max. marks: 180 jee-advance-2014-p1-model · jee-advance-2014-p1-model space for rough work . page...

“un-darken” a darkened bubble.

Master JEE CLASSES Kukatpally, Hyderabad.

IMPORTANT INSTRUCTIONS: 1) This booklet is your Question Paper. 2) Use the Optical Response Sheet (ORS) provided separately for answering the questions 3) Blank spaces are provided within this booklet for rough work. 4) Write your name, roll number and sign in the space provided on the back cover of this booklet. 5) You are allowed to take away the Question Paper at the end of the examination. OPTICAL RESPONSE SHEET: 6) Darken the appropriate bubbles on the ORS by applying sufficient pressure. This will leave an impression at the corresponding place on the Candidate’s sheet. 7) The ORS will be collected by the invigilator at the end of the examination. 8) Do not tamper with or mutilate the ORS. Do not use the ORS for rough work. 9) Write your name, roll number and code of the examination center, and sign with pen in the space provided for this purpose on the ORS. Do not write any of these details anywhere else on the ORS. Darken the appropriate bubble under each digit of your roll number. DARKENING THE BUBBLES ON THE ORS 10) Use a BLACK BALL POINT PEN to darken the bubbles on the ORS. 11) Darken the bubble COMPLETELY. 12) The correct way of darkening a bubble is as: 13) The ORS is machine-gradable. Ensure that the bubbles are darkened in the correct way. 14) Darken the bubbles ONLY IF you are sure of the answer. There is NO WAY to erase or

JEE-ADVANCE-2014-P1-Model Max. Marks: 180

Time: 07:30 AM to 10:30 AM IMPORTANT INSTRUCTIONS Max Marks: 180

PHYSICS:

Section Question Type +Ve Marks

- Ve Marks

No.of Qs

Total marks

Sec – I(Q.N : 1 – 10) Questions with Multiple Correct Choice 3 0 10 30

Sec – II(Q.N : 11 – 20) Questions with Integer Answer Type 3 0 10 30

Total 20 60

CHEMISTRY:


- Ve Marks

No.of Qs

Total marks



Total 20 60

MATHEMATICS:


- Ve Marks

No.of Qs

Total marks



Total 20 60

JEE-ADVANCE-2014-P1-Model

Space for rough work

PHYSICS Max Marks: 60 SECTION-1

(ONE OR MORE OPTIONS CORRECT TYPE) This section contains 10 multiple choice equations. Each question has four choices (A) (B),(C) and (D) out of which ONE or MORE THAN ONE are correct. Marking scheme: +3 for correct answer, 0 if not attempted and 0 in all other cases.

cos2

y A kx t where 1 10.6 , 0.005 , 8.0A m k cm s and is a non-

vanishing constant. Then for this wave,

A) the wavelength of the wave is 8m

B) the maximum velocity vm of a particle of the rope will be ; 7.54 /v m sm

C) magnitudes of velocity and slope of y-x graph for an element of rope are inversely

proportional to each other .

D) magnitudes of velocity and slope of y-x graph for an element of rope are directly

proportional to each other.

2. A wire of 39.8 10 /kg m passes over a frictionless light pulley fixed on the top of a

frictionless inclined plane which makes an angle of 30º with the horizontal. Masses m

and M are tied at the two ends of wire such that ' 'm rests on the plane and M hangs

1. The equation of a transverse wave travelling in a rope is given by


freely vertically downwards. The entire system is in equilibrium and a transverse

wave propagates along the wire with a velocity of 100 m/s.

A) 20m kg B) 5M kg C) 12

mM

D) 2mM

0.8, 24 5 5y x t

x t

represents a moving pulse, where x and y are in the metres

and t is in seconds, then

A) pulse is moving in x direction

B) in 2s it will travel a distance of 2.5 m

C) its maximum displacement is 0.16 m

D) it is a symmetric pulse

4. A transverse wave travelling on a taut string is represented by 0.01sin 2 10Y t x

Y and x are in meters and t in seconds. Then,

A) the speed of the wave is 10m/s.

