Symbols & Units प्रतीक और इकाइयाँ
Table of Symbols / प्रतीकों की तालिका
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| Symbol | Meaning | Symbol | Meaning |
|---|---|---|---|
| $=$ | equal to | $\neq$ | not equal to |
| $\equiv$ | identity | $\approx$ | approximately equal |
| $\cong$ | congruent to | $\propto$ | proportional to |
| $\therefore$ | therefore | $\because$ | since |
| $\infty$ | infinity | $\Sigma / \sigma$ | sigma / summation |
| $\angle$ | angle | $\perp$ | perpendicular |
| $\parallel$ | parallel | $\triangle$ | triangle |
| $\log_b a$ | log base b | $\log_{10}a$ | common log |
| $\ln a$ | natural log | $\mu$ | mean (average) |
| $\cup$ | union | $\cap$ | intersection |
| $\subset$ | subset of | $\supset$ | superset |
| $\phi$ | empty/null set | $!$ | factorial |
| $\exists$ | there exists | $\forall$ | for all |
| $\wedge$ | conjunction (and) | $\vee$ | disjunction (or) |
| $\Rightarrow$ | implication | $\sim$ | negation / equivalence |
| $i$ | imaginary unit | $\{ \}$ | set notation |
Conversion of Units / इकाइयों का रूपांतरण
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Length (लंबाई)
- 10 mm = 1 cm | 10 cm = 1 dm | 10 dm = 1 m
- 10 m = 1 dam | 10 dam = 1 hm | 10 hm = 1 km
- 1 km = 1000 m
- 12 inches = 1 foot (ft) = 0.3048 m
- 3 feet = 1 yard = 0.9144 m
- 22 yards = 1 chain
- 1 km = 0.621 mile; 1 mile = 1.6093 km = 1760 yards = 5280 feet
- 1 inch = 2.54 cm; 1 hectare = 2.471 acres
Area (क्षेत्रफल)
- 100 mm² = 1 cm²; 100 cm² = 1 dm²; 100 dm² = 1 m²
- 100 m² = 1 dam²; 100 dam² = 1 hm²; 100 hm² = 1 km²
- 1 hectare = 10000 m²
- 1 sq ft = 144 sq in = 0.0929 m²
- 1 sq yard = 0.836 m²; 1 acre = 4840 sq yards = 4046.86 m²
- 1 sq mile = 2.59 km² = 640 acres
Volume (आयतन)
- 1000 mm³ = 1 cm³; 1000 cm³ = 1 litre; 1000 litre = 1 m³
- 1 m³ = 10⁶ cm³; 1 L = 10⁻³ m³ = 10³ cm³
Weight (भार)
- 10 mg = 1 cg; 10 cg = 1 dg; 10 dg = 1 g
- 10 g = 1 dag; 10 dag = 1 hg; 10 hg = 1 kg
- 100 kg = 1 quintal; 1000 kg = 1 metric tonne
Time (समय)
- 60 s = 1 min; 60 min = 1 hr; 24 hr = 1 day; 7 days = 1 week
- 28/29/30/31 days = 1 month; 12 months = 1 year
- 365 days = 1 year; 366 days = leap year
- 10 years = decade; 25 = silver jubilee; 50 = golden jubilee
- 60 = diamond jubilee; 75 = platinum/radium jubilee; 100 = century
- 1000 years = millennium
Equivalents (इकाई समतुल्य)
- 1 sq km = 1000 hectares
- 1 litre capacity = 10 decilitres = 100 centilitres = 1000 millilitres
- 10 hectolitres = 1 kilolitre
Geometry ज्यामिति
Lines & Angles / रेखा और कोण
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Types of Angles (कोणों के प्रकार)
| Type | Condition |
|---|---|
| Acute | $0°< \theta < 90°$ |
| Right | $\theta = 90°$, $AB \perp BC$ |
| Obtuse | $90°< \theta < 180°$ |
| Straight/Line | $\theta = 180°$ |
| Reflex | $180°< \theta < 360°$ |
| Complete | $\theta = 360°$ |
Complementary: $\alpha + \beta = 90°$
Supplementary: $\alpha + \beta = 180°$
Supplementary = Complementary + 90°
Supplementary: $\alpha + \beta = 180°$
Supplementary = Complementary + 90°
Transversal Properties (तिर्यक रेखा)
- Corresponding: $\angle1=\angle5,\ \angle4=\angle8,\ \angle2=\angle6,\ \angle3=\angle7$
- Alternate interior: $\angle3=\angle5,\ \angle4=\angle6$
- Co-interior (same side): $\angle4+\angle5=180°,\ \angle3+\angle6=180°$
- Vertically opposite angles are equal
- Linear pair: $x+y=180°$ (supplementary)
- If AB∥CD transversal: $\alpha+\beta+\gamma=360°$
Intercept Theorem
$\dfrac{AB}{BC}=\dfrac{DE}{EF}=\dfrac{m}{n}$, $BE=\dfrac{an+bm}{m+n}$
Triangles — Properties & Types / त्रिभुज
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Basic Properties
- $\angle A+\angle B+\angle C=180°$
- Area $=\tfrac{1}{2}\times\text{base}\times\text{height}$
- $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$, $s=\tfrac{a+b+c}{2}$ (Heron)
- $\Delta=\tfrac{1}{2}ab\sin C=\tfrac{1}{2}bc\sin A=\tfrac{1}{2}ac\sin B$
- $ah_1=bh_2=ch_3=2\Delta=\text{constant}$
- $h_1:h_2:h_3=\tfrac{1}{a}:\tfrac{1}{b}:\tfrac{1}{c}$
- Exterior angle = sum of two non-adjacent interior angles
- Sum of all exterior angles $=360°$
Triangle Inequality (त्रिभुज असमिकाएं)
- $|b-c|<a<b+c$ (for all sides)
- Sum of any two sides > third side
- Difference of any two sides < third side
Types by Angle & Side Relation
| Type | Condition (c = longest) |
|---|---|
| Acute | $c^2<a^2+b^2$ |
| Right | $c^2=a^2+b^2$ |
| Obtuse | $c^2>a^2+b^2$ |
Side-Angle Ratios (भुजा-कोण अनुपात)
45°-45°-90° : sides $=1:1:\sqrt{2}$
30°-60°-90° : sides $=1:\sqrt{3}:2$
15°-75°-90° : sides $=(\sqrt{3}-1):(\sqrt{3}+1):2\sqrt{2}$
120°-30°-30° : $a:b:c=\sqrt{3}:1:1$
30°-60°-90° : sides $=1:\sqrt{3}:2$
15°-75°-90° : sides $=(\sqrt{3}-1):(\sqrt{3}+1):2\sqrt{2}$
120°-30°-30° : $a:b:c=\sqrt{3}:1:1$
Pythagoras Triplets (पाइथागोरस त्रिक)
$(3,4,5)\ (5,12,13)\ (7,24,25)\ (8,15,17)\ (9,40,41)\ (11,60,61)$
$(12,35,37)\ (13,84,85)\ (20,21,29)\ (28,45,53)\ (33,56,65)\ (36,77,85)$
$(39,80,89)\ (65,72,97)\ (20,99,101)$
Multiply or divide any triplet by same constant → still a triplet
$(12,35,37)\ (13,84,85)\ (20,21,29)\ (28,45,53)\ (33,56,65)\ (36,77,85)$
$(39,80,89)\ (65,72,97)\ (20,99,101)$
Multiply or divide any triplet by same constant → still a triplet
Sine Rule & Cosine Rule
Sine Rule:
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$
Cosine Rule:
$a^2=b^2+c^2-2bc\cos A$
$b^2=a^2+c^2-2ac\cos B$
$c^2=a^2+b^2-2ab\cos C$
$a^2=b^2+c^2-2bc\cos A$
$b^2=a^2+c^2-2ac\cos B$
$c^2=a^2+b^2-2ab\cos C$
Special Centres (विशेष केन्द्र)
| Centre | Definition | Key Formula | Position |
