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
“Like a vending machine that gives exactly one snack for each coin you insert.”
A function assigns to every element of its domain exactly one element of the codomain. The rule may leave some codomain elements unused. The common trap is treating a relation that gives two outputs for one input as a function.
Memory trick: One input, one output
“Domain is the set of keys you can turn in a lock; range is the doors those keys can open.”
The domain is the set of all permissible inputs for the rule; the range (image) is the set of values actually produced. It is not the same as the codomain unless every possible output occurs. Students often mistake the codomain for the range, losing marks.
Memory trick: Domain in, range out
“Injective is like assigning each student a unique locker; surjective ensures every locker is occupied.”
Injective (one-one) means different inputs never share the same output; surjective (onto) means every element of the codomain is hit by some input. Both properties together give a bijection, possible only when domain and codomain have equal size. A frequent error is assuming a function is onto just because it is injective.
Memory trick: Injective = no repeats; surjective = no gaps
“Putting a smoothie through a blender then a filter: first g makes the mix, then f refines it.”
The composite (f∘g)(x) means apply g to x first, then feed that result into f. Order matters: (f∘g) ≠ (g∘f) in general. In exams, students often reverse the order, writing f(g(x)) as g(f(x)).
(f∘g)(x)=f(g(x))
Memory trick: Work inside-out
“A weather radar shows bright dots for thunderstorms (real roots, D>0) and fuzzy clouds for fog (complex roots, D<0).”
For ax²+bx+c=0, D=b²−4ac tells root nature: D>0 → two distinct real roots, D=0 → one repeated real root, D<0 → a pair of complex conjugates. The trap is to treat D=0 as ‘no root’ or to ignore the sign of D.
D = b²−4ac
Memory trick: D>0 two, D=0 double, D<0 complex.
“Your phone’s camera lens moves forward and backward; the ± in the quadratic formula captures both focus positions (the two roots).”
The roots are x = (−b ± √(b²−4ac))/(2a). It works only when a≠0; otherwise the equation is linear. A common mistake is plugging a=0 into the formula, which leads to division by zero.
x=(−b±√D)/2a
Memory trick: Minus b, then ±√D, all over 2a.
“A seesaw balances when the total weight (−b/a) sits at the pivot and the product of distances stays fixed (c/a).”
If ax²+bx+c=0 has roots α,β, then α+β = −b/a and α·β = c/a. Remember the sign: the sum carries a minus, the product does not. Students often flip the sign of the product or forget the division by a.
Memory trick: Sum = −b/a, Prod = c/a.
“Multiplying a point (x,y) by i rotates it 90° counter-clockwise, turning (x,y) into (-y,x).”
i is defined by i² = −1, extending real numbers to the complex plane. It acts as a 90° rotation, so two such rotations (i·i) give a 180° turn, i.e., –1. Remember i² is always –1, never +1.
i² = −1
Memory trick: i = 90° turn; i² = 180° → –1
“The modulus is the length of the arrow from the origin to (a,b), just like a GPS shows the straight-line distance to a location.”
|z| = √(a²+b²) is the Euclidean distance of the point (a,b) from the origin, always non-negative. For a real number it reduces to absolute value. Use it to check if a point lies on a circle |z-z₀|=r.
|z|=√(a²+b²)
Memory trick: √(a²+b²) = length of (a,b)
“Reflecting the point (a,b) across the x-axis gives (a,-b); that mirror image is the complex conjugate of a+bi.”
The conjugate of a+bi is a-bi. Multiplying z by its conjugate gives z·z̄ = a²+b² = |z|², a real non-negative number. In calculations always flip the sign of the imaginary part; otherwise the product stays complex.
z·z̄=|z|²
Memory trick: Conjugate = mirror over real axis
“Climbing a staircase where each step is 15 cm higher illustrates an arithmetic progression.”
An AP has a fixed common difference d between successive terms. The nth term is a+(n-1)d. The classic trap is writing a+nd (off by one) or assuming d must be positive—d can be negative too.
aₙ=a+(n−1)d
Memory trick: aₙ = a + (n-1)d → ‘first plus (steps-1)×step’
“Bacteria that double in number every hour follow a geometric progression.”
A GP multiplies each term by a constant ratio r. The nth term is a·r^(n-1). Beware using r=1 as a special case (constant sequence) and confusing r=0, which makes all later terms zero.
aₙ=a·r^(n−1)
Memory trick: aₙ = a·r^(n-1) → start at a, multiply (n-1) times
“Filling a tank by adding half of the remaining empty space each minute converges to a finite level.”
