WBJEE Previous Year - 2012


160 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests | WBJEE Previous Year - 2012


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Attempt WBJEE Previous Year - 2012 | 160 questions in 300 minutes | Mock test for JEE preparation | Free important questions MCQ to study WBJEE Sample Papers, Section Wise & Full Mock Tests for JEE Exam | Download free PDF with solutions
QUESTION: 1

In a mercury thermometer the ice point (lower fixed point) is marked as 10o and the steam point (upper fixed point) is marked as 130o. At 40oC temperature, what will this thermometer read?

Solution:

QUESTION: 2

The magnetic flux linked with a coil satisfies the relation = 4t2 + 6t + 9 Wb, where t is the time in second. The e.m.f. induced in the coil at t = 2 second is

Solution:

QUESTION: 3

Water is flowing through a very narrow tube. The velocity of water below which the flow remains a streamline flow is known as

Solution:
QUESTION: 4

If the velocity of light in vacuum is 3 x 108 ms–1, the time taken (in nanosecond) to travel through a glass of thickness 10 cm and refractive index 1.5 is

Solution:

QUESTION: 5

A charge +q is placed at the orgin O of X – Y axes as shown in the figure. The work done in taking a charge Q from A to B along the straight line AB is

Solution:

QUESTION: 6

What current will flow through 2 k resistor in the circuit shown in the figure?

Solution:

I1= current through 2 kΩ

QUESTION: 7

In a region, the intensity of an electric field is given by  in NC–1. The electric flux through a surface  in the region is

Solution:

QUESTION: 8

A train approaching a railway platform with a speed of 20 ms–1 starts blowing the whistle. Speed of sound in air is 340 ms–1. If the frequency of the emitted sound from the whistle is 640 Hz, the frequency of sound to a person standing on the platform will appear to be

Solution:

QUESTION: 9

A straight wire of length 2 m carries a current of 10 A. If this wire is placed in a uniform magnetic field of 0.15 T making an angle of 45o with the magnetic field, the applied force on the wire will be

Solution:

QUESTION: 10

What is the phase difference between two simple harmonic motions represented by    and x2 = A cos (ωt)?

Solution:

QUESTION: 11

Heat is produced at a rate given by H in a resistor when it is connected across a supply of voltage V. If now the resistance of the resistor is doubled and the supply voltage is made V/3 then the rate of production of heat in the resistor will be

Solution:

QUESTION: 12

Two elements A and B with atomic numbers ZA and ZB are used to produce characteristic x–rays with frequencies A and B respectively. If ZA : ZB= 1 : 2, then vA : vB will be

Solution:

√v = a(z - b) , Ignoring screening effect (i.e. b=0)

QUESTION: 13

The de Broglie wavelength of an electron moving with a velocity c/2 (c = velocity of light in vacuum) is equal to the wavelength of a photon. The ratio of the kinetic energies of electron and photon is

Solution:

QUESTION: 14

Two infinite parallel metal planes, contain electric charges with charge densities + and – respectively and they are separated by a small distance in air. If the permittivity of air is 0 then the magnitude of the field between the two planes with its direction will be

Solution:
QUESTION: 15

A box of mass 2 kg is placed on the roof of a car. The box would remain stationary until the car attains a maximum acceleration. Coefficient of static friction between the box and the roof of the car is 0.2 and g = 10 ms–2.This maximum acceleration of the car, for the box to remain stationary, is

Solution:

amax = μg =0.2x 10 = 2 m/s2

QUESTION: 16

The dimension of angular momentum is

Solution:

[L]= [m r v] = [M1L2T–1]

QUESTION: 17

 have scalar magnitudes of 5, 4 , 3 units respectivley then the angle between A and C is

Solution:

QUESTION: 18

A particle is travelling along a straight line OX. The distance x (in meters) of the particle from O at a time t is given by x = 37 + 27 t – t3 where t is time in seconds. The distance of the particle from O when it comes to rest is

Solution:

t = 3 sec. x = 37 + 27 x 3 – 33, = 37 + 81 – 27 = 91 m

QUESTION: 19

A particle is projected from the ground with kinetic energy E at an angle of 600 with the horizontal. Its kinetic energy at the highest point of its motion will be

Solution:

QUESTION: 20

A bullet on penetrating 30 cm into its target loses its velocity by 50%. What additional distance will it penetrate into the target before it comes to rest?

