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Class 10 · 10th standard · 2026-27

Class 10 maths worksheets, free, with every step on the page

Three free Class 10 maths worksheets, with every question and every step printed on this page rather than behind a download. They cover the three chapters that carry the most board marks and are the most reliably improvable - quadratics, trigonometry and arithmetic progressions. All three are method rather than insight, which is exactly what a board year has time to fix. Every exercise opens with one question worked through in full.

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3 worksheets · 85 questions · every answer on the page · no sign-up · aligned to NCERT's Class 10 maths book is Mathematics, offered at Standard and Basic levels. These sheets are written to the ideas rather than to one book's chapter order and suit both.

What these practise

  • Solving quadratics by splitting the middle term, including with surds
  • The quadratic formula, and the sign of b
  • The discriminant and the nature of the roots
  • Forming a quadratic from a described situation
  • The standard trigonometric values, known cold
  • Proving identities by converting to sine and cosine
  • Complementary angles used to simplify
  • Heights and distances from a labelled diagram
  • The nth term and the sum of n terms of an AP
  • Finding a and d from two given terms, and n from the last term

Last updated 2026-07-29

Free, with answers

The worksheets themselves

Every question is on this page, and so is every answer. Nothing is behind a sign-up, and there is no file to download unless you want one - use the print button on any sheet to save it as a PDF.

practice · medium · 35 min

Quadratic equations

Class 10 Maths · Quadratic equations · 30 questions with answers

Download PDF - Quadratic equations, with the answer key on the last page

Quadratics is the chapter Class 8 factorisation was built for, and it repays that work directly. Three things are asked: solve it, say what kind of roots it has, and turn a described situation into an equation. The third is where marks are actually lost, and it is lost in the first line rather than in the algebra.

  • Solves a quadratic by splitting the middle term
  • Applies the quadratic formula correctly, including the sign of b
  • Uses the discriminant to state the nature of the roots
  • Finds an unknown coefficient given a condition on the roots
  • Forms a quadratic equation from a described situation

Exercise 1 of 3 · 10 questions

Solve by factorisation.

Write the answer · Warm-up

Here’s one done for you

The last one looks alarming and works exactly like the others. The product to aim for is (first coefficient) × (last term) = √2 × 5√2 = 10, and the sum is 7, giving 5 and 2. Split the middle term as 5x + 2x and group. Surds in the coefficients change nothing about the method - only about how the arithmetic looks, and children who realise that stop being frightened by these.

  1. 1)x² - 5x + 6 = 0
  2. 2)2x² + x - 6 = 0
  3. 3)x² - 3x - 10 = 0
  4. 4)6x² - x - 2 = 0
  5. 5)√2x² + 7x + 5√2 = 0
  6. 6)x² - 7x + 12 = 0
  7. 7)3x² - 10x + 8 = 0
  8. 8)x² + 2x - 8 = 0
  9. 9)2x² - 5x - 3 = 0
  10. 10)x² - 16 = 0

Answers

  1. 1) (x - 2)(x - 3) = 0, so x = 2 or x = 3.
  2. 2) (2x - 3)(x + 2) = 0, so x = 3/2 or x = -2. Split as 2x² + 4x - 3x - 6.
  3. 3) (x - 5)(x + 2) = 0, so x = 5 or x = -2.
  4. 4) (3x - 2)(2x + 1) = 0, so x = 2/3 or x = -1/2. Split as 6x² - 4x + 3x - 2.
  5. 5) (√2x + 5)(x + √2) = 0, so x = -5/√2 or x = -√2. The product is √2 × 5√2 = 10 and the sum is 7, so split as 5x + 2x.
  6. 6) (x - 3)(x - 4) = 0, so x = 3 or x = 4.
  7. 7) (3x - 4)(x - 2) = 0, so x = 4/3 or x = 2.
  8. 8) (x + 4)(x - 2) = 0, so x = -4 or x = 2.
  9. 9) (2x + 1)(x - 3) = 0, so x = -1/2 or x = 3.
  10. 10) (x + 4)(x - 4) = 0, so x = 4 or x = -4.

Exercise 2 of 3 · 10 questions

Solve using the quadratic formula, then state the nature of the roots.

