AITS-PT-II

3

Physics

AITS-PT-II-(Paper-2)-PCM-JEE(Advanced)/16

PART – I SECTION – A

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE option is correct. 1.

Consider the shown circuit. The net current supplied as a function of time is (A) (2 2 sin t) A

1H

0.2 mF

(B) (2 2 cos t) A 100 

(C) (2 2 sin t) A (D) (2 2 cos t) A ~

~

1002 sin 100t

2.

The figure shows a system of 2 concentric spheres of radii r1 and r2 and kept at temperatures T1 and T2 respectively. The radial rate of flow of heat in a substance between the two co-centric spheres is proportional to: r1r2 (A) (B) (r2  r1 ) (r2  r1 ) (C)

3.

(r2  r1 ) r1r2

r  (D) log  2   r1 

1002 cos 100t

r1 r2

T1 T2

The rod is free to slide along the two horizontal parallel   rails; the two rails being maintained at a constant   potential difference by the ideal battery. The only    resistance is in the rod itself. A student performs an   experiment with uniform magnetic field B. In first case   when rod is kept at rest, the rod is flow to move with     uniform velocity after a sufficient time. Now in second   case when a force of F0 is applied along with first case, so as to retard the rod, its velocity changes and finally decreases by 20 cm/sec. Now if above experiment is repeated with magnetic field B/3 then velocity would have (A) increased by 60 cm/s (B) decreased by 180 cm/s (C) decreased by 60 cm/s (D) increased by 20/9 cm/s

       

       

B

Space for Rough work

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5.

The graph between two temperature scales A and B is shown in the figure. Between upper fixed point and lower fixed point there are 150 equal divisions on scale A and 100 on scale B. The relationship for conversion between the two scales is given by t  180 t (A) A  B 100 150 t A  30 tB (B)  150 100 tB  180 t (C)  A 150 100 tB  40 tA (D)  100 180

A particle of mass m and charge q is dropped from a height h. In whole space, there exists a constant magnetic field B as shown in figure. Find the minimum value of B such that the particle just grazes the ground. m g 2m g (A) (B) 2q h q h (C)

6.

m g 4q h

(D)

tB = 100 Temperature (B)

m, q

B h

m 2g q h

In the RC circuit shown, switch is closed at t = 0. The resistor in the circuit is very sensitive to heat and as current is flowing, its resistance increases linearly as R = R0 + kt. Find the ratio of heat developed to the work done by the battery upto the moment the resistance becomes thrice its original value. (kC = 1) (A) 1/3 (C) 5/3

tA = 150

4.

4

Temperature (A)


R

C

S V

(B) 2/3 (D) 7/3 Space for Rough work


5

7.


z

A solid conducting cylinder (radius a) is rotated about its symmetric axis at an angular speed . Find the magnitude of induced emf across A(a/2, 0, 0) and B(a/2, a/2, a/3) (m: mass of electron, e : charge of electron) (A)

ma2 2 16e

ma2 2 (C) 32e

(B)



ma2 2 24e

B A

ma2 2 (D) 8e

y

x

8.

An infinitely long wire with triangular cross section (equal sides a) carries a current I uniformly. The magnitude of the magnetic field at O is  I  I (A) 0 (B) 0 6a 4a 0I 0I (C) (D) 12a 3a

                                        O

9.

The potential due to point charge at a point A is 7V and the electric field there is 3 V/m. There is another point B such that electric field at this point has smaller magnitude as compared to that at point A. However, if magnitude of the charge is tripled, the electric field at B becomes 3 V/m. The potential at B now is closest to (A) 7/3 V (B) 7 V (C) 12 V (D) 21 V

10.

