B Pharmacy Sem 2: Remedial Biology / Mathematics

Table of Contents

Subject : Remedial Biology / Mathematics (as applicable)

1.  If Remedial Biology:

Cell Biology & Biomolecules
Basic Genetics & Human Organ Systems Review
Plant Anatomy & Physiology
Microbiology Fundamentals

2.  If Remedial Mathematics:

Algebraic Expressions & Logarithms
Differentiation & Integration Basics
Elementary Statistics (Mean, Median, Mode)
Pharmaceutical Calculations Review

Lets See first the Remedial Biology:

Unit 1: Cell Biology & Biomolecules

This unit introduces the fundamental structural and functional aspects of the cell and the chemical building blocks (biomolecules) that underpin all cellular processes.

1.1 Cell Theory & Cell Types

Cell Theory:
1. All living organisms are composed of one or more cells.
2. The cell is the basic unit of life.
3. All cells arise from pre‑existing cells.

Prokaryotic vs. Eukaryotic Cells:

Feature	Prokaryote	Eukaryote
Nucleus	Absent; DNA in nucleoid	True nucleus with nuclear envelope
Organelles	Few membrane‑bound (ribosomes)	Numerous (mitochondria, ER, Golgi, etc.)
Size	0.1–5 µm	10–100 µm
Examples	Bacteria, Archaea	Animal, plant, fungal, protist cells

1.2 Cell Membrane & Transport Mechanisms

Fluid Mosaic Model: phospholipid bilayer with embedded proteins, cholesterol, glycoproteins.
Membrane Functions: barrier, signal transduction, cell recognition, transport.
Transport Processes:
- Passive:
  - Simple diffusion (small non‑polar molecules)
  - Facilitated diffusion (via channels/carriers)
  - Osmosis (water through aquaporins)
- Active:
  - Primary active transport (e.g., Na⁺/K⁺‑ATPase)
  - Secondary active transport (symporters/antiporters)
- Vesicular: endocytosis (phagocytosis, pinocytosis), exocytosis.

1.3 Membrane‑Bound Organelles & Their Roles

Organelle	Structure & Function
Nucleus	DNA storage; transcription; nuclear pores for traffic
Endoplasmic Reticulum	Rough ER: protein synthesis; Smooth ER: lipid synthesis, detoxification
Golgi Apparatus	Protein/glycoprotein modification, sorting, packaging
Mitochondria	ATP production via oxidative phosphorylation; apoptosis regulation
Lysosomes	Digestive enzymes degrade macromolecules, organelles
Peroxisomes	β‑oxidation of fatty acids; detoxify H₂O₂ via catalase

1.4 Cytoskeleton & Cell Division Overview

Cytoskeleton:
- Microfilaments (actin): cell shape, movement, cytokinesis
- Microtubules (tubulin): vesicle transport, mitotic spindle
- Intermediate Filaments: mechanical support
Cell Cycle Phases:
- G₁ (growth), S (DNA replication), G₂ (preparation), M (mitosis)
- Checkpoints: G₁/S, G₂/M ensure genomic integrity

1.5 Biomolecules: Building Blocks of the Cell

1.5.1 Carbohydrates

Monosaccharides (glucose, fructose), disaccharides (sucrose), polysaccharides (glycogen, cellulose).
Functions: energy source (glycolysis), structural (cell wall), cell recognition (glycocalyx).

1.5.2 Lipids

Neutral lipids: triglycerides for energy storage
Phospholipids & glycolipids: membrane structure
Steroids: cholesterol (membrane fluidity), precursors for steroid hormones
Function: energy, insulation, membrane integrity, signaling (eicosanoids).

1.5.3 Proteins

Amino acids linked by peptide bonds.
Levels of structure: primary → secondary (α‑helix, β‑sheet) → tertiary → quaternary.
Functions: enzymes (catalysis), transport (hemoglobin), signaling (receptors), structural (collagen).

1.5.4 Nucleic Acids

DNA: deoxyribonucleotides store genetic information in double helix.
RNA: ribonucleotides (mRNA, tRNA, rRNA) mediate gene expression.
Functions: replication, transcription, translation.

1.6 Clinical & Pharmaceutical Relevance

Membrane Transporters as drug targets (e.g., P‑gp efflux pumps affecting bioavailability).
Enzymes in drug metabolism (e.g., cytochrome P450 in ER).
Mitochondrial toxicity: some drugs disrupt oxidative phosphorylation (e.g., certain antivirals).
Biomolecule assays: quantification of proteins (Bradford), nucleic acids (UV absorbance) in QC.

1.7 Key Points for Exams

State the cell theory and distinguish prokaryotes from eukaryotes.
Describe the fluid mosaic model and list transport mechanisms.
Match major organelles to their functions.
Outline the cytoskeletal components and cell cycle phases.
Identify the four classes of biomolecules, their subunits, and primary roles.

Unit 2: Basic Genetics & Human Organ Systems Review

This unit provides a foundational understanding of genetic principles relevant to pharmacy (drug response, pharmacogenomics) and a concise review of major human organ systems, emphasizing their physiological functions and relevance to pharmacotherapy.

2.1 Introduction to Genetics in Pharmacy

2.1.1 DNA Structure & Function

Double Helix: two antiparallel strands of deoxyribonucleotides (A–T, G–C base pairing).
Genes: DNA segments encoding proteins or functional RNA.
Chromosomes: organized DNA–protein complexes; humans have 23 pairs.

2.1.2 Gene Expression

Transcription: DNA → pre‑mRNA in nucleus (RNA polymerase II).
RNA Processing: 5′ cap, poly‑A tail, splicing of introns → mature mRNA.
Translation: mRNA → polypeptide at ribosomes; tRNA delivers amino acids.

2.1.3 Genetic Variation & Pharmacogenomics

Polymorphisms: single nucleotide polymorphisms (SNPs) can alter drug‑metabolizing enzymes (e.g., CYP450 variants), transporters, or receptors.
Examples:
- CYP2D6 Poor Metabolizers: risk of toxicity with codeine (reduced conversion to morphine).
- TPMT Deficiency: high risk of myelosuppression with thiopurines.
Clinical Application: genotype‑guided dosing improves safety and efficacy.