Solution:

 From (1) and (2) s = 10 m

QUESTION: 21

When a spring is stretched by 10 cm, the potential energy stored is E. When the spring is stretched by 10 cm more, the potential energy stored in the spring becomes

Solution:

QUESTION: 22

Average distance of the earth from the Sun is L1. If one year of the Earth = D days, one year of another planet whose average distance from the Sun is L2 will be

Solution:

QUESTION: 23

A spherical ball A of mass 4 kg, moving along a straight line strikes another spherical ball B of mass 1 kg at rest.
After the collision, A and B move with velocities v1 ms–1 and v2 ms–1 respectively making angles of 30o and 60o with respect to the original direction of motion of A. The ratio v1/v2 will be

Solution:

Apply conservation of momentum along Normal

QUESTION: 24

The decimal number equivalent ot a binary number 1011001 is

Solution:

(1011001)2= 1 x 26 + 0 x 25 +1x 24 + 1x 23 + 0x 22 + 0x 21 + 1x 20  = 64 + 16 + 8 + 1 =89

QUESTION: 25

The frequency of the first overtone of a closed pipe of length l1 is equal to that of the first overtone of an open pipe of length l2. The ratio of their lenghts (l1 : l2) is

Solution:

QUESTION: 26

The I – V characteristics of a metal wire at two different temperatures (T1 and T2) are given in the adjoining figure. Here, we can conclude that

Solution:

1/V = 1/R

slope of curve, R ∞T greater the slope smaller will be temperature.    

T1 < T2

QUESTION: 27

In a slide calipers, (m + 1) number of vernier divisions is equal to m number of smallest main scale divisions. If d unit is the magnitude of the smallest main scale division, then the magnitude of the vernier constant is

Solution:

QUESTION: 28

From the top of a tower, 80 m high from the ground, a stone is thrown in the horizontal direction with a velocity of 8 ms–1. The stone reaches the ground after a time ‘t’ and falls at a distance of ‘d’ from the foot of the tower. Assuming g = 10 ms–2, the time t and distance d are given respectively by

Solution:

time of flight (t) = 

QUESTION: 29

A Wheatstone bridge has the resistances 10 Ω , 10 Ω , 10Ω and 30 Ω in its four arms. What resistance joined in parallel to the 30 resistance will bring it to the balanced condition?

Solution:

 for a balanced bridge. R = 10 Ω

QUESTION: 30

An electric bulb marked as 50 W–200 V is connected across a 100 V supply. The present power of the bulb is

Solution:

QUESTION: 31

 A magnetic needle is placed in a uniform magnetic field and is aligned with the field. The needle is now rotated by an angle of 600 and the work done is W. The torque on the magnetic needle at this poition is

Solution:

W = MB(1 - cos ) = MB(1 - cos 60o ) = MB/2

QUESTION: 32

In the adjoining figure the potential difference between X and Y is 60 V. The potential difference between the points M and N will be

Solution:

QUESTION: 33

A body when fully immersed in a liquid of specific gravity 1.2 weighs 44 gwt. The same body when fully immersed in water weighs 50 gwt. The mass of the body is

Solution:

mg – σvg = 44, σ = density of liquid,
mg –1.2
ρvg = 44 ..............(1),
ρ = density of water
mg – ρvg = 50.............(2),
(1) – 1.2 x(2)
M = 80 g

QUESTION: 34

When a certain metal surface is illuminated with light of frequency ν , the stopping potential for photoelectric current is νo When the same surface is illuminated by light of frequency  v/2 the stopping potential is v0/4 The threshold frequency for photoelectric emission is

Solution:

solving equation (1) and (2) ,v/3

QUESTION: 35

Three blocks of mass 4 kg, 2 kg, 1 kg respectively are in contact on a frictionless table as shown in the figure. If a force of 14 N is applied on the 4 kg block, the contact force between the 4 kg block, the contact force between the 4 kg and the 2 kg block will be

Solution:

QUESTION: 36

Let L be the length and d be the diameter of cross section of a wire. Wires of the same material with different L and d are subjected to the same tension along the length of the wire. In which of the following cases, the extension of wire will be the maximum?

