Revision Sheets: Formulas and Theorems (Classes VI to X)

What to remember

• Algebra and numbers: (a + b)² = a² + 2ab + b²; quadratic roots x = [−b ± √(b² − 4ac)] / 2a; AP: aₙ = a + (n − 1)d and Sₙ = n/2 [2a + (n − 1)d].
• Geometry: Pythagoras theorem (a² + b² = c²), Basic Proportionality Theorem, angle sum of a triangle = 180°, angle in a semicircle = 90°, tangent ⟂ radius.
• Mensuration, trigonometry and coordinates: area and volume formulas of standard solids; sin²θ + cos²θ = 1; distance formula √[(x₂ − x₁)² + (y₂ − y₁)²].

Sheet 1: Numbers, exponents, percentages

• HCF × LCM of two numbers = product of the two numbers.
• Euclid's division lemma: a = bq + r, 0 ≤ r < b.
• Exponent laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ.
• Percentage: x% of y = xy/100. Percentage change = (change / original) × 100.
• Profit = SP − CP; Profit % = (Profit / CP) × 100; Loss % = (Loss / CP) × 100. SP = CP × (100 + P%)/100.
• Simple interest SI = PRT/100. Amount A = P + SI.
• Compound interest (yearly): A = P (1 + R/100)ⁿ; CI = A − P.
• Ratio and proportion: if a : b = c : d then ad = bc. Direct variation: y/x is constant. Inverse variation: xy is constant.
• Average speed for equal distances at speeds u and v = 2uv/(u + v).
• Speed = distance / time. 1 km/h = 5/18 m/s.

Sheet 2: Algebra

• Quadratic equation ax² + bx + c = 0 (a ≠ 0). Discriminant D = b² − 4ac. If D > 0 two distinct real roots; if D = 0 two equal real roots; if D < 0 no real roots.
• Sum of roots = −b/a. Product of roots = c/a.
• Linear pair in two variables a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: unique solution if a₁/a₂ ≠ b₁/b₂; no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; infinitely many solutions if all three ratios are equal.
• Arithmetic progression: aₙ = a + (n − 1)d. Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l). Sum of first n natural numbers = n(n + 1)/2.
• Polynomial: degree is the highest power. A polynomial of degree n has at most n zeroes. Remainder theorem: remainder of p(x) ÷ (x − a) is p(a). Factor theorem: (x − a) is a factor if p(a) = 0.

Sheet 3: Sets and statistics

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B). A set with n elements has 2ⁿ subsets.
• Mean = Σfx / Σf. Median = l + [(n/2 − cf)/f] × h. Mode = l + [(f1 − f0)/(2f1 − f0 − f2)] × h.
• Mode = 3 Median − 2 Mean.
• Probability P(E) = favourable / total. P(not E) = 1 − P(E). 0 ≤ P(E) ≤ 1.

Sheet 4: Geometry theorems

• Congruence criteria for triangles: SSS, SAS, ASA, AAS, RHS.
• Similarity criteria: AAA (or AA), SSS (sides in proportion), SAS (one angle equal and the sides containing it in proportion).
• Sum of interior angles of a polygon with n sides = (n − 2) × 180°. Each exterior angle of a regular polygon = 360°/n.
• A parallelogram has opposite sides equal and diagonals that bisect each other. A rhombus has equal sides and diagonals that are perpendicular. A rectangle has equal diagonals.
• Mid-point theorem: the line joining the mid-points of two sides is parallel to the third side and half of it.
• Euclid's five postulates and axioms form the base of classical geometry. Axiom: "Things equal to the same thing are equal to one another."

Sheet 5: Mensuration

• Heron's formula: Area = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2.
• Sector of angle θ: arc length = (θ/360) × 2πr; area = (θ/360) × πr².
• Frustum of a cone: volume = (πh/3)(r₁² + r₁r₂ + r₂²). When a solid is melted and recast, the volume stays the same.

Sheet 6: Trigonometry

• sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent. cosec, sec and cot are the reciprocals of sin, cos and tan.
• tan θ = sin θ / cos θ.
• Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
• Complementary angles: sin(90° − θ) = cos θ; tan(90° − θ) = cot θ; sec(90° − θ) = cosec θ.

• Angle of elevation: the angle between the horizontal and the line of sight upwards. Angle of depression: the angle between the horizontal and the line of sight downwards.
• Height problem: height = distance × tan(angle of elevation).

Sheet 7: Coordinate geometry

• Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²]. Distance from the origin = √(x² + y²).
• Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Section formula (internal, ratio m : n) = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
• Area of a triangle = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Three points are collinear if this area is zero.
• Centroid = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).
• Slope of a line through two points = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes.
• Quadrants: I (+, +), II (−, +), III (−, −), IV (+, −).

Exam traps

• (a + b)² is not a² + b². The middle term 2ab is always present.
• Discriminant is b² − 4ac, not b² + 4ac.
• In AP, Sₙ uses n/2, and the nth term uses (n − 1)d, not nd.
• Area of a triangle uses ½; a cone volume uses ⅓; a sphere volume uses 4/3.
• sec 60° = 2, but cos 60° = 1/2. Reciprocals are easy to mix up.
• Slant height l = √(r² + h²), not r + h.
• Similar triangles: areas are in the ratio of the squares of sides; perimeters in the ratio of sides.
• tan 90° and cot 0° are not defined.

One-liners

• 1. HCF × LCM = product of the two numbers.
• 2. SI = PRT/100.
• 3. Roots of a quadratic: x = [−b ± √(b² − 4ac)] / 2a.
• 4. Sum of n terms of an AP = n/2 [2a + (n − 1)d].
• 5. Interior angles of an n-sided polygon sum to (n − 2) × 180°.
• 6. Angle in a semicircle is 90°.
• 7. Tangent lengths from an external point are equal.
• 8. Volume of a cone is one-third of the volume of the cylinder with the same base and height.
• 9. sin²θ + cos²θ = 1.
• 10. sin 30° = cos 60° = 1/2.
• 11. Mid-point of (x₁, y₁) and (x₂, y₂) is the average of the coordinates.
• 12. Mode = 3 Median − 2 Mean.

Practice questions

1. For two positive integers a and b, HCF × LCM equals

   1. a / b
   2. a × b
   3. a + b
   4. a² + b²
   Answer

   B. a × b

   HCF × LCM = product of the two numbers.

2. The discriminant of the quadratic equation ax² + bx + c = 0 is

   1. b² + 4ac
   2. 2b - 4ac
   3. b² - 4ac
   4. 4ac - b
   Answer

   C. b² - 4ac

   D = b² - 4ac decides the nature of roots.

3. If the discriminant of a quadratic equation is negative, then the equation has

   1. exactly one real root
   2. two distinct real roots
   3. two equal real roots
   4. no real roots
   Answer

   D. no real roots

   D < 0 means no real roots.

4. The nth term of an AP with first term a and common difference d is

   1. (n - 1)(a + d)
   2. a + (n - 1)d
   3. a + nd
   4. a - (n - 1)d
   Answer

   B. a + (n - 1)d

   Each step adds d, and n - 1 steps are taken from the first term.

5. The sum of the interior angles of a polygon with n sides is

   1. (n - 2) × 180°
   2. (n - 1) × 180°
   3. (n + 2) × 90°
   4. n × 180°
   Answer

   A. (n - 2) × 180°

   A polygon divides into (n - 2) triangles.

6. The angle in a semicircle is

   1. 180°
   2. 45°
   3. 90°
   4. 60°
   Answer

   C. 90°

   The angle subtended by a diameter at the circle is a right angle.

7. In a cyclic quadrilateral the opposite angles are

   1. equal
   2. supplementary (sum 180°)
   3. complementary (sum 90°)
   4. each 90°
   Answer

   B. supplementary (sum 180°)

   Opposite angles of a cyclic quadrilateral add to 180°.

8. The volume of a cone of base radius r and height h is

   1. ½ πr²h
   2. πr²h
   3. ⅔ πr²h
   4. ⅓ πr²h
   Answer

   D. ⅓ πr²h

   A cone is one-third of the cylinder with the same base and height.

9. The surface area of a sphere of radius r is

   1. 4πr²
   2. 2πr²
   3. 4/3 πr³
   4. 3πr²
   Answer

   A. 4πr²

   Standard formula for a sphere.

10. The curved surface area of a cylinder of radius r and height h is

    1. πr²h
    2. 2πr(r + h)
    3. πrh
    4. 2πrh
    Answer

    D. 2πrh

    CSA is circumference × height.

11. The identity 1 + tan²θ equals

    1. sec²θ
    2. cot²θ
    3. cosec²θ
    4. sin²θ
    Answer

    A. sec²θ

    Divide sin²θ + cos²θ = 1 by cos²θ.

12. Which is NOT a criterion for congruence of triangles?

    1. SAS
    2. SSS
    3. SSA (two sides and a non-included angle)
    4. RHS
    Answer

    C. SSA (two sides and a non-included angle)

    SSA does not fix a triangle in general; RHS is the special right-angle case.

13. The sum of the roots of ax² + bx + c = 0 is

    1. c/a
    2. -b/a
    3. -c/a
    4. b/a
    Answer

    B. -b/a

    Sum of roots is -b/a; product is c/a.

14. A tangent to a circle at a point is ___ to the radius through that point.

    1. parallel
    2. perpendicular
    3. inclined at 45°
    4. equal
    Answer

    B. perpendicular

    Tangent-radius theorem.

15. Heron's formula for the area of a triangle uses s, which equals

    1. abc/2
    2. a + b + c
    3. (a + b + c)/2
    4. (a + b + c)/3
    Answer

    C. (a + b + c)/2

    s is the semi-perimeter.

16. The value of cos 60° is

    1. √3/2
    2. 1/√2
    3. 1
    4. 1/2
    Answer

    D. 1/2

    Standard value.

17. The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to

    1. half of it
    2. twice of it
    3. equal to it
    4. one-third of it
    Answer

    A. half of it

    Mid-point theorem.

18. Euclid's division lemma states that for positive integers a and b there exist unique integers q and r such that

    1. a = bq + r, 0 ≤ r < b
    2. a = bq - r, 0 < r ≤ b
    3. a = b + q + r
    4. b = aq + r, r > a
    Answer

    A. a = bq + r, 0 ≤ r < b

    This lemma is the base of the division algorithm for HCF.

19. The HCF of two numbers is 12 and their LCM is 180. If one number is 36, the other is

    1. 72
    2. 45
    3. 30
    4. 60
    Answer

    D. 60

    Other = 12 × 180 / 36 = 60.

20. Simple interest on 2000 at 5% per year for 4 years is

    1. 500
    2. 440
    3. 400
    4. 200
    Answer

    C. 400

    SI = 2000 × 5 × 4 / 100 = 400.

21. Compound interest on 5000 at 10% per year for 2 years is

    1. 1210
    2. 1050
    3. 1100
    4. 1000
    Answer

    B. 1050

    Amount = 5000 × 1.1 × 1.1 = 6050; CI = 1050.

22. The 12th term of the AP 5, 9, 13, … is

    1. 53
    2. 48
    3. 45
    4. 49
    Answer

    D. 49

    5 + 11 × 4 = 49.

23. The sum of the first 20 natural numbers is

    1. 210
    2. 200
    3. 420
    4. 220
    Answer

    A. 210

    n(n + 1)/2 = 20 × 21 / 2 = 210.

24. The sum of the roots of 2x² - 6x + 1 = 0 is

    1. -3
    2. 6
    3. 1/2
    4. 3
    Answer

    D. 3

    -b/a = 6/2 = 3.

25. The equation x² - 4x + 4 = 0 has

    1. no real roots
    2. two distinct real roots
    3. two equal real roots
    4. three roots
    Answer

    C. two equal real roots

    D = 16 - 16 = 0.

26. The sum of the interior angles of an octagon is

    1. 720°
    2. 1080°
    3. 1440°
    4. 900°
    Answer

    B. 1080°

    (8 - 2) × 180 = 1080.

27. Each exterior angle of a regular polygon is 45°. The number of sides is

    1. 10
    2. 8
    3. 9
    4. 6
    Answer

    B. 8

    360 / 45 = 8.

28. The legs of a right triangle are 9 cm and 12 cm. The hypotenuse is

    1. 14 cm
    2. 21 cm
    3. 15 cm
    4. 13 cm
    Answer

    C. 15 cm

    √(81 + 144) = √225 = 15.

29. The curved surface area of a cylinder of radius 7 cm and height 10 cm (π = 22/7) is

    1. 440 cm²
    2. 1540 cm²
    3. 880 cm²
    4. 220 cm²
    Answer

    A. 440 cm²

    2 × 22/7 × 7 × 10 = 440.

30. The curved surface area of a cone of radius 7 cm and slant height 25 cm (π = 22/7) is

    1. 550 cm²
    2. 1100 cm²
    3. 175 cm²
    4. 1925 cm²
    Answer

    A. 550 cm²

    π r l = 22/7 × 7 × 25 = 550.

31. The total surface area of a solid hemisphere of radius 7 cm (π = 22/7) is

    1. 308 cm²
    2. 462 cm²
    3. 154 cm²
    4. 616 cm²
    Answer

    B. 462 cm²

    3πr² = 3 × 22/7 × 49 = 462.

32. The total surface area of a cube of side 4 cm is

    1. 48 cm²
    2. 64 cm²
    3. 96 cm²
    4. 16 cm²
    Answer

    C. 96 cm²

    6a² = 6 × 16 = 96.

33. The value of 4 sin 30° + 2 cos 60° is

    1. 3
    2. 2
    3. 4
    4. 5
    Answer

    A. 3

    4 × 1/2 + 2 × 1/2 = 2 + 1 = 3.

34. The distance of the point (3, 4) from the origin is

    1. 7
    2. 5
    3. 1
    4. 25
    Answer

    B. 5

    √(9 + 16) = 5.

35. The mid-point of the segment joining (2, -4) and (6, 8) is

    1. (4, 6)
    2. (8, 4)
    3. (2, 2)
    4. (4, 2)
    Answer

    D. (4, 2)

    ((2 + 6)/2, (-4 + 8)/2) = (4, 2).

36. The area of the triangle with vertices (0, 0), (6, 0) and (0, 8) is

    1. 14 square units
    2. 12 square units
    3. 48 square units
    4. 24 square units
    Answer

    D. 24 square units

    ½ × 6 × 8 = 24.

37. Two similar triangles have sides in the ratio 3 : 4. The ratio of their areas is

    1. 27 : 64
    2. 3 : 4
    3. 9 : 16
    4. 6 : 8
    Answer

    C. 9 : 16

    Areas are in the ratio of the squares of the sides.

38. The area of a triangle with sides 3 cm, 4 cm and 5 cm by Heron's formula is

    1. 7.5 cm²
    2. 12 cm²
    3. 5 cm²
    4. 6 cm²
    Answer

    D. 6 cm²

    s = 6; area = √(6 × 3 × 2 × 1) = 6.

39. Consider the statements. 1. The Pythagoras theorem applies only to right-angled triangles. 2. The sum of the exterior angles of any convex polygon is 360°. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both statements are standard results.

40. Consider the statements. 1. a³ + b³ = (a + b)(a² - ab + b²). 2. a³ - b³ = (a - b)(a² - ab + b²). Which is/are correct?

    1. 2 only
    2. Both 1 and 2
    3. Neither 1 nor 2
    4. 1 only
    Answer

    D. 1 only

    a³ - b³ = (a - b)(a² + ab + b²).

41. Consider the statements. 1. The angle in a semicircle is 60°. 2. Opposite angles of a cyclic quadrilateral add to 180°. Which is/are correct?

    1. 2 only
    2. 1 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 2 only

    The angle in a semicircle is 90°.

42. Consider the statements. 1. tan 90° = 1. 2. cos 0° = 0. Which is/are correct?

    1. 1 only
    2. Neither 1 nor 2
    3. 2 only
    4. Both 1 and 2
    Answer

    B. Neither 1 nor 2

    tan 90° is not defined and cos 0° = 1.

43. Match the solid with its volume. P. Cone Q. Sphere R. Cylinder 1. πr²h 2. 4/3 πr³ 3. ⅓ πr²h

    1. P-3, Q-2, R-1
    2. P-2, Q-3, R-1
    3. P-1, Q-2, R-3
    4. P-3, Q-1, R-2
    Answer

    A. P-3, Q-2, R-1

    Standard volume formulas.

44. Match the angle with its value. P. sin 60° Q. cos 60° R. tan 45° 1. 1/2 2. 1 3. √3/2

    1. P-1, Q-3, R-2
    2. P-3, Q-1, R-2
    3. P-3, Q-2, R-1
    4. P-2, Q-1, R-3
    Answer

    B. P-3, Q-1, R-2

    sin 60° = √3/2, cos 60° = 1/2, tan 45° = 1.

45. Match the result with its name. P. Parallel line divides two sides in the same ratio Q. a = bq + r R. Square of hypotenuse = sum of squares of sides 1. Pythagoras theorem 2. Basic Proportionality Theorem 3. Euclid's division lemma

    1. P-1, Q-3, R-2
    2. P-2, Q-1, R-3
    3. P-2, Q-3, R-1
    4. P-3, Q-2, R-1
    Answer

    C. P-2, Q-3, R-1

    BPT, the division lemma and the Pythagoras theorem respectively.