Sri Chaitanya · 2020. 4. 10. · Sri Chaitanya uestionIIT Academy Q 1Paper-_Key & Solutions Sri...

Sri Chaitanya IIT Academy Question Paper-1_Key & Solutions

Sri Chaitanya IIT Academy # 304, Kasetty Heights, Ayyappa Society, Madhapur, Hyderabad – 500081 www. srichaitanya.net, [email protected]

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PHYSICS SECTION-I

SINGLE ANSWER TYPE SECTION 1 (Maximum Marks: 12) • This section contains FOUR (04) questions. • Each question has FOUR options. ONLY ONE of these four options is the correct answer. • For each question, choose the option corresponding to the correct answer. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct option is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : —1 In all other cases. Question ID: 337911112 1. In a radioactive sample, 40

19 K nuclei either decay into stable 4020Ca nuclei with decay

constant 104.5 10 per year of into stable 4018 Ar nuclei with decay constant 100.5 10 per

year. Given that in this sample all the stable 4020Ca and 40

18 Ar nuclei are produced by the 4019 K nuclei only. In time 910t years, if the ratio of the sum of stable 40

20Ca and 4018 Ar

nuclei to the radioactive 4019 K nuclei is 99, the value of r will be [Given: ln10 2.3 ]

1) 4.6 2) 2.3 3) 9.2 4) 1.15 Ans: 3 Sol: 10 10

1 2 4.5 0.5 10 5 10eff per year

0

0

lt

N N e and given that 0

100NN 00

0100tN N e

020

2 1010 t ne t

90 10

2 2.3 9.2 105 10

t

Years 910t Years

Question ID: 337911109 2. Consider a spherical gaseous cloud of mass density r in a free space where r is the

radial distance from its centre. The gaseous cloud is made of particles of equal mass m moving in circular orbits about their common centre with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If r is constant with time. The particle number density n r = r /m is:

[ G is universal gravitational constant]

1) 2 2

Kr m G

2) 2 2

3Kr m G

3) 2 26K

r m G 4) 2 22

Kr m G

Ans: 4

Sol: M(r) = mass of gas in a sphere of radius r. Can any particle, g cpF F 2

2

GM r m mvr r

vm

r

GM (r) m = 2(K) r 2KM r rGm



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On differentiating w.r.t r, 2dM Kdr Gm

................ (1)

Here dM = mass of gas in a spherical shell of radius r and thickness dr. 24dM r dv r r dr ............... (2) Substitute (2) in (1)

24 2r r dr Kdr Gm

22Krr Gm

2 22

r Kn rm r Gm

Question ID: 337911111 3. A current carrying wire heats a metal rod. The wire provides a constant power P to the

rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as T(t)=T0 1/41 t

Where is a constant with appropriate dimensions while T0 is a constant with dimensions of temperature. The heat capacity of metal is

1) 30

4 40

4P T t TT

2) 04 2

0

4P T t TT 3) 2

04 3

0

4P T t TT

4) 40

4 50

4P T t TT

Ans: 1 Sol: dTP ms

dt 1/4

0 1dH T tdt

3/40

4HT t ................ (1) 1/4

0 1T T t

3

1/4 1/ 4 3/40 0

0 0 0

1T TT t t t

T T T T T

.............. (2)

Substitute (2) in (1)

33 300 0

3 4 400

44

P T THT TP HTT T

Question ID: 337911110 4. A thin spherical insulating shell of radius R carries a uniformly distributed charge such

that the potential at its surface is V0. A hole with a small area 24 1R is made on the shell without affecting the rest of the shell. Which one of the following statement(s) is correct?

1) The magnitude of electric field at the centre of the shell is reduced by 0

2VR

2) The ratio of the potential at the centre of the shell to that of the point at 12

R from

centre towards the hole will be 11 2

3) The magnitude of electric field at a point, located on a line passing through the hole and shell’s center, on a distance 2R from the center of the spherical shell will be

reduced by 0

2VR

4) The potential at the center of the shell is reduced by 02 V



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Ans: 2

Sol: 2

04KQ K rv

R R

At center, 1 0v v - (potential due to removed element)

20 0 4Kq Kv v R

R R 0 0v v 0 1v

At mid point between center and hole is

2 0 0 0 02 1 2/ 2

kqV V V V VR

01

2 0

1 1 21 2 1 2

VVV V

is correctOption A : Einitial =0

Efinal = 2

02

4z

Kq k R VR R R

0

i fVE E

R

i.e E will decrease by 0VR i.e. (C) is wrong

Option D: Vinitial = V0Vfinal 2

0 0 04K RV V VR

Vcenter decreases by 0V . i.e. (D) is wrong

Ans: 2

SECTION-II ONE OR MORE TYPE

(Maximum Marks: 32) • This section contains EIGHT (08) questions. • Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s). • For each question, choose the option(s) corresponding to (all) the correct answer(s). • Answer to each question will be evaluated according to the following marking scheme Full Marks : +4 If only (all) the correct option(s) is(are) chosen; Partial Marks : +3 If all the four options are correct but ONLY three options are chosen; Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct; Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : —1 In all other cases. • For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (B) will get +2 marks; choosing ONLY (A) and (D) will get +2 marks; choosing ONLY (B) and (D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY (B) will get +1 mark; choosing ONLY (D) will get +1 mark; choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get —1 mark.



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Question ID: 337911118 1. A thin convex lens is made of two materials with refractive indices n1 and n2, as shown

in figure. The radius of curvature of the left and right spherical surface are equal. f is the focal length of the lens when 1 2n n n . The focal length is f f when

1n n and 2n n n . Assuming 1n n and 1 2n , the correct statement(s) is/are,

1n 2n

1) If 0nn

then 0ff

2) For n=1.5, 310n and 20f cm, the value of f will be 0.02cm (rounded off to 2nd decimal place).

3) The relation between ff and n

n remains unchanged if both the convex surfaces are

replaced by concave surface of the same radius of curvature.

