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
“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
“Rotating a square table, the determinant tells how its area changes – 1 means same area, 2 means double.”
The determinant of an n×n matrix is a single number that represents the scaling factor of n-dimensional volume under the linear map. If det=0 the map collapses space, so the matrix is singular and non-invertible. Remember: a row-swap changes sign, and any dependent rows give det=0.
Memory trick: det = volume scale; zero → collapse.
“Flipping a fair coin, heads appears half the time, so P=1/2.”
Probability equals favourable outcomes divided by total equally likely outcomes. The value always lies between 0 and 1 inclusive. In counting, ensure the denominator includes every possible outcome; otherwise the result can exceed 1, which is wrong.
P=favourable/total
Memory trick: Favourable ÷ Total ≤ 1
“Rolling two dice, the result of one die never changes the other's outcome.”
Independent events do not affect each other's probabilities, so P(A∩B)=P(A)P(B). Mutually exclusive events cannot occur together, giving P(A∩B)=0. They are different concepts; exclusive events are rarely independent unless one has zero probability. Mistaking one for the other leads to wrong calculations.
Memory trick: Exclusive = 0 overlap; Independent = multiply
“In a bag with 3 red and 2 blue balls, after picking a red, the chance the next is blue is conditional.”
Conditional probability P(A|B) = P(A∩B)/P(B) measures the chance of A when B is known to happen. The denominator must be non-zero and represents the reduced sample space. A common slip is to forget the division by P(B) and treat P(A∩B) as the answer.
P(A|B)=P(A∩B)/P(B)
Memory trick: Condition = shrink sample space
“A doctor revises the chance of a disease after a positive test using Bayes' theorem.”
Bayes' theorem relates the reverse conditional: P(A|B)=P(B|A)·P(A)/P(B). The denominator P(B) is found via total probability, summing over all ways B can occur. Ignoring the other possibilities for B gives an inflated posterior.
Memory trick: Posterior = Likelihood × Prior ÷ Evidence
“When you split a pizza among friends, the average slices each gets is the mean.”
Mean = sum of all observations divided by number of observations. Median is the middle value when data are ordered; for even N it is the average of the two central numbers. Mode is the value occurring most often; a set may have no mode or more than one. In skewed data the median better represents central tendency.
Memory trick: Mean = total÷N, Median = middle, Mode = most frequent.
“A sprinter’s lap times vary; the standard deviation shows how tightly the times cluster around the average.”
Standard deviation measures the average spread of data around the mean: σ = √[Σ(xi−μ)² / N] for a population, and s = √[Σ(xi−x̄)² / (n−1)] for a sample. A larger σ means data are more dispersed. Use N for population and (n-1) for sample in JEE problems.
Memory trick: SD = √(average of squared deviations).
“A set is like a drawer where each distinct utensil is stored only once.”
A set is a well-defined collection of distinct objects; order does not matter. Because each element is either in or out, repetitions are ignored. Exam trap: mixing up with a multiset or assuming order matters.
Memory trick: Set = No repeats, no order.
“Union is like a playlist that combines songs from two albums; intersection keeps only the tracks both albums share.”
Union (A∪B) contains every element that belongs to A or B, duplicates removed. Intersection (A∩B) contains only elements common to both. Trap: forgetting that union does not double-count repeated elements.
Memory trick: ∪ = OR (any), ∩ = AND (both).
“If every student in class A also attends class B, then A is a subset of B.”
A⊆B means every element of A is also in B. The empty set is a subset of every set, and A can equal B (improper subset). Trap: assuming ⊂ always means proper subset; check equality.
A⊆B
Memory trick: A fits inside B → A⊆B.
“Turning off all lights (complement) in rooms A or B means you turned off lights in both A and B separately.”
(A∪B)' equals A'∩B' and (A∩B)' equals A'∪B'. Complement flips union to intersection and vice-versa. Trap: swapping the operators or forgetting the complement on each set.
Memory trick: Not (A or B) = Not A and Not B.
“The power set of {red, blue} lists every color combo you could wear: none, red, blue, both.”
Power set P(A) is the set of all subsets of A; if A has n elements, |P(A)| = 2ⁿ. Includes empty set and A itself. Trap: omitting the empty set or the whole set when counting.
|P(A)|=2ⁿ
Memory trick: Power set size = 2ⁿ.