NATIONAL INSTITUTE OF TECHNOLOGY HAMIRPUR

JEE-MAINS MOCK TEST 2015

PAPER -1: PHYSICS, CHEMISTRY& MATHEMATICS Do not open this Test Booklet until you are asked to do so.

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Test Booklet, take out the Answer Sheet and fill in the particulars carefully. 3. The test is of 3 hours duration. 4. The Test Booklet consists of 90 questions. The maximum marks are 360. 5. There are three parts in the question paper A, B, C consisting of, Physics, Chemistry

and Mathematics having 30 questions in each part of equal weightage. Each questionis allotted 4 (four) marks for each correct response.

6. Candidates will be awarded marks as stated above in instruction No. 5 for correct

response of each question. 1/4 (one fourth) marks will be deducted for indicating incorrect response of each question. No deduction from the total score will be made if no response is indicated for an item in the answer sheet.

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Physics PART – 1

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. Two point masses A of mass M and B of mass $M are fixed at the ends of a rod of length 𝑙 and of

negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform

angular speed. For the rod to have the minimum possible kinetic energy of rotation, the axis of

rotation should be at a distance of

(A) 2/5 𝑙 from CM (B) 8/5 𝑙 fom CM

(C) 4/5 𝑙 from CM (D) 𝑙/5 from CM

2. A 10 kg block is gently placed on a plank moving at 1 m/s

on a frictionless plane. Due to friction between the plank

and the block, the block accelerates. The velocity of the

plank is maintained at a constant 1 m/s by an external

fore F acting on it. The block finally reaches a velocity of

1 m/s. The work done by the external force, which

maintains the constant speed of plank, is [the coefficient

of friction between plank and the block is 0.4]. (Take g =

10 m/s2)

(A) 100 J (B) 50 J

(C ) 10 J (D) 5 J

3. Two springs with negligible masses and

force constant of k1 = 200 Nm-1 and k2 =

160 Nm-1 are attached to the block of

mass m = 10 kg as shown in the figure.

Initially the block is at rest, at the

equilibrium position in which both

springs are neither stretched nor

compressed. At time t = 0, sharp

impulse of 50 N-s is given to the block

with hammer

(A) Period of oscillations for the mass m is π

6 s

(B) Maximum velocity of the mass m during its oscillation is 10 ms-1.

(C) Data are insufficient to determine maximum velocity.

(D) Amplitude of oscillation is 5/6 m.

4. The figure below shows the path of white light’s rays which leave in phase from two small

sources S₁ and S₂ and travel to a point X on a screen. The path difference is S₂X - S₁X = 10 x

10⁻⁷ m. What wavelength of light give complete destructive interference at X?

(A) 4 x 10⁻⁷ m (B) 3 x 10⁻⁷ m

(C) 6 x 10⁻⁷ m (D) 5 x 10⁻⁷ m

5. The wavelength of characteristic X-ray Kα line emitted by hydrogen like atom is 0.32 Å.

The wavelength of Kᵦ line emitted by the same amount is

(A) 0.18 Å (B) 0.48 Å

(C) 0.27 Å (D) 0.38 Å

6. An insect of mass m is initially at one end of the stick of length L and mass M, which rests on a

smooth horizontal floor. The coefficient of friction between the insect and stick is k. The

minimum time in which the insect can reach the other end of the stick is t. Mark all the correct

statements.

(A) The centre of mass of the plank has velocity magnitudes [𝑘𝑚𝑔

𝑀] 𝑡 w.r.t. horizontal floor at

time t

(B) t = √2𝐿𝑀

𝑘(𝑀+𝑚)𝑔

(C) The magnitude of the linear momentum of the insect at time ‘t’ is (kmgt) w.r.t horizontal

floor.

(D) All of the above.