Three identical thin uniform charged filaments are fixed along the sides of a cube as shown in the figure. Length of each filament is  and line charge density on each of them is . Determine the magnitude of electric field at the centre of the cube. The cube is a geometrical construct and not made of any matter. (A)

 2 20  3

(C) zero

(B) (D)

 2 40  3  1 40  3




6

Comprehension Type This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only ONE is correct

Paragraph for Questions 11 & 12 One mole of helium gas follows cycle 1-2-3-1 shown in the diagram. During process 3-1, the internal energy (U) of the gas depends on its volume (V) as 2 U = bV , where b is a positive constant. If gas releases the amount of heat Q1 during process 3-1 and gas absorbs the amount of heat Q2 during process 123.

U 4U0

3 2

U0

1 V1

V

Read the passage carefully and answer the following questions 11.

12.

The value of ratio of volume of gas in state-3 (V3) to that of gas in state –1(V1) is 1 2 (A) (B) 2 1 1 (C) (D) none of these 3 The value of Q1 / Q2 is 8 9 (A) (B) 9 8 12 (C) (D) none of these 13 Space for Rough work


7


Paragraph for Questions 13 & 14 An Infinite sheet having surface charge density  is placed in front of a metallic charged sphere of charge Q and radius R, as shown in the figure such that surface of metallic plate is perpendicular to the line CO, where O is the centre of the plate. The separation between the sheet and the centre of the metallic sphere C is 2R. Answer the following questions based on the given situations.

13.

Metallic sphere haveing charge Q

P1

C R

P2

Electric field at point P2 (just outside the sphere)   (A) (B) 20 0 (C)

14.

+ + + + ++ + ++ + ++ + ++ +O + + + ++ + ++ + ++ + ++ + ++ + + 2R +

 kQ  20 R2

(D) greater than

 20

Electric field due to the metallic sphere at point C (A) zero (C)

 towards left 20

(B)

kQ

towards left R2  (D) towards right 20 Space for Rough work



8

Paragraph for Questions 15 & 16 When a current flows through the medium, an electric field exists as well as a potential which varies in space. Suppose that there is a break in a high – voltage transmission line and the free end of a wire of length L is lying on the ground. An electric current flows through the regions of soil adjoining the conductor. If a man happens to be walking near by a potential difference, which is called the step voltage appears between the points where his feet touch the ground. Consequently, an electric current whose strength depends on this potential difference flows through the man. Let us calculate the step voltage. Since the conductor is quite long, we assume that the current flows from it to the ground in a direction perpendicular to the conductor. The equipotential surfaces are the surfaces of semi-cylinders whose axes coincide with the conductor. Suppose that the man is walking in a direction perpendicular to the conductor with a step of length ‘b’ the distance between the conductor and the foot closer to it being d. Assuming that the current flows uniformly from the conductor over the semi cylindrical region we obtain the following expression for the current density at a distance r from the conductor: I j rL In this case, the field strength along the radii perpendicular to the conductor is j I Er    rL Consequently, the step voltage is db

Vst 

I

db . d 

 E dr  Lln  r

d

For example, If I = 500A, d = 1 m, b = 65 cm and L = 30 m we find that Vst = 270 V. Much higher voltages may appear under other conditions and other shapes of conductors. 15.

When a part of a high – voltage transmission line falls on the ground, it creates a hazard (A) because there can be a direct contact between the cable and a human being (B) because of the emergence of step voltages. (C) both (A) and (B) (D) none of (A) and (B)

16.

A hemispherical earth plate is buried into the earth in level with its surface. Find the voltage which may be applied to a woman approaching this earth plate (step voltage). The current passing through the earth plate is equal to 1 amp, the length of the step is b = 1m, and the distance between the plate, the foot closer to it is r0 = 2m and  = 102 /m. (A) 0.9 Volt (B) 1.8 Volt (C) 2.7 Volt (D) 3.6 Volt

I

r0

b

j



9


Match List Type (Only One Option Correct) This section contains four questions, each having two matching lists. Choices for the correct combination of elements from List-I and List-II are given as option (A), (B), (C) and (D) out of which one is correct. 17.

