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QUESTION: 1

**Q. No. 1 -6 carry 3 marks each.**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

**Q.**

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is

given by

The measured masses of the neutron are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065u, respectively. Given that the radii of both the nuclei are same, 1 u = 931.5 MeV/c^{2} (c is the speed of light) and e^{2} /(4πε_{0}) = 1.44 MeV fm. Assuming that the difference between the binding energies of is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10-15m)

Solution:

.........(i)

Now mass defect of N atom = 8 x 1.008665 + 7 x 1.007825 - 15.000109

= 0.1239864 u

So binding energy = 0.1239864 x 931.5 MeV

and mass defect of O atom = 7 x 1.008665 + 8 x 1.007825 - 15.003065

= 0.12019044 u

So binding energy = 0.12019044 x 931.5 MeV

So |B0 - BN| = 0.0037960 x 931.5 MeV .. . . (ii)

from (i) and (ii) we get

R = 3.42 fm.

QUESTION: 2

The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at 10^{0}C. Now the end P is maintained at 10^{0}C, while the end S is heated and maintained at 400^{0}C. The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is 1.2 x 10^{-5 }K^{-1}, the change in length of the wire PQ is

Solution:

Extension in a small element of length dx is

QUESTION: 3

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

Solution:

Required activity

Time required = 6 half lives

= 6 x 18 days

= 108 days.

QUESTION: 4

There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers (C_{1}) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper (C_{2}) has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers C_{1} and C_{2} respectively, are

Solution:

In first; main scale reading = 2.8 cm.

Vernier scale reading = 7 x 1/10 = 0.07 cm

So reading = 2.87 cm ;

In second; main scale reading = 2.8 cm

Vernier scale reading = 7 x - 0.1/10= -0.7/10

=- 0.07 cm

so reading = (2.80 + 0.10 - 0.07) cm = 2.83 cm

QUESTION: 5

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure P_{i} = 10^{5} Pa and volume V_{i} = 10^{-3} m^{3} changes to a final state at P_{f} = (1/32) x 10^{5} Pa and V_{f} = 8 x 10^{-3} m^{3} in an adiabatic quasi-static process, such that P^{3}V^{5} = constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at P_{i} followed by an isochoric (isovolumetric) process at volume V_{f}. The amount of heat supplied to the system in the two-step process is approximately

Solution:

QUESTION: 6

A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle θ = 30^{0} to the axis of the lens, as shown in the figure.

If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the

point (x, y) at which the image is formed are

Solution:

First Image I_{1} from the lens will be formed at 75 cm to the right of the lens.

Taking the mirror to be straight, the image I_{2} after reflection will be formed at 50 cm to the left of the mirror.

On rotation of mirror by 30^{0} the final image is I_{3}.

So x = 50 – 50 cos 60^{0} = 25 cm. and y = 50 sin 60^{0} = 25 √3 cm

*Multiple options can be correct

QUESTION: 7

**Q.No. - 7 - 14 carry 4 marks each**

**Each questions has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.**

**Q.**

While conducting the Young’s double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources (S_{1}, S_{2}) emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3 m from the mid-point of S_{1}S_{2}, as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining S_{1}S_{2}. which of the following is(are) true of the intensity pattern on the screen?

Solution:

Since S_{1}S_{2} line is perpendicular to screen shape of pattern is concentric semicircle

∴ darkness close to O.

*Multiple options can be correct

QUESTION: 8

In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a periodic motion is The values of R and r are measured to be (60 1) mm and (10 1) mm, respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s and 0.59s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true?

Solution:

Error in T

∴

*Multiple options can be correct

QUESTION: 9

A rigid wire loop of square shape having side of length L and resistance R is moving along the x-axis with a constant velocity v_{0} in the plane of the paper. At t = 0, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field B_{0} into the plane of the paper, as shown in the figure. For sufficiently large v_{0}, the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x. Counter-clockwise current is taken as positive.

Which of the following schematic plot(s) is(are) correct? (Ignore gravity)

Solution:

For right edge of loop from x = 0 to x = L

*Multiple options can be correct

QUESTION: 10

Light of wavelength λ_{ph} falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is φ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is λ_{e}, which of the following statement(s) is(are) true?

Solution:

when V is made four times λ_{e} is halved.

*Multiple options can be correct

QUESTION: 11

Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length a through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is ω. The angular momentum of the entire assembly about the point ‘O’ is(see the figure). Which of the following statement(s) is(are) true?

Solution:

*Multiple options can be correct

QUESTION: 12

Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers RC < R/2, which of the following statement(s) about any one of the galvanometers is(are) true?

