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.
“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
MathsRelations & Functions· Class 12
Even and odd functions
“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
MathsRelations & Functions· Class 12
Inverse function
“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.
MathsRelations & Functions· Class 11
Modulus function
“|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.
MathsRelations & Functions· Class 11
Greatest integer function
“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.
MathsRelations & Functions· Class 11
Exponential function
“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.
MathsRelations & Functions· Class 11
Logarithm laws
“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.
MathsTrigonometry· Class 11
Standard trig values
“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.
“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).
MathsTrigonometry· Class 11
Tan of an angle
“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.
MathsTrigonometry· Class 11
Trigonometric ratios of complementary angles
“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.
MathsTrigonometry· Class 11
General solution of trig equations
“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).
MathsInverse Trigonometric Functions· Class 12
Inverse trigonometric functions
“A playground seesaw’s tilt angle = arcsin(rise / board length), so arcsin tells the exact angle from the height ratio.”
sin⁻¹x returns the unique angle θ whose sine equals x, so x must be in [–1, 1]. The principal-value range is restricted to –π/2 ≤ θ ≤ π/2, giving arcsin a single-valued output. In JEE keep the answer in radians unless degrees are explicitly required; mixing ranges costs marks.
Memory trick: Arcsin: ‘S’ for ‘Small’ range –π/2 to π/2.
MathsTrigonometry· Class 11
Heights and distances
“Surveyors point a theodolite at a distant tower; the measured elevation angle lets them compute the tower’s height.”
In height-distance problems, draw the right triangle with the line of sight as the hypotenuse. Use tanθ = opposite/adjacent to relate the unknown height to the measured horizontal distance. A trap is using sin or cos instead of tan for elevation/depression angles, giving a wrong dimension.
Memory trick: Elevation: tanθ = height / base.
MathsComplex Numbers· Class 11
Polar form of a complex number
“A wind-vane’s length tells the speed (r) and its direction the angle θ; together they give the complex number in polar form.”
Any complex number can be written as z = r(cosθ + i sinθ) where r = |z| ≥0 and θ = arg(z) measured from the positive real axis. For multiplication, moduli multiply and arguments add – a quick exam shortcut.
Memory trick: r = radius, θ = direction
MathsComplex Numbers· Class 11
De Moivre's theorem
“Spinning a wheel n times adds up to n times the single turn; De Moivre does the same for angles in complex numbers.”
(cosθ + i sinθ)ⁿ = cos nθ + i sin nθ for any integer n. It lets you raise powers or extract roots by scaling the angle. Do not use it for non-integer exponents unless you first switch to the exponential form.
Memory trick: Power → multiply angle, raise length
MathsComplex Numbers· Class 11
Cube roots of unity
“Place three equally spaced points on a clock at 12, 4, 8 o’clock; those positions represent 1, ω, ω², the cube roots of 1.”
The solutions of z³=1 are 1, ω, ω² with ω = e^(2πi/3). They satisfy ω³=1, 1+ω+ω²=0 and lie on the unit circle at 0°, 120°, 240°. Use ω² = –1–ω to simplify expressions.
1+ω+ω²=0
Memory trick: 120° steps on unit circle sum to zero
MathsQuadratic Equations· Class 11
Common root condition
“A common root is like a single key that can unlock two different locks – the same x-value fits both quadratic equations at zero.”
Two quadratics ax²+bx+c=0 and a' x²+b' x+c'=0 share a root iff their resultant, i.e. the determinant of the 4×4 Sylvester matrix, is zero. This condition is equivalent to the vanishing of the resultant polynomial in a,b,c,a',b',c'. A frequent trap is to use an incorrect simplified formula; always check the resultant or proportionality of coefficients.
Memory trick: Resultant = 0 → common root (R-zero means they’re linked).
MathsQuadratic Equations· Class 11
Nature of a quadratic graph
“A roller-coaster hill (a>0) creates a valley at its lowest point, while a dip (a<0) creates a peak at its highest point.”
A parabola y=ax²+bx+c opens upward when a>0, giving a minimum at the vertex; it opens downward when a<0, giving a maximum. The vertex y-value is −Δ/(4a). The usual error is to reverse the sign of a when deciding max/min.
“Pairing 1 with n, 2 with n-1, … each gives the same sum n+1, giving the total quickly.”
1+2+…+n = n(n+1)/2. The trap is forgetting the division by 2 or writing n(n-1)/2, which is the sum of first (n-1) numbers, not n.
Σn = n(n+1)/2
Memory trick: Sum = n·(first+last)/2
MathsSequences & Series· Class 11
Sum of squares
“Stacking square tiles to form a stepped pyramid uses the sum-of-squares formula.”
1²+2²+…+n² = n(n+1)(2n+1)/6. Students often mix up the denominator with 3 or 12; the correct factor is exactly 6.
n(n+1)(2n+1)/6
Memory trick: Squares sum = n(n+1)(2n+1)/6
MathsSequences & Series· Class 11
Sum of cubes
“Counting every rectangle on an n×n chessboard gives 1³+2³+…+n³, which equals (1+2+…+n)².”
The sum of the first n cubes equals the square of the sum of the first n natural numbers: 1³+2³+…+n³ = [n(n+1)/2]². It follows from combinatorial rectangle counting or algebraic pairing. A frequent mistake is writing n(n+1)/2 without squaring the whole term.
[n(n+1)/2]²
Memory trick: Cubes sum = (sum of 1 to n)² → (n(n+1)/2)².
MathsSequences & Series· Class 11
Harmonic progression
“If you run the same 5 km each day and your time each day grows by 2 min, the daily speeds form a harmonic progression.”
A sequence is a harmonic progression when the reciprocals of its terms form an arithmetic progression. Hence aₙ = 1/(A+(n-1)d). The trap is assuming the HP itself has a constant difference; only its reciprocals do.
Memory trick: HP ⇔ 1/(AP): aₙ = 1/(A+(n-1)d).
MathsSequences & Series· Class 11
Arithmetic mean between two numbers
“Finding the midpoint of two houses on a road gives the arithmetic mean of their distances.”
The arithmetic mean of a and b is (a+b)/2, the exact midpoint on the number line. It balances the two numbers equally. Trap: forgetting parentheses leads to a+b/2, which is wrong.