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Students can Download Chapter 11 Plant Growth and Development Notes, Plus One Botany Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Botany Notes Chapter 11 Plant Growth and Development

Growth
Growth is defined as an irreversible permanent increase in the size of an organ or its parts of an individual cell.
It is accompanied by metabolic processes (both anabolic and catabolic), that occur at the expense of energy.
Eg: expansion of a leaf.

Plant Growth Generally is Indeterminate
Plant growth is unlimited growth due to the presence of meristems.
Root apical meristem and the shoot apical meristem are responsible for the primary growth of the plants and contribute to the elongation of the plants along their axis.

Role of lateral meristem in plants
In dicotyledonous plants and gymnosperms, the lateral meristems, (vascular cambium and cork-cambium) cause an increase in the girth of the organs. This is known as secondary growth.

Growth is Measurable

• Growth is measured in terms of increase in fresh weight, dry weight, length, area, volume, and cell number.
• One single maize root apical meristem can give rise to more than 17,500 new cells per hour, cells in a watermelon increase in size by up to 3,50,000 times.
• In the former, growth is expressed as an increase in cell number.
  latter expresses growth as an increase in the size of the cell.
• While the growth of a pollen tube is measured in terms of its length, an increase in surface area denotes the growth in a dorsiventral leaf.

Phases of Growth
The period of growth is generally divided into three phases.

1. Meristematic: The constantly dividing cells, both at the root apex and the shoot apex, represent the meristematic phase of growth.
2. Elongation: The cells proximal to the meristematic zone represent the phase of elongation.
3. Maturation: Proximal to the phase of elongation represents the phase of maturation.

Growth Rates
The increased growth per unit time is termed as growth rate. The growth rate may be

1. Arithmetic
2. Geometrical

On plotting the length of the organ against time, a linear curve is obtained, it is expressed as

L_t = L₀ + rt
L_t = length at time ‘t’
L₀ = length at time ‘zero’.
r = growth rate/elongation per unit time.

Different phases of the Sigmoid curve

1. lag phase
2. log or exponential phase
3. stationary phase

In most systems, the initial growth is slow (lag phase), and it increases rapidly at an exponential rate (log or exponential phase)
In the end, due to the limited nutrient supply, the growth slows down leading to a stationary phase. It is the typical sigmoid or S-curve.

The exponential growth can be expressed as
W₁ = W₀ e rt
W₁ = final size (weight, height, number etc.)
W₀ = initial size at the beginning of the period
r = growth rate
t = time of growth
e = base of natural logarithms

Quantitative comparisons between the growth of a living system can also be made in two ways:

• Measurement and the comparison of total growth per unit time is called the absolute growth rate.
• The growth of the given system per unit time expressed on a common basis.

In Figure two leaves, A and B, are drawn that are of different sizes but show an absolute increase in area in the given time to give leaves, A1 and B1.

Conditions for Growth
Water, oxygen, and nutrients as very essential elements for growth.
The plant cells grow in size by cell enlargement it requires water.
Turgidity of cells helps in extension growth. Water also provides the medium for enzymatic activities
Oxygen helps in releasing metabolic energy essential for growth activities.
Nutrients (macro and micro essential elements) are required by plants for the synthesis of protoplasm and act as a source of energy.
An optimum temperature range is best suited for plant growth.
Environmental signals such as light and gravity also affect certain phases/stages of growth.

Differentiation, Dedifferentiation, and Redifferentiation
1. The cells derived from root apical and shoot-apical meristems and cambium differentiate and mature to perform specific functions. This is termed as differentiation.
For example, during differentiation, tracheary elements lose their protoplasm and develop a very strong, elastic, lignocellulosic secondary cell wall, to carry water too long distances.

2. The living differentiated cells, that have lost the capacity to divide can regain the capacity of division This phenomenon is termed dedifferentiation.
For example, interfascicular cambium and cork cambium is formed from fully differentiated parenchyma cells.

3. Meristems are able to divide and produce cells that once again lose the capacity to divide but mature to perform specific functions. This is called a redifferentiation.
For example, secondary tissues develop from vascular cambium and cork cambium

Development
It is the stage of the life cycle in which germination of the seed to senescence.

The plant shows a response to the environment to form different kinds of structures. This ability is called plasticity
Heterophylly is an example of plasticity

Types of Heterophylly
1. The leaves of the juvenile plant are different in shape from those in mature plants.
e.g cotton, coriander, and larkspur.
2. Shapes of submerged leaves are different from those produced in the air.
Eg buttercup.

Development is considered as the sum of growth and differentiation.
Development in plants is under the control of intrinsic and extrinsic factors.

A) Intrinsic factors

1. Intracellular (genetic)
2. Intercellular factors (chemicals such as plant growth regulators)

B) Extrinsic factors
Light, temperature, water, oxygen, nutrition, etc.

Plant Growth Regulators

Characteristics
The plant growth regulators (plant hormones or phytohormones) include

1. Indole compounds (indole-3-acetic acid, IAA);
2. Adenine derivatives (N6-furfurylamino purine, kinetin),
3. Derivatives of carotenoids (abscisic acid, ABA);
4. Terpenes (gibberellic acid, GA3) or
5. Gases (ethylene, C2H4).

The PGRs are divided into two groups based on their functions in a living plant body.
One group of PGRs are involved in growth-promoting activities, e.g., auxins, gibberellins, and cytokinins.
The other group mainly involved in growth-inhibiting activities such as dormancy and abscission. Eg-abscisic acid and ethylene.

The Discovery of Plant Growth Regulators
1. Auxin
At first, Charles Darwin and his son Francis Darwin observed that the coleoptiles of canary grass responded to unilateral illumination by growing towards the light source.
After a series of experiments, it was concluded that the tip of the coleoptile was the site of transmittable influence that caused the bending of the entire coleoptile.
Auxin was isolated by F. W. Went from tips of coleoptiles of oat seedlings.

2. Gibberellin
The ‘balance’ (foolish seedling) disease of rice seedlings, was caused by a fungal pathogen Gibberalla fujikuroi.
In this experiment, the uninfected rice seedlings were treated with sterile filtrates of the fungus. It led to the development of the disease. The active substance was gibberellic acid.
It was demonstrated by E. Kurosawa.

3. Cytokinin
The internodal segments of tobacco stems- the callus proliferated in the presence of auxins along with the extracts of vascular tissues, yeast extract, coconut milk or DNA.
Skoog and Miller later identified and crystallized the cytokinesis promoting active substances that they termed kinetin.

4. Abscisic acid(ABA)
During the mid-1960s inhibitory hormones were identified: inhibitor-B, abscission II and dormin.
Later all the three were named abscisic acid (ABA).

5. Ethylene
Ripened oranges that hastened the ripening of stored unripened bananas. Later this volatile substance was identified as ethylene, a gaseous PGR.

Physiological Effects of Plant Growth Regulators
Auxins
Auxins were first isolated from human urine.
They are generally produced by the growing apices of the stems and roots, from where they migrate to the regions of their action.

Two types of auxins

1. Natural (IAA and indole butyric acid (IBA)
2. Synthetic. NAA (naphthalene acetic acid) and 2, 4-D (2, 4-dichlorophenoxyacetic)

1. They help to initiate rooting in stem cuttings
2. Auxins promote flowering e.g. in pineapples.
3. They help to prevent fruit and leaf drop at early stages but promote the abscission of older mature leaves and fruits.
4. In most higher plants, the growing apical bud inhibits the growth of the lateral (axillary) buds, a phenomenon called apical dominance.
5. Removal of shoot tips (decapitation) usually results in the growth of lateral buds Hence it is widely applied in tea plantations, hedge-making, etc.
6. Auxins also induce parthenocarpy.
7. They are widely used as herbicides. 2, 4-D is used to kill dicotyledonous weeds, So it is used to prepare weed-free lawns by gardeners.
8. Auxin controls xylem differentiation and helps in cell division.

Gibberellins
Gibberellic acid (GA3) was one of the first gibberellins to be discovered and remains the most intensively studied form.
GA3 is acidic.

1. GA3 causes an increase in the length of grapes stalks. .
2. Gibberellins, cause fruits like apple to elongate and improve its shape
3. They delay senescence. Hence the fruits are keeping as fresh.
4. GA3 is used to speed up the malting process in the brewing industry.
5. Spraying sugarcane crop with gibberellins increases the length of the stem, thus increasing the yield by as much as 20 tonnes per acre.
6. Spraying juvenile conifers with GAs hastens the maturity period, thus leading to early seed production.
7. Gibberellins also promote bolting(internode elongation just prior to flowering) in beet, cabbages and many plants with rosette habit.

Cytokinins
Cytokinins were discovered as kinetin (a modified form of adenine, a purine) from the autoclaved herring sperm DNA.
Naturally occurring cytokinin-zeatin was isolated from corn-kernels and coconut milk.
Natural cytokinins are synthesised in regions where rapid cell division occurs, for example, root apices, developing shoot buds, young fruits, etc.

1. It helps to produce new leaves, chloroplasts in leaves, lateral shoot growth and adventitious shoot formation.
2. Cytokinins help to overcome apical dominance.
3. They promote nutrient mo8/+9bilisation which helps in the delay of leaf senescence.