4) f nf n

Ans: 1,2,3

Sol: 1 2

n n+dn

____ 12 1

Rfn

In 2nd case, 11 1 n dnnf df R R

1 1 dn

f df f R

Clearly, if 0,dn R.H.S value is less than 1f

0df i.e (1) is correct

Option(2)

20

2 1.5 1Rf R cm

, 1 1 dnf df f R



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1 120 20 20

dndf

= 1 1 12020 1

20

dndf

1 1 2020df dn df dn 320 10 cm

0.02df cm (2) is correct Option(3): If convex surfaces are replaced with concave surfaces, sign of R will change

1 1

1

dnf Rdff

f

1

2 1dn

f f n

2 11 1

2 1

Rfn

R f n

1

1 112 1

dfdnf

f f nn

112 1

df dnf n

n

As this is independent of R, (3) is

correct. Option (4): 112 1

df dnf n

n

For the range 1 < n < 2, 12 1n

is always

greater than 1 i.e, df dnf n

(4) is wrong

Question ID: 337911118 2. A charged shell of radius R carries a total charge Q. Given as the flux of electric field

through a closed cylindrical surface of height h, radius r and with its center same as that

of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is

equidistant from its top and bottom surfaces. Which of the following options is/are

correct?

[ 0 is the permittivity of free space]

1) If 8 / 5h R and 3 / 5r R then 0

2) If 2h R and r R then 0/Q

3) If 2h R and 3 / 5r R then 0/ 5Q

4) If 2h R and 4 / 5r R then 0/ 5Q

Ans: 1,2,3



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Sol. Option-1: If 85Rh and 3

5Rr , the cylindrical surface just fits into the sphere. If

85Rh , the cylindrical surface will be well inside the sphere.

0 0

0 0inq

(1) is correct

35R

45R R

+ +

+

+

++

+

+

Option -2: If h>2R and r>R the sphere will be well inside the cylinder.

0

Q

(2) is correct

++

+ +++

+

r R

2h RR

Option -3: 2in ABq Q 1 cos 22Q

415 5

QQ

0 05inq Q

(3) is correct

2h R

+

+

++

+

+

+

+

++'A

R

3 / 5r RA B

'B

Option -4: 21 cos53º 2 52in

Q Qq 0

25

QQ

(4) is wrong



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Question ID: 337911120 3. Two identical moving coil galvanometers have 10 resistance and full scale deflection

at 2 A current. One of them is converted into a voltmeter of 100mV full scale reading and the order into an Ammeter of 1mA full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm’s law experiment with R=1000 resistor by using an ideal cell. Which of the following statement(s) is/are correct?

1) If the ideal cell is replaced by a cell having internal resistance of 5 then the measured value of R will be more than 1000

2) The measured value of R will be 978 982R 3) The resistance of the Voltmeter will be 100k 4) The resistance of the Ammeter will be 0.02 (round off to 2nd decimal place) Ans: 2,4 Sol: Option 4: g AI G I R 6 32 10 10 1 10 AR

Resistance of ammeter 0.02AR (4) is correct

Option 3: g VV I R1

46

10 5 102 10V

VRIg

50k (3) is wrong

Option 2:

A

1V 2V

RAR

VR

E

V

1 2. ,V

AV

R RR R RR R

Reading of ammeter = I

Reading of voltmeter V = IR

Measured value of resistance is 1mVR RI

3 4

3 4

. 10 5 1010 5 10

V

V

R RR R

45 10M

980.4 2 is correct

Option 1: measured value of R depends on R and RV only, but independent of RA, r

values. Measured value of resistance will again be 980.4 only



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Question ID: 337911114 4. A conducting wire of parabolic shape, initially 2y x is moving with velocity

0V V i in a non – uniform magnetic field

0 1 yB B kL

, as shown in figure. If

0 0. ,V B L and are positive constants and is the potential difference developed between the ends of the wire, then the correct statement(s) is/are

y

x

L

0L

0V V i

B

1) 0 0

12

B V L for 0

2) is proportional to the length of the wire projected on the y-axis. 3) remains the same if the parabola wire is replaced by a straight wire, y=x initially,

of length 2L

4) 0 043

B V L for 2

Ans: 2,3,4

Sol: 0 0 1 .ByV B dy

,

1

0 0

0

4ByB V

, 0 0 1B V

0 021

B V

0 00 2 V , 0 0423

B V

Question ID: 337911119

5. Let consider a system of units in which mass and angular momentum are dimensionless.

If length has dimensions of L, which of the following statement(s) is/are correct?

1) The dimension of linear momentum is 1L

2) The dimension of power is 5L

3) The dimension of energy 2L

4) The dimension of force is 3L

Ans: 1,3,4

Sol: º º ºmass M L T º º ºAngular momentum M L T



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. . º º ºr M V M L T º º ºm rV M L T

1º º º ' ' º º ºM L T L L T M L T 2 1L T

11P mv m v Lr

224

21 11

2mvpower m L

t r r

2

2 21 112 2

Energy mv M Lr

3

21 12

mvForce m Lt L

Question ID: 337911117 6. One mole of a monatomic ideal gas goes through a thermodynamics cycle, as shown in

the volume versus temperature V T diagram. The correct statement(s) is/are: [R is the gas constant]

1

23

4

V

02V

0V

0 / 2T 0T 03 / 2T 02TT

1) Work done in this thermodynamic cycle 1 2 3 4 1 is 0

12

W RT

2) The ratio of heat transfer during processes 1 2 and 2 3 is 1 2

2 3

53

QQ

3) The ratio of heat transfer during processes 1 2 and 3 4 is 1 2

3 4

12

QQ

4) The above thermodynamic cycle exhibits only isochoric and adiabatic processes. Ans: 1,2 Sol: i W Area

0

0

nRTV

0

02nRT

V

1 2

34

1n v

p

00

02nRT V

V

01

2 RT 12

13

53

p

V

nC TQii

Q nC T

112

34 2

2p

p

nC TQiiiQ nC T

iv isochoric and adiabatic



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Question ID: 337911113 7. A cylindrical capillary tube of 0.2mm radius is made by joining two capillaries 1T and

2T of different materials having water contact angles of 0º and 60º, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct?

[ surface tension of water =0.075N/m, density of water =1000kg/m3, take g=10m/s2]

1) For case II, if the capillary joint is 5cm above the water surface, the height of water

column raised in the tube will be 3.75cm. (Neglect the weight of the water in the meniscus)

2) For case I, if the capillary joint is 5cm above the water surface, the height of water column raised in the tube will be more than 8.75cm. (Neglect the weight of the water in the meniscus)

3) The correction in the height of water column raised in the tube, due to weight of water contained in the meniscus, will be different for both cases.