2.2 Mendelian Genetics

Principle	Description
Law of Segregation	Two alleles of a gene segregate during gamete formation; each gamete carries one allele.
Law of Independent Assortment	Genes for different traits assort independently if on different chromosomes.
Dominant & Recessive	Dominant allele expresses phenotype in heterozygote; recessive requires homozygosity.
Genotype vs. Phenotype	Genotype = genetic makeup; phenotype = observable trait influenced by genotype and environment.

2.2.1 Punnett Squares & Inheritance Patterns

Autosomal Dominant: one mutant allele causes disease (e.g., familial hypercholesterolemia).
Autosomal Recessive: two mutant alleles required (e.g., cystic fibrosis).
X‑Linked: mutation on X chromosome (e.g., hemophilia A).

2.3 Human Organ Systems Overview

System	Major Functions	Pharmacological Relevance
Cardiovascular	Pumping blood, nutrient/gas transport, pressure regulation	Antihypertensives, antiarrhythmics, anticoagulants
Respiratory	Gas exchange (O₂/CO₂), acid–base balance	Bronchodilators, steroids, antibiotics for pneumonia
Digestive	Ingestion, digestion, absorption, waste elimination	Proton‑pump inhibitors, antiemetics, antidiarrheals
Nervous	Sensory input, processing, motor output	Analgesics, antiepileptics, psychotropics
Endocrine	Hormone secretion for metabolism, growth, reproduction	Insulin, thyroid hormones, corticosteroids
Renal/Urinary	Filtration, excretion, fluid/electrolyte balance	Diuretics, ACE inhibitors, electrolyte supplements
Musculoskeletal	Support, movement, mineral storage	NSAIDs, muscle relaxants, bone‑active agents (bisphosphonates)
Immune/Lymphatic	Defense against pathogens, fluid balance	Vaccines, immunosuppressants, monoclonal antibodies
Integumentary	Protection, temperature regulation, sensation	Topical corticosteroids, antibacterials, emollients
Reproductive	Gametogenesis, hormone production	Hormonal contraceptives, fertility agents

2.4 Selected System Highlights

2.4.1 Cardiovascular System

Heart: cardiac cycle, contractility; drug targets include β‑adrenergic receptors and calcium channels.
Vasculature: smooth muscle tone regulated by nitric oxide, angiotensin II; antihypertensives act here.

2.4.2 Renal System

Nephron Function: filtration, reabsorption, secretion; diuretics target specific nephron segments (e.g., loop diuretics at ascending loop).

2.4.3 Nervous System

Neurotransmitters: acetylcholine, dopamine, serotonin; psychotropic drugs modulate these pathways.
Blood–Brain Barrier: impacts CNS drug delivery; lipophilicity and transporters are key.

2.5 Integrating Genetics and Physiology

Pharmacokinetics (PK): genetic variation in metabolizing enzymes affects absorption, distribution, metabolism, and excretion.
Pharmacodynamics (PD): receptor polymorphisms alter drug response (e.g., β₂‑adrenergic receptor variants affect asthma therapy).
Personalized Medicine: combining genotype data with organ‑system knowledge to tailor therapy.

2.6 Key Points for Exams

Describe DNA → RNA → protein central dogma and relate to drug target synthesis.
Explain Mendelian inheritance laws and apply Punnett squares to basic scenarios.
Recognize major organ systems, their primary functions, and corresponding drug classes.
Illustrate how pharmacogenomic variants (e.g., CYP450) influence drug metabolism and therapy.
Integrate genetics with physiology to understand personalized pharmacotherapy.

Unit 3: Plant Anatomy & Physiology

This unit provides a foundational overview of plant structure and function, emphasizing aspects relevant to pharmacognosy and herbal drug technology.

3.1 Plant Cell & Tissue Organization

Tissue Type	Location	Characteristics & Functions
Meristematic	Shoot/root apices, cambium	Undifferentiated, actively dividing cells—growth zones
Dermal	Epidermis	Protective outer layer; cuticle reduces water loss
Ground	Cortex, pith	Parenchyma (storage/photosynthesis), collenchyma (support), sclerenchyma (rigidity)
Vascular	Xylem & phloem bundles	Xylem transports water/minerals; phloem translocates photosynthates

3.2 Root Anatomy & Function

Primary Structure:
- Epidermis with root hairs for absorption
- Cortex of parenchyma for storage
- Endodermis with Casparian strips regulating solute entry
- Stele containing xylem (star‑shaped in dicots) and phloem
Physiological Roles:
- Water and mineral uptake via apoplastic and symplastic routes
- Anchoring plant and storage of carbohydrates

3.3 Stem Anatomy & Transport

Externally: nodes (leaf attachment) and internodes; lenticels for gas exchange in woody stems
Vascular Arrangement:
- Dicots: ring of vascular bundles; cambial layer produces secondary growth
- Monocots: scattered vascular bundles; no true secondary growth
Xylem Vessels & Tracheids: thick‑walled conduits for bulk water flow
Phloem Sieve Tubes & Companion Cells: living elements for bidirectional sugar transport (pressure‑flow mechanism)

3.4 Leaf Structure & Photosynthesis

Leaf Region	Features	Role
Epidermis & Cuticle	Waxy cuticle, stomata guard cells	Minimize water loss; gas exchange regulation
Mesophyll	Palisade (chloroplast‑rich) & spongy cells	Light capture and CO₂ diffusion
Veins	Vascular bundles	Deliver water; export photosynthates

Photosynthesis (C₃ pathway):
1. Light Reactions (thylakoid membranes): H₂O → O₂; ATP & NADPH generation
2. Calvin Cycle (stroma): CO₂ fixation by Rubisco → triose phosphates → sucrose/starch
C₄ & CAM Adaptations: spatial (bundle sheath) or temporal (night fixation) separation to minimize photorespiration—important in certain medicinal plants.

3.5 Transpiration & Mineral Nutrition

Transpiration Stream:
- Cohesion‑tension mechanism pulls xylem sap upward
- Regulated by stomatal aperture (ABA hormone triggers closure under drought)

Essential Minerals & Uptake:

Macronutrient	Function	Deficiency Symptom
N (nitrate)	Amino acids, chlorophyll	Chlorosis, stunted growth
P (phosphate)	ATP, nucleic acids	Dark green leaves, delayed maturity
K (potassium)	Osmoregulation, enzyme activation	Marginal chlorosis, weak stems

Mycorrhizae: symbiosis enhancing phosphate and water uptake—relevant in cultivation of medicinal herbs.