Ethylene
The most widely used compound as a source of ethylene is ethephon.
It is readily absorbed and transported within the plant and releases ethylene slowly.
Ethephon hastens fruit ripening in tomatoes and apples and accelerates abscission in flowers and fruits

1. Ethylene is a gaseous hormone that promotes senescence and ripening fruits.
2. It promotes horizontal growth of seedlings, swelling of the axis, and apical hook formation in dicot seedlings.
3. Ethylene promotes senescence and abscission of plant organs, especially of leaves and flowers.
4. Ethylene is highly effective in fruit ripening. It enhances the respiration rate during the ripening of the fruits (respiratory climactic).
5. Ethylene breaks seed and bud dormancy and initiates germination in peanut seeds, sprouting of potato tubers.
6. Ethylene promotes rapid internode/petiole elongation in deepwater rice plants.
7. Ethylene also promotes root growth and root hair formation, thus helping the plants to increase their absorption surface.
8. Ethylene is used to initiate flowering and fruit-set in pineapples.
9. It also induces flowering in mango.
10. It promotes female flowers in cucumbers

Abscisic acid

1. It promotes abscission and dormancy.
2. It acts as an inhibitor of plant metabolism.
3. ABA inhibits seed germination.
4. ABA stimulates the closure of stomata and increases the tolerance of plants to various kinds of stresses. Hence it is called the stress hormone.
5. ABA plays an important role in seed development, maturation, and dormancy.
6. ABA helps the seeds to withstand desiccation
7. ABA acts as an antagonist to GA3.

Photoperiodism
It is the phenomenon of relative day and night length for the initiation of flowering.

Based on the exposure to photoperiod there are three types of plants

1. Long day plants: They require exposure to light for a period greater than critical duration (12 hr).
2. Short-day plants: They require less than critical duration before flowering.
3. Day-neutral plants: In this type, there is no such correlation between exposure to light duration and induction of flowering response.

Which is the organ of a plant perceives light for photoperiodism?
The site of perception of light/dark duration is the leaves. After receiving the required photoperiod, the hormonal substance migrates from leaves to shoot apices for inducing flowering. The shoot apices become changed into flowering apices prior to flowering.

Vernalisation
It is the phenomenon of exposure of low temperature for the initiation of flowering
Some important food plants, wheat, barley, rye have two kinds of varieties: winter and spring varieties.

Nature of spring and winter varieties
The ‘spring’variety are normally planted in the spring and come to flower and produce grain before the end of the growing season.
Winter varieties, planted in spring fail to flower or produce mature grain within a span of a flowering season.
If they are planted in autumn .they germinate and overwinter come out as small seedlings, resume growth in the spring, and are harvested usually around mid-summer.

Biennials and low-temperature treatment
Biennials are monocarpic plants that normally flower and die in the second season. Biennial plants are subjected to a cold treatment, it stimulates photoperiodic flowering response. Sugarbeet, cabbages, carrots are some of the common biennials.

NCERT Supplementary Syllabus

Seed Germination
The seeds germinate under favourable conditions after the period of dormancy.
After dormancy embryo becomes metabolically active and starts growing. This process is known as seed germination.
The conditions necessary for seed germination are the availability of water and oxygen.

A physiological phenomenon in seed germination
The physical phenomenon associated with seed germination is imbibition. It causes the swelling of seed then rupturing of the seed coat, through which radical emerges out.
It develops into a root system but the shoot system arises from the plumule of another end of the embryonal axis.
The metabolic activities require oxygen for breaking down the food reserves such as polysaccharides, proteins and lipid.
It is converted into soluble materials with the help of enzymes and mobilized to the embryonal axis.
The growth of radical and plumule is due to cell extension, cell division, and several biochemical processes.
The seed also needs a suitable temperature (optimum between 25 to35). The rate of respiration increases rapidly during seed germination.

What is the Viviparous type of germination?
Vivipary is the germination of a seed while it is still attached to the parent plant and is nourished by it. The plants grow in marshy land such as Rhizophora and Sonneratia (halophytes)show this type of germination.
During germination, radical elongates, and the weight of the germinating seed increases. As a result, the seedling separates and fail down vertically into the mud and grow into a new plant.

Seed Dormancy
It is the period of rest or a period of suspended growth due to this

1. water content, the metabolic activities become extremely low.
2. the seed coat becomes impermeable to oxygen and moisture and hardens.

The suspension of growth is due to exogenous (environmental conditions) or endogenous control during which metabolic activity of the seed is greatly reduced.

Causes of Dormancy

1. Impermeable or mechanically resistant seed coats.
2. Rudimentary or physiologically immature embryos or
3. Due to the presence of germination inhibitors such as abscisic acid, phenolic acid, short-chain fatty acids, and coumarin.

How can overcome seed dormancy?

1. Mechanical or chemical scarification of the seed coat (scratching of seed coat or seeds soaked in chemicals to break the dormancy)
2. Stratification of seeds or changing environmental conditions such as temperature, light, and pressure. Stratification of seeds is subjecting the moist seeds to oxygen for variable periods of low or high temperatures.

Students can Download Chapter 10 Respiration in Plants Notes, Plus One Botany Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Botany Notes Chapter 10 Respiration in Plants

How are respiration and photosynthesis-related?
Green plants and cyanobacteria can prepare their own food by the process of photosynthesis, they trap light energy and convert it into chemical energy that is stored in the bonds of carbohydrates (Macromolecules) like glucose, sucrose, and starch.
By Cellular respiration, food materials undergo breakdown that release energy and the trapping of this energy for the synthesis of ATP.

ATP is called the energy currency of the cell why?
ATP is broken down whenever (and wherever) energy needs to be utilised. Hence, ATP acts as the energy currency of the cell.

Seat of photosynthesis and respiration
Photosynthesis takes place within the chloroplasts whereas the breakdown of complex molecules to yield energy takes place in the cytoplasm and in the mitochondria (also only in eukaryotes).
The compounds that are oxidized during respiration are known as respiratory substrates.
Eg. Proteins, fats, and even organic acids.

Do Plants Breathe?
Plants require O₂ for respiration and give out CO₂ and H₂O as end products and release energy most of which is given out as heat.

Respiration is least important to plants than animals
Roots, stems, and leaves respire at rates far lower than animals When cells photosynthesize O₂ is released within the cell.
But some cells live where oxygen may or may not be available.
All living organisms retain the enzymatic machinery to partially oxidise glucose without the help of oxygen. This breakdown of glucose to pyruvic acid is called glycolysis.

Glycolysis
The term glycolysis -(Greek words, glycols for sugar, and lysis for splitting).
The scheme of glycolysis was given by Gustav Embden, Otto Meyerhof, and J. Parnas, Hence glycolysis is called an EMP pathway.
In anaerobic organisms, it is the only process in respiration.
Glycolysis occurs in the cytoplasm of the cell and is present in all living organisms.
In this process, Glucose undergoes partial oxidation to form two molecules of pyruvic acid.

Steps lead to end products of glycolysis

1. Glucose and fructose are phosphorylated to give rise to glucose-6-phosphate by the activity of the enzyme hexokinase.

2. This phosphorylated form of glucose then isomerises to produce fructose-6-phosphate.

3. In this pathway, ATP is utilised at two steps: first in the conversion of glucose into glucose 6-phosphate and second in the conversion of fructose 6-phosphate to fructose 1,6 diphosphate).

4. The fructose 1,6-diphosphate is split into dihydroxyacetone phosphate and 3 phosphoglyceraldehydes (PGAL). In this step NADH +H+ is formed from NAD+.

5. 3-phosphoglyceraldehyde (PGAL) is converted to 1,3 bisphosphoglycerate (DPGA).

6. The conversion of DPGA to 3-phosphoglyceric acid (PGA), is also an energy-yielding process; this energy is trapped by the formation of ATP.

7. 3-phosphoglyceric acid (PGA) is converted into 2 phosphoglycerates.

8. 2 phosphoglycerates are converted into 2 phosphoenol pyruvic acid. ATP is synthesized during the conversion of PEP to pyruvic acid.

9. 2 phosphoenol pyruvic acid undergoes dephosphorylation to form 2 molecule of pyruvic acid

The fate of pyruvic acid
It involves

1. Lactic acid fermentation
2. Alcoholic fermentation
3. Aaerobic respiration.

Fermentation takes place under anaerobic conditions in many prokaryotes and unicellular eukaryotes.

The complete oxidation of glucose to CO₂ and H₂O occurs in organisms that adopt Krebs’ cycle which is also called as aerobic respiration. This requires an O₂ supply.

Fermentation
In fermentation, glucose undergoes incomplete oxidation and forms CO₂ and ethanol
The enzymes, pyruvic acid decarboxylase, and alcohol dehydrogenase catalyze these reactions.
Fermentation occurs in the presence of yeast
Yeasts poison themselves to death when the concentration of alcohol reaches about 13 percent. Some organisms like bacteria produce lactic acid from pyruvic acid.

The lactic acid in eukaryotic cell
In animal muscle cells during exercise, when oxygen is inadequate for cellular respiration, pyruvic acid is reduced to lactic acid by lactate dehydrogenase.
The reducing agent is NADH+H* which is re oxidised to NAD+ in both the processes.
In both lactic acid and alcohol fermentation, less than seven percent of the energy in glucose is released.
In eukaryotes second step after glycolysis take place within the mitochondria and this requires O₂.
It is aerobic respiration leads to complete oxidation of carbohydrate in the presence of oxygen and releases CO₂, water and a large amount of energy.
This type of respiration is most common in higher organisms.

Aerobic Respiration
The second step of Aerobic respiration takes place within the mitochondria.
The product of glycolysis- pyruvate is transported from the cytoplasm into the mitochondria.

