12th iit 21st may test paper - solution -...
TRANSCRIPT
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
2
Subject-PHYSICS Part : 1
SECTION-A
(NEXT EIGHT QUESTIONS: SINGLE CHOICE – ONLY ONE CORRECT ANSWER)
Q1 The electric field in a region (relative permittivity = 1) varies with distance “r” from origin according to
relation ,ˆ2rArE where A is a positive constant. The volume charge density in the region is proportional to
(a.) r (b.) 2r (c.) r/1 (d.) 2/1 r
Sol. (a)
dxxxrKQ
rKArE
r2
022
2 4)( [where Q is total charge inside sphere of radius ].r
dxxrr
0
24 x
Q2 A solid hemispherical uniform charged body having charge Q is kept symmetrically along the y-axis as shown
in figure. The electric potential at a distance d from the origin along the x -axis at point P will be
(a.)
dQ
041
(b.)
Less than dQ
041
(c.) More than
dQ
041
and less than dQ
042
(d.)
More thandQ
042
Sol. (a)
By symmetry, potential at P is same due to upper & lower half
Q3 A steel block of mass M is being lifted by a helicopter, which is accelerating upward with an acceleration of
magnitude a. The block is attached to a horizontal bar by two cables, which make angles of θ1 and θ2 w.r.t.
horizontal as shown in the figure. The tension in the cable 1 in terms of M, θ1, θ2, g and a is
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
3
(a.) 21
1
sincos
agM
(b.) 21
1
cossin
agM
(c.) 21
2
cossin
agM
(d.)
21
2
sincos
agM
Sol. (d)
T1cos1 = T2cos2
T1sin1 + T2sin2 – Mg = Ma
T1sin1 + 2
211
cossincos
T– Mg = Ma
T1 = 21
2
sincos
agM
Q4 A block of mass 1 kg is kept over another block of plane. Block of mass 1 kg is being moved continuously with
velocity 1 m/s towards positive X-axis and lower block is moved continuously with velocity 1 m/s towards
positive Z-axis as shown in figure. If coefficient of friction between two blocks is 0.1, the rate at which heat is
generated will be )/10( 2smg
(a.) J/sec1 (b.) J/sec2 (c.) J/sec2 (d.)
J/sec2
1
Sol. (c) Rate of heat generation = power developed by friction
.22)1( RR VmgVf
Q5 The following set of figure show two cases of collision between two balls. Let
1u and
2u be the velocities
before collision and
1V ,
2V be the velocities after collision.
The coefficient of restitution,
e =
12
12
uu
VV
Student-A: Equation holds for both cases as e is property of material of the colliding bodies. Student-B: Equation holds for case II only.
(a.) Student-A is incorrect, Student-B is correct. (b.) Student-A is correct, Student-B is incorrect.
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
4
(c.) Both are correct (d.) Both are incorrect
Sol. (a)
We take velocity of approach along line of collision. In first case, 21app uuv
Q6 A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure
Pa10P 5i and volume 33
i m10V changes to a final state at Pa10321P 5
f
and 33
f m108V
in an adiabatic quasi-static process, such that 53VP constant. Consider another thermodynamic process that
brings the system from the same initial stat to the same final state in two steps: an isobaric expansion at iP
followed by an isochoric (isovolumetric) process at volumes .Vf The amount of heat supplied to the system in
the two step process is approximately (a.) J112 (b.) J294 (c.) J588 (d.) J813
Sol. (c)
Q7 A motor boat is to reach at a point 30 upstream on the other side of a river flowing with velocity .5 1ms The
velocity of motor boat with respect to water is .35 1ms The driver should steer the boat at an angle in
degrees w.r.t the line of destination from the starting point (a.) 30 (b.)
31sin 1
(c.)
31sin 1
(d.)