|---|---|---|---|
| Centroid G | Intersection of medians | Divides median $2:1$; $GO=R/3=H/6$ | Always inside |
| Circumcentre O | Equidistant from vertices | $R=\dfrac{abc}{4\Delta}$ | Acute:in; Right:hyp mid; Obtuse:out |
| Incentre I | Angle bisectors meet | $r=\dfrac{\Delta}{s}=\dfrac{P+B-H}{2}$(right△) | Always inside |
| Orthocentre H | Altitudes meet | $\angle BHC=180°-\angle A$ | Acute:in; Right:vertex; Obtuse:out |
Euler line: O, G, H are collinear. $OG:GH = 1:2$. Distance $d=\sqrt{R^2-2Rr}$
Stewart's Theorem (स्टीवर्ट प्रमेय)
If cevian $CD$ divides $AB$ into $m,n$ (c = m+n):
$a^2n+b^2m=x^2c+mnc$
In isosceles ($a=b$): $x^2=a^2-mn$; Length of angle bisector: $x^2=ab-mn$
$a^2n+b^2m=x^2c+mnc$
In isosceles ($a=b$): $x^2=a^2-mn$; Length of angle bisector: $x^2=ab-mn$
Similarity of Triangles / त्रिभुज की समरूपता
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- AA (Angle-Angle): 2 pairs of equal angles
- SSS (Side-Side-Side): $\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{CA}{FD}$
- SAS (Side-Angle-Side): 2 sides proportional & included angle equal
If $\triangle ABC \sim \triangle DEF$:
$$\frac{BC}{EF}=\frac{AC}{DF}=\frac{AB}{DE}=\frac{h_1}{h_2}=\frac{\text{bisector}_1}{\text{bisector}_2}=\frac{\text{median}_1}{\text{median}_2}=\frac{r_1}{r_2}=\frac{R_1}{R_2}$$
$$\frac{\text{Area }\triangle ABC}{\text{Area }\triangle DEF}=\left(\frac{BC}{EF}\right)^2=\left(\frac{AB}{DE}\right)^2$$
Thales Theorem (थेल्स प्रमेय)
If $DE\parallel BC$: $\dfrac{AD}{DB}=\dfrac{AE}{EC}=\dfrac{DE}{BC}$
Mid-Point Theorem
Line joining midpoints of two sides ∥ third side and equals half of it.
Quadrilaterals / चतुर्भुज
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| Shape | Area | Perimeter | Key Properties |
|---|---|---|---|
| Rectangle | $l\times b$ | $2(l+b)$ | Diagonals equal & bisect; $d=\sqrt{l^2+b^2}$ |
| Square (a) | $a^2$ | $4a$ | Diag $=a\sqrt{2}$; bisect at 90° |
| Parallelogram | $b\times h=ab\sin\theta$ | $2(a+b)$ | Opp. sides/angles equal; diagonals bisect |
| Rhombus ($d_1,d_2$) | $\tfrac{d_1 d_2}{2}$ | $4a$ | $a^2=\tfrac{d_1^2+d_2^2}{4}$; diag ⊥ bisect |
| Trapezium (a,b,h) | $\tfrac{(a+b)h}{2}$ | sum of 4 sides | mid-segment $=\tfrac{a+b}{2}$ |
| Kite | $\tfrac{d_1 d_2}{2}$ | $2(a+b)$ | One diagonal ⊥ bisects other |
Sum of interior angles of any quadrilateral = 360°
Cyclic Quadrilateral (चक्रीय चतुर्भुज)
- Opposite angles supplementary: $\angle A+\angle C=\angle B+\angle D=180°$
- Brahmagupta: $\Delta=\sqrt{(s-a)(s-b)(s-c)(s-d)}$
- Ptolemy: $AC\cdot BD=AB\cdot CD+BC\cdot AD$
- Isosceles trapezium is always cyclic; if $AB\parallel DC$ then $AD=BC$, $AC=BD$
Circles / वृत्त
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Basic Formulas
Area $=\pi r^2$
Circumference $=2\pi r$
Arc length $=\dfrac{\theta°}{360°}\times2\pi r$ or $r\theta$ (radians)
Sector area $=\dfrac{\theta°}{360°}\times\pi r^2=\dfrac{1}{2}lr$
Segment area $=$ Sector area $-$ Triangle area
Circumference $=2\pi r$
Arc length $=\dfrac{\theta°}{360°}\times2\pi r$ or $r\theta$ (radians)
Sector area $=\dfrac{\theta°}{360°}\times\pi r^2=\dfrac{1}{2}lr$
Segment area $=$ Sector area $-$ Triangle area
Chord Properties
- Perpendicular from centre bisects chord
- Equal chords are equidistant from centre
- If two chords $AB$ & $CD$ intersect at $P$: $PA\cdot PB=PC\cdot PD$
- Tangent-secant: $PT^2=PA\cdot PB$
- If 2 chords intersect at 90°: $r=\dfrac{\sqrt{x^2+y^2+z^2+w^2}}{2}$
Angle Theorems
- Angle at centre $=2\times$ inscribed angle (same arc)
- Angles in same segment are equal
- Angle in semicircle $=90°$
- Tangent $\perp$ radius at point of contact
- Two tangents from external point are equal: $PA=PB$
- Alternate segment: tangent-chord angle = inscribed angle in alternate segment
Two Circles
| Situation | Common Tangents |
|---|---|
| External touch ($d=r_1+r_2$) | 3 |
| Internal touch ($d=r_1-r_2$) | 1 |
| Intersect at 2 points | 2 |
| One inside other (no touch) | 0 |
| External, no touch | 4 |
2D Mensuration 2 आयामी क्षेत्रमिति
All 2D Shape Formulas
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| Shape | Area | Perimeter | Other Key Info |
|---|---|---|---|
| Scalene △ (a,b,c) | $\sqrt{s(s-a)(s-b)(s-c)}$ | $a+b+c$ | $s=\tfrac{a+b+c}{2}$ |
| Right △ (P,B,H) | $\tfrac{1}{2}PB$ | $P+B+H$ | $r=\tfrac{P+B-H}{2}$, $R=\tfrac{H}{2}$ |
| Isosceles-right △ (H) | $\tfrac{H^2}{4}$ | $H(1+\sqrt{2})$ | equal legs $=\tfrac{H}{\sqrt{2}}$ |
| Equilateral △ (a) | $\tfrac{\sqrt{3}}{4}a^2$ | $3a$ | $h=\tfrac{\sqrt{3}}{2}a$, $R=\tfrac{a}{\sqrt{3}}$, $r=\tfrac{a}{2\sqrt{3}}$ |
| Rectangle (l,b) | $lb$ | $2(l+b)$ | diag $=\sqrt{l^2+b^2}$ |
| Square (a) | $a^2$ | $4a$ | diag $=a\sqrt{2}$ |
| Parallelogram (b,h) | $bh=ab\sin\theta$ | $2(a+b)$ | — |
| Rhombus ($d_1,d_2$, side a) | $\tfrac{d_1 d_2}{2}$ | $4a$ | $a^2=\tfrac{d_1^2+d_2^2}{4}$ |
| Trapezium (a,b parallel, h) | $\tfrac{(a+b)h}{2}$ | sum of sides | mid-segment $=\tfrac{a+b}{2}$ |
| Circle (r) | $\pi r^2$ | $2\pi r$ | semi: $\tfrac{\pi r^2}{2}+2r$ |
| Sector ($r,\theta°$) | $\tfrac{\theta}{360}\pi r^2$ | $2r+\tfrac{\theta}{360}\cdot 2\pi r$ | arc $l=r\theta$ (rad) |
| Regular Hexagon (a) | $\tfrac{3\sqrt{3}}{2}a^2$ | $6a$ | $R=a$, $r=\tfrac{\sqrt{3}}{2}a$; diagonals=9 |
| Regular Octagon (a) | $2(\sqrt{2}+1)a^2$ | $8a$ | int.∠=135°, ext.∠=45°; diagonals=20 |
Scaling law for similar 2D figures (scale factor K):
Perimeter scales by $K$ | Area scales by $K^2$
Perimeter scales by $K$ | Area scales by $K^2$
Inscribed Square in Triangle
