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QUESTION: 1

In the figure shown the acceleration of A is, , then the acceleration of B is : (A remains in contact with B)

Solution:

From wedge constraint

QUESTION: 2

The speed of sound in hydrogen gas at N.T.P. is 1,328 ms^{-1}. What will be its value in air at N.T.P., if density of hydrogen is 1/16th that of air?

Solution:

Let v_{air} be the velocity of sound in air at N.T.P

QUESTION: 3

A balloon that is initially flat, is inflated by filling it from a tank of compressed air. The final volume of the balloon is 5m^{3}. The barometer reads 95 kPa. The work done in this process is

Solution:

W = P Δ V = 95 × 10^{3} × (5-0)

= 4.75 × 10^{5} J

QUESTION: 4

Two bodies of same mass tied with an inelastic string of length together on a horizontal surface. One a horizontal surface of them is projected vertically upwards with velocity Find the maximum height up to which the centre of mass of system of the two masses rises.

Solution:

Velocity of B when string is just taut

After string in taut both will move up with same speed,

By this time centre of mass in at

Further rise in

Hence maximum height

QUESTION: 5

The electric potential at a point (x, y, z) is given by V = -x^{2}y - xz^{3} + 4 The electric field at that point is

Solution:

The electric potential at a point,

QUESTION: 6

Binding energy per nucleon versus mass number curve for nuclei is shown in figure. W, X, Y and Z are four nuclei indicated on the curve. The process that would release energy is

Solution:

Energy is released in a process when total binding energy of the nucleus (= binding energy per nucleon x number of nucleons) is increased or we can say, when total binding energy of products is more than the reactants.

Binding energy of reactants = 120 x 7. 5 = 900 MeV

and binding energy of products = 2 (60 x 8. 5)

= 1020 MeV > 900 MeV.

QUESTION: 7

A wire of length 1.0 m and radius 10^{-3} m is carrying a heavy current and is assumed to radiate as a black body. At equilibrium, its temperature is 900 K while that of the surroundings is 300 K. The resistivity of the material of the wire at 300 K is π^{2} × 10^{-8} ohm-m and its temperature coefficient of resistance is 7.8× 10^{−3} C^{o}. Find the current in the wire. [Given Stefan's constant = 5.68 × 10^{-8} W/m^{2}K^{4}]

Solution:

According to Stefan's law power radiated by a black body is given by.

Now if I is the current in the resistance R,

According to the given problem Equations (i) and (ii) represent the same.

QUESTION: 8

50 V battery is supplying a steady current of 10 amp when connected to an external resistor. If the efficiency of the battery at this current is 25%, then internal resistance of battery is:

Solution:

QUESTION: 9

A capacitor of capacitance C is charged to a potential difference V from a cell and then disconnected from it. A charge +Q is now given to its positive plate. Now, the potential difference across the capacitor is.

Solution:

Initially

After charge Q is given

Charge distribution is

QUESTION: 10

Energy from the sun falls on the earth at a rate of 1353 W/m^{2}, which is known as solar constant, i.e., the power incident per unit area per second at the top of atmosphere. Find the r.m.s values of the electric and magnetic fields in the sunlight reaching the top of the atmosphere.

Solution:

For an electromagnetic wave of sinusoidal form

The mean value of energy flux is intensity

[The mean value of cos^{2}θ over on cycle is 1/2]

Hence

QUESTION: 11

The size of the image of an object, which is at infinity, is formed by a convex lens of focal length 30 cm is 2 cm. If a concave lens of focal length 20 cm is placed between the convex lens and the image at a distance of 26 cm from the convex lens, calculate the new size of the image

Solution:

Image formed by convex lens at ^ will act as a virtual object for concave lens. For concave lens

Magnification for concave lens

As height of the image at h is 2 cm.

