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  Class 10 CBSE Maths – Chapter 10: Circles 🔹 Key Concepts and Theorems ✅ Circle A circle is the set of all points in a plane that are equidistant from a fixed point (the centre ). The fixed distance is called the radius . ✅ Basic Terms Chord : A line segment joining two points on a circle. Diameter : A chord passing through the centre (longest chord). Tangent : A line that touches the circle at exactly one point . Point of Contact : The point where the tangent touches the circle. Secant : A line that intersects the circle at two distinct points . 📘 Important Theorems 🔸 Theorem 1: The tangent to a circle is perpendicular to the radius at the point of contact. If a line touches a circle at only one point, then: Radius ⊥ Tangent at the point of contact \text{Radius} \perp \text{Tangent at the point of contact} 🔸 Theorem 2: The lengths of tangents drawn from an external point to a circle are equal. If PA and PB are ta...

  Class 10 CBSE Maths – Chapter 9: Some Applications of Trigonometry 🔹 Key Concepts ✅ Line of Sight A line drawn from the eye of an observer to the point being viewed. ✅ Angle of Elevation The angle between the horizontal and the line of sight when the point being viewed is above the horizontal level. ✅ Angle of Depression The angle between the horizontal and the line of sight when the point being viewed is below the horizontal level. 📘 Trigonometric Ratios Used In all questions involving height and distance, use these ratios: sin ⁡ θ = Perpendicular Hypotenuse , cos ⁡ θ = Base Hypotenuse , tan ⁡ θ = Perpendicular Base \sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Perpendicular}}{\text{Base}} Most problems in this chapter use right-angled triangles and involve tan θ . 🧠 Important Notes: Always draw a diagram to represent the situation. Use t...

  Class 10 CBSE Maths – Chapter 7: Coordinate Geometry 🔹 Key Concepts and Formulas ✅ Coordinate Plane A plane with two perpendicular lines (axes): x-axis (horizontal) and y-axis (vertical) The point of intersection is called the origin (0, 0) Coordinates are written as (x, y) ✅ Quadrants I Quadrant: (+x, +y) II Quadrant: (–x, +y) III Quadrant: (–x, –y) IV Quadrant: (+x, –y) ✅ Distance Formula To find distance between two points A ( x 1 , y 1 ) A(x_1, y_1) and B ( x 2 , y 2 ) B(x_2, y_2) : A B = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ✅ Section Formula To find coordinates of a point P which divides the line segment AB in the ratio m : n m:n : P = ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ✅ Midpoint Formula Special case of section formula when m = n: Midpoint = ( x 1 + x 2 2 , y 1 + y 2 2 ) \text{Midpoint} = \left( \frac{x_1 + ...

  Class 10 CBSE Maths – Chapter 6:Triangles 🔹 Key Concepts and Theorems ✅ Similar Figures Figures having the same shape but not necessarily the same size. All congruent figures are similar but not all similar figures are congruent. ✅ Similar Triangles Two triangles are similar if: Their corresponding angles are equal Their corresponding sides are in the same ratio (proportional) Criteria for Similarity of Triangles: AA (Angle-Angle) Criterion : Two triangles are similar if two angles of one triangle are respectively equal to two angles of another triangle. SSS (Side-Side-Side) Criterion : If the sides of two triangles are in the same ratio, then the triangles are similar. SAS (Side-Angle-Side) Criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio. ✅ Basic Proportionality Theorem (Thales Theorem) Statement: If a line is drawn parallel to one side of a triang...

  Class 10 CBSE Maths – Chapter 5: Arithmetic Progressions 🔹 Key Concepts and Formulas ✅ What is an Arithmetic Progression (AP)? An AP is a list of numbers in which the difference between any two consecutive terms is constant. Let the first term be a a and the common difference be d d . Then the AP is: a , a + d , a + 2 d , a + 3 d , … a, a + d, a + 2d, a + 3d, \ldots ✅ Important Formulas General Term (n-th term) of an AP : a n = a + ( n − 1 ) d a_n = a + (n - 1)d Where: a a = First term d d = Common difference n n = Term number Sum of first n terms of an AP : S n = n 2 [ 2 a + ( n − 1 ) d ] or S n = n 2 ( a + l ) S_n = \frac{n}{2} [2a + (n - 1)d] \quad \text{or} \quad S_n = \frac{n}{2} (a + l) Where: a a = First term l l = Last term = a + ( n − 1 ) d a + (n - 1)d n n = Number of terms 🔹 📘 NCERT Solved Examples (Highlights) Example 1: Find the 10th term of the AP: 2, 7, 12, 17, ... a = 2 , d = 5 , n = 10 ⇒ a n = a + ( n − 1 ...

  Class 10 CBSE Maths – Chapter 4: Quadratic Equations 🔹 Key Concepts and Formulas ✅ What is a Quadratic Equation? A quadratic equation in the variable x x is of the form: a x 2 + b x + c = 0 ax^2 + bx + c = 0 where: a , b , c a, b, c are real numbers a ≠ 0 a ≠ 0 It is called a quadratic equation because the highest exponent of the variable x x is 2. ✅ Standard Forms Pure quadratic: a x 2 + c = 0 ax^2 + c = 0 Quadratic trinomials: a x 2 + b x + c = 0 ax^2 + bx + c = 0 Factorable quadratics: can be written as product of binomials. ✅ Methods of Solving Quadratic Equations Factorisation Method Express as a product of two linear factors Set each factor = 0 and solve Completing the Square Method Convert to perfect square by adding/subtracting a constant Take square root on both sides Quadratic Formula: x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} This works for all quadratic equations. ✅ Discriminant (D) ...

  Class 10 CBSE Maths – Chapter 3: Pair of Linear Equations in Two Variables 🔹 Key Concepts and Formulas ✅ Linear Equation in Two Variables: An equation of the form: a x + b y + c = 0 ax + by + c = 0 Where a, b, c are real numbers, and a and b are not both zero. ✅ Solution of a Pair of Linear Equations: A solution is a pair (x, y) that satisfies both equations simultaneously. ✅ Graphical Method: Each equation represents a straight line. The point of intersection gives the solution. Types of Solutions: Consistent and Unique (Lines intersect at one point) Consistent and Infinite Solutions (Lines coincide) Inconsistent (Lines are parallel) Condition Table: Condition Type of Lines Number of Solutions a 1 a 2 ≠ b 1 b 2 \frac{a_1}{a_2} \neq \frac{b_1}{b_2} Intersecting lines Unique solution a 1 a 2 = b 1 b 2 = c 1 c 2 \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} Coincident lines Infinitely many a 1 a 2 = b 1 b 2 ≠ c 1 c 2 \fra...