It's usually around 9 PM, three or four days before the first mid term, that I get the same message from at least two or three students.

Sir, can you send the important formulas?

I send them. They thank me. And then, almost every single time, the same thing happens in the exam hall a few days later — a student stares at a quadratic polynomial question, knows there's a formula for the sum of roots, and just... can't pull it out. Not because they didn't see it. Because they only saw it. They never actually held it in their hands.

This is the part nobody tells you about formula sheets. A list of formulas on its own does almost nothing. I've watched hundreds of students copy formula sheets into their notebooks, highlight them in three colors, paste them on their study table — and still freeze the moment the question doesn't look exactly like the example in the textbook.
So before I give you the formulas (and I will, all of them, clearly), I want to say one thing plainly: this sheet is not your preparation. It's your checkpoint. The real preparation happened — or didn't happen — in the weeks before you opened this page.

Why formulas slip away right when you need them
Here's something most students don't realize. Your brain doesn't store formulas the way it stores, say, your friend's phone number. A formula is stored with context — with the memory of solving a problem, the feeling of getting it wrong once, the moment a teacher pointed at the board and said no, not like that, like this. If you only read a formula, your brain files it under things I saw rather than things I did. And things you saw are the first things to disappear under exam pressure.

This is exactly why a student can watch fifteen YouTube videos explaining the quadratic formula and still draw a blank in the exam hall. Watching is comprehension. Solving is memory. They are not the same activity, even though they feel similar while you're doing them.

Parents often misread this. A child sits with the book open for two hours, looks focused, and still scores poorly. The parent assumes laziness or distraction. Usually, that's not what's happening. The child is reading and re-reading, mistaking familiarity for mastery. I've seen this formula a hundred times feels exactly like I know this formula — until the exam proves otherwise.

Alright. With that out of the way, let's go chapter by chapter.

Chapter 1: Real Numbers

This chapter has fewer formulas than the others, which is exactly why students underestimate it and then lose easy marks on conceptual questions.

Euclid's Division Lemma — the backbone of this chapter:

a = bq + r, where 0 ≤ r < b
Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder. The part students consistently forget is the condition 0 ≤ r < b. It looks like a footnote. It isn't. Examiners specifically test whether you understand that the remainder must be smaller than the divisor and never negative. I've seen full marks lost on a two-mark question purely because a student wrote the lemma without this condition.
HCF × LCM relationship, for any two positive integers:
HCF × LCM = Product of the two numbers
So if the numbers are 12 and 18, then HCF × LCM = 12 × 18 = 216. This single formula quietly solves a whole category of find the LCM if HCF is given questions that otherwise look intimidating.
A quick honest note here — this formula only works for two numbers, not three. Students sometimes try to extend it to three numbers in a hurry and get it wrong. Worth remembering that exception.

Chapter 2: Polynomials

This is where most of the mid-term marks actually live, and where the most common confusion happens.
A polynomial's degree tells you its name:

Linear polynomial: ax + b (degree 1)
Quadratic polynomial: ax² + bx + c (degree 2)
Cubic polynomial: ax³ + bx² + cx + d (degree 3)

For a linear polynomial ax + b, the zero is found by setting it to zero and solving:
ax + b = 0 → x = −b/a
Now here's where it gets interesting, and where the real exam marks are won or lost — the relationship between zeros and coefficients of a quadratic polynomial.
Take ax² + bx + c, with roots α and β. If you actually expand (x − α)(x − β) and compare it to the standard form, you'll see why these relationships exist rather than just memorizing them blind:
Sum of roots: α + β = −b/a
Product of roots: αβ = c/a
Notice the sum formula has a negative sign and the product formula doesn't. This is the single most common mix-up I see — students write αβ = −b/a and α + β = c/a, flipping them by accident under pressure. If you tend to confuse these, try this trick: remember that the product formula has no minus sign — product, plain and simple. It's a small mental anchor, but it works for a surprising number of students.

Chapter 3: Pair of Linear Equations in Two Variables

If your school's first mid term tests one chapter heavily, it's usually this one — and it's also the chapter where careless reading costs the most marks, not lack of knowledge.
Standard form of a pair of linear equations:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Before solving anything, identify a₁, b₁, c₁, a₂, b₂, c₂ correctly. This sounds obvious. It is not. A huge number of mistakes in this chapter happen at this exact step — a student misreads which number is a and which is c, and every calculation after that is doomed from the start, even though the method used was completely correct.
The three consistency conditions — memorize these properly, because vague memory here is worse than no memory at all:

Unique solution (lines intersect at one point):
a₁/a₂ ≠ b₁/b₂
No solution (lines are parallel):
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Infinite solutions (lines coincide):
a₁/a₂ = b₁/b₂ = c₁/c₂

A trick that helps many of my students: read it almost like a sentence. "If the first two ratios are equal to each other but not to the third, there's no solution. If all three ratios are equal, there are infinite solutions." Saying it out loud, not just looking at the symbols, helps it stick differently in memory.