Therefore, height of image at I_{2} will be 2 x 1.25 = 2.5 cm

QUESTION: 12

A network of Four capacitors of capacity equal to C_{1} = C, C_{2} = 2C, C_{3} = 3C and C_{4} = 4C, are connected to a battery as shown in the figure. The ratio of the charges on C_{2} and C_{4} is

Solution:

All the capacitors in upper branch are in series so the charge on each capacitor is

Also charge on capacitor C_{4} is Q = 4CV

QUESTION: 13

A rod AD, consisting of three segments AB, BC and CD joined together, is hanging vertically from a fixed support at A. The lengths of the segments are respectively 0.1 m, 0.2 m and 0.15 m. The cross - section of the rod is uniform 10^{-4 }m^{2}. A weight of 10 kg is hung from D. Calculate the displacements of point D if Y_{AB} = 2.5 x 10^{10} N/m^{2}, Y_{BC} = 4 x 10^{10 }N/m^{2} and Y_{CD} = 1 x 10^{10} N/m^{2}. (Neglect the weight of the rod.)

Solution:

By defination of Young's modulus,

QUESTION: 14

A block of mass m & charge q is released on a long smooth inclined plane. Magnetic field B is constant, uniform, horizontal and out of the plane of paper as shown. Find the time from start when block loses contact with the surface.

Solution:

Block will loose contact with surface when force due to magnetic field will become equal to mg cos θ

(along the inclined plane)

QUESTION: 15

A battery of internal resistance 4Ω is connected to the network of resistances as shown in figure. In order that the maximum power can be delivered to the network, the value of *R* in $$ should be

Solution:

The given circuit is a balanced Wheatstone's bridge.

Thus, no current will flow across 6*R* of the side *CD*. The given circuit will now be equivalent to

For maximum power, net external resistance

= Total internal resistance

or 2*R* = 4

or *R* = 2Ω.

QUESTION: 16

In the circuit shown in diagram, the equivalent resistance between point A and B is

Solution:

The circuit shown in diagram (1) can be redrawn as shown in diagram (2).

∴ R_{AB} = 5R/8

QUESTION: 17

Two trains, which are moving along different tracks in opposite directions, are put on the same track due to a mistake. Their drivers, on noticing the mistake, start slowing down the trains when they are 300 m apart. Graphs given below show their velocities as a function of time as they slow down. The separation between the trains, when both have stopped, is:

Solution:

Initial distance between trains is 300 m. Displacement of 1st train is calculated by area under V-t.

Displacement of train Which means it moves towards left.

∴ Distance between the two is = 300 - 280 = 20 m

QUESTION: 18

A tuning fork sends out waves of wavelength 68.75 cm and 3 m in air and hydrogen gas respectively. If the velocity of sound in air is 330 ms^{-1}, find the velocity of sound in hydrogen. Also, find the frequency of the tuning fork

Solution:

In air : λ_{a} = 68.75 cm = 0.6875 m ; v_{a} = 330 ms^{-1} Let v be the frequency of the tuning fork. Then,

If v_{H} is velocity of sound in hydrogen, then

QUESTION: 19

A particle executes S.H.M. given by x = 0 · 24 cos (400 t - 0.5) in SI units. Find amplitude

Solution:

Here, x = 0.24 cos (4001 - 0.5) ...(i)

The standard equation for S.H.M. is

... (ii)

Comparing the equations (i) and (ii),we have

r = 0.24 m

QUESTION: 20

A lens (μ = 1.5) is coated with a thin film of refractive index 1.2 in order to reduce the reflection from its surface at λ = 4800 Å. Find the minimum thickness of the film which will minimise the intensity of the reflected light. (Assume near normal incidence)

Solution:

*Answer can only contain numeric values

QUESTION: 21

A ray of light from a liquid (μ = √3) is incident on a system of two right angled prism of refractive indices √3 and √2 as shown in the figure. The light suffers zero net deviation when it emerges into air from surface CD. If the angle of incidence (in degrees) is 5n. Find n ?

