physics part 1 - physicaeducator.files.wordpress.com · a disc of radius r is rolling without...

This section contains 8 multiple choice questions numbered 1 to 8. Each question has 4 choices (A), (B),

(C) and (D), out of which ONLY ONE is correct.

1. A disc of radius R is rolling without slipping on a fixed

circular track of radius 2R. Line joining the centre of the disc

and centre (O) of circular track is rotating with constant

angular velocity �� in clockwise sense. A rod AB whose end

A is attached to a small ring which can move only along a

fixed horizontal rod and other end of rod B is pivoted at B,

R/2 distance above centre of disc as shown. Find the angular

velocity of rod AB at the instant when rod AB is making an

angle 30�� with horizontal and

above the centre (O) of the track as shown in figure.(Assume

rod can rotate freely about hinge point B)

(A) Zero

(B) ��

(C) ��

(D) ��

2. Two blocks A and B of masses m and 2m respectively are connected together by a light spring of

stiffness k and then placed on a smooth horizontal surface. The blocks are pushed towards each

other the block A by the spring, by the time the spring acquires its natural length, is

(A) ��

(B) ��

(C) � ��

Physics

Part 1

Straight Objective Type


t of which ONLY ONE is correct.

A disc of radius R is rolling without slipping on a fixed

circular track of radius 2R. Line joining the centre of the disc

and centre (O) of circular track is rotating with constant

in clockwise sense. A rod AB whose end

to a small ring which can move only along a

fixed horizontal rod and other end of rod B is pivoted at B,

R/2 distance above centre of disc as shown. Find the angular

velocity of rod AB at the instant when rod AB is making an

with horizontal and centre of the disc is directly

above the centre (O) of the track as shown in figure.(Assume

rod can rotate freely about hinge point B)

Two blocks A and B of masses m and 2m respectively are connected together by a light spring of

fness k and then placed on a smooth horizontal surface. The blocks are pushed towards each



Two blocks A and B of masses m and 2m respectively are connected together by a light spring of

fness k and then placed on a smooth horizontal surface. The blocks are pushed towards each


(D) ��

3. A cylinder of mass m rests in a supporting block as shown. If

= 60� and θ = 30�, calculate the maximum acceleration a

which the block may be given up the incline so that the

cylinder does not lose contact at B. (neglect friction anywhere)

(A) g/2

(B) g

(C) 2g

(D) g/4

4. A block of mass m is attached to the frame by a light

spring of stiffness k. The frame and block are initially

at rest with x = ��,the uncompressed length of the

spring. If the frame is given a constant horizontal

acceleration �� towards left, determine

maximum velocity �� the frame (block is free to move inside frame). Ignore any friction.

(A) ��

(B) ��

(C) ��

(D) ��

5. A heavy disc with radius R is rolling down

hanging on two non-stretched string wound

around the disc very tightly. The free ends of

the string are attached to a fixed horizontal

support. The strings are always tensed during

the motion. At some instant, the

velocity of the disc is ω, and the angle

between the strings is α. Find the velocity of

A cylinder of mass m rests in a supporting block as shown. If β

, calculate the maximum acceleration a

which the block may be given up the incline so that the

cylinder does not lose contact at B. (neglect friction anywhere)

block of mass m is attached to the frame by a light

spring of stiffness k. The frame and block are initially

,the uncompressed length of the

spring. If the frame is given a constant horizontal

towards left, determine the

�� of the block relative to

the frame (block is free to move inside frame). Ignore any friction.

A heavy disc with radius R is rolling down

stretched string wound

around the disc very tightly. The free ends of

the string are attached to a fixed horizontal

support. The strings are always tensed during

the motion. At some instant, the angular

, and the angle

α. Find the velocity of

centre of mass of the disc at this moment

(A) ��

�� !

(B) ��

"#�� !

(C) �� !

(D) ��"#�� !

