Open any five years of Class 10 Maths papers side by side and a pattern shows up quickly. The numbers change, the stories change, the names of the people in the word problems change, but the underlying question types return. A HCF and LCM problem, a quadratic word problem, a height and distance question, a probability question with two dice: they come back in new clothes.
This article lists fifteen of those recurring types, grouped by unit, with the method for each and a solved example. It is a list of patterns, not a forecast. The exact mix and the marks per type change from year to year and between boards, so use the previous year papers and the sample papers of your own board to see how each type is framed. For a single sheet of formulas to keep beside this article, the Class 10 Maths formula sheet is handy.
How to Use a List of Types Well
A list like this is useful only if you practise by type. Take one type at a time, solve three or four problems of it from different sources, and note the step where you hesitated. The goal is to reach a point where reading the question tells you the method before you have written a line. That is what separates a student who solves a paper in time from one who understands everything but runs out of minutes.
The fifteen types follow the order of the syllabus, starting with number and algebra.
Number, Polynomials and Equations
HCF and LCM, using the product relation. For two positive integers, HCF x LCM = product of the numbers. Given 306 and 657 with HCF 9, the LCM is 306 x 657 / 9 = 22,338. The HCF itself comes from Euclid’s division algorithm: 657 = 2 x 306 + 45, then 306 = 6 x 45 + 36, then 45 = 1 x 36 + 9, then 36 = 4 x 9 with no remainder, so the HCF is 9. The pages on real numbers cover the proofs and the word-problem versions.
Zeroes and coefficients of a quadratic polynomial. For a polynomial with zeroes alpha and beta, the sum is minus b / a and the product is c / a. To build a polynomial from a given sum and product, write x squared minus (sum) x + (product). With sum 1/4 and product minus 1, we get x squared minus x/4 minus 1, and multiplying by 4 gives 4x squared minus x minus 4. See the page on the relationship between zeroes and coefficients.
Pair of linear equations from a word problem. Define two variables, form two equations, solve by substitution or elimination. The sum of the digits of a two-digit number is 9, and on reversing the digits the number increases by 27. Let the tens digit be x and the units digit y. Then x + y = 9 and (10y + x) minus (10x + y) = 27, which gives y minus x = 3. So y = 6 and x = 3, and the number is 36. Check: the reverse is 63, and 63 minus 36 = 27. The algebraic methods page shows both approaches.
Quadratic equation from a word problem. The product of two consecutive positive integers is 306. Let them be x and x + 1. Then x squared + x minus 306 = 0, which factorises as (x + 18)(x minus 17) = 0. Since x is positive, x = 17, and the integers are 17 and 18. Check: 17 x 18 = 306.
Nature of roots using the discriminant. For a x squared + b x + c = 0, the roots are real and equal when b squared minus 4ac = 0. For 2x squared minus 4x + k = 0, the discriminant is 16 minus 8k, which is zero when k = 2. See the quadratic equations important questions for more.
Arithmetic progression: nth term and sum. For the AP 3, 8, 13, and so on, a = 3 and d = 5. The 20th term is 3 + 19 x 5 = 98. The sum of 20 terms is 20 / 2 x (3 + 98) = 10 x 101 = 1,010. Check: the average of the first and last terms is 50.5, and 50.5 x 20 = 1,010. The page on introduction to arithmetic progressions lists the formulas.
Geometry and Coordinate Geometry
Basic proportionality in a triangle. If a line parallel to one side of a triangle meets the other two sides, it divides them in the same ratio. In triangle ABC, with DE parallel to BC, AD = 2, DB = 3 and AE = 4, we get AD / DB = AE / EC, so 2 / 3 = 4 / EC, and EC = 6. The page on similarity of triangles goes into the criteria.
Length of a tangent from an external point. If the point is 13 cm from the centre and the radius is 5 cm, the tangent length is the square root of (169 minus 25) = 12 cm. The usual slips are described in the article on tangent proofs and their common mistakes.
Section formula and distance formula. The point dividing the line joining (1, 2) and (7, 8) in the ratio 1 : 2 has coordinates ((1 x 7 + 2 x 1) / 3, (1 x 8 + 2 x 2) / 3) = (9/3, 12/3) = (3, 4). The distance formula is the square root of the sum of squared differences. The page on the section formula has the derivations.