Solution:

Hence 5*9 = 45

*Answer can only contain numeric values

QUESTION: 22

A drop water of mass *m *= 4.0 g is placed between two clean glass plates, the distance between the plates is 0.01cm. Find the force (10^{3}N) required to pull the plates away. Surface tension of water = 0.08 N/m and density of water is 1000 kg/m^{3}

Solution:

Let R be the radius of the circular layer of water. then πR^{2}.d x r = m(l)

The pressure inside the water.

*Answer can only contain numeric values

QUESTION: 23

A car is accelerating horizontally with constant acceleration a=10m/s^{2}. One end of a light string is attached to the roof of the ceiling and there is a small bob at other end of string. The bob is given an initial velocity such that it continues to move in uniform circular motion with respect to an observer inside the car. The bob moves such the maximum vertical separation between two points of its path is h=1m. The length of the string is and acceleration due to gravity g=10m/s^{2}. If the angular speed of the bob in rad/s is √x .find the value of *x*.

Solution:

First we will drive a result for conical pendulum in stationary car. see figure

Now in acceleration car, with respect to an inside observer, there will be pseudo force acting on the particle opposite

to the acceleration of the car, and hence perpendicular to the weight. since a=g, the effective gravity will be

inclined at an angle of 450 with the verticle axis.

*Answer can only contain numeric values

QUESTION: 24

Find the current (in A) through the battery after the switch S is closed if L/R = RC = 1 ms.

Solution:

Since the battery is across the two branches in parallel the current through the RL branch is unaffected by the current of the RC branch

i.e there will be no transient current through the battery in this case

*Answer can only contain numeric values

QUESTION: 25

Assuming that the law of gravitation is of the form and attractive. A body of mass m revolves in a circular path of radius r around a fixed body of mass M. Find on what power of r will the square of time period depend.

Solution:

The centrepetal acceleration is due to the gravitational force

QUESTION: 26

Place the following alcohols in decreasing order of rate of dehydration with concentrated H_{2}SO_{4} :

1. CH_{3}CH_{2}CH(OH)CH_{2}CH_{2}CH_{3}

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

3. (CH_{3})_{2}C(OH)CH(CH_{3})_{2}

4. CH_{3}CH_{2}CH(OH)CH(CH_{3})_{2}

5. CH_{3}CH_{2}CH_{2}CH_{2}CH_{2}CH_{2}OH

Solution:

The alcohols (3) and (2) are both 3^{o}, but alcohol (3) gives a more substituted alkene. Alcohol (4) and (1) are both 2^{o}, but alcohol (4) can give a more substituted alkene and alcohol (5) is 1^{o}. Rate of dehydration of alcohols with concentrated H_{2}SO_{4} follows the order 3^{o} > 2^{o} > 1^{o}.

QUESTION: 27

1.2 g of a salt with its empirical formula K_{x}H_{y}(C_{2}O_{4})_{z} was dissolved in 50 mL of water and its 10 mL portion required 11 mL of a 0.1 M HCl solution to reach the equivalence point. In a separate titration, 15 mL of the stock solution required 20 mL 0.2475 M KOH to reach the equivalence point. Identify the correct option.

Solution:

∴ y = 3x.

QUESTION: 28

The structure of H_{2}O_{2} is

Solution:

In H_{2}O_{2}, the O - H groups are not in the same plane. So it has non - planar structure. If has a half - opened book structure in which the two O - H groups lie on the two pages of the book. The angle between two pages of the book is 94^{o} and H - O - O bond angle is 97^{o}.

QUESTION: 29

In nitroprusside ion the iron and NO exist as Fe II and NO^{+ }rather than Fe III and NO. These forms can be differentiated by

Solution:

Fe II and Fe III will have different values of magnetic moment due to different number of unpaired electrons in their d-orbitals.

QUESTION: 30

Among the following pair of oxides, which pair cannot be reduced by carbon to give the respective metals ?

Solution:

Potassium and calcium are strong reductant, hence their oxides cannot be reduced by carbon.

