ST. XAVIER’S SR. SEC. SCHOOL, CHANDIGARH

Class 12 English Worksheet-14

ENGLISH LITERATURE

THE TEMPEST

QUESTIONS TO BE ATTEMPTED

1. ‘THE TEMPEST’IS PURELY A ROMANTIC DRAMA. Comment on the given statement and list out the reasons why

it is not a comedy or a tragedy.

2. Do you consider the play as a tragicomedy? Give reasons for your answer.

3. Comment on the handling of the theme of power in the play.

4. Critically examine the theme of forgiveness in the play.

5. ‘The Tempest’ shows us that there are some natures which cannot be improved by nurture. Discuss.

Act 4 Scene 1

Summary

This scene begins with Prospero acknowledging that he has only been testing Ferdinand and Miranda's love. Now that he knows

Ferdinand really loves his daughter, he is willing to bless the wedding. But he severely warns Ferdinand against taking Miranda's

virginity before the wedding and promises "barren hate, sour-eyed disdain, and discord" if he doesn't respect Prospero's wishes.

To celebrate the engagement and to show his powers, Prospero instructs Ariel to go and bring spirit actors to perform a masque. Iris,

Ceres, Jupiter, Venus, Mars, Hymen, and Juno all feature in this dramatic production that honours the contract of marriage.

Ferdinand, amazed by Prospero's powers that can call forth such spirits at his bidding, longs to stay on the island and live with

Prospero as his father and Miranda as his wife.

But Prospero suddenly stops the dancing when he remembers Caliban and his companions plan to kill him and the time is fast

approaching. He disperses the spirits and comforts the disappointed Ferdinand by reminding him that everything eventually comes to

an end.

Our revels now are ended. These our actors,

As I foretold you, were all spirits, and

Are melted into air, into thin air;

And like the baseless fabric of this vision,

The cloud-capped towers, the gorgeous palaces,

The solemn temples, the great globe itself,

Yea, all which it inherit, shall dissolve;

And, like this insubstantial pageant faded,

Leave not a rack behind. We are such stuff

As dreams are made on, and our little life

Is rounded with a sleep. Sir, I am vexed.

Bear with my weakness. My old brain is troubled.

With his troubled mind, Prospero sends Ferdinand and Miranda away so that he can come up with a plan.

Ariel enters and together the two discuss how best to handle "Caliban and his confederates." Ariel recalls what has already been done

to the men. Using music as a lure, Ariel has brought them on an uncomfortable path through "toothed briers, sharp furzes, pricking

gorse, and thorns" and then left them in a filthy, stinking pool of water near Prospero's cell. Prospero sends Ariel off to gather some

glittery apparel. Meanwhile Prospero laments the devilish nature of Caliban, which cannot be broken or corrected even with the

humane treatment Prospero initially offered him.

When Ariel returns they hang the fancy garments on a line near Prospero's cell. When the men enter, soaking wet from the filthy

pond, they are complaining and arguing. Caliban tries to keep them focused on the matter at hand: murdering Prospero. But the men

become distracted by the beautiful clothing, claiming it for "King Stephano" as Caliban becomes more and more upset that they need

to "do the murder first," but the men will have none of it as they try on all the clothes. In the midst of this chaos Prospero sends spirits

in the shape of hunting dogs upon the three men to drive them off.

Now Prospero recognizes that everything he has been working toward is suddenly within his control: "At this hour/Lie at my mercy

all mine enemies." He only has to finish his plans and he can free Ariel.

Analysis

The subject of marriage is central to this scene, which begins with Prospero's assent to his daughter's marriage to Ferdinand and

moves into the lengthy masque. Particularly at the time of the play's writing, marriage was an institution that symbolized order and

structure in society. In fact, many marriages were founded on reasons other than love because the social contract between two

families offered economic or political power for one or both parties.

Ferdinand and Prospero discuss Miranda's virginity and the sexual pleasures of marriage. Modern audience members might extend

the theme of exploitation and power to this relationship, particularly as it was written at a time when women had few rights and a wife

was considered a possession of her husband. Miranda is passed from one "owner," Prospero, to another, Ferdinand, as Prospero says,

"Then as my gift and thine own acquisition/Worthily purchased, take my daughter." However, it is unlikely Shakespeare was pointing

to Miranda and Ferdinand's marriage as an additional type of exploitation since their marriage is a key ingredient and source of

harmony in the happy ending of the play.

Prospero's repeated admonitions to Ferdinand not to give into his passions before the wedding support the ideal of romantic love,

which asserts love is a pure, holy, and honourable condition. At the same time Prospero recognizes the strength of human passion,

whereby "The strongest oaths are straw/To th' fire i' th' blood." Shakespeare, through Prospero, acknowledges two powerful,

sometimes opposing, forces in human nature: passion and reason. Prospero's insistence that Ferdinand and Miranda undergo the full

ceremony of legitimate marriage likewise shows a concern about legitimacy and formality. Despite the fact that Ferdinand has

pledged his love and fidelity and technically now has a legal obligation to Miranda, Prospero wants to ensure that Miranda's future is

secure.

Prospero commands a masque to be performed to celebrate the betrothal of Ferdinand and Miranda. In the 17th century a masque was

an elaborate theatrical production that included costumes, actors, singing, and dancing. The characters and plots were based on story

lines from Greek and Roman mythology. In the masque in The Tempest, the goddesses in the masque symbolize fertility and

prosperity within family and nature.

This scene provides more insight into the humanity of Prospero's character. Despite the murderous nature of Caliban, Stephano, and

Trinculo's plot against Prospero, he does not severely punish them. Instead he uses magic to scare them and run them off, so that they

do not try to kill him again. Prospero wishes to right wrongs, not create new ones. At the same time Ariel's devotion and Prospero's

commitment to freeing Ariel are underscored. In fact there is a true, shared affection between the two, as Ariel asks, "Do you love me,

master?" and Prospero responds, "Dearly, my delicate Ariel."