3.6 Secondary Metabolite Production

Pathways: derived from phenylpropanoid, terpene, and alkaloid biosynthesis
Functions: defense (phytoalexins), attraction (pigments), allelopathy
Pharmaceutical Relevance: source of drugs (e.g., morphine, quinine, flavonoids)

3.7 Key Points for Exams

Distinguish meristematic, dermal, ground, and vascular tissues.
Label root and stem transverse sections; explain the role of the endodermis and cambium.
Outline leaf anatomy and the two stages of photosynthesis.
Describe the cohesion‑tension theory of transpiration and factors affecting stomatal behavior.
List major macronutrients, their roles, and deficiency symptoms in plants.
Explain how secondary metabolites are synthesized and their importance in drug discovery.

Unit 4: Microbiology Fundamentals

This unit provides an essential overview of microbiological principles, focusing on microorganisms relevant to pharmacy, sterilization, and infection control.

4.1 Classification of Microorganisms

Group	Characteristics	Examples in Pharmacy
Bacteria	Prokaryotic; diverse shapes (cocci, bacilli); cell walls (Gram +/–)	Contaminants in sterile products; probiotics (Lactobacillus)
Fungi	Eukaryotic; yeasts (unicellular) and molds (multicellular)	Antibiotic producers (Penicillium); spoilage in oral liquids
Viruses	Acellular; require host cells; DNA or RNA genome	Contamination risk in biologicals; viral vectors in gene therapy
Protozoa	Unicellular eukaryotes, often motile	Rare in pharmaceuticals; models for drug screening
Algae	Photosynthetic eukaryotes	Sources of toxins (cyanobacteria) in natural products

4.2 Microbial Cell Structure & Growth

Bacterial Cell Envelope:
- Gram‑positive: thick peptidoglycan, teichoic acids; retains crystal violet.
- Gram‑negative: thin peptidoglycan, outer membrane with lipopolysaccharide; safranin counterstain.
Fungal Structure: chitin cell wall; ergosterol in membrane (antifungal target).
Viral Structure: protein capsid, sometimes lipid envelope; classification by Baltimore system.
Microbial Growth Curve:
1. Lag Phase: adaptation, no division
2. Log (Exponential) Phase: rapid division, target for antibiotics
3. Stationary Phase: nutrient depletion, waste accumulation
4. Death Phase: decline in viable cells

4.3 Sterilization & Disinfection

Method	Mechanism	Uses in Pharmacy
Moist Heat (Autoclave)	Denatures proteins; steam under pressure	Sterile solutions, surgical instruments
Dry Heat (Oven)	Oxidation and protein denaturation	Glassware, metal instruments
Filtration	Physical removal of microbes (0.22 µm filters)	Heat-labile liquids, vaccines
Radiation (Gamma, UV)	DNA damage	Single-use plastics, surface decontamination
Chemical Sterilants	Alkylation or oxidation (e.g., ethylene oxide, glutaraldehyde)	Medical devices, tubings
Alcohols & Phenolics	Protein denaturation, membrane disruption	Surface disinfection, skin antisepsis

4.4 Aseptic Techniques & Contamination Control

Cleanroom Classifications: ISO 5–8; controlled airflow, HEPA filtration.
Aseptic Practices: gowning, glove use, laminar flow hoods, minimal movement.
Media Preparation & Sterility Testing:
- Culture media (e.g., nutrient agar, Sabouraud’s) for microbial enumeration.
- Bioburden Testing: pre-sterilization microbial load.
- Sterility Test: incubate product samples in fluid thioglycollate and soybean–casein media per pharmacopeia.

4.5 Antimicrobial Agents & Resistance

Antibiotics Production: fermentation organisms (e.g., Penicillium chrysogenum for penicillin).
Mechanisms of Action: inhibit cell wall synthesis, protein synthesis, nucleic acid synthesis, or membrane function.
Resistance Mechanisms: enzymatic degradation (β‑lactamases), efflux pumps, target modification.
Pharmacological Relevance: formulation of stable antibiotic products; stewardship to prevent resistance.

4.6 Quality Assurance in Microbiology

Environmental Monitoring: settle plates, active air sampling in production areas.
Endotoxin Testing: Limulus amebocyte lysate (LAL) for parenterals.
Microbial Limits: specified in pharmacopeias for nonsterile products (total aerobic count, absence of pathogens).
Preservative Efficacy Test: challenge tests to confirm multi-dose container safety.

4.7 Key Points for Exams

Differentiate sterilization vs. disinfection methods and their uses.
Describe the bacterial growth curve and identify the phase most susceptible to antibiotics.
List aseptic techniques critical for sterile manufacturing.
Explain antibiotic mechanisms and basic resistance strategies.
Recall quality control tests in pharmaceutical microbiology (sterility, bioburden, endotoxin, microbial limits).

Lets see the Remedial Mathematics:

Unit 1: Algebraic Expressions & Logarithms

This unit reviews manipulation of algebraic expressions and the fundamentals of logarithms, with applications to pharmaceutical contexts such as concentration calculations and pH.

1.1 Algebraic Expressions

1.1.1 Definitions

Variable: symbol (e.g., x, y) representing a number.
Constant: fixed numerical value (e.g., 5, –2).
Term: product of a constant and one or more variables raised to powers (e.g., 3x², –½ ab).
Polynomial: sum of terms with non‑negative integer exponents (e.g., 2x³ – 5x + 7).

1.1.2 Operations on Expressions

Operation	Rule	Example
Addition/Subtraction	Combine like terms (same variables and exponents)	5x² + 3x – 2x² = (5–2)x² + 3x = 3x² + 3x
Multiplication	Multiply coefficients; add exponents of like bases	(2x)(3x²) = 6x³
Division	Divide coefficients; subtract exponents	(6x³)/(2x) = 3x²
Expansion	Use distributive law: a(b + c) = ab + ac	x(2x + 5) = 2x² + 5x
Factorization	Reverse of expansion; common factor or special products (difference of squares, trinomials)	x² – 9 = (x–3)(x+3); x² + 5x + 6 = (x+2)(x+3)

1.2 Practical Applications in Pharmacy

Formulation Equations: solving for concentrations (e.g., C₁V₁ = C₂V₂ rearranged: V₁ = C₂V₂/C₁).
Batch Calculations: scaling ingredient quantities proportional to batch size.
Rate Equations: half‑life expressions for first‑order kinetics, t½ = (0.693)/k, solved algebraically for k or t½.