B) closest points on the string which differ in phase by 060 are (1/6)m apart.

C) maximum particle velocity is / 4 /m s .

D) the phase of a certain point on the string changes by 120º in (1/20) seconds.


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3.

0.1sinA BY Y t . The correct statement (s) is/are

A) displacement equation of the image of particle A is ' 0.2sin / 2AY t

B) displacement equation of the image of particle B is ' 0.2sinBY t

C) time period of oscillation of both the images is 2s.

D) relative vertical displacement of image of A with respect to image of B is

' 0.4sinABY t

force F. A sudden movement causes a circular loop of radius R to form at one end of

the rope. The loop rolls along the rope with speed ‘v’ as shown in the figure. Then

F F

5. Two identical point particles A and B are placed in front of a concave mirror of focal

length 20 cm, at distances 10 cm and 30 cm respectively. The particles oscillate

perpendicular to the principal axis, such that the displacement equation for both the

particles is given by

6. A long, heavy, flexible rope with mass per unit length ‘’ is stretched by a constant


A) Fv

B) 2Fv

C) the kinetic energy carried by the loop is 2 RF

D) the kinetic energy carried by the loop is Rv

Which of the following functions of x and t represent a progressive wave?

A) sin 4 3y t x B) 2

14 4 3

yt x

C) 14 3

yt x

D) 23 4y x t

’

A) Acceleration of the pulse w.r.t. ground is 34g .

B) Relative to string, acceleration of the pulse on the string will be 3g/4 every where.

C) The time taken by a pulse to reach from bottom to top will be 8 / 3L g .

D) The time taken by a pulse to reach from bottom to top will be 4 / 3L g .

8. One end of a string of length ‘L’ is tied to ceiling of a lift accelerating upwards with

an acceleration ‘2g’. The other end of the string is free. The linear mass density of the

string varies linearly from 0 to ‘ from bottom to top. A transverse pulse is generated

near the bottom of string.


2

7.

53 , as

shown.

A) Transverse velocity of the particle at point A is parallel to positive y direction if the

wave is travelling along positive x-axis.

B) Transverse velocity of the particle at point A is parallel to positive y direction if the

wave is travelling along negative x-direction.

C) Magnitude of transverse velocity of the particle at point A is greater than the wave

speed

D) Magnitude of transverse velocity of the particle at point A is less than the wave

speed

9. y-x curve at an instant for a transverse wave travelling along x axis on a string is

shown. Angle made by tangent with horizontal at the point A on the curve is 0


10. A plane progressive transverse wave is passing through a medium along the positive

x-axis. At the given instant of time, the nature of the displacement(y), position(x)

graph is shown in figure. Which of the following statements are correct? (Consider A,

B, C are three particles of the vibrating string).

A) Elastic potential energy per unit length of the string at the point B is more than that

at points A and C

B) Elastic potential energy per unit length of the string at the point C is more than that

at points A and B

C) Velocity of the particle at A is along negative y direction.

D) Velocity of the particle at C is along positive y direction.


SECTION-II INTEGER TYPE

This section contains 10 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive). Marking scheme: +3 for correct answer, 0 if not attempted and 0 in all other cases.

11. The speed of a transverse wave, going on a wire having a length 50cm and mass 5g is

80 m/s. the area of cross section of the wire is 21.0 mm and its Young’s modulus is

11 28 10 /N m . Find the extension (in 210 mm ) of the wire over its natural length.

12. A uniform rope of mass 0.1kg and length 2.45m hangs from a ceiling. The speed of a

transverse wave in the rope at a point 0.5m distant from the lower end is 2.45 n

where n =

linear density of 319.2 10 kg/m. The speed (with respect to the string) with which a

wave pulse can proceed on the string if the elevator accelerates up at the rate of

22.0 /m s is 12.5n . What is the value of n ? Take 210 /g m s and mstring << 4 kg.