6. A small particle of mass m is attached at B to a hoop as mass m and

radius r, whole system is placed on the rough horizontal ground. The

system is released from rest when B is directly above A and rolls

without slipping. Find the angular acceleration of the system at the

instant when AB becomes horizontal as shown in the f

(A) �$�

(B) �$�%�

(C) �$&�

(D) &$��

7. A uniform rod of mass m and length L is placed on the

fixed cylindrical surface of radius R at a small angular

position ' from the vertical (vertical means line joining

centre and vertex of the cylindrical path) as sho

figure and released from rest. Find the angular velocity

of the rod at the instant when it crosses the horizontal

position (Assume that when it crosses the horizontal position its midpoint and vertex of the

circular surface coincide). Friction

(A) ()3*+�',-.' / 01,'

(B) ()*+�',-.' / 01,' 2

(C) �()*+�',-.' / 01,' 2

(D) ()6*+�',-.' / 01,'

centre of mass of the disc at this moment

A small particle of mass m is attached at B to a hoop as mass m and

radius r, whole system is placed on the rough horizontal ground. The

system is released from rest when B is directly above A and rolls

without slipping. Find the angular acceleration of the system at the

instant when AB becomes horizontal as shown in the figure.

A uniform rod of mass m and length L is placed on the

fixed cylindrical surface of radius R at a small angular

from the vertical (vertical means line joining

centre and vertex of the cylindrical path) as shown in the

figure and released from rest. Find the angular velocity ω

of the rod at the instant when it crosses the horizontal


circular surface coincide). Friction is sufficient to prevent any slipping.

2 1

2 1

2 1

2 1


8. A small ball of mass 100g is attached to a light and

inextensible string of length 50cm. The string is tied to a

support O and the mass m released from point A which is at a

horizontal distance of 30 cm from the support. Calculate the

speed of the ball as its lowest point of the trajectory.

(A) 2.2 m/s

(B) 2.5 m/s

(C) 3.2 m/s

(D) 2.5 m/s

This section contains 3 groups of questions. Each group has 2 multiple choice question based on a

paragraph. Each question has 4 choices (A), (B), (C) and (D), for its

correct.

Paragraph for question 9 to 10

A block of mass 10 kg is placed at the centre of a rough disc

which is at rest on a horizontal surface with its plane horizontal.

The disc starts moving with a constant acceleration � 5 / � 7 m/, as shown in figure. The coefficient of friction

between the block and the disc is μAnswer the following question on the basis of given data.

9. Now if an elongated spring of the spring constant k = 50 N/m is attached so

spring is fixed to periphery of disc and other end to the block as shown in figure, the block just

starts moving in +y (positive y axis) direction with negligible acceleration with respect to disc.

The friction force on block immediately af

(A) 2N(-7�9

(B) 5N(-7�9

(C) 10N(-7�9

A small ball of mass 100g is attached to a light and

inextensible string of length 50cm. The string is tied to a

support O and the mass m released from point A which is at a

horizontal distance of 30 cm from the support. Calculate the

its lowest point of the trajectory.

Comprehensive Type


paragraph. Each question has 4 choices (A), (B), (C) and (D), for its answer, out of which ONLY ONE is

A block of mass 10 kg is placed at the centre of a rough disc

which is at rest on a horizontal surface with its plane horizontal.

The disc starts moving with a constant acceleration �: =

as shown in figure. The coefficient of friction

μ; = μ� = 0.1

Answer the following question on the basis of given data.

Now if an elongated spring of the spring constant k = 50 N/m is attached so that one end of



The friction force on block immediately after block starts moving with respect to disc is


answer, out of which ONLY ONE is

that one end of



ter block starts moving with respect to disc is

(D) 15N(-7�9

10. In the previous problem initial elongation in the spring is

(A) �√�� m

(B) �√ m

(C) �√= m

(D) �√�� m

Paragraph for questions 11 and 12

Three small identical spheres A, B and C each of mass m, are connected to

a small ring D of negligible mass by means of three identical light

inextensible strings of length l each, which are equally spaced as shown.

The spheres may slide freely on a fric

spheres have given same speed >�

are moving n a circle about ring D which is at rest. Suddenly string CD

breaks. After the other two string becomes taut again, determine

11. Speed of ring D

(A) >�

(B) ?�

(C) ?��

(D) ?�

12. The angular speed of A with respect to D is (when string become taut)

(A) �?�

(B) �?�

(C) �?�&

(D) ?�

In the previous problem initial elongation in the spring is


Three small identical spheres A, B and C each of mass m, are connected to

a small ring D of negligible mass by means of three identical light

each, which are equally spaced as shown.