4) For case I, if the joint is kept at 8cm above the water surface, the height of water column in the tube will be 7.5cm. (Neglect the weight of the water in the meniscus)

Ans: 1,3,4

Sol: 1 20º 60º 1 3 3

2 0.075cos0º 0.075 7.50.2 10 10 10

h m cm

2 3 32 0.075 cos60º 3.750.2 10 10 10

h cm

Case -1 : If point is 5 cm above (5 Cm < 7.5 cm)

h = 5cm + 3.75 cm = 8.75 case 2 : 5 cm > 3.75 cm h = 3.75 cm Question ID: 337911115 8. In the circuit shown, initially there is no charge on capacitors and keys S1 and S2 are

open. The values of the capacitors are 1 210 , 30C F C F and 3 4 80C C F .

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



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1) If key S1 is kept closed for long time such that capacitors are fully charged, the voltage across the capacitor C1 will be 4V.

2) At time t=0, the key S1 is closed, the instantaneous current in the closed circuit will be 25mA.

3. If key S1 is kept closed for long time such that capacitors are fully charged, the voltage difference between points P and Q will be 10V.

4) The key S1 is kept closed for long time such that capacitors are fully charged. Now key S2 is closed, at this time, the instantaneous current across 30 resistor (between points P and Q) will be 0.2A (round off to 1st decimal place)

Ans: 1, 2, 4

Sol: C net = 40 10 8 4040 10

F Q C

1

2V

100

3010v

70

4v

12V 30

V across 1

40 410

CC VF

(at steady state)

Just after S1 is closed, 5 2530 70 100

i mA

Long time after S1 is closed, 1

4PQ CV V V Just after 2S is closed

100

1 2i i1i 2i

1 210 30 4 70 , 0i i i 1 2 2 2

1 110 30 5 30 100 02 2i i i i 2 130 100 6i i 2 160 30 6i i 1 2 0.05i i A

SECTION-III DECIMAL INTEGER TYPE

(Maximum Marks: 18) • This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. • For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct numerical value is entered; Zero Marks : 0 In all other cases.



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Question ID: 337911122 1. A block of weight 100N is suspended by copper and steel wires of same cross

sectional area 0.5cm2 and, length 3 m and 1m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30º and 60º, respectively. If elongation in copper wire is Cl and

elongation in steel wire is Sl , then the ratio C

S

ll

is___

[Young’s modulus for copper and steel are 11 21 10 /N m and 11 22 10 /N m , respectively.]

Ans: 2

Sol:

cos60ST cos 30CuT

60º 30sin60ST

sin30CuTST

3060

100

CuT

cos60 cos30S cellT T 32 2S

CuT T

3S CuT T sin 60 sin30 100S CuT T

33 1002 2

CuCu

TT 2 100CuT

100CuT 50CuT N

3 50ST N Elongation in Cu wire Cu Cu

Cu r cel

l Tl A y

CuCu Cu

Cu

Tl lAy

elongation in steel wire

s S

S

l TlsA y

C Cu

C Cu Cu Custell

S SS Cu S S

stell

l Tl A y l T Yl Tl y l T

A y

11

3 50 2 101 10 1 3 50

=2



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Question ID: 337911126 2. A planar structure of length L and width W is made of two different optical media of

refractive indices 1 1.5n and 2 1.44n as shown in figure. If L W , a ray entering from end AB will emerge end CD only if the total internal reflection condition is met inside the structure. For 9.6L m , if the incident angle is varied, the maximum time taken by a ray to exit the plane CD is 910t s, where t is___

Ans: 50 (49 - 51)

Sol:

d C

x

2 4 4K MnO KMnO

w X

24MnO

Y

4MnO

Z

2425

xd

2524

xd distance travelled by light ray 25 9.624

10m

Time taken by ray to travel=1speedin med

dn

5

10 1.5 103 10

1.5C

85 10 S . 950 10 S

Question ID: 337911123 3. A train S1, moving with a uniform velocity of 108 km/h approaches another train S2

standing on a platform. An observer O moves with a uniform velocity of 36km/h towards S2 as shown in figure. Both the trains are blowing whistles of same frequency 120Hz. When O is 600m away from S2 and distance between S1 and S2 is 800m, the number of beats heard by O is_____

[ Speed of the sound =330m/s]

600m

108 km/hS1S2

O36 k

m/h

800m



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Ans: 8.12 (7.00 -10.00) Sol:

2S2 1S

1

800

600

1800 4tan600 3

1 53º 2 37º Freq. Heared form train at 2 2S f

02

V Vf fsV

330 10330 sf

340 1.030330 s sf f

freq. Heared from train at 1 1S f 0 11

2

coscos s

s

V Vf fV V

3330 1054330 305

sf

1336 1.098306 s sf f f

Beat fieq 1.098 1.030s sf f 0.068 sf 0.068 120 8.16 8

Question ID:337911121

4. A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in

presence of a force 2 ,F yi x j N where x and y are in meter and 11Nm .

The work done on the particle by this force F

will be ___ Joule.

Ans: 0.75 Sol: 2F yi x j

:AB y const 2F i x j



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dr dx j

1

0.AB x

W F dr

1

0dx 1ABW

BC:x=const 2F yi j

dr dy j

0.5

12BC y

W dy

1BCW CD: y const

0.5 2F j x j

66dr dx i

0.5

10.5CDW dx 0.25CDW J

DE: x=const F yi j

dr dyi

0

0.5DE yW dy

0.5 EF: Y=0 2F x j

dr dx j

0W FA: W = 0 1 1 0.25 0.5 0.75netW

Question ID: 337911125

5. A liquid at 30ºC is poured very slowly into a Calorimeter that is at temperature of

110ºC. The boiling temperature of the liquid is 80ºC. It is found that the first 5gm of

the liquid completely evaporates. After pouring another 80gm of the liquid the

equilibrium temperature is found to be 50ºC. The ratio of the Latent heat of the liquid

to its specific heat will be ___ºC.

