My question , Here are some challenging math problems for (Javed sir) *Algebra* 1. If a, b, c are roots of x^3 + px^2 + qx + r = 0, then find the value of: ∑(a^2/b + b^2/c + c^2/a) where ∑ denotes summation. 1. Solve for x: |x^2-3x+2| + |x^2+2x-3| = 5 *Calculus* 1. Evaluate: ∫(x^2 + 1) / (x^4 - x^2 + 1) dx 1. Find the maximum value of: f(x) = x^3 - 6x^2 + 9x + 2 on the interval [0, 4] *Geometry* 1. In a triangle ABC, ∠A = 30°, ∠B = 60°, and side AB = 5 cm. Find the length of side BC. 2. A circle is inscribed in a triangle ABC, where ∠A = 90°, ∠B = 45°, and side AB = 10 cm. Find the radius of the circle. *Number Theory* 1. Prove that: 2^(n+1) > (n+1)^2 for all positive integers n. 1. Find the number of positive divisors of: 100000000 = 10^8 *Combinatorics* 1. A committee of 5 members is to be formed from 8 men and 6 women. Find the number of ways to form the committee with exactly 3 men. 2. In a tournament of 7 teams, each team plays every other team exactly once. Find the number of possible outcomes. *Vector Algebra* 1. If vectors a, b, c satisfy: a × b = c, b × c = a, c × a = b then find: |a + b + c| 1. Find the angle between vectors: a = i + 2j + 3k, b = 2i - j + k Good luck! (Reply me 😅😅)
Questions at (9:50) Cell Membrane! *Definition:* The cell membrane, also known as the plasma membrane, is a thin, semi-permeable lipid bilayer that surrounds the cell and separates the interior of the cell from its external environment. *Structure:* 1. Phospholipid Bilayer: Composed of phospholipid molecules with hydrophilic (water-loving) heads and hydrophobic (water-fearing) tails. 2. Embedded Proteins: Integral and peripheral proteins perform various functions, such as transport, signaling, and cell-cell interactions. 3. Cholesterol: Regulates membrane fluidity and permeability. *Functions:* 1. *Selective Permeability:* Controls movement of substances in and out of the cell. 2. *Cell Signaling:* Receives and transmits signals from external environment. 3. *Transport:* Facilitates movement of molecules across the membrane. 4. *Cell-Cell Interactions:* Mediates adhesion, recognition, and communication between cells. 5. *Maintenance of Cell Shape:* Provides structural support and maintains cell morphology. *Types of Transport:* 1. *Passive Transport:* Diffusion, osmosis, and facilitated diffusion. 2. *Active Transport:* Requires energy, e.g., pump proteins. *Membrane Proteins:* 1. *Integral Proteins:* Embedded within the bilayer. 2. *Peripheral Proteins:* Attached to integral proteins or lipids. *Biological Importance:* 1. *Cellular Homeostasis:* Maintains internal environment. 2. *Cell Communication:* Enables signaling and response to stimuli. 3. *Immune Response:* Recognizes and responds to pathogens. 4. *Disease:* Membrane dysfunction contributes to various diseases, e.g., cancer, neurological disorders. *Key Concepts:* 1. Fluid Mosaic Model 2. Lipid Raft 3. Membrane Potential 4. Endocytosis and Exocytosis *Research Applications:* 1. *Cancer Research:* Understanding membrane changes in cancer cells. 2. *Neurological Disorders:* Studying membrane dysfunction in neurodegenerative diseases. 3. *Infectious Diseases:* Developing membrane-targeting therapies. 4. *Synthetic Biology:* Designing artificial membranes for biotechnological applications. *Related Topics:* 1. Cell Wall 2. Cytoplasm 3. Nucleus 4. Mitochondria 5. Endoplasmic Reticulum cell membranes (explained 😅😅)
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My question , Here are some challenging math problems for
(Javed sir)
*Algebra*
1. If a, b, c are roots of x^3 + px^2 + qx + r = 0, then find the value of:
∑(a^2/b + b^2/c + c^2/a)
where ∑ denotes summation.