Solution:

*Multiple options can be correct

QUESTION: 13

In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is (are) true?

Solution:

at t = 0, voltage across each capacitor is zero, so reading of voltmeter is –5 Volt.

at t = ∞ , capacitors are fully charged. So for ideal voltmeter, reading is 5Volt.

at transient state,

where τ = 1 sec

So I becomes 1/e times of the initial current after 1 sec.

The reading of voltmeter at any instant =So at , reading of voltmeter is zero.

*Multiple options can be correct

QUESTION: 14

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x_{0}. Consider two cases: (i) when the block is at x_{0} ; and (ii) when the block is at x = x_{0} + A. In both the cases, a particle with mass m (< M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is (are) true about the motion after the mass m is placed on the mass M?

Solution:

A remains same

QUESTION: 15

**Q. No. 15 -18 carry 3 marks each.**

**This section contains TWO paragraphs**

**Based on each paragraph, there are TWO questions**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct**

**Pragraph 1**

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity ω is an example of a non-inertial frame of reference. The relationship between the force experienced by a particle of mass m moving on the rotating disc and the force experienced by the particle in an inertial frame of reference is

where is the velocity of the particle in the rotating frame of reference and is the position vector of the

particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis . A small block of mass m is gently placed in the slot at and is constrained to move only along the slot.

**Q. The distance r of the block at time t is**

Solution:

where v is the velocity of the block radially outward.

QUESTION: 16

**Pragraph 1**

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity ω is an example of a non-inertial frame of reference. The relationship between the force experienced by a particle of mass m moving on the rotating disc and the force experienced by the particle in an inertial frame of reference is

where is the velocity of the particle in the rotating frame of reference and is the position vector of the

particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis . A small block of mass m is gently placed in the slot at and is constrained to move only along the slot.

**Q.**

**The net reaction of the disc on the block is**

Solution:

QUESTION: 17

**Pragraph 2**

**Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r << h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at +V _{0} and the top plate at –V_{0}. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)**

Q. Which one of the following statements is correct?

Solution:

After hitting the top plate, the balls will get negatively charged and will now get attracted to the bottom plate which is positively charged. The motion of the balls will be periodic but not SHM.

QUESTION: 18

**Pragraph 2**

**Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r << h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at +V _{0} and the top plate at –V_{0}. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)**

**Q. The average current in the steady state registered by the ammeter in the circuit will be**

Solution:

is charge on ball then Q ∝ V0 .....(i)

QUESTION: 19

**Q. No. 19 - 24 carry 3 marks each.**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

**Q.** The correct order of acidity for the following compounds is

Solution:

Stabler the conjugate base stronger the acid.

Conjugate base stabilized by intramolecular H-bond from both the sides.

Conjugate base stabilized by intramolecular H-bond from one side.

QUESTION: 20

The geometries of the ammonia complexes of Ni^{2+}, Pt^{2+} and Zn^{2+}, respectively, are

Solution:

QUESTION: 21

For the following electrochemical cell at 298 K,

Pt(s) | H_{2} (g, 1 bar) | H^{+} (aq, 1M) || M^{4+} (aq), M^{2+} (aq) | Pt(s)

The value of x is

Solution:

QUESTION: 22

The major product of the following reaction sequence is

Solution:

QUESTION: 23

In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are

Solution:

QUESTION: 24

The qualitative sketches I, II and III given below show the variation of surface tension with molar concentration of three different aqueous solutions of KCl, CH_{3}OH and CH_{3}(CH_{2})_{11} OSO_{3}^{-}Na^{+} at room temperature. The correct assignment of the sketches is

Solution:

Strong electrolytes like KCl increase the surface tension slightly. Low molar mass organic compounds usually decrease the surface tension. Surface active organic

compounds like detergents sharply decrease surface tension

*Multiple options can be correct

QUESTION: 25

**Q. No. 25 - 32 carry 4 marks each.**

**Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct.**

**Q.**

For ‘invert sugar’, the correct statement(s) is(are) (Given: specific rotations of (+)-sucrose, (+)-maltose, L-(–)-glucose and L-(+)-fructose in aqueous solution are +66º, +140º, –52º and +92º, respectively)

Solution:

(average is taken as both monomers are one mole each)

*Multiple options can be correct

QUESTION: 26

Among the following, reaction(s) which gives(give) tert-butyl benzene as the major product is(are)

Solution:

*Multiple options can be correct

QUESTION: 27

Extraction of copper from copper pyrite (CuFeS_{2}) involves

Solution:

Refining of blister copper is done by poling technique

*Multiple options can be correct

QUESTION: 28

The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are)

Solution:

The middle layers will have 12 nearest neighbours. The top-most layer will have 9 nearest neighbours.

