Exponents, Surds, Polynomials, Mensuration, Science, and Reasoning Notes

EXPONENTS AND LAWS OF INDICES

If $a$ is a non-zero rational number and $n$ is a natural number, then the product $a \times a \times a \times \dots$ up to $n$ times is denoted by $a^n$ and is read as 'a' raised to the power 'n'. In this expression, the rational number $a$ is called the base and the natural number $n$ is known as the exponent. The form $a^n$ is the exponential representation of the repeated multiplication. For any non-zero rational number, the following specific conditions apply: $a^1 = a$ and $a^0 = 1$.

If $a$ and $b$ are non-zero rational numbers and $m$ and $n$ are natural numbers, the fundamental laws of exponents include:

1. Multiplication with Same Base: $a^m \times a^n = a^{m+n}$
2. Division with Same Base: $a^m \div a^n = a^{m-n}$, provided $m > n$
3. Power of a Power: $(a^m)^n = a^{mn} = (a^n)^m$
4. Power of a Product: $(a \times b)^n = a^n \times b^n$
5. Power of a Quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
6. Negative Exponent Principle: $a^{-m} = \frac{1}{a^m}$

Example Case on Comparison: To determine if $3^4$ or $4^3$ is greater, calculate their values. $3^4 = 3 \times 3 \times 3 \times 3 = 81$. $4^3 = 4 \times 4 \times 4 = 64$. Since $81 > 64$, then $3^4 > 4^3$. Similarly, for $8^2$ and $2^8$, we have $8^2 = 64$ and $2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256$. Thus, $2^8 > 8^2$.

Solving for Unknowns: To find $x$ if $(\frac{2}{3})^5 \times (\frac{2}{3})^7 = (\frac{2}{3})^{4x}$, use the law of multiplication. $(\frac{2}{3})^{5+7} = (\frac{2}{3})^{4x}$, which implies $12 = 4x$, thus $x = 3$.

SURDS AND THEIR OPERATIONS

In mathematics, a surd refers to a number that does not have a rational root. While numbers like $\sqrt{25}$ simplify to $5$, values such as $\sqrt{2}$ or $\sqrt{20}$ do not result in clean rational integers and are termed surds.

Ordering Surds by Magnitude: To arrange surds like $\sqrt[4]{6}$, $\sqrt[3]{7}$, and $\sqrt{5}$ in order, one must convert them to have a common exponent index. This is done by finding the LCM of the roots ($4, 3, 2$), which is $12$. Express each as a power of $\frac{1}{12}$:

• $\sqrt[4]{6} = 6^{1/4} = 6^{3/12} = (6^3)^{1/12} = 216^{1/12}$
• $\sqrt[3]{7} = 7^{1/3} = 7^{4/12} = (7^4)^{1/12} = 2401^{1/12}$
• $\sqrt{5} = 5^{1/2} = 5^{6/12} = (5^6)^{1/12} = 15625^{1/12}$ Based on the internal values, the ascending order is $\sqrt[4]{6} < \sqrt[3]{7} < \sqrt{5}$.

Rationalization: This is the process of removing surds from the denominator of a fraction. To rationalize $\frac{2+\sqrt{5}}{2-\sqrt{5}}$, multiply both the numerator and denominator by the conjugate of the denominator, which is $2+\sqrt{5}$. The calculation results in: $\frac{(2+\sqrt{5})^2}{(2)^2 - (\sqrt{5})^2} = \frac{4 + 4\sqrt{5} + 5}{4 - 5} = \frac{9 + 4\sqrt{5}}{-1} = -(9 + 4\sqrt{5})$.

Finding Square Roots of Surds: To find the square root of $7+4\sqrt{3}$, assume $\sqrt{7+4\sqrt{3}} = \sqrt{x} + \sqrt{y}$. Squaring both sides yields $7+4\sqrt{3} = x + y + 2\sqrt{xy}$. By comparison, $x + y = 7$ and $2\sqrt{xy} = 4\sqrt{3}$, which means $xy = (2\sqrt{3})^2 = 12$. Solving the simultaneous equations gives $x = 4$ and $y = 3$. The root is $\sqrt{4} + \sqrt{3} = 2 + \sqrt{3}$.

