jee-advanced paper-i -...

JEE-ADVANCED_PAPER-I Time: 9.00 AM to 12.00 Noon IMPORTANT INSTRUCTIONS Max Marks: 240 CHEMISTRY

Section Question Type +Ve Marks - Ve

Marks No.of

Qs Total marks

Sec – I (Q.N : 1 – 7) Questions with Single Correct Choice 3 -1 7 21

Sec – II(Q.N : 8 – 11) Questions with Multiple Correct Choice 4 0 4 16

Sec – III(Q.N : 12 – 16) Questions with Comprehension Type (2 Comprehensions – 2 + 3 = 5Q) 3 -1 5 15

Sec – IV(Q.N : 17 – 23) Questions with Integer Answer Type 4 0 7 28 Total 23 80

PHYSICS Section Question Type +Ve Marks

- Ve Marks

No.of Qs

Total marks

Sec – I(Q.N : 24 – 30) Questions with Single Correct Choice 3 -1 7 21



Sec – IV(Q.N : 40 – 46) Questions with Integer Answer Type 4 0 7 28

Total 23 80 MATHEMATICS

Section Question Type +Ve Marks - Ve

Marks No.of

Qs Total marks

Sec – I(Q.N : 47 – 53) Questions with Single Correct Choice 3 -1 7 21



Sec – IV(Q.N : 63 – 69) Questions with Integer Answer Type 4 0 7 28

Total 23 80

JEE-Advanced Syllabus

Chemistry : Total Syllabus

Physics : Total Syllabus

Mathematics : Total Syllabus

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JEE-Advanced_Paper-1_Q’Paper CHEMISTRY Max Marks : 80

SECTION I (Single Correct Answer Type)

This section contains 7 multiple choice questions. Each question has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct

1. Copper reduces 3NO into NO and 2NO depending upon concentration of 3HNO in

solution. Assuming 2 0.1Cu M ,and 2310NO NOP P

bar. At which concentration of

3HNO , thermodynamic tendency for reduction of 3NO into NO and 2NO by copper is

same ? [ Given: 2 | 0.34 ,o

Cu CuE volt

3 |0.96 ,o

NO NOE volt

3 2|0.79o

NO NOE volt ]

A) 1.2310 M

B) 0.5610 M

C) 0.6610 M

D) 0.1210 M

2. A radioactive material 1/2 30t days gets spilled over the floor of a room. If initial

activity is ten times the permissible value, after how many days will it be safe to enter

the room

A) 100

B) 1000

C) 300

D) 10

JEE-Advanced space for rough workPage Page 2

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JEE-Advanced_Paper-1_Q’Paper 3.

2O N

2N Cl

2NO

2NO

+

3CH

3H C 3CH

P[major]

; P is

A) N=N

2NO

2NO

2NO3H C

3CH

3CH

B)

2NO

2NO

2NO3H C

3CH

3CH

C) both in equal proportions

D) Coupling fails


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JEE-Advanced_Paper-1_Q’Paper

4.

Br

O

Br A mixture of all possible stereoisomers from the above structure is subjected to

fractional distillation, which of the following statements is correct ?

A) Two fractions are obtained, each fraction contains a pair of enantiomers

B) Two fractions are obtained, one fraction contains a single meso compound and the

other fraction contains a pair of enantiomers

C) Three fractions are obtained, each fraction contains a pair of enantiomers.

D) Three fractions are obtained, two fractions contain a single meso compound and one

fraction contains a pair of enantiomers

5. Which of the following about 4SF , 4SOF , and 2COF molecules is correct ?

A) Equatorial FSF bond angle in 4SOF will be less than in 4SF molecule

B) Hybridization state of sulphur in 4SF and 4SOF molecule will be different

C) The bond angle FCO will be less than 0120 in molecule 2COF

D) None of these


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JEE-Advanced_Paper-1_Q’Paper 6. 2 3B O substitutes non metal oxides from several metal salts because

A) 2 3B O has more melting point with less volatility than the substituted non-metal

oxide

B) 2 3B O can form solid metal meta borates , by substituting the other non-metal oxide

C) 2 3B O is more stable than the substituted non-metal oxide

D) None of these

7. Which of these is most stable?

A) NF3

B) NCl3

C) NBr3

D) NI3


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JEE-Advanced_Paper-1_Q’Paper SECTION – II