Solution:

QUESTION: 37

An object placed in front of a concave mirror at a distance of x cm from the pole gives a 3 times magnified real image.If it is moved to a distance of (x + 5) cm, the magnification of the image becomes 2. The focal length of the mirror is

Solution:

so f = 30 cm

QUESTION: 38

22320 cal of heat is supplied to 100 g of ice at 0oC If the latent heat of fusion of ice is 80 cal g–1 and latent heat of vaporization of water is 540 cal g–1, the final amount of water thus obtained and its temperature respectively are

Solution:

total energy required to melt and boil at 100oC is

(Q)min = 8000 + 10000 = 18000 cal,

(Qrequired)min < Qgiven  for amount of vapour 22320=18000+ m x 540

= m = 8 gm. temperature = 100oC and water remaining = 92 gm.

QUESTION: 39

A progressive wave moving along x–axis is represented by   The wavelength (λ) at which themaximum particle velocity is 3 times the wave velocity is

Solution:

QUESTION: 40

Two radioactive substances A and B have decay constant 5λ and λ respectively. At t = 0, they have the same number of nuclei the ratio of number of nuclei of A to that of B will be (1/e)2 after a time interval of

Solution:

QUESTION: 41

Li occupies higher position in the electrochemical series of metals as compared to Cu since

Solution:

QUESTION: 42

11Na24 is radioactive and it decays to

Solution:

QUESTION: 43

The paramagnetic behavior of B2 is due to the presence of

Solution:

QUESTION: 44

A 100 ml 0.1 (M) solution of ammonium acetate is diluted by adding 100 ml of water. The pH of the resulting solution will be (pKa of acetic acid is nearly equal to pKb of NH4OH)

Solution:

Hydrolysis of a salt of weak acid and weak base

pH = 7 + ½ (pKa– pKb) = 7 [ pKa = pKb ]

QUESTION: 45

In 2-butene, which one of the following statements is true ? 

Solution:

QUESTION: 46

The well known compounds, (+)- lactic acid and (–) lactic acid have the same molecular formula, C3H6O3. The correct relationship between them is

Solution:

They are enantiomers and correct relationship between them is optical isomerism.

QUESTION: 47

The stability of Me2C=CH2 is more than that of MeCH2CH = CH2 due to

Solution:

Me2C = CH2 is   more stable than MeCH2CH = CHdue to hyperconjugative effect.

QUESTION: 48

Which one of the following characteristics belongs to an electrophile ?

Solution:
QUESTION: 49

Which one of the following methods is used to prepare Me3COEt with a good yield?

Solution:

Williamson’s synthesis (SN2) given by 1° alkyl halide.

QUESTION: 50

58.5 gm of NaCl and 180 gm of glucose were separately dissolved in 1000 ml of water. Identify the correct statement regarding the elevation of boiling point (b.p.) of the resulting solutions.

Solution:

For NaCl (i = 2) for glucose (i = 1), Hence NaCl solution will show higher elevation of bp (Colligative property) as both are one Molar Solution.

QUESTION: 51

Equal weights of CH4 and H2 are mixed in an empty container at 25oC. The fraction of the total pressure exerted by H2 is

Solution:

Let equal weights be w g.

Partial pressure = mole fraction × Total pressure

PH2 = 8/9 x Total pressure

QUESTION: 52

Which of the following will show a negative deviation from Raoult’s law?

Solution:

Acetone – chloroform due to formation of intermolecular Hydrogen bonding

QUESTION: 53

In a reversible chemical reaction at equilibrium, if the concentration of any one of the reactants is doubled, then the equilibrium constant will

Solution:

Equilibrium constant does not depend on conc. It is only a function of temperature

QUESTION: 54

Identify the correct statement from the following in a chemical reaction.

Solution:

Spontaneity of a reaction is decided by a combined effect of suitable change in values of enthalpy and entropy. [ ΔG = ΔH – TΔS]

QUESTION: 55

Which one of the following is wrong about molecularity of a reaction?

Solution:

Molecularity can never be a fraction.

QUESTION: 56

Which of the following does not represent the mathematical expression for the Heisenberg uncertainty principle?

Solution:
QUESTION: 57

The stable bivalency of Pb and trivalency of Bi is

Solution:

Lower oxidation state becomes more stable for group 14 and group 15 elements as we move down the group due to inert pair effect. Hence Pb+2 and Bi+3 are more stable.

