Class 10 (English Medium) Math

This chapter builds the foundation of the real number system for Class 10. It covers Euclid's Division Lemma and Algorithm (retained by many State Boards) for finding HCF, the Fundamental Theorem of Arithmetic and its use in finding HCF and LCM by prime factorisation, proofs of irrationality of numbers like $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{5}$, and the decimal expansions of rational numbers (terminating versus non-terminating recurring). It is a high-scoring, proof-and-numerical chapter that is important for both CBSE and State Board examinations.

This chapter develops the idea of a polynomial in one variable — its degree, types (linear, quadratic, cubic), and the meaning of a zero of a polynomial. It explains the geometrical meaning of zeroes as the points where the graph meets the x-axis, establishes the relationship between the zeroes and the coefficients of quadratic and cubic polynomials, shows how to form a polynomial from its zeroes, and covers the division algorithm for polynomials (retained by many State Boards). It is a high-scoring, concept-plus-computation chapter important for both CBSE and State Board examinations.

This chapter studies pairs of linear equations in two variables and the different ways to solve them. It covers forming equations from word problems, the graphical method (intersecting, coincident and parallel lines), the algebraic methods of substitution and elimination, the cross-multiplication method and the conditions for a pair to be consistent or inconsistent (retained by many State Boards), and equations reducible to a pair of linear equations. It is a high-scoring, application-rich chapter important for both CBSE and State Board examinations.

This chapter introduces quadratic equations of the form $ax^2 + bx + c = 0$ and the methods to solve them: factorisation, completing the square, and the quadratic formula. It covers the discriminant and the nature of roots (real and distinct, real and equal, or no real roots), and a wide range of word problems that reduce to quadratic equations. It is a high-scoring, application-rich chapter important for both CBSE and State Board examinations.

This chapter introduces arithmetic progressions (APs) — lists of numbers in which each term differs from the previous one by a fixed common difference. It covers identifying an AP and its first term and common difference, the formula for the nth term $a_n = a + (n-1)d$, the sum of the first n terms $S_n = \frac{n}{2}[2a + (n-1)d]$, the arithmetic mean, selecting terms in AP, and a wide range of real-life word problems. It is a high-scoring, formula-driven chapter important for both CBSE and State Board examinations.

This chapter studies similar triangles. It explains the difference between congruent and similar figures, the Basic Proportionality Theorem (Thales' theorem) and its converse, the three criteria for similarity of triangles (AAA/AA, SSS, SAS), the theorem on the ratio of areas of similar triangles, and the Pythagoras theorem with its converse (retained by many State Boards). It is a proof-and-application-rich chapter important for both CBSE and State Board examinations.

This chapter connects algebra and geometry through the Cartesian plane. It covers plotting points and quadrants, the distance formula for the distance between two points, the section formula for a point dividing a segment in a given ratio (and the midpoint formula), and the area of a triangle from its vertices together with the collinearity test (area-of-triangle is retained by many State Boards though removed from the rationalised NCERT). Applications include the centroid, identifying types of quadrilaterals, and coordinate-geometry word problems. It is a high-scoring, formula-driven chapter important for both CBSE and State Board examinations.

This chapter introduces trigonometry through the ratios of the sides of a right-angled triangle. It defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) of an acute angle, derives their exact values for the standard angles 0°, 30°, 45°, 60° and 90°, establishes the fundamental trigonometric identities ($\sin^2\theta + \cos^2\theta = 1$ and its companions), and studies trigonometric ratios of complementary angles. It is a formula-rich, high-scoring chapter and a foundation for Applications of Trigonometry (heights and distances), important for both CBSE and State Board examinations.

This chapter applies the trigonometric ratios from the previous chapter to real-world problems of heights and distances. It introduces the line of sight, the angle of elevation and the angle of depression, and shows how to model a situation as a right-angled triangle and solve for an unknown height or distance using sin, cos and tan. It covers single-triangle problems, problems involving two triangles or two observers, angle-of-depression situations, and mixed real-life applications. Calculations rely heavily on the standard angles 30°, 45° and 60°. It is a high-scoring, application-focused chapter important for both CBSE and State Board examinations.

This chapter studies the different ways a line can meet a circle and, in particular, the tangent to a circle — a line that touches the circle at exactly one point. It proves the two fundamental tangent theorems: the tangent at any point of a circle is perpendicular to the radius through the point of contact (Theorem 10.1), and the lengths of the two tangents drawn from an external point are equal (Theorem 10.2). It also develops the length-of-tangent formula $\sqrt{d^2-r^2}$, the number of tangents from a point, and a rich set of board applications — figures circumscribing a circle (quadrilaterals, triangles, parallelograms) and the angle relations between tangents and the centre. It is a short but very high-scoring chapter of the Class 10 Board syllabus.

This chapter measures the parts of a circle. Starting from the circumference $2\pi r$ and area $\pi r^2$ of a circle and the area of a ring, it develops the two central formulae — the area of a sector $\frac{\theta}{360}\pi r^2$ and the length of an arc $\frac{\theta}{360}\times 2\pi r$ — and then the area of a segment as (area of sector $-$ area of triangle). These are applied to a wide range of board favourites: minute and hour hands of clocks, windscreen wipers, grazing animals tied by a rope, umbrellas and fans, and especially areas of combinations of plane figures (shaded regions built from circles, sectors, squares and triangles). It is a formula-rich, high-scoring chapter of the Class 10 Board syllabus.

This chapter measures real solids built from the five basic shapes — cuboid, cube, cylinder, cone, sphere and hemisphere. After recalling every base formula (curved and total surface area, and volume), it shows how to find the surface area of a combination of solids (add only the exposed curved surfaces — joined faces disappear) and the volume of a combination (simply add, or subtract for a cavity). It then covers two high-yield board topics: the conversion of a solid from one shape to another by melting and recasting (volume is conserved) and the frustum of a cone (a bucket/glass shape) with its surface-area and volume formulas. It is a formula-heavy, very high-scoring chapter of the Class 10 Board syllabus.

This chapter finds the three measures of central tendency — mean, median and mode — for grouped data. For the mean it develops all three methods: the direct method $\bar{x}=\frac{\sum f_ix_i}{\sum f_i}$, the assumed-mean method and the step-deviation method. It then gives the mode of grouped data from the modal class, and the median from the cumulative frequency and the median class. Finally it introduces the cumulative frequency distribution and its graph, the ogive (less-than and more-than types), shows how to read the median from an ogive, and states the empirical relationship $3\,\text{Median}=\text{Mode}+2\,\text{Mean}$. It is a formula-driven, high-scoring chapter of the Class 10 Board syllabus.

This chapter measures chance. It develops the theoretical (classical) probability of an event $E$ as $P(E)=\dfrac{\text{number of outcomes favourable to }E}{\text{total number of equally likely outcomes}}$, and explores its consequences: probability always lies between 0 and 1, a sure event has probability 1 and an impossible event probability 0, the probabilities of all elementary events add up to 1, and an event and its complement satisfy $P(\overline{E})=1-P(E)$. It applies these ideas to the standard board settings — tossing coins, throwing dice, drawing cards from a well-shuffled deck of 52, and picking marbles or spinning spinners. It is a concept-light, marks-rich chapter of the Class 10 Board syllabus.