The spheres may slide freely on a frictionless horizontal surface. All three

perpendicular to string, such that, all

are moving n a circle about ring D which is at rest. Suddenly string CD

breaks. After the other two string becomes taut again, determine

The angular speed of A with respect to D is (when string become taut)


Two identical discs A and B of mass m and radius R, each are

placed on the rough horizontal surface. Their

connected with the light spring of spring constant k. Initially

spring is in its natural length and discs are at rest. Now centre

of disc A has given velocity >� in the horizontal direction as

shown in the figure. There is sufficient friction between discs

and ground to prevent the slipping at all instant.

13. Find the maximum compression of the spring

(A) ?� ��

(B) ?� ��

��

(C) ?� ��

(D) ?� ��

14. Find the angular velocity of disc A at the instant of maximum compression in the spring

(A) ?��

(B) ?��

(C) ?��

(D) ?��

This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its

answer, out of which ONE OR MORE

15. A uniform semicircular wire of mass M =

horizontal axis coinciding with the diameter passing through open ends. First the wire is taken

aside such that its plane becomes horizontal and then it is released from rest. Choose the


A and B of mass m and radius R, each are

placed on the rough horizontal surface. Their centres are

connected with the light spring of spring constant k. Initially

spring is in its natural length and discs are at rest. Now centre

in the horizontal direction as

shown in the figure. There is sufficient friction between discs

and ground to prevent the slipping at all instant.

Find the maximum compression of the spring

velocity of disc A at the instant of maximum compression in the spring

Multiple Correct Choice Type


ONE OR MORE is/are correct

A uniform semicircular wire of mass M = �√kg and radius R = 1m is free to rotate about a fixed

zontal axis coinciding with the diameter passing through open ends. First the wire is taken

becomes horizontal and then it is released from rest. Choose the

velocity of disc A at the instant of maximum compression in the spring


kg and radius R = 1m is free to rotate about a fixed

zontal axis coinciding with the diameter passing through open ends. First the wire is taken

becomes horizontal and then it is released from rest. Choose the

correct option(s), when vertical component of velocity of the centre of m

g= 10m/,, @ = 10)

(A) Angular displacement of wire is

(B) Vertical component of force exerted by the axis on the wire is 5

(C) Vertical component of force exerted by the axis on the wire is 10

(D) Horizontal component of force

16. A subway train travels between two of its

station stops with the acceleration schedule

shown in the acceleration verses time

graph. Then

(A) The time interval ∆t during which the

train brakes to a stop with a

deceleration of 2 m/,

(B) The distance between stations is 350 m.

(C) The time interval ∆t during which the


deceleration of 2 m/,

(D) The distance between stations is 416 m.

17. A small sphere of mass m is suspended by a ligh

inextensible string of length

inclined plane of inclination

moving in a circle on the incline plane as shown. If the sphere

has a velocity u at the top most position A. then,

(A) The tension in the string as the sphere passes the

position B equals to m�(B) The tension in the string at the bottom most position

(C) The tension in the string as the sphere passes the

correct option(s), when vertical component of velocity of the centre of mass is maximum (take

Angular displacement of wire is 01BC� √2

Vertical component of force exerted by the axis on the wire is 5√2 N

Vertical component of force exerted by the axis on the wire is 10√2 N

component of force exerted by the axis on the wire is 8 N

A subway train travels between two of its

station stops with the acceleration schedule

shown in the acceleration verses time

t during which the


is 8 sec.

stations is 350 m.

t during which the


is 10 sec.

The distance between stations is 416 m.

A small sphere of mass m is suspended by a light and

inextensible string of length l from a point O fixed on a smooth

inclined plane of inclination ' with the horizontal. The sphere is

moving in a circle on the incline plane as shown. If the sphere

has a velocity u at the top most position A. then,

The tension in the string as the sphere passes the 90�

�F�G 2 2*,-.'!.

The tension in the string at the bottom most position equals to m�F�G / 5*,-.'The tension in the string as the sphere passes the 90� position B equals to

ass is maximum (take

*,-.'!.

position B equals to m�F�G / 2*,-.'!.