32sin 1
Sol. (a)
The velocity of motor boat is given as
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
5
wmwm vvv
302/1sin
120sin35
sin5
Q8 A water cooler of storage capacity 120 litres can cool water at constant rate of P watts. In a closed circulation
system (as shown schematically in the figure), the water from the cooler is used to cool an external device that
generates constantly kW3 of heat (thermal load). The temperature of water fed into the device cannot exceed
C30 and the entire stored 120 litres of water is initially cooled to C.10 The entire system is thermally
insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is:
(Specific heat of water is 11KkgkJ4.2 and the density of water is 3mkg1000 )
(a.) 1600 (b.) 2067 (c.) 2533 (d.) 3933
Sol. (b)
SECTION-B
(MULTIPLE CHOICE – ONE OR MORE THAN ONE CORRECT ANSWER)
Q1 In the presence of stationary charges in the space, line integral of the electric field
dE for a closed path as
shown, is carried out:
(a.) The value of the integral is independent of the path (b.) The value of the integral is independent of where charges may be located relative to the path (c.) The value of the integral is always is constant (d.) The value of the integrals is always negative
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
6
Sol. (a, b, c)
In an electric field dV=
dE for a closed loop, dV = 0
Q2 The figure shows, two point charges )0(2 Q and .Q The charges divide the line joining them in three
parts I, II and III
(a.) Region III has a local maxima of electric field (b.) Region I has a local minima of electric field
(c.) Equilibrium position for a test charge lies in region II (d.) Region II has a local minima of electric field
Sol. (a, d)
Q3 A particle A moves with a velocity 1v and a particle B moves with a velocity ,2v w.r.t. ground
and r is the length of the vector, ,AB then
(a.) 0
dtdr
if |||| 12 vv
(b.) 0
dtdr
if |||| 12 vv
(c.) 0
dtdr
if ABvABv .. 12
(d.)
0dtdr
if 0.)( 21 BAvv
Sol. (c, d)
ABuvvdtdr .)( 12 where ABu is a unit vector along AB
Q4 Two blocks of masses m2 and m are connected as shown in the figure. Now the string between the blocks is
suddenly broken.
(a.) gaA (b.)
2gaA
(c.) gaB (d.)
2gaB
Sol. (b, c) ‘B’ is in free fall, as string is cut, hence gaB tension in string, before it is cut, is Mg. hence net force on A
after cutting string is Mg 2gaA
Q5 A ball tied to the end of a string swings in a vertical circle under the influence of gravity (a.) When the string makes an angle 90 with the vertical the tangential acceleration is zero and radial acceleration
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
7
is somewhere between maximum and minimum (b.) When the string makes an angle 90 with the vertical the tangential acceleration is maximum and radial
acceleration is somewhere between maximum and minimum (c.) At no place in the circular motion, tangential acceleration is equal to radial acceleration in magnitude (d.) Throughout the path whenever radial acceleration has its extreme value, the tangential acceleration is zero
Sol. (b, d) In the shown diagram at position A and C radial acceleration has its maximum values and there
is no component of acceleration in tangential direction.
At position B and D tangential acceleration is maximum )(g and radial acceleration in some
where between maximum and minimum. Q6 Bodies of mass 0.5 kg resting on a horizontal frictionless tabletop are connected with an unstretched spring of
length L = 20 cm and of spring constant 16 N/m. The mass of the spring is negligible. At a certain moment the
bodies are given an initial speed of 0v = 0.36 m/s towards the wall on the right. The body at the right collides
with the wall totally elastically
(a.) There will be 2 collisions with the wall (b.) After 1st collision centre of mass comes to rest (c.) After 2nd collision centre of mass moves to left with speed v0 (d.) After all collisions are over, the system oscillates about the centre of mass
Sol. (a, b, c)
After 1st collision two are moving with speeds 0v in opposite direction. Hence centre of mass is at rest & two
start oscillating with initial position as mean position. So mass on right again collides with wall when at mean
position & both masses have same speed towards left after collision.
SECTION-D
Integer Answer Type
This section Contains 6 questions. The answer to each of the questions is a single digit integer, ranging from 00 to
99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened.
Q1 A tank is filled upto a height 10 m with a liquid and is placed on a platform of height 2m from the ground. The
distance h from the ground where a small hole should be punched so that the liquid stream coming out of the
hole covers maximum horizontal distance in meters is
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
8
Sol. (6)
For maximum horizontal range, hole should be mid point or as close as possible to mid point between ground & liquid surface.