List – I contains four possibilities after removing an infinite cylinder of radius R/2 from an infinite cylinder of radius R and carrying current I of uniform density (I = jR2). List – II contains the magnitude of magnetic field at the centre (O) or a point at the perimeter (P) due to system in List – I. Match list-I with list –II and select the correct answer using the code given below the list List I List II (P) Axis of cavity lies on the 1. Zero axis of cylinder. The magnetic field on the axis of O cylinder (BO) is

(Q)

(R)

(S)

Axis of cavity lies on the axis of cylinder. The magnetic field on the surface of cylinder (BP) is

0I 4R

3.

30I 8R

4.

50I 12R

P

Cavity’s axis is R/2 away from the original cylinder’s axis. The magnetic field on the axis of cylinder (BO) is

Cavity’s axis is R/2 away from the original cylinder’s axis. The magnetic field on the surface of cylinder at P as shown

2.

O

P

Codes: (A) (B) (C) (D)

P 1 1 3 1

Q 3 3 1 2

R 4 2 2 3

S 2 4 4 3 Space for Rough work


10


18.

There are two types of rods: Rod 1: Length L, Thermal conductivity K, Area of cross section A. Rod 2: Length 2L, Thermal conductivity K, Area of cross section A. Four possible arrangements of these rods in steady state are shown in List – I and List – II gives the temperature of the junction. Match the List – I to List – II. List – I List - II (P) (1) 20C 10C

L 100C

L L 10C

(Q)

10C

(2)

27.5C

(3)

28C

(4)

40C

L 2L

100C

L 10C (R)

10C L

80C

L

L

10C

L 10C (S)

10C L L

2L

80C

10C

L 10C Codes: (A) (B) (C) (D)

P 4 3 4 4

Q 3 3 4 3

R 3 2 2 2



11

19.


From an insulating solid hemisphere (radius R) carrying a charge Q, a part of it is removed in the possibilities shown in List-I. List-II gives the magnitude electric field at the centre of the original solid hemisphere for the four combinations shown. Match List I with List II and select the correct  1  answer using the code given below the lists.  given K   40   List – I (P)

List - II (1)

0

(2)

3 KQ 4 R2

(3)

11 KQ 18 R2

(4)

1 KQ 2 R2

O R

(Q) O R

(R) O R/2 R

(S) O R/2 R


P 4 4 4 4

Q 2 3 3 1

R 3 2 1 2




20.

12

In the circuit shown, initially switch S1 is closed while S2 and S3 are open. After a long time, S1 is opened while S2 is closed. Further, after a long time, S2 is opened while S3 is closed. Consider the following parameters a: Total heat generated in 2R0 b : Total heat generated in 4R0 c : Final energy of C0 d : Final energy of 2C0 e : Final energy of 4C0

List – I (P) a/b (Q) b/c (R) c/d (S) d/e Codes: P (A) 4 (B) 3 (C) 1 (D) 3

R 3 1 2 2

R0

4C0

2R0

2C0

4R0

C0

S1

S2

S3

List - II (1) (2) (3) (4)

Q 2 2 4 4

V

2/9 1/2 5 1



13

Chemistry


PART – II

SECTION – A This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE option is correct. 1.

In the following sequence of reaction NC ZnBr2 BuLi PhCOCl    A    B    THF THF 100o C

Br

the major product (C) is: O (A) NC

2.

O

(B)

C

Ph C

Ph

O

O

Ph C

C

(C)

 C Major product 

Br

(D) NC

Ph

CN

In the following sequence of reaction OH O

O O

H2 /Pd   A  

B  Major product 

NO 2

the major product (B) is: O (A) O

C

(B)

OH

CH3

O HN

(C)

NH2 OH

(D)

OH

C

CH3

O C CH3

C NH2

CH3

O

NH2



14


3.

In the following sequence of reaction CH3

 H3C

C

CH2

FeCl3 Cl  

A Major product 

CH3 the major product (A) is: (A)

CH3 CH 2

C

(B)

CH3

CH3

C

CH3

(D)

CH3

CH

CH3 CH2

CH

CH2

HC CH3

CH3

4.