4r = a √2, where ‘a’ is edge length of unit cell and ‘r’ is radius of atom

*Multiple options can be correct

QUESTION: 29

Reagent(s) which can be used to bring about the following transformation is(are)

Solution:

NaBH_{4} and Raney Ni/H_{2} do not react with acid, ester or epoxide entities of an organic compound

*Multiple options can be correct

QUESTION: 30

Mixture (s) showing positive deviation from Raoult’s law at 35^{o}C is (are)

Solution:

Benzene + toluene will form ideal solution.

Phenol + aniline will show negative deviation.

*Multiple options can be correct

QUESTION: 31

The nitrogen containing compound produced in the reaction of HNO_{3} with P_{4}O_{10}

Solution:

*Multiple options can be correct

QUESTION: 32

According to Molecular Orbital Theory

Solution:

QUESTION: 33

**This section contains TWO paragraphs
Based on each paragraph, there will be TWO questions
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct**

**Paragraph 1**

Thermal decomposition of gaseous X_{2} to gaseous X at 298 K takes place according to the following equation:

The standard reaction Gibbs energy, Δ_{r}G0, of this reaction is positive. At the start of the reaction, there is one mole of X_{2} and no X. As the reaction proceeds, the number of moles of X formed is given by β. Thus, β_{equilibrium} is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally. (Given: R = 0.083 L bar K^{-1} mol^{-1})

Q. The equilibrium constant Kp for this reaction at 298 K, in terms of β_{equilibrium}, is

Solution:

Total number of moles at equilibrium

QUESTION: 34

**Paragraph 1**

Thermal decomposition of gaseous X_{2} to gaseous X at 298 K takes place according to the following equation:

The standard reaction Gibbs energy, Δ_{r}G_{0}, of this reaction is positive. At the start of the reaction, there is one mole of X_{2} and no X. As the reaction proceeds, the number of moles of X formed is given by β. Thus, β_{equilibrium} is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally. (Given: R = 0.083 L bar K^{-1} mol^{-1})

Q.

The INCORRECT statement among the following, for this reaction is

Solution:

There is no data given to find the β_{equilibrium} exact value

QUESTION: 35

PARAGRAPH 2

Treatment of compound O with KMnO_{4}/H^{+} gave P, which on heating with ammonia gave Q. The compound Q on treatment with Br_{2}/NaOH produced R. On strong heating, Q gave S, which on further treatment with ethyl

2-bromopropanoate in the presence of KOH followed by acidification, gave a compound T.

Q. The compound R is

Solution:

QUESTION: 36

PARAGRAPH 2

Treatment of compound O with KMnO_{4}/H^{+} gave P, which on heating with ammonia gave Q. The compound Q on treatment with Br_{2}/NaOH produced R. On strong heating, Q gave S, which on further treatment with ethyl

2-bromopropanoate in the presence of KOH followed by acidification, gave a compound T.

Q.

The compound T is

Solution:

QUESTION: 37

**Q. No. 37 - 42 carry 3 marks each.**

**Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

Q.

Let and I be the identity matrix of order 3. If Q= [q_{ij}] is a matrix such that P^{50} – Q = I, then equals

Solution:

QUESTION: 38

Area of the region is equal to

Solution:

Shifting origin to (–3, 0)

Area = Region(OPK) + Region(QLKR) + Region(OLQ) - Tnangle(PQR)

QUESTION: 39

The value of is equal to

Solution:

QUESTION: 40

Let b_{i} > 1 for i = 1, 2, …, 101. Suppose log_{e} b_{1}, log_{e} b_{2}, …, log_{e} b_{101} are in Arithmetic Progression (A. P.) with the common difference log_{e} 2. Suppose a_{1}, a_{2}, …, a_{101} are in A.P. such that a_{1} = b_{1} and a_{51} = b_{51}. If t = b_{1} + b_{2} + … + b_{51} and s = a_{1} + a_{2} + … + a_{51}, then

Solution:

a_{2}, a_{3}, ....., a_{50}

are Arithmetic Means and b_{2}, b_{3}, ....., b_{50} are Geometric Means between a_{1}(=b_{1}) and a_{51}(=b_{51})

Hence b_{2} < a_{2}, b_{3} < a_{3} .....

t < S

Also a_{1}, a_{51}, a_{101} is an Arithmetic Progression and b_{1}, b_{51}, b_{101} is a Geometric Progression

Since a_{1} = b_{1} and a_{51} = b_{51}

b_{101} > a_{101}

QUESTION: 41

The value of is equal to

Solution:

QUESTION: 42

Let P be the image of the point (3, 1, 7) with respect to the plane x – y + z = 3. Then the equation of the plane passing through P and containing the straight line is

Solution:

Mirror image of (3, 1, 7)

Equation of plane passing through line and (-1, 5, 3)

*Multiple options can be correct

QUESTION: 43

**Q. No. 43 - 50 carry 4 marks each.**

**Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s)
is(are) correct. **

**Q.**

**Let and ** be defined by f(x) = a cos(|x^{3 }- x|) + b|x| sin(|x^{3} + x|)Then f is

Solution:

f(x) = a cos(x^{3} - x) + bx sm(x(x^{2} + 1))

It is a differentiable function

*Multiple options can be correct

QUESTION: 44

Let for all x > 0. Then

Solution:

For x ∈ (0, 1) it is increasing function

*Multiple options can be correct

QUESTION: 45

Let be twice differentiable functions such that f" and g" are continuous functions on . Suppose f'(2) = g(2) = 0, f'(2) ≠ 0 and g'(2) ≠ 0. then

Solution:

⇒ f has a local minimum at x = 2.

*Multiple options can be correct

QUESTION: 46

Let be a unit vector in Given that there exists a vector . Which of the following statement(s) is(are) correct?

Solution:

*Multiple options can be correct

QUESTION: 47

Let P be the point on the parabola y^{2} = 4x which is at the shortest distance from the center S of the circle x^{2} + y^{2} – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then

Solution:

Equation of normal of parabola is

y + tx = 2t + t^{3}

Normal passes through S(2, 8)

8 + 2t = 2t + t^{3}

t = 2

Hence P ≡ (4,4) and SQ = eadius = 2

*Multiple options can be correct

QUESTION: 48

Let Suppose and z S, then (x, y) lies on

Solution:

*Multiple options can be correct

QUESTION: 49

Let Consider the system of linear equations

ax + 2y = λ

3x – 2y = μ

Which of the following statement(s) is(are) correct?

Solution:

System has unique solution for

system has infinitely many solutions for

and no solution for

*Multiple options can be correct

QUESTION: 50

Let be functions defined by f(x) = [x^{2} - 3] and g(x) = |x| f(x) + |4x-7| f(x) where [y] denotes the greatest integer less than or equal to y for y ∈ R . Then

Solution:

f(x )= [x^{2} -3]

Which is discontinuous at x = 1,

g(x) = f(x) [|x| + |4x - 7|]

f(x) is non differentiable at

& |x| + |4x - 7| is non differentiable at

Hence g(x) is non differentiable

QUESTION: 51

**This section contains TWO paragraphs**

**Based on each paragraph, there are TWO questions.
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.**

**PARAGRAPH 1**

Football teams T_{1} and T_{2} have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T_{1} winning, drawing and losing a game against T_{2} are 1/2, 1/6 and 1/3, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams T_{1} and T_{2}, respectively, after two games.

**Q.**

**P(X > Y) is**

Solution:

P(X > Y) = P(T1 wins both) + P(T1 wins either of the matches and other is draw)

QUESTION: 52

**PARAGRAPH 1**

Football teams T_{1} and T_{2} have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T_{1} winning, drawing and losing a game against T_{2} are 1/2, 1/6 and 1/3, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams T_{1} and T_{2}, respectively, after two games.

Q.

P (X = Y) is

Solution:

P(X = Y) = P(T_{1} and T_{2} win alternately) + P(Both matches are draws)

QUESTION: 53

PARAGRAPH 2

Let F_{1}(x_{1}, 0) and F_{2}(x_{2}, 0), for x_{1} < 0 and x_{2} > 0, be the foci of the ellipse . Suppose a parabola having vertex at the origin and focus at F_{2} intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

Q.

The orthocentre of the triangle F_{1}MN is

Solution:

For orthocentre : one altitude is y = 0 (MN is

perpendicular)

QUESTION: 54

PARAGRAPH 2

Let F_{1}(x_{1}, 0) and F_{2}(x_{2}, 0), for x_{1} < 0 and x_{2} > 0, be the foci of the ellipse . Suppose a parabola having vertex at the origin and focus at F_{2} intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at

Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF_{1}NF_{2} is

Solution:

Equation of tangent at M and N are

R(6, 0)

Equation of normal

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