(Multiple Correct Answers Type) This section contains 4 multiple choice questions. Each question has four choices (a), (b) , (c) and (d) out of which ONE or MORE may be correct

8. The correct statement regarding various types of molecular speeds are

A) Increasing temperature increases the fraction of molecules having mpsU

B) Increasing temperature increases mpsU

C) In a sample of gas at a given temperature, molecules with extremely low and high

speeds are less

D) At the same temperature lighter gaseous having narrow distribution of molecular

speeds than heavier gaseous

9. The compounds that should be used to prepare glycine and β − alanine by Gabriel

phthalimide synthesis are

A) 3

3

|

|

COOH

Cl C CH

CH

− −

B) 2ClCH COOH C) 2 2Cl CH CH COOH−

D) 2

3

|Cl CH CH COOH

CH

− −


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JEE-Advanced_Paper-1_Q’Paper 10. In which of these compounds, Nitrogen can be estimated by Duma’s method ?

A) Pyridine

B) Aniline

C)Nitro Benzene

D) Pyrrole

11. A black mineral (A) in solid state is fused with KOH and 3KNO and the mixture

extracted with water to get a green coloured solution (B). On passing 2CO gas through

the solution the colour changes to pink with a black residue (C). Which of the

following is/are correct

A) The pink colouration is decolourised by acidified ferrous sulphate solution

B) The black residue is same as compound (A)

C) When 3O is passed through the green colour solution is changes to pink.

D) Shape of the anion of pink substance is tetrahedral


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JEE-Advanced_Paper-1_Q’Paper SECTION – III

(Paragraph Type) This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph 2 multiple choice questions have to be answered.Each of these questions has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct

Passage – I

I

2 3 21 / 1. 1.Mg Et O PBr SOClA B C O 2)NaCH 2COOO t 42. /SnCl 2)

3) H 3) 3 ,H O

then

12. “A” is

A)

O

B)

OH

C) OH

D) O


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JEE-Advanced_Paper-1_Q’Paper 13. Formation of “B” through the attack of first two reagents involve respectively

A) 2 2&SN SN attacks

B) 1 1&SN SN attacks

C) 2 1&SN SN attacks

D) 1 1&SN SN attacks

14. Product “C” is

A)

O

B) O

C) O

D) O


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JEE-Advanced_Paper-1_Q’Paper Passage – II A white substance (A) reacts with dilute 2 4H SO to produce a colourless gas (B) and a

colourless solution (C). The reaction between (B) and acidified 2 2 7K Cr O solution

produces a green solution and a slightly coloured precipitate (D).The substance (D)

burns in air to produce a gas (E), which reacts with (B) to yield (D) and a colourless

liquid. Anhydrous copper sulphate is turned blue on addition of this colourless liquid

.Addition of aqueous 3NH or NaOH to (C) produces first a precipitate which dissolves

in the excess of the respective reagent to produce a clear solution in each case

15. (B) and (D) are respectively

A) 2SO and 4SnSO

B) 2SO and ( )2 4 3Al SO

C) 2H S and BeS

D) 2H S and S


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JEE-Advanced_Paper-1_Q’Paper 16. The precipitate obtained by addition of aqueous 3NH or NaOH to(C) initially

is__________ which dissolves in excess reagent to produce __________

A) The same, the same water soluble complex in each case

B) The same, different water soluble complexes in each case

C) Different, the same water soluble complex in each case

D) Different, different water soluble complexes in each case

SECTION – IV (Integer Answer Type)

This section contains 7 questions. The answer to each of the questions is a single – digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS. 17. The minimum number of moles of solid KCl added to one liter contents of a standard

silver...... silver ion electrode to convert it to standard silver.... silver chloride,

electrode. 0 10/ 0.8 , 10spE Ag Ag V K AgCl are

18. Half life of a reaction is 20 sec . If 2t is second half life of reaction assuming it to be

zero order and 3t is third half life assuming it to be 2nd order reaction, then 3

2

t

t ?

