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
“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].
“A detective notes: “If the window is broken, the alarm rang”; the contrapositive is “If the alarm didn’t ring, the window isn’t broken”.”
The statement ‘If p then q’ is logically identical to ‘If not q then not p’. Both have the same truth table, so they are always true together. In exams, forgetting to negate both parts gives the converse, not the contrapositive, and loses marks.
Memory trick: Flip + Negate both → contrapositive
“A recipe says, “If you add sugar, the cake is sweet”; the converse would be “If the cake is sweet, you added sugar”, which isn’t guaranteed.”
The converse of ‘If p then q’ is ‘If q then p’; the inverse is ‘If not p then not q’. Neither is logically equivalent to the original statement. Students often mistake the inverse for the converse or assume they’re true like the original.
Memory trick: Converse swaps, inverse negates both
“A rubber band stretched around two sticks can’t reach a third longer stick unless you add the third side.”
For any triangle, the sum of any two sides must be greater than the third side; equality occurs only when the points are collinear. This ensures a non-degenerate triangle. Trap: writing ≥ instead of >, which would admit a straight line.
Memory trick: Two sides always outstretch the third
“Three cars travel the same distance at 60, 80 and 100 km/h; their AM speed is highest, HM (based on times) is lowest, and GM lies between.”
For any set of positive numbers, AM ≥ GM ≥ HM, with equality only when all numbers are equal. GM is defined for non-negative numbers, but the chain needs all numbers strictly positive because HM uses reciprocals. A common trap is applying the order when a value is zero or negative, which makes HM undefined or breaks the inequality.
Memory trick: A > G > H for positives – think ‘A-G-Higher’ (HM smallest).
“The number of ways 4 runners can finish a race is 4! = 24.”
n! is the product of all positive integers up to n, defined recursively as n! = n·(n−1)! with 0! = 1. It counts permutations of n distinct items. Trap: using factorial for non-integers or forgetting 0! = 1 in calculations.
n! = n×(n−1)!
Memory trick: 0! = 1, start the product
“Counting steps on a staircase where each step is 2 cm higher, the total height after n steps equals n/2 × (first + last).”
The sum of n terms of an AP is Sₙ = n/2 (first + last) = n/2[2a+(n-1)d]. Remember to use the actual first and last terms, not just a and d. A frequent mistake is mixing up a+(n-1)d with a+nd.
Sₙ=n/2(a+l)
Memory trick: Pair first with last: S = n/2 × (first+last).
“Choosing 3 toppings out of 8 is the same as leaving out the 5 you don’t want.”
nCr equals nC(n−r) because picking r objects is equivalent to deciding which n−r objects to leave out. This symmetry cuts work, especially when r>n/2. Forgetting it can double calculations or cause overflow.
nCr=nC(n−r)
Memory trick: r or n−r, same pick.
“Measuring the gap between two parallel railway tracks with a ruler uses |c1-c2|/√(a²+b²).”
For parallel lines ax+by+c1=0 and ax+by+c2=0, the perpendicular distance is |c1−c2| divided by √(a²+b²). Derive by picking a point on one line and measuring to the other. Trap: drop the absolute value or use √(a²−b²).
Memory trick: Dist = |Δc| / √(a²+b²)
“The steepness of a road that curves like tan x grows as sec² x, just like a shadow lengthening under a moving sun.”
d/dx(tan x)=sec² x because tan x = sin x/cos x and applying the quotient rule yields sec² x. Remember sec² x = 1+tan² x. Trap: writing derivative as sec x or forgetting the square.
d/dx(tanx)=sec²x
Memory trick: tan → sec².
“A chef tasting a soup adds a tiny fraction of spice; the flavor change follows (1+δ)ⁿ and the series works only while the pinch is less than the whole dish (|δ|<1).”
For any real n, (1+x)ⁿ = Σ_(r=0)^∞ C(n,r)x^r, an infinite series that converges for |x|<1. At the endpoints x=±1 convergence is not automatic: at x=1 it converges only if n>-1, at x=-1 only if n>0. In JEE you must state the radius = 1 and check the two endpoints, otherwise the answer is marked wrong.
Memory trick: Radius = 1 → ‘unit circle’; then test x=+1 (need n>-1) and x=-1 (need n>0).
“A piano key can be silent yet still present; D<0 means the equation has two invisible (complex) roots, not zero roots.”
D=b²−4ac < 0 does not mean ‘no solution’; it means the quadratic has two non-real complex conjugate roots. Exams expect you to state ‘complex roots’ rather than ‘no root’. Forgetting this leads to loss of marks.
Memory trick: D<0 → two complex roots, not zero.
“A seesaw balances only when the total torque follows the rule sum = –b/a, just like the two kids on each side.”
For ax²+bx+c=0 the roots α and β satisfy α+β = –b/a and α·β = c/a (Vieta’s relations). The common slip is to forget the minus sign on b, especially when a≠1. Remember the sign when forming equations from given sums or products.
Memory trick: Sum = –b/a, Product = c/a – keep the minus with b.