Prospero, who has recently been engaged with his many plans and the masque itself, stops suddenly because he has to solve a new

problem. He acknowledges he is getting old and in describing the end of the masque makes clear comparisons to life itself. All the

artifice of the masque is like the artifice of life, which, like a play, eventually fades. Our human lives are like short dreams that are

surrounded by sleep. Many scholars view Prospero's speech here as Shakespeare's personal farewell to the theatre, where his own

plays and pageants "now are ended."

Prospero says at the end of the scene that "At this hour/Lies at my mercy all mine enemies," but his confession to Ferdinand a few

minutes earlier that "my old brain is troubled" makes it seem like he does not find the successful completion of his revenge plan as

satisfying as he had anticipated.

Class 12 Physics Worksheet-14

MAGNETISM 1. Bar Magnet and its properties

2. Current Loop as a Magnetic Dipole and Dipole Moment

3. Current Solenoid equivalent to Bar Magnet

4. Bar Magnet and it Dipole Moment

5. Coulomb’s Law in Magnetism

6. Important Terms in Magnetism

7. Magnetic Field due to a Magnetic Dipole

8. Torque and Work Done on a Magnetic Dipole

9. Terrestrial Magnetism

10. Elements of Earth’s Magnetic Field

11. Tangent Law

12. Properties of Dia-, Para- and Ferro-magnetic substances

13. Curie’s Law in Magnetism

14. Hysteresis in Magnetism

Magnetism:

- Phenomenon of attracting magnetic substances like iron, nickel, cobalt, etc.

• A body possessing the property of magnetism is called a magnet.

• A magnetic pole is a point near the end of the magnet where magnetism is concentrated.

• Earth is a natural magnet.

• The region around a magnet in which it exerts forces on other magnets and on objects made

of iron is a magnetic field.

Properties of a bar magnet:

1. A freely suspended magnet aligns itself along North – South direction.

2. Unlike poles attract and like poles repel each other.

3. Magnetic poles always exist in pairs. i.e. Poles can not be separated.

4. A magnet can induce magnetism in other magnetic substances.

5. It attracts magnetic substances.

Repulsion is the surest test of magnetisation: A magnet attracts iron rod as well

as opposite pole of other magnet. Therefore it is not a sure test of magnetisation.

But, if a rod is repelled with strong force by a magnet, then the rod is surely magnetised.

Representation of Uniform Magnetic Field:

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

Uniform field on the plane

of the diagram

Uniform field perpendicular &

into the plane of the diagram

Uniform field perpendicular &

emerging out of the plane of the

diagram

Current Loop as a Magnetic Dipole & Dipole Moment:

Magnetic Dipole Moment is A

B

TIP:

SI unit is A m2.

When we look at any one side of the loop carrying current, if the current is in anti-clockwise

direction then that side of the loop behaves like Magnetic North Pole and if the current is in

clockwise direction then that side of the loop behaves like Magnetic South Pole.

M = I A n

I

Current Solenoid as a Magnetic Dipole or Bar Magnet:

B

x x x x x x x

I I

TIP: Play previous and next to understand the similarity of field lines.

M = m.2l. l

Bar Magnet:

1. The line joining the poles of the magnet is called

magnetic axis.

2. The distance between the poles of the magnet is

called magnetic length of the magnet.

Geographic Length

Magnetic Length

3. The distance between the ends of the magnet is called the geometrical length of the

magnet.

4. The ratio of magnetic length and geometrical length is nearly 0.84.

Magnetic Dipole & Dipole Moment:

A pair of magnetic poles of equal and opposite strengths separated by a finite

distance is called a magnetic dipole.

The magnitude of dipole moment is the product of the pole strength m and the

separation 2l between the poles.

Magnetic Dipole Moment is SI unit of pole strength is A.m

The direction of the dipole moment is from South pole to North Pole along the

axis of the magnet.

S N

Coulomb’s Law in Magnetism:

The force of attraction or repulsion between two magnetic poles is directly

proportional to the product of their pole strengths and inversely proportional to the

square of the distance between them.

F α m1 m2

α r2

k m1 m2 or

F = r2

(where k = μ0 / 4π is a constant and μ0 = 4π x 10-7 T m A-1)

In vector form μ0 m1 m2 r F

= 4π r2

m1 r

m2

F = μ0 m1 m2

4π r2

F = μ0 m1 m2 r

4π r3

Magnetic Intensity or Magnetising force (H):

i) Magnetic Intensity at a point is the force experienced by a north pole of unit pole

strength placed at that point due to pole strength of the given magnet. H = B / μ

ii) It is also defined as the magnetomotive force per unit length.

iii) It can also be defined as the degree or extent to which a magnetic field can magnetise a substance.

iv) It can also be defined as the force experienced by a unit positive charge flowing

with unit velocity in a direction normal to the magnetic field.

v) Its SI unit is ampere-turns per linear metre.

vi) Its cgs unit is oersted.

Magnetic Field Strength or Magnetic Field or Magnetic Induction or

Magnetic Flux Density (B):

i) Magnetic Flux Density is the number of magnetic lines of force passing normally

through a unit area of a substance. B = μ H

ii) Its SI unit is weber-m-2 or Tesla (T).

iii) Its cgs unit is gauss. 1 gauss = 10- 4 Tesla

Magnetic Flux (Φ):

i) It is defined as the number of magnetic lines of force passing

normally through a surface.

ii) Its SI unit is weber.

Relation between B and H:

B = μ H (where μ is the permeability of the medium)

Magnetic Permeability (μ):

It is the degree or extent to which magnetic lines of force can pass

enter a substance.

