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QUESTION: 1

Work done by the system in closed path ABCA, as shown in figure is

Solution:

Work done = area of ΔABC

QUESTION: 2

A body executes S.H.M. of period 20 seconds. Its velocity is 5 cm s^{-1}, 2 seconds after it has passed the mean position. Find amplitude of the bob

Solution:

QUESTION: 3

Calculate the gas constant for 1g of a gas from the following data: c_{p} = 0.245calg^{-1o}C^{-1};c_{v} = 0.165calg^{-1}°C^{-1}

Solution:

QUESTION: 4

A balloon is moving upwards with a speed of 20 m/s. When it is at a height of 14 m from ground in front of a plane mirror in as shown in figure, a boy drops himself from the balloon. Find the time duration for which he will see the image of source S placed symmetrically before plane mirror during free fall.

Solution:

FC = 2 + 10 = 12 m

The boy has dropped himself at point F. So, his velocity is 20 m/s in upward direction.

Let us first find the time to move from F to topmost point and then from topmost point to point C. From

Solving this equation we get, t_{1} = 4.53 s.

Velocity of boy at point Q,

(v^{2} = u^{2} - 2gh)

Time taken to move the boy from Q to topmost point and then from topmost point to Q will be.

∴Time required time is : t = t_{1} - t_{2} = 1.7 s

QUESTION: 5

Prism angle & refractive index for a prism are 60^{o} & 1.414. Angle of minimum deviation will be

Solution:

QUESTION: 6

If a gas is heated at constant pressure then what percentage of total heat supplied is used for doing work ? (Given : γ for gas = 4/3)

Solution:

Heat supplied dQ = nC_{p}dT

Heat used for work = dw = nRdT

= 25%

Thermodynamics Process

The process of change of state of a system involves change of thermodynamic variables such as pressure P, volume V and temperature T of the system. This process is known as thermodynamic process

QUESTION: 7

Four persons K, L, M, N are initially at the four corners of a square of side d. Each person now moves with a uniform speed v in such a way that K always moves directly towards L, L directly towards M, M directly towards N, and N directly towards K. The four persons will meet at a time .....

Solution:

The velocity of *K* throughout the motion towards the centre of the square is v cos45 and the displacement covered by this velocity will be *KO*.

QUESTION: 8

Estimate the speed of sound in air at standard temperature and pressure. The mass of 1 mole of air is 29.0 × 10^{-3} kg

Solution:

At N.T.P., P = 1.013 x 10^{5} N m^{-2}

volume of air, V = 22400 cm^{3} = 2.24 x 10^{-2} m^{3}

mass of 1 mole of air M = 29.0 x 10^{-3} kg Therefore, density of air at N.T.P.,

QUESTION: 9

A pendulum bob is suspended on a flat car that moves with velocity v_{0}. The flat car is stopped by a bumper

(i) What is the angle through which the pendulum swings.

(ii) If the swing angle is θ = 60° and l = 10m, what was the initial speed of the flat car?

Solution:

When the flat car collides with the bumper, due to inertia of motion the bob swings forward No work is done by tension of string on the bob therefore energy is conserved

QUESTION: 10

Two slits, 4 mm apart, are illuminated by light of wavelength 6000Å. What will be the fringe width on a screen placed 2 m from the slits?

Solution:

QUESTION: 11

A car moves uniformly along a horizontal sine curve y = a sin (x/α), where a and α are certain constants. The coefficient of friction between the wheels and the road is equal to k. At what velocity will the car ride without sliding?

Solution:

**Step I :** Determine the radius of curvature of given curve at point P (x, y)

∴ radius of curvature is

**Step II :** Relate the fnction and centnpetal acceleration In this case, centripetal acceleration is provided by force of fnction

From equation (i), for minimum value of radius of curvature,

QUESTION: 12

A large tank filled with water to a height 'h' is to be emptied through a small hole at the bottom. The ratio of times taken for the level of water to fall from h to h/2 and from h/2 to zero is

Solution:

Time taken for the level to fall from H to H'

According to problem the time taken for the level to fall from h to h/2

and similarly time taken for the level to fall from h/2

QUESTION: 13

A bullet of mass 2 g is having a charge of 2μC. The potential difference it must be accelerated through, starting from rest, to acquire a speed of 10 m/s is

Solution:

QUESTION: 14

A wire of length L and 3 identical cells of negligible internal resistance are connected in series. Due to the current, the temperature of the wire is raised by ΔT in a time t. A number of N similar cells are now connected in series with a wire of the same material and area of cross-section but of length 2L. The temperature of the wire is raised by the same amount ΔT in the same time . The value of N is

Solution:

In the first case

When length of the wire is doubled, resistance and mass both are doubled. Therefore, in the second case,

Dividing Eq.(ii) by (i) we get

QUESTION: 15

The intensity of the light coming from one of the slits in YDSE is double the intensity from the other slit. Find the ratio of maximum intensity to minimum intensity in the interference fringe pattern observed.

