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Fundamental OF Mathematics formulas

Master Fundamental OF Mathematics through 30 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Fundamental OF Mathematics, every formula

30 formulas, typeset and free. Print it, or keep it open beside your practice.

Number sets hierarchy
$N \subset W \subset Z \subset Q \subset R$
Q1MCQNumber sets
Which set contains all the others?
- A
  $\mathbb{R}$
- B
  $\mathbb{N}$
- C
  $\mathbb{Z}$
- D
  $\mathbb{Q}$
Closed interval
$[a, b] = {x : a \leq x \leq b}$
Q1MCQClosed interval
$[2,5]$ includes:
- A
  both $2$ and $5$
- B
  neither $2$ nor $5$
- C
  only $2$
- D
  only $5$
Open interval
$(a, b) = {x : a < x < b}$
Q1MCQOpen interval
$(2,5)$ includes:
- A
  neither $2$ nor $5$
- B
  both $2$ and $5$
- C
  only $2$
- D
  only $5$
Modulus definition
$∣ x ∣ = {x - x x \geq 0 x < 0$
|x|<a
$∣ x ∣ < a \Leftrightarrow - a < x < a (a > 0)$
Q1MCQ|x|<a
$|x|<3$ means:
- A
  $-3<x<3$
- B
  $x<-3$ or $x>3$
- C
  $x>3$
- D
  $x=3$
|x|>a
$∣ x ∣ > a \Leftrightarrow x < - a or x > a$
Q1MCQ|x|>a
$|x|>2$ means:
- A
  $x<-2$ or $x>2$
- B
  $-2<x<2$
- C
  $x=2$
- D
  $0<x<2$
Triangle inequality
$∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$
Q1MCQTriangle inequality
$|x+y|$ compares with $|x|+|y|$ as:
- A
  $\le$
- B
  $\ge$
- C
  $=$ always
- D
  no relation
Product/quotient of modulus
$∣ x y ∣ = ∣ x ∣∣ y ∣, \frac{x}{y} = \frac{∣ x ∣}{∣ y ∣}$
Laws of exponents
$a^{m} \cdot a^{n} = a^{m + n}, \frac{a ^{m}}{a ^{n}} = a^{m - n}$
Q1MCQExponent laws
$a^{3}\cdot a^{4}$ equals:
- A
  $a^{7}$
- B
  $a^{12}$
- C
  $a^{1}$
- D
  $2a^{7}$
Power of a power
$(a^{m})^{n} = a^{mn}$
Q1MCQPower of power
$(a^{2})^{3}$ equals:
- A
  $a^{6}$
- B
  $a^{5}$
- C
  $a^{8}$
- D
  $a^{9}$
Zero & negative exponent
$a^{0} = 1, a^{- n} = \frac{1}{a ^{n}}$
Fractional exponent
$a^{m / n} = n a^{m}$
Q1MCQFractional exponent
$8^{2/3}$ equals:
- A
  $4$
- B
  $16$
- C
  $2$
- D
  $6$
Rationalizing a surd
$\frac{1}{a + b} = \frac{a - b}{a - b}$
Q1MCQRationalize
$\dfrac{1}{\sqrt3+\sqrt2}$ equals:
- A
  $\sqrt3-\sqrt2$
- B
  $\sqrt3+\sqrt2$
- C
  $\dfrac{\sqrt3-\sqrt2}{5}$
- D
  $1$
(a±b)²
$(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}$
Q1MCQ(a+b)²
$(x+3)^{2}$ equals:
- A
  $x^{2}+6x+9$
- B
  $x^{2}+9$
- C
  $x^{2}+3x+9$
- D
  $x^{2}+6x+3$
Difference of squares
$a^{2} - b^{2} = (a - b) (a + b)$
Q1MCQDifference of squares
$x^{2}-16$ factors as:
- A
  $(x-4)(x+4)$
- B
  $(x-4)^{2}$
- C
  $(x-16)(x+1)$
- D
  $(x+4)^{2}$
(a±b)³
$(a \pm b)^{3} = a^{3} \pm 3 a^{2} b + 3 a b^{2} \pm b^{3}$
Q1MCQ(a-b)³
$(x-1)^{3}$ equals:
- A
  $x^{3}-3x^{2}+3x-1$
- B
  $x^{3}-1$
- C
  $x^{3}-3x^{2}-3x-1$
- D
  $x^{3}+3x^{2}+3x+1$
Sum/difference of cubes
$a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp ab + b^{2})$
Q1MCQSum of cubes
$x^{3}+8$ factors as:
- A
  $(x+2)(x^{2}-2x+4)$
- B
  $(x+2)^{3}$
- C
  $(x+2)(x^{2}+2x+4)$
- D
  $(x-2)(x^{2}+2x+4)$
a³+b³+c³−3abc
$a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)$
