Algebra identities
- \( (a + b)^2 \)\( a^2 + 2ab + b^2 \)
- \( (a - b)^2 \)\( a^2 - 2ab + b^2 \)
- \( a^2 - b^2 \)\( (a + b)(a - b) \)
- \( (a + b + c)^2 \)\( a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \)
- \( (a + b)^3 \)\( a^3 + 3a^2b + 3ab^2 + b^3 \)
- \( (a - b)^3 \)\( a^3 - 3a^2b + 3ab^2 - b^3 \)
- \( a^3 + b^3 \)\( (a + b)(a^2 - ab + b^2) \)
- \( a^3 - b^3 \)\( (a - b)(a^2 + ab + b^2) \)
- \( a^3 + b^3 + c^3 - 3abc \)\( (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \)
- \( (a + b + c)^3 \)\( a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) \)
- \( a^4 + b^4 \)\( (a^2 + b^2)^2 - 2a^2b^2 \)
- \( a^4 - b^4 \)\( (a^2 + b^2)(a + b)(a - b) \)
- \( 1^2 + 2^2 + \dots + n^2 \)\( \frac{n(n + 1)(2n + 1)}{6} \)
- \( 1^3 + 2^3 + \dots + n^3 \)\( \left[\frac{n(n + 1)}{2}\right]^2 \)
- \( x^2 - (a + b)x + ab = 0 \)Quadratic equation with roots \( a \) and \( b \)
- Quadratic roots\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
- Discriminant\( D = b^2 - 4ac \) (\( D > 0 \): real distinct, \( D = 0 \): equal, \( D < 0 \): imaginary)
- Sum of rootsFor \( ax^2 + bx + c = 0 \implies -\frac{b}{a} \)
- Product of rootsFor \( ax^2 + bx + c = 0 \implies \frac{c}{a} \)
Number system basics
- Prime factor form \( N = p_1^a \cdot p_2^b \cdot p_3^c \dots \)
- Total factors If \( N = p_1^a \cdot p_2^b \cdot p_3^c \dots \) then total factors \( = (a + 1)(b + 1)(c + 1)\dots \)
- Sum of factors Product of \( (1 + p + p^2 + \dots + p^a) \) over all prime factors \( p^a \).
- Factorial \( n! = 1 \cdot 2 \cdot 3 \cdots n \)
- Trailing zeros in n! \( \lfloor n/5 \rfloor + \lfloor n/25 \rfloor + \lfloor n/125 \rfloor + \dots \)
- Sum 1 to n \( 1 + 2 + \dots + n = \frac{n(n + 1)}{2} \)
- Sum of first n evens \( 2 + 4 + \dots + 2n = n(n + 1) \)
- Sum of first n odds \( 1 + 3 + \dots + (2n - 1) = n^2 \)
- Sum of first n squares \( 1^2 + 2^2 + \dots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \)
- Sum of first n cubes \( 1^3 + 2^3 + \dots + n^3 = \left[\frac{n(n + 1)}{2}\right]^2 \)
- AP nth term \( T_n = a + (n - 1)d \)
- AP sum \( S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n(a + l)}{2} \)
- GP nth term \( T_n = a r^{n-1} \)
- GP sum (n terms) \( S_n = \frac{a(1 - r^n)}{1 - r} \) (\( r \neq 1 \))
- Infinite GP sum \( S_\infty = \frac{a}{1 - r} \) (\( |r| < 1 \))
- HCF × LCM For two numbers: \( \text{HCF} \times \text{LCM} = A \times B \)
- HCF from primes Product of common primes with minimum power.
- LCM from primes Product of all primes with maximum power.
- Even / odd form Even \( = 2k \), Odd \( = 2k + 1 \) (where \( k \in \mathbb{Z} \))
- Perfect square test In prime factorization all powers even ⇒ number is a perfect square.
Percentage & profit–loss
- Percentage value \( P\% \text{ of } X = \frac{P \times X}{100} \)
- Fraction to percent \( \frac{x}{y} \implies \left(\frac{x}{y}\right) \times 100\% \)
- Percent increase \( \text{New} = \text{Old} \times \left(1 + \frac{P}{100}\right) \)
- Percent decrease \( \text{New} = \text{Old} \times \left(1 - \frac{P}{100}\right) \)
- Successive x% and y% \( \text{Net}\% = x + y + \frac{xy}{100} \)
- Growth (compound) After \( n \) years: \( \text{Value} = P\left(1 + \frac{R}{100}\right)^n \)
- Depreciation After \( n \) years: \( \text{Value} = P\left(1 - \frac{R}{100}\right)^n \)
- Profit / loss \( \text{Profit} = \text{SP} - \text{CP}, \quad \text{Loss} = \text{CP} - \text{SP} \)
- Profit percent \( \text{Profit}\% = \frac{\text{Profit}}{\text{CP}} \times 100 \)
- Loss percent \( \text{Loss}\% = \frac{\text{Loss}}{\text{CP}} \times 100 \)
- SP for profit% \( \text{SP} = \text{CP} \times \left(1 + \frac{P}{100}\right) \)
- SP for loss% \( \text{SP} = \text{CP} \times \left(1 - \frac{L}{100}\right) \)
- CP from profit% \( \text{CP} = \frac{\text{SP} \times 100}{100 + P} \)
- CP from loss% \( \text{CP} = \frac{\text{SP} \times 100}{100 - L} \)
- Discount \( \text{Discount} = \text{MP} - \text{SP} \)
- Discount percent \( \text{Discount}\% = \frac{\text{Discount}}{\text{MP}} \times 100 \)
- Successive discounts \( \text{Equivalent}\% = x + y - \frac{xy}{100} \)
- A is x% more than B \( A = B\left(1 + \frac{x}{100}\right) \)
- A is x% less than B \( A = B\left(1 - \frac{x}{100}\right) \)
- Reverse percentage If A is \( x\% \) more than B, then B is \( \left[\frac{100x}{100 + x}\right]\% \) less than A.
- Price vs quantity Price increases by \( x\% \implies \text{Quantity} = \text{Original} \times \frac{100}{100 + x} \).
Simple & compound interest
- Simple interest \( \text{SI} = \frac{P \cdot R \cdot T}{100} \)
- Amount (SI) \( A = P + \text{SI} = P\left(1 + \frac{RT}{100}\right) \)
- Rate (SI) \( R = \frac{100 \times \text{SI}}{P \times T} \)
- Time (SI) \( T = \frac{100 \times \text{SI}}{P \times R} \)
- Amount (CI – yearly) \( A = P\left(1 + \frac{R}{100}\right)^T \)
- Compound interest \( \text{CI} = A - P = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right] \)
- CI – half‑yearly \( A = P\left(1 + \frac{R}{200}\right)^{2T} \)
- CI – quarterly \( A = P\left(1 + \frac{R}{400}\right)^{4T} \)
- Different rates each year \( A = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)\dots \)
- CI − SI (2 years) \( \text{Difference} = P\left(\frac{R}{100}\right)^2 \)
- CI − SI (3 years) \( \text{Difference} = P\left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right) \)
Mensuration (2D & 3D)
- Rectangle Perimeter \( = 2(l + b) \), Area \( = l \cdot b \), Diagonal \( = \sqrt{l^2 + b^2} \)
- Square Perimeter \( = 4a \), Area \( = a^2 \), Diagonal \( = a\sqrt{2} \)
- Triangle (base, height) \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
- Triangle (Heron) \( \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s = \frac{a + b + c}{2} \)
- Equilateral triangle \( \text{Area} = \frac{\sqrt{3}}{4}a^2 \), Height \( = \frac{\sqrt{3}}{2}a \)
- Parallelogram \( \text{Area} = \text{base} \times \text{height} \)
- Trapezium \( \text{Area} = \frac{1}{2}(a + b) \times h \)
- Circle Circumference \( = 2\pi r \), Area \( = \pi r^2 \)
- Sector (θ°) \( \text{Area} = \frac{\theta}{360^\circ}\pi r^2 \), Arc length \( = \frac{\theta}{360^\circ} \times 2\pi r \)
- Ring / annulus \( \text{Area} = \pi(R^2 - r^2) \)
- Cuboid Volume \( = l \cdot b \cdot h \), TSA \( = 2(lb + bh + hl) \)
- Cube Volume \( = a^3 \), TSA \( = 6a^2 \), Diagonal \( = a\sqrt{3} \)
- Cylinder Volume \( = \pi r^2 h \), CSA \( = 2\pi rh \), TSA \( = 2\pi r(h + r) \)
- Cone Volume \( = \frac{1}{3}\pi r^2 h \), Slant height \( l = \sqrt{r^2 + h^2} \), CSA \( = \pi r l \)
- Sphere Volume \( = \frac{4}{3}\pi r^3 \), Surface area \( = 4\pi r^2 \)
- Hemisphere Volume \( = \frac{2}{3}\pi r^3 \), CSA \( = 2\pi r^2 \), TSA \( = 3\pi r^2 \)
- Frustum of cone Volume \( = \frac{1}{3}\pi h (R^2 + r^2 + Rr) \)
Ratio, proportion & averages
- Ratio a : b Represents \( \frac{a}{b} \); multiplying terms by same factor keeps ratio same.
- Proportion \( a : b = c : d \implies \frac{a}{b} = \frac{c}{d} \implies ad = bc \)
- Mean proportional Between \( a \) and \( b \): \( x = \sqrt{ab} \)
- Third proportional \( a : b = b : x \implies x = \frac{b^2}{a} \)
- Fourth proportional \( a : b = c : x \implies x = \frac{bc}{a} \)
- Simple average \( \text{Average} = \frac{\sum x}{n} = \frac{\text{Sum of observations}}{\text{Total observations}} \)
- Combined average Combined \( \bar{x} = \frac{n_1 a_1 + n_2 a_2}{n_1 + n_2} \)
- Weighted average \( \bar{x}_w = \frac{\sum x_i w_i}{\sum w_i} \)
- Mixture mean price \( \frac{\text{Cheaper}}{\text{Dearer}} = \frac{D - M}{M - C} \)
- Partnership Profit share ∝ capital × time invested
- Speed vs time For fixed distance: \( \text{Speed} \propto \frac{1}{\text{Time}} \)
Time, work & speed
- Work relation \( \text{Work} = \text{Rate} \times \text{Time} \); rate for unit work \( = \frac{1}{\text{Time}} \)
- A & B together \( \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} \implies T = \frac{T_1 T_2}{T_1 + T_2} \)
- A, B, C together \( \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} + \frac{1}{T_3} \)
- Efficiency vs time \( \text{Time} \propto \frac{1}{\text{Efficiency}} \); if efficiency ratio \( a : b \implies \text{time ratio } b : a \).
- Filling pipe Fill rate \( = \frac{1}{\text{Time to fill}} \); emptying rate is negative.
- Speed basics \( \text{Speed} = \frac{\text{Distance}}{\text{Time}}, \quad \text{Distance} = \text{Speed} \times \text{Time} \)
- Unit conversion \( 1\text{ km/h} = \frac{5}{18}\text{ m/s}; \quad 1\text{ m/s} = \frac{18}{5}\text{ km/h} \)
- Average speed (equal distance) \( V_{\text{avg}} = \frac{2xy}{x + y} \)
- Train crosses pole \( \text{Time} = \frac{\text{Train length}}{\text{Speed}} \)
- Train crosses platform \( \text{Time} = \frac{\text{Train length} + \text{Platform length}}{\text{Speed}} \)
- Two trains opposite \( \text{Relative speed} = S_1 + S_2 \)
- Two trains same direction \( \text{Relative speed} = |S_1 - S_2| \)
- Downstream speed \( v_{\text{down}} = v_{\text{boat}} + v_{\text{stream}} \)
- Upstream speed \( v_{\text{up}} = v_{\text{boat}} - v_{\text{stream}} \)
- Boat speed \( v_{\text{boat}} = \frac{v_{\text{down}} + v_{\text{up}}}{2} \)
- Stream speed \( v_{\text{stream}} = \frac{v_{\text{down}} - v_{\text{up}}}{2} \)
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