13. A 4.0 kg block is suspended from the ceiling of an elevator through a string having a


student starts a sinusoidal transverse wave of wavelength ‘ ’ propagating along the

rope. The wave is in vertical plane. What minimum wave amplitude (in mm) can

make that ant feel weightless momentarily? Assume m is so small that the presence of

the ant has no effect on the propagation of the wave. [Given: 0.5m , 0.1 /kg m ,

3.125F N , take 2g ]

A B The rope is rotating on horizontal plane with constant angular velocity about end A

where as end B is free. &A BS S are the distances covered by pulse A & pulse B

respectively in rope frame before they cross each other. If A

B

SkS

, find the value of

21k .

14. An ant with mass ‘ m ’ is standing peacefully on top of a horizontal, stretched rope.

The rope has mass per unit length ‘ ’ and is under tension F. Without warning, a

15. Two small pulses are created on the rope at points A & B respectively as shown in the

diagram.


length.

16. As shown in figure, string PQR is stretched by force F = (3 –10 Kt) N, where K is a

constant and t is time in seconds. At time t = 0, a small pulse is generated at the fixed

end P of the string. Find the value of K (in N/s) if the value of force becomes zero as

the pulse reaches point Q. (Given mass per unit length of string is 0.03 kg/m)

17. Two small blocks each having a mass of 3.2 kg are connected by wire CD and the

18. The speed of a transverse wave, going on a uniform wire having a length 50 cm and

mass 5.0 g, is 80 m/s. The area of cross-section of the wire is 1.0 mm2 and its Young’s

modulus is 16 × 109 N/m2. Find the extension (in mm) of the wire over its natural

system is suspended from the ceiling by another vertical wire AB with AB wire

connected(fixed) to the mid point of wire CD. The linear mass density of the wire AB

is 10 g/m and that of CD is 80 g/m. If the ratio of speeds of transverse pulse in AB to

CD is 4/n find n.


2

10A

in vertical plane is generated in

the string. Find the least time period of oscillations of string elements so that ant will

not fall from the string. (Neglect the variation of tension in the string due to weight of

the ant 2g 10m/s ).

2 mm along positive Y direction. If wavelength of the wave is 48 mm then find

minimum separation between mean positions of the particles (in mm).

19. An ant sitting on a string can pull the string with a maximum force up to three times

of its own weight. A sine wave of amplitude

20. A simple harmonic wave of amplitude 2mm is travelling along positive x-axis. At a

given instant, displacements for two different particles on the wave are 1 mm and


CHEMISTRY Max Marks: 60

SECTION-1 (ONE OR MORE OPTIONS CORRECT TYPE)

This section contains 10 multiple choice equations. Each question has four choices (A) (B),(C) and (D) out of which ONE or MORE THAN ONE are correct. Marking scheme: +3 for correct answer, 0 if not attempted and 0 in all other cases.

acidification will yield 03 alcohol?

A) 3CH C Cl

O

B)

O

3 2 5CH O C C H

C) D)

O

3CH C O H

21. Which of the following compounds when reacts with PhMgBr (excess) followed by


3H C 3CH

O

3 ;i CH MgBrii H

A

3CH

O

H

3i CH MgBrii H B

A) Victor-Mayer test B) Sodium metal

C) Lucas reagent D) 2 2 7 /K Cr O H

A)

OH

+ CO2(i) NaOH

(ii)H B)

CHO

OH

( )

2 2(i)H O /OH(ii)H

C)

2N Cl OH

( )

OH

D)

OH

2 2 8(i) K S O(ii) H

Which of the following reagent can be used to differentiate between A and B?

23. Which reaction can convert phenol to hydroquinone


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22.

A)

OCH2 CH CH2

B)

2

2 4

(i )O /(ii)dil.H SO

C)

SO3H

(i) fused NaOH(ii)H

D)

Cl

(i)fused KOH(ii)H

25. Which of the following changes are involved in the formation of most stable product

X under given condition?

