Padikam lessons

PADIKAM LEARNING CONTENT PADIKAM NCERT Class 10 Mathematics Class 10 Study Notes 14 chapters Fourteen current chapters. Standard symbols and NCERT-style working are used throughout. Padikam explanations are independently written. The NCERT book is used only for chapter order, standard terminology and coverage. PADIKAM | NCERT CLASS 10 1

Contents 01 Real Numbers 08 Introduction to Trigonometry 02 Polynomials 09 Some Applications of Trigonometry 03 Pair of Linear Equations in Two Variables 10 Circles 04 Quadratic Equations 11 Areas Related to Circles 05 Arithmetic Progressions 12 Surface Areas and Volumes 06 Triangles 13 Statistics 07 Coordinate Geometry 14 Probability How to use this note Read the quick understanding first, recall the key points aloud, then answer the exam prompts without looking back. Official book source https://ncert.nic.in/textbook.php?jemh1=0-14 PADIKAM | NCERT CLASS 10 2

LESSON 1 Real Numbers Quick understanding Use prime factorisation and Euclid's division idea to study divisibility, HCF, LCM and decimal expansions. Must remember Exam practice Euclid form: a = bq + r, where 0 <= r < b. Find the HCF of 135 and 225 by Euclid's algorithm. For two positive integers: HCF x LCM = product of the integers. Decide whether 13/3125 has a terminating decimal. A rational number p/q in lowest form terminates only when q = 2^m x 5^n. Formula and notation a = bq + r HCF(a,b) x LCM(a,b) = a x b Watch this: Reduce p/q to lowest form before testing the denominator. One-line recall Euclid form: a = bq + r, where 0 <= r < b. PADIKAM | NCERT CLASS 10 3

LESSON 2 Polynomials Quick understanding Connect the zeroes of a polynomial with its graph and with the coefficients of a quadratic polynomial. Must remember Exam practice A zero is a value of x for which p(x) = 0. Form a quadratic polynomial whose zeroes are 3 and -2. For ax^2 + bx + c, sum of zeroes = -b/a. Verify the coefficient relations for 2x^2 - For ax^2 + bx + c, product of zeroes = c/a. 7x + 3. Formula and notation alpha + beta = -b/a alpha beta = c/a Watch this: The relations depend on the leading coefficient a; do not assume it is 1. One-line recall A zero is a value of x for which p(x) = 0. PADIKAM | NCERT CLASS 10 4

LESSON 3 Pair of Linear Equations in Two Variables Quick understanding Represent two conditions as two lines and decide whether they meet once, never, or everywhere. Must remember Exam practice Unique solution when a1/a2 is not equal to b1/b2. Solve 2x + 3y = 13 and 3x - 2y = 4. No solution when a1/a2 = b1/b2 but not c1/c2. Convert a word problem on prices into a pair of equations. Infinitely many solutions when all three ratios are equal. Formula and notation a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Watch this: Keep signs consistent when moving terms or using elimination. One-line recall Unique solution when a1/a2 is not equal to b1/b2. PADIKAM | NCERT CLASS 10 5

LESSON 4 Quadratic Equations Quick understanding Solve equations of degree two and use the discriminant to understand the nature of their roots. Must remember Exam practice Write the equation as ax^2 + bx + c = 0, a not equal Solve 2x^2 - 5x - 3 = 0. to 0. Find k if x^2 + kx + 9 = 0 has equal roots. D = b^2 - 4ac: positive gives two real roots, zero gives equal roots, negative gives no real roots. Factorisation is fastest when factors are visible; the formula always works. Formula and notation x = (-b +/- sqrt(b^2 - 4ac)) / 2a D = b^2 - 4ac Watch this: The denominator is 2a, not just 2. One-line recall Write the equation as ax^2 + bx + c = 0, a not equal to 0. PADIKAM | NCERT CLASS 10 6

LESSON 5 Arithmetic Progressions Quick understanding Recognise sequences with a constant difference and calculate any term or a sum without listing every term. Must remember Exam practice Common difference d = a2 - a1. Find the 25th term of 7, 11, 15, ... The nth term is measured from the first term. Find the sum of the first 30 multiples of 3. The sum formula can use either the last term l or the nth-term expression. Formula and notation a_n = a + (n - 1)d S_n = n/2 [2a + (n - 1)d] S_n = n/2 (a + l) Watch this: A decreasing AP has a negative common difference. One-line recall Common difference d = a2 - a1. PADIKAM | NCERT CLASS 10 7

LESSON 6 Triangles Quick understanding Use similarity to compare shapes, divide sides proportionally and relate the areas of similar triangles. Must remember Exam practice AAA, SAS and SSS are similarity criteria. Prove two given triangles similar and find an unknown side. A line parallel to one side divides the other two sides in the same ratio. Use the Basic Proportionality Theorem in a labelled diagram. Area ratio of similar triangles equals the square of the corresponding-side ratio. Formula and notation ar(Delta ABC)/ar(Delta DEF) = (AB/DE)^2 Watch this: Write corresponding vertices in the same order. One-line recall AAA, SAS and SSS are similarity criteria. PADIKAM | NCERT CLASS 10 8