7. A stone is dropped from the top of a tall cliff and n seconds later another stone is thrown

vertically downwards with a velocity u. Then the second stone overtakes the first, below the top

of the cliff at a given distance is given by

(A) 𝑔

2[

𝑛(𝑔𝑛

2−𝑢)

𝑔𝑛−𝑢}^2 (B)

𝑔

2{

𝑛(𝑔𝑛−𝑢

2)

𝑔𝑛−𝑢}^2

(C) 𝑔

2{

𝑛(𝑔𝑛−𝑢

2)

𝑔𝑛−𝑢

2

}^2 (D) 𝑔

2{

𝑛(𝑔𝑛−𝑢)

𝑔𝑛−(𝑢−2)}^2

8. A metal oxide follows the relation V= a√𝐼where a = 1 volt/ (ampere)1/2 . V is

potential drop across the oxide and 𝐼 is current through it. If heat produced in

resistor and metal oxide is same then value of R is

(A) 3 Ω (B) 1/3 Ω

(C) 6 Ω (D) 9 Ω

9. In an LC circuit the capacitor has maximum charge q0. The value of (dI/dt) max is

(A) q0/LC (B) q0/√LC

(C) (q0/LC) - 1 (D) (q0/𝐿𝐶) +1

10. An electron in a hydrogen atom makes a transition from n1 to n2. The time period of the electron

in the initial state is eight times that in the final state. The possible value of n1 and n2 are

(A) n1 = 6, n2 = 4 (B) n1 = 8, n2 = 2

(C ) n1 = 8, n2 = 1 (D) n1 = 6, n2 = 3

11. In a Coolidge tube, the potential difference used to accelerate the electrons is increased from

12.4 kV to 24.8 kV. As a result, the difference between the wavelength of Kα –line and minimum

wavelength increases to thrice its former value. The wavelength of the Kα line is (hc/c = 12.4 kVÅ)

(A) 1 Å (B) 0.5 Å

(C ) 1.5 Å (D) 1.25 Å

12. Wire segments CO and OA are straight current carrying conductors where ABC is a

semi-circular segment placed symmetrically in x-y plane form a closed loop as shown in

the figure. The loop carries a current I. Find the magnitude of magnetic moment of the

loop.

(A) I (π𝑎2

2+

𝑎2

2) (B) I (

π𝑎2

4+

𝑎2

2)

(C ) I (𝜋𝑎2 + 𝑎2 ) (D) I (𝜋𝑎2 +𝑎2

2)

13. A capacitor of capacitance C is charged by a battery of emf V and then disconnected. The work

done by an external agent to slowly insert dielectric of dielectric strength k of half the length of the

capacitor is

(A) 1

2 𝐶𝑉2 [

𝑘−1

𝑘+1] (B)

1

2 𝐶𝑉2 [

1−𝑘

𝑘+1]

(C ) 1

4𝐶𝑉2 (𝑘 − 1) (D) 1/4𝐶𝑉2(1 − 𝑘)

14. There is a conducting ring of radius R. Another ring carrying current I and radius r (r < < R) is kept

on the axis of bigger ring such that it’s centre lies in the axis of bigger ring at a distance x from the

centre of bigger ring and its plane is perpendicular to that axis. The mutual inductance of bigger ring

due to smaller ring

(A) 𝜇𝜋𝑅2 𝑟2

(𝑅2 +𝑋2)3/2 (B)

𝜇𝜋𝑅2 𝑟2

4(𝑅2 +𝑋2)3/2

(C ) 𝜇𝜋𝑅2 𝑟2

16(𝑅2 +𝑋2)3/2 (D)

𝜇𝜋𝑅2 𝑟2

2(𝑅2 +𝑋2)3/2

15. A choke coil is needed to limit the current to operate an arc lamp at 250 V (rms) and 50 Hz. The

lamp has an effective resistance of 15 Ω when running at 10 A (rms). The inductance of the choke

coil is

(A) 10 𝜋 H (B) 1

10𝜋 H

(C ) 1

5𝜋 H (D) 5𝜋 H

16. A trolley is moving horizontally with a velocity of v m/s w.r.t. earth. A man starts running from

one end of the trolley with a velocity 1.5v m/s w.r.t. the trolley. After reaching the opposite end, the

man turn back and continues running with velocity of 1.5v m/s w.r.t. trolley in the backward

direction. If the length of the trolley is L, then the displacement of the man w.r.t. earth measured as

a function of time, will attain a maximum value of

(A) 4/3 L (B) 2/3 L

(C ) 5/3 L (D) 1.5 L

17. An organ pipe P1 closed at one end vibrating in its first harmonic and other pipe P2 open at both

ends vibrating in its third harmonic are in resonance with the given tuning fork, the ratio of length of

P1 to that of P2 is

(A) 8/3 (B) 3/8

(C) 1/6 (D) 1/3

18. A uniform conducting wire of length l= (10±0.2) cm and radius of cross section r=(1±0.01)cm

respectively, is cut to form a resistor. The maximum percentage error in the resistance is( assume

that the resistivity of material of the wire is known accurately)

(A) 1% (B) 2%

(C) 4% (D) 8%

19. As shown in the diagram, there are two blocks of the same mass M, One on top of the other,

lying on a frictionless horizontal surface. Both the blocks are at rest. The upper block is much smaller

than the lower block. A force F is applied on the lower block and both the block Start moving

together without any relative motion. Suddenly, the lower block hits a fixed obstacle and comes to

rest. The upper block continues to slide on the lower block. The upper block just manages to reach

the opposite end of the lower block. What is the coefficient of friction between the two blocks?

(A) F/MG (B) 2F/MG

(C) F/2MG (D) none

20. A conducting flexible distorted loop of total length 22 cm carrying a

current i is placed in a uniform magnetic field B as shown in the figure. The

loop will

(A) Form into a circular loop of radius 3.5 cm (approximately).

(B) Form into a lump.

(C) Remain in its original position.

(D) Nothing can be said.

Q21. A fixed chamber, isolated from surroundings is divided into two equal half A and B as shown in

figure. Part A contains one mole of oxygen and part B

contains one mole of helium, the separator C is

thermally conducting and kept fixed. Initial temperature

of oxygen chamber is 600K and that of helium is 300K.

Specific heat capacity of separator and chamber is

negligible. Choose the correct statement:

(A) Change of temperature of helium is equal to change in temperature of oxygen.

(B) Change in internal energy of helium is equal to 200R joule (R is value of gas constant in SI)

(C) Temperature of oxygen in steady state condition is 487K (approx.)

(D) Temperature in steady state can’t be calculated.

Q22. Two sound waves of slightly different frequency have the amplitude ratio 11/9. What is the

difference of sound levels (in dB) of maximum and minimum intensities heard at a point?

(A) 100

(B) 10

(C) 16

(D) 20

Q23. A shot whose mass is m penetrates a thickness S of a fixed plate of mass M. If M were free to

move, the thickness penetrated will be:

(A) SM/m

(B) Sm/(M+m)

(C) SM/(M+m)

(D) Sm/M

Q24. In Young’s modulus double slit experiment, assume intensity of light on screen due to each

source alone is I0 and K1 is equal to difference of maximum and minimum intensity. Now intensity of

one source is made I0/4 and K2 is again difference of maximum and minimum intensity, then K1/K2=

(A) 4

(B) 3

(C) ¾

(D) 2

Q25. A point charge q is placed at a distance 2r from the centre O of

a conducting uncharged sphere of radius r. Potential of induced

charges at point P (lying in a line joining the point charge and centre

of sphere) is:

(A) Kq/2r

(B) Kq/3r

(C) Kq/6r

(D) None of these

26. A uniform circular ring of radius R, mass m has uniformly distributed charge q. The ring is free to

rotate about its own axis (which is vertical) without friction. In the space, a uniform magnetic field B,

directed vertically downwards, exists in a cylindrical region. Cylindrical region of magnetic field is

coaxial with the ring and has radius ‘r’ greater than R. If magnetic field starts increasing at a constant

rate dB/dt =a, angular acceleration of the ring will be

(A) 𝑞𝑅

2𝑟𝑚

𝑑𝐵

𝑑𝑡 (B)

𝑞

2𝑚(

𝑑𝐵

𝑑𝑡 )

(C ) 𝑞𝑅

𝑟𝑚

𝑑𝐵

𝑑𝑡 (D)

𝑞𝑅

𝑚(

𝑑𝐵

𝑑𝑡 )

27. In figure, infinite conducting rings each having current in the direction

shown are placed concentrically in the same plane as shown in the figure.

The radius of rings are r, 2r, 22r, 23r….∞. The magnetic field at the center

of rings will be

(A) Zero (B) 𝜇𝑖

𝑟

(C) 𝜇𝑖

2𝑟 (D)

𝜇𝑖

3𝑟

28. A rod of length L is pulled with a force L as shown on a smooth horizontal surface. If A is the area

of cross section and Y the Young’s modulus of material of the rod, the elastic potential energy stored

in the rod is

(A) 𝐹2𝐿

3𝑌𝐴 (B)

𝐹2𝐿

2𝑌𝐴

(C) 𝐹2𝐿

6𝑌𝐴 (D)

𝐹2𝐿

𝑌𝐴

29. A uniform circular disc of radius r placed on a rough horizontal surface has initially a velocity V0

and angular velocity 𝜔0 as shown in the figure. The disc comes to rest completely after moving some

distance. Then V0/(r 𝜔0) is equal to

(A) 1 (B) ½

(C) 3/2 (D) 2

30. A cylindrical region of uniform magnetic field, which is increasing at a constant rate dB/dt = α,

exists perpendicular to plane of paper. The diameter of cylindrical region𝑙. A non-conducting rigid

rod of length 𝑙 having two charged particles is kept fixed on the diameter of cylindrical region w.r.t.

inertial frame. If two charged particles having charges q each is kept fixed at the ends of non-

conducting rod. The net force on any one of charges is

(A) 𝑞𝑙𝛼

4 (B)

𝑞𝑙𝛼

2

(C) Zero (D) 𝑞𝑙𝛼

Chemistry PART – II

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. HBr reacts with CH2 = CH – OCH3 under anhydrous conditions at room temperature to give

(A) CH2 CHO and CH3Br (B) BrCH 2CHO and CH 3OH (C) BrCH2 – CH2 – OCH3 (D) H 3C – CHBr – OCH3

2. Which one of the following is the correct statement?

(A) Chlorides of both beryllium and aluminium have bridged chloride structures in solid phase.

(B) B2H6.2NH3 is known as 'inorganic benzene'.

(C) Boric acid is a protonic acid.

(D) Beryllium exhibits coordination number of six.

3. Amount of oxalic acid present in a solution can be determined by its titration with

KMnO 4 solution in the presence of H2SO4. The titration gives unsatisfactory result when

carried out in the presence of HCl, because HCl

(A) Reduces permanganate to Mn2+. (B) Oxidises oxalic acid to carbon dioxide and water. (C) Gets oxidised by oxalic acid to chlorine. (D) Furnishes H+ ions in addition to those from oxalic acid.

4. The half – life of a radioisotope is four hours. If the initial mass of the isotope was 200 g, the mass remaining after 24 hours undecayed is

(A) 1.042 g (B) 4.167 g (C) 3.125 g (D) 2.084 g

5. Amount of oxalic acid present in a solution can be determined by its titration with KMnO4 solution

in the presence of H2SO4. The titration gives unsatisfactory result when carried out in the

presence of HCl, because HCl

(A) Reduces permanganate to Mn2+.

(B) Oxidises oxalic acid to carbon dioxide and water.

(C) Gets oxidised by oxalic acid to chlorine.

(D) Furnishes H+ ions in addition to those from oxalic acid.

6. The organic chloro compound, which shows complete stereochemical inversion during a SN2 reaction, is

i. (CH3)2CHCl

ii. CH 3Cl

iii. (C 2H5)2CHCl

iv. (CH 3)3CCl

7. Toluene is nitrated and the resulting product is reduced with tin and hydrochloric acid. The product

so obtained is diazotized and then heated with cuprous bromide. The reaction mixture so formed contains

(A) mixture of o-; and p-bromoanilines

(B) mixture of o- and m-bromotoluenes

(C) mixture of o- and p-bromotoluenes

(D) mixture of o- and p-dibromobenzenes

8. In context with the industrial preparation of hydrogen from water gas (CO + H2), which of the

following is the correct statement?

(A) H2 is removed through occlusion with Pd.

(B) CO is oxidised to CO2 with steam in the presence of a catalyst followed by absorption of CO2in alkali.

(C) CO and H2 are fractionally separated using differences in their densities.

(D) CO is removed by absorption in aqueous Cu 2Cl 2 solution.

9. The formation of the oxide ion O2-(g) requires first an exothermic and then an endothermic step as shown below

O (g) + e− → O− (g)ΔHo = −142kJmol−1

O− (g) + e− → O2− (g)ΔHo = 844kJmol−1

(A) Oxygen is more electronegative

(B) O- ion has comparatively larger size than oxygen atom

(C) O- ion will tend to resist the addition of another electron

(D) Oxygen has high electron affinity

10. The bond order in NO is 2.5 while that in NO+ is 3. Which of the following statements is true for these two species? (A) Bond length in NO+ is greater than in NO (B) Bond length is unpredictable (C) Bond length in NO+ in equal to that in NO (D) Bond length in NO is greater than in NO+

11. The enthalpies of combustion of carbon and carbon monoxide are -393.5 and -283 kJ mol-1 respectively. The enthalpy of formation of carbon monoxide per mole is (A) 110.5 kJ (B) -110.5 kJ (C) -676.5 kJ (D) 676.5 kJ

12. Excess of KI reacts with CuSO4 solution and then Na2S2O3 solution is added to it. Which of the

statements is incorrect for this reaction?

(A) Cu 2I2 is reduced (B) Evolved I2 is reduced

(C) Na 2S2 O3 is oxidized (D) CuI2 is formed

13. One mole of magnesium nitride on the reaction with an excess of water gives (A) one mole of ammonia (B) two moles of nitric acid (C) two moles of ammonia (D) one mole of nitric acid

14. Which of the following undergoes reaction with 50% sodium hydroxide solution to give the corresponding alcohol and acid?

(A) Phenol (B) Benzoic acid

(C) Butanal (D) Benzaldehyde

15. The oxidation state of Cr in [Cr (NH3)4Cl2]+ is (A) +3 (B) +2 (C) +1 (D) 0

16. Identify the correct statements regarding enzymes (A) Enzymes are specific biological catalysts that can normally function at very high temperature (T ~ 1000 K) (B) Enzymes are specific biological catalysts that the possess well – defined active sites (C) Enzymes are specific biological catalysts that cannot be poisoned (D) Enzymes are normally heterogeneous catalysts that are very specific in their action

17. Which one of the following ores is best concentrated by froth – floatation method? (A) Magnetite (B) Malachite (C) Galena (D) Cassiterite

18. Beryllium and aluminium exhibit many properties which are similar. But the two elements differ in (1) exhibiting maximum covalency in compound (2) exhibiting amphoteric nature in their oxides (3) forming covalent halides (4) forming polymeric hydrides

19. Aluminium chloride exists as dimer, Al 2Cl6 in solid state as well as in solution of non-polar solvents such as benzene. When dissolved in water, it gives (A) Al2+ + 3Cl- (B) Al 2O3 + 6HCl (C) [Al (OH)6]3- (D) [Al(H 2O)6]3+ + 3Cl-

20. The increasing order of the ionic radii of the given isoelectronic species is : (A) Cl–, Ca2+, K+, S2–

(B) S2–, Cl–, Ca2+, K+

(C) Ca2+, K+, Cl–, S2–

(D) K+, S2–, Ca2+, Cl–

21. Iodoform can be prepared from all except: (A) Ethyl methyl ketone (B) Isopropyl alcohol (C) 3–Methyl – 2– butanone (D) Isobutyl alcohol

22. Which one of the following statements is correct? (A) All amino acids except lysine are optically active. (B) All amino acids are optically active. (C) All amino acids except glycine are optically active. (D) All amino acids except glutamic acid are optically active.

23. Very pure hydrogen (99.9%) can be made by which of the following processes? (A) Reaction of methane with steam. (B) Mixing natural hydrocarbons of high molecular weight. (C) Electrolysis of water. (D) Reaction of salt like hydrides with water.

24. Iron exhibits + 2 and +3 oxidation states. Which of the following statements about iron is incorrect? (A) Ferrous oxide is more basic in nature than the ferric oxide. (B) Ferrous compounds are relatively more ionic than the corresponding ferric compounds. (C) Ferrous compounds are less volatile than the corresponding ferric compounds. (D) Ferrous compounds are more easily hydrolysed than the corresponding ferric compounds.

25. For a spontaneous reaction the ΔG, equilibrium constant (K) and o cell E will be respectively (A) -ve, >1, +ve (B) +ve, >1, -ve (C) -ve, <1, -ve (D) -ve, >1, -ve

26. The major product obtained on interaction of phenol with sodium hydroxide and carbon dioxide is: (A) benzoic acid (B) salicylaldehyde (C) salicylic acid (D) phthalic acid

27. Which of the following statements is incorrect regarding physissorptions? (A) It occurs because of vander Waal’s forces. (B) More easily liquefiable gases are adsorbed readily. (C) Under high pressure it results into multi molecular layer on adsorbent surface. (D) Enthalpy of adsorption ΔH is low and positive.

28. A binary liquid solution is prepared by mixing n-heptane and ethanol. Which one of the following statements is correct regarding the behaviour of the solution ? (A) The solution formed is an ideal solution. (B) The solution is non-ideal, showing +ve deviation from Raoult’s law. (C) The solution is non-ideal, showing –ve deviation from Raoult’s law. (D) n-heptane shows +ve deviation while ethanol shows –ve deviation from Raoult’s law.

29. The bond dissociation energy of B – F in BF3 is 646 kJ mol−1 whereas that of C-F in CF4 is 515kJmol−1. The correct reason for higher B-F bond dissociation energy as compared to that of C- F is: (A) smaller size of B-atom as compared to that of C- atom (B) stronger σ bond between B and F in BF3 as compared to that between C and F in CF4 (C) significant pπ - pπ interaction between B and F in BF3 whereas there is no possibility of such interaction between C and F in CF4 . (D) lower degree of pπ - pπ interaction between B and F in 3 BF than that between C and F in CF4 .

30. Two liquids X and Y form an ideal solution. At 300K, vapour pressure of the solution containing 1 mol of X and 3 mol of Y is 550 mm Hg. At the same temperature, if 1 mol of Y is further added to this solution, vapour pressure of the solution increases by 10 mm Hg. Vapour pressure (in mmHg) of X and Y in their pure states will be, respectively : (A) 200 and 300 (B) 300 and 400 (C) 400 and 600 (D) 500 and 600

Mathematics PART – III

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 greatest and least value of log√2 (sin 𝑥 − cos 𝑥 + 3√2) are respectively

(A) 2 and 1 (B) 5 and 3

(C) 7 and 5 (D) 9 and 7

2. The solution of | cos 𝑥| = cos 𝑥 − 2 sin 𝑥 is

(A) 𝑥 = 𝑛𝜋, 𝑛𝜀𝐼 (B) 𝑥 = 𝑛𝜋 +𝜋

4, 𝑛𝜀𝐼

(C) 𝑥 = 𝑛𝜋 + (−1)𝑛 𝜋

4, 𝑛𝜀𝐼 (D) (2n+1)π +π/4

3. Let f, g, h be the length of the perpendicular from the circumcenter of the triangle ABC on the

sides a, b and c respectively. If 𝑎

𝑓+

𝑏

𝑔+

𝑐

ℎ= 𝜆

𝑎𝑏𝑐

𝑓𝑔ℎ then the value of λ is

(A) 1/4 (B) 1

(C) 1 (B) 2

4. If f(x) =Sgn (x3) where Sgn is signum function, then

(A) f is continuous but not derivable at x=0

(B) f’(0+)=2

(C) f’(0-)=2

(D)f is not derivable at x=0

5. If P,P’ represents a complex number z and its additive inverse respectively, then the complex

equation of the circle with PP’ as diameter is

(A) z/z1=conj(z/z1)

(B) z.conj(z)+ z1 .conj(z1)=0

(C) z.conj(z1)+ z1 .conj(z)=0

(D) none of these

6. If one root of the quadratic equation ax2+bx+c=0 is equal to the nth power of the other root, then

the value of

(acn)1/(n+1) + (can)1/(n+1) =

(A) b

(B) –b

(C) 1/bn+1

(D) -1/bn+1

7. The streets of a city are arranged like the lines of a chess board. There are m streets running north

to south and n streets running east to west. The no. of ways in which a man can travel from NW to

SE corner going the shortest distance is

(A) (m2+n2)1/2

(B) {(m-1)2(n-1)2}1/2

(C) (m+n)! / m!.n!

(D) (m+n-2)! / (m-1)!(n-1)!

8. If an = ∑nr=0 1/nCr then ∑n

r=0 r/nCr is equal to

(A) (n-1)an

(B) nan

(C) nan/2

(D) none of these

9. Suppose a, b, c are in A.P. and a2,b2,c2 are in G.P. If a<b<c and a+b+c = 3/2, then the value of a is

(A) 1/2√2

(B) 1/2√3

(C) 1/2 - 1/√3

(D) 1/2 - 1/√2

10. The sum of two positive quantities is 2n. The probability that their not less than ¾ times the

greatest product is

(A) ¾

(B) ½

(C) ¼

(D) None of these

11. The equation of the pair of bisectors of the angle between two straight lines is 12x2-7xy-12y2=0.

If the equation of one of the line is 2y-x=0, then the equation of the other line is

(A) 41x-38y=0

(B) 38x-41y=0

(C) 38x+41y=0

(D) 41x+38y=0

12 The radius of the circle, having centre (2,1) whose one of the chord is a diameter of the circle

x2+y2-2x-6y+6=0 is

(A) 1

(B) 2

(C) 3

(D) √3

13. The locus of middle point of the intercept of the tangents drawn from an internal point to the

ellipse x2+2y2=2, between the co-ordinates axes is

(A) 1/x2 + 1/y2 = 1 (B) 1/4x2 + 1/2y2 = 1

(C) 1/2x2 + 1/4y2 = 1 (D) 1/2x2 + 1/y2 = 1

14. If a and b are two unit vectors such that a + 2b and 5a – 4b are inclined to each other then the

angle between a and b is

(A) 45° (B) 60°

(C) cos-1 (1/3) (D) cos-1 (2/7)

15. Limx→0 (x tan2x – 2x tanx) / (1 – cos2x)2 is

(A) 2 (B) -2

(C) ½ (D) -1/2

16. If (d2y/dx2)(dy/dx)3 + d2y/dx2 = K , then the value of K is equal to

(A) 1 (B) -1

(C) 2 (D) 0

17. The number of real solutions of tan-1 √x(x+1) + sin-1√x2+x+1 = π/2 is

(A) 0 (B) 1

(C) 2 (D) ∞

18. If w is a cube root of unity and 𝐴 = [11

1 1⍵ ⍵2

1 ⍵2 ⍵

]

(A) = [1

⍵2⍵ ⍵2

1 ⍵⍵ ⍵2 1

]

(B) = [11

1 1⍵ ⍵2

1 ⍵2 ⍵

]

(C) = 1/3 [11

⍵ ⍵2

⍵2 ⍵1 1 1

]

(D) = 1/2 [⍵1

1 ⍵2

1 1⍵2 1 ⍵

]

19. The value of R which makes 𝑓(𝑥) = {sin (

1

𝑥 ) , 𝑥 ≠ 0

𝑅 , 𝑥 = 0} is continuous at x=0.

A) 8

B) 1

C) -1

D) None of these

20. Let 𝑓(𝑥) = {𝑥𝑎 log 𝑥 , 𝑥 > 00 , 𝑥 = 0

} , Rolles theorem is applicable to 𝑓 for 𝑥 ∈ [0,1] If a is?

A) -2

B) -1

C) 0

D) 1

2

21. If for a real no. 𝑦, [𝑦] is greatest integer less than or equal to 𝑦 , then the value of integral

∫ [2sin (𝑥)]𝑑𝑥3𝜋/2

𝜋/2 is

A) – 𝜋

B) 0

C) −𝜋/2

D) 𝜋/2

22. If 𝐴1 is the area enclosed by the curve 𝑥𝑦 = 1, 𝑥 𝑎𝑥𝑖𝑠 and the ordinates 𝑥 = 1, 𝑥 = 2 and 𝐴2 is

the area enclosed by the curve 𝑥𝑦 = 1, 𝑥 𝑎𝑥𝑖𝑠 and the ordinates 𝑥 = 2 , 𝑥 = 4 then

A) 𝐴1 = 2𝐴2

B) 𝐴2 = 2𝐴1

C) 𝐴1 = 𝐴2

D) 𝐴1 = 𝐴2/3

23. If (2+sin (𝑥)

1+𝑦)

𝑑𝑦

𝑑𝑥= − cos 𝑥 , 𝑦(0) = 1 then 𝑦(𝜋/2) =

A) 1

B) ½

C) 1/3

D) ¼

24. If AB is double ordinate of the hyperbola 𝑥

𝑎2 +𝑦

𝑏2 = 1 such that ∆OAB (O being origin) is an

equilateral triangle, then the eccentricity ‘e’ of the hyperbola satisfies

A) e > √3

B) 1 < e < 22

√3

C) e = 2

√3

D) e > 2

√3

25. If 𝑎→,

𝑏→ 𝑎𝑛𝑑

𝑐→ are unit vectors, then |

𝑎→ _

𝑏→|

2+|

𝑏→ _

𝑐→|

2+|

𝑐→ _

𝑎→|

2 does not exceeds

A) 4

B) 9

C) 8

D) 6

26. If 𝑔(𝑥) = ∫ cos4 𝑡𝑑𝑡𝑥

0 then If 𝑔(𝑥 + 𝜋) equals

A) 𝑔(𝑥) + 𝑔(𝜋)

B) 𝑔(𝑥) − 𝑔(𝜋)

C) 𝑔(𝑥)𝑔(𝜋)

D) 𝑔(𝑥)/𝑔(𝜋)

27. If 𝑓(𝑥) = 𝑥𝑒𝑥(1−𝑥) then 𝑓(𝑥) is

A) Increasing on (-1/2,1)

B) Decreasing on R

C) Increasing on R

D) Decreasing on [-1/2,1]

28. If (1 + 𝑥)𝑛 = 𝐶0 + 𝐶1𝑥 + 𝐶2𝑥2 + ⋯ + 𝐶𝑛𝑥𝑛 then the value of 𝐶0 − 𝐶2 + 𝐶4 − 𝐶6 + ⋯ is

A) 2𝑛

B) 2𝑛cos (𝑛𝜋

2)

C) 2𝑛sin (𝑛𝜋

2)

D) 2𝑛/2cos (𝑛𝜋

4)

29. Distance between chords of contact of tangents to the circle; 𝑥2 + 𝑦2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0

from the origin and point (g,f) is

A) √𝑔2 + 𝑓2

B) √𝑔2+𝑓2−𝑐

2

C) 𝑔2+𝑓2−𝑐

2 √𝑔2+𝑓2

D) √𝑔2+𝑓2+𝑐

2 √𝑔2+𝑓2

30. If 𝑓(𝑥) =1

2tan (

𝜋𝑥

2) ; (−1 < 𝑥 < 1) and 𝑔(𝑥) = √3 + 4𝑥 − 4𝑥2 then the domain of 𝑔𝑜𝑓 is

A) (-1 , 1)

B) [ -1/2 , 1/2 ]

C) [ -1 , 1/2 ]

D) [ -1/2 , 1 ]