OH

H+

X( most stable)

A) Delocalisation of charge B) 1, 2 shift of Ethyl group C) Aromatisation of ring in which ethyl substituent is present D) Aromatisation of ring in which ethyl substituent is absent

24. In which of the following reactions, a phenolic compound is formed as product ?


OH

OH

NaOH P

CH2-C-Cl

O

Cl

Q R SLiAlH4NaOH(1 eq.) (1 eq.)

select the correct statement(s)

A) ‘Q’ is O

O O

B) ‘S’ is

C) ‘Q’ is

OCl

OH

O

D) ‘R’ is 27. In the reaction the products are

OCH3HBr

A) OCH and H3 2

Br B)

Br and CH Br3

C) Br and CH OH3

D) OH and CH Br3


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26.

A) 3 3CH CO Na + CH3CH2Br B) (CH3)3CBr + 2 5C H O N a

C) C6H5ONa + CH3Br D) C6H5Br + 3CH O N a

A) B)

C) D) O

O

NH

OH

O

NHNaBH4

28. Which of the following will result in the formation of an ether as major product?

29. Select incorrect reaction(s):


30. Diethyl ether is prepared through

A) The reaction between ethanol and conc. 2 4H SO at 0140 C

B) The reaction between ethyl bromide and sodium ethoxide

C) heating of ethyl bromide with dry silver oxide

D) Heating ethanol with conc. 2 4H SO at 0170 C



3NaHCO

to liberate 2CO ?

A) B) C) D)

31. At room temperature how many of the following compounds can decompose


E) F) G)

OH

NO2

H)

2CH C OH

OH

O

(a) (b)

CH2OH

(c)

(d) (e)

OH

CH3 (f)

OH

OH

32. How many of the following compounds give Reimer-Tiemann reaction?


O

O

+

O

O

3 2

2

1. /2.AlCl CS

H O product mixture

How many compounds are exist in the product mixture?

How many of the following are more reactive than Benzene diazonium chloride?

34. The diazonium ions give diazo-coupling with phenol in the presence of dilute NaOH.


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33.

35. Number of bromine atoms in A are X, therefore X is

36. How many of the following ethers cannot be prepared by Williamson’s synthesis?

6 5 2 5 6 5 3 6 5 2 6 5 6 5 6 5

3 3 3 33 3, 3

, , , ,C H OC H C H OCH C H CH OC H C H OC HCH C O C CH CH C O CH

37.

2

2 2

FeH O

OH

3C H C lK O H

OHCHO

2 2(i)alkaline H O(ii)H

X

The number of hydroxyl groups present in the product X is/are


38.

Number of moles of HI consumed per mole of (A) is X. Identify the value of X

2 5

3

C H ClZnPhenol X YDust Anhydrous AlCl

(Monoalkylated product); the number of structural

benzenoid isomers of Y (including Y) are

Five membered ring is major product then the sum of degrees of alcoholic groups

present in P is

40.


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39.

MATHS Max Marks: 60

SECTION-1 (ONE OR MORE OPTIONS CORRECT TYPE)

This section contains 10 multiple choice equations. Each question has four choices (A) (B),(C) and (D) out of which ONE or MORE THAN ONE are correct. Marking scheme: +3 for correct answer, 0 if not attempted and 0 in all other cases.

region in which ‘P’ will move is bounded by parts of parabola/parabolas having

possible equations is/are

A) 2 2 2x a ay B) 2 22y ax a C) 2 2 2y a ax D) 2 2x a ay 42. Let 2 4y ax be a parabola and 2 2 2 0x y bx be a circle. If parabola and circle touch

each other externally then

A) 0, 0a b B) 0, 0a b C) 0, 0a b D) 0, 0a b

43. TP and TQ are tangents to parabola 2 4y x and normal’s at P and Q intersects at a point R on the curve. The locus of the centre of the circle circumscribing triangle TPQ is a parabola whose:

A) Vertex is 1,0 B) foot of directrix 7 ,08

C) Length of latus rectum 14

D) focus is 9 ,08

41. ‘P’ is a point which moves in XY-plane such that the point ‘P’ is nearer to the centre

of a square than any of the sides. The four vertices of the square are a, a . The


