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.
300 fundamentals
“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.
“When two conveyor belts move together, the total speed change equals each belt’s change times the other’s speed.”
If y=uv, then dy/dx = u'v + uv'. Differentiate each factor separately and add the cross-terms. A common trap: applying the rule as (uv)'=u'v' – missing the sum.
(uv)'=u'v+uv'
Memory trick: Derivative = first'·second + first·second'.
“Adjusting a camera zoom (inner) while moving the tripod (outer) changes the final picture size by multiplying both rates.”
For y = f(u) with u = g(x), dy/dx = (dy/du)·(du/dx). Differentiate outer function, then multiply by derivative of inner. Exam trap: forgetting to replace u back with its expression after differentiating.
dy/dx=dy/du·du/dx
Memory trick: Outer' × inner'.
“A mountain climber reaches the summit where the slope flattens before the descent begins.”
At a stationary point f'(x)=0. The second-derivative test decides: f''(x)<0 gives a local maximum, f''(x)>0 a local minimum; if f''=0 the test fails and higher derivatives are needed. Beware assuming every f'=0 is an extremum.
Memory trick: Flat slope + concave down (∩)=max; concave up (∪)=min
“A car’s speedometer needle climbs upward when the vehicle accelerates and drops when it decelerates.”
If f'(x)>0 throughout an interval, the function is strictly increasing there; if f'(x)<0 it is strictly decreasing. The sign must stay unchanged—isolated zeros do not break monotonicity. Forgetting this leads to wrong interval claims.
Memory trick: Up arrow = + slope, down arrow = – slope
“When you add a constant topping to a pizza, the slice’s shape stays the same – just like +C adds to any antiderivative.”
∫xⁿdx = xⁿ⁺¹/(n+1)+C is the reverse of differentiation for any power n except n = –1. When n = –1 the integral becomes ln|x|+C because division by zero is impossible. Forgetting this exception gives a wrong answer, especially in JEE problems with 1/x terms.
∫xⁿ dx=x^(n+1)/(n+1)+C
Memory trick: n≠-1 → power rule; n=-1 → ln|x|+C
“A farmer measures the area of a plot between two fence posts by summing thin strips; the sum is a definite integral.”
A definite integral equals the signed area between the curve and the x-axis from a to b. Area above the axis counts positive, below negative; reversing limits flips the sign.
Memory trick: Top minus bottom gives signed area.
“Your car’s odometer adds tiny speed readings over time; the total distance is the integral of speed.”
The Fundamental Theorem of Calculus states that if F′(x)=f(x), then ∫_aᵇ f(x)dx = F(b)−F(a). It lets you evaluate a definite integral by an antiderivative. Using the wrong antiderivative or mixing limits gives wrong answer.
Memory trick: F(b)-F(a) = total change.
“When you adjust a thermostat, the temperature change depends on the current temperature – a simple differential equation.”
A differential equation contains an unknown function and its derivatives. It expresses a relationship between the function’s value and its rates of change. In exams, ensure the equation actually involves a derivative; algebraic equations are not DEs.
Memory trick: DE = D(ynamics) + Equation.
“When you push a sled up a slope, the work you do equals the dot product of the force and the displacement.”
The dot product a·b = |a||b|cosθ is a scalar. It is positive for angles <90°, zero when vectors are perpendicular, and negative for obtuse angles. The sign comes from cosθ, not from the magnitudes.
a·b=|a||b|cosθ
Memory trick: Work = F·d = |F||d|cosθ; zero → ⟂
“The torque you feel while tightening a bolt is τ = r × F, the cross product of the lever arm and the applied force.”
The cross product a×b is a vector perpendicular to both a and b, magnitude |a||b|sinθ, direction given by the right-hand rule. It vanishes when the vectors are parallel (θ=0 or π). Apply the right-hand rule for direction.
|a×b|=|a||b|sinθ
Memory trick: Torque τ = r×F, right-hand rule, zero when parallel
“Like the grid of pixels on a smartphone screen, a matrix arranges numbers in rows and columns.”
A matrix is a rectangular array of numbers (or symbols) organized in rows and columns. It is the basic tool to store data and represent linear transformations. In JEE, remember that dimensions are written as ‘rows × columns’, not the other way round.
Memory trick: Rows first, then columns: R×C
“Just as turning a Rubik’s cube front then up gives a different pattern than up then front, AB ≠ BA.”
Multiplying matrices corresponds to composing linear maps; the order matters, so in general AB ≠ BA. Only when A and B share special properties (e.g., both diagonal) they commute. In exam problems, check the order before substituting.
Memory trick: AB ≠ BA unless they share eigen-basis