[ Neglect the heat exchange with surrounding)

Ans: 270

Sol: Step 1:Heat lost by calorimeter 30cal calm S

Heat gained by liq 5 .50liqS mL 30 50 5 ___ 1Cal Cal eiq fm S S L

Step 2: Heat lost by calorimeter = 30cal calm S

Heat gained by liq = 80 20liqS 30 80 20 ___ 2Cal Cal eiqm S S

1600 250 5liq liq fS S L 1350 5liq fS L 1350 2705

f

liq

LS



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Question ID:337911124

6. A parallel plate capacitor of capacitance C has spacing d between two plates having

area A. The region between the plates is filled with N dielectric layers, parallel to its

plates, each with thickness dN

. The dielectric constant of the mth layer is

1mmK KN

. For a very large 310N , the capacitance C is 0

ln 2K Ad

. The

value of will be___[ 0 is the permittivity of free space]

Ans: 1 (0.99 – 1.01)

Sol:

x

dx

0K A x Ddcdx m N

01 mnK A

Ndx

01 NxK A

DNdcdx

0

0

1

1

d

xeq

dxxC K AD

0

0

ln 11

1

dxD

K AD

00

1 ln 2eq

DC K A

0

ln 2eqK ACD

K=1



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Chemistry SECTION-I

SINGLE ANSWER TYPE SECTION 1 (Maximum Marks: 12) • This section contains FOUR (04) questions. • Each question has FOUR options. ONLY ONE of these four options is the correct answer. • For each question, choose the option corresponding to the correct answer. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct option is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : —1 In all other cases. Question Id: 337911129 1. Molar conductivity ( m ) of aqueous solution of sodium stearate, which behaves as a

strong electrolyte is recorded at varying concentration(C) of sodium stearate. Which one of the following plots provides the correct representation of micelle formation in the solution?

(Critical micelle concentration (CMC) is marked with an arrow in the figures)

1)

CMC

m

C 2)

CMC

m

C

3)

CMCm

C 4)

CMCm

C Ans: 4 Sol: Conceptual Question Id: 337911128 2. Calamine, malachite, magnetic and cryolite, respectively, are 1) ZnCO3, CuCO3Cu(OH)2, Fe3O4, Na3AlF6 2) ZnSO4, Cu(OH)2, Fe3O4, Na3AlF6 3) ZnSO4,CuCO3, Fe2O3, AlF3 4) ZnCO3,CuCO3,Fe2O3,Na3AlF6 Ans: 1 Sol: Calamine 3ZnCO Malachite 3 2.CuCO Cu OH Magnetite 3 4Fe O Cryalite 3 6Na AlF Question Id: 337911127 3. The green colour produced in the borax bead test of a chromium(III) salt is due to 1) Cr2(B4O7)3 2) Cr2O3 3) Cr(BO2)3 4) CrB Ans: 3



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Sol: 2 4 7 2 2 4 7 2 2 2 3.10Na B O H O Na B O Na NO B O 2 3 2 3 2 3

3 2Cr O B O Cr BO Chromium meta borate. Question Id: 337911130 4. The correct order of acid strength of the following carboxylic acid is

O

OHH

OOH

H

HO

OH

OH

O

MeO H3C

I II III IV 1) I > II > III > IV 2) II > I > IV > III 3) I > III > II > IV 4) III > II > I > IV Ans: 1 Sol: Electronegative order 2 3sp C sp C sp C .

CH C OH

O

> CH2 CH

O

>OH

> OMe

O

OHOHO

SECTION-II ONE OR MORE TYPE

(Maximum Marks: 32) • This section contains EIGHT (08) questions. • Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s). • For each question, choose the option(s) corresponding to (all) the correct answer(s). • Answer to each question will be evaluated according to the following marking scheme Full Marks : +4 If only (all) the correct option(s) is(are) chosen; Partial Marks : +3 If all the four options are correct but ONLY three options are chosen; Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct; Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : —1 In all other cases. • For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (B) will get +2 marks; choosing ONLY (A) and (D) will get +2 marks; choosing ONLY (B) and (D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY (B) will get +1 mark; choosing ONLY (D) will get +1 mark; choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get —1 mark.



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Question Id: 337911137 1. Which of the following statement(s) is(are) true? 1) Oxidation of glucose with bromine water gives glutamic acid 2) Hydrolysis of sucrose gives dextrorotatory glucose and laevorotatory fructose 3) The two six-membered cyclic hemiacetal forms of D-(+)-glucose are called

anomers 4) Monosaccharides cannot be hydrolysed to give polyhydroxy aldehydes and ketones Ans: 2,3,4

Sol: (1) Glucose 2

2

BrH O Gluconic acid. (2)

SucroseH2O Glucose + Fructose

+ -

(3)

O

OH

HOOH

CH2OHOH

and

O

OH

HOOH

CH2OH

OH

are anomers (4) Mono saccharioles do not undergoes further hydrolysis is due to absence of ether

linkage. Question Id: 337911132 2. Fusion of MnO2 with KOH in presence of O2 produces a salt W alkaline solution of W

upon electrolytic oxidation yields another salt X. The manganese containing ions present in W and X, respectively are Y and Z correct statement(s) is(are)

1) In both Y and Z, bonding occurs between p-orbitals of oxygen and d-orbitals of manganese

2) In aqueous acidic solution, Y undergoes disproportionation reaction to give Z and MnO2

3) Both Y and Z are coloured and have tetrahedral shape 4) Y is diamagnetic in nature while Z is paramagnetic Ans: 1,2,3 Sol: 2 3 2 4 22MnO KOH KNO K MnO H O

2 4 4K MnO KMnO

w X

24MnO

Y

4MnO

Z

1,2,3 are correct



Page

-21

Question Id: 337911138

3. Choose the correct option(s) for the following set of reactions

2

) .6 10 )

i MeMgBr conc HClii H O major

C H O Q S

2

2

) . ,) ,

i H Ni Hbr benzoylperoxideii Br hvmajor major major

T R U

3 420% ,360H PO K

1)

H3C

S

CH3

Br

U

Cl

2)

H3C

T

CH3 Br

U

Br

3)

H3C

U

CH3Br

S

Cl

4)

H3C

T

CH3 Br

S

Cl

Ans: 1,2 Sol:

O

i) MeMgBrii) H2O

OH

(Q)

conc.HCl

Cl

(S)

20% H3PO4 / 360 K

Br

(T)

i) H2 - Nii) Br2 /h

HBrBentoyl / peroxide

Br

(R) (U)



Page

-22

Question Id: 337911133 4. Choose the reaction(s) from the following options, for which the standard enthalpy of

reaction is equal to the standard enthalpy of formation

1) 8 2 218

S s O g SO g 2) 2 2 22 2H g O g H O l

3) 2 332

O g O g 4) 2 2 62 3C g H g C H g

Ans: 1,3 Sol: Enthalpy change observed when one mole of substance is formed from it constituent

elements in their reference states. Question Id: 337911131 5. A tin chloride Q undergoes the following reactions(not balanced) Q Cl X 3Q Me N Y 2Q CuCl Z CuCl X is a monoanion having pyramidal geometry. Both Y and Z are neutral compounds.

