NCERT Mathematics Exemplar Problems Class 10th
UNIT I: NUMBER SYSTEMS
1. REAL NUMBER
Fundamental Theorem of Arithmetic—statements after reviewing work done earlier and after illustrating and motivating through examples, proofs of the irrationality of the square roots of 2, 3, and 5.
UNIT II: ALGEBRA
1. POLYNOMIALS
Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials.
2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
A pair of linear equations in two variables and the graphical method of their solution, consistency/inconsistency.
Algebraic conditions for the number of solutions. Solution of a pair of linear equations in two variables algebraically—by substitution, by elimination. Simple situational problems.
3. QUADRATIC EQUATIONS (15) Periods
The standard form of a quadratic equation is ax² + bx + c = 0 (a ≠ 0). Solutions of quadratic equations (only real roots) by factorization and by using the quadratic formula. Relationship between the discriminant and the nature of roots. Situational problems based on quadratic equations related to day-to-day activities to be
incorporated.
4. ARITHMETIC PROGRESSIONS
Motivation for studying arithmetic progression Derivation of the nth term and sum of the first n terms of A.P. and their application in solving daily life problems.
UNIT III: COORDINATE GEOMETRY
Coordinate Geometry
Review: Concepts of coordinate geometry, graphs of linear equations. Distance formula. Section formula (internal division)
UNIT IV: GEOMETRY
1. TRIANGLES
Definitions, examples, and counterexamples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional, and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal, and the two triangles are similar.
5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
2. CIRCLES
Tangent to a circle at the point of contact
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to a circle are equal.
UNIT V: TRIGONOMETRY
1. INTRODUCTION TO TRIGONOMETRY
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values of the trigonometric ratios of 30, 45 and 60. Relationships between the ratios.
2. TRIGONOMETRIC IDENTITIES
Proof and applications of the identity sin²A + cos²A = 1. Only simple identities are to be given.
3. HEIGHTS AND DISTANCES: Angle of elevation, angle of depression.
Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30°, 45°, and 60°.
UNIT VI: MENSURATION
1. AREAS RELATED TO CIRCLES
Area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above-mentioned plane figures. (In calculating the area of a segment of a circle, problems should be restricted to central angles of 60°, 90°, and 120° only.).
2. SURFACE AREAS AND VOLUMES
Surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres, and right circular cylinders/cones.
UNIT VII: STATISTICS AND PROBABILITY
1. STATISTICS
Mean, median, and mode of grouped data (bimodal situation to be avoided).
2. PROBABILITY
Classical definition of probability. Simple problems on finding the probability of an event.