For |r|<1, an infinite GP sums to S∞ = a/(1-r). The exam trap is applying the formula when |r|≥1, which diverges, or forgetting that a negative r with |r|<1 still converges (alternating).
S∞=a/(1−r)
Memory trick: S∞ = a/(1-r) only if |r|<1
“A rectangle’s perimeter-square side equals the AM of its sides, while its area-square side equals the GM; the former is never shorter than the latter.”
For any non-negative numbers, the arithmetic mean (Σx/n) is always ≥ the geometric mean (ⁿ√Πx). Equality holds only when all numbers are identical. In JEE, note that a zero term makes GM zero and negative numbers are invalid for even-order roots.
Memory trick: AM≥GM: ‘Average beats root’ – equal only if all equal.
“Choosing a shirt (5 colors) and a tie (3 styles) gives 5×3 = 15 outfits.”
If one step can be done in m ways and a second independent step in n ways, the total ordered outcomes equal m·n. The principle extends to any number of independent stages. Trap: multiply only when choices do not affect each other; otherwise overcount.
Memory trick: AND → multiply, OR → add
“Arranging 3 books on a shelf in different orders shows 3P3 = 6 possible arrangements.”
A permutation counts ordered selections of r objects from n, given by nPr = n!/(n−r)!. Fix a position, choose any remaining item for each next spot, so order matters. Trap: using nCr when order is required gives a smaller answer.
nPr=n!/(n−r)!
Memory trick: P = Position matters
“Choosing 2 toppings from 4 for a pizza counts combinations, not the order of sprinkling.”
A combination counts unordered selections of r objects from n, formula nCr = n!/[r!(n−r)!]. Since order is irrelevant, divide the permutation count by r! ways to arrange the chosen items. Trap: forgetting the r! division and over-counting.
nCr=n!/(r!(n−r)!)
Memory trick: C = Choose, ignore order
“Opening a soda, each fizz bubble mirrors a term of (a+b)ⁿ expanding.”
(a+b)ⁿ = Σ_(r=0)ⁿ nCr a^(n−r) b^r. Coefficients are nCr and the exponents of a and b always add to n. A common trap is forgetting that r starts at 0, which drops the first term.
Tᵣ₊₁=nCr·a^(n−r)b^r
Memory trick: nCr = ways to choose r from n – think team selection.
“Finding the (r+1)th term is like stepping onto the r-th floor of a building that starts at ground zero.”
The (r+1)th term = nCr·a^(n−r)·b^r. Note the index is r+1 while the binomial coefficient uses r; mixing them shifts the exponents and gives a wrong term.
Tᵣ₊₁
Memory trick: T_(r+1) uses r in nCr and a^(n−r).
“When you rotate a unit stick on a compass, its x- and y-projections always satisfy x²+y²=1.”
For any angle sin²θ+cos²θ=1, a direct result of the unit-circle definition. It holds for acute, obtuse or reflex angles. The common trap is to forget the squares; the identity never changes sign.
sin²θ+cos²θ=1
Memory trick: Sin²+Cos²=1, like a right-triangle on the unit circle.
“A pizza slice of 180° is exactly half a circle, which equals π radians.”
π rad = 180°, so 1° = π/180 rad. Calculus formulas (e.g., d/dx sin x = cos x) are valid only when x is in radians. Using degrees introduces an extra factor π/180, a frequent error in differentiation problems.
π rad = 180°
Memory trick: 180° = π rad; multiply by π/180 to convert to rad.
“A swinging pendulum never tips beyond the walls of its arc, just as sin and cos never exceed ±1.”
For every real θ, –1 ≤ sinθ ≤ 1 and –1 ≤ cosθ ≤ 1 because they are the y- and x-coordinates of a point on the unit circle of radius 1. When solving equations, any root giving |value|>1 must be discarded as extraneous.
Memory trick: Sin, Cos ≤ radius 1 of unit circle.
“Mixing two musical notes of frequencies A and B gives beats described by sin(A±B) formulas.”
sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB. The sign in the cosine formula is opposite to that in the sine formula—mix-up leads to wrong results, especially in product-to-sum conversions.
Memory trick: Sin adds, Cos subtracts signs.
“In a survey, the height of a tower and its shadow form a triangle; the ratio side / sin opposite angle stays constant.”
In any triangle a/sinA = b/sinB = c/sinC. It works for acute, obtuse or scalene triangles. The rule fails when an angle is 0° or 180° because sin 0 = 0, making the ratio undefined.
a/sinA=b/sinB
Memory trick: Side over opposite sine = constant.