Fulltest i\Main\Paper\Question\Paper Aits 2013 Ft i Jeem

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FULL TEST – I

Time Allotted: 3 Hours Maximum Marks: 432 P lease read the inst ruct ions care fu l ly. You are a l lo t ted 5 minutes

spec i f ica l ly for th is purpose. You are not a l lowed to leave the Examinat ion Hal l before the end of

the test .

INSTRUCTIONS

A. General Instructions

1. Attempt ALL the questions. Answers have to be marked on the OMR sheets. 2. This question paper contains Three Parts. 3. Part-I is Physics, Part-II is Chemistry and Part-III is Mathematics. 4. Each part has only one section: Section-A. 5. Rough spaces are provided for rough work inside the question paper. No additional sheets will be

provided for rough work. 6. Blank Papers, clip boards, log tables, slide rule, calculator, cellular phones, pagers and electronic

devices, in any form, are not allowed. B. Filling of OMR Sheet

1. Ensure matching of OMR sheet with the Question paper before you start marking your answers

on OMR sheet. 2. On the OMR sheet, darken the appropriate bubble with black pen for each character of your

Enrolment No. and write your Name, Test Centre and other details at the designated places. 3. OMR sheet contains alphabets, numerals & special characters for marking answers. C. Marking Scheme For All Three Parts. (i) Section-A (01 to 03 and 10 to 12) contains 6 multiple choice questions which have only one

correct answer. Each question carries +8 marks for correct answer and – 2 mark for wrong answer.

Section-A (04 to 09 and 13 to 30) contains 24 multiple choice questions which have only one

correct answer. Each question carries +4 marks for correct answer and – 1 mark for wrong answer.

Name of the Candidate

Enrolment No.

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AITS-FT-I-PCM-JEE(Main)/13


2

Useful Data

PHYSICS

Acceleration due to gravity g = 10 m/s2

Planck constant h = 6.6 ×10−34 J-s

Charge of electron e = 1.6 × 10−19 C

Mass of electron me = 9.1 × 10−31 kg

Permittivity of free space ε0 = 8.85 × 10−12 C2/N-m2

Density of water ρwater = 103 kg/m3

Atmospheric pressure Pa = 105 N/m2

Gas constant R = 8.314 J K−1 mol−1

CHEMISTRY

Gas Constant R = 8.314 J K−1 mol−1 = 0.0821 Lit atm K−1 mol−1 = 1.987 ≈ 2 Cal K−1 mol−1 Avogadro's Number Na = 6.023 × 1023 Planck’s constant h = 6.625 × 10−34 J⋅s = 6.625 × 10–27 erg⋅s 1 Faraday = 96500 coulomb 1 calorie = 4.2 joule 1 amu = 1.66 × 10–27 kg 1 eV = 1.6 × 10–19 J Atomic No: H=1, He = 2, Li=3, Be=4, B=5, C=6, N=7, O=8,

N=9, Na=11, Mg=12, Si=14, Al=13, P=15, S=16, Cl=17, Ar=18, K =19, Ca=20, Cr=24, Mn=25, Fe=26, Co=27, Ni=28, Cu = 29, Zn=30, As=33, Br=35, Ag=47, Sn=50, I=53, Xe=54, Ba=56, Pb=82, U=92.

Atomic masses: H=1, He=4, Li=7, Be=9, B=11, C=12, N=14, O=16, F=19, Na=23, Mg=24, Al = 27, Si=28, P=31, S=32, Cl=35.5, K=39, Ca=40, Cr=52, Mn=55, Fe=56, Co=59, Ni=58.7, Cu=63.5, Zn=65.4, As=75, Br=80, Ag=108, Sn=118.7, I=127, Xe=131, Ba=137, Pb=207, U=238.



3

PPhhyyssiiccss PART – I

SECTION – A

Single Correct Choice Type This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. Which of the following curves correctly represents the variation of capacitive reactance (XC) with

frequency n? (A)

XC

n

(B) XC

n (C)

XC

n

(D) XC

n 2. Three rods of the same mass are placed as shown in the figure. What

will be the co-ordinates of the center of mass of the system?

