Chapter 1: Real Numbers
Essential Formulas:
• HCF × LCM = Product of two numbers
• If d is the HCF of two numbers a and b, then ax + by = d has integral solutions
• For positive integers a and b, HCF(a,b) × LCM(a,b) = a × b
• Prime Factorization: Every composite number can be expressed as a product of prime numbers
Chapter 2: Polynomials
Key Formulas:
• For quadratic ax² + bx + c = 0:
– Sum of zeros (α + β) = -b/a
– Product of zeros (α × β) = c/a
– Sum of squares of zeros (α² + β²) = (b/a)² – 2(c/a)
– Sum of zeros (α + β) = -b/a
– Product of zeros (α × β) = c/a
– Sum of squares of zeros (α² + β²) = (b/a)² – 2(c/a)
• Factor Theorem: p(x) is divisible by (x – a) if and only if p(a) = 0
• Remainder Theorem: When p(x) is divided by (x – a), remainder = p(a)
Chapter 3: Coordinate Geometry
Important Formulas:
• Distance Formula:
d = √[(x2-x1)² + (y2-y1)²]
d = √[(x2-x1)² + (y2-y1)²]
• Section Formula (Internal division m:n):
x = (mx2 + nx1)/(m+n)
y = (my2 + ny1)/(m+n)
x = (mx2 + nx1)/(m+n)
y = (my2 + ny1)/(m+n)
• Section Formula (External division m:n):
x = (mx2 – nx1)/(m-n)
y = (my2 – ny1)/(m-n)
x = (mx2 – nx1)/(m-n)
y = (my2 – ny1)/(m-n)
• Area of Triangle:
A = ½|x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|
A = ½|x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|
• Midpoint Formula:
M = ((x1 + x2)/2, (y1 + y2)/2)
M = ((x1 + x2)/2, (y1 + y2)/2)
Chapter 4: Triangles
Theorems and Formulas:
• Similar Triangles:
– If triangles are similar:
(a1/a2) = (b1/b2) = (c1/c2) = k (ratio of similarity)
– If triangles are similar:
(a1/a2) = (b1/b2) = (c1/c2) = k (ratio of similarity)
• Basic Proportionality Theorem:
– If line l || to side BC of △ABC intersects AB and AC at D and E respectively:
AD/DB = AE/EC
– If line l || to side BC of △ABC intersects AB and AC at D and E respectively:
AD/DB = AE/EC
• Pythagoras Theorem:
In right △ABC, with right angle at C:
a² + b² = c² (where c is hypotenuse)
In right △ABC, with right angle at C:
a² + b² = c² (where c is hypotenuse)
• Area Formulas:
– Area = ½ × base × height
– Area = √s(s-a)(s-b)(s-c)
where s = (a+b+c)/2 (semi-perimeter)
– Area = ½ × base × height
– Area = √s(s-a)(s-b)(s-c)
where s = (a+b+c)/2 (semi-perimeter)
Chapter 5: Arithmetic Progressions
Essential Formulas:
• nth term (an) = a + (n-1)d
where a = first term, d = common difference
where a = first term, d = common difference
• Sum of n terms (Sn):
– Sn = n/2[2a + (n-1)d]
– Sn = n/2(a + l) where l = last term
– Sn = n/2[2a + (n-1)d]
– Sn = n/2(a + l) where l = last term
• Arithmetic Mean (A.M.):
A.M. = (a + b)/2 where a,b are any two terms
A.M. = (a + b)/2 where a,b are any two terms
Chapter 10: Circles
Important Formulas and Theorems:
• Tangent Properties:
– The tangent at any point of a circle is perpendicular to the radius at the point of contact
– The lengths of tangents drawn from an external point to a circle are equal
– The tangent at any point of a circle is perpendicular to the radius at the point of contact
– The lengths of tangents drawn from an external point to a circle are equal
• Angle Properties:
– Angle in a semicircle is 90°
– Angle between tangent and radius at point of contact is 90°
– Angles in the same segment are equal
– Angle in a semicircle is 90°
– Angle between tangent and radius at point of contact is 90°
– Angles in the same segment are equal
• Circle Theorems:
– The perpendicular from the center to a chord bisects the chord
– Equal chords are equidistant from the center
– If two circles intersect, their common chord is bisected by the line joining their centers
– The perpendicular from the center to a chord bisects the chord
– Equal chords are equidistant from the center
– If two circles intersect, their common chord is bisected by the line joining their centers
Chapter 12: Areas
Area Formulas:
• Circle:
– Area = πr²
– Circumference = 2πr
– Length of an arc = (θ/360°) × 2πr
– Area of sector = (θ/360°) × πr²
– Area = πr²
– Circumference = 2πr
– Length of an arc = (θ/360°) × 2πr
– Area of sector = (θ/360°) × πr²
• Combinations with Circles:
– Area of segment = Area of sector – Area of triangle
– Area between two concentric circles = π(R² – r²)
– Area of segment = Area of sector – Area of triangle
– Area between two concentric circles = π(R² – r²)
• Similar Figures:
– Ratio of areas of similar figures = (scale factor)²
– Ratio of perimeters of similar figures = scale factor
– Ratio of areas of similar figures = (scale factor)²
– Ratio of perimeters of similar figures = scale factor
• Special Cases:
– Area of lune = Area of sector – Area of triangle
– Area of shaded region in combinations = Total area – Sum of unshaded areas
– Area of lune = Area of sector – Area of triangle
– Area of shaded region in combinations = Total area – Sum of unshaded areas
Chapter 13: Surface Areas and Volumes
Key Formulas:
• Cube:
– Surface Area = 6a²
– Volume = a³
where a = edge length
– Surface Area = 6a²
– Volume = a³
where a = edge length
• Cuboid:
– Surface Area = 2(lb + bh + hl)
– Volume = l × b × h
where l = length, b = breadth, h = height
– Surface Area = 2(lb + bh + hl)
– Volume = l × b × h
where l = length, b = breadth, h = height
• Cylinder:
– Curved Surface Area = 2πrh
– Total Surface Area = 2πr(r + h)
– Volume = πr²h
where r = radius, h = height
– Curved Surface Area = 2πrh
– Total Surface Area = 2πr(r + h)
– Volume = πr²h
where r = radius, h = height
• Cone:
– Curved Surface Area = πrl
– Total Surface Area = πr(l + r)
– Volume = 1/3πr²h
where r = radius, h = height, l = slant height
– Curved Surface Area = πrl
– Total Surface Area = πr(l + r)
– Volume = 1/3πr²h
where r = radius, h = height, l = slant height
• Sphere:
– Surface Area = 4πr²
– Volume = 4/3πr³
where r = radius
– Surface Area = 4πr²
– Volume = 4/3πr³
where r = radius
• Hemisphere:
– Curved Surface Area = 2πr²
– Total Surface Area = 3πr²
– Volume = 2/3πr³
where r = radius
– Curved Surface Area = 2πr²
– Total Surface Area = 3πr²
– Volume = 2/3πr³
where r = radius
Chapter 14: Statistics
Important Formulas:
• Mean for Grouped Data:
– Direct Method: x̄ = Σ(fx)/Σf
– Assumed Mean: x̄ = A + Σ(fd)/Σf × h
– Direct Method: x̄ = Σ(fx)/Σf
– Assumed Mean: x̄ = A + Σ(fd)/Σf × h
• Mode = l + [(f1 – f0)/(2f1 – f0 – f2)] × h
where l = lower limit of modal class
h = class size
f1 = frequency of modal class
f0 = frequency of pre-modal class
f2 = frequency of post-modal class
where l = lower limit of modal class
h = class size
f1 = frequency of modal class
f0 = frequency of pre-modal class
f2 = frequency of post-modal class
• Median = l + [(n/2 – cf)/(f)] × h
where l = lower limit of median class
n = total frequency
cf = cumulative frequency of class preceding median class
f = frequency of median class
h = class size
where l = lower limit of median class
n = total frequency
cf = cumulative frequency of class preceding median class
f = frequency of median class
h = class size