CH3

CH3

CH3 (C)

CH2

In which of the following reaction the major product formed is incorrect? (A)

H3C

CH2



C 2H5 O  CH3  CH2  CH  CH2 CH3  C2H5 OH

CH

Major product 

F

H3C

(B) H3C

C

Na/Liq.NH3

C

 CH3 

C

H C

H

CH3

Major product  (C)

CH3 H3C

H C

H (D)

Br2 /CCl4  

C

H H

CH3

CH3

Br Br

CH3 Major product  

COOH 1 CrO

Cl /Cs

2 2 2    2  H O 3

Major product  Space for Rough work


15

5.


In the following reaction O

Me N

O

C

H 

C



CN H  

 A Major product 

Me

the major product (A) is: (A)

Me N

O

O

C

C

Me

(B)

(C)

(D)

OH

O

N

CH

C

N

CH

CH

O

OH

C

CH

Me Me Me Me Me

N Me

6.

In the following reaction Ph O C



OH   

+

N

A Major product 

CH3 the major product (A) is: (A)

Ph

O C

(B)

N

C

N

CH3

(C)

Ph

O

CH3 Ph

O C

(D)

Ph

O C

NH




7.

16

In the following sequence of reactions OMe CH OH/H

H O

mCPBA 3 3   A   B  

C


the major product (C) is: (A)

(B)

OMe

OH

OH

(C)

(D)

O

O

O

OH

8. SOCl2 

A Major p r oduct

HO H H3C Which of the following statement is correct regarding this reaction? (A) The reaction takes place by SN1 mechanism. (B) Retention of configuration takes place. (C) In presence of pyridine retention of configuration takes place. (D) The major product (A) of the reaction is Space for Rough work


17

9.


In the following sequence of reaction O O CH3





 2 Ph3 P  CH2    A   B    ii H i I  KOH

C Major product

the major product (C) is (A)

CH3

(B)

CH2

(C)

O

(D)

CH3

O

CH2

10.

CH3

The major product (P) in the following sequence of reaction is: O

O  i H

LDA, THF   A  ii OH ,

Ph (A)

O

P Major product  (B) O

Ph

(C)

Ph

O

O

(D) Ph

Ph

CH3 Space for Rough work



18

Comprehension Type This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only ONE is correct Paragraph for Question Nos. 11 to 12 Addition of water to carbonyl compounds is a nucleophilic addition reaction. The reaction is reversible in nature. Most of the carbonyl compounds form unstable hydrates. However, certain carbonyl compounds form stable hydrates. The stability of hydrates depends upon certain factors like steric effect, intramolecular hydrogen bonding, ring stain, inductive effect, mesomeric effect etc. 11.

Which of the following compound form stable hydrate in aqueous solution? (A)

(B)

O

O

(C)

O

(D)

O C

H3C 12.

H

CH3

Which of the following compounds does not form stable hydrate? (A)

Cl3C – CHO

(C) Ph

O

(B)

O

O

O

C

C

C

CF 3

C O

CF 3

Ph

C

Ph

(D) Ph



19


Paragraph for Question Nos. 13 to 14 Nitrous acid is a weak acid and unstable. It is always prepared in situ, usually by treating NaNO2 with an aqueous solution of strong acid. HCl  aq  NaNO2  aq   HONO  aq  NaCl  aq 

H2 SO4  aq  2NaNO2  aq  2HONO  aq   Na2SO4  aq  Nitrous acid reacts with amines. The product that is obtained from these reaction depends upon the nature of amines. 13.

Which of the following compounds reacts with nitrous acid to form N-Nitrosoamine? NH2 HN CH3 (A) (B)

NH2

(C)

14.

