10 Important Math Questions for Class 10

Class 10 maths builds the base for higher classes, and a few topics carry more weight in exams than others. The questions given below are the ones students meet again and again in school tests, board exams, and competitive exam preparation. Each one is explained in simple language so the idea behind the question becomes clear.

Understanding Real Numbers Through HCF and LCM

This is usually the first chapter and it sets the tone for the rest of the year. An important question here asks you to find the HCF and LCM of two numbers using prime factorisation, and then verify that HCF × LCM equals the product of the two numbers. Another common question uses Euclid’s division lemma to find the HCF of two positive integers step by step.

• The relation HCF × LCM = product of two numbers works only for two numbers.

• Euclid’s division lemma is written as a = bq + r, where r is smaller than b.

• Questions on proving a number irrational, such as √2 or √5, also appear often.

Polynomials and the Relationship Between Zeros and Coefficients

A frequently asked question gives you a quadratic polynomial and asks for its zeros, then asks you to check the sum and product of those zeros against the coefficients. For a polynomial ax² + bx + c, the sum of zeros is –b/a and the product is c/a. Some questions go the other way and give you the zeros, asking you to form the polynomial.

• Division of one polynomial by another and finding the quotient and remainder is a common question type.

• Graphs of polynomials help in identifying the number of zeros.

• Word problems based on zeros and coefficients appear in higher weightage sections.

Solving Pairs of Linear Equations in Two Variables

These questions usually give two equations and ask you to solve them using substitution, elimination, or cross-multiplication. Another version asks whether the pair has a unique solution, no solution, or infinitely many solutions, based on the ratios of the coefficients.

• For a unique solution, the ratio a₁/a₂ is not equal to b₁/b₂.

• Graphical questions ask you to plot both lines and read the intersection point.

• Word problems on ages, speed, and money are often framed from this chapter.

Quadratic Equations and the Nature of Roots

A standard question asks you to find the roots using factorisation or the quadratic formula. A second important question asks you to find the value of the discriminant, b² – 4ac, and state whether the roots are real, equal, or distinct.

• If the discriminant is positive, the roots are real and different.

• If the discriminant is zero, the roots are real and equal.

• Word problems on area, speed, and time often reduce to quadratic equations.

Arithmetic Progressions and Finding Terms and Sums

Questions here ask for the nth term of an AP using a + (n – 1)d, or the sum of the first n terms using n/2 [2a + (n – 1)d]. A common application question asks how many terms of a given AP add up to a particular value.

• The difference between consecutive terms stays the same in an AP.

• The sum formula simplifies to n/2 (a + l) when the last term is known.

• Real-life questions on savings, seating rows, and instalments are common.

Similar Triangles and the Basic Proportionality Theorem

An important question gives two triangles and asks you to prove they are similar using AA, SSS, or SAS similarity. Another question is based on the Basic Proportionality Theorem, also called Thales Theorem, where a line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

• The ratio of areas of two similar triangles equals the square of the ratio of their sides.

• Diagrams should always be drawn clearly before writing the proof.

• Proof-based questions from this chapter carry good marks.

Coordinate Geometry and Finding Distances and Midpoints

Questions ask you to find the distance between two points using the distance formula, or to find the midpoint of a line segment. The section formula is used when a point divides a line in a given ratio.

• Distance formula is √[(x₂ – x₁)² + (y₂ – y₁)²].

• Midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2).

• Questions on proving a triangle is isosceles or right-angled appear often.

Trigonometric Ratios and Their Applications

A common question asks you to prove a given trigonometric identity using standard ratios and the identity sin²θ + cos²θ = 1. Another important question is based on heights and distances, where angles of elevation and depression are used to find the height of a tower or the distance of a ship from a lighthouse.

• Standard values for 0°, 30°, 45°, 60°, and 90° should be memorised.

• Diagrams make height and distance questions much easier to solve.

• Identity-based proofs need step-by-step working for full marks.

Circles and the Properties of Tangents

Questions here usually ask you to prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Another common question involves two tangents drawn from an external point and asks you to prove that they are equal in length.

• The lengths of tangents from an external point to a circle are equal.

• A parallelogram circumscribing a circle is a rhombus, which is a frequent proof.

• Number of tangents from a point inside, on, and outside the circle is also asked.

Surface Areas and Volumes of Combined Solids

These questions give a shape made by joining two solids, such as a cone on a cylinder or a hemisphere on a cylinder, and ask for the total surface area or volume. Another version asks for the volume of a frustum of a cone.

• Always identify which parts of the surfaces are hidden after joining.

• Volume of the combined solid is the sum of the volumes of the parts.

• Questions on conversion of one solid into another are common.

Statistics and Probability as Supporting Topics

Mean, median, and mode questions ask you to calculate the central tendency for grouped data. Probability questions ask for the probability of an event using the formula number of favourable outcomes divided by total number of outcomes.

• The empirical relationship between mean, median, and mode is often tested.

• Probability values always lie between 0 and 1.

• These chapters usually carry direct and scoring questions.

Final Note

These ten areas cover most of the important question types that appear in Class 10 maths papers. Practising them with clear steps, proper diagrams, and correct formulas helps in building confidence before the exam.

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