Page 8 - Mathematics Class - X

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 If a and b are non-zero rational numbers, and m and n are integers, then:

(i) a =1, when n = 0 (ii) a = 1/a m
–m
n
(iii) a × a = a m + n (iv) a ÷a or a /a = a m – n
n
m
m
m
n
n
(v) (a ) = a m × n = a mn (vi) (a ) = a
m n
–1 –1
 Identities:
(i) (a + b) = a + b + 3ab(a + b) (ii) (a – b) = a – b – 3ab(a – b)
3
3
3
3
3
3
(iii) a + b = (a+ b)(a – ab + b ) (iv) a – b = (a– b)(a + ab + b )

2

3

3
2
2
2
3
3
(v) a + b + c – 3abc = (a+ b + c)(a + b + c – ab – bc – ac)

2

2
2
3
3
3
(vi) If a + b+ c = 0, then a + b + c = 3abc
3
3
3

 In a right-angled triangle, Hypotenuse = Perpendicular + Base 2
2
2
 The sum of all the angles in a quadrilateral is 360 .
o
 Angle in a semicircle from a point on its circumference touching the end points of diameter is a right angle.
 Angles formed in the same segment are equal.  Area of a rectangle = l(length) × b (breadth).
 Area of a square = side × side  Perimeter of a square = 4 × side
 Perimeter of a rectangle = 2(l + b)  Surface area of a cuboid = 2(lb + bh + hl )
 Surface area of a cube = 6l [l = side]
2
 Lateral surface area of a cuboid OR Area of four walls = 2(l + b) h.
 Volume of a cuboid = l × b × h.  Volume of a cube = side × side × side.
 Area of a circle = r 2
θ
 Area of a sector = ×r , whenis the angle at the centre.
2
360 
 Diagonal of a cuboid = l + 2 b + 2 h  Curved surface area of cylinder = 2rh
2
2
 Total surface area of cylinder = 2r (h + r)  Volume of cylinder = r h
 Curved surface area of cone = rl, where l is the slant height and is equal to h + r 2
2
1
2
 Total surface area of cone = r (l + r).  Volume of cone = r h
3
 Curved surface area of sphere = 4r  Curved surface area of hemisphere = 2r 2
2
4
2
 Total surface area of hemisphere = 3r  Volume of the sphere = r 3
3
2
 Volume of the hemisphere = r 3
3
6