First step of oxidation of pyruvic acid
In the mitochondrial matrix, pyruvate undergoes oxidative decarboxylation by pyruvic dehydrogenase. The reactions require the participation of several coenzymes, including NAD+ and Coenzyme A.

During this process, two molecules of NADH are produced from the metabolism of two molecules of pyruvic acid.
The acetyl CoA then enters a cyclic pathway, tricarboxylic acid cycle(Krebs’ cycle) after the scientist Hans Krebs who first elucidated it.

Tricarboxylic Acid Cycle
The TCA cycle starts with the condensation of acetyl group with oxaloacetic acid (OAA) and water to yield citric acid.
The reaction is catalysed by the enzyme citrate synthase and a molecule of CoA is released.
It is followed by two successive steps of decarboxylation, leading to the formation of alpha-ketoglutaric acid and then succinyl-CoA. In the remaining steps, Succinic acid is oxidised to OAA.

Which step of the Krebs cycle substrate-level phosphorylation occurs?
During the conversion of succinyl-CoA to succinic acid, a molecule of GTP is synthesised. This is substrate-level phosphorylation.

At three sites in the cycle where NAD+ is reduced to NADH + H+ and one site where FAD+ is reduced to FADH₂.

In the mitochondrial matrix, pyruvate is broken down to release.
8 molecules of NADH + H+
2 molecules of FADH₂
2 molecules of GTP and
3 molecules of CO₂

Electron Transport System (ETS) and Oxidative Phosphorylation

The metabolic pathway through which the electron passes from one carrier to another is called the electron transport system (ETS).
It is present in the inner mitochondrial membrane.
Reduced coenzyme like NADH(complex) in the mitochondrial matrix is oxidised and release 2 electrons and 2protons
Electrons and protons are transferred to FMN, it reduced to FMNH2
It breaks and releases protons and electrons .protons go to intermembrane space but electrons reach ubiquinone.
Ubiquinone also receives reducing equivalents via FADH2 (complex II).
The reduced ubiquinone is then oxidised with the transfer of electrons to cytochrome c via cytochrome bc1 complex (complex III).

Electron Transport System (ETS)
Cytochrome c acts as a mobile carrier for the transfer of electrons between complex III and IV.
Complex IV refers to cytochrome c oxidase complex containing cytochromes a and a3.

Oxidation of one molecule of NADH gives rise to 3 molecules of ATP, while that of one molecule of FADH2 produces 2 molecules of ATP.
Oxygen acts as the final hydrogen acceptor.

Oxidative phosphorylation in mitochondria
In ETS the energy of oxidation-reduction is utilised for the production of proton gradient required for phosphorylation. This process is called oxidative phosphorylation.

Chemiosmosis (proposed by peter Mitchel)
The energy released during the electron transport system is utilised in synthesizing ATP with the help of ATP synthase (complex V) called chemiosmosis.

F1 – F0/exosomes
This complex consists of two major components, F1 and Fo.
The F1 headpiece is a site for synthesis of ATP from ADP and inorganic phosphate.
F0 is an integral membrane protein complex act as a channel through which protons cross the inner membrane.
For each ATP produced, 2H+ passes through F0 from the intermembrane space to the matrix down the electrochemical proton gradient.

The Respiratory Balance Sheet
How many ATP molecules are produced in Aerobic respiration?
In aerobic respiration, the number of ATP molecules produced or utilized in glycolysis, TCA cycle and ETS gives the net gain of 36 ATP molecules
Fermentation accounts for only a partial breakdown of glucose whereas in aerobic respiration it is completely degraded to CO₂ and H₂O.

How many ATP molecules are produced in Fermentation?
In fermentation there is a net gain of only two molecules of ATP for each molecule of glucose degraded NADH is oxidized to NAD+ rather slowly in fermentation.

Amphibolic Pathway
It involves two processes anabolism and catabolism.
For example, fats is broken down into glycerol and fatty acids. Then fatty acids degraded to acetyl CoA and enter the pathway.
Glycerol enters the pathway after being converted to PGAL.
The proteins are degraded by proteases and the individual amino acids enter the pathway at some stage within the Krebs’cycle as pyruvate or acetyl CoA.

Is it true both catabolism and anabolism occur in fat metabolism?
Fatty acids( substrate) are broken down to acetyl CoA before entering the respiratory pathway. But when the organism needs to synthesize fatty acids, acetyl CoA withdrawn from the respiratory pathway for it.
Hence, the respiratory pathway involves the breakdown and synthesis of fatty acids, i.e catabolism, and anabolism respectively. Hence it is considered as an amphibolic pathway.

Respiratory Quotient
Definition: The ratio of the volume of CO₂ evolved to the volume of O2 consumed in respiration is called the respiratory quotient (RQ) or respiratory ratio.

Respiratory quotient of some respiratory substrates
1. Carbohydrates: When carbohydrates are completely oxidised, the RQ is 1, because equal amounts of CO₂ and O₂ are evolved and consumed, respectively

2. Fats: If fats are used in respiration, the RQ is less than 1.

3. Proteins: When proteins are respiratory substrates the ratio is 0.9.

4. Organic acids: When organic acids are respiratory substrates, the ratio is more than one.

Students can Download Chapter 9 Photosynthesis in Higher Plants Notes, Plus One Botany Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Botany Notes Chapter 9 Photosynthesis in Higher Plants

What Do We Know?
Role of light, CO₂, H₂O, and Chlorophyll
Actually, chlorophyll (green pigment of the leaf), light, and CO₂ are required for photosynthesis. A variegated leaf ora leaf that was partially covered with black paper, and one that was exposed to light. On testing these leaves for starch it was clear that photosynthesis occurred only in the green parts of the leaves in the presence of light.

Half leaf experiment and the importance of CO₂ in photosynthesis
In this, a part of a leaf is enclosed in a test tube containing some KOH soaked cotton (which absorbs CO₂), while the other half is exposed to air. The set up is then placed in light for some time. Then conducted the starch test, showed that the exposed part of the leaf tested positive for starch while the portion that was in the tube, tested negative. This showed that CO₂ is required for photosynthesis.

Early Experiments

Historical aspects of photosynthesis
1. Priestley
He observed that a candle burning in a closed space – a bell jar, soon gets extinguished. Similarly, a mouse would soon suffocate in a closed space.

He concluded that a burning candle or an animal that breathes the air, both damage the air. But when he placed a mint plant in the same bell jar, he found that the mouse stayed alive and the candle continued to burn.

2. Jan Ingenhousz
He showed that sunlight is essential to the plant process that purifies the air fouled by burning candles or breathing animals. In aquatic habitat, during bright sunlight, small bubbles were formed around the green parts while in the dark they did not. Later he identified these bubbles are oxygen. So the green part of the plants could release oxygen.

3. Julius von Sachs
Glucose is usually stored as starch. He found that the green parts in plants where glucose is made.

4. T.W Engelmann
By using a prism he split light into its spectral components and then illuminated a green alga, Cladophora, placed in a suspension of aerobic bacteria. The bacteria were used to detect the sites of O₂ evolution. He observed that the bacteria accumulated mainly in the region of blue and red light of the split spectrum.

An empirical equation for photosynthesis

[CH₂O] represent a carbohydrate (e.g., glucose, a six-carbon sugar).

Hydrogen donor in bacteria and green plants
Some organisms do not release O₂ during photosynthesis
When H₂S, instead is the hydrogen donor for purple and green sulphur bacteria, the ‘oxidation’ product is sulphur or sulphate depending on the organism and not O₂. In the green plants, the O₂ evolved from H₂O, not from carbon dioxide.

The modern equation for photosynthesis

C₆H₁₂O₆ represents glucose. The O₂ released is from water

Where Does Photosynthesis Take Place?
It takes place in the chloroplast of leaves that contain grana, the stroma lamellae, and the fluid stroma.

Where is the energy production site in chloroplast?
The energy-rich molecules like ATP and NADPH are synthesized in grana and stroma lamellae by light reactions.

Where is the Glucose production site in chloroplast?
In stroma by dark reactions, CO₂ fixation leading to the synthesis of glucose, which in turn forms starch

Structure of chloroplast

Diagrammatic representation of an electron micrograph of a section of chloroplast

How Many Pigments are Involved in Photosynthesis?
Chromatographic separation of the leaf pigments shows that different types of pigments in leaves i.e
Chlorophyll a (bright or blue-green in the chromatogram)
chlorophyll b (yellow-green)
xanthophylls (yellow)
carotenoids (yellow to yellow-orange)

a) Graph showing the absorption spectrum of chlorophyll a, b, and the carotenoids.

b) Graph showing the action spectrum of photosynthesis.

c) Graph showing action spectrum of photosynthesis superimposed on the absorption spectrum of chlorophyll a.

Wavelengths of light absorbed by pigments
Chlorophyll pigments absorb light, at specific wavelengths of blue and the red regions while carotenoids absorb the blue and green wavelength
Chlorophyll is the major pigment responsible for trapping light, other thylakoid pigments like chlorophyll b, xanthophylls, and carotenoids, which are called accessory pigments, also absorb light and transfer the energy to chlorophyll a. but also protect chlorophyll a from photo-oxidation.

What is Light Reaction?
It is the photochemical phase include

1. light absorption
2. water splitting
3. oxygen release
4. Formation of high-energy rich molecules ATP and NADPH.

The pigments are organised into two light-harvesting complexes(LHC)

1. Photosystem I (PS I)/P700
2. Photosystem II (PS II)/P680

Each photosystem has single chlorophyll a molecule forms the reaction centre, all the pigments except chlorophyll-a forming a light-harvesting system also called antennae.