(D)

H3C

CH3

N

The major product (P) formed in the following reaction is: H3C

N

CH3

NaNO2 ,HCl  

(A)

H3C

N

P Major product 

CH3

(B)

H3C

CH3

N

N

N

(C)

O

O

CH3 +

N

(D) N

H3C

CH3

N

O

CH3 +

N2 Cl



20


Paragraph for Question Nos. 15 to 16 Elimination reactions are the most important means for synthesising alkenes. In Saytzeff’s elimination the most substituted alkene is formed as major product where as in Hoffmann’s elimination the least substituted alkene is the major product. 15.

CH3 C 2H5 ONa  C H OH,  2 5

A


Cl H3C CH3 the major product (A) is: CH3 (A)

(C)

16.

CH3

(B)

CH3

(D)

CH3

In which of the following reaction 2-Butene is the major product? KOH alc. (A) H3C CH2 CH CH3   Major pr oducts 

Br

(B)

H3C

CH2

CH

 3 3 CH3   CH3 3 C  O   Major p r oducts 

(C)

H3C

CH2

Br CH

CH3

H3C

N

+

CH

C OH



OH   Major pr oducts 

CH3

CH3

(D)

H3C

CH2 H3C

CH +

N

CH3

   Major pr oducts

O

H3C



21


(Matching List Type) This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 17.

List-I (Reaction) O (P)

COCH3

List-II (Reagent used in the reaction) (1) m-CPBA

OH

COCH3

(Q)

O

(2)

O

Cold. Aq. KMnO4/ OH

O

(R)

(3)

NOH NH

Anhydrous AlCl3

O

(S)

(4)

HO

PCl5

OH


P 1 2 2 3

Q 2 1 4 1

R 3 4 1 4




18.

22

List-I (Reaction) (P)

List-II (Reagents used in the reaction) OH (1) (i) BH3 – THF (ii) H2 O2 / OH

(Q)

OH

(R)

OH

(2)

(i) OsO4/Pyridine (ii) NaHSO3/H2O

(3)

Dilute H2SO4

(4)

(i) Hg(OAc)2 – THF, H2O (ii) NaBH4/ OH

OH

(S)

OH


P 1 4 2 3

Q 2 1 4 2

R 3 2 1 4



23

19.

List – I (P) Terylene (Q) Nylon – 6, 6 (R) Teflon (S) Buna – N Codes: P Q (A) 1 2 (B) 2 1 (C) 2 4 (D) 3 2

20.

R 3 4 1 4


(1) (2) (3) (4)

List – II Hexamethylene diamine and adipic acid Ethylene glycol and terephthalic acid 1, 3-Butadiene and acrylonitrile Tetrafluoroethene

(1) (2) (3) (4)

List – II (Hydrolysis product) Glucose & Galactose Glucose, Galactose & Fructose Glucose & Fructose Only Glucose

S 4 3 3 1

List – I (P) (Q) (R) (S)

Sucrose Maltose Lactose Raffinose

(A) (B) (C) (D)

P 1 2 3 3

Codes: Q 2 1 4 2

R 3 4 1 4




Mathematics

24

PART – III SECTION – A Straight Objective Type

This section contains 10 multiple choice questions numbered 1 to 10. Each question has 4 choices (A), (B), (C) and (D), out of which only ONE is correct 1.

Let A be the point (8, 6) and D be the point (6, 4). What is the length of the shortest path ABCD, where B is point (x, 3) and C is a point (x, 0). This path consist of three connected segments, with the middle one vertical (A) 53 (B) 53  3 50 (C) (D) none of these 53  3

2.

Suppose points F1 and F2 are the left and right foci of the ellipse

x2 y 2   1 , respectively and 16 4

point P is on the line x  3y  8  2 3  0 . When F1PF2 reaches the maximum then the value of the ratio

PF1 is equal to PF2

(A) 1  2

(B)

2 1

(C)

(D)

3 1

3 1

3.

Let A be the point (–2, 2) and B be the point (3, 3). Number of the integral values of m if the line x + my + m = 0 intersect the line segment AB? (A) 0 (B) 2 (C) 3 (D) none of these

4.