Its SI unit is T m A-1 or wb A-1 m-1 or H m-1

Relative Magnetic Permeability (μr):

It is the ratio of magnetic flux density in a material to that in vacuum.

It can also be defined as the ratio of absolute permeability of the material to that in

vacuum.

μr = B / B0 or μr = μ / μ0

Intensity of Magnetisation: (I):

i) It is the degree to which a substance is magnetised when placed in a magnetic field.

ii) It can also be defined as the magnetic dipole moment (M) acquired per unit volume of the

substance (V).

iii) It can also be defined as the pole strength (m) per unit cross-sectional area (A) of the substance.

iv) I = M / V

v) I = m(2l) / A(2l) = m / A

vi) SI unit of Intensity of Magnetisation is A m-1.

Magnetic Susceptibility (cm ):

i) It is the property of the substance which shows how easily a substance can be magnetised.

ii) It can also be defined as the ratio of intensity of magnetisation (I) in a substance to the

magnetic intensity (H) applied to the substance.

iii) cm = I / H Susceptibility has no unit.

Relation between Magnetic Permeability (μr) & Susceptibility (cm ):

μr = 1 + cm

Magnetic Field due to a Magnetic Dipole (Bar Magnet):

i) At a point on the axial line of the magnet:

If l << x, then

ii) At a point on the equatorial line of the

magnet:

If l << y, then

Magnetic Field at a point on the axial line acts along

the dipole moment vector.

Magnetic Field at a point on the equatorial line acts

opposite to the dipole moment vector.

BN

B Q

θ

θ Q

BS

y

θ θ BP = BN - BS

BS BN

S

l

N P

l

x

BP ≈ μ0 M

4π y3

μ0 2 M x BP

= 4π (x2 – l2)2

BP ≈ μ0 2 M

4π x3

μ0 M

BQ= 4π (y2 + l2)3/2

M

θ

t

Torque on a Magnetic Dipole (Bar Magnet) in Uniform Magnetic Field:

The forces of magnitude mB act

opposite to each other and hence net force acting on the bar

magnet due to external uniform

magnetic field is zero. So, there mB

is no translational motion of the

magnet.

However the forces are along

different lines of action and

constitute a couple. Hence the

magnet will rotate and experience

torque.

Torque = Magnetic Force x distance

t = mB (2l sin θ)

= M B sin θ

N

2l mB

M

θ

S B

B

Direction of Torque is perpendicular and into the plane containing M and B.

t = M x B

Work done on a Magnetic Dipole (Bar Magnet) in Uniform Magnetic Field:

dW = tdθ

= M B sin θ dθ

θ2

W = ∫ M B sin θ dθ θ1

mB

dθ mB

θ1 θ2

mB

mB B

W = M B (cosθ1 - cos θ2)

If Potential Energy is arbitrarily taken zero when the dipole is at 90°, then

P.E in rotating the dipole and inclining it at an angle θ is

Potential Energy = - M B cos θ

Note:

Potential Energy can be taken zero arbitrarily at any position of the dipole.

Terrestrial Magnetism:

i) Geographic Axis is a straight line passing through the geographical poles of the

earth. It is the axis of rotation of the earth. It is also known as polar axis.

ii) Geographic Meridian at any place is a vertical plane passing through the

geographic north and south poles of the earth.

iii) Geographic Equator is a great circle on the surface of the earth, in a plane

perpendicular to the geographic axis. All the points on the geographic equator are at

equal distances from the geographic poles.

iv) Magnetic Axis is a straight line passing through the magnetic poles of the earth.

It is inclined to Geographic Axis nearly at an angle of 17°.

v) Magnetic Meridian at any place is a vertical plane passing through the magnetic north and south poles of the earth.

vi) Magnetic Equator is a great circle on the surface of the earth, in a plane

perpendicular to the magnetic axis. All the points on the magnetic equator are at

equal distances from the magnetic poles.

Declination (θ):

The angle between the magnetic meridian and the

geographic meridian at a place is Declination at that

place.

It varies from place to place.

Lines shown on the map through the places that have

the same declination are called isogonic line.

Line drawn through places that have zero declination

is called an agonic line.

Dip or Inclination (δ):

Geographic

Meridian

BH θ

BV

Magnetic Meridian

The angle between the horizontal component of earth’s magnetic field and the

earth’s resultant magnetic field at a place is Dip or Inclination at that place.

It is zero at the equator and 90° at the poles.

Lines drawn up on a map through places that have the same dip are called isoclinic lines.

The line drawn through places that have zero dip is known as an aclinic line. It is the

magnetic equator.

δ

B

N

tan θ = B2 / B1

Horizontal Component of Earth’s Magnetic Field (BH ):

The total intensity of the earth’s magnetic field does not lie in any horizontal

plane. Instead, it lies along the direction at an angle of dip (δ) to the

horizontal. The component of the earth’s magnetic field along the horizontal at an

angle δ is called Horizontal Component of Earth’s Magnetic Field.

BH = B cos δ

Similarly Vertical Component is BV = B sin δ such

that B = √ BH2 + BV

2

Tangent Law:

If a magnetic needle is suspended in a region where

two uniform magnetic fields are perpendicular to

each other, the needle will align itself along the

direction of the resultant field of the two fields at an

angle θ such that the tangent of the angle is the ratio

of the two fields.

B2 B

B1

Comparison of Dia, Para and Ferro Magnetic materials:

DIA PARA FERRO

1. Diamagnetic substances are those substances which are feebly repelled by a magnet.

Eg. Antimony, Bismuth, Copper, Gold, Silver, Quartz, Mercury, Alcohol, water, Hydrogen, Air, Argon, etc.

Paramagnetic substances are those substances which are feebly attracted by a magnet.

