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
“The curvature of a roller coaster tells you if you’re entering a dip (concave up) or climbing a hill (concave down).”
The second derivative d²y/dx² is the derivative of the slope. Positive d²y/dx² means the graph bends upward (concave up); negative means it bends downward (concave down). It measures how quickly the slope changes, not whether the function itself is increasing.
d²y/dx²
Memory trick: ↑ curvature = + second derivative
“A road’s tangent shows the car’s direction; the normal is the wall built exactly perpendicular to that road.”
At x₀ the tangent line has slope m = f'(x₀). The normal line is perpendicular, so its slope is –1/m (provided m≠0). If the tangent is vertical (undefined slope), the normal is horizontal, and vice-versa. Forgetting the negative reciprocal causes sign errors.
Memory trick: Normal slope = –1/(tangent slope)
“When you turn the faucet knob a tiny bit, the extra water flow ≈ slope (flow rate) × knob rotation.”
For a differentiable function, a tiny change Δx produces a change Δy ≈ f'(x)·Δx. The linear term dominates because higher-order terms contain (Δx)², negligible for small Δx. In exams, ensure Δx is truly small; applying the formula to a large step leads to big error.
Memory trick: Δy ≈ f'·Δx – think “Δy equals slope times tiny step”.
“To find the growth rate of a bacteria colony that multiplies as xˣ, you first take log, turning the exponent into a product.”
Take ln of both sides, use ln(ab)=ln a+ln b and ln(aᵇ)=b ln a, then differentiate. This turns products, quotients or variable powers into sums, simplifying the derivative. Trap: forgetting the factor 1/y when differentiating ln y, so the final dy/dx is off.
Memory trick: ln → bring exponent down.
“The path of a roller-coaster described by x=sin t, y=cos t gives its slope dy/dx = (dy/dt)/(dx/dt).”
For parametric curves x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt) by the chain rule dy/dx = (dy/dt)*(dt/dx). Ensure dx/dt ≠ 0. Trap: swapping numerator and denominator or forgetting to check dx/dt zero.
Memory trick: dy/dx = (dy/dt) ÷ (dx/dt).
“Pouring water into a tub cup-by-cup; as cups get infinitesimally small the total equals the tub’s volume.”
A definite integral is the limit of Riemann sums: Σ f(x_i)Δx as Δx→0. The sum of thin rectangles approaches the exact area. Assuming the sum equals the area without taking the limit is a common error.
Memory trick: Riemann → tiny cups → limit = area.
“A seesaw balanced at the centre: equal opposite weights cancel, leaving zero net torque.”
If f is odd, f(−x)=−f(x) so areas on [−a,0] and [0,a] cancel → integral zero. If f is even, f(−x)=f(x) so total area is twice the half-interval area. The symmetry must be exactly about 0; shifting the interval breaks it.
Memory trick: Odd → 0, Even → 2×half.
“Breaking a chocolate bar into smaller pieces lets you eat each piece separately.”
Partial fraction decomposition rewrites a rational function as a sum of simpler fractions with linear or irreducible quadratic denominators, each easy to integrate. Missing a repeated factor or quadratic term leads to an incorrect decomposition.
Memory trick: PF = split, then integrate each piece.
“A charging capacitor follows dy/dx + (1/RC) y = V/RC, solved by multiplying with e^(∫P dx).”
A first-order linear DE has form dy/dx + P(x) y = Q(x). Multiply by integrating factor μ = e^(∫P dx) to make left side a product derivative: d(μ y)/dx = μ Q. Then integrate. Common mistake: forgetting to simplify μ before integration.
Memory trick: Linear → μ = e^(∫P) → d(μy)=μQ.
“In kinetic energy ½ m (dx/dt)² the squared velocity shows degree 2 after removing the fraction ½ m.”
Degree is the exponent of the highest-order derivative after the equation is cleared of radicals, fractions and denominators, provided the DE is a polynomial in its derivatives. It is undefined if a derivative is inside a transcendental function or a denominator. The exam trap: assuming a degree exists for non-polynomial forms like sin(y') or 1/y'.
Memory trick: Degree = power of the top derivative after polynomialising.
“Three sticks lying flat on a table are coplanar; mathematically their scalar triple product being zero tells the same.”
Three vectors a, b, c are coplanar iff their scalar triple product [a b c]=a·(b×c) equals zero, meaning the volume of the parallelepiped they form is zero. The trap: mixing up order; the product changes sign if the order is permuted, but zero stays zero.
Memory trick: [a b c]=0 → zero volume → coplanar
“Imagine two non-parallel train tracks that never meet; the shortest bridge between them is the common perpendicular.”
For skew lines L₁ and L₂ with direction vectors d₁, d₂ and a point A on L₁, the shortest distance is |(AB·(d₁×d₂))|/|d₁×d₂|, where B is any point on L₂. The numerator projects the connecting vector onto the common normal. A common mistake is to apply the intersecting-lines formula (set numerator to zero).
Memory trick: d₁×d₂ gives common normal; project AB onto it
“A sheet of paper is fixed by a point on it and a needle pointing straight out – the needle is the plane’s normal.”
A plane can be written as (r−r₀)·n=0, where r₀ is a known point on the plane and n is a normal vector. Expanding gives ax+by+cz+d=0 with (a,b,c)=n and d=−n·r₀. Forgetting the sign of d is a frequent error.
Memory trick: Point + normal → (r−r₀)·n=0
“A laser beam hitting a wall makes a complement angle to the wall’s normal – that complement is the line-plane angle.”
If a line has direction vector d and a plane has normal n, the angle φ between the line and the plane satisfies sinφ=|d·n|/(|d||n|); equivalently, φ=90°−θ where θ is the angle between d and n. Students often compute cosφ directly, mixing up line-plane with line-normal angle.
Memory trick: Line-plane φ = 90°−(line-normal θ)
“A gyroscope maintains its orientation; its rotation matrix is orthogonal, keeping vector lengths unchanged.”
A matrix Q is orthogonal when QᵀQ = I, i.e., its transpose equals its inverse. Such matrices represent rotations/reflections and preserve lengths and angles of vectors. In JEE, test orthogonality by checking Qᵀ = Q⁻¹ or QᵀQ = I.
Memory trick: Qᵀ = Q⁻¹ ⇒ QᵀQ = I
“A folded origami that can’t be unfolded is like a singular matrix – its determinant is zero, no undo.”
A singular matrix has det=0, meaning its rows (or columns) are linearly dependent. Such a matrix cannot be inverted and the linear system either has no solution or infinitely many. The trap is assuming a zero determinant still allows a unique solution.
Memory trick: Singular = det zero → no inverse.
“Swapping two rows in a spreadsheet flips the determinant’s sign, just like reversing a wind turbine’s rotation.”
Key determinant properties: swapping two rows (or columns) multiplies det by –1; multiplying a row by k multiplies det by k; adding a multiple of one row to another leaves det unchanged; identical rows make det zero. Forgetting the sign change after multiple swaps is a common mistake.
Memory trick: Swap rows → sign flip; duplicate → zero.
“A three-wire circuit lights a bulb only if the loop isn’t open; likewise a square system ‘lights up’ (has a solution) when its determinant isn’t zero.”
A linear system is consistent if it has at least one solution. For a square coefficient matrix, a non-zero determinant guarantees a unique solution; if the determinant is zero, consistency depends on whether the rank of the coefficient matrix equals the rank of the augmented matrix. The trap: det = 0 does NOT always mean no solution – it may give infinitely many solutions.
Memory trick: Det≠0 → unique; Det=0 → compare ranks (coeff vs aug).
“Choosing a ball from a bag that may contain red or blue boxes uses total probability to add the two ways it can be red.”
If {B_i} partition the sample space, P(A)= Σ P(A|B_i)·P(B_i). It adds contributions from each mutually exclusive case. Mistake: forgetting that the B_i must cover all possibilities; omitting one case gives a smaller total.
Memory trick: Total = Σ (conditional × prior).
“Assigning 0 to tails and 1 to heads when flipping a coin turns the experiment into a random variable.”
A random variable X maps each outcome of an experiment to a real number, enabling probabilistic calculations. It can be discrete or continuous. Pitfall: treating X as the outcome itself; remember X is a function defined on the sample space.
Memory trick: X = outcome → number.
“The spread of students' marks around the class average is measured by variance.”
Variance Var(X)=E[(X−μ)²] quantifies the spread of values around the mean μ. It equals Σ (x_i−μ)²·P(x_i). Common error: forgetting to square the deviations; Σ (x_i−μ) is zero, not the variance.
Memory trick: Var = average of squared deviations.
“When you toss a fair coin, the chance of not getting heads is 1 − P(heads).”
For any event A, its complement A' satisfies P(A')=1−P(A). The same holds conditionally: P(A'|B)=1−P(A|B) provided P(B)>0. A and A' are always mutually exclusive and exhaustive, so subtraction never needs extra checks; the trap is forgetting the conditional denominator must be non-zero.
P(A')=1−P(A)
Memory trick: Complement = 1 − probability
“Comparing two brands of soda, CV tells which has more relative sweetness variation despite different average sugar levels.”
Coefficient of variation (CV) expresses dispersion as a percentage of the mean: CV = (σ / μ) ×100%. It allows comparison of variability across data sets with different units or magnitudes. Do not use CV when the mean is zero or near zero, as it becomes undefined.
Memory trick: CV% = (SD ÷ mean) ×100.
“When the weather gets hotter, ice-cream sales jump; their correlation coefficient is close to +1.”
Correlation r quantifies the linear relationship between two variables: r = Σ[(xi−x̄)(yi−ȳ)] / √[Σ(xi−x̄)² Σ(yi−ȳ)²], ranging from –1 (perfect inverse) to +1 (perfect direct). r = 0 means no linear link. A high |r| does not prove cause-and-effect.
Memory trick: r = Σ[(x-x̄)(y-ȳ)] / √[Σ(x-x̄)² Σ(y-ȳ)²].