QUESTION: 58

The equivalent  weight of K2Cr2O7 in acidic medium is expressed in terms of its molecular weight (M) as

Solution:

Number of electrons gained by one ion = 6
Eq.wt = M/6

QUESTION: 59

Which of the following is correct ?

Solution:

Ca+2, S–2, Clare isoelectronic (consist of 18e) and for isoelectronic species, ionic radii ∞e/z

QUESTION: 60

CO is practically non-polar since

Solution:
QUESTION: 61

The number of acidic protons in H3PO3 are

Solution:

  Number of OH group is 2 hence dibasic in nature.

QUESTION: 62

When H2O2 is shaken with an acidified solution of K2Cr2O7 in presence of ether, the ethereal layer turns blue due to the formation of

Solution:

QUESTION: 63

The state of hybridization of the central atom and the number of lone pairs over the central atom in POCl3 are

Solution:

Number of lone pairs on P atom = 0

QUESTION: 64

Upon treatment with l2 and aqueous NaOH, which of the following compounds will  form iodoform? 

Solution:

Iodoform reaction is given by all alcohols having 

QUESTION: 65

Upon treatment with Al(OEt)3 followed by usual reactions (work up), CH3CHO will produce

Solution:

This reaction is known as Tischenko reaction

QUESTION: 66

Friedel-Craft’s reaction using MeCl and anhydrous AlCl3 will take place most efficiently with

Solution:

As – CH3 group in toluene activates the benzene ring for electrophilic substitution reaction.

QUESTION: 67

Which one of the following properties is exhibited by phenol?

Solution:

Phenol is less acidic than H2CO3. So it cannot liberate CO2 from NaHCO3 (aq).

QUESTION: 68

The basicity of aniline is weaker in comparison to that of methyl amine due to

Solution:

QUESTION: 69

Under identical conditions, the SN1 reaction will occur most efficiently with

Solution:

Reactivity order for SN1 reaction in case of alkyl halide is 3°> 2°> 1°

QUESTION: 70

Identify the method by which Me3CCO2H can be prepared

Solution:

QUESTION: 71

20 ml 0.1 (N) acetic acid is mixed with 10 ml 0.1 (N) solution of NaOH. The pH of the resulting solution is (pKa of acetic acid is 4.74)

Solution:

CH3COOH   +   NaOH ⇒ CH3COONa + H2O

QUESTION: 72

In the brown ring complex [Fe(H2O)5(NO)]SO4, nitric oxide behaves as

Solution:

‘NO’ ligand when attached to Fe behaves as positively charged ligand.

QUESTION: 73

The most contributing tautomeric enol form of MeCOCH2CO2Et is

Solution:

QUESTION: 74

By passing excess Cl2(g) in boiling toluene, which one of the following compounds is exclusively formed?

Solution:

QUESTION: 75

An equimolar mixture of toluene and chlorobenzene is treated with a mixture of conc. H2SO4 and conc. HNO3.Indicate the correct statement from the following

Solution:

 – CH3 is activating group due to +I effect and hyperconjugation.
But for – Cl, – I > + R effect.

∴CH3 is weakly activating and – Cl is weakly deactivating

QUESTION: 76

Among the following carbocations :

Ph2C*CH2Me(I),

PhCH2CH2CH*Ph (II),

Ph2CHCH*Me (III)

and Ph2C(Me)CH2+ (IV),

the order of stability is

Solution:

QUESTION: 77

 Which of the following is correct?

Solution:

Evaparation increases randomness as liquid water is converted to water vapours.

QUESTION: 78

On passing ‘C’ Ampere of current for time ‘t’ sec through 1 litre of 2 (M) CuSO4 solution (atomic weight of Cu = 63.5), the amount ‘m’ of Cu (in gm) deposited on cathode will be

Solution:

1000 ml 2(M) CuSO4 ≡  2(M) CuSO4 solution contains 2 moles Cu+2

Cu2 + 2 →Cu
          2F    63.5g

q = c × t coulomb

  or,    2 × 96500C

   

QUESTION: 79

If the 1st ionization energy of H atom is 13.6 eV, then the 2nd ionization energy of He atom is

Solution:

QUESTION: 80

The weight of oxalic acid that will be required to prepare a 1000 ml (N/20) solution is

Solution:

QUESTION: 81

The maximum value of |z| when the complex number z satisfies the condition |z + Z/2| = 2 is

Solution:

QUESTION: 82

  where x and y are real, then the ordered pair (x, y) is

Solution:

QUESTION: 83

Solution:

QUESTION: 84

There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects

Solution:

n(MUPUB) = n(M) + n(P) + n(B) -n(M ∩ P) - n(PMB) - n(B ∩ P) + n(M ∩ P ∩ B)

given n(M) = 50 , n (P) = 45, n(B) = 40;

n(M∩ P) + n(P∩ B) + n(B∩ M) - 3(M∩ P ∩ B) = 32

99 = 50 + 45 + 40 – (32 + 3 n (M ∩ P∩ B) )+ n(M ∩ P∩ B) ;

2 n(M ∩ P ∩ B) = 36 – 32;

n(M∩ P ∩ B) = 2

QUESTION: 85

A vehicle registration number consists of 2 letters of English alphabets followed by 4 digits, where the first digit is not zero. Then the total number of vehicles with distinct registration numbers is

Solution:

Total No. of ways = 262 x 9 x 103

QUESTION: 86

The number of words that can be written using all the letters of the word ‘IRRATIONAL’ is

Solution:

 IRRATIONAL. There are 2 I, 2 R, 2 A, one T, one O, One N, One L. Total Number of words 

QUESTION: 87

Four speakers will address a meeting where speaker Q will always speak after speaker P. Then the number of ways in which the order of speakers can be prepared is

Solution:

The number of ways in which speaker P and Q can deliver their speach is 4C2.Total number of ways = 4C2 x 2! = 12

QUESTION: 88

The number of diagonals in a regular polygon of 100 sides is

Solution:

Number of Diagonals in n sided Polygon is equal to nC2 – n. Total number of Diagonals = 100C2 – 100=4850

QUESTION: 89

Let the coefficients of powers of x in the 2nd, 3rd and 4th terms in the expansion of (1+x)n, where n is a positive integer, be in arithmetic progression. Then the sum of the coefficients of odd powers of x in the expansion is

Solution:

Co-efficient of 2nd, 3rd and 4th terms are nC1, nC2, nC3. It is given 2.nC2 = nC2 + nC3 = n = 7.

Sum of Co-efficient of odd power of x = 26

QUESTION: 90

Let f(x) = ax2 + bx + c, g(x) = px2 + qx + r, such that f(1) = g(1), f(2) = g(2) and f(3) – g(3) = 2. Then f(4) – g(4) is

Solution:

Let a – p = w, b – q = y and c – r = z

f(1) = g(1) w + y + z = 0;

f(2) = g(2) 4w + 2y + z = 0;

f(3) = g(3) + 2 9w + 2y + z = 2w = 1,

y = –3, z = 2

f(4) – g(4) = 16 w + 4y + z = 6

QUESTION: 91

The sum 1x 1! + 2x 2! + .... + 50x 50! equals

Solution:

QUESTION: 92

Six numbers are in A.P. such that their sum is 3. The first term is 4 times the third term. Then the fifth term is

Solution:

Let α, α - 5d, α - 3d, α - d, + d, + 3d, α + 5d are six numbers in A.P.

QUESTION: 93

The sum of the infinite series 

Solution:

QUESTION: 94

The equations x2 +x +a=0 and x2 +ax +1=0 have a common real root

Solution:

x2 +x +a=0  .....(A) and x2 +ax +1=0 ...(B)

A - B

(1–a)x + (a–1) = 0; (1–a) (x–1) = 0 if a = 1, x = 1, so, only sol is a = -2

QUESTION: 95

If 64, 27, 36 are the Pth , Qth and Rth terms of a G.P., then P + 2Q is equal to

Solution:

tP = 64,

tR = 36,

a . rP–1 = 64.............(1);

tQ = 27, a . rQ–1 = 27.............(2)

a . rR–1 = 36.............(3)

(2)2x (1)/(3)3 ; 2 Q + P = 3R

QUESTION: 96

If (α+√β) and (α-√β) are the roots of the equation x2 + px + q = 0 where α,β p and q are real, then the roots of the equation (p2– 4q) (p2x2 + 4 px) – 16 q = 0 are