Q2 A point mass is subjected to two simultaneous sinusoidal displacements in x -direction, tAtx sin)(1 and
.3
2sin)(2
tAtx A third sinusoidal displacement
34sin)(3
tBtx is added such that its
average kinetic energy of SHM due to combination of the three displacements is four time its average kinetic
energy due to )(1 tx only. If values of B is ,nA write value of :n
Sol. (3) Q3 J600 of heat is added to a monatomic gas in a process in which the gas performs a work of .150 J If molar
heat capacity for the process is ,nR write value of :n
Sol. (2)
Q4 A wave is represented by the equation )02.0100(2sin10)10002.0(2sin10 xttxy on string
of length 1 m, where x and y are in cm and t is time in second. If String vibrates with n loops, write value
of n
Sol. (4) Q5 Velocity versus displacement curve of a particle moving in straight line is shown in the figure. From a point P,
a line is drawn perpendicular to displacement axis and line PR is drawn normal to the curve at P. The
instantaneous acceleration of the particle at point P is
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
9
Sol. (1)
21212
PRofslope12 ms
dxdvva
Q6 n moles of hydrogen gas is mixed with 2 moles in helium gas. The speed of sound in mixture is 12
times the
RMS speed of gas molecules in the mixture. Find .n
Sol. (2)
4233
21
00
frMRT
MrRT
242
)3(2)5(
nn
nf
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
10
Subject-CHEMISTRY Part : 2
SECTION-A
(NEXT EIGHT QUESTIONS: SINGLE CHOICE – ONLY ONE CORRECT ANSWER)
Q1 In the precipitation titration of KCl against AgNO3, K2CrO4 is used as an indicator. End point is detected by
appearance of deep colored precipitate of Ag2CrO4. Determine the concentration of chromate ion required for
detection of end point. Ksp of AgCl = 10–10 and Ksp of Ag2CrO4 = 10–12 (a.) 0.1 M (b.) 0.01M (c.) 0.001M (d.) 0.0001M
Sol. (b)
At end point [Ag+] = SPK =10–5
[CrO42–] =
25
12
10
10
= 0.01
Q2 Normility of “20 vol” H2O2 is (a.) 20 (b.) 10 (c.) 58.3 (d.) 79.1
Q3 The first four ionization energy values of an element are 190, 580, 870 and 5960 kcal. The number of valence
electrons in the element is (a.) 1 (b.) 2 (c.) 3 (d.) 4
Sol. (c)
Differences in first three ionization energies are less but fourth one is very high. This indicates that there are 3
valence electrons
Q4 The correct order of increasing s character of the orbital of B which overlaps with the orbital of F to form B-F
bond in 2BF , BF3 and BF4 is
(a.) 432 BFBFBF (b.) 234 BFBFBF (c.) 423 BFBFBF (d.)
342 BFBFBF
Sol. (b)
%)25(%),33.33(%),50( 34
232 spBFspBFspBF
Q5 Which of the given statement is/are correct for the given reaction knowing the fact that higher the number of
hyperconjugative structures higher the stability of carbocation
CH)(CHC)(CD 3333 CH)(CDC)(CH 3333
(a.) The equilibrium constant should be higher than one for the above reaction
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
11
(b.) The equilibrium constant should be lower than one for the above reaction (c.) C)(CD 33 will be more stable than C)(CH 33 due to higher electron releasing nature of deuterium than
hydrogen (d.) C)(CD 33 will be more stable than C)(CH 33 due to presence of stronger DC bond than HC bond
Sol. (a) cCH 33 )( is more stable than CCD
3)3( due to weaker bond strength of C-H bond
Q6 If the speed of electron in first Bohr orbit of hydrogen be ‘x’, then speed of the electron in second orbit of He+
is: (a.) x / 2 (b.) 2x (c.) x (d.) 4x
Sol. nZV
nZVH
1x
22
HeV
1HeV
1HeV
x xVHe
Q7 The molar entropy of vaporization of acetic acid is 14.4 cal K–1 mol–1 at its boiling point C.118 The latent
heat of vaporization of acetic acid is
(a.) 49 cal g–1 (b.) 64 cal g–1 (c.) 94 cal g–1 (d.) 84 cal g–1
Q8 The crystal system of a compound with unit cell dimensions nm504.0,387.0,387.0 cba and
90 and 120 is:
(a.) Cubic (b.) Hexagonal (c.) Orthorhombic (d.) Rhombohedral
SECTION-B
(MULTIPLE CHOICE – ONE OR MORE THAN ONE CORRECT ANSWER)
Q1 Three metals of alkaline earth metal group (A, B and C) when reacted with a fixed volume of liquid bromine separately gave product (metal bromide) whose mass is plotted against mass of metals taken as shown in the figure:
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
12
From the plot, predict what relation can be concluded between the atomic weights of A, B and C?
