Class 10 maths question patterns: 8 shapes to recognise before you solve

Search for “most repeated questions” and you will find long lists of past questions. A list helps only if a question comes back word for word. A shape helps whenever it comes back, whether the numbers change, the story changes or the figure is turned round. This guide covers eight shapes that Class 10 questions are built on, with the trap inside each, so that you can recognise them however they are dressed up.

1. Prove that a number is irrational

Looks like: “Prove that $7 - 4\sqrt{3}$ is irrational”, given that $\sqrt{3}$ is irrational.

The move: assume it is rational and isolate the root.

2. HCF and LCM with a missing piece

Looks like: given the HCF and one number, find the LCM; or “can two numbers have HCF 9 and LCM 100?”

The move: for two numbers, $\text{HCF} \times \text{LCM} = \text{product of the numbers}$. And the HCF always divides the LCM.

3. Quadratic word problems

Looks like: speed and time, the age of two people, the sides of a rectangular field, the number of articles made in a day.

The move: name the unknown, write the equation from the story, bring it to standard form, solve, and then go back to the story.

4. Arithmetic progressions: terms and sums

Looks like: “Which term of the AP $5, 14, 23, 32, \ldots$ is 90 more than its 20th term?”

The move: two terms differ by the number of steps between them, times $d$. Neither term needs to be calculated.

5. Basic Proportionality Theorem and similar triangles

Looks like: a line parallel to one side of a triangle cuts the other two sides; find a length. Or prove BPT itself.

The move: if $DE \parallel BC$ in $\triangle ABC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.

6. Tangents from an external point

Looks like: two tangents from a point, a quadrilateral drawn around a circle, or a chord of the larger of two concentric circles that touches the smaller one.

The move: two facts do almost all the work. Tangents from an external point are equal in length, and a tangent is perpendicular to the radius at the point of contact.

7. Heights and distances with two angles

Looks like: a tower seen from two points; angles of elevation that are complementary; a person on a building looking down at two cars.

The move: write one ratio from each right triangle and connect them. For complementary angles from points 8 m and 18 m from the base:

8. Grouped data and probability

Looks like: find the median of a frequency table with class intervals; or a probability question with cards, dice or numbered discs.

The move: for the median, $\text{Median} = l + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h$, where $cf$ is the cumulative frequency of the class before the median class. For probability, count the favourable outcomes carefully, then divide by the total.

How to practise with past papers properly

• Name before you solve. For every past-paper question, write the pattern name in the margin first. If you cannot name it, that is the gap to work on.
• One plain, one twisted. For each pattern, solve one standard past-paper question and then one with a trap. The challenges linked above are built as twisted versions.
• Mix the chapters. Once a pattern is familiar, practise it mixed in with others. In a mixed set nobody tells you the chapter, so recognition is the skill you need, and chapter-wise practice hides it.

The eight patterns above are not the whole subject. Coordinate geometry, polynomials, linear equations and mensuration have shapes of their own. The same habit works for them: name the shape first, then write the first line. For the quadratic pattern in particular, see how to choose the right method in ten seconds.