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Number Systems

• 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
• (1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1) / 6
• (1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/ 2)²
• Sum of first n odd numbers = n²
• Sum of first n even numbers = n (n + 1)

Algebra

• (a – b)² = (a² + b² – 2ab)
• (a + b)² = (a² + b² + 2ab)
• (a + b)(a – b) = (a² – b²)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ – b³) = (a – b)(a² + ab + b²)
• (a³ + b³) = (a + b)(a² – ab + b²)
• (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc
• (a + b)n = an + (nC1)an-1 b + (nC2)an-2 b² + … + (nCn-1)abn-1 + bn

Percentage

• Percentage Change = New Value - Old Value ×100

Old Value

• Percentage Decrease = (Original value-Decreased Value)/Original Value × 100

• What percentage of x is y: (y/x) × 100
• Increase N by S % = N( 1+ S/100)
• Decrease N by S % = N (1 – S/100)

Profit, Loss & Discount

• Profit (or) Gain = SP – CP
• Profit % = (Profit/CP) × 100
• SP = (100 + gain % )/100 × CP
• CP = 100/(100 + gain %) × SP
• Loss = CP – SP
• Loss % = Loss/(CP) × 100
• SP = (100 – loss %)/100 × CP
• CP = 100/(100 – loss %) × SP

HCF & LCM

• HCF × LCM = Product of the two numbers
• LCM of co-prime numbers = Product of the numbers
• HCF of co-prime numbers = 1
• HCF of two numbers a and b = HCF of (a-b, b)
• LCM of two numbers a and b = (a x b) / HCF(a, b).
• HCF of three numbers a, b, and c = HCF of (HCF(a, b), c)

Speed, Time, and Distance

• Distance = Speed × Time

• Time = Distance/Speed

• Speed= Distance/Time

• Average Speed = (Total distance travelled) / (Total time taken)

Time and Work

• If A can do a piece of work in n days, then A’s 1 day’s work = 1/n
• If A’s 1 day’s work =1/n, then A can finish the work in n days

Averages

Sum of observations / Number of observations

Mixture & Alligation

Rule of Alligation: If two ingredients are mixed, then

(Quantity of cheaper / Quantity of dearer) = (C.P. of dearer – Mean Price / Mean price – C.P. of cheaper)

Trigonometry

• cosec θ = 1/sin θ
• sec θ = 1/ cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

Coordinate Geometry

Distance Between two Points A(x1, y1) and B(x2, y2): AB² = (x2 – x1)² + (y2 – y1)²

Probability

P(E) = n(E) / n(S)