POLYNOMIALS AND ALGEBRAIC IDENTITIES

An identity is an equality true for all values of the variables involved. Essential algebraic identities include:

• Identity I: $a^2 - b^2 = (a-b)(a+b)$
• Identity II: $(a+b)^2 = a^2 + 2ab + b^2$
• Identity III: $(a-b)^2 = a^2 - 2ab + b^2$
• Identity IV: $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
• Identity V: $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$
• Identity VI: $(a-b)^3 = a^3 - b^3 - 3ab(a-b)$
• Identity VII: $a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca)$
• Special Case: If $a+b+c = 0$, then $a^3 + b^3 + c^3 = 3abc$.

Division Algorithm: If $p(x)$ and $g(x)$ are two polynomials where $g(x) \neq 0$, there exist unique polynomials $q(x)$ and $r(x)$ such that $p(x) = g(x) \times q(x) + r(x)$, where either $r(x) = 0$ or the degree of $r(x) <$ degree of $g(x)$.

Remainder Theorem and Factor Theorem:

• The Remainder Theorem states that if a polynomial $p(x)$ of degree $\geq 1$ is divided by $(x-a)$, the remainder is $p(a)$.
• The Factor Theorem states that if $p(a) = 0$, then $(x-a)$ is a factor of $p(x)$. Conversely, if $(x-a)$ is a factor, then $p(a) = 0$.

Factorization Methods:

1. Common Factor: $am + bm + cm = m(a + b + c)$.
2. Perfect Square: $25x^2 + 20xy + 4y^2 = (5x+2y)^2$.
3. Trinomial Factorization: Based on $(x+a)(x+b) = x^2 + (a+b)x + ab$. To factorize $x^2 - 7x + 12$, find two numbers whose sum is $-7$ and product is $12$. These are $-3$ and $-4$, giving $(x-3)(x-4)$.

MENSURATION OF PLANE AND SOLID FIGURES

Plane figures are two-dimensional geometries. Key area and perimeter formulas include:

• Triangle: Area $= \frac{1}{2} \times \text{Base} \times \text{Altitude}$. For an equilateral triangle, Area $= \frac{\sqrt{3}}{4} \times (\text{side})^2$. For an isosceles triangle with base $b$ and equal sides $a$, Area $= \frac{b}{4} \sqrt{4a^2 - b^2}$.
• Rectangle: Area $= \text{Length} \times \text{Breadth}$. Diagonal $= \sqrt{l^2 + b^2}$.
• Square: Area $= a^2$. Diagonal $= a\sqrt{2}$.
• Trapezium: Area $= \frac{1}{2} \times (\text{sum of parallel sides}) \times h$.
• Circle: Circumference $= 2\pi r$. Area $= \pi r^2$.
• Rhombus: Area $= \frac{1}{2} \times d_1 \times d_2$. Side $= \frac{1}{2} \sqrt{d_1^2 + d_2^2}$.

Solid figures or three-dimensional shapes include:

• Cuboid: Total Surface Area (TSA) $= 2(lb + bh + hl)$. Lateral Surface Area (LSA) $= 2h(l + b)$. Volume $= l \times b \times h$.
• Cube: TSA $= 6a^2$. LSA $= 4a^2$. Volume $= a^3$.
• Cylinder: Curved Surface Area (CSA) $= 2\pi rh$. TSA $= 2\pi r(r+h)$. Volume $= \pi r^2 h$.
• Cone: CSA $= \pi rl$. TSA $= \pi r(l+r)$. Volume $= \frac{1}{3} \pi r^2 h$. Slant height $l = \sqrt{r^2 + h^2}$.
• Sphere: Surface Area $= 4\pi r^2$. Volume $= \frac{4}{3} \pi r^3$.
• Hemisphere: CSA $= 2\pi r^2$. TSA $= 3\pi r^2$. Volume $= \frac{2}{3} \pi r^3$.

Practical Applications: If a cube with a side of $9\,cm$ is immersed in water in a rectangular vessel with a base area of $15\,cm \times 12\,cm$, the rise in water level ($x$) is found by equating the volume of the cube to the volume of water displaced: $9^3 = 15 \times 12 \times x$, resulting in $x = \frac{729}{180} = 4.05\,cm$.

PHYSICS: LIGHT, SHADOWS, AND REFLECTION

Light is a form of electromagnetic energy responsible for the sensation of vision. Objects that produce their own light are Luminous (Sun, Stars), while those that reflect light to become visible are Non-luminous (Moon, Table).

Transmission of Light through Media:

• Transparent objects allow light to pass through fully (Air, Glass).
• Translucent objects transmit light partially (Paraffin wax, Ground glass).
• Opaque objects do not allow light to pass at all (Wood, Metal).

