Real Numbers<\/strong><\/h3>\n\n\n\nNatural Numbers<\/td> N ={ 1, 2,3,4,5 \u2026 } <\/td><\/tr> Whole Numbers <\/td> W={ 0, 1, 2, 3, 4, 5\u2026 }<\/td><\/tr> Rational Numbers<\/td> Those numbers which can be presented in the form of a\/b are called Rational Numbers. <\/td><\/tr> Real Numbers <\/td> Real Numbers can be found on a number line <\/td><\/tr> LCM (P, Q, R)<\/td> P.Q.R.H.C.F(P, Q, R) \/ [HCF ( P, Q) . HCF( Q, R) . HCF ( P, R)]<\/td><\/tr> HCF (P, Q, R)<\/td> P.Q.R.L.C.M(P, Q, R) \/ [LCM ( P, Q) . LCM ( Q, R) . LCM ( P, R)]<\/td><\/tr><\/tbody><\/table> Real Numbers<\/strong><\/figcaption><\/figure>\n\n\n\nPolynomials<\/strong><\/h3>\n\n\n\n\n- (a+b)2<\/sup> = a2<\/sup>+2ab+b2<\/sup><\/li>\n\n\n\n
- (a\u2212b)2<\/sup>=a2<\/sup>\u22122ab+b2<\/sup><\/li>\n\n\n\n
- (x+a)(x+b) = x2<\/sup>+(a+b)x+ab<\/li>\n\n\n\n
- a2<\/sup>\u2212b2<\/sup> = (a+b)(a\u2212b)<\/li>\n\n\n\n
- a3<\/sup>\u2212b3<\/sup> = (a\u2212b)(a2<\/sup>+ab+b2<\/sup>)<\/li>\n\n\n\n
- a3<\/sup>+b3<\/sup> = (a+b)(a2<\/sup>\u2212ab+b2<\/sup>)<\/li>\n\n\n\n
- (a+b)3<\/sup> = a3<\/sup>+3a2<\/sup>b+3ab2<\/sup>+b3<\/sup><\/li>\n\n\n\n
- (a\u2212b)3<\/sup> = a3<\/sup>\u22123a2<\/sup>b+3ab2<\/sup>\u2212b3<\/sup><\/li>\n<\/ol>\n\n\n\n
![\"\"\/](\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/math-polynomials.jpg\")
<\/figure>\n\n\n\nClass 10 Algebra Formulas<\/h3>\n\n\n\n\n- (a + b)2<\/sup> = a2<\/sup> + 2ab + b2<\/sup><\/li>\n\n\n\n
- (a – b)2<\/sup> = a2<\/sup> – 2ab + b2<\/sup><\/li>\n\n\n\n
- (a + b) (a – b) = a2<\/sup> – b2<\/sup><\/li>\n\n\n\n
- (x + a)(x + b) = x2<\/sup> + (a + b)x + ab<\/li>\n\n\n\n
- (x + a)(x – b) = x2<\/sup> + (a – b)x – ab<\/li>\n\n\n\n
- (x – a)(x + b) = x2<\/sup> + (b – a)x – ab<\/li>\n\n\n\n
- (x – a)(x – b) = x2<\/sup> – (a + b)x + ab<\/li>\n\n\n\n
- (a + b)3<\/sup> = a3<\/sup> + b3<\/sup> + 3ab(a + b)<\/li>\n\n\n\n
- (a – b)3<\/sup> = a3<\/sup> – b3<\/sup> – 3ab(a – b)<\/li>\n\n\n\n
- (x + y + z)2<\/sup> = x2<\/sup> + y2<\/sup> + z2<\/sup> + 2xy + 2yz + 2xz<\/li>\n\n\n\n
- (x + y – z)2<\/sup> = x2<\/sup> + y2<\/sup> + z2<\/sup> + 2xy – 2yz – 2xz<\/li>\n\n\n\n
- (x – y + z)2<\/sup> = x2<\/sup> + y2<\/sup> + z2<\/sup> – 2xy – 2yz + 2xz<\/li>\n\n\n\n
- (x – y – z)2<\/sup> = x2<\/sup> + y2<\/sup> + z2<\/sup> – 2xy + 2yz – 2xz<\/li>\n\n\n\n
- x3<\/sup> + y3<\/sup> + z3<\/sup> – 3xyz = (x + y + z)(x2<\/sup> + y2<\/sup> + z2<\/sup> – xy – yz – xz)<\/li>\n<\/ul>\n\n\n\n
Pair of Linear Equations in Two Variables<\/strong><\/h3>\n\n\n\n\n- Linear equation in one variable: ax +b =0, a\u22600 and a&b are real numbers<\/li>\n\n\n\n
- Linear equation in two variables: ax+ by+ c =0 , a\u22600 & b\u22600 and a,b & c are real numbers<\/li>\n\n\n\n
- Linear equation in three variables: ax+ by+ cz= 0, a\u22600 , b\u22600, c\u22600 and a,b,c,d are real numbers<\/li>\n\n\n\n
- a1x+b1y+c1=0<\/li>\n\n\n\n
- a2x+b2y+c2=0<\/li>\n<\/ul>\n\n\n\n
Where a1, b1, c1, a2, b2, c2 are all real numbers and a12+ b12 \u2260 0, a22+ b22 \u2260 0<\/p>\n\n\n\n
Quadratic Equations<\/strong><\/h3>\n\n\n\nx = (\u03b1, \u03b2) = [-b \u00b1 \u221a(b2<\/sup> \u2013 4ac)]\/2a provided b2<\/sup> \u2013 4ac >= 0 <\/p>\n\n\n\nA quadratic equation ax2<\/sup> + bx + c = 0 has<\/p>\n\n\n\n(i) two distinct real roots, if b2<\/sup> \u2013 4ac > 0,<\/p>\n\n\n\n(ii) two equal roots (i.e., coincident roots), if b2<\/sup> \u2013 4ac = 0, and<\/p>\n\n\n\n(iii) no real roots, if b2<\/sup> \u2013 4ac < 0<\/p>\n\n\n\nArithmetic Progression Formulas<\/h3>\n\n\n\n
