College Mathematics

Textbook Solutions

Maths at College and Pre-Engineering Level

This page contains summary of formula and concepts in mathematics at college level

कक्षा 11 और 12 में भौतिक विज्ञान, रसायन विज्ञान और गणित के विषयों का पाठ्यक्रम, कक्षा 10 में पढ़ाए जाने वाले विषयों की तुलना में अधिक विस्तृत होता है। इससे छात्रों पर बहुत अधिक दबाव पड़ता है। हालाँकि, इनमें कुछ बातें समान होती हैं। उदाहरण के लिए, गणित की अवधारणाओं का उपयोग भौतिक विज्ञान में सीखने की प्रक्रिया को बेहतर बनाने के लिए किया जा सकता है। सलाह दी जाती है कि इन तीनों विषयों के पाठ्यक्रम (सिलेबस) का प्रिंट निकलवाकर उसे दीवारों या दरवाज़ों पर लगा लें। साथ ही, उन अध्यायों को चिह्नित करें जो आपको कठिन, रोचक / आकर्षक या समझने में आसान लगते हैं।

• Chapter 01: Physical World
• Chapter 02: Units and Measurements -- Conservation of Basic Variables
• Chapter 03: Motion in a Straight Line -- Vectors, Kinematics, Relative Velocity
• Chapter 04: Motion in a Plane -- Projectiles, Circular Motion, Motion along a Curved Path
• Chapter 05: Laws of Motion -- Forces and Newton's Laws, Pulleys, Force Equilibrium, Linear Momentum, Free-Body Diagrams
• Chapter 06: Work, Energy and Power -- Friction, Conservation of Energy
• Chapter 07: System of Particles and Rotational Motion -- Center of Mass, Linear Momentum, Collision, Conservation of Momentum, Rotational Mechanics, Angular Momentum
• Chapter 08: Gravitation -- Laws of Gravity, Escape Velocity, Centre of Gravity
• Chapter 09: Mechanical Properties of Solids -- Stress, Strain, Hooke's Law
• Chapter 10: Mechanical Properties of Fluids -- Pressure, Streamlines, Bernoulli's principle
• Chapter 11: Thermal Properties of Matter -- Temperature, Heat, Coefficient of Thermal Expansion, Newton's Law of Cooling
• Chapter 12: Thermodynamics -- Heat, Internal Energy, Thermal Equilibrium, First Law of Thermodynamics
• Chapter 13: Kinetic Theory -- Mean Free Path, Behaviour of Gases, Specific Heat Capacity
• Chapter 14: Oscillations -- Simple Harmonic Motions, Wave Motion and Waves on a String
• Chapter 15: Waves -- Sound Waves, Light Waves, Superposition of Waves, Reflection of Waved, Doppler Effect, Waves in Moving Media

Permutation and Combinations: 6 categories

1. Basic Arrangements (Linear Permutations)
   • The Concept: Arranging distinct objects in a straight line where the order of arrangement is critical.
   • Common Scenarios: Arranging books on a shelf, Forming words from a given set of letters, Creating PIN numbers or passwords from digits.
   • Formula: \(n!\) (for all objects) or \(^nP_r = \frac{n!}{(n-r)!}\) (for selecting and arranging \(r\) objects out of \(n\)).
2. Group Selection (Combinations)
   • The Concept: Picking a subset of items from a larger pool where the order of selection does not matter.
   • Common Scenarios: Choosing a committee of 3 people out of 10, Selecting a hand of cards from a deck, Picking colours to paint a house from a catalog.
   • Formula: \(^nC_r = \frac{n!}{r!(n-r)!}\)
3. Conditional Permutations (Constraints)
   • The Concept: Linear arrangements that include specific rules or restrictions on how items can sit next to one another.
   • Common Sub-types: Tie-Together Method such as "Objects X and Y must always be together." (Treat X and Y as a single giant element, arrange the group, then arrange X and Y internally). Gap Method such as "No two vowels can stand next to each other." (Arrange the consonants first, then place vowels in the spaces/gaps between them).
4. Arrangements with Identical Items (Multiset Permutations)
   • The Concept: Arranging items where some of the objects are completely indistinguishable from one another.
   • Common Scenarios: Arranging the letters of the word "MISSISSIPPI" (where letters like 'I', 'S', and 'P' repeat), Arranging a line of colored flags where 3 are red, 2 are blue, and 1 is green.
   • Formula: \(\frac{n!}{p!\cdot q!\cdot r!}\) (where \(p, q, r\) are the frequencies of the repeating items).
5. Circular Permutations
   • The Concept: Arranging objects in a closed loop or circle. Because a circle has no fixed starting point, shifting everyone one seat to the right results in the exact same arrangement.
   • Common Scenarios: People sitting around a round dinner table, Beads threaded onto a necklace or keys on a keyring.
   • Formula: \((n-1)!\) (for seated people) or \(\frac{(n-1)!}{2}\) (for necklaces/keyrings, because flipping it over makes clockwise and counter-clockwise arrangements identical).
6. Distribution & Partitioning (Stars and Bars)
   • The Concept: Distributing identical objects into distinct bins, or dividing a total number into smaller integer groups.
   • Common Scenarios: Distributing 10 identical apples among 3 children. Finding the number of non-negative integer solutions to \(x + y + z = 10\).
   • Formula: \(^{n+r-1}C_{r-1}\) (where \(n\) is identical items and \(r\) is the number of recipients/bins).

                  [ Read the Problem ]
                           │
            Is it a SELECTION or an ARRANGEMENT?
            ┌──────────────┴──────────────┐
     [ Selection ]                [ Arrangement ]
     (Order doesn't matter)       (Order matters)
            │                             │
    Does it have identical        Are the items in a
      items or groups?            LINE or a CIRCLE?
     ┌──────┴──────┐               ┌──────┴──────┐
   [No]          [Yes]          [Line]        [Circle]
    │              │               │             │
  Basic       Distribution       Does it      Circular
Combination   (Stars & Bars)   have repeat   Permutation
                                 letters?
                               ┌───┴───┐
                             [No]     [Yes]
                              │         │
                            Basic    Identical
                         Permutation   Items

Step 1: The "Order" Litmus Test - Ask yourself - If I swap the position of two chosen items, does it change the outcome?

• Yes \(\rightarrow \) Permutation. (e.g. Changing "1-2-3" to "3-2-1" gives a completely different lock combination).
• No \(\rightarrow \) Combination. (e.g. A committee of "A and B" is identical to a committee of "B and A").

Step 2: Check for Repetition/Identical items: Are all items distinct (like unique people)? Are some items identical duplicates (like standard coins or letters)? If yes, use Category 4 or Category 6.

Step 3:Identify Constraints Look for keyword triggers: "Together", "Never together", "Alternating", "At least one". If these exist, you are dealing with a Conditional Problem (Category 3), which requires adjusting basic rules using the Gap or Tie-Together methods.