Choose the correct option(s). 1) The oxidation state of the central atom in Z is +2 2) The central atom in Z has one lone pair of electrons 3) The central atom in X is sp3 hybridized 4) There is a coordinate bond in Y Ans: 3, 4 Sol: 2 3SnCl Cl SnCl X 2 3 2 3SnCl Me N SnCl NMe Y Dative bond between N and Sn 2 2 4SnCl CuCl SnCl CuCl Z Question Id: 337911135 6. Each of the following options contains a set of four molecules. Identify the option(s)

where all four molecules posses permanent dipole moment at room temperature 1) 3 3 6 6, , ,BF O SF XeF 2) 2 3 3 3, , ,NO NH POCl CH Cl 3) 2 6 5 2 5, , ,SO C H Cl H Se BrF 4) 2 2 3 3, , ,BeCl CO BCl CHCl Ans: 2, 3 Sol: value of 3 6 2 2, , , , 3BF SF BeCl CO BCl Zero Question Id: 337911134 7. Which of the following statement(s) is(are) correct regarding the root mean square

speed(Urms) of a molecule in a gas at equilibrium? 1) av at a given temperature does not depend on its molecular mass 2) Urms is inversely proportional to the square root of its molecular mass 3) Urms is doubled when its temperature is increased four times 4) av is doubled when its temperature is increased four times



Page

-23

Ans: 1,2,3

Sol: 32avE RT 1

rmsVM

rmsV T (absolute temp) qvE T (absolute temp)


8. In the decay sequence, 31 2 4

234234 234 23023892 90 91 90

xx x xTh Pa Z Th 1 2 3, ,x x x and 4x are particles/radiation emitted by the respective isotopes. The correct

option(s) is(are) 1) Z is an isotope of uranium

2) x1 will deflect towards negatively charged plate

3) x3 is ray

4) x2 is

Ans: 1,2,4

Sol: 238 234 492 90 2 1U Th He x

234 234 090 91 1 2Th Pa e x

234 234 091 92 1 3Pa Z e x

234 230 492 90 2 4Z Th He x

SECTION-III

DECIMAL INTEGER TYPE (Maximum Marks: 18) • This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. • For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct numerical value is entered; Zero Marks : 0 In all other cases. Question Id: 337911141

1. For the following reaction, the equilibrium constant Kc at 298 K is 1.6×1017

2 2Fe aq S aq FeS s

When equal volumes of 0.06 M Fe+2 (aq) and 0.2M S2-(aq) solutions are mixed, the

equilibrium concentration of Fe2+ (aq) is found to be 1710Y M . The value of Y is



Page

-24

Ans: 8.93

Sol:

2 2aq aq sFe S FeS

0.03 0.1

x 0.07

Intial(after mixing)

at equlibrium 172

11.6 107 10x

17

172

10 100 10 8.9285 8.931.6 7 10 11.2

x

Question Id: 337911140 2. At143K, the reaction of XeF4 with O2F2 produces a xenon compound Y. The total

number lone pair(s) of electron present on the whole molecule of Y is___ Ans: 19.00 Sol: 4 2 2 6 2XeF O F XeF O

Shape of 6XeF is distorted octahedral contain one lone pair e s on central atom 3 lone

pair e s on each F atom surrounded by Xe.

Total no.of lone pairs: 1 + 18 = 19.00


3. Consider the kinetic data given in the following table for the reaction

A B C product.

Experiment No.

[A] (mol dm-3)

[B] (Mol dm-3)

[C] (Mol dm-3)

Rate of reaction (Mol dm-3)

1 0.2 0.1 0.1 6.0×10-5 2 0.2 0.2 0.1 6.0×10-5

3 0.2 0.1 0.2 1.2 × 10-4 4 0.3 0.1 0.1 9.0×10-5

The rate of the reaction for[A] = 0.15 mol-3, [B] = 0.25 mol dm-3 and [C] = 0.15 mol dm-3 is found to be Y × 10-5 mol dm-3 s-1. The value of Y is _____

Ans: 6.75 (6.70 – 6.80) Sol: rate = x y 2K A B C From 1 & 2 Y = 0 From 1 & 3 Z = 1 From 1 & 4 X = 1 56 10 0.2 0.1K 133 10K Rate = 3 53 10 0.15 1 0.15 6.75 10



Page

-25

Question Id: 337911142 4. On dissolving 0.5 g of a non-volatile non-ionic solute to 39 g of benzene, its vapour

pressure decreases from 650 mm Hg to 640 mm Hg. The depression of freezing point of Benzene (in K) upon addition of the solute is_____

(Given data : Molar mass and the molal freezing point depression constant of benzene are 78 g mol-1 and 5.12 K kg mol-1, respectively)

Ans: 1.03 (0.97 – 1.06)

Sol: 10650

B

A B

nn n

10Bn

640An 10 1000640 78

m

5.12 10 1000 1.025 1.03640 78ft

Question Id: 337911139 5. Among 2 6 3 3 6 2 2 4 2 2 3 2 2 8, , , , ,B H B N H N O N O H S O H S O , the total number of polecules

containing covalent bond between two atoms of the same kind is _____ Ans: 4.00

Sol: B

HH

HHB

H

HB2H6

N2O N N O or N = N = O

N2O4 N N

O

O O

O

H2S2O3 S

S

HO OH

H2S2O8 S

O

HO OO

OS

O

OHO

B3N3H6

N

BN

B

NB H

H

H

H

H

H

Total no.of molecules containing covalent bond between two atoms of the same kind

are 4. Question Id: 337911144 6. Schemes 1 and 2 describes the conversion of P to Q and R to S, respectively, scheme 3

describes the synthesis of T from Q and S. The total number of Br atoms in a molecule of T is _______