Solution:

2α = -p, α2 - β = q,

so the given equation is 4β(4α2x2 - 8αx) - 16α2 16+b = 0

QUESTION: 97

The number of solutions of the equation log2(x2 + 2x –1) = 1 is

Solution:

x2 + 2x –1 = 2

x2 + 2x – 3=0

(x + 3) (x – 1) = 0;

x = 1, –3

QUESTION: 98

The sum of the series 

Solution:

QUESTION: 99

Solution:

QUESTION: 100

   then the value of the determinant of Q is equal to

Solution:

QUESTION: 101

The remainder obtained when 1!+2!+...+95! is divided by 15 is

Solution:

Starting from 5! all the other numbers are divisible by 15 since 15 will be a factor So 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33 So the remainder on dividing 33 by 15 is 3.

QUESTION: 102

If P,Q,R are angles of triangle PQR, then the value of 

Solution:

On expanding the determinant,

Δ = – 1 + cos2p + cos2Q + cos2R + 2cosP cosQ cosR

= – 1 cos2 P + cos2Q + cos2Q + cos2R + (cos(P+Q) + cos(P – Q)) cosR

now P + Q = π – R

So Δ = – 1+ cos2P + cos2Q + cos2R – cos2R – cos(P – Q) cos(P + Q)

= – 1 + cos2P + cos2Q – cos.2P + sin2Q = – 1 + 1 = 0

QUESTION: 103

The number of real values of for which the system of equations

x + 3y+5z = αx
5x+y+3z =α y    
3x+5y+z = αz

Solution:

QUESTION: 104

The total number of injections (one-one into mappings) from {a1, a2, a3 , a4} to {b1,b2 ,b3 ,b4 ,b5 ,b6 , b7 is}

Solution:

So total = 7 × 6 × 5 × 4 = 840 one-one into functions

QUESTION: 105

Solution:

QUESTION: 106

Two decks of playing cards are well schuffled and 26 cards are randomly distributed to a player. Then the probability that the player gets all distinct cards is

Solution:

 Total no. of way of distribution = 104 C26 Favourable no. of ways of ditribution =

(selecting any one decki) × (selecting 26 cards out of it) = 2 x 52 C26

QUESTION: 107

An urn contains 8 red and 5 white balls. Three balls are drawn at random. Then the probability that balls of both colours are drawn is

Solution:

Total no. of selection = 13 C3

QUESTION: 108

Two coins are avilable, one fair and the other two-headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probalility 3/4. Given that the outcome is head, the probability that the two-headed coin waschosen is

Solution:

B is the event of selecting a biased coin
UB is the event of selecting an unbiased coin

QUESTION: 109

Let be the set of real numbers and the funtions f: R → R and g: R → R be defined by f(x)=x2 + 2x-3 and g(x) = x+1. Then the value of x for which f(g(x)) = g(f(x)) is

Solution:

 fog(x) = (x + 1)2 + 2(x + 1) – 3 = x2 + 4x

= x2 + 4x

gof(x) = x2 + 2x – 3 + 1 = x2 + 2x – 2

fog(x)  = gof(x)  

x2 + 4x  = x2 + 2x – 2 x = – 1

QUESTION: 110

If a,b,c are in arithmetic progression, then the roots of the equation ax2 - 2bx + c = 0 are

Solution:

a – 2b + c = 0

So x = 1 is a root

Now product of roots = c/a

So the other root is = c/a

QUESTION: 111

 If sin-1x + sin-1y + sin-1z = 3π/2, then the value of 

Solution:

sin-1x + sin-1y + sin-1z = π/2

 x = y = z = 1

QUESTION: 112

Let p,q,r be the sides opposite to the angles P,Q,R respectively in a triangle PQR, if r2sin P sin Q = pq, then the triangle is

Solution:

QUESTION: 113

Let p,q,r be the sides opposite to the angles P,Q,R respectively in a triangle PQR. Then 

Solution:

QUESTION: 114

Let P (2, 3) , Q (2,1) be the vertices of the triangle PQR. If the centroid of PQR lies on the line 2x+3y = 1, then the locus of R is

Solution:

Let R(h, k)

QUESTION: 115

Solution:

QUESTION: 116

If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, then

Solution:

 f(x) f (x) < 0  ....(i)
f(x) < 0 ; f (x) > 0   or f(x) > 0 ; f (x) < 0      ............... (ii)

So possible graphs of f(x) are

|f(x)| is decreasing function in both cases.