The graph of y = sin x, y = cos x, y = tan x and y = cosec x are drawn on the same axes from  x = 0 to x  . A vertical line is drawn through the point where the graph of y = cos x and y = tan 2 x cross, intersecting the other two graphs at point A and B. The length of line segment AB is 1 (A) 0 (B) 2 (C) 1 (D) 2 Space for rough work


25


5.

Tangent are drawn from (–4, 0) the circle x2 + y2 – 6x – 10y + 32 = 0 which touches it at point A having integral coordinate (both abscissa and ordinate are integer). Two points B and C are taken on the circle whose distance from A is 2 units, then the common chord of member of the system of circles passing through points B and C and the circle x 2 + y2 – 2x + 4y – 14 = 0 passes through a fixed point, that fixed point is (A) (2, 3) (B) (2, 4) (C) (1, 3) (D) 4, 6)

6.

A parabola touches the bisectors of the angles between the lines 7x – y + 3 = 0 and x + y – 3 = 0 at the points (3, 4) and (2, –3), then the length of latus rectum of the parabola is 16 2 (A) 16 2 (B) 5 (C) 4 2

(D)

4 2 5

7.

P lies inside the triangle ABC, right angle at B and each side subtend an angle 120º at P. If PA = 10, PB = 6, then PC is equal to (A) 11 (B) 22 (C) 33 (D) 44

8.

If p, q, r are the length of perpendicular from the vertices of a ABC upon a straight line meeting the sides AB, AC and CB externally in D, E and F respectively (a, b, c are length of sides BC, AC and AB), then a2(p – q)(p – r) + b2(q – r)(q – p) + c2(r – p)(r – q) is equal to (A) 2 (B) 22 2 (C) 3 (D) 42

9.

10.

 1 The number of x-intercept on the graph of y  sin   in the interval (0.001, 0.01), is x (A) 283 (B) 285 (C) 287 (D) cannot be determined If the lines 2x + y + 5 = 0 and 2px + y + 1 = 0 are the asymptotes of a hyperbola, the possible value of p are (A) R (B) R – {1} (C) R – {2} (D) R – {3} (Where R is set of real numbers) Space for rough work



26

Comprehension Type This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only ONE is correct Paragraph for Question Nos. 11 to 12 Read the following write up carefully and answer the following questions: Let A(4, –2) and B(0, 1) are two points on the parabola, normals at which intersect at the point P(2, 2) 11.

12.

Equation of directrix of the parabola is (A) y + x + 1 = 0 (C) y = –3

(B) x = 2 (D) 2x + y – 1= 0

Focus of the parabola is 6 6 (A)  ,   5 5  12 6  (C)  ,   5 5

 16 7  (B)  ,   5  5  6 6 (D)   ,   5 5 Paragraph for Question Nos. 13 to 14

Read the following write up carefully and answer the following questions: Let ABCD be any quadrilateral and E, F, G, H are mid-points of the sides AB, BC, CD and DA respectively. The bisectors of the internal angles of quadrilateral EFGH form a quadrilateral PQRS. If coordinates of E, F and G are (1, 2), (2, 1) and (3, 8) respectively, then 13.

14.

Area of quadrilateral PQRS is 61 (A) sq. units 5 63 (C) sq. units 5

62 sq. units 5 64 (D) sq. units 5

(B)

Equation of the circumcircle of quadrilateral PQRS is (A) x2 + y2 – 4x – 10y + 21 = 0 (B) x2 + y2 – 4x – 8y + 12 = 0 2 2 (C) x + y – 4x – 10y + 25 = 0 (D) x2 + y2 – 4x – 8y + 16 = 0 Space for rough work


27


Paragraph for Question Nos. 15 to 16 Read the following write up carefully and answer the following questions: Let D, E, F be the mid-points of the sides BC, CA and AB respectively of a ABC and C1, C2 be the circumcircles of ABC and DEF respectively 15.