19. Ka for weak acid HA is 52 10 at 298K what is the pH value of 0.2M aqueous solution

of its salt with a strong base KOH? (log2=0.3)


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20. How many of the following reactions are eliminations [majorly]

a) 3Me C

3N Me OH

; b) 3

Me C

3N Me OH

c) IMeO d)

Br O

e)

OCl

f) Br

ACO

g) Br

EtO

h) Br

CN

21. How many of the following are reducing carbohydrates Glucose, Fructose, Sucrose,

Lactose, Maltose, Galactose, Cellobiose, Cellulose, Starch, Arabinose, Ribose

22. Find the value of ‘x’ in the tremolite asbestos ( ) ( )2 4 11 2 2Ca Mg Si O OHx

23. The mass in grams of available oxygen per litre from a solution of 2 2H O 10ml of

which when titrated with N/20 4KMnO solution required 25 ml in acid medium


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JEE-Advanced_Paper-1_Q’Paper PHYSICS Max Marks : 80



24. A washer is made of metal having resistivity 10-7 Ω m. The washer has inner radius 1

cm, outer radius 3 cm and thickness 1 mm. A magnetic field, oriented normal to the

plane of the washer, has the time dependent magnitude B = (2t) tesla /sec. Find the

current (in ampere) around the washer

A) 2 amp

B) 4 amp

C) 6 amp

D) 8 amp


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JEE-Advanced_Paper-1_Q’Paper 25. Ends of two wires A and B having resistivity 53 10A mρ

−= × Ω and 56 10B mρ−= × Ω of

same cross section area joined in series together to form a single wire. If the resistance

of the joined wire does not change with temperature, then find the ratio of their lengths

A

B

l

l, given that temperature coefficient of resistivity of wire A and B is 54 10 /A Cα

−= × °

and 66 10 /B Cα−= − × ° . Assume that mechanical dimensions do not change with

temperature

A) 37

B) 103

C) 310

D) 12


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JEE-Advanced_Paper-1_Q’Paper 26. A cone of radius = height = r is under a liquid of density d. Its base is parallel to the

free surface of the liquid at a depth H from it as shown in the figure. What is the net

force due to liquid on its curved surface. (neglect atmospheric pressure)

H

A) 23r

r dg Hπ −

B) 22

23r

r dg Hπ −

C) 2 23r

r dg Hπ −

D) 22

23r

r dg Hπ −


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JEE-Advanced_Paper-1_Q’Paper 27. A spherical black body has a radius R and steady surface temperature T, heat sources

ensure the heat evolution at a constant rate and distributed uniformly over its volume.

What would be the new steady surface temperature of the object if the radius is

decreased by half ? Assume surrounding to be at absolute zero and heat evolution rate

through unit volume remain same.

A) 4 2T

B) 412

T

C) 2T

D) 12

T

28. The smallest length scale known in physics is the Planck length. It is an important

ingredient in some current cosmological theories. Which of the following expressions

could represent this Planck length ? (Symbols has usual meaning)

A) ( )2e

hc

B) hc G

C) Ghc

D) 3Gh c


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JEE-Advanced_Paper-1_Q’Paper 29. In the figure shown two motors P1 & P2 fixed on a plank which is hanging with light

string passing over fixed Pulley P. If the motors starts winding the thread with angular

velocity 1 2&ω ω then velocity of plank V is (here R1 & R2 are the radii of motor rotor

respectively)[Given : 1ω = 2 rad/s, R1 = 2m, 2ω = 2 rad/s, R2 = 3m]

P

P1P2

1ω 2ω

Plank

V

A) 20 m/s

B) 10 m/s

C) 5 m/s

D) 4 m/s


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JEE-Advanced_Paper-1_Q’Paper 30. Five identical balls each of mass m and radius r are strung like beads at random and at

rest along a smooth, rigid horizontal thin rod of length L, mounted between

immovable supports. Assume 10r < L and that the collision between balls or between

balls and supports are elastic. If one ball is struck horizontally so as to acquire a speed

v, the average force felt by the support is

A) 25

5mv

L r−

B) 2

10mv

L r−

C) 25

10mv

L r−

D) 2

5mv

L r−


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JEE-Advanced_Paper-1_Q’Paper SECTION – II

(Multiple Correct Answers Type) This section contains 4 multiple choice questions. Each question has four choices (a), (b) , (c) and (d) out of which ONE or MORE may be correct

31. A double star is a system of two stars rotating about their centre of mass only under

their mutual gravitational attraction. Let the stars have masses m and 2m and let their

separation be l. Their time period of rotation about their centre of mass will be

proportional to

A) 3

2l

B) l

C) 1

2m

D) 1

2m−


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JEE-Advanced_Paper-1_Q’Paper 32. If white light is used in a Young’s double – slit experiment. Point C represents centre

of a screen.