Square inside any triangle (base $x$, height $y$): Side $a=\dfrac{xy}{x+y}$
Square in right angle triangle (legs $a,b$): Side $=\dfrac{ab}{a+b}$
Square in right angle triangle (legs $a,b$): Side $=\dfrac{ab}{a+b}$
Polygons — Interior & Exterior Angles
$$\text{Sum of interior} = (n-2)\times180° \qquad \text{Each interior (regular)} = \frac{(n-2)\times180°}{n}$$
$$\text{Each exterior (regular)} = \frac{360°}{n} \qquad \text{Diagonals} = \frac{n(n-3)}{2}$$
3D Mensuration 3 आयामी क्षेत्रमिति
All 3D Shape Formulas
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| Shape | LSA / CSA | TSA | Volume |
|---|---|---|---|
| Cube (a) | $4a^2$ | $6a^2$ | $a^3$; diag $=a\sqrt{3}$ |
| Cuboid (l,b,h) | $2h(l+b)$ | $2(lb+bh+hl)$ | $lbh$; diag $=\sqrt{l^2+b^2+h^2}$ |
| Cylinder (r,h) | $2\pi rh$ | $2\pi r(r+h)$ | $\pi r^2h$ |
| Hollow Cylinder (R,r,h) | $2\pi h(R+r)$ | $2\pi(R+r)(h+R-r)$ | $\pi h(R^2-r^2)$ |
| Cone (r,h,l) | $\pi rl$ | $\pi r(r+l)$ | $\tfrac{1}{3}\pi r^2h$ |
| Sphere (r) | — | $4\pi r^2$ | $\tfrac{4}{3}\pi r^3$ |
| Hemisphere (r) | $2\pi r^2$ | $3\pi r^2$ | $\tfrac{2}{3}\pi r^3$ |
| Frustum (R,r,h,l) | $\pi(R+r)l$ | $\pi[R^2+r^2+(R+r)l]$ | $\tfrac{\pi h}{3}(R^2+r^2+Rr)$ |
| Square Pyramid (base a, h) | $2al$ | $a^2+2al$ | $\tfrac{1}{3}a^2h$ |
| Tetrahedron (a) | $\tfrac{3\sqrt{3}}{2}a^2$ | $\sqrt{3}a^2$ | $\tfrac{a^3}{6\sqrt{2}}$ |
Cone: slant $l=\sqrt{h^2+r^2}$. Frustum: $l=\sqrt{h^2+(R-r)^2}$.
Sphere-Cube Relations
Sphere in cube (side $a$): $r=\tfrac{a}{2}$; $\dfrac{V_{cube}}{V_{sphere}}=\dfrac{21}{11}$
Cube in sphere (radius $R$): $a=\dfrac{2R}{\sqrt{3}}$; diag of cube $=2R$
Cube in sphere (radius $R$): $a=\dfrac{2R}{\sqrt{3}}$; diag of cube $=2R$
Hollow Sphere (R,r)
Vol. of metal $=\tfrac{4}{3}\pi(R^3-r^3)$; Thickness $t=R-r$
Sphere in Cone (r,h)
Sphere radius $=\dfrac{hr}{l+r}$ where $l=\sqrt{h^2+r^2}$
Cutting a Sphere
- 1 cut → 2 hemispheres: TSA each $=3\pi r^2$
- 2 cuts → 4 pieces: TSA each $=2\pi r^2$
- 3 cuts (x,y,z axes) → 8 pieces: TSA each $=\tfrac{5}{4}\pi r^2$
- $n$ cuts through centre: $2n$ pieces
Special Prisms
| Prism | Base Area | Volume |
|---|---|---|
| Triangular equil. (a) | $\tfrac{\sqrt3}{4}a^2$ | $\tfrac{\sqrt3}{4}a^2h$ |
| Pentagonal (a) | $\sqrt3a^2$ | $\sqrt3a^2h$ |
| Hexagonal (a) | $\tfrac{3\sqrt3}{2}a^2$ | $\tfrac{3\sqrt3}{2}a^2h$ |
Number System संख्या प्रणाली
Types & Properties of Numbers
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- Natural (N): 1,2,3… | Whole (W): 0,1,2… | Integers (Z): …−2,−1,0,1,2…
- Rational: $p/q$, $q\neq0$; includes terminating & recurring decimals
- Irrational: non-terminating non-recurring: $\sqrt{2},\pi,e$
- Real = Rational ∪ Irrational
- Prime: exactly 2 factors. Smallest = 2 (only even prime). 1 is NOT prime.
- Composite: more than 2 factors. 1 is neither prime nor composite.
- Co-prime: HCF = 1. e.g. (2,3), (4,9), (11,13)
- Twin primes: differ by 2: (3,5),(5,7),(11,13),(17,19)
- Only prime triplet: 3, 5, 7. Smallest 3-digit prime: 101. Largest: 997.
- Perfect numbers: sum of proper factors = number: 6, 28, 496, 8128…
Divisibility Rules / विभाज्यता नियम
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| ÷ by | Rule |
|---|---|
| 2 | Last digit even (0,2,4,6,8) |
| 3 | Sum of all digits divisible by 3 |
| 4 | Last 2 digits divisible by 4 |
| 5 | Last digit 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 7 | Double last digit, subtract from rest; repeat. e.g. 1071→107−2=105→10−10=0 ✓ |
| 8 | Last 3 digits divisible by 8 |
| 9 | Sum of all digits divisible by 9 |
| 10 | Last digit is 0 |
| 11 | Alternating sum of digits (odd − even positions) = 0 or multiple of 11 |
| 12 | Divisible by 3 and 4 |
| 25 | Last 2 digits divisible by 25 |
| 125 | Last 3 digits divisible by 125 |
HCF & LCM
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$\text{HCF}\times\text{LCM}=a\times b$ (for two numbers)
For fractions: $\text{HCF}=\dfrac{\text{HCF(numerators)}}{\text{LCM(denominators)}}$, $\text{LCM}=\dfrac{\text{LCM(numerators)}}{\text{HCF(denominators)}}$
For fractions: $\text{HCF}=\dfrac{\text{HCF(numerators)}}{\text{LCM(denominators)}}$, $\text{LCM}=\dfrac{\text{LCM(numerators)}}{\text{HCF(denominators)}}$
- Largest number dividing $a,b,c$ leaving same remainder $r$: HCF of $(a-b),(b-c),(a-c)$
- Largest number dividing $a,b,c$ leaving remainders $r_1,r_2,r_3$: HCF of $(a-r_1),(b-r_2),(c-r_3)$
- Smallest number divisible by $a,b,c$: LCM of $a,b,c$
- Smallest number that leaves remainder $r$ when divided by $a,b,c$: LCM$(a,b,c)+r$
Surds & Indices घातांक और करणी
Laws of Indices & Surds
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Laws of Indices:
$a^m\times a^n\times a^p=a^{m+n+p}$
$\dfrac{a^m}{a^n}=a^{m-n}$; $a^0=1$; $a^{-n}=\dfrac{1}{a^n}$
$(a^m)^n=a^{mn}$; $(abc)^n=a^n b^n c^n$
$\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$; $\left(\dfrac{a}{b}\right)^{-m}=\left(\dfrac{b}{a}\right)^m$
$a^{p/q}=\sqrt[q]{a^p}=(\sqrt[q]{a})^p$
$(-1)^n=+1$ (n even); $-1$ (n odd)
$a^m\times a^n\times a^p=a^{m+n+p}$
$\dfrac{a^m}{a^n}=a^{m-n}$; $a^0=1$; $a^{-n}=\dfrac{1}{a^n}$
$(a^m)^n=a^{mn}$; $(abc)^n=a^n b^n c^n$
$\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$; $\left(\dfrac{a}{b}\right)^{-m}=\left(\dfrac{b}{a}\right)^m$
$a^{p/q}=\sqrt[q]{a^p}=(\sqrt[q]{a})^p$
$(-1)^n=+1$ (n even); $-1$ (n odd)
Laws of Surds:
$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$
$\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$
$(\sqrt[n]{a})^m=a^{m/n}$
$\sqrt{a}\times\sqrt{a}=a$; $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$