(D) The tension in the string at the bottom most position

18. In the adjacent figure, a uniform disc of mass 2m and radius

and radius l/2 is lying at rest on a smooth horizontal surface. A

particle ‘A’ of mass m is connected to a light string of length

whose other end is attached to the circumference of the disc.

Initially string is just taut and tangential to the disc, particle A is

at rest. In the same horizontal plane another particle B of same

mass m moving with velocity

elastically with A. just after impact which of the following

statements will be true

(A) Tension in the string is

(B) Acceleration of the centre of the disc is

(C) Tension in the string is

(D) Acceleration of the centre of the disc is

19. In the adjacent figure a block A of mass m is hanging vertically with the help of a light

inextensible string which is passi

to block C of mass 2M as shown in the figure

to the block B of mass 2M as shown. Then, (ignore any

friction)

(A) Tension in the string immediately after system is

released from rest is �I$�J

(B) Tension in the string immediately after system is released from rest is

(C) The acceleration of block of mass m immediately after system is released from rest is

(D) The acceleration of block of mass m immediately after system is released from rest is

The tension in the string at the bottom most position equals to m�F�G 2 5*,-.'In the adjacent figure, a uniform disc of mass 2m and radius l/2

/2 is lying at rest on a smooth horizontal surface. A

particle ‘A’ of mass m is connected to a light string of length l ,

whose other end is attached to the circumference of the disc.

Initially string is just taut and tangential to the disc, particle A is

at rest. In the same horizontal plane another particle B of same

mass m moving with velocity >� perpendicular to string collides

elastically with A. just after impact which of the following

Tension in the string is �?��=G

leration of the centre of the disc is ?��=G

Tension in the string is �?��=G

Acceleration of the centre of the disc is ?��=G

In the adjacent figure a block A of mass m is hanging vertically with the help of a light

inextensible string which is passing over massless and frictionless pulley its other end is attached

to block C of mass 2M as shown in the figure. Pulley is fixed

to the block B of mass 2M as shown. Then, (ignore any

Tension in the string immediately after system is �I$JI

Tension in the string immediately after system is released from rest is �I$�JI

The acceleration of block of mass m immediately after system is released from rest is

The acceleration of block of mass m immediately after system is released from rest is

*,-.'!.

In the adjacent figure a block A of mass m is hanging vertically with the help of a light

ng over massless and frictionless pulley its other end is attached

�I$I

The acceleration of block of mass m immediately after system is released from rest is �$�JI

The acceleration of block of mass m immediately after system is released from rest is �$�JI

20. A rod CD of length L and mass M is placed horizontally on a

frictionless horizontal surface as shown. A second identical rod

AB which is also placed horizontally (perpendicular to CD) on the

same horizontal surface is moving along the surface with a

velocity v in a direction perpendicular to rod CD and its end B

strikes the rod CD at end C and sticks to it rigidly. Then,

(A) Velocity of centre of mass of the system just after impact is

(B) The ω(angular speed) of system just after collision is

(C) Velocity of centre of mass of the system just after impact is

(D) The ω(angular speed) of system just after collision is

This section contains 8 multiple choice questions numbered 1 to 8. Each question has 4 choices

(C) and (D), out of which ONLY ONE

1. Which observation did not contribute to the development of Bohr’s model of the atom?

(A) Photons have specific frequency and wavelength

(B) Electrons have dual nature (wave

(C) Light is emitted by K gas at low pressure when electricity is passed through it

(D) The nuclear model proposed by

Rutherford

2. What wavelength of light is required to

convert

Given bond enthalpies: N=N (711 kj/mole)

N-N (163 kj/mole)

(A) 218 nm

A rod CD of length L and mass M is placed horizontally on a

frictionless horizontal surface as shown. A second identical rod

AB which is also placed horizontally (perpendicular to CD) on the

same horizontal surface is moving along the surface with a

ty v in a direction perpendicular to rod CD and its end B

strikes the rod CD at end C and sticks to it rigidly. Then,

Velocity of centre of mass of the system just after impact is ?

The ω(angular speed) of system just after collision is �?=(

of centre of mass of the system just after impact is ?

