West Bengal Joint Entrance Examination (WBJEE) is must for a student to clear to get admissions into premier institutes of West Bengal. WBJEE 2020 was held on 2nd February 2020, for all the three subjects. In WBJEE 2020, Biological Sciences Paper will be conducted between 9.00 a.m. to 11.30 a.m. (2 ½ hours) Physics + Chemistry Paper will be conducted between 12.45 p.m. to 2.45 p.m. (2 hours) & Mathematics conducted between 3.30 p.m. to 5.30 p.m. (2 hours). The WBJEE 2020 Answer keys will be published after the WBJEE Exam is conducted.

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## Here are the WBJEE answer keys for 2016 –

### Physics and Chemistry: WBJEE Answer Key 2016

Subject: Physics

Category – I (Q.1 to Q.30)

Only one answer is correct. The correct answer will fetch full marks 1. Incorrect answer or any

combination of more than one answer will fetch – 1⁄4 marks.

1. The velocity of sound in air at 20^{o}C and 1 atm pressure is 344.2 m/s. At 40^{o}C and 2 atm pressure, the velocity of sound in air is approximately

(A) 350 m/s

(B) 356 m/s

(C) 363 m/s

(D) 370 m/s

Solution : (B)

2. The perfect gas equation for 4 gm of hydrogen gas is

(A) PV = RT

(B) PV = 2RT

(C) PV = ½RT

(D) PV = 4RT

Solution : (B)

3. If the temperature of the Sun gets doubled, the rate of energy received on the Earth Will increase by a factor of

(A) 2

(B) 4

(C) 8

(D) 16

Solution : (D) ; Comments: Temperature must be in absolute scale.

4. A particle vibrating simply harmonically has an acceleration of 16 cms when it is at a distance of 4 cm from the mean position. Its time period is

(A) 1s

(B) 2.572 s

(C) 3.142 s

(D) 6.028 s

Solution : (C)

5. Work done for a certain spring when stretched through 1 mm is 10 Joule. The amount of work that must be done on the spring to stretch it further by 1 mm is

(A) 30 J

(B) 40 J

(C) 10 J

(D) 20 J

Solution : (A)

6. If the r.m.s velocity of Hydrogen gas at a certain temperature is c, then the r.m.s velocity of Oxygen gas at the same temperature is

(A) c/8

(B) c/10

(C) c/4

(D) c/2

Solution : (C)

7.

For air at room temperature the atmospheric pressure is 1.0 × 10 5 Nm2 and density of air is 1.2 Kg/m3 . For a tube of length 1.0 m closed at one end the lowest frequency generated is 84 Hz. The value of (ratio of two specific heats) for air is

(A) 2.1

(B) 1.5

(C) 1.8

(D) 1.4

Solution : (D)

8. A gas bubble of 2 cm diameter rises through a liquid of density 1.75 gm/cm3 with a fixed speed of 0.35 cm/s. Neglect the density of the gas. The coefficient of the viscosity of the liquid is

(A) 870 poise

(B) 1120 poise

(C) 982 poise

(D) 1089 poise

Solution : (B)

9. The temperature of the water of a pond is 0^{o}C while that of the surrounding atmosphere is 20^{o}C. If the density of ice is, the coefficient of thermal conductivity is k and latent heat of melting is L then the thickness Z of ice layer formed increases as a function of time t as

(A) Z^2 = 60kt/ρL

(B) Z = √(40kt/ρL)

(C) Z = 40k√t/ρL

(D) Z^2 = 40kt/ρL

Solution : (D)

10. 1000 droplets of water having 2 mm diameter each coalesces to form a single drop. Given the surface tension of water is 0.072 Nm 1. The energy loss in the process is

(A) 8.146 × 104 J

(B) 4.4 × 104 J

(C) 2.108 × 105 J

(D) 4.7 × 10 J

Solution : (A)

11. A zener diode having break-down voltage 5.6 V is connected in reverse bias with a battery of emf 10 V and a resistance of 100 in series. The current flowing through the Zener is

(A) 88 mA

(B) 0.88 mA

(C) 4.4 mA

(D) 44 mA

Solution : (D)

12. In case of a bipolar transistor = 45. The potential drop across the collector resistance of 1 k is 5 V. The base current is approximately

(A) 222 A

(B) 55 A

(C) 111 A

(D) 45 A

Solution : (C)

13. An electron enters an electric field having intensity E = 3i+6j+2k Vm and magnetic field having induction B = 12i+3j T with a velocity V = 3j m/s (Given e = 1.6 × 10-19 C). The magnitude of the force acting on the electron is

(A) 2.02 × 10-18 N

(B) 5.16 × 10-16 N

(C) 3.72 × 10-17 N

(D) 4.41 × 10-18N

Solution : (Does not match); Probable Ans : 1.12 × 10–18 N

14. Two coils of self inductances 6mH and 8mH are connected in series and are adjusted for the highest coefficient of coupling. Equivalent self-inductance L for the assembly is approximately

(A) 50 mH

(B) 36 mH

(C) 28 mH

(D) 18 mH

Solution : (C)

15. A 1 F capacitor C is connected to a battery of 10 V through a resistance 1M. The voltage across C after 1 sec is approximately