$(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{ab}$
$(\sqrt{a}-\sqrt{b})^2=a+b-2\sqrt{ab}$
$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$
$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$
$\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$
$(\sqrt[n]{a})^m=a^{m/n}$
$\sqrt{a}\times\sqrt{a}=a$; $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$
$(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{ab}$
$(\sqrt{a}-\sqrt{b})^2=a+b-2\sqrt{ab}$
$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$
Square Root Shortcut (वर्गमूल शॉर्टकट)
$\sqrt{a+b+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$
e.g. $\sqrt{7+4\sqrt{3}}=\sqrt{4+3+2\sqrt{4\times3}}=2+\sqrt{3}$
$\sqrt{28+10\sqrt{3}}=\sqrt{25+3+2\times5\times\sqrt{3}}=5+\sqrt{3}$
e.g. $\sqrt{7+4\sqrt{3}}=\sqrt{4+3+2\sqrt{4\times3}}=2+\sqrt{3}$
$\sqrt{28+10\sqrt{3}}=\sqrt{25+3+2\times5\times\sqrt{3}}=5+\sqrt{3}$
Useful Results
- If $x=\dfrac{4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}$ then $\dfrac{x+2\sqrt{a}}{x-2\sqrt{a}}+\dfrac{x+2\sqrt{b}}{x-2\sqrt{b}}=2$
- If $x=\dfrac{2\sqrt{ab}}{\sqrt{a}+\sqrt{b}}$ then $\dfrac{x+\sqrt{a}}{x-\sqrt{a}}+\dfrac{x+\sqrt{b}}{x-\sqrt{b}}=2$
- If $\dfrac{a-b}{\sqrt{ab}}=k$: $\left(\dfrac{a}{b}\right)^3-\left(\dfrac{b}{a}\right)^3=k^3+3k$
Algebra बीजगणित
Algebraic Identities / बीजगणितीय सर्वसमिकाएं
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$(a+b)^2=a^2+2ab+b^2$
$(a-b)^2=a^2-2ab+b^2$
$(a+b)(a-b)=a^2-b^2$
$(a+b)^3=a^3+3a^2b+3ab^2+b^3$
$(a-b)^3=a^3-3a^2b+3ab^2-b^3$
$a^3+b^3=(a+b)(a^2-ab+b^2)$
$a^3-b^3=(a-b)(a^2+ab+b^2)$
$(a-b)^2=a^2-2ab+b^2$
$(a+b)(a-b)=a^2-b^2$
$(a+b)^3=a^3+3a^2b+3ab^2+b^3$
$(a-b)^3=a^3-3a^2b+3ab^2-b^3$
$a^3+b^3=(a+b)(a^2-ab+b^2)$
$a^3-b^3=(a-b)(a^2+ab+b^2)$
$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$
$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
If $a+b+c=0$: $a^3+b^3+c^3=3abc$
$(a+b)^2-(a-b)^2=4ab$
$(a+b)^2+(a-b)^2=2(a^2+b^2)$
$a^4-b^4=(a^2+b^2)(a+b)(a-b)$
$a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$
$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
If $a+b+c=0$: $a^3+b^3+c^3=3abc$
$(a+b)^2-(a-b)^2=4ab$
$(a+b)^2+(a-b)^2=2(a^2+b^2)$
$a^4-b^4=(a^2+b^2)(a+b)(a-b)$
$a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$
Higher Power Formulae (उच्च घात सूत्र)
$x^4-\dfrac{1}{x^4}=\left(x^2+\dfrac{1}{x^2}\right)\!\left(x+\dfrac{1}{x}\right)\!\left(x-\dfrac{1}{x}\right)$
$x^5+\dfrac{1}{x^5}=\left(x^2+\dfrac{1}{x^2}\right)\!\left(x^3+\dfrac{1}{x^3}\right)-\left(x+\dfrac{1}{x}\right)$
$x^7-\dfrac{1}{x^7}=\left(x^4+\dfrac{1}{x^4}\right)\!\left(x^3-\dfrac{1}{x^3}\right)+\!\left(x-\dfrac{1}{x}\right)$
$x^8-\dfrac{1}{x^8}=\left(x^4+\dfrac{1}{x^4}\right)\!\left(x^2+\dfrac{1}{x^2}\right)\!\left(x+\dfrac{1}{x}\right)\!\left(x-\dfrac{1}{x}\right)$
$x^5+\dfrac{1}{x^5}=\left(x^2+\dfrac{1}{x^2}\right)\!\left(x^3+\dfrac{1}{x^3}\right)-\left(x+\dfrac{1}{x}\right)$
$x^7-\dfrac{1}{x^7}=\left(x^4+\dfrac{1}{x^4}\right)\!\left(x^3-\dfrac{1}{x^3}\right)+\!\left(x-\dfrac{1}{x}\right)$
$x^8-\dfrac{1}{x^8}=\left(x^4+\dfrac{1}{x^4}\right)\!\left(x^2+\dfrac{1}{x^2}\right)\!\left(x+\dfrac{1}{x}\right)\!\left(x-\dfrac{1}{x}\right)$
Formulae based on $x + \tfrac{1}{x}$
▼
$\left(x+\dfrac{1}{x}\right)^2=x^2+\dfrac{1}{x^2}+2$ | $\left(x-\dfrac{1}{x}\right)^2=x^2+\dfrac{1}{x^2}-2$
$\left(x+\dfrac{1}{x}\right)^3=x^3+\dfrac{1}{x^3}+3\left(x+\dfrac{1}{x}\right)$
$\left(x-\dfrac{1}{x}\right)^3=x^3-\dfrac{1}{x^3}-3\left(x-\dfrac{1}{x}\right)$
$\left(x+\dfrac{1}{x}\right)^3=x^3+\dfrac{1}{x^3}+3\left(x+\dfrac{1}{x}\right)$
$\left(x-\dfrac{1}{x}\right)^3=x^3-\dfrac{1}{x^3}-3\left(x-\dfrac{1}{x}\right)$
| If $x+\tfrac{1}{x}=$ | Then |
|---|---|
| $2$ | $x=1$ |
| $-2$ | $x=-1$ |
| $1$ | $x^3=-1$; $x^2-x+1=0$ |
| $-1$ | $x^3=1$; $x^2+x+1=0$ |
| $\pm\sqrt{3}$ | $x^3+\tfrac{1}{x^3}=0$; $x^6=-1$ |
Power 2 & Power 4 Relations
$(x^2+y^2)^2=x^4+y^4+2x^2y^2$
$(x^2+y^2)^2-(xy)^2=x^4+y^4+x^2y^2$
If $x^2+y^2+xy=A$ and $x^2+y^2-xy=B$: $x^2+y^2=\dfrac{A+B}{2}$, $xy=\dfrac{A-B}{2}$
$(x^2+y^2)^2-(xy)^2=x^4+y^4+x^2y^2$
If $x^2+y^2+xy=A$ and $x^2+y^2-xy=B$: $x^2+y^2=\dfrac{A+B}{2}$, $xy=\dfrac{A-B}{2}$
Quadratic Equations & Coordinate Geometry
▼
Quadratic Equation
For $ax^2+bx+c=0$:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
$\alpha+\beta=-\dfrac{b}{a}$, $\alpha\beta=\dfrac{c}{a}$
$D=b^2-4ac$: >0 real distinct; =0 equal; <0 complex
$D=b^2-4ac$: >0 real distinct; =0 equal; <0 complex
Coordinate Geometry
Distance $=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Midpoint $=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$
Section $m:n$ $=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$
Area of △ $=\tfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$
Slope $=\dfrac{y_2-y_1}{x_2-x_1}=\tan\theta$
Midpoint $=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$
Section $m:n$ $=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$
Area of △ $=\tfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$
Slope $=\dfrac{y_2-y_1}{x_2-x_1}=\tan\theta$
Sequences & Series अनुक्रम और श्रृंखला
AP, GP, HP
▼
Arithmetic Progression (AP)
$a_n=a+(n-1)d$
$S_n=\dfrac{n}{2}[2a+(n-1)d]=\dfrac{n}{2}(a+l)$
$\text{AM}=\dfrac{a+b}{2}$
If $a_m=n$, $a_n=m$: $a_{m+n}=0$, $a_{mn}=1$, $a_p=\dfrac{mn}{p}$
$S_n=\dfrac{n}{2}[2a+(n-1)d]=\dfrac{n}{2}(a+l)$
$\text{AM}=\dfrac{a+b}{2}$
If $a_m=n$, $a_n=m$: $a_{m+n}=0$, $a_{mn}=1$, $a_p=\dfrac{mn}{p}$
Middle term of AP = AM. If △ angles in AP → middle angle = 60°.