Solution:

QUESTION: 16

Consider α-particles, β-particles and γ- rays each having an energy of 0.5 MeV. In increasing order of penetrating powers, the radiations are

Solution:

Penetrating power is maximum for γ - rays, then of β -particles and then α - particles because basically it depends on the velocity. However, ionization power is in reverse order.

QUESTION: 17

A battery is supplying power to a tape-recorder by cable of resistance of 0.02 Ω . If the battery is generating 50 W power at 5V, then power received by tape-recorder is :

Solution:

P = VI,

50 = 5 × I

I = 10 A

Power lost in cable= I^{2}R = 10 × 10 × 0·02 = 2W

Power supplied to T. R= 50 W - 2 W = 48 W** **

QUESTION: 18

A uniform rod of mass m and length ℓ is rotating with constant angular velocity ω about an axis which passes through its one end and perpendicular to the length of rod. The area of cross section of the rod is A and its Young's modulus is Y. Neglect gravity. The strain at the mid point of the rod is :

Solution:

⇒

∴ Tension at mid point is :

QUESTION: 19

Initially car A is 10.5 m ahead of car B. Both start moving at time t = 0 in the same direction along a straight line. The velocity time graph of two cars is shown in figure. The time when the car B will catch the car A, will be ;

Solution:

QUESTION: 20

Two masses m_{1} and m_{2} are connected by a string of length *l*. They are held in a horizontal plane at a height H above two heavy plates A and B made of different material placed on the floor. Initially distance between two masses is a < *l*. When the masses are released under gravity they make collision with A and B co - efficient of restitution 0.8 and 0.4 respectively. The time after the collision when the string becomes tight is

Solution:

String will become tight when

∴

*Answer can only contain numeric values

QUESTION: 21

The difference between (n + 2)^{th} Bohr radius and n^{th}** **Bohr radius is equal to the (n – 2)^{th}** **Bohr radius. The value of n is ?

Solution:

*Answer can only contain numeric values

QUESTION: 22

The figure shows an RC circuit with a parallel plate capacitor. Before switching on the circuit, plate A of the capacitor has a charge –Q_{0}** **while plate B has no net charge. Now, at t = 0, the circuit is switched on. How much time (in second) will elapse before the net charge on plate A becomes zero.

(GivenC=1μF, Q_{0}=1mC, ε=1000V and

Solution:

Let at any time t charge flown through the plate B to plate A is Q and instantaneous current is I

Now for charge on plate A to zero q = Q_{0}

Putting value of C_{1 },Q_{0} ,∈ and R , we get t = 2 seconds

*Answer can only contain numeric values

QUESTION: 23

A thin non conducting disc of mass M = 2kg, charge Q = 2x10^{–2} C and radius R = 1/6m is placed on a frictionless horizontal plane with its centre at the origin of the coordinate system. A non uniform, radial magnetic field exists in space, where B_{0} = 10T and is a unit vector in the radially outward direction. The disc is set in motion with an angular velocity w = x*10^{2} rad/sec, about an axis passing through its centre and perpendicular to its plane, as shown in the figure. At what value of x, the disc will lift off from the surface.

Solution:

The force an any small part of the disc is in the vertically upward direction

*Answer can only contain numeric values

QUESTION: 24

All springs, string and pulley shown in figure are light. Initially when both the springs were in their natural state, the system was released from rest. The maximum displacement of block m is calculate x

Solution:

*Answer can only contain numeric values

QUESTION: 25

In Young’s Double slit experiment, a thin glass mica strip of thickness (t = 8λ) is pasted infront of slit S_{1}. If the same strip is now shifted in front of other slit S_{2}. If the number of fringes which will cross the central point on the screen is N. Find N? (d >> D, λ << d and n_{mica}** **= 1.5)

Solution:

(μ - 1)t = nλ

n = number of fringes shifted

(1.5 - 1)8λ = nλ

n = 4

If the number of fringes which will cross the central point on the screen is N

N, then

N = 2 n = 8

Hence the total number of fringes shifted is

8

QUESTION: 26

A saturated aqueous solution of sparingly soluble salt AB_{3} has the vapour pressure 0.08 mm lesser than the vapour pressure 17.33 mm of solvent at 25^{o}C. The solubility product of AB_{3} is :

Solution:

QUESTION: 27

Photoelectric emission is observed from a surface for frequencies v_{1} and v_{2} of the incident radiation (v_{1} > v_{2}). If the maximum kinetic energies of the photoelectrons in the two cases are in the ration l : k then the threshold frequency v_{0} is given by :

Solution:

QUESTION: 28

In the following electron dot structure calculate the formal charge from left to right nitrogen atom ?

Solution:

1.Left N

⇒ 5 - 4 - 1/2 (4) = - 1

2.Central N

⇒ 5 - 0 - 1/2 (8) = + 1

3.Right N

⇒ 5 - 4 - 1/2 (4) = - 1

QUESTION: 29

A 50 litre vessel is equally divided into three parts with the help of two stationary semi permeable membrane (SPM). The vessel contains 60 g H_{2}_{ }gas in the left chamber, 160 g O_{2} in the middle and 140 g N_{2} in the right one. The left SPM allows transfer of only H_{2} gas while the right one allows the transfer of both H_{2} and N_{2}. The final ratio of pressure in the three chambers.

Solution:

QUESTION: 30

Acetamide is treated separately with the following reagents. Which of these would give methylamine ?

Solution:

QUESTION: 31

White fumes. Metal (M) can be

Solution:

Li, Mg and Al when burn in air produces nitrides whose hydrolysis will liberate NH_{3}. NH_{3} gives white fumes with HCl.

QUESTION: 32

Copper ore is heated in a blast furnace after mixing with silica. Iron oxide slags off as iron silicate and copper is produced in the form of

Solution:

During smelting, the roasted ore is mixed with coke and silica in a small blast furnace. The changes occurring are

Most of

FeS is oxidised to ferrous oxide

FeO removed as slag

(slag) at the lowest point of furnace, motion mass is Cu_{2}S plus little FeS.

QUESTION: 33

The ionization isomer of [Cr(H_{2}O)_{4}Cl(NO_{2}_{ })]Cl is

Solution:

Ionisation isomers are the complexes that produce different ions in solution, i.e, they have ions interchanged inside and outside the coordination sphere.

Cr(H_{2}O)_{4}CI(NO_{2})]Cl and [Cr(H_{2}O)_{4}Cl_{2}](NO_{2}) have different ions outside the coordination sphere and they are isomers. Therefore, they are ionisation isomers.

QUESTION: 34

What is the pH of 0.01 M glycine solution ? For glycine

Solution:

For glycine H_{2}N - CH_{2} - COOH which can act as zwitter ion

QUESTION: 35

Which one of the following cannot be prepared from B_{2}H_{6} ?

Solution:

In diborane (B_{2}H_{6}) molecule, only four terminal hydrogen atoms are replaceable. The two bridge H - atoms can not be substituted. Hence B_{2}(CH_{3})_{6} cannot be prepared from B_{2}H_{6}.

QUESTION: 36

Titration of I_{2} produced from 0.1045 g of primary standard KIO_{3} required 30.72 mL of sodium thiosulphate as shown below :

The molarity of sodium thiosulphate ion is :

Solution:

QUESTION: 37

The standard reduction potentials of Cu^{2+ }/ Cu and Cu^{2+ }/ Cu^{+} are 0.337 V and 0.153 V respectively. The standard electrode potential of Cu^{+ }/ Cu half cell is

Solution:

QUESTION: 38

The electron energy in hydrogen atom is given by Calculate the energy required to remove an electron completely from the n = 2 orbit. What is the longest wavelength (in cm) of light that can be used to cause this transition?