QUESTION: 31

Two liquids A and B are mixed. The partial vapour pressures of A and B in pure state are 100 and 200 mm respectively. If they are mixed in 1 : 4 mole ratio, assuming that mixture obeys Raoult's law, the mole fractions of A and B present in gaseous state in equilibrium of above solution are :

Solution:

QUESTION: 32

Identify the final product (Z) in the following sequence of reactions :

Solution:

QUESTION: 33

In a face centred cubic lattice, atom A occupies the corner positions and atom B occupies the face centre positions. If one atom of B is missing from one of the face centred points, the formula of the compound is

Solution:

In FCC arrangement

after removal of atom from face centre.

No. of A atoms per unit cell = 1/8 × 8 = 1

No. of B atoms per uni cell = = 1/2 ×5= 5/2

so formula is A_{2}B_{5}

QUESTION: 34

The standard e.m.f. of a cell, involving one electron change is found to be 0.591 V at 25. The equilibrium constant of the reaction is-

(F = 96500 C mol^{-1} ,R = 8.314 JK^{-1} mol^{-1}

Solution:

∆G° = ∆G + RT lnQ

-nFE° = -nFE + RT lnQ

E_{Cell} = E°_{Cell} - RT/nF lnQ

At equilibrium, E = 0

0 = 0.591 - (0.0591/1)× logK_{C}

-0.591 = -0.0591 logK_{C}

log K_{C} = 0.591/0.0591 = 10

K_{C} = antilog 10 = 1×10^{10}

QUESTION: 35

For the indicator HIn the ratio [In_{-}]/[HIn] is 7.0 at pH of 4.3. K_{eq} for the indicator is [Given log 7 = 0.845 and Antilogo (0.545) = 3.5

Solution:

QUESTION: 36

Match the hybrid bond orbitals of list I with the species of the list II and pick out the correct ?

Solution:

QUESTION: 37

The electrochemical cell shown below is a concentration cell.

M|M^{2+} (saturated solution of a sparingly soluble salt, MX_{2}) ||M^{2+} (0.001 mol dm^{- }^{3}) | M

The emf of the cell depends on the difference in concentrations of M^{2}^{+} ions at the two electrodes. The emf of the cell at 298 K is 0.059 V.

The value of Δ G (kJ mol^{-1}) for the given cell is (Take 1 F = 96500 C mol^{-1})

Solution:

QUESTION: 38

First and second ionization energies of Magnesium are 7.646 and 15.035 eV respectively. The amount of energy in KJ needed to convert all the atoms of Magnesium into Mg^{2}^{+} ions present in 12 mg of Magnesium vapours is :

[Given : 1 eV = 96.5 kJ mol^{-1}]

Solution:

QUESTION: 39

The enthalpy change of the reaction is -57.3 kJ mol^{-1}. If the enthalpies of Formation of are zero and -285.84 kJ mol^{-1} respectively, then the enthalpy of formation of

Solution:

QUESTION: 40

Amongst the halides

(1) BCl_{3}

(2) AlCl_{3}

(3) GaCl_{3}

(4) InCl_{3}

The order of decreasing Lewis acid character is

Solution:

Lewis acid strength of group 13 halides follow the order : BCl_{3} > AlCl_{3} > GaCl_{3} > InCl_{3}.

As we move down the group, the size of atom increases, and as a result, the tendency to attract electrons decreases. This leads to a decrease in Lewis acid nature down the group** **

QUESTION: 41

A cationic colloidal electrolyte forms micelle at 10^{-4} M concentration in water. If 1 mm^{3} solution contains 10^{12} micelle structure, then the no. of cations involved in one micelle are NA = 6 × 10^{23}.

Solution:

No. of particles of cationic colloidal electrolyte/litre before micelle formation = 10^{-4} × 6 × 10^{23} = 6 × 10^{19}

∴ No. of particles of cationic colloidal electrolyte/mm^{3} = 6 × 10^{19} × 10^{-6} = 6 × 10^{13}

Number of micelles formed = 10^{12} /mm^{3}

∴ Number of cations in one micelle = 6 × 10^{13}/10^{12} = 60

QUESTION: 42

The half-life of a radioactive isotope is three hours. If the initial mass of the isotope were 256 g, the mass of it remaining undecayed after 18 hours would be

Solution:

QUESTION: 43

Which of the above compounds are enantiomers?