(a.) C > B (b.) B > A
(c.) C < A < B (d.) Data is inadequate to predict
Sol. (a, b)
Since mass of bromine taken is constant, same moles of 2MBr will be produced in each case.
Since, moles = (mass) / molecular weight.
Heaviest metal will produce maximum mass of product. Hence, correct order of atomic mass of A, B, C is
CBA
Q2 Two liquids A and B form an ideal solution. The solution has a vapor pressure of 700 Torr at .C80 It is
distilled till 2/3rd of the solution is collected as condensate. The composition of the condensate is 75.0' Ax
and that of the residue is .30.0" Ax If the vapor pressure of the residue at C80 is 600 Torr, which of the
following is/are true?
(a.) The composition of the original liquid was 6.0Ax (b.) The composition of the original liquid was 4.0Ax
(c.) Torr3/25000 AP (d.) Torr5000 BP
Sol. (a, c, d)
700 BBAA PxPx … (i)
6007.030.0"" BABBAA PPPxPx … (ii)
If moles of A & B initially are x & y then
)(31
3.0)(32
75.0 yxyxx
yxxxA
or yx
yxB
Solving given
6.0Ax 4.0Bx
torr500&torr3
2500 BA PP
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
13
Q3 A open ended mercury manometer is used to measure the pressure exerted by a trapped gas as shown in the
figure. Initially manometer shows no difference in mercury level in both columns as shown in diagram.
After sparking '' A dissociates according to following reaction
)(3)()( gCgBgA
If pressure of Gas "" A decreases to 0.9 atm. Then (Assume temperature to be constant and is 300 K)
(a.) Total pressure increased to 1.3 atm (b.) Total pressure increased by 0.3 atm
(c.) Total pressure increased by 22.3 cm to Hg (d.) Difference in mercury level is 228 mm
Sol. (a, b, d)
atm.3.11.031)31( TP
atm2.0P or cm3.076 of Hg or mm3.0760 of Hg
Q4 According to Maxwell’s distribution of molecular speeds: (a.)
mpavrms vvv
(b.) Only a small fraction of molecules are having either very high speeds or very low speeds.
(c.) As temperature increases, the fraction of molecules having velocity equal to mpv decreases.
(d.) For two gas samples at same temperature, the gas having higher molar mass will have larger fraction of
molecules possessing velocity equal to .vmp
Q5 Which of the following molecules posses zero dipole moment (a.) BF3 (b.) NF3 (c.) ClF3 (d.) CF4
Sol. (a, d)
Q6 Which of the following is(are) deactivating group(s) by mesomeric effect for electrophilic substitution reaction
when attached to benzene ring? (a.)
3RN
(b.) COOH (c.) RCHCH (d.) CN
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
14
Sol. (b, d)
(a) is electron withdrawing group by inductive effect.
(c) is electron releasing group when attached to benzene ring.
SECTION-D
Integer Answer Type
This section Contains 6 questions. The answer to each of the questions is a single digit integer, ranging from 00 to
99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened.
Q1 4.48 L of an ideal gas at STP requires 12 cal to raise the temperature by C15 at constant volume. The pC of
the gas is ______ cal.
Sol. (6)
4.48 L required 12 cal
22.4 L required cal6048.4
124.22
For C15 60 cal is required
for C1 cal415
601
is required
cal.4vC RCC vp cal6pC
Q2 One mole of van der waal’s gas, having compressibility factor (Z) = 0.8 exerts 2 atm pressure in a 10 L vessel.
Find the intercept along Y-axis when a graph plotted between mPV vs. P where P = pressure and mV Molar
volume of the real gas. (Assume at this condition pressure of ideal gas is 2.02 bar).
Sol. (25)
2.02 bar 2 atm
Based on given information force at attraction in real gas is negligible because gasidealgasreal PP
RTb)(VnaP 2
is RTb)(VP RTPbPV
RTPbPV
For PPV slope is b
Value of intercept is RT
RTPVZ
RT1028.0
250.8
102RT
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
15
Q3 Mixture of equal molecules of 2N and 5PCl is kept at 6 atm. Pressure. At equilibrium total pressure of the
vessel is 7.5 atm. Find the % degree of dissociation of .PCl5 Assume constant temperature.