1.3 Introduction to Logarithms

1.3.1 Definition

The logarithm base b of a number N is the exponent x to which b must be raised to yield N:

$\log_b N = x \quad\Leftrightarrow\quad b^x = N.$ $log_{b} N = x \Leftrightarrow b^{x} = N .$
Common Logarithm: base 10 (log N).
Natural Logarithm: base e ≈ 2.718 (ln N).

1.3.2 Logarithm Properties

Property	Formula
Product Rule	$\log_b(MN) = \log_b M + \log_b N$ $log_{b} (MN) = log_{b} M + log_{b} N$
Quotient Rule	$\log_b\bigl(\tfrac{M}{N}\bigr) = \log_b M – \log_b N$ $log_{b} (N M) = log_{b} M - log_{b} N$
Power Rule	$\log_b(M^k) = k\,\log_b M$ $log_{b} (M^{k}) = k log_{b} M$
Change of Base	$\log_b N = \tfrac{\log_k N}{\log_k b}$ $log_{b} N = l o g ^{k} b l o g ^{k} N$
Inverse Relationship	$b^{\log_b N} = N,\quad \log_b(b^x) = x$ $b^{l o g b N} = N, log_{b} (b^{x}) = x$

1.4 Applications of Logarithms in Pharmacy

pH Calculation:

$\text{pH} = -\log_{10}[{\rm H}^+].$ $pH = - log_{10} [H^{+}] .$
- Example: if [H⁺] = 1.0 × 10⁻⁷ M, pH = –(–7.00) = 7.00.
First‑Order Kinetics:

$\ln\!\bigl(\tfrac{C_t}{C_0}\bigr) = -k\,t,$ $ln (C ^{0} C ^{t}) = - k t,$
solved for k or t using natural logs.
Dilution Series: plotting log concentration vs. response for calibration curves in spectrophotometry (linear relationship).

1.5 Worked Examples

Factor & Simplify:
Simplify
$4x²y – 2xy + 6x²y – 3xy$ $4 x^{2} y -2 x y + 6 x^{2} y -3 x y$ .
- Combine like terms:
  $(4+6)x²y + (–2 –3)xy = 10x²y –5xy = 5xy(2x –1)$ $(4 + 6) x^{2} y + (-2-3) x y = 10 x^{2} y -5 x y = 5 x y (2 x -1)$ .
Solve for Volume:
Given C₁V₁ = C₂V₂: 10 M stock, need 250 mL of 0.2 M.

$V_1 = \frac{0.2\times250}{10} = 5\,\text{mL}.$ $V_{1} = 100.2 \times 250 = 5 mL .$
pH Determination:
If an acid solution has [H⁺] = 3.16 × 10⁻⁴ M, find pH.

$\text{pH} = -\log(3.16\times10^{-4}) \approx 3.50.$ $pH = - log (3.16 \times 1 0^{-4}) \approx 3.50.$
First‑Order Decay:
A drug concentration decreases from 100 mg/L to 25 mg/L in 4 h. Find k.

$\ln\!\frac{25}{100} = -k\times4 \;\Longrightarrow\; -1.386 = -4k \;\Longrightarrow\; k=0.3465\,\text{h}^{-1}.$ $ln 10025 = - k \times 4 ⟹ - 1.386 = - 4 k ⟹ k = 0.3465 h^{-1} .$

1.6 Key Points for Exams

Combine like terms, expand, and factor basic polynomials.
Apply C₁V₁=C₂V₂ algebraically for dilution calculations.
Know log rules: product, quotient, and power.
Calculate pH from hydrogen ion concentration using log 10.
Solve first‑order kinetics equations using natural logarithms.

Unit 2: Differentiation & Integration Basics

This unit introduces the fundamentals of differential and integral calculus with applications in pharmaceutical rate processes, formulation kinetics, and dosage profile analysis.

2.1 Overview of Calculus in Pharmacy

Differentiation: measures how a quantity changes—critical for reaction rates, dissolution rates, and pharmacokinetic rate constants.
Integration: computes the accumulation of a quantity—used to determine total drug released over time, area under the concentration–time curve (AUC), and cumulative dose.

2.2 Differentiation

2.2.1 Definition

If
$y = f(x)$ $y = f (x)$ , the derivative
$\dfrac{dy}{dx}$ $d x d y$ or
$f'(x)$ $f^{'} (x)$ is the instantaneous rate of change of
$y$ $y$ with respect to
$x$ $x$ :

$f'(x) = \lim_{\Delta x \to 0}\frac{f(x + \Delta x) – f(x)}{\Delta x}.$ $f^{'} (x) = Δ x \to 0 lim Δ x f ( x + Δ x ) - f ( x ) .$

