Square:

12=1 112=121 212=441 312=961

22=4 122=144 222=484 322=1024
32=9 132=169 232=529 352=1225
42=16 142=196 242=576
52=25 152=225 252=625
62=36 162=256 262=676
72=49 172=289 272=729
82=64 182=324 282=784
92=81 192=361 292=841
102=100 202=400 302=900
Cube: Square root:
13=1 113=1331 √1=1.000 √10=3.162
23=8 123=1728 √2=1.414
33=27 153=3375 √3=1.732
43=64 213=9261 √4=2.000
53=125 253=15625 √5=2.236
63=216 √6=2.449
73=343 √7=2.645
83=512 √8=2.828
93=729 √9=3.000
Number system:
1. 1+2+3+4+……………+n={n(n+1)} ÷2
2. 1+3+5+7+……………+n=n2
3. 2+4+6+8+……………+n=n(n+1)
4. 12+22+32+………..+n2={n(n+1)(2n+1)} ÷6
5. 12+32+52+………..+ n2={n(2n+1)(2n-1)} ÷3
6. 22+42+32+………..+n2={2n(n+1)(2n+1)} ÷3
7. 13+23+33+………..+n3=[{n(n+1)}÷2]2
8. 14+24+34+………..+n4= {n(n+1)(2n+1)(3n2+3n-1)}÷30
* 0 to 100 number of prime numbers is 25
* 1-50 prime no. 15 &51-100 prime no. 10
Nth term: a+d(n-1)
Sum of Nth term: n/2 * (a+l) or n/2{2a+(n-1) d}, Where a=1st term, d= Difference between two term, n= Last term &
a= 1st term, l= Last term.

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Algebraic Formula:

• a2 – b2 = (a – b)(a + b)
• (a + b)2 = a2 + 2ab + b2
• a2 + b2 = (a + b)2 – 2ab
• (a – b)2 = a2 – 2ab + b2
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
• (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
• (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 – ab + b2)
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
• (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
• a4 – b4 = (a – b)(a + b)(a2 + b2)
• If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
• If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
• If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1)
• (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
• Laws of Exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn
1 1 −m 1𝑚
• Fractional Exponents X0 = 1 ; xa ÷xb=x(a-b), x-1= 𝑋, x-2 = x2, x1/2=√x, a =𝑎

• Roots of Quadratic Equation

o For a quadratic equation ax2 + bx + c = 0 where a ≠ 0, the roots will be given by the equation as
−𝑏±√𝑏2 −4𝑎𝑐
x= 2𝑎
o Δ = b2 − 4ac is called the discriminant
o For real and distinct roots, Δ > 0
o For real and coincident roots, Δ = 0
o For non-real roots, Δ < 0
o If α and β are the two roots of the equation ax2 + bx + c = 0 then, α + β = (-b / a) and α × β = (c / a).
o If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

• Factorials

o n! = (1).(2).(3)…..(n − 1).n
o n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
o 0! = 1
n n n−1 n(n−1) n−2 2 n(n−1)(n−2) n−3 3 n
o (a+b) =a +na b+ 2! a b + 3!
a b +….+b ,where,n>1

1.(a+b)3 + (a-b)3 = 2a (a2 +3b2) 2. (a+b)3 - (a-b)3 = 2b (b2 +3a2)

3. (a+b) (b+c) (c+a)+abc = (a+b+c) (ab+bc+ca)
Indices:
xa * xb=xa+b, ax * bx=(ab)x, ax/bx=(a/b)x
Law of Surds:
Addition and subtraction of surds
a√b + c√b = (a + c)√b
a√b - c√b = (a - c)√b
Multiplication and Division
√ab = √a × √b
√(a/b) = √a/√b

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HCF &LCM:
1. HCF= HCF of numerators/LCM of denominators
2. LCM= LCF of numerators/HCM of denominators
3. Product of two number = their HCF* Their LCM
Arithmetic:
SI &CI:

1.Simple interest (I) = (P*R*T)/100 P = Principal, R = rate of Interest

2.In Compound interest Amount (A) = P(1+R/100) T, T= Time

3. CI – SI = P(R/100)2 (2nd years) 4.CI – SI = P(R/100)2(R/100+3) ( for 3rd years)

Profit & Loss:
Gain = S.P. – C.P. S.P. = Selling price C.P. = Cost price
With respect to price:
Gain % = {(SP-CP)/CP} * 100 Loss % = {(CP-SP)/CP} * 100
With respect to article:
Gain % = {(SP ~ CP)/SP} * 100 Loss % = {(CP ~ SP)/SP} * 100
Dishonest Business man:
Gain % = {Stolen Weight / Given Weight} *100
Successive profit % = (X + Y+ X*Y)/100
Earnings ratio: Ratio of ratio of savings/ Ratio of percentage of savings
100↔50↔33.33↔25↔20↔16.66↔14.285↔12.5↔11.111↔09.99↔08.33
Time & Distance:
Distance = speed * time

1.X k/h = (X *5/18) m/s 2. X m/s = (X * 18/5) k/h

Average Speed = Total distance / Total time

Suppose a man covers a distance at x k/h and an equal distance at y k/h then, the average speed is (2xy/ x+y) k/h.

Boat & stream:

1. speed of downstream = (Boat + stream)

2. speed of upstream = (Boat – stream)
3. Boat speed (still water) = (upstream + downstream)/2
4. Stream speed = (upstream - downstream)/2

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Mensuration List of important formula:

1. i) Area of a rectangle = Length × Breadth, ii) Diagonal2 = Length2 + Breadth2 iii) Perimeter =2× (Length +Breadth).

2. i) Area of Square = Side2 or ½ (Diagonal2) ii) Perimeter of Square = 4× side.

3. i) area of 4 walls of a room =2× (Length +Breadth) × Height.
4. Area of Parallelogram = (Base + Height). *½ × Sum of diagonals square = sum of sides square.
5. i) Area of Rhombus = ½ × product of diagonals ii) side of rhombus=½ √ (sum of diagonals Square)
6. i) Area of equilateral triangle = √3/4 * side^2 ii) Perimeter =3* side
7 i) Area of isosceles triangle = b/4 √ (4a^2 – b^2) ii) Height =1/2√ (4a^2 – b^2) (when a= equal side, b= unequal side)
8. Area of triangle = , [when a, b, c are the lengths of sides of that triangle and s=1/2 (a+b+c)]
√(𝑠 (𝑠 − 𝑎) (𝑠 − 𝑏) (𝑠 − 𝑐))
9. i) Area of triangle = ½ × Base×Height ii) Hypotenuse of right-angle triangle =
√(Base2 + Height2).
10. Area of trapezium = ½ × sum of parallel sides × perpendicular distance between them.
Perpendicular distance (h) =2/k× Area = (a+b)/k *
√(s (s − k) (s − d) (s − c)) √(s (s − k) (s − d) (s − c))
{Where a& b are parallel sides, k = (a-b), c&d are nonparallel sides, s= (k+c+d)/2}