Eg. Aluminium, Chromium, Alkali and Alkaline earth metals, Platinum, Oxygen, etc.

Ferromagnetic substances are those substances which are strongly attracted by a magnet.

Eg. Iron, Cobalt, Nickel, Gadolinium, Dysprosium, etc.

2. When placed in magnetic field, the lines of force tend to avoid the substance.

N S

The lines of force prefer to pass through the substance rather than air.

S N

The lines of force tend to crowd into the specimen.

S N

2. When placed in non- uniform magnetic field, it moves from stronger to weaker field (feeble repulsion).

When placed in non- uniform magnetic field, it moves from weaker to stronger field (feeble attraction).

When placed in non- uniform magnetic field, it moves from weaker to stronger field (strong attraction).

3. When a diamagnetic rod is freely suspended in a uniform magnetic field, it aligns itself in a direction perpendicular to the field.

N S

When a paramagnetic rod is freely suspended in a uniform magnetic field, it aligns itself in a direction parallel to the field.

N S

When a paramagnetic rod is freely suspended in a uniform magnetic field, it aligns itself in a direction parallel to the field very quickly.

N S

4. If diamagnetic liquid taken in a watch glass is placed in uniform magnetic field, it collects away from the centre when the magnetic poles are closer and collects at the centre when the magnetic poles are farther.

If paramagnetic liquid taken in a watch glass is placed in uniform magnetic field, it collects at the centre when the magnetic poles are closer and collects away from the centre when the magnetic poles are farther.

If ferromagnetic liquid taken in a watch glass is placed in uniform magnetic field, it collects at the centre when the magnetic poles are closer and collects away from the centre when the magnetic poles are farther.

5. When a diamagnetic When a paramagnetic When a ferromagnetic

substance is placed in a substance is placed in a substance is placed in a

magnetic field, it is magnetic field, it is magnetic field, it is

weakly magnetised in the weakly magnetised in the strongly magnetised in

direction opposite to the direction of the inducing the direction of the

inducing field. field. inducing field.

6. Induced Dipole Moment (M) is a small – ve value.

Induced Dipole Moment (M) is a small + ve value.

Induced Dipole Moment (M) is a large + ve value.

7. Intensity of Magnetisation (I) has a small – ve value.

Intensity of Magnetisation (I) has a small + ve value.

Intensity of Magnetisation (I) has a large + ve value.

8. Magnetic permeability μ is always less than unity.

Magnetic permeability μ is more than unity.

Magnetic permeability μ is large i.e. much more than unity.

9. Magnetic susceptibility

cm has a small – ve value. Magnetic susceptibility cm

has a small + ve value.

Magnetic susceptibility cm

has a large + ve value.

10. They do not obey Curie’s Law. i.e. their properties do not change with temperature.

They obey Curie’s Law. They lose their magnetic properties with rise in temperature.

They obey Curie’s Law. At a certain temperature called Curie Point, they lose ferromagnetic properties and behave like paramagnetic substances.

Curie’s Law:

Magnetic susceptibility of a material varies inversely with

the absolute temperature.

I α H / T or I / H α 1 / T I

cm α 1 / T

cm = C / T (where C is Curie constant)

Curie temperature for iron is 1000 K, for cobalt 1400 K and for nickel 600 K.

H / T

Hysteresis Loop or Magnetisation Curve:

Intensity of Magnetisation (I) increases with increase in

Magnetising Force (H) initially through OA and reaches saturation

at A.

When H is decreased, I decreases but it does not come to zero at H

= 0.

The residual magnetism (I) set up in the material represented by

OB is called Retentivity.

To bring I to zero (to demagnetise completely), opposite (negative)

magnetising force is applied. This magetising force represented by

OC is called coercivity.

After reaching the saturation level D, when the magnetising force

is reversed, the curve closes to the point A completing a cycle.

The loop ABCDEFA is called Hysteresis Loop.

The area of the loop gives the loss of energy due to the cycle of

magnetisation and demagnetisation and is dissipated in the form

of heat.

The material (like iron) having thin loop is used for making

temporary magnets and that with thick loop

(like steel) is used for permanent magnets. Animating Hysteresis Lo

I B

C O

Class 12 Chemistry

Worksheet-14

HALOARENES

Haloarenes are named by prefixing ‘HALO’(flouro, cholro, bromo, iodo)to the name

of parent hydrocarbon

PREPARATION OF HALOARENES

1.SANDMEYER’S REACTION

2.GATTERMANN REACTION

CHEMICAL PROPERTIES OF HALOARENES

1.NUCLEOPHILIC SUBSTITUTION (replacement of chlorine by -OH, -NH2 )

Replacement of chlorine by -OH,

Replacement of chlorine by -NH2

2. ELECTROPHILIC SUBSTITUTION(chlorination,nitration & sulphonation)

a.Chlorination

b.Nitration

c.Sulphonation

3.REDUCTION

4.WURTZ FITTIG REACTION

5.FITTIG REACTION

6.FORMATION OF GRIGNARD’S REAGENT

Class 12 Maths

Worksheet-14

sin−1 𝑥 + cos−1 𝑥 = 𝜋

2 tan−1 𝑥 + cot−1 𝑥 =

𝜋

2 sec−1 𝑥 + cosec−1 𝑥 =

𝜋

2

sin−1 𝑥 + sin−1 𝑦 = sin−1( 𝑥√1 − 𝑦2 + 𝑦 √1 − 𝑥2 )

sin−1 𝑥 - sin−1 𝑦 = sin−1( 𝑥√1 − 𝑦2 − 𝑦 √1 − 𝑥2 )

cos−1 𝑥 + cos−1 𝑦 = cos−1( 𝑥 𝑦 − √1 − 𝑥2 √1 − 𝑦2 )

cos−1 𝑥 - cos−1 𝑦 = cos−1( 𝑥 𝑦 + √1 − 𝑥2 √1 − 𝑦2 )

tan−1 𝑥 + tan−1 𝑦 = tan−1( 𝑥+𝑦

1−𝑥𝑦 ) , if x y < 1

= π + tan−1( 𝑥+𝑦

1−𝑥𝑦 ) , if xy > 1 , x>0, y>0

tan−1 𝑥 - tan−1 𝑦 = tan−1( 𝑥−𝑦

1+𝑥𝑦 ) , xy > -1

Illustration 9: Prove that tan−1 1 + tan−1 2 + tan−1 3 = 𝜋

Solution:

𝜋 + tan−1 1+2

1−(1)(2) + tan−1 3 = tan−1(−3) + tan−1 3

[ Applying tan-1x + tan-1y formula when x y > 1]

Again applying the same formula we get,

= 𝜋 + tan−1 −3+3

1−(−3)(3) = 𝜋 + tan−1 0 = 𝜋

Illustration 10: Prove that sin−1 3

5 - sin−1 8

17 = cos−1 84

85

Solution: Since the right hand side is in terms of cosine, therefore we will first convert sine

inverse into cos inverse.

3 5 8 17

Also Remember:

Since sin 2x = 2 𝑡𝑎𝑛𝑥

1+ 𝑡𝑎𝑛2𝑥 therefore 2 tan−1 𝑥 = sin−1 2𝑥

1+𝑥2

Since cos 2x = 1− 𝑡𝑎𝑛2 𝑥

1+ 𝑡𝑎𝑛2𝑥 therefore 2 tan−1 𝑥 = cos−1 1−𝑥2

1+𝑥2

Since tan 2x = 2 𝑡𝑎𝑛𝑥

1− 𝑡𝑎𝑛2𝑥 therefore 2 tan−1 𝑥 = tan−1 2𝑥

1−𝑥2

Similarly: 3 sin-1 x = sin-1 ( 3x – 4x3 ) , |x| ≤ 1

2

3 cos-1x = cos-1 ( 4x3 -3x ) , 1

2 ≤ x ≤ 1

3 tan-1x = tan-1 ( 3𝑥− 𝑥3

1−3𝑥2 ) , |x| <

1

√3

4 15

So sin−1 3

5 = cos−1 4

5 and sin−1 8

17 = cos−1 15

17

Therefore we get cos−1 4

5 - cos−1 15

17 = cos−1(

4

5 x

15

17 + √1 −

225

289 √1 −

16

25 )

⇒ cos−1{ 60

85+ (

8

17 ) (

3

5 )} = cos−1(

84

85 )

Illustration 11: Prove that sin ( 2 tan−1 3

5 - sin−1 7

25 ) =

304

425

Solution: Since 2 tan−1 𝑥 = sin−1 2𝑥

1+𝑥2 , therefore 2 tan−1 3

5 = sin−1(

6

5

1+ 9

25

)

= sin−1( 6

5 𝑥

25

34 ) = sin−1(

15

17 )

Hence we have sin ( 𝑠𝑖𝑛−1 15

17 - sin−1 7

25 ) = sin { sin−1(

15

17 √1 −

49

625 -

7

25 √1 −

225

289 )

⇒ sin { sin−1( 15

17 .

24

25 -

7

25 .

8

17 ) } = sin { sin−1

360−56

425 } = sin { sin-1

304

425 } =

304

425

SOLVE YOURSELVES:

1. Prove that tan−1 1

7 + tan−1 1

13 = tan−1 2

9

2. Prove that tan−1 1 + tan−1 1

2 + tan−1 1

3 =

𝜋

2

3. Prove that 2 tan−1 1

3 + cot−1 4 = tan−1 16

13 [ cot-1 4 = tan-1

𝟏

𝟒

]

4. Prove that tan−1 1

5 + tan−1 1

7 + tan−1 1

3 + tan−1 1

8 =

𝜋

4

5. Prove that cos−1 3

5 + sin−1 12

13 = sin−1 56

65

6. Prove that tan−1 1

3 + sec−1 √5

2 =

𝜋

4

7. Prove that sin−1 4

5 + cos−1 2

√5 = cot−1 2

11

8. Evaluate tan ( 2 tan-1 1

2 - cot-13 ) Ans:

9

13

Illustration 12: Write the following in simplest form: tan−1 1

√𝑥2−1

Solution: Let x = cosec 𝜃 , it means 𝜃 = cosec-1x

⇒ tan−1 1

√𝑥2−1 = tan−1 1

√𝑐𝑜𝑠𝑒𝑐2𝜃−1 = tan-1

1

cot 𝜃 = tan-1 ( tan 𝜃) = 𝜃 = cosec-1 x

Illustration 13: Write tan−1( √1+𝑥2 −1

𝑥 ) , x ≠ 0 in the simplest form.

Solution: Let x = tan 𝜃, it means 𝜃 = tan-1 x

⇒ tan−1( √1+𝑥2 −1

𝑥 ) = tan-1 (

√1+𝑡𝑎𝑛2𝜃 −1

tan 𝜃 ) = tan-1 (

sec 𝜃−1

tan 𝜃 ) = tan-1 (

1

𝑐𝑜𝑠𝜃−1

𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃

)

⇒ tan-1 ( 1−𝑐𝑜𝑠𝜃

𝑠𝑖𝑛𝜃 ) = tan-1 (

2𝑠𝑖𝑛2 𝜃/2

2 sin 𝜃/2 𝑐𝑜𝑠𝜃/2 ) = tan-1 ( tan 𝜃/2) = 𝜃/2 =

tan−1 𝑥

2

Illustration 14: Write tan−1 (3𝑎2𝑥−𝑥3

𝑎3−3𝑎𝑥2 )

Solution: As we know that tangent has a characteristic formula, so we divide

numerator and denominator by a3

⇒ tan-1 ( 3

𝑥

𝑎 − (

𝑥

𝑎 )3

1−3 ( 𝑥

𝑎 )2

) Now we can put 𝑥

𝑎 = tan 𝜃 or in other words, 𝜃 = tan-1 (

𝑥

𝑎 )

⇒ tan-1 ( 3 𝑡𝑎𝑛𝜃− 𝑡𝑎𝑛3𝜃

1−3𝑡𝑎𝑛2𝜃 ) = tan-1 ( tan 3𝜃) = 3 𝜃 = 3 tan-1 (

𝑥

𝑎 )

Illustration 15: Prove that sin [ tan−1 (1−𝑥2

2𝑥 ) + cos−1(

1−𝑥2

1+𝑥2 ) ]

Solution: Put x = tan𝜃 or 𝜃 = tan-1x

⇒ sin [ tan-1 ( 1−𝑡𝑎𝑛2𝜃

2𝑡𝑎𝑛𝜃 ) + cos-1 (

1−𝑡𝑎𝑛2𝜃

1+𝑡𝑎𝑛2𝜃 ) ] = sin [ tan-1

1

tan 2𝜃 + cos-1 ( cos 2𝜃)]

⇒ sin [ tan-1 (cot 2𝜃) + 2𝜃 ] = sin [ tan-1 { tan ( 𝜋

2 - 2𝜃) + 2𝜃 } ]

⇒ sin [ 𝜋

2 - 2𝜃 + 2𝜃 ] = sin

𝜋

2 = 1

Illustration 16: Prove that tan-1 [ √1+𝑥− √1−𝑥

√1+𝑥+√1−𝑥 ]

Solution: Put x = cos 2𝜃 [ ∵ we know 1 + cos 2𝜽= 2 cos2 𝜽 and 1- cos2𝜽 =

2sin2𝜽 ]

∴ 2𝜃 = cos-1 x or 𝜃 = 1

2 cos-1 x ------------------------------------(1)

⇒ tan-1 [ √2 cos 𝜃− √2 𝑠𝑖𝑛𝜃

√2 𝑐𝑜𝑠𝜃+ √2 𝑠𝑖𝑛𝜃 ] = tan-1 [

cos 𝜃−sin 𝜃

cos 𝜃+sin 𝜃 ]

= tan-1 [ 1−𝑡𝑎𝑛𝜃

1+𝑡𝑎𝑛𝜃 ] [Dividing Nr & Dr by

cos𝜽]

= tan-1 [ tan ( 𝜋

4 – 𝜃 ) ] =

𝜋

4 – 𝜃 =

𝜋

4 –

1

2 cos-1 x [ using (1)]

WRITE THE FOLLOWING IN SIMPLEST FORM:

9. tan−1( 1−cos 𝑥

1+cos 𝑥 ) [Hint:Use 1- cosx = 2 sin2 x/2 and 1+ cos x = 2 cos2 x/2] Ans:

𝑥

2

10. tan-1 ( 2√𝑥

1−𝑥 ) [ Hint: Put x = tan2 𝜽 ] Ans: 2 tan-1

√𝑥

11. cos-1 ( 1-2x2) [Hint: Put x = sin2 𝜽 ] Ans: 2 sin-

1x

12. tan-1 ( √1−𝑥

1+𝑥 ) [ Hint: Put x = cos 2𝜽 ] Ans:

1

2 cos-1

x

13. tan-1 ( 𝑥

√𝑎2−𝑥2 ) [ Hint: put x = a sin𝜽 ] Ans: sin-1

𝑥

𝑎

14. tan-1 ( 𝑎 cos 𝑥−𝑏 𝑠𝑖𝑛𝑥

𝑏 cos 𝑥+𝑎 sin 𝑥 ) Ans: tan-1

𝑎

𝑏 –x

[Hint: Divide Nr & Dr by bcosx first and then apply tan-1 𝒙−𝒚

𝟏+𝒙𝒚 = tan-1x – tan-1y ]

Class 12 Biology Worksheet-14

CHAPTER-Reproduction in Organisms

Life Span

The period from birth to the natural death of an organism is called its life span.

However, lifespan of different organisms is different. The life spans of organisms are

not correlated with their size and body mass.

MORE QUESTIONS BASED ON PROPERTIES OF INVERSE TRIG. FUNCTIONS CONTD…… IN WORKSHEET-15

Reproduction

It is defined as a biological process in which an organism gives rise to young one

similar to itself. It is an important characteristic feature of all living organisms.

Reproduction in organism is of two main types:

1) Asexual reproduction

2) Sexual reproduction

Note:

As a result of asexual reproduction, the newly formed individual is the exact copy of

its parent. The morphologically and genetically similar copy of an organism is called a

clone

As a result of sexual reproduction, the new formed individual is not the exact copy of

its parents. Hence it is not called clone. It is better be called as Hybrid.

Modes of Asexual reproduction

1) Fission

In this process the organism divide into two or more parts. Each part is able to

produce the new individual. It is mainly of following type.

A) Binary Fission

In this process, the parental organism divides into two halves. Each half

receives equal genetic material and is able to grow into a new individual. It

may be simple, transverse binary fission.

a) Simple binary fission: When the plane of division passes through any

direction it is called Simple binary fission.Eg. Amoeba

b) Transverse binary fission: When the direction of division plane coincides

with the transverse axis of the animal, it is called transverse binary fission Eg.

Paramecium

2) Budding

In this process the parent animal produces a small projection out form its body.

This projection is called a bud. It grows in size and get separated from the

parent body. It becomes a new individual.eg. Hydra, Yeast,

3) Gemmule Formation

In some fresh water sponges, many internal buds are developed inside the

parental body during unfavorable conditions. The internal buds are called

gemmules.On the return of favorable period each gemmule gives rise to new

individual.

4) Fragmentation

In this process, the body of the parent is broken down into many small

fragments. Each fragment under suitable condition may give rise to new

individual by regeneration.eg Planaria, Seastar, Hydra.

5) Sporulation

Amoeba during unfavorable conditions withdraws pseudopodia and secretes

three layered covering i.e.Cyst around itself this is called Encystation.

On the return of favourable period, the amoeba (Encysted) divides by multiple

fission to produce many small psedupodiospores. Now the cyst bursts to release

these psedupodiospores. This Process id called Sporulation.

6) Spore Formation

Especially in fungi, the special structures are present for asexual reproduction.

These structures are as follows.

a) Zoospores : These spores are motile spores as they have flagella for

locomotion.

These develop into specialized structures called sporangium.eg

Chlamydomonas.

b) Sporangiospores:Thes are also developed in the sporangium but these are

non motile as they do not have flagella.eg Rhizopus.

c) Conidia: These are also non motile spores but these do not develop inside the

sporangium.The spores develop freely on terminal portion of Hyphae called

conidiophores.eg Penicillium

Note: Learn

Definitions of Life Span, Clone and Hybrids.

Differences between asexual and sexual reproduction

Life span of all organisms

Types of Asexual Reproduction

Class 12 Physical Education Worksheet-13

CHAPTER 3: TOPIC 1: CAREER ASPECTS IN PHYSICAL EDUCATION.

INTRODUCTION: career opportunities in physical education have never been

greater. Traditional careers of teaching have expanded from schools, colleges and

universities to non-school settings, such as community centre’s (e.g. YMCA AND

YWC) commercial clubs) i.e. tennis, golf, cricket, gymnastics etc.) Many physical

educationists are pursuing careers in the physical fitness field, working in health clubs

or corporate fitness centers.

During the past 30 years, career aspects in physical education have expanded

tremendously. The trend is a result of several factors, such as:

First, millions people from all segments of society engage in physical fitness activities

on a regular basis. They participate in a variety if activities, including working out at

health and fitness clubs and engaging in community and commercial fitness

programmes.

Secondly, people seeking to use their leisure time in enjoyable and constructive

manner have sought out physical activities and sport.

Thirdly, the increased interest in competitive sports by all segments of the population

has served as the impetus for growth of competitive sports programmes, sports clubs,

and the leagues or IPLs and the associated career aspects or opportunitie4s in

coaching sports management, officiating, and the athletics training.

Fourthly and finally, the incre3ase in the depth and breadth of knowledge in physical

education has led to the further development of sub-disciplines and expanded career

opportunities such as biomechanists, sports psychologists, exercise physiologists, and

adapted physical activity specialists.

Keeping in view the requirements of the students of physical education from the

examination point of view the career aspects mentioned in the syllabus are

mentione3d and discussed below:

Professional sportsmen: professional players are selected from amongst those who

show record performance in a number of tournaments played at various levels e.g.

institutional level, district level, regional level and National level. Outstanding

performers represent the country at the national and international level.

Nationalized banks, public sector undertaking, corporate houses and some private

groups recruit outstanding sportspersons. TATA, BPCS, MRF, IOC etc have their

own sports team to compete in matches/tournaments.

Sports manager: in physical education, now days, emphasis is given on the

development of desirable management competencies or skill so that an individual is

able to manage office, finances, human resources, sports facilities, sports

competitions, sports persons, sports teams, events etc.

Various institutes in India have set up sports management courses with the aim of

training individuals in the different aspects related with field. A job as a sports

manager in a reputed sports management team like Tigers from India is a very

lucrative job. Many private and government sponsored event. Clubs, hotels,

gymnasiums and sport centers also hire such people.

Sports Teacher/lecture (Teaching job): physical education occupies a dignifies

position in the higher education sector apart from teaching in training colleges

physical education has been included as an elective subject in the degree course (B.A.

B. SC) level. Physical education professional, who so ever qualifies NET, SLET, or

having M. Phil, or Ph.D degree may get job as Asst. Professor or Lecturer in colleges.

Sports coach (coaching job): Coaching - Trained coaches are in demand at all the

training centers and by state and central sports bodies. An individual proficient in

coaches training in the National Institute of Sports and become certified coach persons

with a degree or master’s degree in physical Education are considered to be more

suitable for this profession.

Gym instructor (Fitness Sector):Today fitness training is booming business and there

are many opportunities the fitness sector, career opportunities in this field are that of

fitness instructor, personal trainer, fitness consultant, fitness specialist, physical

therapy assistant, health and life style coordinator. They work in local health clubs, for

professional sports teams, college athletic departments etc..

Sports officials: As the sports competitions are becoming more intense,

professionalism taking the top and front seat win and lose decide the earning of

professional players, the officiating has also become a profession of lucrative earning.

The officials for cricket, tennis, football or rugby league, boxing wrestling etc are as

important part of the sports any other (training coaching, selection, competition,

nutrition etc.)

Sports event coordinator: some of the universities in India and abroad provide training

to become qualified event manager. Now a day, even organizing a small birthday

party of a kid needs expertise to manage it efficiently.

An event involves many coordinative activities- preparation of sports facilities,

arrange3ment of finances, controlling of finances, arrangement of accommodation for

sportsmen and officials, arrangement of official, arrangement of refreshment and

hospitality, arrangement of catering, arrangement of cleanliness, arrangement of

medical facility etc….

Sports journalist and commentators: sports journalism is an area where the Physical

education and sports science experts can be worthwhile. Those with communicating

skils can take on sports commentating, or sports journalism, working with television,

newspapers or magazines that have a sports column or programe. Sports channel such

as star sports, ESPN TEN SSPORTS, ZEE SPORTS and DD sports have

commentators, directors and presenters to prepare and present sports programmes.