(A) (a/2, a/2) (B) a a,2 2

(C) 2a 2a,3 3

(D) a a,3 3

O X

Y

(0, a)

(a, 0)

Rough work



4

3. If switch S is closed at t = 0 then the time at which power supplied by battery is

equal to rate of energy storage in capacitor is (A) t = 0 (B) t = 4RC (C) t = 5RC (D) It never happens (except t → ∞) because resistor always consume energy

R C

Sε 4. Two conducting spheres of radii a and b are separated

by a large distance. The capacity of this system between points A and B is a A bB

(A) 041 1a b

πε

+ (B) 4πε0(a + b)

(C) 2πε0(a + b) (D) None of the above 5. Suppose Electric field between plate (A) and (B) is E1 and

between (B) and (C) is E2 then

(A) 1 1

2 2

EE

=

(B) 1 2

2 1

EE

=

(C) E1 1 > E2 2 (D) none of these

E1 E2

1 2 3++++++++++

−

−

l1 l2

++++++++++

−

−

A B C

6. In the given circuit diagram, the key K is switched on at t = 0.

The ratio of the current i through the cell at t = 0 and t = ∞ will be

(A) 3 : 1 (B) 1 : 3 (C) 1 : 2 (D) 2 : 1

L

ε K

A BC

6R i

R3R

Rough work



5

7. The density ρ of a liquid varies with depth h from the free surface as ρ = kh. A small body of density ρ1 is released from the surface of liquid. The body will

(A) come to a momentary rest at a depth 12kρ

from the free surface

(B) execute simple harmonic motion about a point at a depth 1

kρ

from the surface

(C) execute simple harmonic motion of amplitude 1

kρ

(D) all of the above 8. Dimensional formula of modulus of rigidity is (A) 2 1 2M L T− − (B) 2 2ML T− − (C) 1MLT− (D) 1 2ML T− − 9. At the center of a non-uniform ring of radius R, made up of two uniform halves of mass 2M and M

(G : Newton’s gravitational constant) (A) field and potential both are zero

(B) field is zero but potential is 3GMR

−

(C) field is zero but potential is GMR

−

(D) magnitude of field is 22GM

Rπ and potential is 3GM

R−

10. For a satellite of mass m orbiting the earth very close to earth’s surface (mass of

earth = M, radius of earth = R) total energy is (A) zero (B) greater than zero

(C) GMmR

(D) GMm2R

−

11. An ideal gas has initial volume V and pressure P. In doubling its volume the minimum work done

will be in the following process (of given processes) (A) Isobaric process (B) Isothermal process (C) Adiabatic process (D) None of the above

Rough work



6

12. A particle moving in the positive x-direction has initial velocity v0. The particle undergoes retardation kv2, where v is its instantaneous velocity. The velocity of the particle as a function of time is given by

(A) v = v0/(1 + kv0t) (B) 02vv

1 kt=

+

(C) 0vv

kt= (D) 0

2 20

vv

(1 k v t)=

+

13. Velocity of a particle at any instant is given by the equation ( )2ˆ ˆv 2t i 3t j= + m/s and radius of the

curvature of the path is 2m. Centripetal acceleration of the particle at t = 2 s is (A) 80 m/s2 (B) 160 m/s2 (C) 40 m/s2 (D) 100 m/s2

14. A car accelerates on a horizontal road due to the force exerted (A) by the engine of the car (B) by the driver of the car (C) due to the earth’s gravity (D) by the friction of the road 15. A current carrying wire frame is in the shape of digit eight (8). It is

carrying current i0. If the radius of each loop is R0, then the net magnetic dipole moment of the figure is

(A) 20 0(i R ) 2π

(B) zero (C) 2

0 0i 2 R× π

(D) 0 0i 4 R× π

R0

x-axis

y-axis

i0

Rough work



7

16. Suppose that a material emits X–rays of wavelengths: K K L, ,α β α

λ λ λ when it is excited by fast

moving electrons; the wavelengths corresponding to K , K , Lα β α X–rays of the material respectively. Then we can write :

(A) K K Lβ α αλ = λ + λ (B) K K Lβ α α

λ = λ + λ

(C) K K L

1 1 1

β α α= +

λ λ λ (D)

K K L

1 1 1

β α α

= +λ λ λ

.