In PS I the reaction centre, chlorophyll a has an absorption peak at 700 nm while in PS II it has absorption maxima at 680 nm, and is called P680.

The Electron Transport
How does electron flow in electron carriers that connect two photosystems?
Initially, excitation of chlorophyll molecule occurs due to light, then electrons are emitted from Ps II (uphill) that are accepted by electron acceptor, electron flows through electron carriers cytochromes, (downhill) and (Loss of electrons of PSII is compensated by electrons coming from water and loss of electrons of PS I is compensated by electrons coming from PS II).

PS I is also excited due to light and electrons are emitted (uphill), it transfers an electron to another accepter, and finally down the hill to NADP+ causing it to be reduced to NADPH + H+ is called the Z scheme.

Result of Z-scheme

1. production of ATP and NADPH
2. O₂ evolution

Splitting of Water
Photolysis
It is the splitting of water into H+, [O] and electrons in the presence of light and these electrons are available to PSII.
This process takes place on the inner side of the membrane of the thylakoid.
Oxygen released is one of the net products of photosynthesis.
2H₂O → 4H⁺ + O₂ + 4e^–

Cyclic and Non-cyclic Photo-phosphorylation
Phosphorylation
The process of which ATP from ADP and inorganic phosphate in the presence of light (in mitochondria and chloroplasts) is named phosphorylation.

Electron in a cyclic process
When only PS I is functional, the cyclic flow of electrons within the photosystem and the phosphorylation occurs in the stroma lamellae.
Cyclic photophosphorylation also occurs when only light of wavelengths beyond 680 nm are available for excitation i.e at 700 nm

Result of cyclic photophosphorylation
ATP is produced.

Where does the light-harvesting complex work for acyclic and noncyclic processes?
The membrane or lamellae of the grana have both PS I and PS II so a noncyclic process occurs,
The stroma lamellae membranes lack PS II as well as NADP reductase enzyme So a cyclic process occurs.

Chemiosmotic Hypothesis
It is the ATP synthesis linked to the development of a proton gradient across a membrane
In chloroplast, the proton accumulation is towards the inside of the membrane, i.e., in the lumen. In respiration, protons accumulate in the intermembrane space of the mitochondria when electrons move through the ETS.
The proton gradient develops due to,

a. Splitting of the water molecule takes place on the inner side of the membrane, the protons accumulate within the lumen of the thylakoids

b. As electrons move through the photosystems, protons are transported across the membrane moves into the lumen side of the membrane

c. The NADP reductase enzyme is located on the stromal side of the membrane. Along with electrons that come from the accepter of electrons of PS I, protons are necessary for the reduction of NADP+ to NADPH+ H+. These protons are also removed from the stroma.

In chloroplast, protons in the stroma decrease in number, while in the lumen there is an accumulation of protons. This creates a proton gradient across the thylakoid membrane The gradient is broken down due to the movement of protons across the membrane to the stroma through the transmembrane channel of the F₀ of the ATP.
ATPase have a channel that allows diffusion of protons back across the membrane; this releases enough energy to activate the ATPase enzyme that catalyses the formation of ATP.

Where are the ATP and NADPH Used?
It is used in the biosynthetic phase of photosynthesis. This process does not directly depend on the presence of light but is dependent on the products of the light reaction, i.e., ATP and NADPH.
Melvin Calvin studied the algal photosynthesis by using radioactive 14C led to the discovery that the first CO₂ fixation product was identified as 3-phosphoglyceric acid or PGA.

The Primary Acceptor of CO₂
The studies showed that the accepter molecule was a 5-carbon sugar -ribulose bisphosphate (RuBP) in the Calvin cycle.

The Calvin Cycle
It involves three stages:

1. carboxylation
2. reduction
3. regeneration

1. Carboxylation:
Carboxylation is the fixation of CO₂ into a stable compound catalysed by the enzyme RuBisCO that results in the formation of two molecules of 3-PGA.

2. Reduction: These are a series of reactions that lead to the formation of glucose.
The steps involve the utilization of 3 molecules of ATP for phosphorylation and two NADPH for reduction per CO₂ molecule fixed. For the fixation of six molecules of CO₂, 6 turns of the cycle are required and one molecule of glucose is generated

3. Regeneration: Regeneration of the CO₂ acceptor molecule require one ATP for phosphorylation to form RuBP.
Hence for every CO₂ molecule entering the Calvin cycle, 3 molecules of ATP and molecules of NADPH are required

The Calvin cycle proceeds in three stages:
1. carboxylation, during which CO₂ combines with ribulose- 1, 5- bisphosphate
2. reduction, during which carbohydrate is formed at the expenses of the photochemically made ATP and NADPH; and
3. regeneration during which the CO₂ acceptor ribulose- 1, 5-bisphosphate has formed again so that the cycle continues

The C₄ Pathway (Hatch and Slack Pathway)
Plants that are adapted to dry tropical regions have the C₄ pathway, C₄ plants have a special type of leaf anatomy. They tolerate higher temperatures. They lack a process called photorespiration and have greater productivity of biomass.

Special leaf anatomy-kranz anatomy
Large cells around the vascular bundles are centripetally arranged bundle sheath cells such anatomy is called ‘Kranz’ anatomy. Eg- maize or sorghum

Primary CO₂, accepter, first stable product and Enzyme of C₄ Pathway
The primary CO₂ acceptor is a 3-carbon molecule- phosphoenolpyruvate (PEP) present in the mesophyll cells. The enzyme responsible for this fixation is PEP carboxylase or PEPcase.
The first stable product C₄ acid OAA is formed in the mesophyll cells.

Diagrammatic representation of a Hatch arid Slack Pathway
OAA converted into 4-carbon compounds like malic acid or aspartic acid in the mesophyll cells .which are transported to the bundle sheath cells. In the bundle sheath cells, these C4 acids are broken down to release CO₂ and a 3-carbon molecule. The 3-carbon molecule is transported back to the mesophyll where it is converted to PEP again, thus, completing the cycle.
Thus the basic pathway that results in the formation of the sugars, the Calvin pathway, is common to the C₃ and C₄ plants.

Photorespiration
In C₃ plants, under high concentration of O₂ and low CO₂ concentration, RUBP binds with O₂ to form one molecule of PGA and phosphoglycolate and a large quantity of CO₂ is released.

Can you say photorespiration is a wasteful process?
This process utilise ATP but neither synthesis of sugars, nor of ATP. Hence photorespiration is a wasteful process.

The specialty of C₄ plants to avoid Photorespiration
In C₄ plants photorespiration does not occur because C₄ acid from the mesophyll is broken down in the bundle cells to release CO₂ – this results in increasing the intracellular concentration of CO₂. Here RuBisCO functions as a carboxylase minimizing the oxygenase activity, productivity, and yields are better in these plants.

Factors Affecting Photosynthesis
Photosynthesis is influenced by several factors, both internal (plant) and external. The plant factors include the number, size, age, and orientation of leaves, mesophyll cells and chloroplasts, internal CO₂ concentration, and the amount of chlorophyll. The external factors include the availability of sunlight, temperature, CO₂ concentration, and water.

Blackman s Law of Limiting Factors
If a chemical process is affected by more than one factor, then its rate will be determined by the factor which is nearest to its minimal value: it is the factor that directly affects the process if its quantity is changed.
For example, In the green leaf, the light and CO₂ conditions are optimum but the plant does not photosynthesize if the temperature is very low.

Light
The availability of light shows a direct relationship with CO₂ fixation rates at low light intensities At higher light intensities the rate does not show further increase because other factors are not in optimal amount. The intensity of light beyond a point causes the breakdown of chlorophyll and a decrease in photosynthesis.

Carbon dioxide Concentration
The concentration of CO₂ is very low in the atmosphere (between 0.03 and 0.04 percent). An increase in concentration up to 0.05 percent can cause an increase in CO₂ fixation rates. The C₃ and C₄ plants respond differently to CO₂ concentrations.

Graph of light Intensity on the rate of photosynthesis

C₄ plants show saturation at about 360µL^-1 while C₃ responds to increased CO₂ concentration and saturation is seen only beyond 450µL^-1 Thus, the current availability of CO₂ levels is limiting to the C₃ plants.

C₃ plants respond to higher CO₂ concentration by showing increased rates of photosynthesis leading to higher productivity The above concept is used for some greenhouse crops such as tomatoes and bell pepper.

Temperature
The dark reactions that take place in stoma are enzymatic and temperature controlled. C₄ plants respond to higher temperatures and show a higher rate of photosynthesis while C₃ plants have a much lower temperature optimum.

Water
Water stress causes the closure of stomata and it is difficult to receive CO₂ for photosynthesis. The stress condition also makes leaves wilt and reducing the surface area of the leaves and their metabolic activities.

Students can Download Chapter 8 Mineral Nutrition Notes, Plus One Botany Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Botany Notes Chapter 8 Mineral Nutrition

Methods to Study the Mineral Requirements of Plants
In 1860, Julius von Sachs a German botanist demonstrated that plants could be grown in a nutrient solution in the complete absence of soil.

Hydroponics and its Importance

• This technique of growing plants in a nutrient solution is known as hydroponics.
• The nutrient solutions must be aerated to obtain optimum growth.
• In this method, essential elements are used and their deficiency symptoms can be studied.
• Hydroponics is used in the commercial production of vegetables such as tomato, seedless cucumber, and lettuce.