Rectilinear Propagation of Light: This principle states that light travels in straight lines. This is demonstrated by the ability to see a candle flame through a straight pipe but not a bent one. The speed of light in vacuum is approximately $3 \times 10^8\,m/s$.

Shadow Formation: A shadow occurs when an opaque object blocks the path of light. Essentials for a shadow include a light source, an opaque object, and a screen. The size of the shadow depends on the distance between the source, the object, and the screen. Smaller/paraxial rays cast sharp shadows, while distant sources or parallel sunlight cast shadows equivalent to the object's profile.

Reflection of Light: When light hits a surface and bounces back into the same medium, it is called reflection. Key terms include:

• Incident Ray: The ray approaching the mirror.
• Reflected Ray: The ray bouncing off.
• Normal: Perpendicular line at the point of incidence.
• Laws of Reflection: 1. Angle of incidence ($i$) is equal to the angle of reflection ($r$). 2. The incident ray, reflected ray, and the normal all lie in the same plane.

Plane Mirrors: Images formed are virtual, erect, of the same size as the object, and laterally inverted (left appears as right). To see a full image of a person, the mirror must be at least half the person's height.

Devices Using Mirrors:

• Periscope: Uses two plane mirrors at $45^{\circ}$ to see over obstacles.
• Kaleidoscope: Uses three plane mirrors at $60^{\circ}$ to create symmetrical patterns via multiple reflections.

PHYSICS: BASICS OF ELECTRICITY

Conductors allow electric charges to flow through them easily (Metals like Copper, Silver), whereas Insulators resist the flow of current (Rubber, Wood, Plastic).

Electric Cells and Batteries: A cell converts chemical energy into electrical energy.

• Primary Cells: Non-rechargeable (Dry cell, Voltaic cell).
• Secondary Cells: Rechargeable (Lead-acid accumulator, Lithium-ion).
• Battery: A combination of two or more cells.

Electric Bulb: Invented by Thomas Alva Edison, it contains a thin filament made of tungsten, chosen for its high melting point. The bulb is typically filled with a mixture of $90\%$ Argon and $10\%$ Nitrogen to prevent oxidation and extend filament life.

Electric Circuits: A complete path for electricity to flow is a circuit.

• Closed Circuit: Electricity flows continuously.
• Open Circuit: The path is broken, and no current flows.
• Symbols: A long line and a short line represent a cell (positive and negative terminals respectively); a circle with a cross represents a bulb.

CHEMISTRY: MATTER AND ITS CLASSIFICATION

Matter is anything that occupies space and has mass. It exists in three primary states:

1. Solids: Tightly packed particles, fixed shape and volume, high density, and low compressibility. Particles only vibrate in place.
2. Liquids: Less tightly packed, fixed volume but no fixed shape (takes the shape of container), and higher diffusion rate than solids.
3. Gases: Particles are far apart, no fixed shape or volume, highly compressible, and possess high kinetic energy.

Chemical Classification:

• Pure Substances: Made of only one kind of particle (elements or compounds).
• Mixtures: Physical combination of two or more substances.     - Homogeneous: Uniform composition (Salt in water, Air).     - Heterogeneous: Non-uniform composition (Sand in water, Oil and water).

Separation Techniques:

• Magnetic Separation: Uses magnets to extract magnetic materials (Iron filings from sand).
• Sublimation: Separates substances that change directly from solid to gas (Ammonium chloride, Iodine).
• Sedimentation/Decantation: Settling heavy insoluble solids from liquid.
• Filtration: Using a porous medium (filter paper) to remove insoluble solids from a liquid.
• Distillation: Boiling a mixture and then condensing the vapor to recover the solvent (obtaining distilled water).
• Centrifugation: High-speed rotation to separate particles based on density (separating cream from milk).
• Fractional Distillation: Separating miscible liquids with different boiling points (Refining petroleum).
• Crystallization: Purifying a solid by forming crystals from a solution.
• Loading: Using Alum to speed up the sedimentation of fine particles in water.

CHEMISTRY: ATOMIC THEORIES AND ELEMENTS

Atomic Theory Progression:

• Ancient Philosophers: Maharishi Kanad proposed "Paramanu" (indivisible particles) around $500\,BC$. Democritus used the term "atoms."
• Dalton's Atomic Theory (1808): Stated that matter consists of small indivisible atoms; all atoms of an element are identical; atoms combine in simple ratios.
• Modern Atomic Theory: Atoms are divisible into subatomic particles (protons, neutrons, electrons). Atoms of the same element can have different masses (Isotopes).