17. An excited hydrogen atom in the state n (principal quantum number of electron : n) is moving with

a velocity v (v << c, the velocity of light in vacuum). If this atom emits a photon of frequency ν opposite to the direction of its motion, while making a transition to the ground state, then

(A) 0v1c

ν = ν +

(B) 0v1c

ν = ν −

(C) 2

0 2v1c

ν = ν −

(D) ν = ν0, independent of v

where ν0 is the frequency of radiation emitted when the atom is at rest. 18. A thin lens forms a virtual, diminished image of an object placed at 4 m from it. The size of image

is half of the object. Which one of the following statements is correct regarding the nature and focal length of the lens :

(A) diverging of focal length = 4 m (B) converging of focal length = 4 m (C) converging of focal length = 2 m (D) diverging of focal length = 2m 19. A thin convex lens of focal length 2 m has an object placed at a distance of 0.5 m from it. The

image formed is (A) Real, at a distance of 2/3 m from the lens (B) Virtual, at a distance of 2/3 m from the lens (C) Real, at a distance of 0.4 m from the lens (D) Virtual, at a distance of 0.4 m from the lens 20. The elongation of a freely hanging uniform steel rope, if its length is doubled, will increase in the

ratio of (A) 2 : 1 (B) 4 : 1 (C) 8 : 1 (D) 16 : 1

Rough work



8

21. Which one of the graphs shown in the option represents the

variation of potential difference V across the inductor L with time t, the key K being plugged at t = 0, in the circuit shown in adjacent figure ?

L

( )

R

K

V

(A)

V

t O

(B)

V

t O (C)

V

t O

(D)

V

t O 22. A uniform ring of mass m and radius R is doing uniform pure rolling

motion on a horizontal surface. The velocity of the centre of the ring is V0. The kinetic energy of the segment AOB is

(A) 2 20 0mV mV

2−

π (B)

2 20 0mV mV

2+

π

(C) 20mV

2 (D) 2

0mV .

O

A B C

D

V0

M, R

23. A rope of mass ‘m’ is looped in a circle of radius R and rotated with a constant angular velocity ω0

about its axis in gravity free space. Find the tension in the rope ? (A) 2

0T mR= ω (B) 2 m R πω2

(C) 20mR

T2ω

=π

(D) 204mRπω

Rough work



9

24. The displacement function of a oscillating body is given by x = 0.3 sin 10 t6π π +

where x and t

are measured in meter and second respectively (A) the period of oscillation is 5 s (B) the body starts its motion from the equilibrium (C) the minimum time the body takes to reach the equilibrium is 1/12 sec (D) all of these 25. A simple pendulum is attached to the roof of a stationary elevator. If the time period of oscillation

is T when it is stationary, what will be time period of oscillation when the elevator falls freely (A) 0 (B) T/2 (C) 2T (D) none of these 26. A uniform rope of mass 0.1 kg and length 2.45 m hangs from the ceiling. Time taken by a

transverse wave to travel the full length of the rope is (g = 9.8 m/s2) (A) 0.5 s (B) 1.0 s (C) 1.5 s (D) cannot be calculated until a mass is suspended at lower end of string. 27. A radar wave (speed 3 × 108 m/s) has a frequency 7.9 × 109 Hz, is reflected from an aeroplane,

shows a frequency difference of 2.7 × 103 Hz on the higher side (if we assume that Doppler’s effect can be applied on light wave similar as sound wave)

(A) velocity of aeroplane is 1.87 × 103 km/m (B) velocity of aeroplane in the line of sight is 1.87 × 103 km/m (C) velocity of aeroplane is 1.35 × 103 km/m (D) velocity of aeroplane in the line of sight is 1.35 × 103 km/m.

