. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j (t) at any point in the material?
18
Two inductors L1 (inductance 1mH, internal resistance 3$$\Omega$$) and L2 (inductance 2 mH, internal resistance 4$$\Omega$$), and a resistor R (resistance 12$$\Omega$$) are all connected in parallel across a 5V battery. The circuit is switched on at time t = 0. The ratio of the maximum to the minimum current (Imax / Imin) drawn from the battery is