2.2.2 Basic Rules

Rule	Formula	Example
Constant Rule	$\dfrac{d}{dx}[C] = 0$ $d x d [C] = 0$	$\dfrac{d}{dx}[5] = 0$ $d x d [5] = 0$
Power Rule	$\dfrac{d}{dx}[x^n] = n\,x^{n-1}$ $d x d [x^{n}] = n x^{n - 1}$	$\dfrac{d}{dx}[x^3] = 3x^2$ $d x d [x^{3}] = 3 x^{2}$
Constant Multiple	$\dfrac{d}{dx}[a\,f(x)] = a\,f'(x)$ $d x d [a f (x)] = a f^{'} (x)$	$\dfrac{d}{dx}[4x^2] = 4·2x = 8x$ $d x d [4 x^{2}] = 4 \cdot 2 x = 8 x$
Sum/Difference	$\dfrac{d}{dx}[f + g] = f’ + g’$ $d x d [f + g] = f^{'} + g^{'}$	$\dfrac{d}{dx}[x^2 + x] = 2x + 1$ $d x d [x^{2} + x] = 2 x + 1$
Product Rule	$\dfrac{d}{dx}[f\,g] = f’\,g + f\,g’$ $d x d [f g] = f^{'} g + f g^{'}$	$\dfrac{d}{dx}[x·e^x] = 1·e^x + x·e^x = e^x(1 + x)$ $d x d [x \cdot e^{x}] = 1 \cdot e^{x} + x \cdot e^{x} = e^{x} (1 + x)$
Quotient Rule	$\dfrac{d}{dx}\Bigl[\tfrac{f}{g}\Bigr] = \tfrac{f’\,g – f\,g’}{g^2}$ $d x d [g f] = g ^{2} f ^{'} g - f g ^{'}$	$\dfrac{d}{dx}\bigl[\tfrac{x}{t}\bigr]$ $d x d [t x]$
Chain Rule	$\dfrac{d}{dx}[f(g(x))] = f'(g(x))·g'(x)$ $d x d [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$	$\dfrac{d}{dx}[e^{3x}] = e^{3x}·3$ $d x d [e^{3 x}] = e^{3 x} \cdot 3$

Rule

Formula

Example

Constant Rule

$\dfrac{d}{dx}[C] = 0$ $d x d [C] = 0$

$\dfrac{d}{dx}[5] = 0$ $d x d [5] = 0$

Power Rule

$\dfrac{d}{dx}[x^n] = n\,x^{n-1}$ $d x d [x^{n}] = n x^{n - 1}$

$\dfrac{d}{dx}[x^3] = 3x^2$ $d x d [x^{3}] = 3 x^{2}$

Constant Multiple

$\dfrac{d}{dx}[a\,f(x)] = a\,f'(x)$ $d x d [a f (x)] = a f^{'} (x)$

$\dfrac{d}{dx}[4x^2] = 4·2x = 8x$ $d x d [4 x^{2}] = 4 \cdot 2 x = 8 x$

Sum/Difference

$\dfrac{d}{dx}[f + g] = f’ + g’$ $d x d [f + g] = f^{'} + g^{'}$

$\dfrac{d}{dx}[x^2 + x] = 2x + 1$ $d x d [x^{2} + x] = 2 x + 1$

Product Rule

$\dfrac{d}{dx}[f\,g] = f’\,g + f\,g’$ $d x d [f g] = f^{'} g + f g^{'}$

$\dfrac{d}{dx}[x·e^x] = 1·e^x + x·e^x = e^x(1 + x)$ $d x d [x \cdot e^{x}] = 1 \cdot e^{x} + x \cdot e^{x} = e^{x} (1 + x)$

Quotient Rule

$\dfrac{d}{dx}\Bigl[\tfrac{f}{g}\Bigr] = \tfrac{f’\,g – f\,g’}{g^2}$ $d x d [g f] = g ^{2} f ^{'} g - f g ^{'}$

$\dfrac{d}{dx}\bigl[\tfrac{x}{t}\bigr]$ $d x d [t x]$

Chain Rule

$\dfrac{d}{dx}[f(g(x))] = f'(g(x))·g'(x)$ $d x d [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$

$\dfrac{d}{dx}[e^{3x}] = e^{3x}·3$ $d x d [e^{3 x}] = e^{3 x} \cdot 3$

2.2.3 Pharmaceutical Applications

Dissolution Rate: if
$C(t)$ $C (t)$ is drug concentration at time
$t$ $t$ ,
$\dfrac{dC}{dt}$ $d t d C$ gives the instantaneous dissolution rate.
Pharmacokinetics: for first‑order elimination,
$C(t) = C_0 e^{-k t}$ $C (t) = C_{0} e^{- k t}$ , differentiation yields
$\dfrac{dC}{dt} = -k C_0 e^{-k t}$ $d t d C = - k C_{0} e^{- k t}$ .
Reaction Kinetics: rate laws expressed as
$\dfrac{d[A]}{dt} = -k [A]^n$ $d t d [ A ] = - k [A]^{n}$ .

2.3 Integration

2.3.1 Definition

The integral
$\int f(x)\,dx$ $\int f (x) d x$ represents the accumulation (area under the curve) of
$f(x)$ $f (x)$ with respect to
$x$ $x$ .
Indefinite Integral:

$\int f(x)\,dx = F(x) + C,$ $\int f (x) d x = F (x) + C,$
where
$F'(x) = f(x)$ $F^{'} (x) = f (x)$ and
$C$ $C$ is the constant of integration.
Definite Integral:

$\int_{a}^{b} f(x)\,dx = F(b) – F(a),$ $\int_{a b} f (x) d x = F (b) - F (a),$
the net accumulation from
$x=a$ $x = a$ to
$x=b$ $x = b$ .

2.3.2 Basic Rules

Rule	Formula	Example
Power Rule	$\int x^n\,dx = \tfrac{x^{n+1}}{n+1} + C$ $\int x^{n} d x = n + 1 x ^{n + 1} + C$	$\int x^2\,dx = \tfrac{x^3}{3} + C$ $\int x^{2} d x = 3 x ^{3} + C$
Constant Multiple	$\int a\,f(x)\,dx = a \int f(x)\,dx$ $\int a f (x) d x = a \int f (x) d x$	$\int 5x\,dx = 5·\tfrac{x^2}{2} + C$ $\int 5 x d x = 5 \cdot 2 x ^{2} + C$
Sum/Difference	$\int [f + g]\,dx = \int f\,dx + \int g\,dx$ $\int [f + g] d x = \int f d x + \int g d x$	$\int (x + 1)\,dx = \tfrac{x^2}{2} + x + C$ $\int (x + 1) d x = 2 x ^{2} + x + C$
Definite Integral	$\int_{a}^{b} f(x)\,dx = F(b) – F(a)$ $\int_{a b} f (x) d x = F (b) - F (a)$	$\int_{0}^{1} 2x\,dx = [x^2]_0^1 = 1$ $\int_{01} 2 x d x = [x^{2}]_{01} = 1$

Rule

Formula

Example

Power Rule

$\int x^n\,dx = \tfrac{x^{n+1}}{n+1} + C$ $\int x^{n} d x = n + 1 x ^{n + 1} + C$