Solution:

The required transition is n_{1} = 2to n_{2} = ∞ and corresponding transition energy is 1erg = 10^{-7} joule

As the electron is to be ejected from the atom, the final state will have 0 energy as n_{2} tends to infinity Hence, the amount of energy required for t

1 erg = 10^{-7} joule

The longest wavelength that can cause above transition can be determined as

QUESTION: 39

An element has fcc structure with edge length 200 pm, calculate the density if 200g of this element contains 24 × 10^{23} atoms. (N_{A = }6 × 10^{23})

Solution:

QUESTION: 40

A compound that gives a positive iodoform test is

Solution:

QUESTION: 41

1.44 g of Titanium (Ti) reacted with an excess of O_{2} and produced x gm of a non-stoichiometric compound Ti_{1.44}O_{1}. The value of x is :

Solution:

QUESTION: 42

A solution of a metal ion when treated with KI gives a red precipitate which dissolves in excess KI to give a colourless solution. Moreover, the solution of metal ion on treatment with a solution of cobalt (II) thiocyanate gives rise to a deep blue crystalline precipitate. The metal ion is

Solution:

QUESTION: 43

One desires to prepare a positively charged solution of silver iodide. This can be achieved by

Solution:

KI + AgNO_{3} → AgI + KNO_{3}

AgNO_{3} → Ag^{+} + NO_{3}^{- }

AgI_{(s)} + Ag^{+} → [AgI] + Ag^{+}.

QUESTION: 44

Anhydrous AlCl_{3}. How will it behave in Aqueous solution?

Solution:

It will behave ionic.

QUESTION: 45

The ionisation energy of solid NaCl is 180 kcal per mol.The dissolution of the solid in water in the form of ions is endothermic to the extent of 1 kcal per mol. If the solvation energies of Na^{+} and Cl^{-} ions are in the ratio 6 : 5, what is the enthalpy of hydration of sodium ion ?

Solution:

*Answer can only contain numeric values

QUESTION: 46

In gas phase what is the maximum number of coplanar atoms in X.

Solution:

*Answer can only contain numeric values

QUESTION: 47

A 200 ml aqueous solution of KCl was electrolysed for 16 min and 5 sec. If the pH of the final solution was 13 and volume of solution remains practically unchanged, then the average current used in ampere is.

Solution:

*Answer can only contain numeric values

QUESTION: 48

Reaction of hydrated ferric chloride (FeCl_{3}. 6H_{2}O) with thionyl chloride gives anhydrous ferric chloride with evolution of hydrochloric acid (HCl) and sulphur dioxide (SO_{2}) gases. The number of sulphur dioxide (SO_{2}) molecules involved in the balanced chemical equation is.

Solution:

*Answer can only contain numeric values

QUESTION: 49

A complex compound is represented as CoCl_{3}. xNH_{3}. Its 0.1 M solution in water shows depression in freezing point equal to 0.0558K. Assuming 100% ionisation of complex and coordination number of Co to be six, calculate the value of x. K_{f} for H_{2}O is 1.86 K kgmol^{–1}.

Solution:

*Answer can only contain numeric values

QUESTION: 50

How many stereoisomers of Pt(gly)_{2}Cl_{2} are possible?

Solution:

QUESTION: 51

Solution:

QUESTION: 52

ABC is an acute - angled triangle with circumcentre 'O' and orthocentre H. If AO = AH, then angle A is

Solution:

As we know in triangle, the distance of the ortho-centre from vertex A is 2R cos A

In the question it is given distance of the circum-centre and ortho-centre from the vertex A is equal.

QUESTION: 53

The locus of the mid point of the line segment joining the focus to a moving point on the parabola y^{2} = 4ax is another parabola with directrix

Solution:

Let p (h, k) be the mid point of the line segment joining the focus (a, 0) and a general point Q (x, y) on the parabola. Then

Put these values of x and y in y^{2} = 4ax, we get

QUESTION: 54

The eccentricity of the ellipse which meets the straight line on the axis of x and the straight line on the axis of y and whose axes lie along the axes of coordinates, is :

Solution:

Let ellipse be

∵ Ellipse meets the straight line on the axis of x.

and meets the straight line on the axis of y,

i.e., -b = -5

⇒ b = 5

∴ b^{2} = a^{2} (1 - e^{2})

⇒

∴

∴

QUESTION: 55

A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y^{2} = 4ax. The common chord of the circle and parabola is

Solution:

is the required circle ......(1)

now given parabola is

⇒ x = a/2 common chord of circle and parabola.

QUESTION: 56

A (3, 2, 0), B (5, 3, 2) and C (-9, 6, -3) are three points forming a triangle and AD is bisector of the angle AD meets BC at the point :

Solution:

The bisector of ∠BAC i.e., AD divides the side BC in the ratio AB : AC

Let co-ordinates of D (x: y, z)

QUESTION: 57

If p, q ∈{1,2,3,4} the number of equations of the form px^{2 }+ qx + 1 = 0 having real roots is

Solution:

For real roots, we must have

Thus, there are 7 values of (p,q).

