Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“Filling a tank by adding half of the remaining empty space each minute converges to a finite level.”
For |r|<1, an infinite GP sums to S∞ = a/(1-r). The exam trap is applying the formula when |r|≥1, which diverges, or forgetting that a negative r with |r|<1 still converges (alternating).
S∞=a/(1−r)
Memory trick: S∞ = a/(1-r) only if |r|<1
“A rectangle’s perimeter-square side equals the AM of its sides, while its area-square side equals the GM; the former is never shorter than the latter.”
For any non-negative numbers, the arithmetic mean (Σx/n) is always ≥ the geometric mean (ⁿ√Πx). Equality holds only when all numbers are identical. In JEE, note that a zero term makes GM zero and negative numbers are invalid for even-order roots.
Memory trick: AM≥GM: ‘Average beats root’ – equal only if all equal.
“Choosing a shirt (5 colors) and a tie (3 styles) gives 5×3 = 15 outfits.”
If one step can be done in m ways and a second independent step in n ways, the total ordered outcomes equal m·n. The principle extends to any number of independent stages. Trap: multiply only when choices do not affect each other; otherwise overcount.
Memory trick: AND → multiply, OR → add
“Arranging 3 books on a shelf in different orders shows 3P3 = 6 possible arrangements.”
A permutation counts ordered selections of r objects from n, given by nPr = n!/(n−r)!. Fix a position, choose any remaining item for each next spot, so order matters. Trap: using nCr when order is required gives a smaller answer.
nPr=n!/(n−r)!
Memory trick: P = Position matters
“Choosing 2 toppings from 4 for a pizza counts combinations, not the order of sprinkling.”
A combination counts unordered selections of r objects from n, formula nCr = n!/[r!(n−r)!]. Since order is irrelevant, divide the permutation count by r! ways to arrange the chosen items. Trap: forgetting the r! division and over-counting.
nCr=n!/(r!(n−r)!)
Memory trick: C = Choose, ignore order
“Opening a soda, each fizz bubble mirrors a term of (a+b)ⁿ expanding.”
(a+b)ⁿ = Σ_(r=0)ⁿ nCr a^(n−r) b^r. Coefficients are nCr and the exponents of a and b always add to n. A common trap is forgetting that r starts at 0, which drops the first term.
Tᵣ₊₁=nCr·a^(n−r)b^r
Memory trick: nCr = ways to choose r from n – think team selection.
“Finding the (r+1)th term is like stepping onto the r-th floor of a building that starts at ground zero.”
The (r+1)th term = nCr·a^(n−r)·b^r. Note the index is r+1 while the binomial coefficient uses r; mixing them shifts the exponents and gives a wrong term.
Tᵣ₊₁
Memory trick: T_(r+1) uses r in nCr and a^(n−r).
“When you rotate a unit stick on a compass, its x- and y-projections always satisfy x²+y²=1.”
For any angle sin²θ+cos²θ=1, a direct result of the unit-circle definition. It holds for acute, obtuse or reflex angles. The common trap is to forget the squares; the identity never changes sign.
sin²θ+cos²θ=1
Memory trick: Sin²+Cos²=1, like a right-triangle on the unit circle.
“A pizza slice of 180° is exactly half a circle, which equals π radians.”
π rad = 180°, so 1° = π/180 rad. Calculus formulas (e.g., d/dx sin x = cos x) are valid only when x is in radians. Using degrees introduces an extra factor π/180, a frequent error in differentiation problems.
π rad = 180°
Memory trick: 180° = π rad; multiply by π/180 to convert to rad.
“A swinging pendulum never tips beyond the walls of its arc, just as sin and cos never exceed ±1.”
For every real θ, –1 ≤ sinθ ≤ 1 and –1 ≤ cosθ ≤ 1 because they are the y- and x-coordinates of a point on the unit circle of radius 1. When solving equations, any root giving |value|>1 must be discarded as extraneous.
Memory trick: Sin, Cos ≤ radius 1 of unit circle.
“Mixing two musical notes of frequencies A and B gives beats described by sin(A±B) formulas.”
sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB. The sign in the cosine formula is opposite to that in the sine formula—mix-up leads to wrong results, especially in product-to-sum conversions.
Memory trick: Sin adds, Cos subtracts signs.
“In a survey, the height of a tower and its shadow form a triangle; the ratio side / sin opposite angle stays constant.”
In any triangle a/sinA = b/sinB = c/sinC. It works for acute, obtuse or scalene triangles. The rule fails when an angle is 0° or 180° because sin 0 = 0, making the ratio undefined.
a/sinA=b/sinB
Memory trick: Side over opposite sine = constant.
“A GPS triangulation uses c² = a²+b²−2ab·cosC to compute the distance between two points.”
c² = a² + b² − 2ab cosC generalises Pythagoras; when C = 90° the cosine term vanishes and the formula reduces to a²+b²=c². Forgetting the minus sign or using ‘+’ gives a wrong distance, especially for obtuse angles.
c²=a²+b²−2ab·cosC
Memory trick: c² = a²+b²−2ab·cosC (Pythagoras + correction).
“Measuring the straight-line distance between two GPS points on a map uses the same √(Δx²+Δy²) rule.”
The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²] by the Pythagorean theorem. Both differences are squared, so order doesn’t matter. A common trap is forgetting to square each Δ before adding.
d=√(Δx²+Δy²)
Memory trick: Distance = hypotenuse of Δx-Δy right triangle.
“The tilt of a skateboard ramp is its slope m = rise over run.”
Slope m = (y₂−y₁)/(x₂−x₁) equals tan θ, measuring how steep a line is. It is undefined for a vertical line (Δx=0). Students often mix up rise and run, giving the reciprocal.
m=Δy/Δx
Memory trick: Rise over run = tanθ.
“Railway tracks stay parallel because they share the same slope.”
Parallel lines have equal slopes (m₁=m₂). Perpendicular lines satisfy m₁·m₂ = −1, meaning their directions differ by 90°. Beware when one line is vertical (undefined slope) and the other horizontal (m=0); they are still perpendicular.
Memory trick: ⊥ → m₁·m₂ = -1, ∥ → m₁ = m₂.
“When you draw a bike wheel on graph paper, the points satisfying (x−h)²+(y−k)²=r² trace its rim.”
All points whose distance from (h,k) equals r satisfy (x−h)²+(y−k)²=r², derived from the distance formula. The radius r must be real, so r²≥0. In exams, forgetting that a negative r² gives no circle is a common error.
(x−h)²+(y−k)²=r²
Memory trick: Center-Radius = distance² → (x−h)²+(y−k)²=r²
“A satellite dish’s cross-section follows y²=4ax, reflecting signals toward its focus.”
A parabola is the locus of points equidistant from focus (a,0) and directrix x=−a, giving y²=4ax which opens to the right. The parameter a must be positive for rightward opening; if a is negative the curve opens left. Mistaking the orientation by reading the equation incorrectly is a frequent trap.
y²=4ax
Memory trick: Focus (a,0), directrix x=−a → y²=4ax
“A satellite in a circular orbit has e=0, while a Hohmann-transfer orbit around Earth has 0<e<1.”
Eccentricity e measures deviation from a circle. Circle: e=0. Ellipse: 0<e<1 (e=√(1-b²/a²)). Parabola: e=1. Hyperbola: e>1 (e=√(1+ b²/a²)). Trap: hyperbola always has e>1 regardless of which variable is called a; use the transverse axis a.
Memory trick: 0-circle, 1-parabola, >1-hyperbola; ellipse lives between.
“When you approach a traffic light, the speed you’re about to drive at is the limit, even if you stop at the light.”
A limit is the value f(x) approaches as x gets arbitrarily close to a point. It may exist even when f at that point is undefined or different. In JEE, never substitute directly when you get 0/0; simplify or use standard limits first.
Memory trick: Limit = approaching value, not f(at point).
“A smooth river without waterfalls represents a continuous function – you can follow it without lifting the boat.”
A function is continuous at a point when the left-hand limit, right-hand limit and the function’s value are all equal. Both sides must exist and match. Exam trap: check continuity for piecewise definitions at the joining x-value.
Memory trick: Continuous ⇔ limit = value at that point.
“A spinning wheel’s angle (in radians) and its arc length become equal as the angle gets tiny – that’s sin x ≈ x.”
lim (x→0) sin x / x = 1, but only when x is measured in radians; in degrees the limit equals π/180. This result is used repeatedly in JEE limits. Forgetting the radian condition gives a wrong answer.
lim(x→0)sinx/x=1
Memory trick: sin x≈x only in radians.
“Reading the speedometer needle at an exact moment gives the car’s instantaneous velocity – the derivative of distance.”
The derivative dy/dx is the limit of Δy/Δx as Δx→0, giving the slope of the tangent at that point. It exists only if the function is smooth there; a corner or cusp makes it undefined. In JEE, watch out for points where left-hand and right-hand limits differ.
dy/dx
Memory trick: dy/dx = lim Δx→0 Δy/Δx
“Accelerating a car: speed ∝ xⁿ, its change follows n·x^(n-1).”
For any constant exponent n, d/dx(xⁿ)=n·x^(n-1). It follows from the limit definition of derivative. Trap: n must be constant; the rule fails where xⁿ isn’t smooth, e.g., |x| at x=0 or non-integer n with x≤0.
d/dx(xⁿ)=nx^(n−1)
Memory trick: n down, power down – bring exponent down one.