LESSON 7 Coordinate Geometry Quick understanding Turn a geometric problem into arithmetic using coordinates. Must remember Exam practice Distance comes from the Pythagorean theorem. Find the distance between (-2, 3) and (4, -5). The section formula gives the point dividing a segment in a stated ratio. Find the point dividing A(1,2), B(7,8) in the ratio 2:1. The midpoint is the special case 1:1. Formula and notation d = sqrt((x2-x1)^2 + (y2-y1)^2) P = ((mx2+nx1)/(m+n), (my2+ny1)/(m+n)) Watch this: In AP:PB = m:n, multiply B by m and A by n. One-line recall Distance comes from the Pythagorean theorem. PADIKAM | NCERT CLASS 10 9

LESSON 8 Introduction to Trigonometry Quick understanding Describe a right triangle through ratios and use standard identities to move between them. Must remember Exam practice sin theta = perpendicular/hypotenuse; cos theta = Evaluate sin 60 cos 30 + cos 60 sin 30. base/hypotenuse. If tan theta = 3/4, find sin theta and cos tan theta = sin theta/cos theta. theta. Learn exact values at 0, 30, 45, 60 and 90 degrees. Formula and notation sin^2 theta + cos^2 theta = 1 1 + tan^2 theta = sec^2 theta 1 + cot^2 theta = cosec^2 theta Watch this: Ratios depend on the chosen angle; label opposite and adjacent sides first. One-line recall sin theta = perpendicular/hypotenuse; cos theta = base/hypotenuse. PADIKAM | NCERT CLASS 10 10

LESSON 9 Some Applications of Trigonometry Quick understanding Use angles of elevation or depression to find inaccessible heights and distances. Must remember Exam practice Draw the horizontal through the observer before Find the height of a tower from a 30 marking the angle. degree elevation at 20 m. Angle of depression equals the corresponding angle Solve a two-position observation of elevation. problem. Add observer height only after solving the right triangle. Formula and notation tan theta = vertical height / horizontal distance Watch this: Keep eye level separate from the object's full height. One-line recall Draw the horizontal through the observer before marking the angle. PADIKAM | NCERT CLASS 10 11

LESSON 10 Circles Quick understanding Use the special right-angle and equal-length properties of tangents. Must remember Exam practice A tangent is perpendicular to the radius at the point Prove PA = PB for tangents from P. of contact. Find a tangent length using a right Tangents drawn from one external point are equal. triangle. Join the centre to points of contact before proving results. Formula and notation OA perpendicular PA PA = PB Watch this: A tangent touches at exactly one point; a secant cuts at two. One-line recall A tangent is perpendicular to the radius at the point of contact. PADIKAM | NCERT CLASS 10 12

LESSON 11 Areas Related to Circles Quick understanding Break a figure into sectors, segments and familiar polygons before calculating area or perimeter. Must remember Exam practice A sector is a fraction theta/360 of a circle. Find the area of a 90 degree sector of radius 14 cm. A segment equals sector minus triangle for a minor segment. Find the shaded area formed by a chord and arc. Use the same value of pi throughout a problem. Formula and notation Sector area = theta/360 x pi r^2 Arc length = theta/360 x 2 pi r Watch this: Perimeter of a sector includes two radii as well as the arc. One-line recall A sector is a fraction theta/360 of a circle. PADIKAM | NCERT CLASS 10 13

LESSON 12 Surface Areas and Volumes Quick understanding Model a combined solid, identify exposed surfaces and keep units cubic or square as required. Must remember Exam practice CSA means curved surface only; TSA includes all Find the volume of a cone mounted on a exposed faces. cylinder. For a combination, add volumes but count only outer Find the number of small spheres made surface areas. from a larger sphere. When a solid is recast, volume is conserved. Formula and notation Cylinder V = pi r^2h Cone V = 1/3 pi r^2h Sphere V = 4/3 pi r^3 Watch this: Do not include a joined circular face in the external surface area. One-line recall CSA means curved surface only; TSA includes all exposed faces. PADIKAM | NCERT CLASS 10 14

LESSON 13 Statistics Quick understanding Summarise grouped data with mean, median and mode, and interpret what each measure represents. Must remember Exam practice Class mark xi is the midpoint of a class interval. Find the mean by the step-deviation method. Median uses cumulative frequency and the median class. Find the median of a grouped frequency table. The modal class has the highest frequency. Formula and notation Mean = sum(fi xi)/sum(fi) Median = l + [(n/2 - cf)/f]h Mode = l + [(f1-f0)/(2f1-f0-f2)]h Watch this: Use the cumulative frequency before the median class, not the class total. One-line recall Class mark xi is the midpoint of a class interval. PADIKAM | NCERT CLASS 10 15

LESSON 14 Probability Quick understanding Measure chance as favourable equally likely outcomes divided by all equally likely outcomes. Must remember Exam practice Probability lies from 0 to 1. Find the probability of a sum of 8 with two dice. P(not E) = 1 - P(E). A card is drawn: find P(not a face card). List the sample space carefully before counting. Formula and notation P(E) = favourable outcomes / total outcomes P(not E) = 1 - P(E) Watch this: Theoretical probability requires equally likely outcomes. One-line recall Probability lies from 0 to 1. PADIKAM | NCERT CLASS 10 16