44. Variable chords of the parabola 2 4y ax subtend a right angle at the vertex. Then:

A) Locus of feet of the perpendicular from the vertex on these chords is a circle

B) Locus of the middle points of the chords is a parabola

C) Variable chords passes through a fixed point on the axis of the parabola

D) None

45. A parabola which touches x-axis at 1,0 and y-axis at 0, 2 . Then

A) Vertex is 16 2,25 25

B) Vertex is 16 2,5 5

C) Focus is 4 2,5 5

D) Focus is 4 2,25 25

46. Equation of chord joining at 1 3,A t B t on 2 4y ax subtends a right angle at 2C t

and 4D t on the same parabola. Then

A) 2 4

1 3

1t tt t

B) 2 4 1 3 0 t t t t C) 2 4

1 3

12

t tt t

D) 2 4 1 3 4 t t t t

47. If two distinct chords of a parabola 2 4y ax passing through the point , 2a a are

bisected by the line 1x y , then the length of the latus rectum cannot be

A) 5 B) 6 C) 9 D) 11


2 4y ax at a point P on it meet its axis

at points T and N respectively. The locus of the centroid of the triangle PTN is a

parabola whose 0a

A) Vertex is 2 ,03a

B) focus is 2 ,0a

C) Length of latus rectum is 43a D) directrix is

3ax

49. Normal’s drawn at points P,Q and R lying on the parabola 2 4y x intersect at 3,0 .

Then

A) Centroid of le PQR = 3 ,02

B) Circumcentre of le PQR = 5 ,03

C) Area of le PQR =2 D) Radius of circumcircle of le PQR is 52

50. Through a point 2,0P , tangents PQ and PR are drawn to the parabola 2 8y x . Two

circles each passing through the focus of the parabola and one touching parabola at

Q and other at R are drawn. Which of the following points(s) with respect to the

triangle PQR lie (s) on the common chord of the two circles?

A) Centroid B) orthocenter C) incentre D) Circumcentre

48. The tangent PT and normal PN to the parabola




51. The radius of circle passing through the focus of parabola 2 4x y and touches it at

6,9 is equal to 10 then ........

52. Abscissa of two points P and Q on the parabola 2 8y x are roots of

equation 2 17 11 0x x . Tangents at P and Q meet at point T. Distance of T from the

focus of the parabola.

53. A line ‘L’ with slope 2 cuts parabola 2 8y x to form a chord. If the midpoint of chord

lies on 4x then the equation of L is y=2x+k. Then .........3k

54. Let 1 2: 0 : 0L x y and L x y are tangent to a parabola whose focus is 1,2S . If the

length of latus-rectums of the parabola can be expressed as mn

(m and n are co

prime). Then the value of 4m n is…..... 55. A pair of variable straight lines 2 25 3 0x y xy R , cut 2 4y x at two points P

and Q. If the locus of the point of intersection of tangents to the given parabola at P

and Q is 12Kx K N . Then ........K


1 2 3, ,m m m are the slope of normal’s 1 2 3m m m drawn through the point 9, 6 to the

parabola 2 4 ; ijy x and A a is square matrix of order 3 such that

1..........

iji

if i ja Then A

m if i j

57. A tangent is drawn to the parabola 2 4y x at the point ‘P’ whose abscissa lies in the

interval 1,4 . The maximum possible area of the triangle formed by the tangent at ‘P’,

ordinate of the point ‘P; and the x-axis is equal to K. Then the value of ..........K is

58. If the co-ordinate of the point on the parabola 2 7 2y x x , which is nearest to the

straight line 3 3y x is represented as 1 2,K K . Then the value of 1 2 ..........2

K K

59. If the lengths of two perpendicular tangents from a point to a parabola are of 3 and 4

units and the length of latus rectum of the parabola is K units. Then ......K

. . .denotesG I F

60. Let S be the focus of the parabola 2 8y x and PQ be the common chord of the circle 2 2 2 4 0x y x y and the given parabola. The area of triangle PQS is…….