If C1 and C2 touch each other then cos 2A + cos 2B + cos 2C is equal to 1 (A) 0 (B) 2 (C) 1 (D) –1

16.

If C1 and C2 intersect orthogonally, then cos 2A + cos 2B + cos 2C is equal to 1 (A) 0 (B) 2 (C) 1 (D) –1 (Match List Type)

This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which only ONE is correct 17.

Match the following List-I with List-II (P)

(Q)

(R)

(S)

List – I Normals are drawn at points P and Q to the parabola y2 = 4x, which meet at a point R(9, 6) on the parabola. Tangents at P and Q to the parabola meet at T. Then the length of the tangents drawn from (–1, –1) to the circumcircle of the quadrilateral PTQR is The radius of smaller circle that touches the parabola 6 8 75y2 = 64(5x – 3) at the point P  ,  and the x-axis is 5 5 If P(, ) lies inside or on the triangle formed by the lines 2x + y = 2, x – y = 1 and x + 2y = 4, then the least value of 3 + 2 is A tangent is drawn at any point (x 1, y1) on the parabola y2 = 4ax from any point on this tangent, tangents are drawn to the circle x2 + y2 = a2. The chord of contact pass through a fixed point 2

List – II

1.

1

2.

2

3.

3

4.

4

y  x  (x2, y2) if  1   2  1  ,  > 0, then the value of  is  y2   x2  Codes: (A) (B) (C) (D)

P 4 1 4 4

Q 1 2 3 1

R 3 3 2 2

S 2 4 1 3 Space for rough work



18.

28

Match the following List-I with List-II (P)

(Q) (R)

(S)

List – I 2 2 If the circle x + y + 2gx + 2fy + c = 0 is touched by the line y = x at point p such that OP = 6 2 , where O is origin then the value of c is Number of integers in the range of 3 2 y = cos  – 6 cos  + 11 cos  – 6 is If tangent at point P(4t2 , 8t3) to the curve x 3 = y2 is also normal 6 to the curve at point Q, then the value of 2 is t Let (sin x – cos x)(tan x + cot x) = 2 and (sin x + cos x)(tan x – cot x) = a  b , (a, b  N), then 2 2 |a – b | is equal to

List – II 1.

72

2.

25

3.

27

4.

24

Codes: (A) (B) (C) (D) 19.

P 4 1 2 1

Q 3 2 3 2

R 2 4 4 3

S 1 3 1 4

Match the following List-I with List-II

(P) (Q) (R)

List – I  r  r  If in a triangle ABC,  1  1  1  1   2 the ABC is r r  2  3  B C If in a ABC, c  a  b  cos  b  a  c  cos , then ABC is 2 2 2 2 sin  A  B  a b  If in a ABC, 2 , then ABC is 2 sin  A  B  a b acos2

(S)

If in a ABC,

A B C  bcos 2  c cos2 2 2 2  3 , then ABC is abc 4

List – II 1.

equilateral

2.

right-angled

3.

isosceles

4.

either right angle or isosceles


P 4 1 2 2

Q 3 2 3 3

R 2 3 1 4



29

20.


Consider the hyperbola 9x2 – 4y2 – 126x – 24y + 369 = 0. A variable point P(t + 7, t2 – 4),  t  R exists in xy-plane. Let HL and HR be the left and right branches of given hyperbola, then match the following List-I with List-II List – I

List – II 1.

(P)

The set of values of t for which two distinct real tangents can be drawn to HL from P

(Q)

The set of values of t for which real tangents can be drawn to both HL and HR from P

(R)

The set of values of t for which only one real tangent can be drawn to HL only from point P The set of values of t for which two real and distinct tangents can be drawn to hyperbola from point P

(S)

1 1 ,    2,    2 2 1   2,  2     1    ,  2  2 

 ,  2     2. 3. 4.

1 1   R   2,  , , 2  2 2  


P 2 1 2 4

Q 1 2 1 3

R 4 3 3 2



AITS-PT-II

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