Geometric Progression (GP)
$a_n=ar^{n-1}$
$S_n=\dfrac{a(r^n-1)}{r-1}$ $(r>1)$ or $\dfrac{a(1-r^n)}{1-r}$ $(r<1)$
$S_\infty=\dfrac{a}{1-r}$ $\left(|r|<1\right)$
$\text{GM}=\sqrt{ab}$
$S_n=\dfrac{a(r^n-1)}{r-1}$ $(r>1)$ or $\dfrac{a(1-r^n)}{1-r}$ $(r<1)$
$S_\infty=\dfrac{a}{1-r}$ $\left(|r|<1\right)$
$\text{GM}=\sqrt{ab}$
Harmonic Progression (HP)
$\text{HM}=\dfrac{2ab}{a+b}$ (two numbers)
$\dfrac{1}{H}=\dfrac{1}{n}\!\left(\dfrac{1}{a_1}+\dfrac{1}{a_2}+\cdots+\dfrac{1}{a_n}\right)$
If $a_p=q$, $a_q=p$: $a_{pq}=1$
If H is HM of $a,b$: $(H-2a)(H-2b)=H^2$; $\dfrac{H+a}{H-a}+\dfrac{H+b}{H-b}=2$
$\dfrac{1}{H}=\dfrac{1}{n}\!\left(\dfrac{1}{a_1}+\dfrac{1}{a_2}+\cdots+\dfrac{1}{a_n}\right)$
If $a_p=q$, $a_q=p$: $a_{pq}=1$
If H is HM of $a,b$: $(H-2a)(H-2b)=H^2$; $\dfrac{H+a}{H-a}+\dfrac{H+b}{H-b}=2$
Key Inequality: $\text{AM}\geq\text{GM}\geq\text{HM}$
$A=\dfrac{a+b}{2}$, $G=\sqrt{ab}$, $H=\dfrac{2ab}{a+b}$, $G^2=AH$
$A=\dfrac{a+b}{2}$, $G=\sqrt{ab}$, $H=\dfrac{2ab}{a+b}$, $G^2=AH$
Summation Formulae & Special Series
▼
$$\sum_{r=1}^n r=\frac{n(n+1)}{2} \qquad \sum_{r=1}^n r^2=\frac{n(n+1)(2n+1)}{6}$$
$$\sum_{r=1}^n r^3=\left[\frac{n(n+1)}{2}\right]^2 \qquad \sum_{r=1}^n r^4=\frac{n(n+1)(6n^3+9n^2+n-1)}{30}$$
Sum of first $n$ even naturals $=n(n+1)$ | Sum of first $n$ odd naturals $=n^2$
Alternating Series $1^2-2^2+3^2\cdots$
$n$ odd: $\dfrac{n(n+1)}{2}$
$n$ even: $\dfrac{-n(n+1)}{2}$
$n$ even: $\dfrac{-n(n+1)}{2}$
Exponential Series
$e=1+\dfrac{1}{1!}+\dfrac{1}{2!}+\cdots$ ($2<e<3$, irrational)
$e^x=\sum_{n=0}^\infty\dfrac{x^n}{n!}=1+x+\dfrac{x^2}{2!}+\cdots$
$e^{-x}=1-x+\dfrac{x^2}{2!}-\dfrac{x^3}{3!}+\cdots$
$e^x=\sum_{n=0}^\infty\dfrac{x^n}{n!}=1+x+\dfrac{x^2}{2!}+\cdots$
$e^{-x}=1-x+\dfrac{x^2}{2!}-\dfrac{x^3}{3!}+\cdots$
Trigonometry त्रिकोणमिति
Ratios, Values & Identities
▼
$\sin\theta=\dfrac{P}{H}$, $\cos\theta=\dfrac{B}{H}$, $\tan\theta=\dfrac{P}{B}$, $\csc\theta=\dfrac{H}{P}$, $\sec\theta=\dfrac{H}{B}$, $\cot\theta=\dfrac{B}{P}$
| θ | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
|---|---|---|---|---|---|---|---|---|---|
| sin | 0 | ½ | $\tfrac{1}{\sqrt2}$ | $\tfrac{\sqrt3}{2}$ | 1 | $\tfrac{\sqrt3}{2}$ | $\tfrac{1}{\sqrt2}$ | ½ | 0 |
| cos | 1 | $\tfrac{\sqrt3}{2}$ | $\tfrac{1}{\sqrt2}$ | ½ | 0 | $-\tfrac{1}{2}$ | $-\tfrac{1}{\sqrt2}$ | $-\tfrac{\sqrt3}{2}$ | −1 |
| tan | 0 | $\tfrac{1}{\sqrt3}$ | 1 | $\sqrt3$ | ∞ | $-\sqrt3$ | −1 | $-\tfrac{1}{\sqrt3}$ | 0 |
Pythagorean Identities:
$\sin^2\theta+\cos^2\theta=1$
$1+\tan^2\theta=\sec^2\theta$
$1+\cot^2\theta=\csc^2\theta$
$\sin^2\theta+\cos^2\theta=1$
$1+\tan^2\theta=\sec^2\theta$
$1+\cot^2\theta=\csc^2\theta$
Compound Angles:
$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$
$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$
$\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}$
$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$
$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$
$\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}$
$\sin2A=2\sin A\cos A$; $\cos2A=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1$
$\tan2A=\dfrac{2\tan A}{1-\tan^2A}$
ASTC: Quadrant I—All; II—sin,csc; III—tan,cot; IV—cos,sec positive.
$\tan2A=\dfrac{2\tan A}{1-\tan^2A}$
ASTC: Quadrant I—All; II—sin,csc; III—tan,cot; IV—cos,sec positive.