The formulas that get forgotten the most, in my experience
After years of correcting mid-term papers, the same four things come up again and again as silent mark-killers:
The relationship between zeros and coefficients — specifically, the sign confusion between sum and product. The consistency conditions for linear equations — specifically, mixing up no solution with infinite solutions, which are visually almost identical on paper but mean opposite things. The condition 0 ≤ r < b in Euclid's lemma, which students drop entirely because it feels unimportant. And simply misidentifying a₁, b₁, c₁ from the equation before starting the problem.

None of these are hard concepts. They're small, sharp-edged details that get smoothed over during casual reading and then trip you up exactly when marks are on the line.

A revision method that actually works (because reading doesn't)
If you've got three or four days left, here's what to do instead of re-reading this page ten times.
Cover the formula with your hand. Try writing it from memory on rough paper. Uncover it and check what you got wrong — not just whether you got it wrong, but which part. Then immediately solve one question that uses that exact formula. That last step matters more than people think. A formula without an attached problem is just trivia. A formula attached to a solved problem becomes a tool you can actually reach for.
Do this for maybe twenty minutes a day rather than two hours the night before. Your hand needs to remember the formula as much as your eyes do. Your brain remembers what your hands practice — not what your eyes scan.
For parents reading this
If your child is sitting with formula sheets spread across the table looking focused, please don't assume that focus equals progress. Ask them to close the book and write three formulas from memory. That's the only honest test. It takes two minutes and tells you more than an hour of watching them study.
Also — and this is important — please don't say "but I sent you the formula list three times already" if they forget something on exam day. That comment, however well-meant, lands as blame, not help.

The child already knows they forgot. What helps is asking, calmly, which one slipped your mind? and then quietly working through that one formula together. Maths anxiety in Class 10 is rarely about intelligence.

It's almost always about a child who froze once, got criticized for it, and now associates the subject itself with that feeling of failure.
One honest truth before you close this page
A formula sheet cannot replace understanding, and board exams are built specifically to expose students who tried to make it do exactly that. The examiner isn't testing whether you memorized x = [−b ± √(b² − 4ac)] / 2a. They're testing whether you know when to reach for it, what each symbol stands for, and whether you can substitute correctly under time pressure with three other students' pencils scratching around you.
Maths does not forgive fake understanding. It's one of the few subjects where you can't write your way around not knowing something — there's no clever paragraph that disguises a wrong derivation.

But here's the encouraging part, and it's not a hollow one: every single formula on this page is learnable by anyone, including a student who's currently scoring 40 out of 80. None of this requires natural talent. It requires sitting down, writing the same three formulas badly a few times, and writing them slightly less badly the next day. That's genuinely the whole method. Confidence in this subject doesn't arrive in a single burst of motivation — it arrives quietly, one correctly solved problem at a time.
Go over this list once a day for the next few days. Not by reading it. By writing it.

FAQ

Which chapters are usually included in the CBSE Class 10 Maths first mid term exam?

Most CBSE schools cover Real Numbers, Polynomials, and Pair of Linear Equations in Two Variables for the first mid term, though this can vary by school. Some schools that move faster also include the early part of Quadratic Equations — it's worth confirming the exact syllabus with your school rather than assuming.
What is the most important formula in the Polynomials chapter?

The relationship between zeros and coefficients of a quadratic polynomial — sum of roots equals −b/a, and product of roots equals c/a — is the most frequently tested and most frequently confused formula in this chapter.
Why do students forget the condition 0 ≤ r < b in Euclid's Division Lemma?

Because it looks like a minor technical detail attached to the main formula, students often skip writing it. But examiners specifically check for this condition, and omitting it can cost marks even when the rest of the answer is correct.
How can I quickly tell the difference between no solution and infinite solutions in linear equations?

Compare all three ratios — a₁/a₂, b₁/b₂, and c₁/c₂. If the first two are equal but the third is different, there's no solution. If all three are equal, there are infinite solutions. The safest way to avoid confusion is writing out all three ratios every time rather than trying to judge it by eye.
Is memorizing formulas enough to score well in the mid term?

No. CBSE board exams test application and understanding, not just recall. A memorized formula only becomes useful once you've practiced identifying which question requires it and how to substitute the correct values into it.