$\int x^2\,dx = \tfrac{x^3}{3} + C$ $\int x^{2} d x = 3 x ^{3} + C$

Constant Multiple

$\int a\,f(x)\,dx = a \int f(x)\,dx$ $\int a f (x) d x = a \int f (x) d x$

$\int 5x\,dx = 5·\tfrac{x^2}{2} + C$ $\int 5 x d x = 5 \cdot 2 x ^{2} + C$

Sum/Difference

$\int [f + g]\,dx = \int f\,dx + \int g\,dx$ $\int [f + g] d x = \int f d x + \int g d x$

$\int (x + 1)\,dx = \tfrac{x^2}{2} + x + C$ $\int (x + 1) d x = 2 x ^{2} + x + C$

Definite Integral

$\int_{a}^{b} f(x)\,dx = F(b) – F(a)$ $\int_{a b} f (x) d x = F (b) - F (a)$

$\int_{0}^{1} 2x\,dx = [x^2]_0^1 = 1$ $\int_{01} 2 x d x = [x^{2}]_{01} = 1$

2.3.3 Pharmaceutical Applications

Area Under Curve (AUC):
$\mathrm{AUC} = \int_{0}^{\infty} C(t)\,dt$ $AUC = \int_{0 \infty} C (t) d t$ quantifies total drug exposure.
Cumulative Release: if release rate
$r(t)$ $r (t)$ , cumulative amount released
$Q(t) = \int_{0}^{t} r(\tau)\,d\tau$ $Q (t) = \int_{0 t} r (τ) d τ$ .
Mass Balance: integrate concentration profiles to calculate total mass in a compartment over time.

2.4 Worked Examples

Differentiate:
$f(x) = 5x^3 – 2x + 7$ $f (x) = 5 x^{3} - 2 x + 7$ .

$f'(x) = 15x^2 – 2.$ $f^{'} (x) = 15 x^{2} - 2.$
Integrate:
$\int (4t^2 – 3)\,dt$ $\int (4 t^{2} - 3) d t$ .

$\int 4t^2\,dt = \tfrac{4t^3}{3},\quad \int -3\,dt = -3t;\quad \text{so } \tfrac{4t^3}{3} – 3t + C.$ $\int 4 t^{2} d t = 3 4 t ^{3}, \int - 3 d t = - 3 t; so 3 4 t ^{3} - 3 t + C .$
Compute AUC for
$C(t) = C_0 e^{-k t}$ $C (t) = C_{0} e^{- k t}$ from
$t=0$ $t = 0$ to
$\infty$ $\infty$ .

$\mathrm{AUC} = \int_{0}^{\infty} C_0 e^{-k t}\,dt = \frac{C_0}{k}.$ $AUC = \int_{0 \infty} C_{0} e^{- k t} d t = k C ^{0} .$

2.5 Key Points for Exams

Apply power, sum, and chain rules to differentiate typical functions.
Recognize that the derivative represents rate of change, and integral gives accumulated total.
Use definite integrals to calculate AUC and cumulative release in pharmacokinetics.
Memorize basic differentiation and integration formulas and practice simple examples.

Unit 3: Elementary Statistics (Mean, Median, Mode)

This unit introduces basic statistical measures used in pharmaceutical analysis and quality control to summarize and describe data sets: the arithmetic mean, median, and mode. You’ll learn their definitions, calculation methods, interpretation, and practical examples relevant to pharmacy.

3.1 Importance of Descriptive Statistics in Pharmacy

Quality Control: assess uniformity of tablet weights, potency assay results, content uniformity.
Experimental Analysis: summarize replicate measurements (e.g., pH readings, dissolution times).
Data Interpretation: provide quick insights into central tendency and variability before inferential analysis.

3.2 The Arithmetic Mean (Average)

3.2.1 Definition

The arithmetic mean is the sum of all observations divided by the number of observations. It represents the “center of gravity” of the data.

$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$ $x ˉ = n 1 i = 1 \sum n x_{i}$

$x_i$ $x_{i}$ : each individual value
$n$ $n$ : total number of values

3.2.2 Properties

Sensitive to every data point, including outliers.
Useful for normally distributed data.
The sum of deviations
$(x_i – \bar{x})$ $(x_{i} - x ˉ)$ equals zero.

3.2.3 Worked Example 1: Tablet Weight

Five tablets of a new formulation are weighed (mg): 198.2, 201.5, 199.8, 200.1, 199.4. Calculate the mean weight.

$\bar{x} = \frac{198.2 + 201.5 + 199.8 + 200.1 + 199.4}{5} = \frac{998.0}{5} = 199.6\;\text{mg}$ $x ˉ = 5198.2 + 201.5 + 199.8 + 200.1 + 199.4 = 5998.0 = 199.6 mg$

Thus, the average tablet weight is 199.6 mg.

3.2.4 Worked Example 2: Assay Results

Assay of active ingredient in five samples yields (% of label claim): 98.5, 99.0, 98.8, 99.2, 98.9. Mean assay:

$\bar{x} = \frac{98.5 + 99.0 + 98.8 + 99.2 + 98.9}{5} = \frac{494.4}{5} = 98.88\;\%$ $x ˉ = 598.5 + 99.0 + 98.8 + 99.2 + 98.9 = 5494.4 = 98.88 %$

Average potency = 98.88 % of label claim.

3.3 The Median

3.3.1 Definition

The median is the middle value of an ordered data set. It divides the data into two equal halves.

For odd
$n$ $n$ : median is the
$\tfrac{n+1}{2}$ $2 n + 1$ th value.
For even
$n$ $n$ : median is the average of the
$\tfrac{n}{2}$ $2 n$ th and
$\tfrac{n}{2} + 1$ $2 n + 1$ th values.

3.3.2 Properties

Resistant to outliers and skewed data.
Better measure of central tendency when data are not symmetrically distributed.

3.3.3 Worked Example 1: Dissolution Times

Eight tablets’ dissolution times (minutes) sorted: 12, 13, 14, 15, 16, 17, 18, 19.

$n=8$ $n = 8$ , even.

