Basic Maths | Foundation Physics | #neet #jeeadvanced

Main ideas & lessons (basic maths + trig/graphs + conics foundations)

1) Coordinate geometry basics

• Plotting points and lines
  • A line is formed by connecting two points on the coordinate axes.
• Distance between two points
  • For points (A(x_1, y_1)) and (B(x_2, y_2)), the distance formula comes from the Pythagoras theorem:
    • Horizontal difference: ((x_2 - x_1))
    • Vertical difference: ((y_2 - y_1))
  • Thus, [ AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]

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2) Slope / inclination of a line

• Inclination angle and slope

  • The angle (\theta) is measured anticlockwise from the positive x-axis.
  • The slope (often (m)) is the rate of change: [ m=\tan\theta=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x} ]

• Change notation

  • (\Delta x) means “change in (x)” and (\Delta y) means “change in (y)”.
  • (\Delta x) can be any finite measurable change (not necessarily small).
• Interpreting slope
  • If two lines have the same (\Delta x), then the one with larger (\Delta y) has larger slope.
  • Magnitude
    • Slope (>1): for a given change in (x), (y) changes even more.
    • Slope between (0) and (1): (y) changes less than (x) proportionally.
  • Sign
    • Positive slope: as (x) increases, (y) increases.
    • Negative slope: as (x) increases, (y) decreases.
• Special angle convention
  • A “negative angle” corresponds to clockwise rotation relative to the positive x-axis (since positive is anticlockwise).

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3) Trigonometry: quadrant rules + unit-circle logic

• Quadrant-based sign rules
  • Angles are split into four quadrants:
    • I: (0^\circ) to (90^\circ)
    • II: (90^\circ) to (180^\circ)
    • III: (180^\circ) to (270^\circ)
    • IV: (270^\circ) to (360^\circ)
  • Mnemonic provided: S, T, C
    • S = (\sin)
    • T = (\tan)
    • C = (\cos)

• Unit circle interpretation

  • On a radius-1 circle: [ \sin\theta=\frac{\text{perpendicular}}{1},\quad \cos\theta=\frac{\text{base}}{1} ]

  • Signs (positive/negative) depend on the quadrant.

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4) Graphs of (\sin), (\cos), (\tan) + periodicity

• Sin graph
  • (y=\sin x) is curved and bounded between (-1) and (+1).
  • Periodicity: (\sin) repeats every (360^\circ) (or (2\pi) radians).
  • Symmetry/mirror behavior across quadrants is described.
• Cos graph
  • (y=\cos x) is also bounded and periodic with period (360^\circ).
  • Difference from sine: phase shift (cos starts at (1) when (x=0)).
• Graph shifting (transformations)
  • For a known graph (y=f(x)):
    • (y=f(x-x_0)): shift right by (x_0)
    • (y=f(x+x_0)): shift left by (x_0)
    • (y=f(x)+y_0): shift up by (y_0)
    • (y=f(x)-y_0): shift down by (y_0)
• Tan graph
  • (y=\tan x):
    • Undefined at odd multiples of (90^\circ) (vertical asymptotes)
    • Range is unbounded: from (-\infty) to (+\infty)
    • Period: repeats every (180^\circ) (or (\pi) radians)

  • Asymptotes

    • Straight lines the graph approaches at infinity.

    • For (\tan): asymptotes at [ x=(2n+1)\frac{\pi}{2} ] (odd multiples of (90^\circ))

    • Discontinuous at asymptotes (described like a “pen lift”).

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5) Ordering/monotonic behavior of (\tan)

• Within continuous intervals

  • (\tan) is increasing between asymptotes.
  • So if (\theta_2>\theta_1) within the same continuous interval: [ \tan\theta_2 > \tan\theta_1 ]

• Sign vs magnitude

  • Values can be negative depending on the interval, so “greater” should be interpreted algebraically (not only by magnitude).

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6) Trig approximations for small angles

• For small angles (up to about (10^\circ) taught): [ \sin\theta \approx \theta,\quad \tan\theta \approx \theta ] where (\theta) is in radians.

• Extra note: (\sin\theta \approx \theta) tends to work better than (\tan\theta \approx \theta) as the angle increases.

• Example taught: (\sin 3^\circ \approx \theta), with (\theta) converted into radians.

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7) Binomial approximation (used in physics)

• Condition

  • Use the approximation when: [ |x| \le 0.05 ]

  • (a “very small compared to 1” rule, about a 5% guideline)

  • Approximation rule [ (1+x)^n \approx 1+nx ]

• Physics use-case

  • Gravitational field at height (h) where (h \ll r), via binomial expansion.

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Methodology / step-by-step instructions explicitly taught

A) Distance formula from two points

• Input: ((x_1,y_1)), ((x_2,y_2))
• Steps:
  • Compute ((x_2-x_1)^2)
  • Compute ((y_2-y_1)^2)
  • Add them
  • Take the square root
• Output: [ \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]

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B) Slope of a line

• Input: ((x_1,y_1)), ((x_2,y_2))

• Steps:

  • Compute (\Delta y=y_2-y_1)
  • Compute (\Delta x=x_2-x_1)
  • Divide: [ m=\frac{\Delta y}{\Delta x} ]

• Interpretation:

  • Sign of (m): direction (positive/negative)
  • Magnitude of (m): steepness (with caution about angles/sign intervals)

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C) Graph shifting (key transformation rules)

If (y=f(x)), then:

• Left shift by (a): [ y=f(x+a) ]

• Right shift by (a): [ y=f(x-a) ]

• Up shift by (b): [ y=f(x)+b ]

• Down shift by (b): [ y=f(x)-b ]

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D) Using trig identities for sums/differences

Core formulas: [ \sin(a+b)=\sin a\cos b+\cos a\sin b ] [ \sin(a-b)=\sin a\cos b-\cos a\sin b ] [ \cos(a+b)=\cos a\cos b-\sin a\sin b ] [ \cos(a-b)=\cos a\cos b+\sin a\sin b ]

Approach used:

• Rewrite the target angle (e.g., (75^\circ) as (30^\circ+45^\circ))
• Substitute known (\sin/\cos) values
• Simplify

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E) Graph selection: independent vs dependent variables

• Rule:
  • Independent variable (\rightarrow) x-axis
  • Dependent variable (\rightarrow) y-axis
• Example:
  • If plotting (v) vs (t^2), then (x=t^2) and (y=v).

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F) Binomial approximation in physics (gravity at height)

• Goal: [ g(h)=\frac{GM}{(r+h)^2} ]

• Steps:

  • Replace (r) by (r+h)

  • Rewrite: [ \frac{1}{(r+h)^2}=\frac{1}{r^2\left(1+\frac{h}{r}\right)^2} ]

  • Use binomial approximation for small (\frac{h}{r}): [ (1+x)^{-2}\approx 1-2x ]

  • Substitute (x=h/r)

  • Use (g=\frac{GM}{r^2}) at the surface to simplify

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Major equations/results highlighted later in the video (conics + parametrics)

Circle (standard form)

[ (x-x_0)^2+(y-y_0)^2=r^2 ]

Ellipse (standard form)

[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ]

• with (a>b) (major axis length related to (a))

Parametric equation of a circle (taught derivation)

General idea: [ x=x_0+r\cos\theta,\quad y=y_0+r\sin\theta ]

• Eliminate (\theta) to return to the circle equation.

Rectangular hyperbola definition (taught)

• When the product of two variables is constant: [ xy=C ]

• Quadrant behavior:

  • If (C>0): in I and III
  • If (C<0): in II and IV
• Asymptotes: coordinate axes

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Speakers / sources featured

• “Sir” / teacher: referenced repeatedly by students; no name provided in subtitles.
• Students: multiple respondents (e.g., “We/yes sir”, “Do Wei”, “Molly and Nishit”, “Paresh”, “Poojan Bor”, “Mudra”, “Anna Sheel”), not presented as distinct named experts with full roles beyond responding.