QUESTION: 58

Solution:

QUESTION: 59

The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is

Solution:

Let the equation of the circle is (x - 1)^{2} + (y - k)^{2} = k^{2}

It passes through (2, 3) ⇒ (2 - 1)^{2} + (3 - k)^{2} = k^{2}

QUESTION: 60

If f(x) = a - (x - 3)^{8/9}, then maximum value of f(x) is:

Solution:

∵

∴

At x = 3: f' (x) is not defined,

hence sign scheme of f' (x) is

∴ f(x) is maximum at x = 3

Hence maximum value of f (x) is equal to "a"

QUESTION: 61

The number of terms in the expansion of which are integers, is given by

Solution:

should be integers so R be even

⇒ R = 0,2,4,........,250

∴ Number of integers terms = 251.

QUESTION: 62

In the Argand plane if O, P, Q represent respectively the origin, the complex number z and z + iz, then angle OPQ is

Solution:

QUESTION: 63

There are unlimited number of identical balls of four different colours. How many arrangements of at most 8 balls in row can be made by using them?

Solution:

The number of arrangements of one ball = 4,

because there are only four different balls.

The number of arrangements of two balls = 4 x 4 = 4^{2} , etc.

∴ the required number of arrangements

QUESTION: 64

The value of i^{i} is :

Solution:

QUESTION: 65

For what value of 'a' is the area bounded by the curve y = a^{2}x^{2} + ax + 1 and the straight line y = 0, x = 0 & x = 1 the least ?

Solution:

upward parabola with vertex

QUESTION: 66

If a_{1}, a_{2}, a_{3},....a_{20} are AMs between 13 and 67, then the maximum value of product a_{1} a_{2} a_{3} ....a_{20} is.

Solution:

∴ Maximum value of a_{1} a_{2} a_{3} .......a_{20} is (40)^{20}.

QUESTION: 67

The area bounded by the curvesy = (x - 1)^{2} , y = (x+1) ^{2} and y = 1/4 is

Solution:

The curves y = (x - 1)^{2}, y = (x + 1)^{2} and y = 1/4 are shown as

where point of intersection are

QUESTION: 68

Let z and ω be two non-zero complex numbers such that z = ω and arg z + arg ω = π, then z equals

Solution:

QUESTION: 69

If two distinct chords, drawn from the point (p, q) on the circle x^{2} + y^{2} = px + qy (where pq ≠ 0) are bisected by the x-axis, then

Solution:

Let (h,0) be the mid-point of chord then its equation is

T = S,

this line passes through

(p,q)

D > 0 (Since these are two such chords)

QUESTION: 70

If two vertices of a triangle are (5, -1), (-2, 3) and the orthocentre of the triangle lies at the origin, then the third vertex is

Solution:

Let the third vertex C be (x_{1}. y_{1}) since 0(0, 0) is the orthocenter

⇒ x_{1} = - 4

y_{1} = - 7

Hence the third vertex is

(-4, -7)

*Answer can only contain numeric values

QUESTION: 71

The summation of series upto infinite terms is equal to length of latus rectum of standard hyperbola with e = 2. If r is radius of director circle of its conjugate hyperbola, then r^{2}** **=

Solution:

⇒ radius of director circle of conjugate hyperbola is given by

r^{2}= b^{2}-a^{2} = 2

*Answer can only contain numeric values

QUESTION: 72

Let Find tr(A^{2012})

Solution:

A^{3} = 0, using CHT

*Answer can only contain numeric values

QUESTION: 73

Let P, Q be points of the ellipse 16x^{2} + 25y^{2} = 400 so that PQ = 96/25 and P, Q lie above major axis. Circle drawn with PQ as diameter touch major axis at positive focus. If m is slope of PQ then find the value of 1/|m|

Solution:

*Answer can only contain numeric values

QUESTION: 74

Solution:

*Answer can only contain numeric values

QUESTION: 75

if sin^{−1}x + sin^{−1}y + sin^{−1}z =π/2 then the value of x^{2}+y^{2}+z^{2}+2xyz is equal to

Solution:

sin^{−1}x + sin^{−1}y + sin^{−1}z =π/2

put

sin^{−1 }x = α, sin^{−1}y=β, sin^{−1}z=γ

(Given) or

cos α cos β - sin α sin β = sin γ ... (1)

and, we have

sin α = x ,

Similarly,

∴ From equation (i), we get

Squaring both sides, we have

x^{2} + y^{2} + z^{2} + 2xyz = 1

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