The ω(angular speed) of system just after collision is =?�(

Chemistry

Part 2


This section contains 8 multiple choice questions numbered 1 to 8. Each question has 4 choices

ONLY ONE is correct.

Which observation did not contribute to the development of Bohr’s model of the atom?

Photons have specific frequency and wavelength

Electrons have dual nature (wave-particle duality)

gas at low pressure when electricity is passed through it

The nuclear model proposed by

What wavelength of light is required to

Given bond enthalpies: N=N (711 kj/mole)

N (163 kj/mole)


Which observation did not contribute to the development of Bohr’s model of the atom?

gas at low pressure when electricity is passed through it

(B) 310 nm

(C) 425 nm

(D) 612 nm

3. Boron nitride is reffered to as white graphite because it is lubricious material with the same

platy hexagonal structure as black graphite (Carbon graphite).

Which of the following properties of hBN is untrue?

(A) Hcp unit cell

(B) High thermal conductivity

(C) High electrical resistance

(D) Chemically inert

4. Lanthanum (III) chloride is a white hygroscopic powder which attains the following equilibrium

in a sealed vessel:

LaCL� (s)+ KM (g) ⇌ LaCIO (s)+ 2HCI(g)

More water vapour is added and the equilibrium is allowed to re-establish. If at new

equilibrium, OPQ has doubled, OPRS increases:

(A) 2 times

(B) √2 times

(C) 3 times

(D) √3 times

5. Electron capture accomplishes the same end result for the nucleus as:

(A) α emission

(B) TC� emission

(C) TJ emission

(D) U emission

6. If the reaction A + B → P is exothermic to the extent of 30 kcal/mole and V� for the forward

reaction is 294 kj/mole, V� for the backward reaction in kcal/mole is:

(A) 324

(B) 264

(C) 100

(D) 40

7. Which physical constant for KM has higher magnitude than WM:

(A) Boiling point

(B) Temperature of maximum density

(C) Dielectric constant

(D) Bond dissociation energy

8. Ratio of frequency of revolution of electron in second excited state of HXJ and second state of H

is Y�Z. x+y is

(A) 6

(B) 7

(C) 8

(D) 9

Comprehensive Type


paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is

correct.

Paragraph for question Nos. 9 to 10

Orthophosphoric acid �K�OM� is a weak tribasic mineral acid

K�OM ⇌ KJ + KOMC [�\ = 10C�

K�OMC ⇌ KJ + KOMC [�� = 10C]

K�OMC ⇌ KJ + KOMC� [�^ = 10C��

9. If 1,2,3,4 moles of K�OM, NaKOM , N�KOMand N��OM respectively are mixed together to

form an aqueous solution, resulting pH is:

(A) 8

(B) 9

(C) 10

(D) 12

10. If 2.5 mole each of K�OM, NaKOM , N�KOM and N��OM are mixed together to form an

aqueous solution, resulting pH is

(A) 5

(B) 7

(C) 9

(D) 11

Paragraph for questions Nos. 11 to 12

The isomerisation of cyclopropane ��K%� to propene is believed to occur by the following mechanism:

Step-1: ��K% + ��K% → ��K% + ��K%∗

Step-1: ��K%∗ → CK� - CH = CK

11. At low pressure, step-1 unindirectional and slow whereas step-2 is bidirectional and fast. Order

of the reaction is:

(A) 0

(B) 1

(C) 2

(D) 3

12. At high pressure, step-1 is bidirectional and fast whereas step-2 is unidirectional and slow. Order

of the reaction is:

(A) 0

(B) 1

(C) 2

(D) 3

Paragraph for Question Nos. 13 to 14

Redox reaction involves transfer of electrons between 2 chemical species. An unbalanced and

incomplete example is shown below:

+ K�OM → H � + OM%

13. If the above reaction were possible in either direction, maximum equivalent weight would

belong to:

(A)

(B) K�OM

(C) K �

(D) OM%

14. If the above reaction were unidirectional towards right, what fraction of OM% would be left

unreacted if reaction were started with stoichiometric amount of reactants.

(A) �

(B) �

(C) ��

(D) �

Multiple Correct Choice Type


answer, out which ONE OR MORE is/are correct.