Page

-26

2 2) ,i Br excess H O2) , ,273ii NaNO HCl K

) /iii CuCN KCN

3) ,iv H O 2) ,v SOCl pyridine

P

Q(major)

Scheme 1:

R

(major)

Scheme 2:

)i Oleum) ,ii NaOH )iii H

2 2) , ,273iv Br CS K

S

(major)

Scheme 3:

S )i NaOH)ii Q

T

Ans: 4.00 Sol: Scheme – I

H2N NH2

Br

Br

Br

i) Br2 excess

H2ON+N

Br

Br

Br

Cl-

ii) NaNO2

HCl, 273 K

iii) CuCN/KCN

Br

Br

Br

Niv) H3O+

O

OH

Br

Br

Br

v) SoCl2 / Py

O

Cl

Br

Br

Br

(Q)



Page

-27

Scheme – II

i) OleumS

O

O

OHii) NaOH /

iii) H+

HO

NaO

iv) Br2

CS2 / 273 KHO Br

(S)

HO Br

(S)

i) NaOHii) Q Br

Br

C

BrO

O

Br(T)

Mathematics SECTION – I

SECTION 1 (Maximum Marks: 12) • This section contains FOUR (04) questions. • Each question has FOUR options. ONLY ONE of these four options is the correct answer. • For each question, choose the option corresponding to the correct answer. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If ONLY the correct option is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : —1 In all other cases. Question ID : 337911146

1. Let 4 2

12 4

sin 1 sin1 cos cos

M I M

Where and are real numbers, and I is the 2 2 identity matrix. If * is the minimum of the set : [0,2 ) and

* is the minimum of the set : [0,2 ) . Then the value of * * is

1) 2916

2) 3716

3) 3116

4) 1716

Key : 1

Sol : 4 2

12 4

sin 1 sin1 cos cos

M I M



Page

-28

Where and are real numbers

4 2 2sin cos 1 sin 1 cosM

2 2 4 42 sin cos sin cos 1M I M

4 2 4 2

2 42 4

0sin 1 sin cos 1 sin0 1 cos sin1 cos cos M

Comparing on both side

4 4sin cosM

___ (1)

2 21 sin 1 sin MM

_____ (2)

So 2 2 4 42 sin cos sin cos

37*16

Minimum of

at 4

From (1) and (2) 4 4sin cos

2 21 2sin cos 211 sin 22

* = minimum of 1 112 2

37 1 37 8 29* *16 2 16 16

Question ID : 337911148 2. The area of the region 2, : 8,1x y xy y x is

1) 148log 23e 2) 78log 2

3e 3) 16log 2 6e 4) 1416log 23e

Key : 4 Sol : The area of the region 2, : 8,1x y xy y x



Page

-29

Required area 4

1

8 y dyy

443/228ln

3y y

3/22 28ln 4 4 8 03 3

16 216ln 23 3

= 16 ln 2 143

Question ID : 337911147 3. A line 1y mx intersects the circle 2 23 2 25x y at the points P and Q. If

the midpoint of the line segment PQ has x-coordinate 35

, then which one of the

following options is correct ? 1) 4 6m 2) 3 1m 3) 6 8m 4) 2 4m Key : 4 Sol :

3 5 25 13 35

m

m

3 15 1

18mm

23 15 18 0m m

2 5 6 0m m 2 3 0 2,3m m m

Question ID : 337911145

4. Let S be the set of all complex numbers z satisfying 2 5z i . If the complex

number 0z is such that 0

11z

is the maximum of the set 1 ;

1z S

z

, then the

principal argument of 0 0

0 0

42

z zz z i

is

1) 2

2) 4 3)

2 4) 3

4

Key : 1



Page

-30

Sol : 2 22 5 2 1 5Z i x y where Z = x + iy

For 0 1Z to be minimum, where 0 0 eZ x iy is at point P on the circle (figure)

Arg 0 0

0 0

42

Z ZZ Z i

0

0

4 22 2

xArgiy i

0

0

21

i xArg

y

Arg i where 0

0

21

xy

2

(where 0 )

SECTION - 2 (Maximum Marks: 32) • This section contains EIGHT (08) questions. • Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s). • For each question, choose the option(s) corresponding to (all) the correct answer(s). • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +4 If only (all) the correct option(s) is(are) chosen; Partial Marks : +3 If all the four options are correct but ONLY three options are chosen; Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct; Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : —1 In all other cases. • For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (B) will get +2 marks; choosing ONLY (A) and (D) will get +2 marks; choosing ONLY (B) and (D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY (B) will get +1 mark; choosing ONLY (D) will get +1 mark; choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get —1 mark. Question ID : 337911153 1. Define the collections 1 2 3, , ,......E E E of ellipses and 1 2 3, , ,.......R R R of rectangles as

follows : 2 2

1 : 19 4x yE ;

1R : rectangle of largest area, with sides parallel to the axes, inscribed in 1E ;

nE : ellipse 2 2

2 2 1n n

x ya b

of largest area inscribed in 1, 1nR n ;

nR : rectangle of largest area, with sides parallel to the axes, inscribed in , 1nE n . Then which of the following options is/are correct?

1) The distance of a focus from the centre in 9E is 532

2) The length of latus rectum of 9E is 16

3) The eccentricities of 18E and 19E are NOT equal

4) 1

24N

nn

area of R

, for each positive integer N

Key : 2, 4



Page

-31

Sol : 2 2

1 : 19 4x yE (ellipse), 5

3e 1 :R Max. Rectangle inscribed having sides parallel to axes

R1

B

C

A

D

OE1

For this 3cos , 2sinA , where

4

i.e 3 3,2 2

A

Sides: 3 2, 2 2 Area of 1 3 2 2 2R

If a,b are semi axes of ellipse then 2, 2a b are sides.

For 2 2

2 2 2: 13 22 2

x yE

i.e. semi axes are ,2 2

a b .