1. Solve for x:
|x^2-3x+2| + |x^2+2x-3| = 5
*Calculus*
1. Evaluate:
∫(x^2 + 1) / (x^4 - x^2 + 1) dx
1. Find the maximum value of:
f(x) = x^3 - 6x^2 + 9x + 2
on the interval [0, 4]
*Geometry*
1. In a triangle ABC, ∠A = 30°, ∠B = 60°, and side AB = 5 cm. Find the length of side BC.
2. A circle is inscribed in a triangle ABC, where ∠A = 90°, ∠B = 45°, and side AB = 10 cm. Find the radius of the circle.
*Number Theory*
1. Prove that:
2^(n+1) > (n+1)^2
for all positive integers n.
1. Find the number of positive divisors of:
100000000 = 10^8
*Combinatorics*
1. A committee of 5 members is to be formed from 8 men and 6 women. Find the number of ways to form the committee with exactly 3 men.
2. In a tournament of 7 teams, each team plays every other team exactly once. Find the number of possible outcomes.
*Vector Algebra*
1. If vectors a, b, c satisfy:
a × b = c, b × c = a, c × a = b
then find:
|a + b + c|
1. Find the angle between vectors:
a = i + 2j + 3k, b = 2i - j + k
Good luck! (Reply me 😅😅)
t.me/+FhSjfK-k-ps3ODRl
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Questions at (9:50) Cell Membrane!
*Definition:*
The cell membrane, also known as the plasma membrane, is a thin, semi-permeable lipid bilayer that surrounds the cell and separates the interior of the cell from its external environment.
*Structure:*
1. Phospholipid Bilayer: Composed of phospholipid molecules with hydrophilic (water-loving) heads and hydrophobic (water-fearing) tails.
2. Embedded Proteins: Integral and peripheral proteins perform various functions, such as transport, signaling, and cell-cell interactions.
3. Cholesterol: Regulates membrane fluidity and permeability.
*Functions:*
1. *Selective Permeability:* Controls movement of substances in and out of the cell.
2. *Cell Signaling:* Receives and transmits signals from external environment.
3. *Transport:* Facilitates movement of molecules across the membrane.
4. *Cell-Cell Interactions:* Mediates adhesion, recognition, and communication between cells.
5. *Maintenance of Cell Shape:* Provides structural support and maintains cell morphology.
*Types of Transport:*
1. *Passive Transport:* Diffusion, osmosis, and facilitated diffusion.
2. *Active Transport:* Requires energy, e.g., pump proteins.
*Membrane Proteins:*
1. *Integral Proteins:* Embedded within the bilayer.
2. *Peripheral Proteins:* Attached to integral proteins or lipids.
*Biological Importance:*
1. *Cellular Homeostasis:* Maintains internal environment.
2. *Cell Communication:* Enables signaling and response to stimuli.
3. *Immune Response:* Recognizes and responds to pathogens.
4. *Disease:* Membrane dysfunction contributes to various diseases, e.g., cancer, neurological disorders.
*Key Concepts:*
1. Fluid Mosaic Model
2. Lipid Raft
3. Membrane Potential
4. Endocytosis and Exocytosis
*Research Applications:*
1. *Cancer Research:* Understanding membrane changes in cancer cells.
2. *Neurological Disorders:* Studying membrane dysfunction in neurodegenerative diseases.
3. *Infectious Diseases:* Developing membrane-targeting therapies.
4. *Synthetic Biology:* Designing artificial membranes for biotechnological applications.
*Related Topics:*
1. Cell Wall
2. Cytoplasm
3. Nucleus
4. Mitochondria
5. Endoplasmic Reticulum
cell membranes (explained 😅😅)
Very good
@ExAmdoo please solve the problem of maths