JEE Main 2012 Paper -1 with Solutions

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed ω, as shown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v, with respect to the
rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains ω throughout. The magnitude of the torque about O, as a function of time is best represented by which plot?

A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40
amu) is kept at 300 K in a container. The ratio of the rms speeds   is

Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage
source of potential difference X. A proton is released at rest midway between the two plates. It is found to
move at 450 to the vertical JUST after release. Then X is nearly

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal
black surfaces and have very high thermal conductivity. The first and third plates are maintained at
temperatures 2T and 3T respectively. The temperature of the middle (i.e. second) plate under steady state
condition is

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of
the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is
stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular
frequency ω = π/3 rad/s. Simultaneously at t = 0, a small pebble is projected with speed v form point P at
an angle of 450 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits
the block at t = 1 s, the value of v is (take g = 10 m/s²)

Young’s double slit experiment is carried out by using green, red and blue light, one color at a time. The
fringe widths recorded are  respectively. Then

Consider a thin spherical shell of radius R with centre at the origin, carrying uniform positive surface
charge density. The variation of the magnitude of the electric field  and the electric potential V(r)
with the distance r from the centre, is best represented by which graph?

SECTION II

Q. No. 11 -15 carry 4 mark each

Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

Q.

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric
and magnetic fields  At time t = 0, this charge has velocity   in the x-y plane, making an angle q with the x-axis. Which of the following option(s) is (are) correct for time t > 0?

A cubical region of side a has its centre at the origin. It encloses three fixed point charges, -q at
(0, -a/4, 0), +3q at (0, 0, 0) and -q at (0, +a/4, 0). Choose the correct options(s)

A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe.
When this pulse reaches the other end of the pipe,

A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle θ with the
horizontal. A horizontal force of 1 N acts on the block through its centre of mass as shown in the figure.
The block remains stationary if (take g = 10 m/s²)

For the resistance network shown in the figure, choose the correct option(s)

SECTION III

Q. No. 16 - 20 carry 4 marks each

The answer to each question is single digit integer, ranging from 0 to 9 (both inclusive).

A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side a(a<<R) having two turns is placed with its centre at z =  along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of 45° with respect to the z-axis. If the mutual inductance between the loops is given by  then the value of p is

An infinitely long solid cylinder of radius R has a uniform volume charge density r. It has a spherical
cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the
electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the
expression  The value of k is

A proton is fired from very far away towards a nucleus with charge Q = 120 e, where e is the electronic
charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of
the proton at its start is: (take the proton mass, m_p = (5/3) x 10^-27 kg; h/e = 4.2 x 10^-15 J.s / C;

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and
radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing though O and P
is I_O and I_P respectively. Both these axes are perpendicular to the plane of the lamina. The ratio I_P / I_O to
the nearest integer is

A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the
cylinder and the cavity are infinity long. A uniform current density J flows along the length. If the
magnitude of the magnetic field at the point P is given by  then the value of N is

SECTION – I

Q. No. 21- 30 caeey 3 marks each and 1 mark is deducted for every wrong answer

Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Q.

A compound M_pX_q has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below.
The empirical formula of the compound is

The carboxyl functional group (–COOH) is present in

As per IUPAC nomenclature, the name of the complex [Co(H₂O)₄(NH₃)₂]Cl₃ is

In allene (C₃H₄), the type(s) of hybridization of the carbon atoms is (are)

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a₀ is Bohr radius]

Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. 1/V plot is shown below. The
value of the van der Waals constant a (atm.litre²mol^–2) is

The number of aldol reaction(s) that occurs in the given transformation is

The colour of light absorbed by an aqueous solution of CuSO₄ is

The number of optically active products obtained from the complete ozonolysis of the given compound is

SECTION II

Q. No. 31 -35 carry 4 marks each

Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

Q.

Which of the following hydrogen halides react(s) with AgNO₃(aq) to give a precipitate that dissolves in Na- 2S₂O₃(aq)?

Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as
shown in the given scheme

For an ideal gas, consider only P-V work in going from an initial state X to the final state Z. The final state
Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is(are)
correct? [Take DS as change in entropy and w as work done]

Which of the following molecules, in pure form, is (are) unstable at room temperature?

Choose the correct reason(s) for the stability of the lyophobic colloidal particles.

SECTION III

Q. No. 36 - 40 carry 4 marks each

The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).

Q.

29.2 % (w/w) HCl stock solution has density of 1.25 g mL^-1. The molecular weight of HCl is 36.5 g mol^-1.
The volume (mL) of stock solution required to prepare a 200 mL solution of 0.4 M HCl is

The substituents R₁ and R₂ for nine peptides are listed in the table given below. How many of these
peptides are positively charged at pH = 7.0?

An organic compound undergoes first-order decomposition. The time taken for its decomposition to 1/8 and
1/10 of its initial concentration are t_1/8 and t_1/10 respectively. What is the value of

0.3

When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its
pyranose form is

The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a
nuclear reaction yielding element X as shown below. To which group, element X belongs in the periodic
table?

SECTION I

Q. No 42- 50 carry 3 marks each and 1 mark is deducted for every wrong answer.

Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct

Q.

if  , then

Let P = [a_ij] be a 3 × 3 matrix and let  If the determinant of P is 2, then the determinant of the matrix Q is

The locus of the mid–point of the chord of contact of tangents drawn from points lying on the straight line
4x – 5y = 20 to the circle x² + y² = 9 is

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that
each person gets at least one ball is

The integral   equals (for some arbitrary constant K)

The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, –1, 4) with the plane
5x – 4y – z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length
of the line segment PS is

Let 

Let z be a complex number such that the imaginary part of z is nonzero and a = z² + z + 1 is real. Then a
cannot take the value

The ellipse  is inscribed in a rectangle R whose sides are parallel to the coordinate axes.
Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the
ellipse E₂ is

The function f : [0, 3] → [1, 29], defined by f(x) = 2x³ – 15x² + 36x + 1, is

SECTION II

Q. No. 51 -55 carry 4 marks each.

Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

Q.

Tangents are drawn to the hyperbola   parallel to the straight line 2x - y = 1. The points of contact of the tangents on the hyperbola are

Let  be such that   and   Then φ cannot satisfy

If y(x) satisfies the differential equation  y' - ytanx = 2x secx and y(0) = 0, then

A ship is fitted with three engines E₁, E₂ and E₃. The engines function independently of each other with
respective probabilities 1/2, 1/4,  and 1/4 For the ship to be operational at least two of its engines must
function. Let X denote the event that the ship is operational and let X₁, X₂ and X₃ denote respectively the
events that the engines E₁, E₂ and E₃ are functioning. Which of the following is(are) true ?

Let S be the area of the region enclosed by 

Then

SECTION III

Q. No. 56 - 60 carry 4 marks each

The answer to each question is single digit integer, ranging from 0 to 9 (both inclusive).

Q.

if  are unit vectors satisfying   is

Let f : IR → IR be defined as   The total number of points at which f attains either a local
maximum or a local minimum is

Let S be the focus of the parabola y² = 8x and let PQ be the common chord of the circle x² + y² - 2x - 4y =
0 and the given parabola. The area of the triangle PQS is

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x
= 3. If p(1) = 6 and p(3) = 2, then p'(0) is

The value of