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56.

KEY SHEET

PHYSICS 1 ABD 2 AD 3 BCD 4 AB 5 BCD

6 AC 7 AB 8 BC 9 BC 10 BCD

11 4 12 2 13 4 14 2 15 2

16 2 17 1 18 2 19 1 20 2

CHEMISTRY 21 ABC 22 ACD 23 BD 24 ABCD 25 ABC

26 BCD 27 D 28 C 29 ABC 30 ABC

31 3 32 4 33 4 34 2 35 1

36 2 37 2 38 0 39 4 40 4

MATHS 41 ABC 42 AC 43 ABD 44 ABC 45 AC

46 AD 47 ABCD 48 ACD 49 CD 50 ABCD

51 5 52 7 53 2 54 7 55 5

56 8 57 4 58 5 59 4 60 4

Master JEE CLASSES Kukatpally, Hyderabad.

JEE-ADVANCE-2014-P1-Model Max.Marks:180

SOLTUIONS PHYSICS

1. Conceptual

2. Tv

410 39.8 10T

98T

T = Mg M = 10 , 0sin30T mg

20m

2mM

3. Conceptual

4. Conceptual

5. Displacement of image of particle A is in phase with the displacement of particle A

hence the displacement equation of particle A’ s image is

' 0.2 sin ..... 1AY t

Displacement of image of particle B is out of phase with the displacement of particle

B therefore the displacement of equation of particle 'B s image is

' 0.2sin ... 2AY t

Relative vertical displacement of A.w.r.t. to B is

' ' 'AB A BY Y Y

0.2sin 0.2sint t

0.4sin ... 3t

6. FV

2 21 12 2 CE MV I

7. Conceptual

8. Conceptual

9. Conceptual

10. Conceptual

11. T LVm

Page 2

Extension FLeAY

12. MxT gL

So, T Mgxv gxMLL

hence 0.5x 2n

13. 4 10 212.5 319.2 10n

4n

14. Conceptual

15. At any distance

2 2 22 2

m L x mT L x L xL L

But 2T v in rope frame

22 2 22

v L x

pulse will be meeting at a distance 2

AB from A.

12 2 12 1

2

AB

k ABAB

21 2k

16. 2

3 (10K)t dxv3 10 dt

0tL

00 0

10 3dx 3 10kt dt where t10K3

17. 64 6400 80 /310 10

TABv m sABAB

32380 10

TCDvCDDC

80 4 :120

vABvCD

Page 3

18. The linear mass density is

3

22

5 10 1.0 1050 10

kg/m

The wave speed is Tv

Thus, the tension is 2T v 2(1.0 10 ) 6400 64 N

The Young’s modulus is given by //

T AYL L

The extension is, therefore, TLLAY

6 9

64 0.501.0 10 16 10

2 mm.

19. At the highest point

max

2

3

4

4 2

2

mg mg m a

mg m A

gA

T

20. 1

11

1

sin 2;

1, 2,12 sin2 6

2, 2,

y t XA Ty A

t x radT

y A

12

1

12 sin

421

12

xt radT

xtT

2

1 2

1 2

18

124

24

t xT

x xsubstract

x x

CHEMISTRY 21. Conceptual

22. Conceptual

23. Conceptual

Page 4

24. Conceptual

25.

26.

O

OH

+

O

Cl

Cl

O

O Cl

O O

O

O

LiAlH4

OH

O CH2 CH2OH 27. Conceptual

28. Conceptual

29.

30. Conceptual

31. A)

OH

PKa 9.31 B)

Page 5

C) PKa 7.15 D)

OH

NO2

NO2

PKA 3.96

E) PKA 0.38 F) PKA 18

G) PKa 7.12 H) PKa 3.83

32. Conceptual

33. Conceptual

34. Electron withdrawing groups will increase the dizocupling

35. Conceptual

36. Conceptual

37. Conceptual

38. Conceptual

39.

