CBSE Class 10 Board Exam Mathematics (Standard) Syllabus

Mathematics (Standard) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (Standard) in CBSE Class 10 Board Exam, not a summary of it.

1. 1

   Number Systems

   4 topics

   • Euclid's Division Lemma and Algorithm
   • Fundamental Theorem of Arithmetic
     • HCF and LCM by prime factorisation
   • Irrational Numbers
     • Proofs of irrationality (root 2, root 3, root 5)
   • Decimal expansions of rational numbers (terminating vs non-terminating)
2. 2

   Algebra

   4 topics

   • Polynomials
     • Zeroes of a polynomial and geometric meaning
     • Relationship between zeroes and coefficients of quadratics
   • Pair of Linear Equations in Two Variables
     • Graphical method of solution
     • Substitution and elimination methods
     • Consistency and number of solutions
     • Word problems and equations reducible to linear form
   • Quadratic Equations
     • Solution by factorisation
     • Quadratic formula and discriminant
     • Nature of roots
     • Situational word problems
   • Arithmetic Progressions
     • nth term of an AP
     • Sum of first n terms
     • Application problems
3. 3

   Coordinate Geometry

   3 topics

   • Distance Formula
     • Distance between two points
   • Section Formula
     • Internal division of a line segment
     • Midpoint formula
   • Plotting points and verifying geometric figures on a graph
4. 4

   Geometry

   3 topics

   • Triangles
     • Similarity criteria (AAA, SAS, SSS)
     • Basic Proportionality Theorem (Thales)
     • Theorems on similar triangles
   • Circles
     • Tangent to a circle at a point
     • Number of tangents from an external point
     • Length of tangents theorem
   • Constructions
     • Division of a line segment in a given ratio
     • Tangents to a circle from an external point
5. 5

   Trigonometry

   3 topics

   • Trigonometric Ratios
     • Trigonometric ratios of an acute angle
     • Ratios of specific angles (0, 30, 45, 60, 90)
   • Trigonometric Identities
     • Proving and applying standard identities
   • Heights and Distances
     • Angle of elevation and depression
     • Real-life application problems
6. 6

   Mensuration, Statistics and Probability

   4 topics

   • Areas Related to Circles
     • Area of sector and segment
     • Perimeter and areas of plane figures combinations
   • Surface Areas and Volumes
     • Combination of solids (cone, cylinder, sphere, hemisphere)
     • Frustum of a cone
   • Statistics
     • Mean of grouped data (direct, assumed mean, step-deviation)
     • Mode and median of grouped data
   • Probability
     • Classical definition of probability
     • Simple problems on single events

Mathematics (Standard) flashcards for CBSE Class 10 Board Exam

21 of 56 cards from the Mathematics (Standard) deck — real questions with worked answers.

1. State Euclid's Division Lemma.

   For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.

2. Describe the steps of Euclid's Division Algorithm to find the HCF of two positive integers a and b (a > b).

   1) Apply the division lemma to get a = bq + r. 2) If r = 0, the HCF is b. 3) If r ≠ 0, apply the lemma to b and r, and continue until the remainder is 0. The last nonzero divisor is the HCF.

3. State the Fundamental Theorem of Arithmetic.

   Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur.

4. For two positive integers a and b, what is the relationship between their HCF, LCM, and product?

   HCF(a, b) × LCM(a, b) = a × b.

5. How do you find the LCM and HCF of numbers using prime factorisation?

   HCF = product of the smallest power of each common prime factor. LCM = product of the greatest power of each prime factor involved in the numbers.

6. Prove-style fact: What is the standard method to prove that √2 is irrational?

   Use contradiction: assume √2 = p/q (p, q coprime, q ≠ 0). Then 2q² = p², so 2 divides p², hence 2 divides p. Writing p = 2c gives q² = 2c², so 2 divides q too. This contradicts p, q being coprime, so √2 is irrational.

7. A rational number p/q (in lowest terms) has a terminating decimal expansion under what condition on q?

   When the prime factorisation of q is of the form 2^n × 5^m (i.e., q has no prime factors other than 2 and 5).

8. A rational number p/q (in lowest terms) has a non-terminating repeating decimal expansion under what condition on q?

   When the denominator q has at least one prime factor other than 2 or 5.

9. What is the general form of a polynomial, and what is meant by its degree?

   A polynomial in x is p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ with aₙ ≠ 0. The degree is the highest power of x (n) with a nonzero coefficient.

10. Define a zero of a polynomial p(x). Geometrically, what does it represent?

    A zero is a value k such that p(k) = 0. Geometrically, the zeroes are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

11. For a quadratic polynomial ax² + bx + c with zeroes α and β, what are the sum and product of the zeroes?

    Sum of zeroes: α + β = −b/a. Product of zeroes: αβ = c/a.

12. For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β, γ, give the three relations between zeroes and coefficients.

    α + β + γ = −b/a; αβ + βγ + γα = c/a; αβγ = −d/a.

13. State the Division Algorithm for polynomials.

    Given polynomials p(x) and g(x) (g(x) ≠ 0), there exist unique polynomials q(x) and r(x) such that p(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

14. What is the general form of a pair of linear equations in two variables?

    a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, where a₁, b₁, a₂, b₂ are not all zero.

15. Give the conditions on the ratios a₁/a₂, b₁/b₂, c₁/c₂ for a pair of linear equations to have: a unique solution, infinitely many solutions, or no solution.

    Unique solution (consistent, intersecting lines): a₁/a₂ ≠ b₁/b₂. Infinitely many solutions (consistent, coincident lines): a₁/a₂ = b₁/b₂ = c₁/c₂. No solution (inconsistent, parallel lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

16. Name the algebraic methods to solve a pair of linear equations in two variables.

    Substitution method, Elimination method, and Cross-multiplication method (graphical method is the non-algebraic one).

17. State the cross-multiplication formula for solving a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

    x / (b₁c₂ − b₂c₁) = y / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁).

18. What is the standard form of a quadratic equation?

    ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

19. State the quadratic formula for the roots of ax² + bx + c = 0.

    x = [−b ± √(b² − 4ac)] / (2a), provided b² − 4ac ≥ 0.

20. What is the discriminant of a quadratic equation, and how does it determine the nature of the roots?

    Discriminant D = b² − 4ac. If D > 0: two distinct real roots. If D = 0: two equal (real and equal) roots. If D < 0: no real roots (roots are imaginary).

21. Name the methods used to find the roots of a quadratic equation.

    Factorisation (splitting the middle term), completing the square, and the quadratic formula.

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