$\tfrac{n}{2} = 4$ $2 n = 4$ → 4th value = 15
5th value = 16

$\text{Median} = \frac{15 + 16}{2} = 15.5\;\text{minutes}$ $Median = 215 + 16 = 15.5 minutes$

3.3.4 Worked Example 2: Skewed Content Uniformity

Content uniformity results (% of label) sorted for seven units: 92, 95, 96, 97, 98, 99, 105.

$n=7$ $n = 7$ , odd → median is 4th value = 97 %.
Note the outlier (105 %) does not affect median.

3.4 The Mode

3.4.1 Definition

The mode is the value(s) that occur most frequently in a data set. A set may be:

Unimodal: one mode
Bimodal: two modes
Multimodal: more than two modes
No mode: all values occur with equal frequency

3.4.2 Properties

Applicable to nominal, ordinal, and numerical data.
Indicates the most typical or frequent observation.

3.4.3 Worked Example 1: Particle Size Distribution

Particle size measurements (µm): 50, 55, 55, 60, 60, 60, 65, 70.

Frequency: 60 µm appears 3 times → mode = 60 µm.

3.4.4 Worked Example 2: Excipients in Market Survey

A survey of 12 pharmacies on preferred diluent: Lactose (5), MCC (4), DCP (3). Mode = Lactose (most frequent choice).

3.5 Comparing Mean, Median, and Mode

Characteristic	Mean	Median	Mode
Sensitivity to Outliers	High	Low	None
Data Type	Interval/ratio	Interval/ratio	Any (including nominal)
Calculation Complexity	Moderate (sum/division)	Simple (ordering)	Simple (frequency count)
Best Use	Symmetric distributions	Skewed distributions	Categorical data

3.6 Pharmaceutical Applications

Stability Data: use median to summarize degradation times when outliers (accelerated failures) occur.
Quality Control Charts: track mean and mode of process parameters to detect shifts.
Formulation Surveys: mode identifies most commonly used excipient or packaging type.

3.7 Key Points for Exams

Mean: sum of values ÷ number of values; affected by every observation.
Median: middle value in ordered list; robust against extremes.
Mode: most frequent value; the only measure applicable to nominal data.
Selection: choose median or mode for skewed or categorical data; mean for normally distributed measurement data.
Example Calculations: practice with small data sets in contexts like tablet weights, assay results, and survey frequencies.

Unit 4: Pharmaceutical Calculations Review

This unit consolidates key calculation techniques used throughout pharmacy practice—including strength expressions, dilution, allegation, dose calculations, and rate/time computations—with detailed examples to reinforce proficiency.

4.1 Expression of Strength

4.1.1 Percent Strengths

% w/v: grams of solute per 100 mL of solution

$\%\,w/v = \frac{\text{mass of solute (g)}}{\text{volume of solution (mL)}} \times 100$ $% w / v = volume of solution (mL) mass of solute (g) \times 100$
% w/w: grams of solute per 100 g of preparation

$\%\,w/w = \frac{\text{mass of solute (g)}}{\text{mass of preparation (g)}} \times 100$ $% w / w = mass of preparation (g) mass of solute (g) \times 100$

Example 4.1: Prepare 500 mL of 2 % w/v sodium chloride.

Required NaCl =
$2 \text{g}/100 \text{mL} × 500 \text{mL}/100 = 10 \text{g}$ $2 g /100 mL \times 500 mL /100 = 10 g$ .

4.1.2 Ratio Strength

Represents one part of drug per n parts of total

$\text{Ratio} = 1 : n \quad\Longrightarrow\quad \%\,w/v = \frac{1}{n} × 100$ $Ratio = 1 : n ⟹ % w / v = n 1 \times 100$

Example 4.2: A 1 : 200 solution of dextrose.

% w/v =
$1/200 × 100 = 0.5\%$ $1/200 \times 100 = 0.5%$ ; thus, 0.5 g per 100 mL.

4.2 Dilution Calculations

4.2.1 Basic Dilution Formula

$C_1 V_1 = C_2 V_2$ $C_{1} V_{1} = C_{2} V_{2}$

$C_1, V_1$ $C_{1}, V_{1}$ : concentration and volume of stock
$C_2, V_2$ $C_{2}, V_{2}$ : desired concentration and final volume

Example 4.3: From a 5 M stock, prepare 250 mL of 0.1 M solution.

$V_1 = \frac{C_2 V_2}{C_1} = \frac{0.1 × 250}{5} = 5 \text{mL}.$ $V_{1} = C ^{1} C ^{2} V ^{2} = 50.1 \times 250 = 5 mL .$

4.2.2 Serial Dilutions

Stepwise dilutions to reach low concentrations.
Example 4.4: To achieve 1:1 000 000 from a 1:1 000 stock, perform three successive 1:100 dilutions.

4.3 Alligation

4.3.1 Alligation Medial & Alternate

Medial: find mean strength when ingredients are combined in known weights.
Alternate: determine proportions of two strengths to obtain a target strength.

Example 4.5: Mix 20 % and 5 % solutions to get 12 % final, total 100 mL.

Difference: |20 – 12| = 8; |12 – 5| = 7 → ratio 8 : 7.
Volume of 20 % =
$\tfrac{8}{15} × 100 = 53.3 \text{mL}$ $15 8 \times 100 = 53.3 mL$ , 5 % = 46.7 mL.

4.4 Dose Calculations

4.4.1 Weight‑Based Dosing

$\text{Patient dose} = \text{Dose per kg} × \text{Weight (kg)}$ $Patient dose = Dose per kg \times Weight (kg)$

Example 4.6: 5 mg/kg amoxicillin for a 20 kg child.

Dose = 5 mg/kg × 20 kg = 100 mg.

4.4.2 Body Surface Area (BSA) Dosing

Mosteller formula:

$\text{BSA} = \sqrt{\frac{\text{Wt (kg)} × \text{Ht (cm)}}{3600}}$ $BSA = 3600 Wt (kg) \times Ht (cm)$
Example 4.7: Patient 70 kg, 170 cm:
$\text{BSA} ≈ \sqrt{\tfrac{70×170}{3600}} ≈ 1.8 m^2$ $BSA \approx 3600 70 \times 170 \approx 1.8 m^{2}$ .
If drug 50 mg/m²: 50 mg/m² × 1.8 m² = 90 mg.