Hydroponic plant production. Plants are grown in a tube or trough placed on a slight incline. A pump circulates a nutrient solution from a reservoir to the elevated end of the tube. The solution flows down the tube and returns to the reservoir due to gravity. Inset shows a plant whose roots are continuously bathed in the aerated nutrient solution. The arrows indicate the direction of the flow.

Essential Mineral Elements

Some minerals are not essential to plants

• More than sixty elements are found in different plants.
• Some plant species absorb selenium, some others gold, while some plants growing near nuclear test sites take up radioactive strontium.

Criteria for Essentiality
The criteria for the essentiality of an element are given below:

• The element must be supporting normal growth and production.
  In the absence of the element, the plants do not complete their life cycle or set the seeds.
• The requirement of the element must be specific and not replaceable by another element.
• The element must be directly involved in the metabolism of the plant.

Based upon the above criteria 17 elements are essential for plant growth and metabolism. They are
i. Macronutrients:
Carbon, hydrogen, oxygen, nitrogen, phosphorous, sulphur, potassium, calcium, and magnesium
They are present in plant tissues in large amounts(in excess of 10 m mole/ Kg of dry matter).

ii. Micronutrients:
Iron, manganese, copper, molybdenum, zinc, boron, chlorine, and nickel
They are needed in very small amounts (less than 10 m mole /Kg of dry matter).

In addition to the essential elements, sodium, silicon, cobalt, and selenium are required by higher plants. Essential elements are grouped into four broad categories on the basis of their diverse functions.
i. Essential elements as components of biomolecules (e.g., carbon, hydrogen, oxygen, and nitrogen).

ii. Essential elements that are components of energy-related chemical compounds (e.g, magnesium in chlorophyll and phosphorous in ATP).

iii. Essential elements that activate or inhibit enzymes, (Mg²⁺ is an activator for both ribulose bisphosphate carboxylase oxygenase and phosphoenolpyruvate carboxylase, both of which are critical enzymes in photosynthetic carbon fixation
Zn²⁺ is an activator of alcohol dehydrogenase and Mo of nitrogenase during nitrogen metabolism.

iv. Essential elements alter the osmotic potential of a cell.
Potassium plays an important role in the opening and closing of stomata.

Role of Macro- and Micro-nutrients
Essential elements participate in various metabolic processes in the plant cells. The various forms and functions of mineral elements are given below.

Nitrogen

• It is absorbed mainly as NO₃^– Some taken up as NO₂ or NH₄⁺
• Nitrogen is required in meristematic tissues and the metabolically active cells.
• It is one of the major constituents of proteins, nucleic acids, vitamins and hormones

Phosphorus

• It is absorbed in the form of phosphate ions (either as HPO₄^2- or H₂PO₄^–)
• Phosphorus is a constituent of cell membranes, certain proteins, all nucleic acids, and nucleotides.
• It is required for all phosphorylation reactions.

Potassium

• It is absorbed as a potassium ion (K⁺).
• It is required in abundant quantities for meristematic tissues, buds, leaves, and root tips.
• Potassium helps to maintain an anion-cation balance in cells
• It is involved in protein synthesis
• It is involved in the opening and closing of stomata and activation of enzymes
• It helps in the maintenance of the turgidity of cells.

Calcium

• It is absorbed in the form of calcium ions (Ca²⁺).
• Calcium is required by meristematic and differentiating tissues.
• It is important in the formation of calcium pectate in the middle lamella.
• It is also needed during the formation of the mitotic spindle.
• It activates certain enzymes and plays an important role in regulating metabolic activities.

Magnesium

• It is absorbed by plants in the form of divalent Mg²⁺
• It activates the enzymes of respiration, photosynthesis, and are involved in the synthesis of DNA and RNA.
• Magnesium is a constituent of the ring structure of chlorophyll
• It helps to maintain the ribosome structure.

Sulphur

• It is absorbed in the form of sulphate (SO₄^2-)ion.
• Sulphur is present in two amino acids – cysteine and methionine
• It is the main constituent of several coenzymes, vitamins (thiamine, biotin, Coenzyme A), and ferredoxin.

Iron

• It is absorbed in the form of ferric ions (Fe³⁺)
• It is an important constituent of proteins involved in the transfer of electrons like ferredoxin and cytochromes.
• It activates the catalase enzyme and is essential for the formation of chlorophyll.

Manganese

• It is absorbed in the form of manganous ions (Mn²⁺).
• It activates many enzymes involved in photosynthesis, respiration, and nitrogen metabolism.
• It is also involved in the splitting of water to liberate oxygen during photosynthesis.

Zinc

• Plants obtain zinc as Zn²⁺ ions.
• It activates various enzymes, especially carboxylases.
• It is also needed in the synthesis of auxin.

Copper

• It is absorbed as cupric ions (Cu²⁺).
• It is essential for the certain enzymes involved in redox reactions

Boron

• It is absorbed as BO₃^3- or B₄O₇^2-
• It is required for uptake and utilisation of Ca²⁺
• it helps in membrane functioning
• it helps pollen germination
• it helps cell elongation and cell differentiation
• it is involved in carbohydrate translocation.

Molybdenum

• Plants obtain it in the form of molybdate ions (MoO₂²⁺).
• It is a component of nitrogenase and nitrate reductase both of which participate in nitrogen metabolism.

Chlorine

• It is absorbed in the form of chloride anion (Cl^–).
• Along with Na⁺ and K⁺, it helps in determining the solute concentration and the anion cation balance in cells.
  It is essential for the water-splitting reaction in photosynthesis, a reaction that leads to oxygen evolution.

Deficiency Symptoms of Essential Elements
If the concentration of the essential element below the critical concentration plants shows certain morphological changes. These are indications of deficiency symptoms.

Mobility of element determines deficiency symptoms
Deficiency symptoms in older tissues
Deficiency symptoms also depend on the mobility of the element in the plant. It first appears in the older tissues.
For example, the deficiency symptoms of nitrogen, potassium, and magnesium are visible first in the senescent leaves.
In the older leaves, biomolecules containing these elements are broken down and available for mobilising to younger leaves.

Deficiency symptoms in younger tissues
Sometimes the deficiency symptoms appear first in the young tissues. If the elements are immobile, they are not transported from mature organs to younger organs.
For example, Elements like sulphur and calcium are structural components of the cell and hence are not easily released.

The deficiency symptoms are

1. Chlorosis
2. Necrosis
3. stunted plant growth
4. premature fall of leaves and buds
5. and inhibition of cell division.

Chlorosis is the loss of chlorophyll leading to yellowing in leaves. It is due to the deficiency of elements like N, K, Mg, S, Fe, Mn, Zn, and Mo.
Necrosis, or death of tissue, particularly leaf tissue. It is due to the deficiency of Ca, Mg, Cu, K.
Lack or low level of N, K, S, Mo causes inhibition of cell division.
Deficiency of elements like N, S, Mo delay flowering

Toxicity of Micronutrients
If the supply of micronutrients at a moderate decreased level shows deficiency symptoms but the moderate increase causes toxicity, i.e the excess of an element inhibits the uptake of another element.

Symptoms and other effects of Manganese toxicity

• Symptom of manganese toxicity is the appearance of brown spots surrounded by chlorotic veins.
• Manganese competes with iron and magnesium for uptake and for binding with enzymes.
• Manganese also inhibits calcium translocation in the shoot apex.
• Symptoms of manganese toxicity induce
• Deficiency symptoms of iron, magnesium, and calcium.

Mechanism of Absorption of Elements
The process of absorption occurs in two main phases-

1. Apoplast (passive). The passive movement of ions into the apoplast occurs through ion- channels and the transmembrane proteins.
2. Symplast(active) The inward movement of ions into the cells is called influx and the outward movement efflux. This movement occurs by using metabolic energy.

Translocation of Solutes

• Mineral salts are pulled up through the plant by the transpirational pull.
• Analysis of xylem sap shows the presence of mineral salts in it.
• Radioisotopic studies support the xylem transport of mineral elements.

Soil as a Reservoir of Essential Elements

• Soil consist of a variety of minerals, nitrogen-fixing bacteria, and other microbes holds water and supplies air to the roots, and acts as a matrix that stabilises the plant.
• If the amount of nutrients in the soil is decreased, it is supplied from outside as fertilizers in the form of macronutrients (N, P, K, S, etc.) and micronutrients (Cu, Zn, Fe, Mn, etc.)

Metabolism of Nitrogen
Nitrogen Cycle
Nitrogen is a constituent of amino acids, proteins, hormones, chlorophyll, and many vitamins.
Atmospheric nitrogen consists of two nitrogen atoms joined by a very strong triple covalent bond main nitrogen pools-atmospheric soil, and biomass.

1. N₂ Fixation: The process of conversion of atmospheric nitrogen (N₂) to ammonia is termed nitrogen fixation.

2. Nitrification:

• Ammonia is converted into nitrate.
• Ammonia is first oxidized to nitrite by Nitrosomonas or Nitrococcus.
• The nitrite is further oxidized to nitrate with the help of the bacterium Nitrobacter

3. Ammonification: Decomposition of organic nitrogen of dead plants and animals into ammonia is called ammonification

4. Denitrification: It is the conversion of soil nitrate into molecular N₂ by Thiobacillus and pseudomonas

Formation of nitrogen oxides

• In nature, lightning and UV provide energy to convert nitrogen to nitrogen oxides (NO, NO₂, N₂O).
• Industrial combustions, forest fires, automobile exhausts, and power generating stations are also sources of atmospheric nitrogen oxides.