The nth term of AP = nth term = a + (n-1) d<\/p>\n\n\n\n
Sum of n terms in AP = Sn = n\/2[2a + (n \u2212 1) \u00d7 d]<\/p>\n\n\n\n
Sum of all terms in AP with the last term \u2018l\u2019 = n\/2(a + l)<\/p>\n\n\n\n
Triangles<\/strong><\/h3>\n\n\n\nHere,<\/p>\n\n\n\n
A = Area of Triangle<\/p>\n\n\n\n
B = Base of Triangle<\/p>\n\n\n\n
H = Height of a Triangle<\/p>\n\n\n\n
Area of Triangle = A = \u00bd (b \u00d7 h) square units <\/p>\n\n\n\n
Area of an Isosceles Triangle = 1\/4 b\u221a(4a2<\/sup> \u2013 b2<\/sup>)<\/p>\n\n\n\nArea of a Right Triangle = A = 1\/2 \u00d7 Base \u00d7 Height<\/p>\n\n\n\n
Area of an Equilateral Triangle = A = (\u221a3)\/4 \u00d7 side2<\/sup><\/p>\n\n\n\nCoordinate Geometry<\/strong><\/h3>\n\n\n\n\n- Distance Formula to find distance between two points P(x1,y1) and Q(x2,y2) is = \u221a[(x2 \u2013 x1)2 + (y2 \u2013 y1)2 ]<\/li>\n\n\n\n
- Distance of a point P(x, y) from the origin is = \u221ax2<\/sup> + y2<\/sup><\/li>\n\n\n\n
- The coordinates of the point P(x, y) which divides the line segment joining the points A(x1 , y1 ) and B(x2 , y2 ) internally in the ratio m1 : m2 = Section Formula = ((m1x2 + m2x1)\/m1+ m2 , (m1y2 + m2y1)\/m1+ m2)<\/li>\n\n\n\n
- The mid-point of the line segment joining the points P(x1, y1) and Q(x2, y2 ) = [(x1+x2\/2), (y1+y2\/2)<\/li>\n<\/ul>\n\n\n\n
Trigonometry Formulas<\/h3>\n\n\n\n
| Natural Numbers<\/td> | N ={ 1, 2,3,4,5 \u2026 } <\/td><\/tr> |
| Whole Numbers <\/td> | W={ 0, 1, 2, 3, 4, 5\u2026 }<\/td><\/tr> |
| Rational Numbers<\/td> | Those numbers which can be presented in the form of a\/b are called Rational Numbers. <\/td><\/tr> |
| Real Numbers <\/td> | Real Numbers can be found on a number line <\/td><\/tr> |
| LCM (P, Q, R)<\/td> | P.Q.R.H.C.F(P, Q, R) \/ [HCF ( P, Q) . HCF( Q, R) . HCF ( P, R)]<\/td><\/tr> |
| HCF (P, Q, R)<\/td> | P.Q.R.L.C.M(P, Q, R) \/ [LCM ( P, Q) . LCM ( Q, R) . LCM ( P, R)]<\/td><\/tr><\/tbody><\/table>Polynomials<\/strong><\/h3>\n\n\n\n<\/figure>\n\n\n\nClass 10 Algebra Formulas<\/h3>\n\n\n\nPair of Linear Equations in Two Variables<\/strong><\/h3>\n\n\n\nWhere a1, b1, c1, a2, b2, c2 are all real numbers and a12+ b12 \u2260 0, a22+ b22 \u2260 0<\/p>\n\n\n\n Quadratic Equations<\/strong><\/h3>\n\n\n\nA quadratic equation ax2<\/sup> + bx + c = 0 has<\/p>\n\n\n\n (i) two distinct real roots, if b2<\/sup> \u2013 4ac > 0,<\/p>\n\n\n\n (ii) two equal roots (i.e., coincident roots), if b2<\/sup> \u2013 4ac = 0, and<\/p>\n\n\n\n (iii) no real roots, if b2<\/sup> \u2013 4ac < 0<\/p>\n\n\n\n The nth term of AP = nth term = a + (n-1) d<\/p>\n\n\n\n Sum of n terms in AP = Sn = n\/2[2a + (n \u2212 1) \u00d7 d]<\/p>\n\n\n\n Sum of all terms in AP with the last term \u2018l\u2019 = n\/2(a + l)<\/p>\n\n\n\n Here,<\/p>\n\n\n\n A = Area of Triangle<\/p>\n\n\n\n B = Base of Triangle<\/p>\n\n\n\n H = Height of a Triangle<\/p>\n\n\n\n Area of Triangle = A = \u00bd (b \u00d7 h) square units <\/p>\n\n\n\n Area of an Isosceles Triangle = 1\/4 b\u221a(4a2<\/sup> \u2013 b2<\/sup>)<\/p>\n\n\n\n Area of a Right Triangle = A = 1\/2 \u00d7 Base \u00d7 Height<\/p>\n\n\n\n Area of an Equilateral Triangle = A = (\u221a3)\/4 \u00d7 side2<\/sup><\/p>\n\n\n\n |
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