Solution:

(i),(ii),(iv) have plane of symmetry, so they'll be meso compound. So its mirror image is identical.

Non-Mirror image to each other so III and V are enantiomers.

QUESTION: 44

Give the IUPAC name of :

Solution:

QUESTION: 45

When H_{2}O_{2} is added to ice cold solution of acidified potassium dichromate in ether and the contents are shaken and allowed to stand

Solution:

When H_{2}O_{2} is added acidified potassium dichromate in ether solution, potassium dichromate is oxidised to blue peroxide of chromium (CrO_{5}) which is soluble in ether and produces blue coloured solution.

K_{2}Cr_{2}O_{7} + H_{2}SO_{4} + 4H_{2}O_{2} 2CrO_{5} + K_{2}SO_{4} + 5H_{2}O

*Answer can only contain numeric values

QUESTION: 46

Identify the compounds in given set which can form enol.

Solution:

Only 3, 4, 5, 7 and 10 can form enol.

*Answer can only contain numeric values

QUESTION: 47

A** **reaction takes place at 300K. When catalyst is added, rate of reaction increases. How much temperature should be increased (in ^{∘}C) which can create same affect as produced by catalyst. (Experimentally it is known that catalyst change the activation energy by20%).

Solution:

*Answer can only contain numeric values

QUESTION: 48

The vapour pressure of a certain liquid is given by the equation Where P is vapour pressure in mmHg and T is temperature in K. Calculate latent heat of vaporisation (in cal mol^{−1}) at 75 .

K(R=2calmol^{−1}K^{−1})

Solution:

*Answer can only contain numeric values

QUESTION: 49

How many of total isomers are possible for the complex [Co(NH_{3})_{4}(NO_{2})_{2}]NO_{3}?

Solution:

*Answer can only contain numeric values

QUESTION: 50

How many atoms will have positive charge due to delocalization in given structure. Give answer including carbon on which charge is shown.

Solution:

QUESTION: 51

If the area bounded by y = ax^{2} and x = ay^{2}, a > 0, is 1, then a is equal to

Solution:

solving simultaneously

∴ (0,0) and are the points of intersection

∴ area between the given curves

QUESTION: 52

A single letter is selected at random from the word 'PROBABILITY'. The probability that it is a vowel, is :

Solution:

Out of 11 letters in the word PROBABILITYO,A,I and I are the vowels.

So probability that the letter selected is a vowel, is 4/11

QUESTION: 53

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number.

Solution:

Let the events be :

A : Number on the card drawn is even.

B : Number on the card is more than 3.

We are to find P(A/B).

We have : S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Then A = {2, 4, 6, 8,10}

and B = {4, 5, 6, 7, 8, 9, 10}

QUESTION: 54

Solution:

QUESTION: 55

Solution:

QUESTION: 56

A point R with x-co-ordinate 4 lies on the line segment joining the points P (2, - 3, 4) and Q (8, 0,10). Find the co-ordinates of the point R.

Solution:

QUESTION: 57

A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). The coordinates of the point A are

Solution:

Let the coordinates of A be (a. 0). Then the slope of the reflected ray is

Then the slope of the incident ray

From Eqs. (i) and (ii),

Thus, the coordinate of A is

QUESTION: 58

The points A, B and C represent the complex numbers z_{1}, z_{2}, (1−i)z_{1}+iz_{2} (where ) respectively on the complex plane. The triangle ABC is :

Solution:

Hence the triangle is isosceles and right angled.

QUESTION: 59

In the expansion of the term containing same powers of a and b is -

Solution:

∴ 13^{th} term contains same power of a & b.