Rough work



10

28. An external agent moves the block m slowly from A to B, along a

smooth hill such that every time he applies the force tangentially. Find the work done by the agent in this interval.

(A) 2 2 2m g H

L (B)

2mgHL

(C) mg (H + L) (D) mgH

B

H

LA

SmoothHill

Fagent

29. An agent applies force of constant magnitude F0 always in the

tangential direction as shown in the figure. Find the speed of the bob when string becomes horizontal, assuming that it is at rest at its lowest point.

(A) ( )0F 2mgm

π − (B) g

(C) ( )0F 4mgm

π − (D) 0Fm

mF0

l

F0

F0

×

30. Three bodies A, B and C of masses m, m and 3 m respectively are

supplied heat at a constant rate. The change in temperature θ versus time t graph for A, B and C are shown by I, II and III respectively. If their specific heat capacities are SA, SB and SC respectively then which of the following relation is correct? (Initial temperature of body is 0ºC)

(A) SA > SB > SC (B) SB = SC < SA (C) SA = SB = SC (D) SB = SC > SA

θ

tπ/3 π/4 π/6

III

III

Rough work



11

CChheemmiissttrryy PART – II

SECTION – A

Single Correct Choice Type This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1.

H

HOOC H

COOH

( ) ( )4

Icold alkaline KMnO B→

( ) ( )4 3

IIOsO /NaHSO C→

( ) ( )3 2

IIIPhCO H/H O D→

( ) ( )2 5 2

IVMMPP C H OH/H O E+→

( ) ( )2 2 4

VH O OsO F+→

( ) ( )2 2 2

VIH O SeO G+→

Which of the following reaction will yield racemic mixture? (A) (I), (III), (V) (B) (I), (II), (V) (C) (I), (II), (IV) (D) (III), (IV), (V) 2. A gas behaving ideally was allowed to expand reversibly and adibatically to twice its volume. Its

initial temperature was 25oC and V5C R2

= . Calculate ∆H (Given 22/5 = 1.32)

(A) 1499.84 J (B) 2102.4 J (C) 8671.5 J (D) Zero 3. Which of the following ions are expected to be paramagnetic to the same extent? (I) [FeF6]-3 (II) [MnCl4]-2 (III) [NiCl4]-2 (A) I, II and III (B) I and II (C) I and III (D) II and III

Rough work



12

4.

( )

3H O2 5

ExcessC H MgBr Product

+

+ →O OO

Me The major product is:

(A) COOH

Me

O

H5C2

(B) COOH

Me

C2H5

O

(C) COOH

Me

OH

H5C2

C2H5

(D)COOH

Me

OHCH3

Me

5.

( ) ( ) ( )3

3 2

OLi in liquidNH Zn/H OA B C→ → +

NO2

The compounds (B) and (C) are respectively:

(A)

OO

H

NO2

H

H

O

H

O(B)

OO

H

NH2

H

H

O

H

O

(C)

O

NO2

O H

O

H

OH (D)

O

NH2

O H

O

H

OH

Rough work



13

6. Consider a reaction:

Me

Me

Me MeMe

( ) ( )2 2 7K Cr OH

B C+∆→ →

The product (C) is:

(A) COOH

COOH

Me MeMe

(B)

Me

O

Me Me

O

O

(C) Both (A) and (B) (D) None of these 7. Which among the following compounds will not give effervescence with sodium bicarbonate: (A) C6H5CO2H (B) C6H5SO3H (C) C6H5OH (D) picric acid 8. When glycolic acid is subjected to reduction with HI, the product formed is: (A) Acetic acid (B) Formic acid (C) Iodoacetic acid (D) None of these 9. Which one is the correct statement: (A) The boiling points of alkyl halides are more than those of the corresponding alkanes. (B) In water, solubility of C2H5OH > CH3OH > C6H5OH (C) C6H5NH2 is stronger base than NH3 (D) All of the above statements are correct