11. i) Circumference of a circle = 2пr ii) Area of cicle = пr2 iii) Arc AB = 2пr ϴ
360
12.i) In a parallelogram Area = 2 √(s (s − a) (s − b) (s − d)) (where a & b are the two adjacent sides and d is the diagonal connecting
the ends of the two sides)
ii) Area = B * H, where B = base & H = height.
iii) Perimeter = 2(a + b) where a & b are the two adjacent sides.
1 2 2 2 2
iV) 2 (d1 +d2 ) = a + b where a & b are the two adjacent sides & d1, d2are diagonal.
13. Cuboid:
i) Volume = l × b× h ii) Lateral Surface Area = 2(l + b) × h iii) Total Surface Area = 2(l×b + b×h + h×l) iv)
diagonal = √(l2 + b2 + h2 ) [Where l = Length, b = Breadth, h = Height]
14. Cube:
Volume = a3, Lateral Surface Area = 4a2, Total Surface Area = 6a2, Diagonal = a√3 [Where a = each side of cube]
15. Cylinder:
Volume = пr2h, Curved Surface Area = 2пrh, Total Surface Area = 2пrh +2пr2 [Where r = is the radius of the base of the cylinder]
16.Sphere:
4
Volume = 3пr 3, Surface Area = 4пr2 [Where r = is the radius of the Sphere]
2
Volume = 3пr 3, Curved Surface Area = 2пr2, Total Surface Area = 3пr2 ( for Half sphere)
17. Right Circular Cone:
Lateral height =√ (h2 + r2), Volume =1/3 пr2h, Curved Surface Area = пrl=пr√ (h2 + r2)
[Where l = slant height.]
18. Frustum of a Right Circular Cone:
If a cone is cut by a plane parallel to the base so as to divide the come into two parts as shown in the figure. The lower
part is called the frustum of the cone. The radius of the base of the frustum = R, the radius of the top = r, height = h &
slant height = l units.
i) Slant height =√ (h2 + (R- r)2) units ii) Curved Surface Area = п (r + R) l Sq. units iii) Total Surface Area = п {(r
+R) l + r2 + R2} Sq. units iv) Volume = 1/3 п (r2+ R2 +Rr) ×h Cu. Units.

19. i) For an equilateral triangle of side ‘a’ radius of the inscribed circle = a/2√3 and the radius of Circumcircle = a/ √3
3√3
ii) Area of regular polygon = ½ (number of sides) × (radius of inscribe circle). iii) Area of regular hexagon = × side 2
2
iv) area of regular octagon = 2(√2 + 1) × side 2

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 𝑛 (𝑛−3)
20. n = Number of side (i) Number of diagonal of a polygon = (ii) the sum of internal angles of a polygon = 2
2
0 0
(n - 2) × 90 (iii) Externals angles of the polygon = 360 /n (iv) The sum of internal angles and corresponding
external angles = 1800 (v) Number of sides = 3600/ External angles.
Trigonometry

Angles (In Degrees) 0° 30° 45° 60° 90° 180° 270° 360°
Angles (In Radians) 0° π/6 π/4 π/3 π/2 π 3π/2 2π
sin 0 1/2 1/√2 √3/2 1 0 -1 0
cos 1 √3/2 1/√2 1/2 0 -1 0 1
tan 0 1/√3 1 √3 ∞ 0 ∞ 0
cot ∞ √3 1 1/√3 0 ∞ 0 ∞
csc ∞ 2 √2 2/√3 1 ∞ -1 ∞
sec 1 2/√3 √2 2 ∞ -1 ∞ 1

Co-function Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = csc x
• csc(90°−x) = sec x

Sum & Difference Identities

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)

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 • cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

Double Angle Identities

• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]

• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
• cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
• tan(2x) = [2tan(x)]/ [1−tan2(x)]
• sec (2x) = sec2 x/(2-sec2 x)
• csc (2x) = (sec x. csc x)/2

Triple Angle Identities

• Sin 3x = 3sin x – 4sin3x

• Cos 3x = 4cos3x-3cos x
• Tan 3x = [3tanx-tan3x]/[1-3tan2x]

Product identities

• sinx⋅cosy=sin(x+y)+sin(x−y)2
• cosx⋅cosy=cos(x+y)+cos(x−y)2
• sinx⋅siny=cos(x−y)−cos(x+y)2

Sum to Product Identities

• sinx+siny=2sinx+y2cosx−y2
• sinx−siny=2cosx+y2sinx−y2
• cosx+cosy=2cosx+y2cosx−y2
• cosx−cosy=−2sinx+y2sinx−y2

Inverse Trigonometry Formulas

• sin-1 (–x) = – sin-1 x

• cos-1 (–x) = π – cos-1 x
• tan-1 (–x) = – tan-1 x
• cosec-1 (–x) = – cosec-1 x
• sec-1 (–x) = π – sec-1 x
• cot-1 (–x) = π – cot-1 x

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 One Step Closer

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