Biological Nitrogen Fixation
The nitrogen-fixing microbes are free-living or symbiotic. ‘Free-living nitrogen-fixing aerobic microbes are Azotobacter, Beijernickia Rhodospirillum Bacillus Anabaena Nostoc.

Development of root nodules in soyabean

Development of root nodule sin soyabean:

• Rhizobium bacteria contact susceptible root hair, divide near it.
• Upon successful infection of the root hair cause it to curl.
• Infected thread carries the bacteria to the inner cortex. The bacteria get modified into rod-shaped bacteroids and cause inner cortical and pericycle cells to divide. Division and growth of cortical and pericycle cells lead to nodule formation.
• A mature nodule is complete with vascular tissues continuous with those of the root.

Basic steps are given below

• Rhizobium bacteria attach the root hair.
• Root hair curls.
• Infected thread carries the bacteria to the inner cortex.

The bacteria get modified into rod-shaped bacteroids and cause inner cortical and pericycle cells to divide. Division and growth of cortical and pericycle cells lead to nodule formation, d) A mature nodule is complete with vascular tissues continuous with those of the root.

Overall equation for N2 fixation
N₂ + 8e^– + 8H⁺ +16ATP → 2NH₃ + H₂ + 16ADP + 16Pt

Fate of ammonia

• At first, ammonia protonated to form NH⁴⁺.
• This ammonium ion is used to synthesise amino acid in plants

There are two ways for the synthesis of amino acids in plants

1. Reductive animation
In this, ammonium ion reacts with alpha-ketoglutaric acid and forms glutamic acid.

2. Transamination
It involves the transfer of an amino group from one amino acid to the keto group of a keto acid.
Glutamic acid is the main amino acid from which the transfer of amino groups takes place and other amino acids are formed in the presence of transaminase.

Amides

• The important amides are asparagine and aspartate.
• Amide is formed when the hydroxyl group of one amino acid is replaced by an amino group.
• Since amide contains more nitrogen than amino acids. They are transported through xylem vessels.

Kerala State Board New Syllabus Plus One Maths Notes Chapter 16 Probability.

Kerala Plus One Maths Notes Chapter 16 Probability

I. Random Experiments
An experiment is called a random experiment if it satisfies the following two conditions:

• It has more than one outcome.
• It is not possible to predict the outcome in advance.

Sample space: The set of all possible outcomes of a random experiment is called sample space. Generally denoted by S.

Event: Any subset E of a sample space S is called an event.

Types of Events:
1. Impossible event and sure event: The empty set φ and the sample space S describe the impossible event and sure event respectively.

2. Simple event: An event E having only one sample point of a sample space.

3. Compound event: An event having more than one sample point of a sample space.

Algebra of events:

1. Event ‘not A’ = A’
2. Event ‘A or B’ = A ∪ B
3. Event ‘A and B’ = A ∩ B
4. Event ‘A but not B’ = A ∩ \(\bar{B}\) = A – B

If A ∩ B = φ, then A and B are mutually exclusive events or disjoint events.

If E₁ ∪ E₂ ∪ E₃ ∪ …… ∪ E_n = S, then we say that E₁, E₂, E₃, …….., E_n are exhaustive events.

If E₁ ∪ E₂ ∪ E₃ ∪ …… ∪ E_n = S, and E_i ∩ E_j = φ, i ≠ j then we say that E₁, E₂, E₃,…….., E_n are mutually exclusive events and exhaustive events.

II. Probability of an Event
Let S is a sample space and E be an event, such that n(S) = n and n(E) = m. If each outcome is equally likely, then it follows that P(E) = \(\frac{m}{n}\).

P(Impossible event) = 0 and P(Sure event) = 1, hence 0 ≤ P(E) ≤ 1.

If A and B are any two events, then P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive events, then P(A ∪ B) = P(A) + P(B)

If A is any events, then P(A’) = 1 – P(A)

P(A ∩ \(\bar{B}\)) = P(A) – P(A ∩ B)

Kerala State Board New Syllabus Plus One Maths Notes Chapter 15 Statistics.

Kerala Plus One Maths Notes Chapter 15 Statistics

Statistics deals with data collected for specific purposes and making decisions about the data by analyzing and interpreting it.

I. Measure of Dispersion
This gives a measure of the dispersion of the observation around the measure of central tendency of the data collected.

1. Range = Maximum value – Minimum value.
2. Mean Deviation.

Where,
x_i – observations
a – Any measure of central tendency.

Grouped data:
i. Discrete frequency distribution.
ii. Continuous frequency distribution.

Where,
x_i – Observations/midpoints of class intervals
a – Any measure of central tendency.

Median class is the class in which the \(\left(\frac{N}{2}\right)^{t h}\) observation lies.

l – The lower limit of the median class.
f₀ – Cumulative frequency of the class preceding the median class.
f₁ – Frequency of the median class.
C – Width of the class interval.

3. Variance and Standard Deviations.
Standard Deviation (σ) = √Variance
Ungrouped data:

Where, x_i – observations
\(\bar{x}\) – Mean
n – number of observations

Grouped data:
i) Discrete frequency distribution.
ii) Continuous frequency distribution.

Where,
x_i – Observations/mid points of class intervals.
\(\bar{x}\) – Mean
f_i – Frequency.

Short cut method of finding variance and standard deviation:
Let A be the assumed mean and the scale be reduced to \(\frac{1}{h}\) times (h being the width of class intervals). Let the new value be y_i and prepare the required tables using y_i. i.e; y_i = \(\frac{x_{i}-A}{h}\)
Find the variance and standard deviation of y_i using the above-mentioned method, let it

II. Coefficient of Variation

The distribution having greater CV has more variability around the central value than the distribution having a smaller value of the CV.

Less the CV more consistent is the data.

For distributions with equal means, the distribution with lesser standard deviation is more consistent or less scattered.

Kerala State Board New Syllabus Plus One Maths Notes Chapter 14 Mathematical Reasoning.

Kerala Plus One Maths Notes Chapter 14 Mathematical Reasoning

I. Statement
The basic unit involved in mathematical reasoning is a mathematical sentence.

A sentence is called a mathematically acceptable statement if it is either true or false but not both. Usually denoted by small letters p, q, r, ……..

Denial of a statement is called the negation of the statement. While forming the negation of a statement, phrases like, “It is not the case” or “it is false that” are also used. The negation of a statement p is denoted by ~p.

II. Compound Statement
Many mathematical statements are obtained by combining one or more statements using some connective words like “and”, “or”, etc.

Contrapositive statement: the contrapositive of a statement p ⇒ q is the statement ~q ⇒ ~p.

Converse of a statement: Converse of a statement p ⇒ q is the statement q ⇒ p.

III. Validity of Statement
A statement is said to be valid or invalid according to it is true or false.

Kerala State Board New Syllabus Plus One Maths Notes Chapter 13 Limits and Derivatives.

Kerala Plus One Maths Notes Chapter 13 Limits and Derivatives

Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain changes.

I. Limit
Limit of a function f(x) at x = a is the behaviors of f(x) at x = a.

x → a^–: Means that ‘x’ takes values less than ‘a’ but not ‘a’.

x → a⁺: Means that ‘x’ takes values greater than ‘a’ but not ‘a’.

x → a: Read as ‘x’ tends to ‘a’, means that ‘x’ takes values very close to ‘a’ but not ‘a’.

\(\lim _{x \rightarrow a^{-}} f(x)=A\): Read as left limit of f(x) is ‘A’, means that f(x) → A as x → a^–. To evaluate the left limit we use the following substitution \(\lim _{x \rightarrow a^{-}} f(x)=\lim _{h \rightarrow 0} f(a-h)\)

\(\lim _{x \rightarrow a^{+}} f(x)=B\): Read as right limit of f(x) is ‘B’, means that f(x) → B as x → a⁺. To evaluate the left limit we use the following substitution \(\lim _{x \rightarrow a^{+}} f(x)=\lim _{h \rightarrow 0} f(a+h)\).

If left limit and right limit of f(x) at x = a are equal, then we say that the limit of the function f(x) exists at x = a and is denoted
by lim \(\lim _{x \rightarrow a} f(x)\). Otherwise we say that \(\lim _{x \rightarrow a} f(x)\) does not exist.

II. Evaluation Methods

1. Direct substitution method
2. Factorisation method
3. Rationalisation method
4. Using standard results.

III. Algebra of Limits:
For functions f and g the following holds;

IV. Standard Results

\(\lim _{x \rightarrow a} k=k\), where k is constant.

\(\lim _{x \rightarrow a} f(x)=f(a)\), if f(x) is a polynomial function.

1. \(=\frac{0}{0}\), if possible we can factorise the numerator and denominator and then, cancel the common factors and again put x = a. This factorization method is not possible in all cases so we are studying some standard limits.

V. Derivatives
A derivative of f at a: Suppose f is a real-valued function and a is a point in its domain of definition. The derivative of f at a is defined by \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)
Provided this limit exists. A derivative of f (x) at a is denoted by f'(a).
Derivative of f at x. Suppose f is a real-valued function, the function defined by \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)

Wherever this limit exists is defined as the derivative of f at x and is denoted by f”(x) |\(\frac{d y}{d x}\)| |y₁| y’. This definition of derivative is also called the first principle of the derivative.

VI. Algebra of Derivatives
For functions f and g are differentiable following holds;

VII. Standard Results

Kerala State Board New Syllabus Plus One Maths Notes Chapter 12 Introduction to Three Dimensional Geometry.

