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
“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.
“When two exam scores are equal, each score equals itself, you can swap them, and if score A = B and B = C, then A = C.”
A relation R on a set is reflexive if every element relates to itself (∀x, xRx). It is symmetric if xRy implies yRx. It is transitive if xRy and yRz imply xRz. When all three properties hold, R is an equivalence relation. In JEE, forgetting any one of the three makes the relation non-equivalent.
Memory trick: R S T → Equivalence
“Even function mirrors a river’s banks about the y-axis; odd function flips a seesaw about the origin.”
Even functions satisfy f(−x)=f(x) and are symmetric about the y-axis; odd functions satisfy f(−x)=−f(x) and are symmetric about the origin. Both properties are preserved under addition, but multiplication follows specific rules. A common slip is labeling a function odd just because it passes through the origin.
Memory trick: Even = y-axis mirror; odd = origin flip
“Finding the inverse of f(x)=2x+3 is like rewinding a video to the exact frame you started from.”
An inverse function f⁻¹ undoes f: f⁻¹(f(x))=x. It exists only when f is one-to-one (injective) and onto (surjective), i.e., bijective. In JEE, ensure the given relation passes the horizontal line test before writing f⁻¹.
Memory trick: Invert only bijections – H-line test.
“|x| is like the distance a runner covers, regardless of whether they run forward or backward on a track.”
The modulus gives the non-negative distance of x from zero, so |x|≥0 for any real x. It equals x if x≥0 and –x if x<0. Remember: when solving |x|<a, split into –a<x<a; for |x|>a, two separate rays.
|x|
Memory trick: Abs = distance, never negative.
“The floor [x] is like the number of whole pizza slices you can take from a partially cut pizza.”
[x] returns the greatest integer ≤ x, i.e., it rounds down to the nearest integer. For negative numbers it moves further left on the number line (e.g., [-2.3]=-3). In JEE, watch that [x]≠x unless x is integer.
[x]
Memory trick: Floor = round down, even for negatives.
“aˣ behaves like a bank account where the balance multiplies by a fixed factor a each year.”
For a>0, f(x)=aˣ grows if a>1 and decays if 0<a<1, maintaining a constant ratio between successive values: aˣ⁺¹=a·aˣ. In JEE, remember that a⁰=1 and a⁻ˣ=1/aˣ; mis-applying sign is a common trap.
aˣ
Memory trick: Exp = constant ratio per unit step.
“Logarithms turn multiplication of ingredients into addition of their quantities, like adding teaspoons instead of mixing whole spoons.”
log_b(mn)=log_b m+log_b n, log_b(m/n)=log_b m−log_b n, log_b(m^p)=p·log_b m. These follow from the definition b^(log_b x)=x. In JEE, ensure the base is same for all terms; mixing bases leads to wrong simplification.
Memory trick: Log turns ×→+, ÷→−, power→coeff.
“A ladder leaning at 30° rises half its length vertically, because sin30° = ½.”
sin30°=½, sin45°=1/√2, sin60°=√3/2, sin90°=1. These follow from the 30-60-90 and 45-45-90 right triangles, the basis of all trig tables. In exams, remember these values apply to acute angles; using them in other quadrants without adjusting the sign gives wrong answers.
Memory trick: 30-60-90: sides 1,√3,2 → sin30=½, sin60=√3/2; 45-45-90: sides 1,1,√2 → sin45=1/√2.
“When a swing swings through twice the angle, its vertical rise follows sin2θ = 2sinθcosθ.”
sin2θ=2sinθcosθ; cos2θ can be written as cos²θ−sin²θ, 1−2sin²θ or 2cos²θ−1. Choose the form that matches the given data to avoid extra steps. A common trap is using cos2θ=cos²θ−sin²θ and then substituting cos²θ=1−sin²θ but forgetting the sign change.
sin2θ=2sinθcosθ
Memory trick: cos2θ = 1-2sin²θ (drop the cos²).
“Looking straight up at the sky gives a line with infinite slope; tan 90° is undefined because cos 90° = 0.”
tanθ = sinθ / cosθ, so it measures the slope of the terminal side. At θ = 90° (π/2 rad) cosθ = 0, making tan undefined (vertical line). Students often write tan90° = 0 or 1; the correct answer is “does not exist”. Also remember tan repeats every π, not 2π.
tanθ=sinθ/cosθ
Memory trick: tan = rise/run → sin ÷ cos.
“Rotate a right-triangle 90° and the side that was opposite becomes adjacent, turning sin into cos.”
For complementary angles (sum 90°), sin(90°-θ)=cosθ and cos(90°-θ)=sinθ. This comes from the right-triangle definition where the two acute angles swap opposite and adjacent sides. Ensure the angle is in degrees (or convert to radians) before applying the co-function identity.
Memory trick: Co-function: sin(90°-θ)=cosθ.
“A clock’s hour hand repeats every 12 h; solving sinθ = sinα adds full cycles and a mirror position, like the hand returning after a turn.”
If sinθ = sinα, solutions are θ = α + 2πn or θ = (π−α) + 2πn, compactly θ = nπ + (−1)ⁿα. For cosθ = cosα, θ = ±α + 2πn. Missing the second family or the (−1)ⁿ factor loses half the answers, a frequent exam mistake.
Memory trick: sinθ=sinα → θ=α or π−α (plus 2πk).