2 2

9 2 2

8 8

:

3 2

2 2

x yE

distance from centre to foucs of 9E = ae =

8

3 5 53 162

L.L.R =

2

82

8

22 4222 1 16 12563 3 32 3 6

162

ba

Eccentricities for all ellipses are equal. Also sum of areas 1 2 3........R R R 1=4 Areassumin Q

2 2 3 3

3 2 3 2 3 2=4 ......2 2 2 2 2 2

1 1 14 6 .......2 4 8

1224 2411

2

n1

area of R 24N

n



Page

-32

Question ID : 337911156 2. Let 1L and 2L denote the lines 2 2 ,r i i j k R

and

2 2 ,r i j k R respectively. If 3L is a line which is perpendicular to both 1L

and 2L and cuts both of them, then which of the following options describe(s) 3L ?

1) 2 2 2 2 2 ,9

r i j k t i j k t

2) 2 2 ,r t i j k t

3) 2 4 2 2 ,9

r i j k t i j k t

4) 1 2 2 2 ,3

r i k t i j k t

Key : 1, 3, 4 Sol : 1 :L 2 2r i i j k .............. (1) R 2 : 2 2L r i j k ...................... (2) R given lines Drs if required line 3L

P

Q

900

900

1, 2,2 2, 1, 2 i.e. 2, 2, -1 Let P on 1L be 1 ,2 , 2 Q on be 2 , ,2 2, 2, 1PQ

1 2 2 2 22 2 1

Solving 3 3 1 ............. (1) 1/ 9 6 3 0 ......... (2) 2 / 9

8 2 2 4 2 4, , , , ,9 9 9 9 9 9

P Q

From choices 1, 3, 4 are correct. Question ID : 337911149 3. Let and be the roots of 2 1 0x x , with . For all positive integers n,

define , 1n n

na n

1 1b and 1 1, 2n n nb a a n . Then which of the following options is/are correct?



Page

-33

1) 1

1010 89

nn

n

a

2) n nnb for all 1n

3) 1

810 89

nn

n

b

4) 1 2 3 2...... 1n na a a a a for all 1n

Key : 1, 2, 4 Sol : Let , roots of 2 1 0x x Clearly 2 1 0 ........... (1) 2 1 2 1 0 .......... (2) 2 1

Solving 1 5 1 5,2 2

Given , 1n n

na n

Given 1 1b , Given 1 1n n nb a a 1 1 1 1n n n n

1 2 1 21 1n n

1 12 2n n

1 15 5 5 52 2

5

n n

1 15 1 5 52 2

n n

1 1n n n n n nnb

Also 1 1 1 110 10 10 10

n nn nnn n

n n n n

b

10 210 1010 10 100 101 1

10 10

10 1 21 1 12100 10 1 89

1 1 1 1

110 10 10 105

n nn nnn n

n n n n

a

1

10 105

101 1 10 5 10100 10 89 895 5

Also 1 2 3 ....... na a a a

2 2 3 31 ....... n n

2 21 ..... ......n n

2 21 1 ...... 1 .......n n

1 1 2 21 1 1

11 1

n n n n

2 1na



Page

-34

Question ID : 337911154 4. Let :f be given by

5 4 3 2

2

3 2

5 10 10 3 1, 0;1, 0 1;

2 84 7 , 1 3;3 3

102 log 2 , 3.3e

x x x x x xx x x

f x x x x x

x x x x

Then which of the following options is/are correct? 1) f is onto 2) f is increasing on ,0 3) 'f has a local maximum at x = 1 4) 'f is NOT differentiable at x = 1 Key : 1, 3, 4 Sol : Let :f R R

5 4 3 2

2

3 2

5 10 10 3 1, 01,0 12 8

4 7 ,1 33 3, 3102 ln 2

3

x x x x xxx x

xf x x x x x

xx x x

5

2

3 2

1 2 , 01, 0 1

2 84 7 , 1 33 3

102 ln 2 , 33

x x xx x x

f x x x x x

x x x x

4

12

5 1 2, 02 1, 0 1

2 8 7, 1 31 ln 2 1, 3

x xx xf x

x x xx x

3

11

20 1 , 02 0 1

4 8 1 31 3

2

x xx

f x x x

xx



Page

-35

ff

Range = R f is onto

Clearly 1 1 5 2 02 32

f 12

f atx

Also 11 1 2 0f 11 1 0f 1f x has local maximum at 1x

11

1

1 4. .

1 2

fL D R D

f

1f is not differentiable at x = 1

Question ID : 337911151 5. There are bags 1 2,B B and 3B . The bag 1B contains 5 red and 5 green balls, 2B contains

3 red 5 green balls, and 3B contains 5 red and 3 green balls, Bags 1 2,B B and 3B have

probabilities 3 3,10 10

and 410

respectively of being chosen. A bag is selected at random

and a balls is chosen at random from the bag. Then which of the following options is/are correct?

1) Probability that the chosen balls is green equals 3980

2) Probability that the selected bag is 3B , given that the chosen ball is green, equals 513

3) Probability that the chosen ball is green, given that the selected bag is 3B , equals 38

4) Probability that the selected bag is 3B and the chosen ball is green equals 310

Key : 1, 3

Sol :

5R5G

3R3G

1B 2B3B

5G5R

iB = Event for choosing bag , 1,2,3iB i . G = Event for choosing a green ball.

1 2 31 2 3

G G GP G P B P P B P P B PB B B

3 5 3 5 4 310 10 10 8 10 8

3 3 3 39

20 16 20 80 Option 1 correct



Page

-36

33

4 3410 8

39 1380

P B GBPG P G

option 2 incorrect

3

38

GPB

option 3 is correct

3 33

4 3 3.10 8 20

GP B G P B PB

option 4 incorrect

Question ID : 337911155 6. Let B denote a curve y y x which is in the first quadrant and let the point 1,0 lie

on it. Let the tangent to B at a point P intersect the y axis at PY . If PPY has length 1 for each point P on B, then which of the following options is/are correct?

