Class 9 & 10 Maths Formula Sheet: Last-Minute Revision for Board Exams

Board exams are close. You have studied everything, but now you need a quick revision. You don’t have time to read full chapters again. You just want all the important formulas in one place.

This article is your Class 9 & 10 Maths Formula Sheet. It covers all the main formulas from both classes. Keep it with you for last-minute revision. Read it again and again before the exam. It will help you remember everything fast.

1. Algebra

Algebraic Identities (Class 9 & 10)

These are very important. They come in many questions.

• (a + b)^2 = a^2 + 2ab + b^2

• (a – b)^2 = a^2 – 2ab + b^2

• a^2 – b^2 = (a + b)(a – b)

• (a + b)^3 = a^3 + b^3 + 3ab(a + b)

• (a – b)^3 = a^3 – b^3 – 3ab(a – b)

• a^3 + b^3 = (a + b)(a^2 – ab + b^2)

• a^3 – b^3 = (a – b)(a^2 + ab + b^2)

• a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)

• If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc

Linear Equations in Two Variables (Class 9 & 10)

For equations: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

• Unique solution: a1/a2 is not equal to b1/b2

• No solution: a1/a2 = b1/b2 and b1/b2 is not equal to c1/c2

• Infinite solutions: a1/a2 = b1/b2 = c1/c2

Quadratic Equations (Class 10)

For ax^2 + bx + c = 0:

• Quadratic formula: x = [-b ± sqrt(b^2 – 4ac)] / 2a

• Discriminant: D = b^2 – 4ac

  • If D > 0: two distinct real roots

  • If D = 0: two equal real roots

  • If D < 0: no real roots

• Sum of roots: alpha + beta = -b/a

• Product of roots: alpha * beta = c/a

Arithmetic Progressions (Class 10)

• nth term: an = a + (n – 1)d

• Sum of n terms: Sn = n/2 [2a + (n – 1)d]

• Sum when last term is known: Sn = n/2 (a + l)

Polynomials (Class 9 & 10)

For a quadratic polynomial ax^2 + bx + c:

• Sum of zeroes = -b/a

• Product of zeroes = c/a

For a cubic polynomial ax^3 + bx^2 + cx + d:

• Sum of zeroes = -b/a

• Sum of product of zeroes taken two at a time = c/a

• Product of zeroes = -d/a

2. Geometry

Triangles (Class 9 & 10)

• Pythagoras theorem: In a right triangle, hypotenuse^2 = base^2 + height^2

• Similar triangles: If two triangles are similar, ratio of sides = ratio of corresponding sides, and ratio of areas = square of ratio of sides.

• Mid-point theorem: The line joining midpoints of two sides is parallel to the third side and half of it.

Circles (Class 9 & 10)

• Angle in a semicircle is 90°.

• Equal chords subtend equal angles at the centre.

• Perpendicular from centre to a chord bisects the chord.

• Tangent is perpendicular to radius at point of contact.

• Length of tangent from external point: sqrt(d^2 – r^2), where d = distance from centre, r = radius.

• Cyclic quadrilateral: Opposite angles sum to 180°.

Coordinate Geometry (Class 9 & 10)

• Distance formula: d = sqrt((x2 – x1)^2 + (y2 – y1)^2)

• Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)

• Section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

• Area of triangle: 1/2 * |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|

3. Trigonometry (Class 10)

Trigonometric Ratios

• sin theta = Perpendicular / Hypotenuse

• cos theta = Base / Hypotenuse

• tan theta = Perpendicular / Base

• cosec theta = 1 / sin theta

• sec theta = 1 / cos theta

• cot theta = 1 / tan theta

Trigonometric Identities

• sin^2 theta + cos^2 theta = 1

• 1 + tan^2 theta = sec^2 theta

• 1 + cot^2 theta = cosec^2 theta

Trigonometric Table (in text form)

For theta = 0°:

• sin 0° = 0

• cos 0° = 1

• tan 0° = 0

For theta = 30°:

• sin 30° = 1/2

• cos 30° = sqrt(3)/2

• tan 30° = 1/sqrt(3)