QUESTION: 60

All the spades are taken out from a pack of cards. From these cards, cards are drawn one by one without replacement till the ace of spades comes. The probability that the ace comes in the 4th draw is

Solution:

So the required probability that are comes in 4^{th} draw

QUESTION: 61

Fifteen coupons are numbered 1, 2, 3, ....., 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupons is at most 9, is :

Solution:

Total coupons = 15

1 __<__ selected coupon number __<__ 9

i.e., 1, 2, 3, 4, 5, 6,, 7, 8, 9

∴ Probability of one selected coupon to have number

Hence, the required probability

= (3/5) x (3/5) x ....... 7 times = (3/5)^{7} (Using multiplicative principle for independent events)

QUESTION: 62

Find the value(s) of the parameter 'a'(a>0) for each of which the area of the figure bounded by the straight line, the parabola is the greatest

Solution:

point of intersection of (i) and (ii)

f(a) is max is when

QUESTION: 63

Solution:

QUESTION: 64

The parameter on which the value of the determinant

does not depend upon, is

Solution:

⇒ Δ = (1 + a^{2} - 2a cos dx) [sin (p + d)x cos px - sin px cos (p + d) x]

⇒ Δ = (1 + a2 - 2a cos dx) sin dx

Which is independent of p.

QUESTION: 65

If n is even positive integer, then the condition that the greatest term in the expansion of (1 + x)^{n} may have the greatest coefficient also is

Solution:

⇒

and

⇒

Hence,

QUESTION: 66

The equation of the common tangent to the curves y^{2} = 8 x and xy = -1 is

Solution:

Tangent to the curve y^{2} = 8x is

So, it must satisfy xy = - 1.

⇒

⇒

Since, it has equal roots.

∴ D = 0

Hence, equation of common tangent is y = x + 2

QUESTION: 67

If ABC be a triangle with and AB = x such that (AB) (AC) = 1. If x varies, then the longest possible length of the angle bisector AD is

Solution:

⇒ y_{max} = 1/2 since the minimum value of the denominator is 2 if x > 0.

QUESTION: 68

The locus of the midpoint of the portion of the line x cos α + y sin α = p intercepted between the axes is

Solution:

If M(h, k) is the mid-point then

Hence the requried locus is

QUESTION: 69

Find the derivative with respect to x of the function :

Solution:

QUESTION: 70

A lady wants to select one cotton saree and one polyester saree from a textile shop. If there are 10 cotton varieties and 12 polyester varieties, in how many ways can she choose the two sarees ?

Solution:

There are 10 cotton sarees. The lady can select the cotton saree in 10 different ways. Corresponding to each of the above 10 different ways, she can select the polyester saree in 12 different ways because there are 12 polyester sarees.. Hence the no. of different ways of selecting the two sarees

= 10 × 12 = 120

*Multiple options can be correct

QUESTION: 71

Two lines and are coplanar. then find the value of ** α**

Solution:

Direction ratios of L_{1} are (0, α – 3, 2) and it passes through the point (5, 0, 0) and direction ratios of L_{2} are (0, 1, α – 2) and it passes through the point (α, 0, 0) If L_{1} & L_{2} are coplanar then

so α = 1,4,5

*Answer can only contain numeric values

QUESTION: 72

A man will take 3 steps forward or 3 steps backward he will fall in the well. Given the probability of steps forward is 1/2 and steps backward is 1/2. If probability that he fells in well in the 5^{th} steps is p/2q when p and q are co-prime, then find the value of q+p.

Solution:

FFBFF X 3 + BBFBB X 3

*Answer can only contain numeric values

QUESTION: 73

The sum of the areas of *n* squares in n^{2}. If the areas of the squares can be put in A.P. What is the length of the side of the 25^{th} square?

Solution:

*Answer can only contain numeric values

QUESTION: 74

If where *p* and *q* are co-prime, then find the value of p+q.

Solution:

*Answer can only contain numeric values

QUESTION: 75

If A is a square matrix of order *n** *such that then find the possible value of

Solution:

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