SCREEN

C

A) bright white fringe is formed at the centre of the screen

B) fringe closest to either side of centre of the screen is green

C) fringe closest to either side of centre of the screen is violet

D) after few fringes no clear fringe pattern is seen


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JEE-Advanced_Paper-1_Q’Paper 33. Consider a thermodynamic cycle in a PV diagram shown in the figure performed by

one mole of a monoatomic gas. The temperature at A is T0 and volume at A and B are

related as VB = VC = 2VA. Choose the correct option(s) form the following

A

B

C

P

V

A) The maximum temperature during the cycle is 4T0

B) Net work done by the gas during the cycle is 0.5 RT0

C) The heat capacity of the process AB is 2R

D) Heat capacity of process AB is 3R

34. If the plank’s constant would be double the present value, in the Bohr’s model for

hydrogen atom

A) the speed of the electron in the ground state would be halved

B) the radius of electronic orbit in the ground state would be four times

C) the energy required to ionize the hydrogen atom in the ground state would be

doubled

D) the angular momentum of the electron in the ground state would be doubled JEE-Advanced space for rough workPage Page 21

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SECTION – III (Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph 2 multiple choice questions have to be answered.Each of these questions has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct

Passage-1

In 1906, Robert Millikan devised an experiment that allowed him to determine the

charge of an electron. A schematic of Millikan’s set – up is shown below:

Two metal plates are connected by a series of batteries to form a capacitor. There is an

electric field between the plates. The metal plates are inside an insulated cylindrical

container.

Oil drops are introduced into the container through a small hole in the top. The oil

drops acquire a negative charge as they pass through the nozzle of the oil can. Some of

the drops fall through a hole in the upper plate. By adjusting the voltage between the

plates, certain drops can be suspended between them. The relationship between the

electric field between the plates and the voltage across the plates is: V EL∆ =

Where E is the electric field and L is the plate separation.

Millikan chose oil because of its relatively low vapour pressure and high charge

holding ability. (To answer the following question assume oil drops as to be non

conducting tiny spheres)


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Oil Can

35. In order for an oil drop of mass m, radius r and volume charge density ρ , to be

suspended between the plates, the magnitude and direction of the electric field must

be:

A) 33

4

mg

rπ ρ; downward

B) 34

3

r

mg

π ρ ; downward

C) 33

4

mg

rπ ρ; upward

D) 34

3

r

mg

π ρ ; upward


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JEE-Advanced_Paper-1_Q’Paper 36. Suppose the original oil droplet were replaced with a positively charged one that had

twice the charge and three times the mass of the original droplet, How would the

magnitude of the electric field have to be changed in order for the drop to remain

suspended?

A) no change is necessary

B) it must be increased by a factor of 2

C) It must be decreased by a factor of 2/3

D) it must be increased by a factor of 3/2

Passage:2

In a hypothetical atom, a negatively charged particle having a charge of magnitude 3e

and mass 3m revolves around a proton. Here, e is the electronic charge and m is the

electronic mass. Mass of proton may be assumed to be much larger than that of the

negatively charged particle, thus the proton is at rest. This “ atom” obeys Bohr’s

postulate of quantization of angular momentum, that is 2

hmvr n = π

It is given that for

the first Bohr orbit of hydrogen atom: radius of orbit is 0r speed of electron is 0V , and

total energy is 0E−


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Now answer the following questions.

37. Speed of the revolving particle is, in the first Bohr orbit.

A) 0V

B) 09V

C) 03V

D) 09

V

38. Radius of hypothetical atom is

A) 0r

B) 09

r

C) 027

r

D) 081

r


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JEE-Advanced_Paper-1_Q’Paper 39. The momentum of an emitted photon when it makes a transition from the second

excited state to ground state, is

A) 08Ec

B) 0Ec

C) 0814

E

c

D) 024Ec


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JEE-Advanced_Paper-1_Q’Paper SECTION – IV

(Integer Answer Type) This section contains 7 questions. The answer to each of the questions is a single – digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS. 40. When a certain metallic surface is illuminated with monochromatic light of

wavelength λ , the stopping potential for photoelectric current is 3V0 and when the

same surface is illuminated with light of wavelength 2 λ , the stopping potential is V0.

