| NCERT Exemplar Solutions | ||||||
|---|---|---|---|---|---|---|
| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
Class 10 Mathematics NCERT Exemplar Solutions
Chapter 1 Real Numbers
This chapter revisits the properties of real numbers and their relationship with rational and irrational numbers. Students learn about the Euclid’s Division Lemma which states that for any two integers $a$ and $b$ ($b > 0$), there exist unique integers $q$ and $r$ such that $a = bq + r$. It also explores the concept of HCF and LCM using the Euclidean algorithm and proves the irrationality of numbers like $\sqrt{2}$ and $\sqrt{3}$.
Chapter 2 Polynomials
Learn about the structure of a polynomial, its degree, and coefficients. The chapter covers the relationship between zeros and coefficients using the identities: For a quadratic polynomial $ax^2 + bx + c = 0$, if $\alpha$ and $\beta$ are the zeros, then: $\alpha + \beta = -\frac{b}{a}$ and $\alpha \beta = \frac{c}{a}$. Graphical representation of polynomials helps visualize their behavior and the number of real roots.
Chapter 3 Pair of Linear Equations in Two Variables
This chapter introduces methods to solve a pair of equations in two variables using substitution, elimination, and cross-multiplication. Graphical interpretation explains when lines are intersecting, parallel, or coincident based on the ratio $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$.
Chapter 4 Quadratic Equations
A quadratic equation is of the form $ax^2 + bx + c = 0$. Solutions can be found by factorization, completing the square, or using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The nature of the roots depends on the discriminant $\Delta = b^2 - 4ac$.
Chapter 5 Arithmetic Progressions
Explore the pattern of numbers where each term increases by a constant difference $d$. The $n^{th}$ term is given by $a_n = a + (n-1)d$, and the sum of the first $n$ terms by $S_n = \frac{n}{2}[2a + (n-1)d]$. Real-life problems involving installment payments and sequences are discussed.
Chapter 6 Triangles
Understand the criteria for similarity of triangles: AAA, SAS, and SSS. The Pythagoras theorem and its converse are explored with geometric proofs: In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
Chapter 7 Coordinate Geometry
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Learn to find the coordinates of the midpoint, section points, and the area of a triangle using determinants.
Chapter 8 Introduction to Trigonometry & Its Identities
Discover the trigonometric ratios for acute angles in a right triangle: $\sin \theta = \frac{P}{H}$, $\cos \theta = \frac{B}{H}$, and $\tan \theta = \frac{P}{B}$. The fundamental identity $\sin^2 \theta + \cos^2 \theta = 1$ forms the base for many trigonometric equations and applications.
Chapter 9 Circles
Study tangent properties, including the theorem that the tangent to a circle is perpendicular to the radius at the point of contact. The length of tangents drawn from an external point to a circle are equal.
Chapter 10 Constructions
Learn step-by-step geometric constructions using a compass and ruler, including dividing a line segment in a given ratio and constructing tangents to a circle from an external point.
Chapter 11 Areas Related to Circles
Calculate the area of sectors and segments of a circle using the formulas: Area of a sector $= \frac{\theta}{360^\circ} \times \pi r^2$, and Perimeter of a sector $= \frac{\theta}{360^\circ} \times 2\pi r + 2r$.
Chapter 12 Surface Areas & Volumes
Explore formulas for surface area and volume of 3D solids like: Cylinder – $V = \pi r^2 h$, Cone – $V = \frac{1}{3}\pi r^2 h$, and Sphere – $V = \frac{4}{3}\pi r^3$. Practical problems involve combining or converting one solid into another.
Chapter 13 Statistics & Probability
Learn to calculate mean, median, and mode for grouped data. Understand basic probability using the formula $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.
Design of Question Paper – Set I
Provides an understanding of the structure, typology, and marks distribution in Class 10 Mathematics question paper (Set I).
Design of Question Paper – Set II
Includes additional models and practice sets following the CBSE exam pattern for enhanced preparation.