15. Bubbling aK% through a solution of alkali metal (M) in liquid NK� produces which of the

following chemical species:

(A) MBK

(B) `KaK

(C) K

(D) baK

16. If equatorial plane in PCL= molecule is the X-Y plane, the orbitals hybridizing to produce axial

bonds will be:

(A) cd

(B) e �Cf�

(C) cf

(D) ed�

17. Which of the following anions is/are bigger than hydride ion(KC)

(A) gC

(B) �LC

(C) ahC

(D) LC

18. Which of the following changes can be experimentally realized in the laboratory:

(A) Combustion of Mg ribbon in CM

(B) Reduction of FeCL� (aq.) on adding Zn granules

(C) Dry heating of hydrated MgCL to get anhydrous MgCL

(D) Evolution of brown gas on heating LiNM�

19. Which digits would appear as stoichiometric coefficient of either reactant or product when the

following skeletal redox reaction is balanced:

ib.M + Kj + KjM → iKjM + b.jM + b.jM + j + KM

(A) 2

(B) 4

(C) 6

(D) 8

20. Identify the correct sequencing of hydrides based on the parameter specified in bracket:

(A) NK� > OK� > l,K� > jmK� (bond angle)

(B) SbK� > `K� > l,K� > OK� (boiling point)

(C) NK� > OK� > l,K� > jmK� (dipole moment)

(D) NK� > OK� > l,K� > jmK� (basic nature)

Mathematics

Part 3



(C) and (D) out of which ONLY ONE is correct.

1. The value of lim →rs�t.��01BC��u is

(A) − ��

(B) ��

(C) �

(D) − �

2. The value of v t.|2,-.� + 1|x� e�, is

(A) Is equal to -2πln2

(B) Is equal to − x t.2

(C) Is equal to 0

(D) Does not exist

3. Let f be a differentiable function such that |y��| ≤ 1∀� ∈ }−1,1� and g(x) = |y′��|, g(0) = 4,

then choose the correct statement

(A) There is no point x in the interval (-1,0) at which g(x) ≤ 2

(B) g(x) > 2 ∀ � ∈ (0,1)

(C) there is a point of local maxima of g(x) in (-1,1)

(D) x=0 is a point of local maxima of g(x)

4. v "#� J√��J �� J�

^!J;�� J�^! dx, is

(A) ��;

�J;�� J�^! + 0

(B) ;��

�J;�� J�^! + 0

(C) ;��

�J;�� J�^! + 0

(D) � B�.C� �1 + 2,-. �� + x

�!� + 0

5. Solution of the differential equation �f� = = ^C f�C

� �fCf^ , is

(A) In|� − 4�� + 5� + 2� + 2| − B�.C� �f�C �C� �C� ! = 0

(B) In|� − 4�� + 5� + 2� + 2| + B�.C� �f�C �C� �C� ! = 0

(C) In|� − 4�� + 5� + 2� + 2| + 2B�.C� �f�C �C� �C� ! = 0

(D) In|� − 4�� + 5� + 2� + 2| − 2B�.C� �f�C �C� �C� ! = 0

6. The value of v �J�� J�^/� e�

� , is

(A) )2√2

(B) )3√2

(C) )2√3

(D) )3√3

7. Let �� = 0, m� = 8, ��J� = �� + � m� �.e m�J� = �

m� ∀ - = 1, 2, 3, … ….. let �� be area of the loop

formed by |� − ��| + |�| = m�. If all the loops are plotted on the same X – Y plane, then the

value of ⋃ l�� is

(A) =�

� ,�. �.-B,. (B)

�� ,�. �.-B,.

(C) �&

� ,�. �.-B,. (D)

&�� ,�. �.-B,.

8. The number of real roots of the equation 54� − 36�� + 18� − 6� + 1 = 0, is

(A) 0

(B) 2

(C) 3

(D) 4

Comprehension type

This section contains three groups of questions. Each group has two multiple choice question based on a

paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only one is

correct.

Paragraph for question Nos. 9 and 10

Read the following write up carefully and answer the following questions:

Let y�� = �� , � ∈ �0, @��.e *��be inverse of y��

9. If ‘A’ is the area of the region bounded by � = *��, x − axis, the line � = x and the line � = �

x,

then the value of ‘A’ lies in the interval

(A) �0, �%!