Sports software engineers: conducting mega sports events in which more than ten

thousand sports competitors and thousands officials participate from the globe has

forced the professional physical educationist to open up another area of study to

prepare sports software engineers to manage and conduct the mega sports events like

Olympic games, commonwealth Games, Asian Games, World Cups.

Marketing and manufacturing sports equipment: In physical Education courses, the

trainees are prepared for sport marketing (Equipment and players) also input is given

for manufacturing of sports equipment and preparing sports apparels. The aspirants

get advance training in the field and they get in to sports marketing.

Questions regarding above topic 1 IS given below:

1. Write in detail career aspect in physical education. Explain any two.

2. List various career options available in the field of physical education. Discuss any

two of them?

3. Write short notes on the following:

Role of sports journalist

Role of sports coach

Role of a sports manager

Role of soft ware engineer

Role of sports coordinator or event manager

Sports officials / officiating

Marketing and manufacturing sports equipment

Professional sportsmen

Gym instructor

Sports Teacher or Lecturer (Teaching job)

Class 12 Computer Science Worksheet-13

What is Logic?

Logic is the basis of all mathematical reasoning, and of all automated reasoning.

The rules of logic specify the meaning of mathematical statements.

These rules help us understand and reason with statements

Importance of Mathematical Logic

The rules of logic give precise meaning to mathematical statements. These rules

are used to distinguish between valid and invalid mathematical arguments.

Apart from its importance in understanding mathematical reasoning, logic has

numerous applications in Computer Science, varying from design of digital circuits,

to the construction of computer programs and verification of correctness of

programs.

Propositional Logic

What is a proposition?

A proposition is the basic building block of logic. It is defined as a declarative

sentence that is either True or False, but not both.

The Truth Value of a proposition is True(denoted as T) if it is a true statement,

and False(denoted as F) if it is a false statement. For Example,

1. The sun rises in the East and sets in the West.

2. 1 + 1 = 2

3. 'b' is a vowel.

All of the above sentences are propositions, where the first two are Valid(True) and

the third one is Invalid(False).

Some sentences that do not have a truth value or may have more than one

truth value are not propositions. For Example,

1. What time is it?

2. Go out and play.

3. x + 1 = 2.

The above sentences are not propositions as the first two do not have a truth

value, and the third one may be true or false.

To represent propositions, propositional variables are used. By

Convention, these variables are represented by small alphabets such as

The area of logic which deals with propositions is called propositional

calculus or propositional logic.

It also includes producing new propositions using existing ones. Propositions

constructed using one or more propositions are called compound

propositions.

The propositions are combined together using Logical Connectives or Logical Operators.

Following are some basic facts about propositional logic:

• Propositional logic is also called Boolean logic as it works on 0 and 1.

• In propositional logic, we use symbolic variables to represent the logic, and we

can use any symbol for a representing a proposition, such A, B, C, P, Q, R, etc.

• Propositions can be either true or false, but it cannot be both.

• Propositional logic consists of an object, relations or function, and logical

connectives.

• These connectives are also called logical operators.

• The propositions and connectives are the basic elements of the propositional

logic.

• Connectives can be said as a logical operator which connects two sentences.

• A proposition formula which is always true is called tautology, and it is also

called a valid sentence.

• A proposition formula which is always false is called Contradiction.

• A proposition formula which has both true and false values is called

• Statements which are questions, commands, or opinions are not propositions

such as "Where is Harsh", "How are you", "What is your name", are not

propositions.

Syntax of propositional logic:

The syntax of propositional logic defines the allowable sentences for the

knowledge representation. There are two types of Propositions:

1. Atomic Propositions

2. Compound propositions

• Atomic Proposition: Atomic propositions are the simple propositions. It

consists of a single proposition symbol. These are the sentences which must be

either true or false.

Example:

a) 2+2 is 4, it is an atomic proposition as it is a true fact.

b) "The Sun is cold" is also a proposition as it is a false fact.

c)

• Compound proposition: Compound propositions are constructed by

combining simpler or atomic propositions, using parenthesis and logical

connectives.

Example:

( a) "It is raining today, and street is wet."

(b) "Apoorav is a doctor, and his clinic is in Mumbai."

Truth Table

Since we need to know the truth value of a proposition in all possible

scenarios, we consider all the possible combinations of the propositions

which are joined together by Logical Connectives to form the given

compound proposition. This compilation of all possible scenarios in a

tabular format is called a truth table.

Logical Connectives:

Logical connectives are used to connect two simpler propositions or representing a sentence logically.

We can create compound propositions with the help of logical connectives.

There are mainly five connectives, which are given as follows:

1. Negation: A sentence such as ¬ P is called negation of P. A literal can be either Positive literal or negative literal.

2. Conjunction: A sentence which has ∧ connective such as, P ∧ Q is called a conjunction. Example: Tushar is intelligent and hardworking. It can be

written as, P= Tushar is intelligent,

Q= Tushar is hardworking. → P∧ Q.

3. Disjunction: A sentence which has ∨ connective, such as P ∨ Q. is called disjunction, where P and Q are the propositions.

Example: “Apoorav is a doctor or Engineer",

Here P= Apoorav is Doctor. Q= Apoorav is Engineer, so we can write it as P ∨ Q.

1. Implication: A sentence such as P → Q, is called an implication.

Implications are also known as if-then rules. It can be represented as

If it is raining, then the street is wet.

Let P= It is raining, and Q= Street is wet, so it is represented as P → Q

2. Biconditional: A sentence such as P⇔ Q is a Biconditional sentence, example If I am breathing, then I am alive

P= I am breathing, Q= I am alive, it can be represented as P ⇔ Q.

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