Kerala Plus One Maths Notes Chapter 12 Introduction to Three Dimensional Geometry

Introduction
To refer to a point in space we require a third axis (say z-axis) which leads to the concept of three-dimensional geometry. In this chapter, we study the basic concept of geometry in three-dimensional space.

I. Octant
Consider three mutually perpendicular planes meet at a point O. Let these three planes intercept along three lines XOX’, YOY’ and ZOZ’ called the x-axis, y-axis, and z-axis respectively. These three planes divide the entire space into 8 compartments called Octants. These octants could be named as XOYZ, XOYZ’, XOYZ, X’OYZ, XOY’Z’, X’OYZ, X’OYZ’, X’OYZ’.

Distance between two points: The distance between the points (x1, y1, z1) and (x2, y2, z2) is

Section formula:
1. Internal: The coordinate of the point R which divides the line segment joining the points (x1, y1, z1) and (x2, y2, z2) internally in the ratio l : m is

2. External: The coordinate of the point R which divides the line segment joining the points (x1, y1, z1) and (x2, y2, z2) externally in the ratio l : m is

3. Midpoint: The coordinate of the point R which is the midpoint of the line segment joining the points (x1, y1, z1) and (x2, y2, z2) is

4. Centroid: The coordinate of the centroid of a triangle whose vertices are given by the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is

Kerala State Board New Syllabus Plus One Maths Notes Chapter 11 Conic Sections.

Kerala Plus One Maths Notes Chapter 11 Conic Sections

I. Circle
A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. The fixed point is the centre and the fixed distance is the radius.
Equation of a circle with centre origin and radius r is x² + y² = r².

Equation of a circle with centre (h, k) and radius r is (x – h)² + (y – k)² = r².

General form of the equation of a circle is x² + y² + 2gx + 2fy + c = 0 with centre (-g, -f) and radius \(\sqrt{g^{2}+f^{2}-c}\).

II. Conic
A conic is the set of all points in a plane which moves so that the distance from a fixed point is in a constant ratio to its distance from a fixed-line. The fixed point is the focus and fixed line is directrix and the constant ratio is eccentricity, denoted by ‘e’.

III. Parabola (e = 1)

y² = 4ax

Vertex: (0, 0)
Focus(S): (a, 0)
Length of Latusrectum: (LL’) = 4a
Equation of directrix (DD’) is x = -a

y² = -4ax

Vertex: (0, 0)
Focus(S): (-a, 0)
Length of Latusrectum (LL’) = 4a
Equation of directrix (DD’) is x = a

x² = 4ay

Vertex: (0, 0)
Focus(S): (0, a)
Length of Latusrectum (LL’) = 4a
Equation of directrix (DD’) is y = -a

x² = -4ay

Vertex: (0, 0)
Focus(S): (0, -a)
Length of Latusrectum (LL’) = 4a
Equation of directrix (DD’) is y = a

IV. Ellipse (e < 1)

\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), a > b

1. Eccentricity, e = \(\frac{\sqrt{a^{2}-b^{2}}}{a}\)
(ae)² = a² – b² ⇒ c² = a² – b²
2. b² = a²(1 – e²)
3. Length of Latusrectum (LL’) = \(\frac{2 b^{2}}{a}\)
4. Focii, S(ae, 0) and S'(-ae, 0) or S(c, 0), S'(-c, 0)
5. Centre (0, 0)
6. Vertices A(a, 0) and A'(-a, 0)
7. Equation of directrix (DD’) is x = \(\frac{a}{e}\) and x = \(-\frac{a}{e}\)
8. Length of major axis (AA’) = 2a
9. Length of minor axis'(BB’) = 2b

\(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), a > b

1. Eccentricity, e = \(\frac{\sqrt{a^{2}-b^{2}}}{a}\)
(ae)² = a² – b² ⇒ c² = a² – b²
2. b² = a²(1 – e²)
3. Length of Latus rectum (LL’) = \(\frac{2 b^{2}}{a}\)
4. Focii, S(0, ae) and S'(0, -ae) or S(0, c), S'(0, -c)
5. Centre (0, 0)
6. Vertices A(0, a) and A'(0, -a)
7. Equation of directrix (DD’) is y = \(\frac{a}{e}\) and y = \(-\frac{a}{e}\)
8. Length of major axis (AA’) = 2a
9. Length of minor axis (BB’) = 2b

V. Hyperbola (e > 1)

\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)

1. Eccentricity, e = \(\frac{\sqrt{a^{2}+b^{2}}}{a}\)
(ae)² = a² + b² ⇒ c² = a² + b²
2. b² = a²(e² – 1)
3. Length of Latus rectum (LL’) = \(\frac{2 b^{2}}{a}\)
4. Focii, S(ae, 0) and S'(-ae, 0) or S(c, 0), S'(-c, 0)
5. Centre (0, 0)
6. Vertices A(a, 0) and A'(-a, 0)
7. Equation of directrix (DD’) is x = \(\frac{a}{e}\) and x = \(-\frac{a}{e}\)

\(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\)

1. Eccentricity, e = \(\frac{\sqrt{a^{2}+b^{2}}}{a}\)
(ae)² = a² + b² ⇒ c² = a² + b²
2. b² = a²(e² – 1)
3. Length of Latus rectum (LL’) = \(\frac{2 b^{2}}{a}\)
4. Focii, S(0, ae) and S'(0, -ae) or S(0, c), S'(0, -c)
5. Centre (0, 0)
6. Vertices A(0, a) anti A'(0, -a)
7. Equation of directrix (DD’) is y = \(\frac{a}{e}\) and y = \(-\frac{a}{e}\)

Kerala State Board New Syllabus Plus One Maths Notes Chapter 10 Straight Lines.

Kerala Plus One Maths Notes Chapter 10 Straight Lines

I. Slope of Line
The slope of a line is the ‘tan’ of the angle the line makes with the positive direction of the x-axis. If θ is the angle then, slope = tan θ.

The slope of the x-axis is zero and that of the y-axis is not defined.

Parallel lines have the same slope.

The product of the slopes of perpendicular lines is -1.

The slope is positive if θ < 90°. The slope is negative if θ > 90°.

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

If three points A, B, and C are collinear, then AB and BC have the same slope.

If m₁ and m₂ be slopes of two lines then, θ the angle between is given by tan θ = \(\left|\frac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|\), 1 + m₁m₂ ≠ 0

II. Equation of a Line
Equation of x-axis is y = 0.

Equation of y-axis is x = 0.

The equation of a horizontal line is y = a. If ‘a’ is positive then the line is above the x-axis and if negative it will be below the x-axis.

The equation of a vertical line is x = a. If ‘a’ is positive then the line is to the right of the x-axis and if negative it will be to the left of the x-axis.

Point-slope form: y – y₁ = m(x – x₁), where ‘m’ is the slope and (x₁, y₁) is a point on the line.

Two-Point form:
y – y₁ = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) (x – x₁) where (x₁, y₁) and(x₂, y₂) are two point on the line.

Slope intercept form:
1. y = mx + c, where ‘m’ is the slope and ‘c’ is the y-intercept.
2. y = m(x – d), where ‘m’ is the slope and ‘d’ is the x-intercept.

Intercept form: \(\frac{x}{a}+\frac{y}{b}=1\) = 1, where ‘a’ and ‘b‘ are x and y intercept respectively.

Normal form: x cos θ + y sin θ = p, where ‘p’ is the length of the normal from the origin to the line and ‘θ’ is the angle the normal makes with the positive direction of the x-axis.

General equation of a Line: ax + by + c = 0, where a, b and c are real constants.
1. Slope of the line ax + by + c = 0 is \(-\frac{a}{b}\)

2. Parallel lines differ in constant term, i.e; a line parallel to ax + by + c = 0 is ax + by + k = 0.

3. A line perpendicular to ax + by + c = 0 is bx – ay + k = 0.

4. The equation of the family of lines passing through the intersection of the lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is of the form a₁x + b₁y + c₁ + k(a₂x + b₂y + c₂) = 0.

5. The perpendicular distance of a point (x1, y1) from the line ax + by + c = 0 is \(\left|\frac{a x_{1}+b y_{1}+c}{\sqrt{a^{2}+b^{2}}}\right|\)

6. The distance between the parallel lines ax + by + c = 0 and ax + by + k = 0 is \(\left|\frac{c-k}{\sqrt{a^{2}+b^{2}}}\right|\)

7. Normal form of the equation ax + by + c = 0 is x cos θ + y sin θ = p;
Where cos θ = \(\pm \frac{a}{\sqrt{a^{2}+b^{2}}}\); sin θ = \(\pm \frac{b}{\sqrt{a^{2}+b^{2}}}\) and p = \(\pm \frac{c}{\sqrt{a^{2}+b^{2}}}\)

Proper choice of signs is made so that p should be positive.

III. Shifting of Origin
An equation corresponding to a set of points with reference to a system of coordinate axes by shifting the origin is shifted to a new point is called a translation of axes.

Let us take a point P (x, y) referred to the axes OX and OY. Let (h, k) be the coordinates of origin and P(X, Y) be the coordinate of P(x, y) with respect to the new axis. Then, the transformation relation between the old coordinates (x, y) and the new coordinates (X, Y) are given by X = x + h and Y = y + k.

Kerala State Board New Syllabus Plus One Maths Notes Chapter 7 Permutation and Combinations.

Kerala Plus One Maths Notes Chapter 7 Permutation and Combinations

I. Fundamental Principle of Counting
If an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total number of occurrences of the events in the given order is m × n.