QUESTION: 117

Rolle’s theorem is applicable in the interval [-2,2] for the function.

Solution:

For f(x) = 4x4 f(–2) = f(2) & f'(0) exist

QUESTION: 118

The solution of   

y(0) = 1,            
y(1) = 2e1/5 is

Solution:

Let y = emx is a trial solution

dy/dx = m

25m2 - 10m - 1 = 0

m = 1/5(equal roots)

The differential solution y = (1 + x)ex/5 (A & B are two arbitrary constant)

Initially y (0) = 1 ( A + 0) eo

1 ⇒ A = 1

QUESTION: 119

Let P be the midpoint of a chord joining the vertex of the parabola y2 = 8x to another point on it. Then the locus of P is

Solution:

 Let P(h, k) be the midpoint of chord AB through vertex (0, 0) of parabola y2 = 8x

Co-ordinate of B= (2h - 0, 2k - 0) = (2h, 2k )  
B (2h, 2k) lies on y2 = 8x
⇒ 4k2 = 8 × 2h
⇒ k2 = 4h

QUESTION: 120

The line x = 2y intersects the ellipse x2/4 + y2 = 1 at the points P and Q. The equation of the circle with PQ as diameter is

Solution:

Solving the given equations then common points are 

Eqn. of Circle PQ as
diameter 

QUESTION: 121

The eccentric angle in the first quadrant of a point on the ellipse x2 /10 + y2/8 = 1 at a distance 3 units from the centre of the ellipse is

Solution:

QUESTION: 122

The transverse axis of a hyperbola is along the x-axis and its length is 2a. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is

Solution:

QUESTION: 123

A point moves in such a way that the difference of its distance from two points (8, 0) and (–8, 0) always remains 4.Then the locus of the point is

Solution:

Let P(h, k) be the moving point

Conjugate equation of (i)

QUESTION: 124

The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is

Solution:

From given equation x = 5/3+4m, x is an integer

3 + 4m = ± 5
or ± 1 m = – 2
or m = – 1

QUESTION: 125

If a straight line passes through the point (α,β ) and the portion of the line intercepted between the axes is divided equally at that point, then x/α + y/β

Solution:

Let, the equation of line be x/a + y/b = 1 , passes through P(α,β )

(α,β ),is the mid point of the portion intercepted between the axes

QUESTION: 126

The equation y2 + 4x + 4y + k = 0 represents a parabola whose latus rectum is

Solution:

y2 + 4x + 4y + k = 0

⇒ (y + 2)2 = 4 (–x + 1 – k/4)

Latus rectum  = 4

QUESTION: 127

If the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect orthogonally, then k is equal to

Solution:

2g1g2 + 2f1f2

= c1 + c2  
⇒ 2k2 – k – 6 = 0
⇒ k = 2,  –3/2

QUESTION: 128

If four distinct points (2k, 3k), (2, 0), (0, 3) (0, 0) lie on a circle, then

Solution:

Equation of circle x(x – 2) + y (y – 3) = 0
(2k, 3k) will satisfy
So, K = 1

QUESTION: 129

The line joining A (b cosα , b sinα ) and B (a cosβ , a sinβ ), where a ≠ b , is produced to the point M(x, y) so that AM : MB = b : a. 

Solution:

QUESTION: 130

Let the foci of the ellipse x2/9+ y2 = 1 subtend a right angle at a point P. Then the locus of P is

Solution:

P(h, k) be a point on the ellipse,, e = 2√2/3

= h2+ k2 = 8

QUESTION: 131

The general solution of the differential equation 

Solution:

 Put x + y = θ

QUESTION: 132

The value of the integral 

Solution:

QUESTION: 133

the value of the integral 

Solution:

I = π/4

QUESTION: 134

The integrating factor of the differential equation 

Solution:

 

QUESTION: 135

Number of solutions of the equation tan x + sec x = 2 cos x, x ∈ [0, π] is

Solution:

tanx  + secx = 2cosx
2sin2x + sinx – 1 = 0
sinx = 1/2

QUESTION: 136

The value of the integral 

Solution:

QUESTION: 137


Solution:

QUESTION: 138

Maximum value of function f(x) = x/8 + 2/x on the interval [1, 6] is

Solution:

QUESTION: 139

Solution:

QUESTION: 140

The value of the integral 

Solution:

QUESTION: 141

The points representing the complex number z for which 

 

Solution:

QUESTION: 142

Let a, b, c, p,q, r be positive real numbers such that a, b, c are in G.P. and ap = bq = cr. Then

Solution:

b2 = ac ; 2logb =  loga + log c

ap = bq

loga/logb = q/p....(ii)

QUESTION: 143

Let Sk be the sum of an infinite G.P. series whose first term is k and common ratio is k/k+1.Then the value of 

Solution:

 

QUESTION: 144

 The quadratic equation 2x2 – (a3+ 8a – 1)x + a2 – 4a = 0 possesses roots of opposite sign. Then

Solution:

QUESTION: 145

If loge(x2 – 16) loge(4x – 11), then

Solution:

x2 – 16 > 0

QUESTION: 146

The coefficient of x10 in the expansion of 1 + (1 + x) + ......... + (1 + x)20 is

Solution:

1 + (1 + x) + ......... + (1 + x)20

QUESTION: 147

The coefficient of x10 in the expansion of 1 + (1 + x) + ......... + (1 + x)20 is

Solution:

1 + (1 + x) + ......... + (1 + x)20

QUESTION: 148

The system of linear equations

λx + y + z = 3
x – y – 2z = 6
– x + y + z = μ has

Solution:

if  λ = – 1
then Δ = 0 so equation has infinite number of solution for  λ = – 1 and μ = 3

QUESTION: 149

Let A and B be two events with P(AC) = 0.3, P(B) = 0.4 and P(A ∩ BC) = 0.5. Then P(B/A  U BC) is equal to

Solution:

P(A ∩ BC) = P(A) - P(A ∩ B) ; P(A ∩ B) = 0.2

QUESTION: 150

Let p, q, r be the altitudes of a triangle with area S and perimeter 2t. Then the value of 1/p + 1/q +1/r is

Solution:

QUESTION: 151

Let C1 and C2 denote the centres of the circles x2 + y2 = 4 and (x – 2)2 + y2 = 1 respectively and let P and Q be their points of intersection. Then the areas of triangles C1PQ and C2PQ are in the ratio

Solution:

Equation of chord S2 – S1 = 0   ; x2 + y2 – 4x + 3 = 0

x = 7/4

C1T = Perpendicular distance of point (0, 0) from x = 7/4 is 7/4

C2T = Perpendicular distance of point (0, 0) from x = 7/4 is 1/4

 

QUESTION: 152

A straight line through the point of intersection of the lines x + 2y = 4 and 2x + y = 4 meets the coordinate axes at  A and B. The locus of the midpoint of AB is

Solution:

Point of intersection of given lines x = 4/3, y = 4/3

QUESTION: 153

Let P and Q be the points on the parabola y2 = 4x so that the line segment PQ subtends right  angle at the vertex.If PQ intersects the axis of the parabola at R, then the distance of the vertex from R is

Solution:

 PQ subtants right angle at vertex.

so t1,t2 = -4

equation of PQ is

it intersect the x axis so, put y  = 0 we get x = 4

QUESTION: 154

The incentre of an equilateral triangle is (1, 1) and the equation of one side is 3x + 4y + 3 = 0. Then the equation of the circumcircle of the triangle is

Solution:

For equalateral triangle R = 2r  = 2 × 2 = 4 equation of circumcircle (x – 1)2 + (y – 1)2 = 42 ;  x2 + y2 – 2x – 2y – 14 = 0

QUESTION: 155

Solution:

QUESTION: 156

The area of the region bounded by the curves y = x3 , y = 1/x, x = 2 is

Solution:

QUESTION: 157

Let y be the solution of the differential equation    satisfying y(1) = 1. Then y satisfies

Solution:

QUESTION: 158

The area of the region, bounded by the curves y = sin –1x +x (1 – x) and y = sin –1x – x(1 – x) in the first quadrant, is

Solution:

QUESTION: 159

The value of the integral 

Solution:

QUESTION: 160

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

Solution:

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