The threshold wavelength of this surface for photoelectric effect is Kλ . Calculate the

value of K.

41. A monochromatic beam of electrons accelerated by a potential difference V falls

normally on the plane containing two narrow slits separated by a distance d. The

interference pattern is observed on a screen parallel to the plane of the slits and at a

distance of D from the slits. Fringe width is found to be w1. When electron beam is

accelerated by the potential difference 4V the fringe width becomes w2. Find the

ratio 12

w

w. (Given d

JEE-Advanced_Paper-1_Q’Paper 42. A target element A is bombarded with electrons and the wavelengths of the

characteristic spectrum are measured. A second characteristic spectrum is also

obtained, because of an impurity in the target. The wavelength of the Kα lines are 196

pm (element A) and 169 pm (impurity). If the atomic number of impurity is

z = (10 x – 1). Find the value of x. (atomic number of element A is 27).

43. What is the minimum height (in 102 m) of a brick column of uniform cross section for

which column breaks due to its own weight.

[Patmospheric = 100 kPa, ρ = 1.8 × 103kg/m3. Breaking stress σ = 3.7 M Pa]

44. A sound source emits frequency of 175 Hz when moving towards a rigid wall with

speed 5 m/s and observer is moving away from wall with same speed 5 m/s. Both

source and observer moves on a straight line which is perpendicular to the wall. The

number of beats per second heard by the observer will be [Speed of sound = 355 m/s]

Source is in between observer and wall.


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45. Gravitational acceleration on the surface of planet is 611

g ; where g is the gravitational

acceleration on the surface of earth. The average mass density of the planet is 2/3

times that of the earth. If escape speed on the surface of the earth is taken to be 111km s− ,

escape speed on the surface of the planet in 1km s− , will be:

46. A point object ‘O’ is placed in a medium of refractive index 1 1.4.µ = 1 2S and S are two

concentric spherical surfaces of radii 1m and 2m. To the right of ‘O’ contains a

medium of refractive index 2 1.5µ = between the interfaces 1 2S and S . Find the object

distance of O form 1S (in meter) so that an image of ‘O’ as seen by observer from

air, coincides with O.

O

a p

1S 2S

1µ 2µair


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JEE-Advanced_Paper-1_Q’Paper MATHEMATICS Max Marks:80



47. Let ,a b and c be positive constants. The value of ' 'a in terms of ' 'c if the value of

integral ( )1

1 3 3 5

0

b bacx a bx dx+ ++∫ is independent of ‘b ’ equals

A) 32

c

B) 23

c

C) 3

c

D) 32c

48. Let a, b, c

are three vectors along the adjacent edges of a tetrahedron, if

a | b | | c | 2= = =

and a.b b.c c.a 1,= = =

then volume of tetrahedron is

A) 12

B) 23

C) 32

D) 2 23


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JEE-Advanced_Paper-1_Q’Paper 49. Let ω be a complex cube root of unity with 1ω ≠ . A fair die is thrown three times. If

r1,r2 and r3 are the numbers obtained on the die, then the probability that 31 2 0rr rω ω ω+ + = is

A) 118

B) 19

C) 29

D) 136

50. A ray of light travels along a line y=4 and strikes the surface of a curves

2 4( ),y x y= + then equations of the line along which reflected ray travel is

A) x 0=

B) x 2=

C) x y 4+ =

D) 2x y 4+ =


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JEE-Advanced_Paper-1_Q’Paper 51. A triangle ABC having vertices A(5, 1), B (-1, - 7) and C(1, 4) respectively. If 1L be

the line mirror passing through C and parallel to AB, a light ray emanating from point

A and goes along the direction of internal bisector of the angle A, which meets the

mirror at E and BC at D. Then sum of the area of ∆ACE and ∆ ABC is

A) 192

sq. units

B) 38 sq. units

C) 19 sq. units

D) 752

sq. units

52. The number of different ways in which the persons A, B, C having 6 one rupee coins,

7 one rupee coins, 8 one rupee coins respectively donate 10 one rupee coins

collectively is equal

A) 44

B) 45

C) 46

D) 47


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JEE-Advanced_Paper-1_Q’Paper 53. Let ( ) ( )1 1cos cos 20 , cos cos30a b− −= = and ( )1sin sinc a b−= + then maximum value of