40.

O

CH3

CH2

CH3

H

O+

CH3

CH2

CH3

H

CH3 CH3

OH+

+2H O

CH3 CH3

CH2OH

H

CH2OH

OH

CH3CH3

Page 6

MATHS 41. Let ‘O’ be the centre of square ABCD and

Let ,P x y

By the given data 2 2 2 2 2 2, ,x y a x x y a x x y a b and 2 2x y a y

42. For externally touching a and b must have same sign

43. Locus of centre of triangle circumscribing triangle TPQ is

2 1 12

1,01. . 42

y x

Vertex V

L L R a

44. Conceptual

45. Conceptual

46. 2 2 21 1 2 2 1, 2 , 2 , ,2A at at B at at P at at

0

1 3

490 1 1APBt t t t

Let 2 3,t t be the roots of (1)

2 4 1 3t t t t

2 4

2 4 1 31 3

2 4 1 3

1 4

4

t t and t t t tt tt t t t

47. Given point , 2P a a let other entity be 2 , 2Q at at

Midpoint of 2

, 12

a atPQ a t

lies as 1x y

2

1 12

a at a t

2 2 3 2 0at at a

Here

0 0,1

4 0, 4

a

a

48. Let ,G h k

2 2 22 2,3 3

at at a at ath k

Page 7

233 2

2kh a aa

2

293 2 9 4 3 24kh a k a h aa

Returns 2 4 23 3a ay x

2,03 3 3 3x a a aV directs x x

49. Normal at ‘t’ is 32y xt at at

(3,0) lies on it 3 0 0,1, 1t t t

1, 2 , 0,0 , 1, 2P Q R

1 1 4 02

Area

2 5 5,0 ,03 2 2

Centroid S R

50. Conceptual

51. Tangent at 6,9 is 3 9 0x y

Equation of circle is 2 26 9 3 9 0x y x y

0,1 lies on it 10

5 10 5circle is r

52. Use result .ST SP SQ

53. 4, y be the mid point

equation of chord is 1 11S S

21 14 16yy x y

Given slope =2 1 2y

54. Feet of the perpendicular from focus upon any tangent to parabola lies on the tangent

at the vertex.

Equation of 1SN is 1x y k

1,2 lies on it 1 3x y

Page 8

Solve 13 33 ,2 2

x y and x y N

Equation of 2SN is 2x y t

1,2 lies on it 2 1y x

Solve 21 11 0 ,

2 2y x and x y N

Equation of tangent at vertex = equation of 1 2N N

2 4 3 0x y

L.L.R = 4(distance from S(1,2) to tangent at vertex)

L.L.R = 2 8 3 4 3 644 16 2 5 5

55. Equation of a chord of contact is 1 0S

i.e. 1 12yy x x 1

1

2 1 12

yy xx

Horizontal curve 2 4 1y x with

2 1

1

24 02

yy xy xx

2 21 12 4 0y x y xy x

2 21 14 2 0 1x y xy y x

Equation of 2 25 2 3 0 2x xy y

1&2 are identical 1 1245 3

y x

1 112 5 12 5 125

x x x

56. Equation is 32y mx m m

9, 6 lies on it 3 7 6 0m m

Roots are -1,-2,3

2 1 1

1 1 1 81 1 3

A A

57. Area of triangle 2 31 2 2 22

PTQ t t t

Page 9

Given 2 1,4 2, 1 1,2t t

max 16 2for t

16 16 4K k

58. Parabola is 37 1 414

2 4 4x y

Slope of tangent parallel to given live =m=3

21 2

7 41. 2 , 2, 8 ,2 4

P C am am k k

1 2 52

k k

59. 2 22 22 2

2 2

1 19 16t ta at and a att t

2 9 4 1444 4 416 125

t a a

60. 22 ; 4x t y t , substitute in circle

4 2 24 16 4 16 0 0 1t t t t t or

0,0 ; 2, 4 ; 2,0P Q S

Area 1 0 8 42

units

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