Sol. (50)
5PCl 23 ClPCl
P3 P P
Total pressure 7.53P3
1.5P
degree of dissociation is 50% Q4 62HB excess of 3NH at high temperature give Product .x Hybridisation of B in P is .ysp Find the value
of .x
Sol. (2)
xtemphighNHofexcess
62 (BN)HB 3 (Boron nitride)
Hybridization is 2sp Ans. 2
Q5 Find the total number of isomeric alkynes possible of FClBrI.C3
Sol. (8)
F C C
Cl
Br
IC 2 Br C C
Cl
IC 2
F
F
C CCl Br
I
C 2
F
C C Cl
Br
I C 2
Q6 1 M 1000 ml aqueous solution of HCl is diluted to 2 litre. 1 litre of this solution is again diluted to 2 litre this
process is repeated for 3 more times. 20 ml of final solution is titrated with strong base .(OH)X 2 5 ml of
2(OH)X is required. Then normality of strong base is .YN
Find the value of .Y
Sol. (8)
Molarity of HCl is changed by
3216842MMMMMM
2211 VNVN
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
16
NNN 220
32
82NN
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
17
Subject-Mathematics Part : 3
SECTION-A
(NEXT EIGHT QUESTIONS: SINGLE CHOICE – ONLY ONE CORRECT ANSWER)
Q1 The equation ||)3(|1| )23(log 2)1( xxx xxx has
(a.) Unique solution (b.) Two solutions (c.) No solution (d.) More than two solutions
Sol. (c)
Q2 The coefficient of 49x in the product )99(...)3()1( xxx is
(a.) 299 (b.) 2500 (c.) 1 (d) 5000
Sol. (b)
Here, number of factors 50
The coefficient of 99...53149 x
)991(2
50
2500
Q3 If the number of ways of selecting n cards out of unlimited number of cards bearing the number .3,9,0 So
that they cannot be used to write the number 903 is ,93 then n is equal to
(a.) 3 (b.) 4 (c.) 5 (d.) 6
Sol. (c)
We can not write 903
If in the section of n cards, we get either 9( or 9(),3 or 0(),0 or ),3 (only ),0 (only )3 or (only )9
Can be selected n....222 times n2
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
18
Similarly, 9( or )0 or 3( or )0 can be selected n2
In the above selection (only )0 or (only )3 or (only )9 is repeated twice
Total ways 933222 nnn
962.3 n
52322 n
5n
Q4 The number of distinct values of a 22 determinant whose entries are from the set }1,0,1{ is
(a.) 3 (b.) 4 (c.) 5 (d.) 6
Sol. (c)
Possible values are 2,1,0,1,2
i.e., 111
10,0
0011
,11101
21111
,21111
Q5
If is a non real cube root of ,2 then the value of 122
31121
22
is equal to
(a.) 11 (b.) 12 (c.) 13 (d.) 0
Sol. (c)
12231121
22
applying 133 RRR
Then 131)2(616122
31121
322
)2,)2(( 33/1
Q6 If
233033
2 yx and ,441
5142
xy then
(a.)
230103
yx (b.)
021112
x
(c.)
212321
yx (d.)
211211
y
Sol. (b)
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
19
Q7
If ,122212221
A then Aadj is equal to
(a.) A (b.) TA (c.) A3 (d.) TA3
Sol. (d)
122212221
A
366636663
333231
232221
131211
CCCCCCCCC
Ac
.3122212
2213
366636
663')(adj Tc AAA
Q8 The decimal parts of the logarithms of two numbers taken at random are found to six places. Probability that
second can be subtracted from one first without borrowing is (a.) 6
21
(b.) 6
209
(c.) 6
2011
(d.) 6
203
Sol. (c)
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
20
Let
6
16543211
ii xxxxxxxD and
6
16543212
ii yyyyyyyD
Digit after decimal can be .9,...,3,2,1,0
Now, 2D subtract from 1D without borrowing.
ii yx
Let )90( iy
9,...,1, ix
After choosing ii xy , can be choosen in )10( ways.