(B) ��% , �

!

(C) �� ,

�!

(D) �� , 1!

10. The slope of tangent to the curve � = *��, � ∈ �0, @� at the point where it meets the curve

�� = 1, is

(A) − x��

(B) − x�

(C) − x�

(D) − �x�


Let f: R → R; f�x� = 2x� − 3�k + 2�x + 12kx − 7, −4 ≤ k ≤ 6, k ∈ l, then

11. The number of values of k for f(x) to be invertible, is

(A) 0

(B) 1

(C) 2

(D) 3

12. The number of values of k for f(x) to have three real and distinct roots, is

(A) 2

(B) 4

(C) 5

(D) 6


Let a tangent be drawn at a point on the locus f(x, y) = 0 and it meets the positive X and Y – axis at points

F and Q. let A and G be respectively the arithmetic and geometric means of the segments OP and OQ.

Now,

13. If A = 1 then the locus f(x, y) = 0, is

(A) � = f� +

�C�, (where k is a parameter)

(B) � + f

�C� = 2, (where k is a parameter)

(C) � = �� + �

�C�, (where k is a parameter)

(D)

�� + f�C� = 1, (where k is a parameter)

14. If G = 1, then the locus of f(x, y) = 0, is

(A) [� − 2�� + [� + 1 = 0, (where k is a parameter)

(B) �1 − [�� − 2�� + [� + 1 = 0, (where k is a parameter)

(C) [� − 2�� + �� + 1 = 0, (where k is a parameter)

(D) �1 − [�� − 2�� + �� + 1 = 0, (where k is a parameter)

Multiple correct choice type

This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) for

its answer, out of which one or more is/are correct.

15. Let y�� = √YC√YJ. The graphs of y = f(x) and y = f(4x) are symmetrical about the points (a1, 0)

and (a2, 0) respectively, than

(A) a1 = 1, a2 = 4

(B) a1 = 2, a2 = �

(C) v y�2��e� = 1�

(D) v y�2��e� = 0�

16. Let g(x) = f(x) + f(1-x) where f(x) = � sin�@�� , � © �0,1� then

(A) g(x) = � y �

! y ��C !

(B) g(x) > π ∀ x∈ (0,1)

(C) g(x) ≤ 4 ∀ x ∈ (0,1)

(D) x < v ;��x �

��C ��/

� e� < 2

17. Let f(x) = 5x tan x + 8sin(tan x) + 5ln(cos x), then in the interval �Cx , 0!

(A) F(x) is increasing

(B) F(x) has a point of local maxima

(C) F(x) has a root

(D) F(x) is always negative

18. Let f(x) = 01,�. X�� , x ∈ �Cx , x

! then

(A) F’(x) has a point of local maxima at x=x

(B) F’(x) has a point of local maxima in the interval �Cx , 0!

(C) F’(x) has exactly two points of local maxima/minima in �Cx , x

!

(D) F’(x) has no root in �Cx , x

!

19. Let f be any twice differentiable function ∀ x∈ + and x = b be its points of local maxima and local

minima respectively then

(A) y = [f(x)], ([.] denotes the G.I.F) must be continuous at x = a

(B) y = [f(x)], ([.] denotes the G.I.F) must be continuous at x = b

(C) f”(a) must be negative

(D) f”(b) must be negative

20. let | = v ��2 �!�/

� dx, then

(A) | = v ��1 − �� e��/�

(B) | = �v ��1 − �� e��

�

(C) | = v ,-.�' 01,�' e'x/�

(D) | = ��%�

ANSWER KEY

Q. No. Physics Chemistry Mathematics

1 A B A

2 D A C

3 B A C

4 C B C

5 B C D

6 C C B

7 D C A

8 A C A

9 C C B

10 A B B

11 B C B

12 A B B

13 C C A

14 A D C

15 A,B,D A,B,C B,D

16 C,D A,D A,B,C,D

17 B,C B,C,D A,D

18 A,B A,B,D A,B,C

19 A,C A,B,D B,D

20 B,C A,B,C,D A,B,C,D

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