Mathematics is one such subject which often gives nightmares to students. While Maths is a little tricky, it is not difficult. It just requires a thorough understanding of the concepts, regular practice and a good hold on all important formulas to score high in the Maths subject.
To help students get all important formulas, theorems and properties at one place, we have collated the chapter-wise formulas along with important terms & properties occurring in Class 10 Maths. Students must grasp all the formulas and theorems included in chapters like Triangles, Polynomials, Coordinate Geometry, Trigonometry and Mensuration as these
1. Real Numbers:
Euclid’s Division Algorithm (lemma): According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r such that a = bq + r, where 0 ≤ r ≤ b. (Here, a = dividend, b = divisor, q = quotient and r = remainder.)
2. Polynomials:
(i) (a + b)2 = a2 + 2ab + b2
(ii) (a – b)2 = a2 – 2ab + b2
(iii) a2 – b2 = (a + b) (a – b)
(iv) (a + b)3 = a3 + b3 + 3ab(a + b)
(v) (a – b)3 = a3 – b3 – 3ab(a – b)
(vi) a3 + b3 = (a + b) (a2 – ab + b2)
(vii) a3 – b3 = (a – b) (a2 + ab + b2)
(viii) a4 – b4 = (a2)2 – (b2)2 = (a2 + b2) (a2 – b2) = (a2 + b2) (a + b) (a – b)
(ix) (a + b + c) 2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
(x) (a + b – c) 2 = a2 + b2 + c2 + 2ab – 2bc – 2ca
(xi) (a – b + c)2 = a2 + b2 + c2 – 2ab – 2bc + 2ca
(xii) (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
(xiii) a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
3. Linear Equations in Two Variables:
For the pair of linear equations
a1 + b1y + c1 = 0 and a2 + b2y + c2 = 0,
the nature of roots (zeroes) or solutions is determined as follows:
(i) If a1/a2 ≠ b1/b2 then we get a unique solution and the pair of linear equations in two variables are consistent. Here, the graph consists of two intersecting lines.
(i) If a1/a2 ≠ b1/b2 ≠ c1/c2, then there exists no solution and the pair of linear equations in two variables are said to be inconsistent. Here, the graph consists of parallel lines.
(iii) If a1/a2 = b1/b2 = c1/c2, then there exists infinitely many solutions and the pair of lines are coincident and therefore, dependent and consistent. Here, the graph consists of coincident lines.
4. Quadratic Equation:
For a quadratic equation, ax2 + bx + c = 0
- Sum of roots = –b/a
- Product of roots = c/a
- If roots of a quadratic equation are given, then the quadratic equation can be represented as:
x2 – (sum of the roots)x + product of the roots = 0
- If Discriminant > 0, then the roots the quadratic equation are real and unequal/unique.
- If Discriminant = 0, then the roots the quadratic equation are real and equal.
- If Discriminant 0, then the rots the quadratic equation are imaginary (not real).
- Important Formulas - Boats and Streams
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(i) Downstream In water, the direction along the stream is called downstream. (ii) Upstream In water, the direction against the stream is called upstream. (iii) Let the speed of a boat in still water be u km/hr and the speed of the stream be v km/hr, then 5. Arithmetic Progression:
an = a + (n−1)×d
6. Similarity of Triangles:
7. Coordinate Gemetry:
8. Trigonometry: In a right-angled triangle, the Pythagoras theorem states (perpendicular )2 + ( base )2 = ( hypotenuse )2 Important trigonometric properties: (with P = perpendicular, B = base and H = hypotenuse)
Trigonometric Identities:
Relations between trigonometric identities are given below:
Trigonometric Ratios of Complementary Angles are given as follows:
Values of Trigonometric Ratios of 0° and 90° are tabulated below: 9. Circles: Important properties related to circles:
Important formulas related to circles:
Here, Area of the segment APB = Area of the sector OAPB – Area of ∆ OAB 10. Mensuration: Check below the important formulas for areas and volumes of solids:
11. Statistics: For Ungrouped Data: Mean: The mean value of a variable is defined as the sum of all the values of the variable divided by the number of values.
Median: The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.
For Grouped Data: Mean: If x1, x2, x3,......xn are observations with respective frequencies f1, f2, f3,.....fn then mean is given as:
Median: For the given data, we need to have class interval, frequency distribution and cumulative frequency distribution. Then, median is calculated as
Where Mode: Modal class: The class interval having highest frequency is called the modal class and Mode is obtained using the modal class.
Where 12. Probability:
Understanding the basic concepts and learning all the important formulas is extremely sufficient to pass the Maths exam with flying colours. If you know the formulas very well then it will not take much time for you to solve questions in exam paper. So, keep practicing with the list of important formulas given above in this article.
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