Favourable ways
9
0
)10(
9
0
9
0
10
5521091010
And total number of ways 1001010
Required probability (for six digits) 66
2011
10055
SECTION-B
(MULTIPLE CHOICE – ONE OR MORE THAN ONE CORRECT ANSWER)
Q1 A bag contains four tickets marked with ,222,211,121,112 one ticket is drawn at random from the bag. Let
)3,2,1( iEi denote the event that thi digit on the ticket is .2 Then
(a.) 1E and 2E are independent (b.)
2E and 3E are independent
(c.) 3E and 1E are independent (d.)
21 , EE and 3E are independent
Sol. (a, b, c)
We have, 21
42)( iEP for ,3,2,1i
Also, for
).()(41)(, jiji EPEPEEPji
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
21
Therefore, iE and jE are independent for ,ji
Also, )()()(41)( 321321 EPEPEPEEEP
321 ,, EEE are not independent.
Q2
The determinant 22
22
22
xcbcacbcxbabacabxa
is divisible by
(a.) x (b.) 2x (c.) 3x (d.) 4x
Sol. (a, b, c, d)
Taking cba ,, commons from 321 ,, CCC respectively and then multiplying from cba ,, in 321 ,, RRR
respectively.
2222
2222
2222
xcccbxbbaaxa
Applying ,3211 RRRR then
2222
2222
222222222222
xcccbxbb
cbaxcbaxcbax
Applying 122 CCC and ,133 CCC then
)(0
000
22224
22
22
0222
cbaxxxc
xbcbax
Q3
Let ,122212221
A then
(a.) OIAA 32 54 (b.)
)4(51
31 IAA
(c.) 3A is not invertible (d.) 2A is invertible
Sol. (a, b, d)
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
22
Q4 Two roads are represented by the equations 6 xy and .8 yx An inspection bunglow has to be so
constructed that it is at a distance of 100 from each of the roads. Possible location of the bunglow is given by
(a.) )7,12100( (b.) )7,21001( (c.) )21007,1( (d.) )21007,1(
Sol. (a, b, c, d)
Q5 From the point )3,0(A on the circle ,0)3(4 22 yxx a chord AB is drawn and extended to a point
,M such that .2ABAM An equation of the locus of M is
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
23
(a.) 0)2(6 22 yxx (b.) 0)3(8 22 yxx
(c.) 096822 yxyx (d.) 044622 yxyx
Sol. (b, c)
Q6 The normals to the parabola axy 42 from the point )2,5( aa are
(a.) axy 3 (b.) axy 122 (c.) axy 333 (d.) axy 3
Sol. (a, b)
Any normal to parabola axy 42 may be taken as
)(2 2atxtaty or 02 3 atattxy … (i)
If Eq. (i) passes through ),2,5( aa then
0252 3 atatata
or 0233 tt
or 0)2()1( 2 ttt
1,2,1 t
From Eq. (i) normals are 03 axy and 0122 axy
SECTION-D
Integer Answer Type
This section Contains 6 questions. The answer to each of the questions is a single digit integer, ranging from 00 to
99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened.
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
24
Q1 If 7
2 i
e
and ,)(20
10
k
kk xAAxf then the value of
6
0
220 )()(
r
nn
nn
r xAxAAnxf then n
must be equal to
Sol. (7)
Q2 Let CBA ,, be three events. If the probability of occurring exactly one event out of A and B is ,1 a out
of B and C is ,21 a out of C and A is a1 and that of occurring three events simultaneously is .2a If
the probability that atleast one out of CBA ,, will occur is greater than then the value of 10 must be
Sol. (5)
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
25
Q3 Suppose a matrix A satisfies .752 OIAA If ,5 bIaAA then the value of
59
ab must be
Sol. (4)
Q4
The value of 4log4log3log3log
9log8log3log512log
33
82
43
43 is 2 then must be
Sol. (5)
4log4log3log3log
9log8log3log512log
33
82
43
43
23
23
222
23
3
29
3
2log2log3log3log
3log2log3log2log 3
2
2
2log22log2
3log313log
3log222log3
3log212log9
33
22
23
23
322
2319
1034
415
Q5 The trigonometric equation xxx 4cosec2coseccosec has a solution nn (,
7)12(
integer), then n
should not be an integer of the type pp (),60(,7 integer). The numerical quantity should be
Sol. (3)
The given equation is not defined for
12th IIT 21st May Test Solution
101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232
26
Q6
If ,2
2tan50
142
1
r rr
rS then the value of
Stan
12752552
must be
Sol. (2)