Write the answer · Guided

Here’s one done for you

For 2x² - 7x + 3, note that b = -7 and not 7, so -b is +7. That sign is the commonest error in the whole chapter, and it survives because the answer still looks like a number. Write a, b and c on their own line before substituting - three seconds, and it removes the error entirely.

  1. 1)2x² - 7x + 3 = 0
  2. 2)3x² - 5x + 2 = 0
  3. 3)x² + 4x - 5 = 0
  4. 4)2x² - 4x + 3 = 0
  5. 5)x² - 6x + 9 = 0
  6. 6)5x² - 6x + 1 = 0
  7. 7)4x² - 4x + 1 = 0
  8. 8)x² - 4x + 4 = 0
  9. 9)3x² + 2x + 5 = 0
  10. 10)2x² + 5x - 3 = 0

Answers

  1. 1) D = 49 - 24 = 25, so x = (7 ± 5)/4, giving x = 3 or x = 1/2. Two distinct real roots.
  2. 2) D = 25 - 24 = 1, so x = (5 ± 1)/6, giving x = 1 or x = 2/3. Two distinct real roots.
  3. 3) D = 16 + 20 = 36, so x = (-4 ± 6)/2, giving x = 1 or x = -5. Two distinct real roots.
  4. 4) D = 16 - 24 = -8, which is negative. No real roots.
  5. 5) D = 36 - 36 = 0, so x = 3 is a repeated root. Two equal real roots.
  6. 6) D = 36 - 20 = 16, so x = (6 ± 4)/10, giving x = 1 or x = 1/5. Two distinct real roots.
  7. 7) D = 16 - 16 = 0, so x = 4/8 = 1/2 twice. Two equal real roots.
  8. 8) D = 16 - 16 = 0, so x = 2 twice. Two equal real roots.
  9. 9) D = 4 - 60 = -56, which is negative, so there are no real roots.
  10. 10) D = 25 + 24 = 49, so x = (-5 ± 7)/4, giving x = 1/2 or x = -3. Two distinct real roots.

Exercise 3 of 3 · 10 questions

Form the equation and solve it. Check the answer against the question.

Write the answer · Practice

Here’s one done for you

The train question is the one worth studying, because the algebra is easy and the first line is not. Time equals distance over speed, so the two journey times are 360/x and 360/(x + 5), and the faster one takes an hour less - which gives 360/x - 360/(x + 5) = 1. Once that line is written the rest is routine. In every word problem here, the marks are decided by whether the child can write that first line, so it is worth practising the setting-up separately from the solving.

  1. 1)Find k so that kx² - 2kx + 6 = 0 has two equal roots.
  2. 2)The sum of a number and its reciprocal is 10/3. Find the number.
  3. 3)The product of two consecutive positive integers is 306. Find them.
  4. 4)A rectangular plot has an area of 528 m², and its length is one metre more than twice its breadth. Find both.
  5. 5)A train covers 360 km at a uniform speed. Had the speed been 5 km/h more, it would have taken one hour less. Find the speed.
  6. 6)Find the value of k for which x² + kx + 9 = 0 has equal roots.
  7. 7)The sum of the squares of two consecutive positive integers is 313. Find them.
  8. 8)The perimeter of a rectangle is 28 m and its area is 48 m². Find its sides.
  9. 9)A two-digit number is such that the product of its digits is 18, and when 27 is subtracted the digits interchange. Find the number.
  10. 10)The difference of two numbers is 3 and their product is 88. Find them.