Trigonometry
Evaluating trigonometric expressions. Use the standard values. For example, sin 60 cos 30 + sin 30 cos 60 = (root 3 / 2)(root 3 / 2) + (1/2)(1/2) = 3/4 + 1/4 = 1. A related type gives one ratio and asks for the others: if sin theta = 3/5, then cos theta = 4/5 and tan theta = 3/4. Revise with the page on trigonometric ratios of specific angles.
Heights and distances. From a point 30 m from the foot of a tower, the angle of elevation of the top is 60 degrees. Then height = 30 x tan 60 = 30 root 3, which is about 51.96 m when root 3 is taken as 1.732. A clear figure with the angle marked at the right place is half the answer. The applications of trigonometry page covers the variations.
Mensuration, Statistics and Probability
Area of a sector. A sector of radius 7 cm with a central angle of 90 degrees has area (90 / 360) x (22/7) x 49 = 38.5 square cm. Segment questions subtract a triangle, as explained in the article on sector and segment confusion.
Combined solids and melting and recasting. A solid cone of radius 6 cm and height 24 cm is melted and recast into spheres of radius 3 cm. Volume of the cone is one third x pi x 36 x 24 = 288 pi, and each sphere is four thirds x pi x 27 = 36 pi, so there are 288 / 36 = 8 spheres. Volume is conserved, not surface area. The page on surface areas and volumes has the formulas for every solid.
Mean or median of grouped data. Build the table, find class marks or cumulative frequencies, and apply the formula. For a table with total frequency 50, cumulative frequencies 5, 13, 28, 44, 50 and classes of width 10 starting at 0, the median class is 20 to 30, and the median is 20 + ((25 minus 13) / 15) x 10 = 28. The choice between measures is explained in the page on median of grouped data.
Probability with dice or cards. When two dice are thrown, there are 36 equally likely outcomes. The sum 8 occurs for (2,6), (3,5), (4,4), (5,3) and (6,2), which is 5 outcomes, so the probability is 5/36. From a pack of 52 cards, the probability of a king or a queen is 8/52 = 2/13. The probability page lists the standard sample spaces.
A Quick Self-Check Across All Fifteen
Before leaving any numerical, ask three things. Is the answer the right kind: a length should be positive, a probability must lie between 0 and 1, a count of objects must be a whole number? Is the unit right and stated? Does the answer pass a quick reasonableness test, such as the average of the first and last terms for an AP sum, or the product test for consecutive integers? These checks take a few seconds and catch the commonest slips.
What Your Board Expects
Chapter names are the safe way to refer to these topics, because numbering differs between boards and between editions. Constructions has been removed from the current CBSE Class 10 syllabus, while the Karnataka SSLC textbook keeps it, so practice for that topic depends on your board. Boards also differ on how many marks go to each unit, on whether the sector and segment questions include only certain angles, and on how heavily the competency-based questions weigh on word problems. Confirm all of this against your board’s current syllabus and the important questions for Class 10 Maths set for your board.
Frequently Asked Questions
Q1. Do the same types really appear every year in Class 10 Maths?
The core types recur, but the exact selection, the marks and the framing change. Treat any list, including this one, as a set of patterns to practise rather than a prediction.
Q2. How should I practise by type?
Take one type, solve three or four problems from different sources, and note where you hesitated. Revisit that step before moving to the next type.
Q3. Which types are usually quickest to score on?
Types with a fixed method, such as HCF and LCM, arithmetic progressions, the section formula and probability, are usually reliable once the formulas are secure, because the steps rarely vary.
Q4. Is the CBSE and SSLC pattern the same?
The core numerical types are shared, but the syllabus and the marking pattern differ in places, for example Constructions. Check your own board’s syllabus and sample papers.
Q5. How do I avoid careless arithmetic mistakes?
Write every substitution on a separate line, use the self-check of sign, unit and reasonableness, and practise the arithmetic of squares, cubes and fractions regularly.
Conclusion: Learn the Pattern, Then Vary the Numbers
The numbers in a Class 10 Maths paper are new every year, but the question types are old friends. Learn each type once, practise it under time, and keep a short check at the end of every answer.
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