Q1MCQa³+b³+c³−3abc
If $a+b+c=0$, then $a^{3}+b^{3}+c^{3}$ equals:
- A
  $3abc$
- B
  $0$
- C
  $abc$
- D
  $-3abc$
Logarithm definition
$lo g_{a} x = y \Leftrightarrow a^{y} = x, a > 0, a \neq = 1$
Log product/quotient
$lo g (mn) = lo g m + lo g n, lo g \frac{m}{n} = lo g m - lo g n$
Q1MCQLog product
$\log(mn)$ equals:
- A
  $\log m+\log n$
- B
  $\log m\cdot\log n$
- C
  $\log m-\log n$
- D
  $\dfrac{\log m}{\log n}$
Log power & base change
$lo g_{a} m^{n} = n lo g_{a} m, lo g_{a} m = \frac{lo g _{b} m}{lo g _{b} a}$
Q1MCQLog power
$\log_a m^{n}$ equals:
- A
  $n\log_a m$
- B
  $\log_a m+n$
- C
  $(\log_a m)^{n}$
- D
  $\dfrac{\log_a m}{n}$
Componendo–dividendo
$\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a + b}{a - b} = \frac{c + d}{c - d}$
Q1MCQComponendo-dividendo
If $\dfrac{a}{b}=\dfrac{c}{d}$, then $\dfrac{a+b}{a-b}$ equals:
- A
  $\dfrac{c+d}{c-d}$
- B
  $\dfrac{c-d}{c+d}$
- C
  $\dfrac{a}{b}$
- D
  $1$
Quadratic roots
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Q1MCQQuadratic roots
The roots of $x^{2}-5x+6=0$ are:
- A
  $2$ and $3$
- B
  $-2$ and $-3$
- C
  $1$ and $6$
- D
  $-1$ and $-6$
Discriminant
$D = b^{2} - 4 a c$
D>0 real distinct, =0 equal, <0 complex
Sign of a quadratic
$a > 0, D < 0 \Rightarrow a x^{2} + b x + c > 0 \forall x$
Q1MCQSign of quadratic
For $x^{2}+x+1$ ($a>0,\ D<0$), the expression is:
- A
  positive for all $x$
- B
  negative for all $x$
- C
  zero somewhere
- D
  changes sign
Wavy-curve (sign scheme)
$mark roots, alternate signs from the rightmost +$
Q1MCQWavy curve
In the wavy-curve method, the sign in the rightmost interval (all factors linear, positive leading) is:
- A
  $+$
- B
  $-$
- C
  $0$
- D
  undefined
AM ≥ GM
$\frac{a + b}{2} \geq ab$
a,b>0
Q1MCQAM≥GM
For $a,b>0$, $\dfrac{a+b}{2}$ compared with $\sqrt{ab}$ is:
- A
  $\ge$
- B
  $\le$
- C
  $=$ always
- D
  no relation
Factorial
$n! = 1 \cdot 2 \dots n, 0! = 1$
Remainder theorem
$f (x) \div (x - a) leaves remainder f (a)$
Factor theorem
$(x - a) is a factor \Leftrightarrow f (a) = 0$
Q1MCQFactor theorem
$(x-2)$ is a factor of $f(x)$ iff:
- A
  $f(2)=0$
- B
  $f(0)=2$
- C
  $f(2)=2$
- D
  $f'(2)=0$

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Fundamental OF Mathematics formulas

Fundamental OF Mathematics, every formula

Number sets hierarchy

Closed interval

Open interval

Modulus definition

|x|<a

|x|>a

Triangle inequality

Product/quotient of modulus

Laws of exponents

Power of a power

Zero & negative exponent

Fractional exponent

Rationalizing a surd

(a±b)²

Difference of squares

(a±b)³

Sum/difference of cubes

a³+b³+c³−3abc

Logarithm definition

Log product/quotient

Log power & base change

Componendo–dividendo

Quadratic roots

Discriminant

Sign of a quadratic

Wavy-curve (sign scheme)

AM ≥ GM

Factorial

Remainder theorem

Factor theorem

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

Preparing students for India’s top institutes

Join QuestPix, Today!