Height & Distance / ऊँचाई और दूरी
▼
| # | Situation | Formula |
|---|---|---|
| 1 | Two observers same side: angles $\theta_1,\theta_2$ at distance $a$ | $d=h(\cot\theta_1-\cot\theta_2)$ |
| 2 | Observers opposite sides: angles $\theta_1,\theta_2$, base $a$ | $a=h(\cot\theta_1+\cot\theta_2)$ |
| 3 | Elevation changes 30°→60° | $h=\dfrac{d\sqrt{3}}{2}$ |
| 4 | Elevation $\theta$→$2\theta$ (moved $x$, tower far $y$) | $h^2=y^2-x^2$ |
| 5 | Complementary elevations ($\theta+\alpha=90°$), dist $x$ | $h=\sqrt{x\cdot d}=\sqrt{H_1 H_2}$ |
| 6 | Two towers $H_1,H_2$ across road of width $a$ | $\dfrac{1}{h}=\dfrac{1}{H_1}+\dfrac{1}{H_2}$ (cross-height) |
| 7 | Elevation changes 0°→$2\theta$ (base $x$, far $y$) | $h^2=y^2-x^2$ |
Percentage प्रतिशत
Core Formulas & Shortcuts
▼
$x\%$ of $y=\dfrac{xy}{100}$ | % increase $=\dfrac{\text{increase}}{\text{original}}\times100$
New (increase by $r\%$) $=\text{original}\times\dfrac{100+r}{100}$
Net change for $x\%$ then $y\%$: $\left(x+y+\dfrac{xy}{100}\right)\%$
New (increase by $r\%$) $=\text{original}\times\dfrac{100+r}{100}$
Net change for $x\%$ then $y\%$: $\left(x+y+\dfrac{xy}{100}\right)\%$
A is $x\%$ more than B:
B is $\dfrac{x}{100+x}\times100\%$ less than A
A is $x\%$ less than B:
B is $\dfrac{x}{100-x}\times100\%$ more than A
B is $\dfrac{x}{100+x}\times100\%$ less than A
A is $x\%$ less than B:
B is $\dfrac{x}{100-x}\times100\%$ more than A
Price–Quantity:
Price↑ $x\%$ → reduce qty by $\dfrac{x}{100+x}\times100\%$
Price↓ $x\%$ → increase qty by $\dfrac{x}{100-x}\times100\%$
Price↑ $x\%$ → reduce qty by $\dfrac{x}{100+x}\times100\%$
Price↓ $x\%$ → increase qty by $\dfrac{x}{100-x}\times100\%$
Sequential Deductions / Increases
After $a\%$, $b\%$, $c\%$ deductions: Initial $=x\cdot\dfrac{100}{100-a}\cdot\dfrac{100}{100-b}\cdot\dfrac{100}{100-c}$
After $a\%$, $b\%$, $c\%$ increases: Initial $=x\cdot\dfrac{100}{100+a}\cdot\dfrac{100}{100+b}\cdot\dfrac{100}{100+c}$
After $a\%$, $b\%$, $c\%$ increases: Initial $=x\cdot\dfrac{100}{100+a}\cdot\dfrac{100}{100+b}\cdot\dfrac{100}{100+c}$
Exam-Based (परीक्षा आधारित)
Failed in Maths $a\%$, English $b\%$, both $c\%$:
Passed both $=(100-(a+b-c))\%$ | Failed either $=(a+b-c)\%$
Spending $x\%,y\%,z\%$, saving ₹R: Monthly income $=\dfrac{100}{100-(x+y+z)}\times R$
Passed both $=(100-(a+b-c))\%$ | Failed either $=(a+b-c)\%$
Spending $x\%,y\%,z\%$, saving ₹R: Monthly income $=\dfrac{100}{100-(x+y+z)}\times R$
Profit & Loss लाभ और हानि
All Profit & Loss Formulas
▼
$P=SP-CP$; $L=CP-SP$
$P\%=\dfrac{P}{CP}\times100$; $L\%=\dfrac{L}{CP}\times100$
$SP=CP\times\dfrac{100+P\%}{100}$ (profit) | $SP=CP\times\dfrac{100-L\%}{100}$ (loss)
$CP=\dfrac{SP\times100}{100+P\%}$ (profit) | $CP=\dfrac{SP\times100}{100-L\%}$ (loss)
$P\%=\dfrac{P}{CP}\times100$; $L\%=\dfrac{L}{CP}\times100$
$SP=CP\times\dfrac{100+P\%}{100}$ (profit) | $SP=CP\times\dfrac{100-L\%}{100}$ (loss)
$CP=\dfrac{SP\times100}{100+P\%}$ (profit) | $CP=\dfrac{SP\times100}{100-L\%}$ (loss)
Discount (छूट)
Discount $=MP-SP$; $D\%=\dfrac{D}{MP}\times100$
$SP=MP\times\dfrac{100-D\%}{100}$
Two successive discounts $d_1\%,d_2\%$: Net $=\left(d_1+d_2-\dfrac{d_1 d_2}{100}\right)\%$
Three equal discounts giving net $65.7\%$: $\sqrt[3]{100}:\sqrt[3]{100-D\%}$
$SP=MP\times\dfrac{100-D\%}{100}$
Two successive discounts $d_1\%,d_2\%$: Net $=\left(d_1+d_2-\dfrac{d_1 d_2}{100}\right)\%$
Three equal discounts giving net $65.7\%$: $\sqrt[3]{100}:\sqrt[3]{100-D\%}$
False Weight:
$P\%=\dfrac{\text{True wt}-\text{False wt}}{\text{False wt}}\times100$
$P\%=\dfrac{\text{True wt}-\text{False wt}}{\text{False wt}}\times100$
Same SP, one $P\%$ profit one $P\%$ loss:
Always a loss $=\left(\dfrac{P}{10}\right)^2\%$
Always a loss $=\left(\dfrac{P}{10}\right)^2\%$
Simple & Compound Interest साधारण और चक्रवृद्धि ब्याज
Simple Interest (SI) / साधारण ब्याज
▼
$SI=\dfrac{P\times R\times T}{100}$; $P=\dfrac{SI\times100}{RT}$; $R=\dfrac{SI\times100}{PT}$; $T=\dfrac{SI\times100}{PR}$
$A=P+SI$; $SI=A-P$
$A=P+SI$; $SI=A-P$
| Compounding | Rate Used | Time Used |
|---|---|---|
| Annual | $r\%$ | $t$ years |
| Half-yearly | $r/2\%$ | $2t$ half-years |
| Quarterly | $r/4\%$ | $4t$ quarters |
| Monthly | $r/12\%$ | $12t$ months |
SI ∝ T (P,R const) · SI ∝ P (R,T const) · SI ∝ R (P,T const)
Compound Interest (CI) / चक्रवृद्धि ब्याज
▼
$A=P\!\left(1+\dfrac{r}{100}\right)^{\!n}$; $CI=A-P$
Half-yearly: $A=P\!\left(1+\dfrac{r}{200}\right)^{\!2n}$
Quarterly: $A=P\!\left(1+\dfrac{r}{400}\right)^{\!4n}$
Half-yearly: $A=P\!\left(1+\dfrac{r}{200}\right)^{\!2n}$
Quarterly: $A=P\!\left(1+\dfrac{r}{400}\right)^{\!4n}$
CI vs SI:
2 yrs: $CI-SI=P\!\left(\dfrac{r}{100}\right)^2$
3 yrs: $CI-SI=P\!\left(\dfrac{r}{100}\right)^2\!\left(3+\dfrac{r}{100}\right)$
2 yrs: $CI-SI=P\!\left(\dfrac{r}{100}\right)^2$
3 yrs: $CI-SI=P\!\left(\dfrac{r}{100}\right)^2\!\left(3+\dfrac{r}{100}\right)$
Population / Depreciation:
Growth: $P_n=P_0\!\left(1+\dfrac{r}{100}\right)^n$
Decay: $V_n=V_0\!\left(1-\dfrac{r}{100}\right)^n$
Growth: $P_n=P_0\!\left(1+\dfrac{r}{100}\right)^n$
Decay: $V_n=V_0\!\left(1-\dfrac{r}{100}\right)^n$
Time, Speed & Distance समय, गति और दूरी
Core TSD Formulas
▼
$S=\dfrac{D}{T}$; $D=S\times T$; $T=\dfrac{D}{S}$
km/h→m/s: $\times\dfrac{5}{18}$ | m/s→km/h: $\times\dfrac{18}{5}$
km/h→m/s: $\times\dfrac{5}{18}$ | m/s→km/h: $\times\dfrac{18}{5}$
Average Speed:
Equal distance at $u,v$: $\text{Avg}=\dfrac{2uv}{u+v}$
Equal time at $u,v$: $\text{Avg}=\dfrac{u+v}{2}$
Equal distance at $u,v$: $\text{Avg}=\dfrac{2uv}{u+v}$
Equal time at $u,v$: $\text{Avg}=\dfrac{u+v}{2}$
Relative Speed:
Same dir: $|u-v|$; Opposite: $u+v$
Meet time (opp): $\dfrac{D}{u+v}$; (same): $\dfrac{D}{u-v}$
Same dir: $|u-v|$; Opposite: $u+v$
Meet time (opp): $\dfrac{D}{u+v}$; (same): $\dfrac{D}{u-v}$
Trains & Boats
▼
Trains (रेलगाड़ी)
- Pass pole/person: $T=\dfrac{L}{S}$
- Pass platform: $T=\dfrac{L_\text{train}+L_\text{platform}}{S}$
- Two trains same dir: $T=\dfrac{L_1+L_2}{|S_1-S_2|}$; opposite: $T=\dfrac{L_1+L_2}{S_1+S_2}$
- Equal trains, same dir, cross man at $t_1,t_2$: cross time $=\dfrac{2t_1 t_2}{t_1-t_2}$
- Equal trains, opp dir, cross man at $t_1,t_2$: cross time $=\dfrac{2t_1 t_2}{t_1+t_2}$
- Stoppage time/hr $=\dfrac{u-v}{u}$ (u=without, v=with stoppage speed)
- If train covers distance $d$ more than another: $D=S\!\left(\dfrac{t_1 t_2}{t_1-t_2}\right)$
- Train crosses bridge $x$ in $t_1$, another $y$ in $t_2$: new time $=\left(\dfrac{L+y}{L+x}\right)t_1$
Boats & Streams (नाव और धारा)
Downstream $=u+v$; Upstream $=u-v$
Boat speed $u=\dfrac{\text{DS}+\text{US}}{2}$; Stream speed $v=\dfrac{\text{DS}-\text{US}}{2}$
Time downstream: $t_1=\dfrac{D}{u+v}$; Upstream: $t_2=\dfrac{D}{u-v}$
Boat speed $u=\dfrac{\text{DS}+\text{US}}{2}$; Stream speed $v=\dfrac{\text{DS}-\text{US}}{2}$
Time downstream: $t_1=\dfrac{D}{u+v}$; Upstream: $t_2=\dfrac{D}{u-v}$
Time & Work समय और कार्य
All Time & Work Formulas
▼
Work by A in 1 day $=\dfrac{1}{\text{days}}$
A in $m$ days, B in $n$ days, together: $T=\dfrac{mn}{m+n}$
A+B+C together: $T=\dfrac{1}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$
MDH law: $M_1D_1H_1=M_2D_2H_2$; $\dfrac{M_1D_1}{W_1}=\dfrac{M_2D_2}{W_2}$
A in $m$ days, B in $n$ days, together: $T=\dfrac{mn}{m+n}$
A+B+C together: $T=\dfrac{1}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$
MDH law: $M_1D_1H_1=M_2D_2H_2$; $\dfrac{M_1D_1}{W_1}=\dfrac{M_2D_2}{W_2}$
Wages (वेतन)
A in $m$, B in $n$: total wage $R$
A's share $=\dfrac{n}{m+n}\times R$; B's $=\dfrac{m}{m+n}\times R$
A's share $=\dfrac{n}{m+n}\times R$; B's $=\dfrac{m}{m+n}\times R$
Efficiency
$E\propto\dfrac{1}{D}$; $E_1:E_2=\dfrac{1}{D_1}:\dfrac{1}{D_2}$
B is $R\%$ more efficient than A: $B=\dfrac{100}{100+R}\times D_A$ days
B is $R\%$ more efficient than A: $B=\dfrac{100}{100+R}\times D_A$ days
Pipes & Cisterns
Fill $A$ (x hr) + Fill $B$ (y hr) together: $\dfrac{xy}{x+y}$
Fill $A$ (x hr) − Drain $B$ (y hr): $\dfrac{xy}{y-x}$ hr
Required food time: $\dfrac{M_1(D-d)}{M_1+M_2}$
Fill $A$ (x hr) − Drain $B$ (y hr): $\dfrac{xy}{y-x}$ hr
Required food time: $\dfrac{M_1(D-d)}{M_1+M_2}$
Men-Days
- A,B men or C boys in $x$ days; $A_1$ men $B_1$ boys: Time $=\dfrac{x}{\frac{A_1}{A}+\frac{B_1}{B}}$
- 16M=21W=18C: if 32M+35W+27C: $T=\dfrac{93}{\frac{32}{16}+\frac{35}{21}+\frac{27}{18}}=18$ days
Ratio & Proportion अनुपात तथा समानुपात
Core Concepts & Alligation
▼
$a:b=\dfrac{a}{b}$; Product of extremes $=$ Product of means: $ad=bc$
$\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\cdots\Rightarrow$ each $=\dfrac{a+c+e+\cdots}{b+d+f+\cdots}$
$\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\cdots\Rightarrow$ each $=\dfrac{a+c+e+\cdots}{b+d+f+\cdots}$
R divided in $m:n$:
Part A $=\dfrac{m}{m+n}\times R$; Part B $=\dfrac{n}{m+n}\times R$
If diff $=R$: Part A $=\dfrac{m}{m-n}\times R$ $(m>n)$
Part A $=\dfrac{m}{m+n}\times R$; Part B $=\dfrac{n}{m+n}\times R$
If diff $=R$: Part A $=\dfrac{m}{m-n}\times R$ $(m>n)$
Adding $x$ to $a:b$: new $=\dfrac{a+x}{b+x}$
Three parts $A:B:C=m:n:p$, total $R$:
$A=\dfrac{m}{m+n+p}R$; $B=\dfrac{n}{m+n+p}R$
Three parts $A:B:C=m:n:p$, total $R$:
$A=\dfrac{m}{m+n+p}R$; $B=\dfrac{n}{m+n+p}R$
Mixture & Alligation (मिश्रण और सन्धि)
$$\frac{\text{cheaper qty}}{\text{dearer qty}}=\frac{d-m}{m-c}$$
where $c$=cheaper price, $d$=dearer price, $m$=mean price (alligation rule)
Three-glass mixture ratio (milk:water):
$=\left(\dfrac{m}{m+n}+\dfrac{p}{p+q}\right):\left(\dfrac{n}{m+n}+\dfrac{q}{p+q}\right)$
If $\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\cdots$ then each $=\dfrac{a+c+e}{b+d+f}$
$=\left(\dfrac{m}{m+n}+\dfrac{p}{p+q}\right):\left(\dfrac{n}{m+n}+\dfrac{q}{p+q}\right)$
If $\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\cdots$ then each $=\dfrac{a+c+e}{b+d+f}$
Partnership साझेदारी
Partnership Formulas
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Profit $\propto$ Capital $\times$ Time
A, B, C profit ratio $=C_1T_1:C_2T_2:C_3T_3$
A, B, C profit ratio $=C_1T_1:C_2T_2:C_3T_3$
| Type | Condition | Profit Ratio |
|---|---|---|
| Simple partnership | All invest same time, diff capital | $C_1:C_2:C_3$ |
| Simple partnership | Same capital, diff time | $T_1:T_2:T_3$ |
| Compound partnership | Diff capital, diff time | $C_1T_1:C_2T_2:C_3T_3$ |