4.5 Flow Rate & Infusion Time

4.5.1 IV Infusion Rate

$\text{Rate (mL/h)} = \frac{\text{Total volume (mL)}}{\text{Infusion time (h)}}$ $Rate (mL/h) = Infusion time (h) Total volume (mL)$

Example 4.8: 500 mL D5W over 4 h.

Rate = 500 mL ÷ 4 h = 125 mL/h.

4.5.2 Drops per Minute

Formula:

$\text{gtt/min} = \frac{\text{Volume (mL)} × \text{Drop factor (gtt/mL)}}{\text{Time (min)}}$ $gtt/min = Time (min) Volume (mL) \times Drop factor (gtt/mL)$

Example 4.9: 200 mL infusion, 20 gtt/mL set, over 2 h (120 min).

gtt/min =
$200 × 20 ÷ 120 ≈ 33 gtt/min$ $200 \times 20 \div 120 \approx 33 g tt / min$ .

4.6 Percentage Error & Yield

4.6.1 Percentage Error

$\%\,\text{Error} = \frac{\lvert \text{Measured} – \text{True}\rvert}{\text{True}} × 100$ $% Error = True ∣ Measured - True ∣ \times 100$

Example 4.10: Tablet assay found 98 mg vs. label 100 mg.

% Error =
$\tfrac{2}{100}×100 = 2\%$ $100 2 \times 100 = 2%$ .

4.6.2 Percentage Yield

$\%\,\text{Yield} = \frac{\text{Actual mass obtained}}{\text{Theoretical mass}} × 100$ $% Yield = Theoretical mass Actual mass obtained \times 100$

Example 4.11: Theoretical 50 g granules, actual yield 45 g.

% Yield =
$45/50×100 = 90\%$ $45/50 \times 100 = 90%$ .

4.7 Key Points for Exams

Express concentrations as % w/v, % w/w, and ratio strengths.
Apply C₁V₁ = C₂V₂ for dilutions and alligation for mixing.
Calculate patient doses using weight and BSA formulas.
Determine infusion rates (mL/h, gtt/min) accurately.
Compute percentage error and yield for quality assessments.

🧠 Tip for Students

Start with Human Anatomy & Physiology II to build on your Semester 1 foundation—understanding the body’s systems is key for all subsequent pharma subjects.

🔔 Stay updated! New unit‑wise content and practice materials added weekly.

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B Pharmacy Sem 2: Remedial Biology / Mathematics

Subject : Remedial Biology / Mathematics (as applicable)

1. If Remedial Biology:

2. If Remedial Mathematics:

Lets See first the Remedial Biology:

Unit 1: Cell Biology & Biomolecules

1.1 Cell Theory & Cell Types

1.2 Cell Membrane & Transport Mechanisms

1.3 Membrane‑Bound Organelles & Their Roles

1.4 Cytoskeleton & Cell Division Overview

1.5 Biomolecules: Building Blocks of the Cell

1.5.1 Carbohydrates

1.5.2 Lipids

1.5.3 Proteins

1.5.4 Nucleic Acids

1.6 Clinical & Pharmaceutical Relevance

1.7 Key Points for Exams

Unit 2: Basic Genetics & Human Organ Systems Review

2.1 Introduction to Genetics in Pharmacy

2.1.1 DNA Structure & Function

2.1.2 Gene Expression

2.1.3 Genetic Variation & Pharmacogenomics

2.2 Mendelian Genetics

2.2.1 Punnett Squares & Inheritance Patterns

2.3 Human Organ Systems Overview

2.4 Selected System Highlights

2.4.1 Cardiovascular System

2.4.2 Renal System

2.4.3 Nervous System

2.5 Integrating Genetics and Physiology

2.6 Key Points for Exams

Unit 3: Plant Anatomy & Physiology

3.1 Plant Cell & Tissue Organization

3.2 Root Anatomy & Function

3.3 Stem Anatomy & Transport

3.4 Leaf Structure & Photosynthesis

3.5 Transpiration & Mineral Nutrition

3.6 Secondary Metabolite Production

3.7 Key Points for Exams

Unit 4: Microbiology Fundamentals

4.1 Classification of Microorganisms

4.2 Microbial Cell Structure & Growth

4.3 Sterilization & Disinfection

4.4 Aseptic Techniques & Contamination Control

4.5 Antimicrobial Agents & Resistance

4.6 Quality Assurance in Microbiology

4.7 Key Points for Exams

Lets see the Remedial Mathematics:

Unit 1: Algebraic Expressions & Logarithms

1.1 Algebraic Expressions

1.1.1 Definitions

1.1.2 Operations on Expressions

1.2 Practical Applications in Pharmacy

1.3 Introduction to Logarithms

1.3.1 Definition

1.3.2 Logarithm Properties

1.4 Applications of Logarithms in Pharmacy

1.5 Worked Examples

1.6 Key Points for Exams

Unit 2: Differentiation & Integration Basics

2.1 Overview of Calculus in Pharmacy

2.2 Differentiation

2.2.1 Definition

2.2.2 Basic Rules

2.2.3 Pharmaceutical Applications

2.3 Integration

2.3.1 Definition

2.3.2 Basic Rules

2.3.3 Pharmaceutical Applications

2.4 Worked Examples

2.5 Key Points for Exams

Unit 3: Elementary Statistics (Mean, Median, Mode)

3.1 Importance of Descriptive Statistics in Pharmacy

3.2 The Arithmetic Mean (Average)

3.2.1 Definition

3.2.2 Properties

3.2.3 Worked Example 1: Tablet Weight

3.2.4 Worked Example 2: Assay Results

3.3 The Median

3.3.1 Definition

B Pharmacy Sem 2: Remedial Biology / Mathematics

Subject : Remedial Biology / Mathematics (as applicable)

1.  If Remedial Biology:

2.  If Remedial Mathematics:

Unit 1: Cell Biology & Biomolecules

Unit 2: Basic Genetics & Human Organ Systems Review

Unit 3: Plant Anatomy & Physiology

Unit 4: Microbiology Fundamentals

Unit 1: Algebraic Expressions & Logarithms

Unit 2: Differentiation & Integration Basics

Unit 3: Elementary Statistics (Mean, Median, Mode)

Unit 4: Pharmaceutical Calculations Review