Atomic Structure:

• Nucleus: Center of the atom, contains protons (positive) and neutrons (neutral). It holds the atom's mass.
• Extra-nuclear region: Electrons revolve around the nucleus in specific orbits.
• Atomic Radius: Half the distance between adjacent atoms, measured in nanometers ($1\,nm = 10^{-9}\,m$).

Chemical Symbols: Berzelius introduced the modern system of using letters. Symbols are derived from:

• Single letters (Oxygen: O, Hydrogen: H).
• Double letters (Calcium: Ca, Nickel: Ni).
• Latin names (Iron: Ferrum/Fe, Sodium: Natrium/Na, Potassium: Kalium/K).
• Named after scientists (Einsteinium: Es, Curium: Cm) or places (Germanium: Ge, Polonium: Po).

Classification of Elements:

• Metals: Lustrous, malleable, good conductors (Iron, Aluminum).
• Non-metals: Dull, brittle, poor conductors (Sulfur, Oxygen). Bromine and Mercury are the only elements liquid at room temperature (non-metal and metal respectively).
• Metalloids: Properties of both (Silicon, Germanium).
• Noble Gases: Unreactive gases (Argon, Neon).

BIOLOGY: HUMAN HEALTH AND DISEASES

Definition of Health: WHO (1948) defines health as a state of complete physical, mental, and social well-being. It is not merely the absence of disease. Community health involves improving the health of the entire population through sanitation and public services.

Distinction Between Healthy and Disease-Free: A disease-free person may not be healthy (mentally or socially), but a healthy person is inherently disease-free.

Symptoms and Signs:

• Symptom: Felt by the patient but cannot be seen by others (Headache, nausea).
• Sign: Observable change in the body (Rashes, swelling). Signs and symptoms together facilitate Diagnosis (identification of disease).

Causes of Diseases:

• Intrinsic/Internal: Genetic disorders, hormonal imbalance, or organ failure (Diabetes, Cancer, Heart Attack).
• Extrinsic/External: Microbes, unbalanced diet, pollutants, or drugs.

Types of Diseases:

• Congenital: Present since birth due to genetic defects (Sickle cell anemia).
• Acquired: Developed after birth. Can be Communicable (infectious) or Non-communicable.
• Duration-based:     - Acute: Sudden onset, short duration (Cold, Malaria).     - Chronic: Long-lasting, often persists for years (Tuberculosis, Diabetes).
• Occurrence-based:     - Endemic: Restricted to a specific area (Goitre in Himalayas).     - Epidemic: Spreads over a large area rapidly (Plague).     - Pandemic: Worldwide spread (AIDS, COVID-19).

Pathogens and Spreading:

• Protozoans: Plasmodium causes Malaria (via Anopheles mosquito); Leishmania causes Kala-azar (via sand fly).
• Viruses: HIV causes AIDS; Influenza virus causes Common Cold; Rabies virus (via dog bite).
• Bacteria: Vibrio cholerae causes Cholera; Mycobacterium causes TB.
• Helminths: Ascaris causes Ascariasis; Wuchereria causes Elephantiasis.

Treatment and Prevention:

• Treatment focuses on reducing symptoms and killing the infectious agent. Antibiotics target bacterial biochemical pathways. Antivirals are harder to produce because viruses use host cell machinery.
• Prevention: General methods include safe drinking water, vector control (killing mosquitoes), and proper waste disposal. Specific methods involve Immunization. Edward Jenner developed the first vaccine against smallpox using cowpox pus.

REASONING AND APTITUDE

Number Series: Solving for missing or wrong terms in a sequence. Patterns include:

• Arithmetic Progression: Adding/subtracting a constant difference.
• Geometric Progression: Multiplying/dividing by a constant.
• Square/Cube Series: Using powers of natural numbers or primes (e.g., $2^2, 3^2, 5^2, 7^2, 11^2$).
• Multiple Series: A combination of two different sequences interspersed.

Alphabet Series: Identifying patterns based on the numerical position of letters ($A=1, Z=26$). Patterns include skipping letters forward or backward or reversing the order of letter clusters.

Missing Terms in Figures: Involves finding the logic in a grid or diagram. Operations often involve cross-multiplication, sum of squares, or row/column-wise mathematical relations. For example, if a grid shows $6$ and $9$ in the top row and $15$ in the bottom, the operation is addition ($6+9=15$).