II. Permutation
A permutation is the arrangement of some or all of a number of different objects.

Factorial notation: The notation n! represents the product of first n natural numbers,
ie; n! = n(n – 1 )(n – 2) ….. 3.2.1
1. 1! = 1
2. 0! = 1

The number of permutation of ‘n’ different objects taken ‘r’ at a time, where the objects do not repeat is n(n – 1)(n – 2)……(n – r + 1) which is denoted by ⁿP_r.

The number of permutation of ‘n’ different objects taken ‘r’ at a time, where repetition is allowed is n^r.

Permutation when all the objects are not distinct.
1. The number of permutations of ‘n’ objects, where ‘p’ objects are of the same kind and rest all different = \(\frac{n !}{p !}\)

2. The number of permutations of ‘n’ objects, where ‘p1’ objects are of one kind, ‘p2’ objects are of the second kind, …….., ‘p_k‘ objects are of a kth kind and rest all different = \(\frac{n !}{p_{1} ! p_{2} ! \ldots p_{k} !}\)

III. Combinations
A combination is a selection of some or all of a number of different objects (the order of selection is not important). The number of selection of ‘n’ things taken ‘r’ at a time is ⁿC_r.

Kerala State Board New Syllabus Plus One Maths Notes Chapter 5 Complex Numbers and Quadratic Equations.

Kerala Plus One Maths Notes Chapter 5 Complex Numbers and Quadratic Equations

we have studied linear equations in one and two variables and quadratic equations in one variable. We have seen that the equation x² + 1 = 0 has no real solution since the root of a negative number does not exist in a real number. So, we need to extend the real number system to a larger number system to accommodate such numbers.

I. Complex Numbers
A number of the form a + ib, where a and b are real numbers and i = √-1.
Usually, a complex number is denoted by z, a is the real part of z denoted by Re(z) and b is the imaginary part of z denoted by Im(z).

II. Algebra of Complex Numbers

Addition: Let z₁ = a + ib and z₂ = c + id be two complex numbers. Then the sum z₁ + z₂ is obtained by adding the real and imaginary parts.

1. z₁ + z₂ = z₂ + z₁, commutative.
2. z₁ + (z₂ + z₃) = (z₁ + z₂) + z₃, associative.
3. 0 + i0 is the identity element.
4. -z is the inverse of z.

Multiplication: Let z₁ = a + ib and z₂ = c + id be two complex numbers.
Then the product z₁z₂ is defined as follows:
z₁z₂ = (ac – bd) + i(ad + bc).

1. z₁z₂ = z₂z₁, commutative.
2. z₁(z₂z₃) = (z₁z₂)z₃, associative.
3. 1 + i0 is the identity element.
4. \(\frac{1}{z}\) is the inverse of z.
5. z₁(z₂ + z₃) = z₁z₂ + z₁z₃, distributive law.

Power of ‘i’: i³ = -i, i⁴ = 1
In general i^4k = 1, i^4k+1 = i, i^4k+2 = -1, i^4k+3 = -i

Identities:

The Modulus and Conjugate of a complex number:
Consider a complex number z = a + ib . Then, the conjugate of z is denoted by \(\bar{z}\), defined as \(\bar{z}\) = a – ib and the modulus of z is denoted by |z|, defined as \(\sqrt{a^{2}+b^{2}}\).

Properties:

III. Representation of Complex Number

Argand Plane:

A complex number z = a + ib which corresponds to the ordered pair (a, b) can be represented geometrically as the unique point P(a, b) in the XY-plane, where the real part is taken along the x-axis and the imaginary part along the y-axis. Such a plane is called the Argand Plane or Complex plane.

Polar Form:

Let the point P represent the non-zero complex number z = x + iy. Let the directed line segment OP be of length ‘r’ and be the angle which OP makes with the positive direction of the x-axis. Then, P is determined by the unique ordered pair of a real number (r, θ) called polar coordinate of the point P, where x = r cos θ, y = r sin θ and therefore the polar form of z can be represented as z = r(cos θ + i sin θ).
The principle argument of z is value ‘θ’ such that -x ≤ θ ≤ π, denoted by arg z.
To find the principle argument, we find tan α = |\(\frac{y}{x}\)|, 0 ≤ α ≤ \(\frac{\pi}{2}\)

Kerala State Board New Syllabus Plus One Maths Notes Chapter 4 Principle of Mathematical Induction.

Kerala Plus One Maths Notes Chapter 4 Principle of Mathematical Induction

Induction means the generalization from a particular case or facts. In contrast to deductive reasoning, inductive depends on working with each case and developing a conjecture by observing incidences till we have observed each and every case. In algebra or in another discipline of mathematics, there are certain results or statements that are formulated in terms of n, where n is a positive integer. To such statements, the well-suited principle that is based on the specific technique is known as the principle of mathematical induction.

The Principle of Mathematical Induction:
Suppose there is a statement P(n) involving the natural number n such that

1. The statement is true for n = 1, i.e; P(1) is true, and

2. If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e; the truth of P(k) implies the truth of P(k+1). Then, P(n) is true for all natural numbers n.

Kerala State Board New Syllabus Plus One Maths Notes Chapter 3 Trigonometric Functions.

Kerala Plus One Maths Notes Chapter 3 Trigonometric Functions

I. Angles
The measure of an angle is the amount of rotation performed to get the terminal side from the initial side.

1. Degree measure: If a rotation from the initial side to terminal side is \(\left(\frac{1}{360}\right)^{t h}\) of a revolution, the angle is said to have a measure of one degree, written as 1°. 1° = 60′ and f = 60″.

2. Radian measure: An angle subtended at the center by an arc of length 1 unit in a unit circle is said to be of 1 radian. Radian measure is a real number corresponding to degree measure.

180° = π radians

Radian measure = \(\frac{\pi}{180}\) × Degree measure

Degree measure = \(\frac{180}{\pi}\) × Radian measure

l = rθ, where l = arc length, r = radius of the circle and θ = angle in radian measure.

II. Trigonometric Function
Consider a unit circle with centre at the origin of the coordinate axis.
Let P (a, b) be any point on the circle which makes an angle θ° with the x-axis. Let x be the corresponding radian measure of the angle θ°, i.e; x is the arc length corresponding to θ°.

From the ∆OMP’m the figure we get;
sin θ = sin x = \(\frac{b}{1}\) = b and cos θ = cos x = \(\frac{b}{1}\) = a
This means that for each real value of x we get corresponding unique ‘sin’ and ‘cosine’ value which is also real. Hence we can define the six trigonometric functions as follows.

1. f : R → [-1, 1] defined by f(x) = sin x

2. f : R → [-1, 1] defined by f(x) = cos x

3. f : R – {nπ, n ∈ Z} → R – (-1, 1) defined by f(x) = \(\frac{1}{\sin x}\) = cosec x

4. f : R – {(2n + 1) \(\frac{\pi}{2}\)} → R – (-1, 1) defined by f(x) = \(\frac{1}{\cos x}\) = sec x

5. f : R – {(2n + 1)π, n ∈ Z} → R defined by f(x) = \(\frac{\sin x}{\cos x}\) = tan x

6. f : R – {nπ, n ∈ Z} → R defined by f(x) = \(\frac{\cos x}{\sin x}\) = cot x

Sign of trigonometric functions in different quadrants;

For odd multiple of \(\frac{\pi}{2}\) trignometric functions changes as given below.
sin → cos
cos → sin
sec → cosec
cosec → sec
tan → cot
cot → tan

The value of trigonometric functions for some specific angles;

III. Compound Angle Formula

sin(x + y) = sin x cos y + cos x sin y

sin(x – y) = sin x cos y – cos x sin y

cos(x + y) = cos x cos y – sin x sin y

cos(x – y) = cos x cos y + sin x sin y

sin(x + y) sin(x – y) = sin² x – sin² y = cos² x – cos² y

cos(x + y) cos(x – y) = cos² x – sin² y

IV. Multiple Angle Formula

cos2x = cos² x – sin² x
= 1 – 2sin² x
= 2 cos² x – 1
= \(\frac{1-\tan ^{2} x}{1+\tan ^{2} x}\)

V. Sub-Multiple Angle Formula

VI. Sum Formula

VII. Product Formula

2 sin x cos y = sin(x + y) + sin(x – y)

2 cos x sin y = sin(x + y) – sin(x – y)

2 cos x cos y = cos(x + y) + cos(x – y)

2 sin x sin y = cos(x – y) – cos(x + y)

VIII. Solution of Trigonometric Equations

sin x = 0 gives x = nπ, where n ∈ Z

cos x = 0 gives x = (2n + 1)π, where n ∈ Z

tanx = 0 gives x = nπ, where n ∈ Z

sin x = sin y ⇒ x = nπ + (-1)ⁿ y, where n ∈ Z

cos x = cos y ⇒ x = 2nπ ± y, where n ∈ Z

tan x = tan y ⇒ x = nπ + y, where n ∈ Z

Principal solution is the solution which lies in the interval 0 ≤ x ≤ 2π.

IX. Sine and Cosine formulae

Let ABC be a triangle. By angle A we mean the angle between the sides AB and AC which lies between 0° and 180°. The angles B and C are similarly defined. The sides AB, BC, and CA opposite to the vertices C, A, and B will be denoted by c, a, and b, respectively.

Theorem 1 (sine formula): In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC
\(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\)

Theorem 2 (Cosine formulae): Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B, and C, respectively, then
a² = b² + c² – 2bc cos A
b² = c² + a² – 2ca cos B
c² = a² + b² – 2ab cos C

A convenient form of the cosine formulae, when angles are to be found are as follows: