Chapter 1 real numbers

1. Decimal representation of rational numbers:
- Every rational number can be written as a terminating or repeating decimal.

2. Operations on real numbers:

- Addition: (a + b) = c, where a, b, and c are real numbers.

- Subtraction: (a - b) = c, where a, b, and c are real numbers.

- Multiplication: (a * b) = c, where a, b, and c are real numbers

- Division: (a / b) = c, where a, b, and c are real numbers and b ≠ 0.

3. Euclid's Division Algorithm:
- For any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b.

4. Fundamental Theorem of Arithmetic:

- Every composite number can be expressed as a unique product of prime numbers, up to the order of the factors.

5. Squares and square roots:

- The square of a number 'a' is denoted as (a^2).

- The square root of a number 'a' is denoted as (√a).

6. Law of exponents:

- Product law: a^m * a^n = a^(m+n), where 'a' is a non-zero real number and 'm' and 'n' are any real numb ers.

- Quotient law: a^m / a^n = a^(m-n), where 'a' is a non-zero real number and 'm' and 'n' are any real numb ers.

- Power law: (a^m)^n = a^(m*n), where 'a' is a non-zero real number and 'm' and 'n' are any real numbers.

7. Irrational numbers:

- A number is irrational if it cannot be expressed as a fraction of two integers, and its decimal representati on never terminates or repeats.

8. The decimal expansion of irrational numbers:

- The decimal expansion of an irrational number is non-terminating and non-repeating.

9. Converting recurring decimals into fractions:

- To convert a recurring decimal into a fraction, let 'x', the recurring decimal, be written as x = 0.abbb..., w here 'a' and 'b' are non-recurring and recurring digits, respectively.

Then, multiply 'x' by a suitable power o f 10 to eliminate the recurring part and subtract the original number from it, representing it algebraically. Fi nally, simplify the resulting equation to obtain the fraction.

10. Finding the square root of a real number:

- The square root of a real number 'x' can be found using the long division method or using the iterative pr ocess of the Babylonian method.

Chapter 2 polynomials

1. Polynomial: A polynomial is an expression consisting of variables and coefficients combined using addit ion, subtraction, and multiplication operations.

2. Degree of a Polynomial: The degree of a polynomial is the highest power of the variable in the polynom ial.

3. Linear Polynomial: A polynomial of degree one is called a linear polynomial. It can be represented as a x + b, where a and b are constants.

4. Quadratic Polynomial: A polynomial of degree two is called a quadratic polynomial. It can be represente d as ax^2 + bx + c, where a, b, and c are constants.

5. Cubic Polynomial: A polynomial of degree three is called a cubic polynomial. It can be represented as a x^3 + bx^2 + cx + d, where a, b, c, and d are constants.

6. Monomial: A polynomial with only one term is called a monomial. Example: 3x^2.

7. Binomial: A polynomial with two terms is called a binomial. Example: 2x + 5.

8. Trinomial: A polynomial with three terms is called a trinomial. Example: 3x^2 + 2x + 7.

9. Coefficient: The coefficient is the numerical factor of each term in a polynomial. In the polynomial 5x^2 + 3x - 2, the coefficients are 5, 3, and -2.

10. Constant Term: The constant term is the term in a polynomial that does not have any variable. In the p olynomial 4x^2 + 3x - 2, the constant term is -2.

11. Zeroes/Roots of a Polynomial: The zeroes or roots of a polynomial are the values of the variable that s atisfy the polynomial equation and make it equal to zero.

12. Factor Theorem: According to the factor theorem, if a polynomial f(x) has a factor (x - a), then f(a) = 0.

13. Remainder Theorem: According to the remainder theorem, if a polynomial f(x) is divided by (x - a), the n the remainder will be f(a).

14. Fundamental Theorem of Algebra: The fundamental theorem of algebra states that any polynomial eq uation of degree greater than zero has at least one complex root.

15. Division Algorithm: The division algorithm states that any polynomial f(x) can be divided by a polynomi al g(x) where the degree of g(x) is less than or equal to the degree of f(x), resulting in a quotient q(x) and r emainder r(x) such that f(x) = g(x) * q(x) + r(x).

Chapter 3 pair of linear equations in two variables

1. The general form of a pair of linear equations in two variables:

ax + by + c = 0 dx + ey + f = 0

2. Formulas for finding the solution to a pair of linear equations:

- Substitution Method:

- Solve one equation for one variable in terms of the other variable.

- Substitute the expression from step 1 into the other equation and solve for one variable.

- Substitute the value from step 2 into the expression found in step 1 and solve for the other variable.

- The values obtained in step 2 and 3 will be the solution to the pair of equations.

- Elimination Method:

- Multiply both equations by suitable numbers so that the coefficients of one variable are the same or ar e additive inverses.

- Add or subtract the equations to eliminate one variable and solve for the other variable.

- Substitute the value found in step 2 into one of the original equations and solve for the first variable.

- The values obtained in step 2 and 3 will be the solution to the pair of equations.

- Cross Multiplication Method for substitution:

- Solve one equation for one variable in terms of the other variable.

- Multiply both sides of the other equation by the denominator of the variable obtained in step 1.

- Equate the two expressions found in step 1 and 2 and solve for one variable.

- Substitute the value found in step 3 into the expression obtained in step 1 and solve for the other varia ble.

- The values obtained in step 3 and 4 will be the solution to the pair of equations.

3. Special cases:

- Consistent Equations: When the equations have a unique solution.

- Inconsistent Equations: When the equations have no solution.

- Dependent Equations: When the equations have infinitely many solutions.

Chapter 4 quadratic equations

1. Quadratic Formula:

This formula helps in finding the roots of a quadratic equation: ax^2 + bx + c = 0 The formula is: x = (-b ± √(b^2 - 4ac))/2a

2. Discriminant:

The discriminant helps in determining the nature of the roots of a quadratic equation. It is given by: D = b^2 - 4ac where D represents the discriminant, and a, b, and c are coefficients of the quadratic equation.

3. Nature of the Roots:

i) If D > 0, the quadratic equation has two distinct real roots.

ii) If D = 0, the quadratic equation has two equal real roots.

iii) If D < 0, the quadratic equation has no real roots.

4. Sum and Product of Roots:

If α and β are the roots of a quadratic equation ax^2 + bx + c = 0, then

i) Sum of Roots (α + β) = -b/a

ii) Product of Roots (α * β) = c/a

Chapter 5 arithmetic progressions

1. Formula to find the nth term of an Arithmetic Progression (AP):

an = a + (n - 1)d where, an = nth term, a = first term, n = number of terms, d = common difference.

2. Formula to find the sum of the first n terms of an AP:

Sn = (n/2)(2a + (n - 1)d) where, Sn = sum of the series, a = first term, n = number of terms, d = common difference.

3. Formula to find the number of terms in an AP, given the first term, last term, and the common differenc e:

n = [(l - a)/d] + 1 where, l = last term, a = first term, d = common difference.

4. Formula to find the sum of an AP, given the first term, last term, and the number of terms:

Sn = (n/2)(a + l) where, Sn = sum of the series, a = first term, l = last term, n = number of terms.

5. Formula to find the arithmetic mean of two different APs having a common difference:

AM = (a1 + b1) / 2 where, AM = arithmetic mean, a1 = first term of the first AP, b1 = first term of the second AP.

6. Formula to find the sum of the even terms of an AP:

Se = (n/2)(a + l) where, Se = sum of the even terms, a = first term, l = last term, n = number of terms.

7. Formula to find the sum of the odd terms of an AP:

So = (n/2)(2a + (n - 1)d) where, So = sum of the odd terms, a = first term, d = common difference, n = number of terms.

Chapter 6 triangles

1. Area of a triangle:

- Using Heron's formula:

- Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter and a, b, c are the lengths of the sides of the triangle.

- Using the base and height:

- Area = 1/2 * base * height

- Using SAS (Side-Angle-Side) Formula:

- Area = 1/2 * a * b * sin(C), where a and b are the lengths of two sides of the triangle and C is the included angle.

2. Pythagorean theorem:

- In a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the square s of the other two sides.

- a² + b² = c², where a, b are the lengths of the two shorter sides (legs) and c is the length of the hypotenuse.

3. Congruence criteria:

Two triangles are congruent if any of the following conditions are satisfied:

- SSS (Side-Side-Side)

- SAS (Side-Angle-Side)

- ASA (Angle-Side-Angle)

- RHS (Right-Angle-Hypotenuse-Side)

4. Similarity criteria:

Two triangles are similar if any of the following conditions are satisfied:

- AAA (Angle-Angle-Angle)

- SSS (Side-Side-Side)

- SAS (Side-Angle-Side)

5. Median of a triangle:

- The median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.

- In a triangle ABC, the medians are AD, BE, and CF.

- The medians divide each other in the ratio 2:1.

6. Altitude of a triangle:

- The altitude is a perpendicular line segment drawn from a vertex of a triangle to the opposite side (or a n extension of it).

- An acute triangle has three altitudes, each starting from a different vertex and intersecting the opposite side.

- An obtuse triangle has one altitude outside the triangle.

7. Similarity of triangles and Pythagoras theorem with a proof.

Chapter 7 coordinate geometry

1. Distance Formula:

The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is given by the formula:

√((x2 - x1)^2 + (y2 - y1)^2)

2. Section Formula:

The coordinates of a point dividing a line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n a re given by: [(mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)]

3. Mid-point Formula:

The coordinates of the mid-point of a line segment joining two points (x1, y1) and (x2, y2) are given by: [(x1 + x2)/2, (y1 + y2)/2]

4. Slope of a Line:

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by: (y2 - y1)/(x2 - x1)

5. Equation of a Line:

- Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept.

- Point-slope form: (y - y1) = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

- Two-point form: (y - y1)/(y2 - y1) = (x - x1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on the lin e.

- Intercept form: x/a + y/b = 1, where a and b are the x and y-intercepts respectively.

6. Parallel and Perpendicular Lines:

- Two lines with slopes m1 and m2 are parallel if m1 = m2.

- Two lines with slopes m1 and m2 are perpendicular if m1 x m2 = -1.

7. Area of a Triangle:

The area of a triangle formed by three points (x1, y1), (x2, y2) and (x3, y3) is given by: |((x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|

Chapter 8 introduction to trigonometry

1. Trigonometric Ratios:

- sin A = Perpendicular / Hypotenuse

- cos A = Base / Hypotenuse

- tan A = Perpendicular / Base

2. Reciprocal Trigonometric Ratios:

- cosec A = 1 / sin A = Hypotenuse / Perpendicular

- sec A = 1 / cos A = Hypotenuse / Base

- cot A = 1 / tan A = Base / Perpendicular

3. Pythagorean Identities:

- sin² A + cos² A = 1

- 1 + tan² A = sec² A

- 1 + cot² A = cosec² A

4. Complementary Angle Relationship:

- sin (90° - A) = cos A

- cos (90° - A) = sin A

- tan (90° - A) = cot A

- sec (90° - A) = cosec A

- cosec (90° - A) = sec A

- cot (90° - A) = tan A

5. Trigonometric Ratios of Some Specific Angles:

- sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3

- sin 45° = cos 45° = 1/√2, tan 45° = 1

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

6. Evaluation of Trigonometric Ratios:

- sin (-A) = -sin A

- cos (-A) = cos A

- tan (-A) = -tan A

- sec (-A) = sec A

- cosec (-A) = -cosec A

- cot (-A) = -cot A

7. Trigonometric Identities:

- sin (Π/2 ± A) = ± cos A

- cos (Π/2 ± A) = ± sin A

- tan (Π/2 ± A) = ± cot A

- sec (Π/2 ± A) = ± cosec A

- cosec (Π/2 ± A) = ± sec A

- cot (Π/2 ± A) = ± tan A

Chapter 9 some applications of trigonometry

1. Sine rule: sin(A)/a = sin(B)/b = sin(C)/c Where A, B, and C are angles of a triangle and a, b, and c are the lengths of the opposite sides respecti vely.

2. Cosine rule: c² = a² + b² - 2ab * cos(C)

Where a, b are the lengths of two sides of a triangle and C is the angle between these two sides.

3. Area of a triangle:

- Using sides: A = (1/2) * a * b * sin(C)

- Using base and height: A = (1/2) * b * h

- Using vertices: A = (1/2) * | x (y - y ) + x (y - y ) + x (y - y ) |

4. Height and Distance:

- Finding height using tangent: h = d * tan(θ)

- Finding distance using sine: d = h/sin(θ)

- Finding distance using cosine: d = h/cos(θ)

5. Simple Pendulum:

- Time period: T = 2π * √(L/g)

Where T is the time period, L is the length of the pendulum, and g is acceleration due to gravity.

6. Heights and Distances (Angle of Elevation and Angle of Depression):

- Angle of Elevation: If a person is looking up at an object, the angle between the line of sight and the ho rizontal is called the angle of elevation.

- Angle of Depression: If a person is looking down at an object, the angle between the line of sight and t he horizontal is called the angle of depression.

Chapter 10 circles

1. Area of a Circle:

- A = πr²

- Where A represents the area and r represents the radius of the circle

2. Circumference of a Circle:

- C = 2πr

- Where C represents the circumference and r represents the radius of the circle

3. Diameter of a Circle:

- d = 2r

- Where d represents the diameter and r represents the radius of the circle

4. Length of an Arc:

- L = (θ/360) × 2πr

- Where L represents the length of the arc, θ represents the angle subtended by the arc at the center of the circle, and r represents the radius of the circle

5. Area of a Sector:

- A = (θ/360) × πr²

- Where A represents the area of the sector, θ represents the angle subtended by the sector at the cent er of the circle, and r represents the radius of the circle

6. Chord Length:

- c = 2r × sin(θ/2)

- Where c represents the chord length, θ represents the angle subtended by the chord at the center of t he circle, and r represents the radius of the circle

7. Tangent Length:

- t = √(d² - 4r²)

- Where t represents the length of the tangent, d represents the distance between the center of the circl e and the point of tangency, and r represents the radius of the circle

8. Properties of Tangents:

- The tangent to a circle is perpendicular to the radius drawn at the point of tangency

9. Properties of Chords:

- Perpendicular bisector of a chord passes through the center of the circle

- The angle between a chord and a tangent at the point of contact is equal to the angle in the alternate s egment

Chapter 11 constructions

1. Constructing a perpendicular bisector:

Given a line segment AB, the perpendicular bisector is constructed as follows:

- Step 1: Place the compass point on point A and extend the compass to a distance greater than half the l ength of AB.

- Step 2: With the compass still open to the same width, place the compass point on point B and draw an arc above and below AB.

- Step 3: Without changing the compass width, place the compass point on one of the intersections of the arcs and draw arcs that intersect the other arc.

- Step 4: Connect the two intersection points of the arcs to construct the perpendicular bisector of AB.

2. Constructing an angle bisector:

Given an angle with vertex at point A, the angle bisector is constructed as follows:

- Step 1: Place the compass point on point A and draw an arc that intersects both sides of the angle.

- Step 2: Without changing the compass width, place the compass point on each intersection point and dr aw arcs that intersect the other side of the angle.

- Step 3: Draw a straight line connecting point A to the intersection of the arcs. This is the angle bisector.

3. Constructing a perpendicular from a point to a line:

Given a line and a point not on the line, the perpendicular from the point to the line is constructed as follo ws:

- Step 1: Place the compass point on the given point and extend the compass to a distance greater than t he distance from the point to the line.

- Step 2: With the compass still open to the same width, draw arcs that intersect the line at two points.

- Step 3: Without changing the compass width, place the compass point on each intersection point and dr aw arcs that intersect each other.

- Step 4: Connect the two intersection points of the arcs to construct the perpendicular from the point to th e line.

4. Constructing a triangle when given its three sides:

Given the lengths of the three sides of a triangle, the triangle can be constructed as follows:

- Step 1: Draw a line segment with a length equal to the first given side.

- Step 2: From one of the endpoints of the first side, draw an arc with a radius equal to the second given si de.

- Step 3: From the other endpoint of the first side, draw another arc with a radius equal to the third given si de.

- Step 4: Connect the two intersection points of the arcs to form the triangle.

Chapter 12 areas related to circles

1. Circumference of a Circle: C = 2πr, where r is the radius of the circle.

2. Area of a Circle: A = πr², where r is the radius of the circle.

3. Diameter of a Circle: d = 2r, where r is the radius of the circle.

4. Length of an Arc: L = (θ/360) × 2πr, where θ is the central angle (in degrees) and r is the radius of the c ircle.

5. Area of a Sector: A = (θ/360) × πr², where θ is the central angle (in degrees) and r is the radius of the ci rcle.

6. Area of a Segment: A = A[sector] - A[triangle], where A[sector] is the area of the corresponding sector a nd A[triangle] is the area of the corresponding triangle.

7. Chord Length Formula: l = 2r × sin(θ/2), where r is the radius of the circle and θ is the central angle (in degrees) formed by the chord.

8. Perpendicular from the Center to a Chord: The perpendicular drawn from the center of a circle to a chor d bisects the chord.

9. Tangent to a Circle: A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

10. Length of a Tangent: LT = √(LS × LT'), where LS is the length of the secant segment and LT' is the len gth of the external segment.

Chapter 13 surface areas and volumes

1. Surface Area of Cube = 6a² (where 'a' is the length of a side)

2. Volume of Cube = a³ (where 'a' is the length of a side)

3. Surface Area of Cuboid = 2(lw + lh + wh) (where 'l' is the length, 'w' is the width, and 'h' is the height)

4. Volume of Cuboid = lwh (where 'l' is the length, 'w' is the width, and 'h' is the height)

5. Surface Area of Right Circular Cylinder = 2πrh + 2πr² (where 'r' is the radius and 'h' is the height)

6. Volume of Right Circular Cylinder = πr²h (where 'r' is the radius and 'h' is the height)

7. Surface Area of Sphere = 4πr² (where 'r' is the radius)

8. Volume of Sphere = (4/3)πr³ (where 'r' is the radius)

9. Surface Area of Hemisphere = 2πr² (where 'r' is the radius)

10. Volume of Hemisphere = (2/3)πr³ (where 'r' is the radius)

11. Curved Surface Area of Cone = πrl (where 'r' is the radius and 'l' is the slant height)

12. Total Surface Area of Cone = πr(r + l) (where 'r' is the radius and 'l' is the slant height)

13. Volume of Cone = (1/3)πr²h (where 'r' is the radius and 'h' is the height)

14. Surface Area of Frustum of Cone = π(R + r)(l) + πR² + πr² (where 'R' is the radius of the upper base, 'r ' is the radius of the lower base, and 'l' is the slant height)

15. Volume of Frustum of Cone = (1/3)πh(R² + r² + Rr) (where 'R' is the radius of the upper base, 'r' is the radius of the lower base, and 'h' is the height)

Chapter 14 statistics

1. Mean:

Mean = Sum of all values / Total number of values Symbolically: = Σx / N

2. Median:

1) If the number of observations is odd:

Median = Value of (N + 1) / 2 th observation Symbolically: Median = X[(N + 1) / 2]

2) If the number of observations is even:

Median = (Value of (N / 2)th observation + Value of ((N / 2) + 1)th observation) / 2 Symbolically: Median = (X[N / 2] + X[(N / 2) + 1]) / 2

3. Mode:

Mode = Value with the maximum frequency Symbolically: Mode = L + ((f1 - f0) / ((2 * f1) - f0 - f2)) * c, where:

L = Lower boundary of the modal class

f1 = Frequency of the modal class

f0 = Frequency of the class before the mode class

f2 = Frequency of the class after the mode class

c = Width of the modal class

4. Range:

Range = Maximum value − Minimum value Symbolically: Range = Xmax - Xmin

5. Quartiles:

1st Quartile (Q ) = (n + 1) / 4 th term

3rd Quartile (Q ) = 3(n + 1) / 4 th term

6. Interquartile Range:

Interquartile Range = Q - Q

7. Variance:

Variance = Sum of (Xi - )² / N

Symbolically: σ² = Σ(Xi - )² / N

8. Standard Deviation:

Standard Deviation = √Variance

Symbolically: σ = √(Σ(Xi - )² / N)

9. Correlation Coefficient:

Correlation Coefficient = Covariance / (Standard Deviation of X ×

Standard Deviation of Y)

Symbolically: r = Σ((Xi - ) × (Yi - y )) / √(Σ(Xi - )² × Σ(Yi - y )²)

Chapter 15 probability

1. Formula for Probability:

P(A) = Number of favorable outcomes / Total number of possible outcomes.

2. Formula for Probability of an Event not occurring:

P(A') = 1 - P(A)

3. Formula for Addition Rule of Probability:

P(A or B) = P(A) + P(B) - P(A and B)

4. Formula for Independent Events:

P(A and B) = P(A) * P(B)

5. Formula for Conditional Probability:

P(A|B) = P(A and B) / P(B)

6. Formula for Empirical Probability:

P(A) = Number of times A occurs / Total number of trials

7. Formula for Theoretical Probability:

P(A) = Number of favorable outcomes for A / Total number of possible outcomes

8. Formula for Complementary Probability: