Algebraic identities
| Name | Identity |
|---|---|
| Square of a sum | (a + b)2 = a2 + 2ab + b2 |
| Square of a difference | (a - b)2 = a2 - 2ab + b2 |
| Difference of squares | a2 - b2 = (a + b)(a - b) |
| Cube of a sum | (a + b)3 = a3 + 3a2b + 3ab2 + b3 |
| Cube of a difference | (a - b)3 = a3 - 3a2b + 3ab2 - b3 |
| Sum of cubes | a3 + b3 = (a + b)(a2 - ab + b2) |
| Difference of cubes | a3 - b3 = (a - b)(a2 + ab + b2) |
| Square of three terms | (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx |
| Three cubes | If x + y + z = 0, then x3 + y3 + z3 = 3xyz |
Real numbers and equations
| Idea | Formula |
|---|---|
| Euclid division lemma | a = bq + r, where 0 ≤ r < b |
| HCF and LCM of two numbers | HCF × LCM = product of the two numbers |
| Quadratic equation | ax2 + bx + c = 0 (a is not 0) |
| Quadratic formula | x = (-b ± √(b2 - 4ac)) ÷ 2a |
| Discriminant | D = b2 - 4ac |
| Nature of roots | D > 0: two distinct real roots. D = 0: two equal real roots. D < 0: no real roots |
| Sum and product of roots | Sum = -b ÷ a, product = c ÷ a |
| Pair of linear equations | a1/a2 is not b1/b2: one solution. a1/a2 = b1/b2 is not c1/c2: no solution. All three equal: infinitely many solutions |
Arithmetic progression (AP)
| Idea | Formula |
|---|---|
| nth term | an = a + (n - 1)d |
| Sum of first n terms | Sn = (n ÷ 2)[2a + (n - 1)d] |
| Sum when the last term l is known | Sn = (n ÷ 2)(a + l) |
| Common difference | d = a2 - a1 |
a is the first term and d the common difference.
Coordinate geometry
| Idea | Formula for points (x1, y1) and (x2, y2) |
|---|---|
| Distance between two points | √[(x2 - x1)2 + (y2 - y1)2] |
| Midpoint | ((x1 + x2) ÷ 2, (y1 + y2) ÷ 2) |
| Section formula (ratio m : n, internal) | ((m x2 + n x1) ÷ (m + n), (m y2 + n y1) ÷ (m + n)) |
| Area of a triangle with a third point (x3, y3) | ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| |
Trigonometry
| Idea | Rule |
|---|---|
| sin θ | Opposite ÷ hypotenuse |
| cos θ | Adjacent ÷ hypotenuse |
| tan θ | Opposite ÷ adjacent = sin θ ÷ cos θ |
| cosec θ, sec θ, cot θ | 1 ÷ sin θ, 1 ÷ cos θ, 1 ÷ tan θ |
| Identity 1 | sin2θ + cos2θ = 1 |
| Identity 2 | 1 + tan2θ = sec2θ |
| Identity 3 | 1 + cot2θ = cosec2θ |
| Complementary angles | sin(90° - θ) = cos θ, tan(90° - θ) = cot θ |
Trigonometric values to remember
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | Not defined |
Mensuration
| Solid | Curved surface area | Total surface area | Volume |
|---|---|---|---|
| Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | lbh |
| Cube (a) | 4a2 | 6a2 | a3 |
| Cylinder (r, h) | 2πrh | 2πr(r + h) | πr2h |
| Cone (r, h, slant height l) | πrl | πr(l + r) | ⅓πr2h |
| Sphere (r) | 4πr2 | 4πr2 | (4/3)πr3 |
| Hemisphere (r) | 2πr2 | 3πr2 | (2/3)πr3 |
| Frustum of a cone (r1, r2, h, slant height l) | π(r1 + r2)l | π(r1 + r2)l + π(r12 + r22) | ⅓πh(r12 + r1r2 + r22) |
Slant height of a cone: l = √(r2 + h2). Slant height of a frustum: l = √(h2 + (r1 - r2)2).
Triangles, circles and sectors
| Idea | Formula |
|---|---|
| Heron's formula | Area = √[s(s - a)(s - b)(s - c)], where s = (a + b + c) ÷ 2 |
| Area of an equilateral triangle (side a) | (√3 ÷ 4) × a2 |
| Length of an arc (angle θ in degrees) | (θ ÷ 360) × 2πr |
| Area of a sector | (θ ÷ 360) × πr2 |
| Diagonal of a cuboid | √(l2 + b2 + h2) |
| Diagonal of a cube | √3 × a |
| Similar triangles | Ratio of areas = (ratio of corresponding sides)2 |
Statistics and probability
| Idea | Formula |
|---|---|
| Mean of grouped data (direct method) | Σfx ÷ Σf |
| Mean (assumed mean a, deviations d = x - a) | a + Σfd ÷ Σf |
| Median of grouped data | l + [(n/2 - cf) ÷ f] × h |
| Mode of grouped data | l + [(f1 - f0) ÷ (2f1 - f0 - f2)] × h |
| Empirical relation | Mode = 3 × Median - 2 × Mean |
| Probability of an event E | P(E) = favourable outcomes ÷ total outcomes |
| Complementary events | P(E) + P(not E) = 1 |
In the median and mode formulas, l is the lower limit of the median or modal class, h its class size, cf the cumulative frequency before it, f its frequency, and f0, f2 the frequencies of the classes before and after the modal class.