Answers

  1. 1) Equal roots require D = 0, so 4k² - 24k = 0, giving 4k(k - 6) = 0 and k = 0 or 6. k = 0 makes it non-quadratic, so k = 6.
  2. 2) x + 1/x = 10/3 gives 3x² - 10x + 3 = 0. D = 64, so x = (10 ± 8)/6, giving x = 3 or x = 1/3.
  3. 3) n(n + 1) = 306 gives n² + n - 306 = 0. D = 1225 = 35², so n = 17 (rejecting the negative root). The integers are 17 and 18.
  4. 4) b(2b + 1) = 528 gives 2b² + b - 528 = 0. D = 4225 = 65², so b = 16 and the length is 33 m.
  5. 5) 360/x - 360/(x + 5) = 1 gives x² + 5x - 1800 = 0. D = 7225 = 85², so x = 40. The speed is 40 km/h.
  6. 6) Equal roots need D = 0, so k² - 36 = 0 and k = ±6.
  7. 7) n² + (n + 1)² = 313 gives 2n² + 2n - 312 = 0, so n² + n - 156 = 0 and (n + 13)(n - 12) = 0. Taking the positive value, the integers are 12 and 13.
  8. 8) Let the sides be l and 14 - l. Then l(14 - l) = 48, so l² - 14l + 48 = 0 and (l - 6)(l - 8) = 0. The sides are 6 m and 8 m.
  9. 9) Let the digits be x and y with xy = 18 and 10x + y - 27 = 10y + x, so x - y = 3. Then x = 6 and y = 3, giving 63. Check: 63 - 27 = 36.
  10. 10) Let them be n and n + 3. Then n(n + 3) = 88, so n² + 3n - 88 = 0 and (n + 11)(n - 8) = 0. The numbers are 8 and 11 (or -11 and -8).

While your child works

  • Write a, b and c on their own line before using the formula. The sign of b is the commonest error in the chapter.
  • Reject impossible roots explicitly - a negative length or a negative speed. Saying why earns the mark.
  • In word problems, practise writing the first line without solving. That is where the marks actually are.

Open the Quadratic equations worksheet on its own page

Class 10 Maths - the guide

Why these three chapters?

Because they carry a large share of the paper and they are the three most improvable in the time a board year actually has.

Quadratics rewards Class 8 factorisation directly. Trigonometry is a small table of values plus one technique - convert to sine and cosine - repeated. Arithmetic progressions is two formulas and a fixed method, with questions that repeat year after year.

None of the three needs insight. They need the method drilled until it is automatic, which is the one kind of improvement a board year can reliably deliver. The wider argument about what the paper rewards is on our Class 10 maths tuition page.

How should we use these at home?

Thirty to thirty-five minutes each, and then again under time pressure closer to the exam. Every exercise opens with one question worked through in full.

  • In quadratics, write a, b and c on their own line before using the formula. The sign of b is the commonest error.
  • In trigonometry, draw the diagram first for every height and distance question.
  • In AP, check the '+1' when counting terms. Twenty-nine gaps means thirty terms.
  • Let your child mark their own work. Every answer is on the page for exactly that reason.

Where do Class 10 children usually stall?

Three places, and all three are careless rather than conceptual - which is the good news.

  • The sign of b in the quadratic formula. For 2x² - 7x + 3, b is -7, so -b is +7. The wrong answer still looks like a number, which is why it survives.
  • Squaring only part of a trigonometric ratio. sec² 30° is 4/3, not 2/√3.
  • The first line of a word problem. The algebra is usually fine; translating the situation is not. Practise writing the equation without solving it.

All three respond to a fortnight of deliberate checking, which is why presentation and method are the first things worth working on in a board year - see the Class 10 board year.

Questions parents ask

Class 10 Maths worksheets - your questions

Are these Class 10 maths worksheets free?

Yes. Every question and every answer is on this page, with no sign-up, no email wall and no account. The printable version puts the answers on a separate final sheet.

Do these suit Maths Standard and Maths Basic?

Yes. Both levels cover the same syllabus at different depths, and these three chapters appear in both. The word problems are the part where Standard demands more.

What does the discriminant tell us?

Positive means two distinct real roots, zero means two equal real roots, negative means no real roots. It answers 'what kind of roots' without solving the equation at all.

How do I prove a trigonometric identity?

Work on one side only and convert everything to sine and cosine before simplifying. That one approach handles almost every identity in the Class 10 syllabus.

Why does my child lose marks in word problems?

Almost always in the first line rather than in the algebra. Practise writing the equation from the situation without solving it - the setting-up is the skill the marks are for.

Are these matched to NCERT?

They are written to the ideas the Class 10 paper tests rather than to one book's chapter order, so they work whether your school uses NCERT or its own text.

How long should a Class 10 child spend on one sheet?

Thirty to thirty-five minutes first time. Closer to the exam, do them again under time pressure - working to time is a separate skill from working correctly.

Can I print these worksheets?

Yes. They print cleanly on A4 with the answers on a separate final page, and nothing needs to be downloaded or signed up for.

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