For theta = 45°:

• sin 45° = 1/sqrt(2)

• cos 45° = 1/sqrt(2)

• tan 45° = 1

For theta = 60°:

• sin 60° = sqrt(3)/2

• cos 60° = 1/2

• tan 60° = sqrt(3)

For theta = 90°:

• sin 90° = 1

• cos 90° = 0

• tan 90° = Not defined

4. Mensuration

2D Shapes (Area & Perimeter)

• Rectangle: Area = l * b, Perimeter = 2(l + b)

• Square: Area = a^2, Perimeter = 4a

• Triangle: Area = 1/2 * base * height

• Heron’s formula: Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2

• Circle: Area = pi * r^2, Circumference = 2 * pi * r

• Parallelogram: Area = base * height

• Trapezium: Area = 1/2 * (a + b) * h

3D Shapes (Surface Area & Volume)

• Cube:

  • Curved Surface Area = 4a^2

  • Total Surface Area = 6a^2

  • Volume = a^3

• Cuboid:

  • Curved Surface Area = 2h(l + b)

  • Total Surface Area = 2(lb + bh + hl)

  • Volume = l * b * h

• Cylinder:

  • Curved Surface Area = 2 * pi * r * h

  • Total Surface Area = 2 * pi * r * (r + h)

  • Volume = pi * r^2 * h

• Cone:

  • Curved Surface Area = pi * r * l

  • Total Surface Area = pi * r * (l + r)

  • Volume = 1/3 * pi * r^2 * h

  • For cone: l = sqrt(r^2 + h^2)

• Sphere:

  • Curved Surface Area = 4 * pi * r^2

  • Total Surface Area = 4 * pi * r^2

  • Volume = 4/3 * pi * r^3

• Hemisphere:

  • Curved Surface Area = 2 * pi * r^2

  • Total Surface Area = 3 * pi * r^2

  • Volume = 2/3 * pi * r^3

5. Statistics & Probability (Class 9 & 10)

Statistics

• Mean (Direct method): x bar = (sum of fi * xi) / (sum of fi)

• Mean (Assumed mean method): x bar = a + (sum of fi * di) / (sum of fi), where di = xi – a

• Mean (Step-deviation method): x bar = a + h * (sum of fi * ui) / (sum of fi), where ui = (xi – a) / h

• Median (grouped data): Median = l + ((n/2 – cf) / f) * h

  • l = lower limit of median class

  • cf = cumulative frequency of class before median class

  • f = frequency of median class

  • h = class size

• Mode (grouped data): Mode = l + ((f1 – f0) / (2f1 – f0 – f2)) * h

  • l = lower limit of modal class

  • f1 = frequency of modal class

  • f0 = frequency of class before modal class

  • f2 = frequency of class after modal class

  • h = class size

Probability

• P(E) = Number of favourable outcomes / Total number of outcomes

• 0 <= P(E) <= 1

• P(not E) = 1 – P(E)

6. Real Numbers (Class 9 & 10)

• Euclid’s Division Lemma: a = bq + r, where 0 <= r < b

• Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorisation is unique.

• HCF * LCM = Product of two numbers

  • For two numbers a and b: HCF(a, b) * LCM(a, b) = a * b

7. Quick Tips for Last-Minute Revision

1. Write formulas on a small card. Keep it in your pocket. Look at it whenever you have free time.

2. Practice one example for each formula. This helps you remember how to use it.

3. Don’t just read. Write. Writing formulas again and again makes them stick in your memory.

4. Focus on weak areas. If you forget a formula, revise that chapter first.

5. Take deep breaths. Stay calm. You have prepared well. Trust yourself.

8. Conclusion

This formula sheet covers all the important formulas for Class 9 and 10 Maths. Use it for quick revision before your board exams. Remember, formulas are the tools. You need to know when and how to use them. So, practice solving problems with these formulas.

If you need more help, Lumens Academy offers expert online maths tutoring. We provide live classes, personalised guidance, and comprehensive study material. Book a free demo class today and boost your confidence for the exams.

All the best for your exams! You can do it!