(A) 5.6 V

(B) 7.8 V

(C) 6.3 V

(D) 10 V

Solution : (C)

16. Two equal resistances, 400 each, are connected in series with an 8 V battery. If the resistance of the first one increases by 0.5 %, the change required in the resistance of the second one in order to keep the potential difference across it unaltered is to

(A) increase it by 1

(B) increase it by 2

(C) increase it by 4

(D) decrease it by 4

Solution : (B)

17. The angle between an equipotential surface and electric lines of force is

(A) 0^{o}

(B) 90^{o}

(C) 180^{o}

(D) 270^{o}

Solution : (B)

18. The equivalent capacitance between A & B in the figure is

(A) 20 F

(B) 8 F

(C) 12 F

(D) 16 F

Solution : (B)

19. Two wires of same radius having lengths l1 and l2 and resistivities ρ1 and ρ2. The equivalent resistivity will be

Solution : (B)

20. A hollow metal sphere of radius R is charged with a charge Q. The electric potential and intensity inside the sphere is respectively

(A) Q/4πεR^2 and Q/4πεR

(B) Q/4πεR and Zero

(C) Zero and Zero

(D) 4πεQ/R and Q/4πεR^2

Solution : (B)

21. The potential difference V required for accelerating an electron to have the de Broglie wavelength of 1Å is

(A) 100 V

(B) 125 V

(C) 150 V

(D) 200 V

Solution : (C)

22. The work function of Cesium is 2.27 eV. The cut-off voltage which stops the emission of electrons from a cesium cathode irradiated with light of 600 nm wavelength is

(A) 0.5 V

(B) 0.2 V

(C) 0.5 V

(D) 0.2 V

Solution :

23. The number of Broglie wavelengths contained in the second Bohr orbit of a Hydrogen atom is

(A) 1

(B) 2

(C) 3

(D) 4

Solution : (B)

24. The wavelength of the second Balmer line in Hydrogen spectrum is 600 nm. The wavelength for its third line in Lymann series is

(A) 800 nm

(B) 600 nm

(C) 400 nm

(D) 200 nm

Solution :

25.

A ray of light strikes a glass plate at an angle of 60o. If the reflected and refracted rays are perpendicular

to each other, the refractive index of glass is

(A) √3/2

(B) 3/2

(C) 1/2

(D) √3

Solution : (D)

26.

Light travels through a glass plate of thickness t and having refractive index . If c be the velocity of light in vacuum, time taken by the light to travel through this thickness of glass is

(A) t/μc

(B) tc/μ

(C) μt/c

(D) μtc

Solution : (C)

27.

If x = at + bt^2 where x is in metre (m) and t is in hour (hr) then unit of b will be

(A) m^2/hr

(B) m

(C) m/hr

(D) m/hr^2

Solution : (D)

28.

The vectors A & B are such that **|A+B|**=**|A-B|** . The angle between the two vectors will be

(A) 0^{o}

(B) 60^{o}

(C) 90^{o}

(D) 45^{o}

Solution : (C)

29.

At a particular height, the velocity of an ascending body is **u** . The velocity at the same height while the body falls freely is

(A) 2**u**

(B) –**u**

(C) **u**

(D) –2**u**

Solution : (B)

30.

Two bodies of masses m1 & m2 are separated by a distance R. The distance of the centre of mass of the bodies from the mass m is

(A) m2R/(m1+m2)

(B) m1R/(m1+m2)

(C) m1m2/(m1+m2) R

(D) m1+m2/m1 R

Solution : (A)

Category – II (Q.31 to Q.35)

Only one answer is correct. Correct answer will fetch full marks 2. Incorrect answer or any

combination of more than one answer will fetch – 1⁄2 marks.

31.

A mass of 1 kg is suspended by means of a thread. The system is (i) lifted up with an acceleration of 4.9 ms^{2}. (ii) lowered with an acceleration of 4.9 ms^{2}. The ratio of tension in the first and second case is

(A) 3 : 1

(B) 1 : 2

(C) 1 : 3

(D) 2 : 1

Solution : (A)

32.

The effective resistance between A and B in the figure is 7/12Ω if each side of the cube has 1 resistance. The effective resistance between the same two points, when the link AB is removed, is

(A) 7/12Ω

(B) 5/12Ω

(C) 7/5Ω

(D) 5/7Ω

Solution : (C)

33.

A current I = I_{0}e^{-λt} is flowing in a circuit consisting of a parallel combination of resistance R and capacitance C. The total charge over the entire pulse period is.

(A) I_{0}/λ

(B) 2I_{0}/λ

(D) I_{0}λ

(C) e^{Ioλ}

Solution : (A)

34.

For Fraunhofer diffraction to occur,

(A) Light source should be at infinity

(B) Both source and screen should be at infinity

(C) Only the source should be a finite distance

(D) Both source and screen should be at a finite distance.

Solution : (B)

35.

The temperature of blackbody radiation enclosed in a container of volume V is increased from 100^{o}C to 1000^{o}C. The heat required in the process is

(A) 4.79 × 10^{–}^{4} cal

(B) 9.21 × 10^{–}^{5} cal

(C) 2.17 × 10^{–}^{4} cal

(D) 7.54 × 10^{–}^{4} cal

Solution :

Category – III (Q.36 to Q.40)

One or more answer(s) is (are) correct. Correct answer(s) will fetch marks 2. Any combination containing one or more incorrect answer will fetch 0 marks. If all correct answers are not marked and also no incorrect answer is marked then score = 2 × number of correct answers marked / actual number of correct answers.

36.

A drop of water detaches itself from the exit of a tap when ( = surface tension of water, = density of water, R = radius of the tap exit, r = radius of the drop)

37.

A rectangular coil carrying current is placed in a non-uniform magnetic field. On that coil the total

(A) force is non-zero

(B) force is zero

(C) torque is zero

(D) torque is non-zero

Solution : (A,D)

38.

A charged particle of mass m 1 and charge q 1 is revolving in a circle of radius r. Another charged particle of charge q 2 and mass m 2 is situated at the centre of the circle. If the velocity and time period of the revolving particle be v and T respectively, then

39. The distance between a light source and a photoelectric cell is d. If the distance is decreased to d/2 then

(A) The emission of electron per second will be four times.

(B) Maximum kinetic energy of photoelectrons will be four times.

(C) Stopping potential will remain the same.

(D) The emission of electrons per second will be doubled.

Solution : (A,C)

40.

A train moves from rest with acceleration and in time t_{1} covers a distance x. It then decelerates to

rest at constant retardation for distance y in time t_{2}. Then

(A) x/y = β/α

(B) β/α = t_{1}/t_{2}

(C) x = y

(D) x/y = βt_{1}/αt_{2}

Solution : (A,B)

Subject: Chemistry

Category – I (Q41 to Q70)

Only one answer is correct. Correct answer will fetch full marks 1. Incorrect answer or any

combination of more than one answer will fetch – 1⁄4 marks.

41.

An element X belong to the fourth period and a fifteenth group of the periodic table. Which of the following statements is true?

(A) It has a completely filled s-orbital and a partially filled d-orbital.

(B) It has completely filled s- and p-orbitals and a partially filled d-orbital

(C) It has completely filled s- and p-orbitals and a half filled d-orbital.

(D) It has a half filled p-orbital, and completely filled s- and d-orbitals.

Solution : (D)

42.

Which of the following plots represent an exothermic reaction ?

Solution : (A)

43.

If P^{0} and P are the vapour pressure of the pure solvent and solution and n1 and n2 are the moles of solute

and solvent respectively in the solution then the forrect relation between P and P^{0} is

(A) P^{0 = }P[n1/(n1+n2)]

(B) P^{0 = }P[n2/(n1+n2)]

(C) P = P^{0}[n2/(n1+n2)]

(D) P = P^{0}[n1/(n1+n2)]

Solution : (C)

44.

Ionic solids with Schottky defect may contain in their structure

(A) cation vacancies only (B) cation vacancies and interstitial cations

(C) equal number of cation and anion vacancies (D) anion vacancies and interstitial anions

Solution : (C)

45.

The condition for a reaction to occur spontaneously is

(A) ΔH must be negative

(B) (ΔH-TΔS) must be negative

(C) ΔS must be negative

(D) (ΔH + TΔS) must be negative

Solution : (C)

46.

The order of equivalent conductances at infinite dilution for LiCl, NaCl and KCl is

(A) LiCl > NaCl > KCl

(B) KCl > NaCl > LiCl

(C) NaCl > KCl > LiCl

(D) LiCl > KCl > NaCl

Solution : (B)

47.

The molar solubility (in mol L 1 ) of a sparingly soluble salt MX 4 is S . The corresponding solubility product

is is Ksp . S in terms of Ksp is given by the relation

(A) S

Ksp

128

1/4

(B) S

Ksp

256

1/5

(C) S = (256 Ksp) 1/5

(D) S = (128 Ksp) 1/4

Solution : (B)

48.

Ozonolysis of an alkene produces only one dicarbonyl compound. The structure of the alkene is :

Solution : (B)

49.

From the following compounds choose the one which is not aromatic :

Solution : (B)

50.

Amongest the following compounds, the one that will not respend to cannizzaro reaction upon treatment

with alkali is

(A) Cl_{3}CCHO

(B) Me_{3}CCHO

(C) C_{6}H_{5}CHO

(D) HCHO

Solution : (A)

51.

Which of the following compounds would not react with Lucas reagent at room temperature ?

(A) H_{2}C=CHCH_{2}OH

(B) C_{6}H_{5}CH_{2}OH

(C) CH_{3}CH_{2}CH_{2}OH

(D) (CH_{3})_{3}COH

Solution : (C)

52.

Amongest the following compounds the one which would not respond to iodoform test is

(A) CH_{3}CH(OH)CH_{2}CH_{3}

(B) ICH_{2}COCH_{2}CH_{3}

(C) CH_{3}COOH

(D) CH_{3}CHO

Solution : (C)

53.

Which of the following will be dehydrated most readily in alkaline medium ?

Solution : (B)

54.

The correct order of basicity of the following compounds is

(A) 1 < 2 < 3 < 4

(B) 1 < 2 < 4 < 3

(C) 2 < 1 < 3 < 4

(D) 4 < 3 < 2 < 1

Solution : (C)

55.

Which of the following reactions will not result in the formation of carbon carbon bonds ?

(A) Cannizaro reaction

(B) Wurtz reaction

(C) Reimer-Tiemann reaction

(D) Friedel-Crafts acylation

Solution : (A)

56.

Point out the false statement.

(A) Colloidal sols are homogeneous

(B) Colloids carry +ve or ve charges

(C) Colloids show Tyndall effect

(D) The size range of colloidal particles is 10-1000 Å

Solution : (A)

57.

The correct structure of the drug paracetamol is

Solution : (B)

58.

Which of the following statements regarding Lanthanides is false ?

(A) All lanthanides are solid at room temperature.

(B) Their usual oxidation state is +3.

(C) They can be separated from one another by ion-exchange method.

(D) Ionic radii of trivalent lanthanides steadily increases with increase in atomic number.

Solution : (D)

59.

Nitrogen dioxide is not produced on heating

(A) KNO_{3}

(B) Pb(NO_{3})_{2}

(C) Cu(NO_{3})_{2}

(D) AgNO_{3}

Solution : (A)

60.

The boiling points of HF, HCl, HBr and HI follow the order

(A) HF > HCl > HBr > H I (B) HF > H I > HBr > HCl

(C) H I > HBr > HCl > HF (D) HCl > HF > HBr > H I

Solution : (B)

61.

In the solid state PCl 5 exists as

(A) [PCl 4 ] and [PCl 6 ] (B) covalent PCl 5 molecules only

(C) [PCl 4 ] + and [PCl 6 ] ions (D) covalent P 2 Cl 10 molecules only

Solution : (C)

62.

Which statement is not correct for ortho and para hydrogen ?

(A) They have different boiling points

(B) Ortho-form is more stable than para-form

(C) They differ in their nuclear spin

(D) The ratio of ortho to para hydrogen changes with change in temperature

Solution : (B)

63.

The acid in which O O bonding is present is

(A) H_{2}S_{2}O_{3}

(B) H_{2}S_{2}O_{6}

(C) H_{2}S_{2}O_{8}

(D) H_{2}S_{4}O_{6}

Solution : (C)

64.

The metal which can be used to obtain metallic Cu from aqueous CuSO_{4} solution is

(A) Na

(B) Ag

(C) Hg

(D) Fe

Solution : (D)

65.

If radium and chlorine combine to form radium chloride, the compound would be

(A) half as radioactive as radium

(B) twice as radioactive

(C) as radioactive as radium

(D) not radioactive

Solution : (C)

66.

Which of the following arrangements is correct in respect of solubility in water ?

(A) CaSO 4 > BaSO 4 > BeSO 4 > MgSO 4 > SrSO 4

(B) BeSO 4 > MgSO 4 > CaSO 4 > SrSO 4 > BaSO 4

(C) BaSO 4 > SrSO 4 > CaSO 4 > MgSO 4 > BeSO 4

(D) BeSO 4 > CaSO 4 > MgSO 4 > SrSO 4 > BaSO 4

Solution : (B)

67.

The energy required to break one mole of hydrogen-hydrogen bonds in H 2 is 436 kJ. What is the longest

wavelength of light required to break a single hydrogen-hydrogen bond ?

(A) 68.5 nm

(B) 138 nm

(C) 274 nm

(D) 548 nm

Solution : (C)

68.

The correct order of O-O bond length in O_{2} , H_{2}O_{2} and O_{3} is

(A) O_{2} > O_{3} > H_{2}O_{2}

(B) H_{2}O_{2} > O_{3} > O_{2}

(C) O_{3} > O_{2} > H_{2}O_{2}

(D) O_{3} > H_{2}O_{2} > O_{2}

Solution : (B)

69.

The number of σ bonds and π bonds between two carbon atoms in calcium carbide are

(A) one σ, one π

(B) one σ, two π

(C) two σ, one π

(D) one σ, 1 1⁄2 π

Solution : (B)

70.

An element E loses one α and two β particles in three successive states. The resulting element will be

(A) An isobar of E (B) An isotone of E

(C) An isotope of E (D) E itself

Solution : (C)

Category – II (Q.71 to Q.75)

Only one answer is correct. Correct answer will fetch full marks 2. Incorrect answer or any

combination of more than one answer will fetch – 1⁄2 marks.

71.

Among the following, which should have the highest r.m.s. speed at the same temperatrue ?

(A) SO_{2}

(B) CO_{2}

(C) O_{2}

(D) H_{2}

Solution : (D)

72.

The major products obtained during ozonolysis of 2,3 dimethyl-1-butene and subsequent reductions with Zn and H_{2}O

(A) Methanoic acid and 2-methyl-2-butanone

(B) Methanal and 3-methyl-2-butanone

(C) Methanol and 2,2-dimethyl-3-butanone

(D) Methanoic acid and 2-methyl-3-butanone

Solution : (B)

73.

Identify X in the following sequence of reactions :

Solution : (B)

74.

Solution : (A)

75.

The time taken for an electron to complete one revolution in Bohr orbit of hydrogen atom is

Solution : (C)

Category – III (Q.76 to Q.80)

One or more answer(s) is (are) correct. Correct answer(s) will fetch marks 2. Any

combination containing one or more incorrect answer will fetch 0 marks. If all correct

answers are not marked and also no incorrect answer is marked then score = 2 × number

of correct answers marked / actual number of correct answers.

76.

In which of the following mixed aqueous solutions pH=pKa at equilibrium ?

(1) 100 ml of 0.1 M CH 3 COOH + 100 ml of 0.1 M CH 3 COONa

(2) 100 ml of 0.1 M CH 3 COOH + 50 ml of 0.1 M NaOH

(3) 100 ml of 0.1 M CH 3 COOH + 100 ml of 0.1 M NaOH

(4) 100 ml of 0.1 M CH 3 COOH + 100 ml of 0.1 M NH 3

(A) (1) is correct (B) (2) is correct

(C) (3) is correct (D) both (1) and (2) are correct

Solution : (A,B,D)

77.

Amongst the following compounds, the one(s) which readily react with ethanolic KCN ?

(A) Ethyl chloride

(B) Chloro benzene

(C) Benzaldehyde

(D) Salicylic acid

Solution : (A,C)

78.

Choose the correct statement(s) among the following

Solution : (B,D)

79.

Which of the following statement(s) is (are) correct when a mixture of NaCl and K2Cr2O7 is gently warmed with conc. H2SO4

(A) A deep red vapour is evolved

(B) The vapour when passed through NaOH solution, gives a yellow solution

(C) Chlorine gas is also evolved

(D) Chromyl chloride is formed

Solution : (A,B,C,D)

80.

Of the following molecules, which have shape similar to CO 2 ?

(A) HgCl2

(B) SnCl2

(C) C2H2

(D) NO2

Solution : (A,C)

## Mathematics : WBJEE Answer Key 2016

Category

I (Q.1 to Q.50)

Only one answer is correct. Correct answer will fetch full marks 1. Incorrect answer or any

combination of more than one answer will fetch

1.

The number of ways in which the letters of the word ARRANGE can be permuted such that the R s occur

together is

(A) 7!/(2!2!)

(B) 7!/2!

(C) 6!/2!

(D) 5!/2!

Solution : (C)

2.

If, 1/^{5}C_{r }_{+ }

6

1

4

C r

C r

, then the value of r equals to

(A) 4

(B) 2

(C) 5 (D) 3

(C) 5 (D) 6

Solution : (B)

3.

For +ve integer n, n^{3} + 2n is always divisible by

(A) 3

(B) 7

Solution : (A)

4.

In the expansion of (x

1) (x

(A) 684

(B)

2) …. (x

18), the coefficient of x 17 is

171

(C) 171

(D) 342

(D) 2cos 2

Solution : (C)

5.

1 + n C 1 cos

+ n C 2 cos 2 + … + n C n cos n equals

n

(A)

2cos

2

cos

n

2

2

(B) 2cos

n

2

2n

(C) 2cos

2

n

2

Solution : (A)

6.

If x, y and z be greater than 1, then the value of

1

log x y log x z

log y x

1

log y z

log z x log z y

is

1

(A) log x . log y . log z (B) log x + log y + logz

(C) 0 (D) 1

{(log x). (log y) . (log z)}

Solution : (C)

7.

Let A is a 3 × 3 matrix and B is its adjoint matrix. If B = 64, then A =

(A) ± 2

(B) ± 4

(C) ± 8

(D) ± 12

Solution : (B)

cos

8.

Let Q

sin

sin

4

4

cos

1

4

2

1

and x

4

then Q 3 x is equal to

2

1

0

1

(A)

1

2

(B)

1

0

(C)

1

2

1

(D)

2

2

Solution : (D)

9.

Let R be a relation defined on the set Z of all integers and xRy when x + 2y is divisible by 3.

Then

(A) R is not transitive (B) R is symmetric only

(C) R is an equivalence relation (D) R is not an equivalence relation

Solution : (D)

10.

If A = {5 n

4n

1 : n N} and B = {16(n

(A) A = B

(B) A

1) : n N}, then

B =

(C) A

(D) B

B

A

Solution : (C)

11.

If the function f :

is defined by f(x) = (x 2 + 1) 35

x

R , then f is

(A) one-one but not onto (B) onto but not one-one

(C) neither one-one nor onto (D) both one-one and onto

Solution : (C)

12.

Standard Deviation of n obervations a 1 , a 2 , a 3 , … a n is . Then the standard deviation of observations

a 1 , a 2 , …, a n is

(B)

(A)

(C)

(D)

11

Solution : (C)

13.

Let A and B be two events such that P A

(A) A and B are independent

(C) P

A

B

B

1

,P A

6

B

31

and P B

45

7

then

10

(B) A and B are mutually exclusive

1

6

(D) P

B

A

1

6

Solution : (A)

14.

The value of cos 15ocos 7

(A)

1

2

1o

1o

is

sin7

2

2

(B)

1

8

(C)

1

4

(D)

1

16

Solution : (B)

15.

The smallest positive root of the equation tan x

(A) (0, /2)

(B) ( /2, )

x = 0 lies in

,

(C)

3

2

(D)

3

,2

2

Solution : (C)

16.

If in a triangle ABC, AD, BE and CF are the altitudes and R is the circumradius, then the radius of the

circumcircle of DEF is

(A)

R

2

(B)

2R

3

(C)

1

R

3

(D) None of these

Solution : (A)

17.

The points ( a, b), (a, b), (0, 0) and (a 2 , ab), a

0, b

0 are always

(A) collinear (B) vertices of parallelogram

(C) vertices of a rectangle (D) lie on a circle

Solution : (A)

18.

The line AB cuts off equal intercepts 2a from the axes. From any point P on the line AB perpendiculars

PR and PS are drawn on the axes. Locus of mid-point of RS is

(A) x

y

a

2

(B) x + y = a

(C) x 2 + y 2 = 4a 2

(D) x 2

y 2 = 2a 2

Solution : (B)

19.

x + 8y 22 = 0, 5x + 2y 34 = 0, 2x 3y + 13 = 0 are the three sides of a triangle. The area of the triangle

is

(A) 36 square unit

(B) 19 square unit

(C) 42 square unit

(D) 72 square unit

Solution : (B)

20.

The line through the points (a, b) and ( a, b) passes through the point

(A) (1, 1)

(B) (3a, 2b)

(C) (a 2 , ab)

(D) (a, b)

Solution : (C)

21.

The locus of the point of intersection of the straight lines

x

a

y

b

K and

x

a

y

b

1

, where K is a non-

K

zero real variable, is given by

(A) a straight line

(B) an ellipse

(C) a parabola

(D) a hyperbola

Solution : (D)

22.

The equation of a line parallel to the line 3x + 4y = 0 and touching the circle x 2 + y 2 = 9 in the first quadrant

is

(A) 3x + 4y = 15

(B) 3x + 4y = 45

(C) 3x + 4y = 9

(D) 3x + 4y = 27

Solution : (A)

23.

A line passing through the point of intersection of x + y = 4 and x y = 2 makes an angle tan

the x-axis. It intersects the parabola y 2 = 4(x

x 1 x 2 is equal to ;

(A)

16

9

(B)

32

9

1

3

4

with

3) at points (x 1 , y 1 ) and (x 1 , y 2 ) respectively. then

(C)

40

9

(D)

80

9

Solution : (C)

24.

The equation of auxiliary circle of the ellipse 16x 2 + 25y 2 + 32x

(A) x 2 + y 2 + 2x 4y

(C) (x + 1) 2 + (y 2) 2 = 400

20 = 0

100y = 284 is

(B) x 2 + y 2 + 2x 4y = 0

(D) (x + 1) 2 + (y 2) 2 = 225

Solution : (A)

25.

If PQ is a double ordinate of the hyperbola

centre. Then the eccentricity e satisfies

2

(A) 1 e

y 2

a 2 b 2

2

(B) e

3

x 2

1 such that OPQ is equilateral, O being the

(C) e

2

3

2

2

(D) e

3

Solution : (D)

26.

If the vertex of the conic y 2

2x + 2y 1 = 0 then

(A) 2 < a < 4

4y = 4x

1

2

(B)

a

4a always lies between the straight lines; x + y = 3 and

2

(C) 0 < a < 2

1

2

(D)

a

3

2

Solution : (B)

27.

A straight line joining the points (1, 1, 1) and (0, 0, 0) intersects the plane 2x + 2y + z = 10 at

(A) (1, 2, 5)

(B) (2, 2, 2)

(C) (2, 1, 5)

(D) (1, 1, 6)

Solution : (B)

28.

Angle between the planes x + y + 2z = 6 and 2x

(A)

(B)

4

y + z = 9 is

(C)

6

(D)

3

2

Solution : (C)

29.

If y = (1 + x) (1 + x 2 ) (1 + x 4 ) …. (1 + x 2n ) then the value of

(A) 0

(B)

1

dy

dx

at x = 0 is

(C) 1

(D) 2

Solution : (C)

30.

If f(x) is an odd differentiable function defined on (

(A) 0

, ) such that f (3) = 2, then f ( 3) equal to

(B) 1 (C) 2

(B) does not exist (C) is

(D) 4

Solution : (C)

1

31.

1 x

lim

x 1 2 x

x

1 x

(A) is 1

2

3

(D) is ln 2

Solution : (C)

log

32.

If f x

tan

1

(A) x 2

e

x 2

log ex

2

tan

1

(B) x

3 2log x

then the value of f (x) is

1 6log x

(C) 1

(D) 0

Solution : (D)

log x

dx is equal to

3x

33.

(A)

1

log x

3

2

c

(B)

2

log x

3

2

c

(C)

2

log x

3

2

c

(D)

1

log x

3

2

c

Solution : (A)

2 x f x

34.

f x log2 dx is equal to

(A) 2 x f (x) + c

(B) 2 x log 2 + c (C) 2 x f(x) + c (D) 2 x + c

(B) 0 (C) 2 (D) None of these

Solution : (B)

1

1

1 dx

x

log

35.

0

(A) 1

Solution : (B)

36.

The value of lt

n 1

n 2

2

2 2

3

2n 1

is

n

n

(1)

…..

3/2

1

(2)

2

3

2 1

(3)

2

3

2

1

(4)

2

2 2

3

1

Solution : (A)

37.

If the solution of the differential equation x

dy

dx

xe x , xy = e x

y

x + c then

x is

equal to

(A) x + 1

(B) x

1

(C) 1

x

(D) x

Solution : (D)

38.

The order of the differential equation of all parabolas whose axis of symmetry along x-axis is

(A) 2

(B) 3

(C) 1

(D) None of these

Solution : (C)

39.

The line y = x +

(A)

is tangent to the ellipse 2x 2 + 3y 2 = 1. Then

2

(B) 1

is

5

6

(C)

(D)

2

3

Solution : (C)

40.

The area enclosed by y

5 x 2 and y = x

1 is

(A) 5

4 2 sq. units

s (B)

(C) 5

4 1

2 (D)

sq. unitss

5

2

2

2

sq. unitss

5 sq. units

s

Solution : (C)

41.

Let S be the set of points whose abscissas and ordinates are natural numbers. Let P S such that the

sum of the distance of P from (8, 0) and (0, 12) is minimum among all elements in S. Then the number

of such points P in S is

(A) 1

(B) 3

(C) 5

(D) 11

Solution : (B)

42.

Math Ques with answer keys

Time period T of a simple pendulum of length l is given by T 2

l

. If the length is increased by 2%,

g

then an approximate change in the time period is

(A) 2%

(B) 1%

(C)

1

%

2

(D) None of these

Solution : (B)

43.

The cosine of the angle between any two diagonals of a cube is

1

3

(A)

(B)

1

2

(C)

2

3

1

(D)

3

Solution : (A)

44.

If x is a positive real number different from 1 such that log a x, log b x, log c x are in A.P., then

(A) b

a c

2

(B) b ac

(C) c 2 ac

log a b

(D) None of (A), (B), (C) are correct

Solution : (D)

45.

If a, x are real numbers and a < 1, x < 1, then 1 + (1 + a)x + (1 + a + a 2 )x 2 + …… is equal to

1

1

a

1 a x

(A)

(B)

1

1 a 1 x

(C)

1

1

x

1 a x

(D)

1

1

a

x

1 a

Solution : (C)

46.

If log 0.3 (x

(A)

1) < log 0.09 (x

2,

1), then x lies in the interval

(B) (1, 2)

(C) ( 2, 1) (D) None of these

(C) 1 (D) 0

Solution : (A)

i n i n 1 , i

13

47.

The value of

1 , is

n 1

(A) i

(B) i

1

Solution : (B)

48.

If z 1 , z 2 , z 3 are imaginary numbers such that z 1 = z 2 = z 3

(A) equal to 1

(B) less than 1

1

1

1

1 then z 1 + z 2 + z 3 is

z 1 z 2 z 3

(C) greater than 1

(D) equal to 3

Solution : (A)

49.

If p, q are the roots of the equation x 2 + px + q = 0, then

(A) p = 1, q = 2

(B) p = 0, q = 1

(C) p = 2, q = 0

(D) p = 2, q = 1

Solution : (A)

50.

The number of values of k for which the equation x 2 3x + k = 0 has two distinct roots lying in the interval

(0, 1) are

(A) three (B) two

(C) infinitely many (D) no value of k satisfies the requirement

Solution : (D)

Category II (Q.51 to Q.65)

Only one answer is correct. Correct answer will fetch full marks 2. Incorrect answer or any

combination of more than one answer will fetch

51.

1⁄2 marks.

is an imaginary cube root of unity, then the value of

If

2 2 2 2 3 3 2 …… n 1 n n 2

(A)

n 2

2

n 1 n

4

(B)

n 2

2

n 1 n

4

(C)

is

n 2

2

n 1

4

(D)

n 2

n 1 n

4

Solution : (A)

52.

If n C r 1 36 , n C r 84 and n C r 1 126 then the value of n C 8 is

(A) 10

(B) 7

(C) 9

(D) 8

Solution : (C)

53.

In a group 14 males and 6 females, 8 and 3 of the males and females respectively are aged above 40

years. The probability that a person selected at random from the group is aged above 40 years, given

that the selected person is a female, is

(A)

2

7

(B)

1

2

(C)

1

4

(D)

5

6

Solution : (B)

54.

The equation x 3

yx 2 + x

y = 0 represents

(A) a hyperbola and two straight lines (B) a straight line

(C) a parabola and two straight lines (D) a straight line and a circle

Solution : (B)

55.

The locus of the midpoints of chords of the circle x 2 + y 2 = 1 which subtends a right angle at the origin

is

(A) x 2

y 2

1

4

2

(B) x

y 2

1

2

(C) xy = 0

(D) x 2

y 2 = 0

Solution : (D)

56.

The locus of the midpoints of all chords of the parabola y 2 = 4ax through its vertex is another parabola

with directrix

(A) x =

a

(B) x = a

(C) x = 0

(D) x

a

2

Solution : (D)

57.

If [x] denotes the greatest integer less than or equal to x, then the value of the integral

2

x 2 x dx equals

0

(A)

5

3

(B)

7

3

(C)

8

3

(D)

4

3

Solution : (B)

58.

The number of points at which the function f(x) = max {a x, a + x, b},

differentiable

(A) 0

(B) 1

< x < , 0 < a < b cannot be

(C) 2

(D) 3

Solution : (C)

59.

For non-zero vectors a and b if a b

a

b , then a and b are

(A) collinear (B) perpendicular to each other

(C) inclined at an acute angle (D) inclined at an obtuse angle

Solution : (D)

60.

General solution of y

dy

dx

by 2

acos x , 0 < x < 1 is

(A) y 2 = 2a (2b sin x + cos x) + ce

2bx

(B) (4b 2 + 1)y 2 = 2a(sin x + 2b cos x) + ce

(C) (4b 2 + 1)y 2 = 2a(sin x + 2b cos x) + ce 2bx

(D) y 2 = 2a (2b sin x + cos x) + ce

2bx

2bx

Here c is an arbitrary constant

Solution : (B)

61.

The points of the ellipse 16x 2 + 9y 2 = 400 at which the ordinate decreases at the same rate at which

the abscissa increases is/are given by

(A) 3, 16

3 &

(C) 1 1

,

16 9 &

16

3

16

3 (B) 3,

1

1

,

16 9 (D) 1

1

,

16 9

3,

3,

&

16

3

1 1

,

16 9

&

Solution : (A)

62.

The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical order

as in an English dictionary. The number of words that appear before the word COCHIN is

(A) 96

(B) 48

(C) 183

(D) 267

Solution : (A)

63.

If the matrix A

2 0 0 a 0 0

0 2 0 ,then A n 0 a 0 ,n N where

b 0 a

2 0 2

(A) a = 2n, b = 2 n

(B) a = 2 n , b = 2n

(C) a = 2 n , b = n2 n

1

(D) a = 2 n , b = n2 n

Solution : (D)

64.

The sum of n terms of the following series; 1 3 + 3 3 + 5 3 + 7 3 + ……. is

(A) n 2 (2n 2

1)

(B) n 3 (n

1)

(C) n 3 + 8n + 4

(D) 2n 4 + 3n 2

Solution : (A)

65.

If

and

are roots of ax 2 + bx + c = 0 then the equation whose roots are

(A) a 2 x 2 (b 2

2ac)x + c 2 = 0

(C) a 2 x 2 + (b 2 +ac)x +c 2 = 0

(B) a 2 x 2 + (b 2

2 and

2

is

ac) x + c 2 = 0

(D) a 2 x 2 + (b 2 + 2ac)x +c 2 = 0

Solution : (A)

Category III (Q.66 to Q.75)

One or more answer(s) is (are) correct. Correct answer(s) will fetch marks 2. Any

combination containing one or more incorrect answer will fetch 0 marks. If all correct

answers are not marked and also no incorrect answer is marked then score = 2 × number

of correct answers marked / actual number of correct answers.

66.

Let f : X

X be such that f(f(x)) = x for all x

X and X

then

(A) f is one-to-one (B) f is onto

(C) f is one-to-one but not onto (D) f is onto but not one-to-one

Solution : (A,B)

67.

If A, B are two events such that P A

B

3

1

and

4

8

P A

B

3

then

8

(A) P A P B 11

8 (B) P A .P B

(C) P A PB 7

8 (D) None of these

3

8

Solution : (A,C)

68.

If the first and the (2n +1) th terms of an AP, GP and HP are equal and their n th terms are respectively

a, b, c then always

(A) a = b = c

(B) a

b

c

(C) a + c = b

(D) ac

b 2 = 0

Solution : (B,D)

69.

The coordinates of a point on the line x + y + 1 = 0 which is at a distance

1

unit from the line

5

3x + 4y + 2 = 0 are

(1) (2,

3)

(2) ( 3, 2)

(3) (0, 1)

(4) ( 1, 0)

Solution : (B,D)

70.

If the parabola x 2 = ay makes an intercept of length

(1) 1

(2)

2

40 unit on the line y

(3)

1

2x = 1 then a is equal to

(4) 2

Solution : (A,B)

71.

If f(x) is a function such that f (x) = (x 1) 2 (4

x), then

(1) f (0) = 0 (2) f(x) is increasing in (0, 3)

(3) x = 4 is a critical point of f(x) (4) f(x) is decreasing in (3, 5)

Solution : (B,C)

72.

On the ellipse 4x^{2} + 9y^{2} = 1, the points at which the tangents are parallel to the line 8x = 9y are

2 1

,

5 5

(1)

2 1

,

5 5

(2)

2 1

,

5 5

(3)

(4)

2 1

,

5 5

Solution : (B,D)

73.

Solution : (A,B)

74.

If the equation x^{2} + y^{2} -10x + 21 = 0 has real roots x = α and y = β

(1) 3 <= x <= 7

(2) 3 <= y <= 7

(3) -2 <= y <= 2

(4) -2 <=x <= 2

Solution : (A,C)

75.

If z = sinΘ – icosΘ then for any integer in,

Solution : (A,C)

_______________

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