- Active partner: invests + works → gets salary/commission before profit sharing
- Sleeping partner: only invests money
- If B invests twice in diff periods: $(B_1t_1+B_2t_2)$ vs A's $(A\times T)$
Permutation & Combination क्रमपरिवर्तन और संयोजन
Factorial, Permutation & Combination
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Factorial:
$n!=n(n-1)(n-2)\cdots1$; $0!=1!=1$
$n!=n(n-1)!$; $\dfrac{n!}{r!}=n(n-1)\cdots(r+1)$
Exponent of prime $p$ in $n!$:
$E_p=\left[\dfrac{n}{p}\right]+\left[\dfrac{n}{p^2}\right]+\left[\dfrac{n}{p^3}\right]+\cdots$
$n!=n(n-1)(n-2)\cdots1$; $0!=1!=1$
$n!=n(n-1)!$; $\dfrac{n!}{r!}=n(n-1)\cdots(r+1)$
Exponent of prime $p$ in $n!$:
$E_p=\left[\dfrac{n}{p}\right]+\left[\dfrac{n}{p^2}\right]+\left[\dfrac{n}{p^3}\right]+\cdots$
Counting Principles:
Multiplication: Op1($m$) × Op2($n$) $=mn$ ways
Addition (mut. excl.): Op1($m$) or Op2($n$) $=m+n$ ways
Multiplication: Op1($m$) × Op2($n$) $=mn$ ways
Addition (mut. excl.): Op1($m$) or Op2($n$) $=m+n$ ways
Permutation: $^nP_r=\dfrac{n!}{(n-r)!}$; $^nP_n=n!$; $^nP_0=1$
Combination: $^nC_r=\dfrac{n!}{r!(n-r)!}=\dfrac{^nP_r}{r!}$
$^nC_r=^nC_{n-r}$; $^nC_r+^nC_{r-1}=^{n+1}C_r$
If $^nC_a=^nC_b$: $a=b$ or $a+b=n$
Combination: $^nC_r=\dfrac{n!}{r!(n-r)!}=\dfrac{^nP_r}{r!}$
$^nC_r=^nC_{n-r}$; $^nC_r+^nC_{r-1}=^{n+1}C_r$
If $^nC_a=^nC_b$: $a=b$ or $a+b=n$
Circular permutations: $(n-1)!$
Necklace/bracelet: $\dfrac{(n-1)!}{2}$
With $p$ alike, $q$ alike: $\dfrac{n!}{p!\,q!}$
Necklace/bracelet: $\dfrac{(n-1)!}{2}$
With $p$ alike, $q$ alike: $\dfrac{n!}{p!\,q!}$
Total subsets $=2^n$ (incl. empty set)
Diagonals in $n$-gon $=\dfrac{n(n-3)}{2}$
△ from $n$ points $=^nC_3$
$^nC_0+^nC_1+\cdots+^nC_n=2^n$
Diagonals in $n$-gon $=\dfrac{n(n-3)}{2}$
△ from $n$ points $=^nC_3$
$^nC_0+^nC_1+\cdots+^nC_n=2^n$
Probability प्रायिकता
Core Probability Formulas
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$P(E)=\dfrac{n(E)}{n(S)}$; $0\leq P(E)\leq1$; $P(E')=1-P(E)$
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Mutually exclusive: $P(A\cup B)=P(A)+P(B)$
Independent: $P(A\cap B)=P(A)\cdot P(B)$
Conditional: $P(A|B)=\dfrac{P(A\cap B)}{P(B)}$
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Mutually exclusive: $P(A\cup B)=P(A)+P(B)$
Independent: $P(A\cap B)=P(A)\cdot P(B)$
Conditional: $P(A|B)=\dfrac{P(A\cap B)}{P(B)}$
Coins (सिक्का)
- 1 coin: $n(S)=2$; 2 coins: 4; $n$ coins: $2^n$
- P(exactly $r$ heads in $n$ coins) $=\dfrac{^nC_r}{2^n}$
Dice (पासा)
- 1 die: $n(S)=6$; 2 dice: 36; $n$ dice: $6^n$
- Sum = 7 with 2 dice: 6 ways, $P=\tfrac{1}{6}$ (max)
- Sum = 2 or 12: 1 way each, $P=\tfrac{1}{36}$ (min)
- Doublet of even: $(2,2),(4,4),(6,6)$; $P=\tfrac{1}{12}$
- Same number: $P=\tfrac{6}{36}=\tfrac{1}{6}$
Cards (ताश) — 52 total
- 4 suits × 13 = 52; Red=Black=26 each
- Hearts=Diamonds=Spades=Clubs=13 each
- Face cards: 12 (4K+4Q+4J); Aces: 4
- Non-face: 40; Number cards (2–10): 36
- Red King=Red Queen=Red Jack=Red Ace = 2 each
- Total King=Queen=Jack=Ace = 4 each
Odds (बाधा)
Odds in favour $=\dfrac{P(E)}{P(E')}=\dfrac{m}{n}$
$P(E)=\dfrac{m}{m+n}$
$P(E)=\dfrac{m}{m+n}$
Binomial Probability
$P(X=r)=^nC_r\cdot p^r\cdot q^{n-r}$ where $q=1-p$
Mean $=np$; Variance $=npq$; SD $=\sqrt{npq}$
Mean $=np$; Variance $=npq$; SD $=\sqrt{npq}$
Statistics सांख्यिकी
Mean, Median, Mode
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Mean (माध्य)
$\bar{x}=\dfrac{\sum x}{n}=\dfrac{\sum fx}{\sum f}$ (weighted)
Combined: $\bar{x}=\dfrac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}$
Combined: $\bar{x}=\dfrac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}$
Median (मध्यिका)
Ungrouped: middle value when sorted
Even $n$: $\dfrac{(n/2)^\text{th}+(n/2+1)^\text{th}}{2}$
Grouped: $M=l+\dfrac{\tfrac{n}{2}-cf}{f}\times h$
Even $n$: $\dfrac{(n/2)^\text{th}+(n/2+1)^\text{th}}{2}$
Grouped: $M=l+\dfrac{\tfrac{n}{2}-cf}{f}\times h$
Mode (बहुलक)
Ungrouped: most frequent value
Grouped: $Z=l+\dfrac{f_1-f_0}{2f_1-f_0-f_2}\times h$
Grouped: $Z=l+\dfrac{f_1-f_0}{2f_1-f_0-f_2}\times h$
Empirical relation:
$\text{Mode}=3\times\text{Median}-2\times\text{Mean}$
$\text{Mode}=3\times\text{Median}-2\times\text{Mean}$
Variance, SD & Frequency Distribution
▼
$\sigma^2=\dfrac{\sum(x-\bar{x})^2}{n}=\dfrac{\sum x^2}{n}-\bar{x}^2$ (Variance)
$\sigma=\sqrt{\sigma^2}$ (Standard Deviation)
Coefficient of variation $=\dfrac{\sigma}{\bar{x}}\times100\%$
$\sigma=\sqrt{\sigma^2}$ (Standard Deviation)
Coefficient of variation $=\dfrac{\sigma}{\bar{x}}\times100\%$
Frequency Distribution
- Class mark $=\dfrac{\text{upper limit}+\text{lower limit}}{2}$
- Class width $h=$ upper limit $-$ lower limit
- Cumulative frequency: running total of frequencies
- Frequency polygon: line graph joining class midpoints
- Ogive (cumulative freq. curve): used to find median, quartiles
Variance (example): Data 3,4,5,5,8,9,9,13,15; mean=8
Variance $=\dfrac{\sum(x-\bar{x})^2}{n}=\dfrac{136}{10}=13.6$
$\sigma=\sqrt{13.6}\approx3.69$; CV $=\dfrac{3.69}{8}\times100=46.1\%$
Variance $=\dfrac{\sum(x-\bar{x})^2}{n}=\dfrac{136}{10}=13.6$
$\sigma=\sqrt{13.6}\approx3.69$; CV $=\dfrac{3.69}{8}\times100=46.1\%$