class 10 cbse mathematics all chaptee formula
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Answer:
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
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 are real and equal.
If Discriminant < 0, then the roots the are imaginary (not real).
Important Formulas - Boats and Streams
(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
Speed downstream = (u + v) km/hr
Speed upstream = (u - v) km/hr.
5. Arithmetic Progression:
nth Term of an Arithmetic Progression: For a given AP, where a is the first term, d is the common difference, n is the number of terms, its nth term (an) is given as
an = a + (n−1)×d
Sum of First n Terms of an Arithmetic Progression, Sn is given as:
6. Similarity of Triangles:
If two triangles are similar then ratio of their sides are equal.
Theorem on the area of similar triangles: If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.
7. Coordinate Gemetry:
Distance Formulae: Consider a line having two point A(x1, y1) and B(x2, y2), then the distance of these points is given as:
Section Formula: If a point p divides a line AB with coordinates A(x1, y1) and B(x2, y2), in ratio m:n, then the coordinates of the point p are given as:
Mid Point Formula: The coordinates of the mid-point of a line AB with coordinates A(x1, y1) and B(x2, y2), are given as:
Area of a Triangle: Consider the triangle formed by the points A(x1, y1) and B(x2, y2) and C(x3, y3) then the area of a triangle is given as-
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)
SinA = P / H
CosA = B / H
TanA = P / B
CotA = B / P
CosecA = H / P
SecA = H/B
Trigonometric Identities:
sin2A + cos2A=1
tan2A +1 = sec2A
cot2A + 1= cosec2A
Relations between trigonometric identities are given below:
Trigonometric Ratios of Complementary Angles are given as follows:
sin (90° – A) = cos A
cos (90° – A) = sin A
tan (90° – A) = cot A
cot (90° – A) = tan A
sec (90° – A) = cosec A
cosec (90° – A) = sec A
9. Circles:
Important properties related to circles:
Equal chord of a circle are equidistant from the centre.
The perpendicular drawn from the centre of a circle, bisects the chord of the circle.
The angle subtended at the centre by an arc = Double the angle at any part of the circumference of the circle.
Angles subtended by the same arc in the same segment are equal.
To a circle, if a tangent is drawn and a chord is drawn from the point of contact, then the angle made between the chord and the tangent is equal to the angle made in the alternate segment.
The sum of opposite angles of a cyclic quadrilateral is always 180o.
Area of a Segment of a Circle: If AB is a chord which divides the circle into two parts, then the bigger part is known as major segment and smaller one is called minor segment.
Here, Area of the segment APB = Area of the sector OAPB – Area of ∆ OAB