Rough work



14

10. Which of the following compounds can undergo nucleophilic substitution easily?

C

O

(I)

CC

OO

O

(II)

CCl3C

O

(III)

C

O

NH2

(IV)

(V)

C

OF

(A) Only (II) (B) (I), (II), (III) and (IV) (C) Only (IV) (D) (II), (III) and (V) 11. When conc. HNO3 acts on our skin, the skin becomes yellow, because (A) HNO3 acts as an oxidizing agent (B) HNO3 acts as a dehydration agent (C) Nitro-cellulose is formed (D) the proteins are converted into xanthoproteins 12. Which one of the following is a correct statement (A) When a liquid is transferred from a small container to large container at the same

temperature; vapour pressure remains constant (B) Viscosity increases with increase in temperature (C) Addition of detergent to water increases its surface tension (D) Surface tension increases with increase in temperature

Rough work



15

13. Consider the following reaction: ( ) ( )dry HI P

distillationBarium adipate A B+→ → Compound (B) is:

(A)

(B) OH

(C) OH

(D)

14. What is the de-Broglie wavelength associated with He atom at room temperature 25oC (A) 6.34 × 10-11 m (B) 6.34 × 10-8 m (C) 7.34 × 10-11 m (D) 7.34 × 10-8 m 15. Entropy of system depends upon (A) Volume and pressure only (B) Pressure and temperature only (C) Temperature and volume only (D) Pressure, volume and temperature 16. Which forces of attraction are responsible for liquefication of H2? (A) Coloumbic forces (B) Hydrogen bonding (C) Dipole forces (D) All of these 17. Which of the following salt is basic? (A) HOCl (B) NaOCl (C) NaHSO4 (D) NH4NO3 18. For the reaction:

( ) 34 2 36

K Fe CN Fe CO NO−

+ → + + The n-factor is: (A) 1 (B) 11 (C) 5/3 (D) 61 19. Dopping of AgCl crystals with CdCl2 results in: (A) Schottky defect (B) Frenkel defect (C) Substitutional cation vacancy (D) Formation of F-centres

Rough work



16

20. Freshly obtained SnO2 is added to water containing a little HCl. The solution obtained would be……..due to preferential absorption of ……ions.

(A) positively charged, H+ (B) negatively charged, 23SnO −

(C) positively charged, Sn+4 (D) positively charged Sn2+

21. The product formed, when Mg(NH4)PO4 is heated (A) Mg(NH4)2PO4 (B) MgO (C) PbO (D) Mg2P2O7 22. Carbon cannot be used in the reduction of Al2O3 because (A) the enthalpy of formation of CO2 is more than that of Al2O3 (B) pure carbon is not easily available (C) the enthalpy of formation of Al2O3 is very high (D) both (B) and (C) 23.

( )3 2 2SolubleA CH COOH B CO H O+ → + +

( )4 2 42B NH C O White ppt.+ →

A and B may contain (A) Ni2+ (B) Ba2+ (C) Sr2+ (D) Ca2+ 24. Racemic acid + optically active alcohol then the product will be: (A) Optically inactive mixture (B) Meso compound (C) Diastereomeric mixture (D) Racemic mixture

25. Bad conductor of electricity is (A) H2F2 (B) HCl (C) HBr (D) HI

Rough work



17

26. Cryolite is: (A) Na3AlF6 and is used in the electrolysis of alumina for decreasing electrical conductivity (B) Na3AlF6 and is used in electrolysis of alumina for lowering the melting point of alumina (C) Na3AlF6 and is used in the electrolytic purification of alumina (D) Na3AlF6 and is used in the electrolysis of alumina 27. An organic compound consumes 4 moles of periodic acid to form following compounds per mole

of the starting compound HCHO, 3HCOOH and CHOCOOH. The organic compound is: (A) Glucose (B) Fructose (C) Gluconic acid (D) Sorbitol 28. For which of the following species d – d transition does not account for its colour? (A) 2

2 7Cr O− (B) 24CrO−

(C) CrO2Cl2 (D) All of the above 29. Which one is the correct statement: (A) Lactose is a disaccharide and is a non reducing sugar (B) methyl-α-D-glucopyranoside has a acetal structure and a reducing sugar (C) α-D-glucopyranose has a hemiacetal structure and is reducing sugar (D) All of the above 30. Which one of the following is a correct relation on the basis of Bohr’s theory

(A) velocity of electron 2

1n

∝ (B) radius of orbit 2

Zn

∝

(C) frequency of revolution 3

1n

∝ (D) All of the above

Rough work



18

MMaatthheemmaattiiccss PART – III

SECTION – A

Single Correct Choice Type This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. The points represented by a , b , c , d are coplanar and ( ) ( )sinA a 2sin2B b+ +

( ) − =3sin3C c 4d 0 then, the least value of ( )2 2 221 sin A sin 2B sin 3C8

+ + is

(A) 1 (B) 2 (C) 4 (D) 3 2. A and B are two non–singular square matrices of each 3 × 3 such that AB = A and BA = B and

|A + B| ≠ 0 then (A) |A + B| = 0 (B) |A + B| = 8 (C) |A – B| = 1 (D) |A + B| = 2

3. For integer n > 1, the digit at units place in the number n

202

r 0

r! 2=

+∑ is

(A) 0 (B) 4 (C) 6 (D) 9

4. If ( )=

+ ≤ + + +∑4

2 2i i 1 3 2 4 2 3 1 4

i 1

x y 2x x 2x x 2y y 2y y then points (x1, y1), (x2, y2), (x3, y3), (x4, y4) are

(A) vertices of a rectangle (B) collinear (C) concyclic (D) none of these 5. If (a cos θ1, a sin θ1), (a cos θ2, a sin θ2), (a cos θ3, a sin θ3) represent the vertices of an

equilateral triangle inscribed in a circle, the (A) isec 0θ =∑ (B) θ =∑ isin 0

(C) θ =∑ itan 0 (D) θ =∑ icot 0

Rough work



19

6. If = + xy xlog

a bx then

23

2d yxdx

is equal to

(A) −

2dyx ydx

(B) ( )

2

2a x

a bx+

(C) −

2dy ydx

(D) +

2dyx ydx

7. If the line x + 2y + 4 = 0, cutting the ellipse + =2 2

2 2x y 1a b

in points whose eccentric angles are 30º

and 60º subtends right angle at the origin then its equation is

(A) + =2 2x y 1

8 4 (B) + =

2 2x y 116 4

(C) + =2 2x y 1

4 16 (D) none of these

8. If the two roots of the equation (c – 1)(x2 + x + 1)2 – (c + 1)(x4 + x2 + 1) = 0 are real and distinct

and −=

+1 xf(x)1 x

then ( )( ) + 1f f x f fx

is

(A) –c (B) c (C) 2c (D) –2c 9. Let W denotes the set of words in the English dictionary. Define the relation R by R = {(x, y) ∈ W

× W; the words x and y have as least one letter in common}. Then R is (A) reflexive, symmetric not transitive (B) reflexive, symmetric and transitive (C) reflexive, not symmetric, and transitive (D) Not reflexive, symmetric and transitive

10. If α and β are the roots of the equation tan x = 2x; then the value of 1

1

sin x sin x dx( , 0)−

α β α β >∫ is

(A) 0 (B) 1 (C) 2 (D) 3

Rough work



20

11. Let A = {1, 2, 3, 4, 5}. Then the number of one–one onto functions from A to A in which at least three elements have self image, is equal to

(A) 10 (B) 11 (C) 120 (D) 55 – 5! 12. Let f and g be two functions with f″ and g″ existing everywhere. If f(x)g(x) = 2 ∀ x and

f '(x), g'(x) 0≠ . Then f "(x) g"(x)f '(x) g'(x)

=

(A) has at least one real root (B) may have real roots (C) has at most one real root (D) has no real roots 13. If 2x + 3y = 7 and x – y = 1 are two normals of a parabola y2 = 4ax from a point then third normal

may be (A) x – 3y + 1 = 0 (B) x + 3y = 5 (C) x + 3y + 1 = 0 (D) x – 3y – 7 = 0 14. If [.] denotes the greatest integer function then domain of the real valued function

[ ]+ + −2x 1/ 2log x x 6 is

(A) ∪ ∞

1, 1 (1, )2

(B) ∪ ∞ 30, (2, )2

(C) ∪ ∞

3 , 2 (2, )2

(D) ( ] ∪ ∞

30, 1 , 2

15. The length of sub tangent, ordinate and subnormal to the parabola y2 = 4ax at a point (different

from origin) are in (A) AP (B) GP (C) HP (D) AGP

16. If p, q ∈ R and p2 + q2 – pq – p – q + 1 ≤ 0 then − −1 2p 3

3q 1 5pp q p

is equal to

(A) 1 (B) –1 (C) –2 (D) 0 17. If the standard deviation of x1, x2 ….., xn is 3.5, the standard deviation of –2x1 – 3, –2x2 – 3, ….., –2xn – 3 is (A) –7 (B) –4 (C) 7 (D) 1.75

Rough work



21

18. The line x 2 y 1 z 13 2 1− + −

= =−

intersects the curve = 2xy c , z = 0 if c is equal to

(A) ±1 (B) ± 13

(C) ± 5 (D) ±2 19. Sum of first n terms of an AP (having positive terms) is given by ( )( )n n nS 1 2T 1 T= + − (where Tn

is the nth term of the series). Then the value of 212T is

(A) 2 (B) 4 (C) 1 (D) 6 20. If p = ∆ABC is equilateral and q = each angle is 60º. Then symbolic form of statement (A) p ∨ q (B) p Λ q (C) p ⇒ q (D) p ⇔ q 21. The locus of the middle points of chords of hyperbola 3x2 – 2y2 + 4x – 6y = 0 parallel to y = 2x is (A) 3x – 4y = 4 (B) 3y – 4x + 4 = 0 (C) 4x – 4y = 3 (D) 3x – 4y = 2 22. If A and B represent the complex numbers z1 and z2 such that |z1 – z2| = |z1 + z2| the circumcentre

of ∆AOB, O being origin is

(A) 0 (B) 1 2z 2z3+

(C) 1 2z z2+

(D) 1 2z z2−

23. If ∠A = 90º in ∆ABC, then 1 1c btan tana b a c

− − + + + is equal to

(A) 0 (B) 1

(C) 4π (D)

6π

24. The number of real values of a such that a2 – 2a sin x + 1 = 0 for any x, is (A) 1 (B) 2 (C) 3 (D) 4

Rough work



22

25. ( )1/ 44

5x x dx

x−

∫ is equal to

(A) 5 / 4

34 11 c

15 x − +

(B) 5 / 4

34 11 c5 x − +

(C) 5 / 4

34 11 c

15 x + +

(D) 5 / 4

34 11 c5 x + +

26. If x, y, z are angles of a triangle and tan x = 2 cot y and tan x + tan y + tan z = 6 then value of z is

(A) n4π

π + ; n ∈ I (B) 1n tan 2−π + ; n ∈ I

(C) 1n tan 3−π + ; n ∈ I (D) nπ; n ∈ I

27. ( )3 3

x a

x xlim a 0a a−→

− > ; where [.] represent G.I.F.

(A) a2 – 3 (B) a2 – 1 (C) a2 (D) a2 + 1 28. Out of 40 consecutive integers two are chosen at random then the probability that their sum is

odd, is

(A) 1429

(B) 2039

(C) 12

(D) 13

29. The area bounded by the curves y = x2 – 2x + 2 and its inverse is given by

(A) 13

(B) 23

(C) 53

(D) 43

30. Given a triangle with sides 17, 15 and 19 cm long. If a circle touching two smaller sides has its

centre on the largest side, then the radius of the circle is

(A) 51 91 (B) 2 2 219 17 15+ −

(C) 51 9164

(D) 17 91

Rough work

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