( )( ) ( )( )2sin 2 cosa b c x a b c x+ + + + + is

A) 5 12

+

B) 5 12

−

C) 3 12

−

D) 1

SECTION – II (Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (a), (b) , (c) and (d) out of which ONE or MORE may be correct

54. If ' 'iA is the area bounded by i i| x a | | y | b ,i N,− + = ∈ where i 1 i i3

a a b2+

= + and

ii 1

bb

2+= , 1 10, 32a b= = then

A) 3 128A =

B) 3 256A =

C) ( )n

2

ini 1

8lim A 32

3→∞ ==∑

D) ( )n

2

ini 1

4lim A 16

3→∞ ==∑


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JEE-Advanced_Paper-1_Q’Paper 55. Let ˆˆ ˆ, ,a b c be unit vectors such that ˆˆ ˆ 0a b c and x, y, z be distinct integers then

ˆˆ ˆx a y b zc cannot be equal to

A) 1

B) 2

C) 2

D) 3

56. A fore cast is to be made for five games each of which can be a win or a draw or a loss. Let

A = number of fore casts with exactly one error

B = number of fore casts with exactly three errors

C = number of fore casts with all five errors

A) 2B 5C=

B) 8A B=

C) 8A 5C=

D) 2(A C) B+ >


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57. If conjugate of a complex number z a ib= + is defined as z b ai= + then which of the

following is/are true

A) 1 2 1 2z z i z z= −

B) z z z z+ = +

C) 1 2 1 23 3

z z z z

z z

+ =

D) None of these

SECTION – III (Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph 2 multiple choice questions have to be answered.Each of these questions has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct

Passage-1

A rectangle ABCD of dimensions r and 2r is folded along diagonal BD such that

planes ABD and CBD are perpendicular to each other. Let the position of the vertex A

remains unchanged and 1C is the new position of C .


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JEE-Advanced_Paper-1_Q’Paper 58. The distance of 1C from A is equal to

A) 2 55

r

B) 2 105

r

C) 855

r

D) 45

r

59. If 1ABC θ∠ = , then cosθ is equal to

A) 15

B) 25

C) 25

D) 45


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JEE-Advanced_Paper-1_Q’Paper 60. The shortest distance between the edge AB and 1C D is equal to..... (in units)

A) 53

r

B) 2r

C) r

D) 45

r

Passage-2

Let continuous function ( )2

2

2

10 8, 2

, 2 0, 0.

2 , 0

x x x

f x px qx r x where p

x x x

+ + ≤ −= + + − < < ≠ + ≥

If a line ‘L’

touches the graph of y = f(x) at 3 points, then

61. The value of p + q + r is equal to

A) 5 2

B) 5

C) 6

D) 7


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JEE-Advanced_Paper-1_Q’Paper 62. Range of ( )f x in [ ]2,0− is

A) [ ]8,0−

B) [ ]9,1−

C) [ ]10,3−

D) none of these

SECTION – IV (Integer Answer Type)

This section contains 7 questions. The answer to each of the questions is a single – digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.

63. y 1

1Given that Sec x dx and for y ),1(− = λ ≥∫

( ) ( )1 y1 1 2 1 1 2y 1if {Sec x Tan x 1 }dx {Sec x Tan x 1 }dx− − − − −− − − + − − =∫ ∫ ( )y a b , then a+b ___π − − λ =

64. In ABC∆ Orthocentre is (2, 3) Circum centre is (6,10) and equation of side BC

is

2 17x y+ = .Then the radius of the Circum circle of ABC∆ is


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65. C is the centre of the hyperbola 2 2

14 1

x y− = and ‘A’ is any point on it. The tangents at

A to the hyperbola meet the line 2 0x y− = and 2 0x y+ = at Q and R respectively. The

value of .CQ CR .

66. Total number of even divisors of ‘1323000’ which are divisible by 105 is 2 10k − ,

then k is

67. The area bounded by 2 2y x= − − ; 3yx

= is 3ln 32

k − , then k =_______

68. If 2 20, , 4sec 9cos2

and k ecπθ θ θ ∈ = +

, then the number of solutions of the system of

equations 3 4 3, 2 3 2,6 5 3x y z x y z x y kz− + = + − = − + + = − is

69. The shortest distance between the two opposite edges of a regular tetrahedron of edge

8 units is


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