1) 2

21 1log 1exy x

x

2)

221 1log 1e

xy xx

3) 2' 1 0xy x 4) 2' 1 0xy x Key : 2,3

Sol : Equation of tangent at ,P x y

PY ,P x y

Xdy Ydx xdy ydx .So pxdyY ydx

2

2 2 1PdyPY x xdx

21dy xdx x

So 21 xdy dx

x

2

2

11

x dxx x

2 2 21 1

xdx xdxx x x

Let 2 21 x t xdx tdt

So 21 .tdt tdty

tt t

1 1ln

2 1t tt

2

21 1ln 1xy x cx

Or

221 1ln 1xy x C

x



Page

-37

Putting 1,0 0c but as function lies in 1st quadrant so 2

21 1ln 1xy xx

& 21dy x

dx x

Question ID : 337911152 7. In a non-right angled triangle PQR , let p, q, r denote the lengths of the sides opposite

to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If

3, 1p q and the radius of the circumcircle of the PQR equals 1, then which of the following options is/are correct?

1) Length of 16

OE 2) Radius of incircle of 3 2 32

PQR

3) Area of 312

SOE 4) Length of 72

RS

Key : 1, 2, 4 Sol :

3 32 sinsin 2

pp so P= 60º, 120º 1 12 sin

sin 2Q

Q so Q = 30º, 150º

Clearly P = 60º not possible. So P = 120º, Q = 30º, R = 30º

31,2

PQ PR QE RE

O must be centroid so 1 1 3 1. 13 3 4 6

OE PE option 1 is correct

Area of 1 1 332 2 4

PQR Semiperimeter 3 23 1 1 / 22

So in radius of 3

3 2 3423 2

2

PQR

option 2 is correct



Page

-38

2 2 22

2PR QR PSRS

11 3 24

2

= 7

2 option 4 is correct

Area of 13

SOE . Area of SPE 1 1 53 4

area of PQR

1 1 3 36 2 4 48

option 3 is incorrect


8. Let 0 11 2 33 1

aM

b

and adj 1 1 1

8 6 25 3 1

M

where a and b are real numbers.

Which of the following options is/are correct? 1) 3a b 2) 2det 81adjM

3) 1 1adjM adjM M 4) If 123

M

, then 3

Key : 1, 3, 4

Sol : 2 3 8 6 2 3 1 3 2

1 3 3 8 33 2 1 6 3 1

Tb b b ab aAdjM ab a a a

a a b

Comparing 6 5 1b b 3 2 1 2a a 3a b . Option 1 is correct 8 2 1 6 2M

2 42 2 16adj M M M option 2 is correct

1 1 2. Madj M adj M MM

option 3 is correct

2 12 3 2

3 3M

So 1 2 2 1 1, 1

So 3 option 4 is correct

SECTION - 3 (Maximum Marks: 18) • This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE. • For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +3 If



Page

-39

ONLY the correct numerical value is entered; Zero Marks : 0 In all other cases. Question ID : 337911162 1. Three lines are given by ,r i

,r i j and ,r v i j k v

.

Let the lines cut the plane 1x y z at the points A, B and C respectively. If the area

of the triangle ABC is then the value of 26 equals______ Key : 0.75 Sol : Let ,0,0A , ,0B , ,C A, B, C are in the plane 1x y z 1,0,0A

1 1, ,02 2

B

1 1 1, ,3 3 3

C

Area of 12

ABC AB AC

1 1 32 6

26 3 / 4 0.75 Question ID : 337911160 2. Let the point B be the reflection of the point 2,3A with respect to the line

8 6 23 0x y . Let AC and BC be circles of radii 2 and 1 with centers A and B respectively. Let T be a common tangent to the circles AC and BC such that both the circles are on the same side of T. If C is the point of intersection of C and the line passing through A and B, then the length of the line segment AC is ______

Key : 10 Sol :

2,3A DB C

12

8 6 23 0x y distance from A to the line 8 6 23 0x y is 5

2

5AB , Let BC x5 2 5

1x x

x

5 10AC x

Question ID : 337911158 3. Let ;AP a d denote the set of all the terms of an infinite arithmetic progression with

first term a and common difference d > 0. If 1;3 2;5 3;7 ;AP AP AP AP a d then a + d equals_______



Page

-40

Key : 157 Sol : 1;3 1,4,7,10,..........AP say 1S

2;5 2,7,12,17,..........AP say 2S

3;5 3,10,17,24,..........AP say 3S

1 2 7,22,37,52,.......S S i.e., 7;15AP To find the term common to AP (7; 15) and AP ( 3; 7)

7 15 3 7l m 4 27

lm l where ,l m W l can be 3 m = 7

1 2 3 . 52;105S S S A P 157a d Question ID : 337911161

4. If /4

sin/4

21 2 cos2x

dxIe x

then 227 I equals ____

Key : 4

Sol : 4

sin

4

21 2 cos2x

dxIe x

Apply

b b

a a

f x dx f a b x dx

4

sin

4

21 2 cos2x

x

dxIe x

4

4

222 cos2x

dxIx

4

2

1 22 2cos 1b

dxIx

24

20

2 sec3sec 2

xdxx

2sec xdx dt 1

20

23 1 2

dtt

1

1

0

2 1 3tan13

t

2 2033 3 3

2 24 27 427

I I


5. Let S be the sample space of all 3 3 matrices with entries from the set 0,1 . Let the

events 1E and 2E be given by

1 ; 0E A S Det A and 2 ; 7E A S sum of entries of A is .

If a matrix is chosen at random from S, then the conditional probability 1 2|P E E

equals_____

Key : 0.5



Page

-41

Sol : 1 : 0E A 2 1 2 ....... 7aE a a a

1 2 1 21

2 2 2

8 E E n E EEPE P E n E

9 22 7 2: 71' ,20' 36E s s C C

2 1

1 1 1 1 1 1 1 1 1: 1 1 1 1 1 1 1 1 1 3 3

1 0 0 0 1 0 0 0 1E nE

0 1 1 0 1 1 1 1 10 1 1 1 1 1 0 1 1 3 31 1 1 0 1 1 0 1 1

2 1 236, 18n E n E E 1

2

18 136 2

EPE

Question ID : 337911157 6. Let 1 be a cube root of unity. Then the minimum of the set

22 : , , inta b c a b c distinct non zero egers equals _________

Key : 3

Sol : 2

22 1 3 1 32 2 2 2

i ia b c a b C

2

32 2 2b c ia b c

2

232 2 4b ca b c

2 21 2 34

a b c b c , , 1,0,1a b c

Min value : 1, 1, 0a b c 2 21 122 1 3 1 34 4

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