topper smart guide-2010 class-x math
TRANSCRIPT
Chapter : Real Numbers
Key Concepts
1. An Algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
2. A lemma is a proven statement used for proving another statement.
3. Euclid’s Division Lemma: Given positive integers a and b, there
exists unique integers q and r satisfying a bq r, where0 r b= + ≤ <
4. Euclid’s Division Algorithm states that HCF of any two positive
integers a and b, with a>b is obtained as follows:
Step 1: Apply Euclid’s division lemma, to a and b, to find q and r
where a = bq+r, 0 ≤ r<b.
Step 2: If r=0, the HCF is b. If r ≠ 0, apply Euclid’s division lemma to
b and r.
Step 3: Continue the process till the remainder is zero. The divisor at
this stage will be HCF of (a, b). Also, HCF (a,b) = HCF (b,r)
5. Euclid’s Division Algorithm is stated for only positive integers but it can be extended for all integers except zero, i.e, b ≠ 0.
6. The numbers which can be represented in the form of p/q where
q 0≠ and p and q are integers are called Rational numbers.
7. Irrational numbers are the numbers which are non-terminating and non-repeating.
8. Irrational numbers are used in
i. Finding the length of diagonal of a square whose sides are given.
ii. Finding the hypotenuse of a right triangle. iii. Deducing the circumference of a circle whose radius is known.
9. Rational and irrational numbers together constitute Real numbers.
10. Terminating fractions are the numbers which leaves remainder 0 on normal division.
11. Recurring fractions are the numbers which never leave a remainder
0 on normal division.
12. If p is a prime and p divides a2, then p divides q where a is a positive
integer.
13. If p is a prime, then p is an irrational number.
14. The decimal expansion of rational number is either terminating or non-terminating recurring (repeating).
15. The decimal expansion of an irrational number is non-terminating,
non-recurring.
16. If the decimal expansion of rational number terminates, then we can
express the number in the form ofp
q, where p and q are coprime, and
the prime factorization of q is of the form 2n5m, where n and m are non
negative integers.
17. A number ends with the digit zero if and only if it has either 2 or 5 as
its prime factors.
18. The sum, difference, product and quotient of two irrational numbers need not always be irrational number.
19. There are more irrational numbers than rational numbers between
two consecutive numbers.
20. Sum and product of a rational number and an irrational number is an
irrational number.
21. Fundamental Theorem of Arithmetic: Every composite number can
be expressed (factorised) as a product of primes, and this factorisation
is unique, apart from the order in which the prime factors occur.
22. The prime factorisation of a composite number is unique, except for
the order of its factors.
23. Highest Common Factor (HCF) is the product of the smallest power
of each common prime factor in the numbers.
24. Lowest Common Multiple (LCM) is the product of the greatest power of each prime factor, involved in the numbers.
Key Formulae
1. HCF (a,b) x LCM (a,b) = a x b
where a and b are positive integers.
2. LCM (p, q, r)=( )
( ) ( ) ( )
p.q.rHCF p,q,r
HCF p,q .HCF q,r .HCF p,r
where p, q and r are three numbers.
3. HCF (p, q, r) = ( )
( ) ( ) ( )
p.q.r.LCM p,q,r
LCM p,q .LCM q,r .LCM p,r
where p, q and r are three numbers.
Chapter : Polynomials
Top Definitions
1. A polynomial p(x) in one variable x is an algebraic expression in x of the form
p(x) = 1 2 2
1 2 2 1 0
n n n
n n na x a x a x ........ a x a x a− −
− −+ + + + + + , where
(i) 0 1 2 na ,a ,a ......a are constants
(ii)x is a variable
(iii)0 1 2 na ,a ,a ......a are respectively the coefficients of xi.
(iv) Each of 1 2 2
1 2 2 1 0
n n n
n n na x a x ,a x ,........a x ,a x,a ,− −
− −+ with 0na ,≠ is called a
term of a polynomial.
2. The highest exponent of the variable in a polynomial is called the degree
of the polynomial.
3. A polynomial of degree one is called a linear polynomial. It is of the
form ax + b. Examples: x-2, 4y+89, 3x-z.
4. A polynomial of degree two is called a quadratic polynomial. It is of the
form ax2 + bx + c. where a, b, c are real numbers and a≠ 0 Examples: x2-2x+5, x2-3x etc.
5. A polynomial of degree 3 is called a cubic polynomial and has the
general form ax3 + bx2 + c x +d. For example: 3 2
2 2 5+ − +x x x etc.
6. A real number k is said to be the zero of the polynomial p(x) if p (k) = 0.
Top Concepts:
1. The graph of a polynomial p(x) of degree n can intersects or touch the x axis at atmost n points.
2. A polynomial of degree n has at most n distinct real zeroes.
3. The zero of the polynomial p(x) satisfies the equation p(x) = 0.
4. For any linear polynomial ax+b, zero of the polynomial will be given by the expression (-b/a).
5. The number of real zeros of the polynomial is the number of times its graph touches or intersects x axis.
6. A polynomial p(x) of degree n will have atmost n real zeroes
7. A linear polynomial has atmost one real zero.
8. A quadratic polynomial has atmost two real zeroes.
9. A cubic polynomial has atmost three real zeroes.
10. Division algorithm can also be used to find the zeroes of a polynomial.
If ‘a’ and ‘b’ are two zeroes of a fourth degree polynomial f(x), then other two zeroes can be found out by dividing f(x) by (x-a)(x-b)
11. If f(x) = q(x) g(x) + r(x), and r(x) = 0 then polynomial g(x) is a factor of polynomial f(x).
12. Process of dividing a polynomial f(x) by another polynomial g(x) is as
follows:
Step1: To obtain the first term of the quotient, divide the highest degree
term of the dividend by the highest degree term of the divisor. Then carry out the division process.
Step2: To obtain the second term of the quotient, divide the highest degree term of the new dividend by the highest degree term of the
divisor. Then again carry out the division process Step3: Continue the process till the degree of the new dividend is less
that the degree of the divisor. This will be called the remainder.
Top Formulae
1. Relationship between zeroes and coefficients of a linear polynomial:
a) If k is a zero of the linear polynomial ax + b, then p (k) = ak+b = 0.
i.e., k=b
a
−. Thus, the zero of the linear polynomial ax + b is
b (cons tant term)
a Coefficient of x
−= −
b) For a quadratic polynomial ax2+bx+c, a ≠ 0,
Sum of the zeroes = 2
b (coefficient of x)
a Coefficient of x
−= −
Product of the zeroes = 2
c cons tant term
a coefficient of x=
c) For a cubic polynomial ax3 + bx2 + cx + d = 0, a ≠ 0 then
Sum of zeroes =2
3
b (coefficient of x )
a Coefficient of x
−= − ,
Sum of the product of zeroes taken 2 at a time=3
c Coefficient of x
a Coefficient of x= ,
Product of zeroes = 3
d cons tant term
a Coefficient of x
−=
2. The quadratic polynomial whose sum of the zeroes = (α+β) and product
of zeroes = (αβ) is given by:
k (x2 - (α+β) x + (αβ)), where k is real.
3. Division algorithm for polynomials: If f(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can always find polynomials q(x) and
r(x) such that
f(x) = q(x) g(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x)
Top Diagrams
1. Linear Polynomial having no zero.
2. Linear Polynomial having 1 zero.
3. Quadratic Polynomial having no zeroes.
4. Quadratic Polynomial having 1 zero.
5. Quadratic Polynomial having 2 zeroes.
6. Cubic Polynomial having no zeroes.
7. Cubic Polynomial having 1 zero.
8. Cubic Polynomial having 2 zeroes.
9. Cubic Polynomial having 3 zeroes.
Chapter : Pair of Linear Equations in two Variables
Top Definitions
1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation
in two variables. 2. Two linear equations in same two variables x and y are called pair of
linear equations in two variables.
3. The solution of pair of linear equations a1x+b1y+c1= 0 and a2x+b2y+c2= 0 is the ordered pair (x, y) which satisfies both the equations.
Top Concepts
1. A linear equation in two variables is represented geometrically by a
straight line. 2. Each solution of a linear equation in two variables, ax + by + c = 0,
corresponds to a point on the line representing the equation and vice versa.
3. The general form of a pair of linear equations in two variables is
1 1 1
2 2 2
a x b y c 0
a x b y c 0
+ + =
+ + =
where a1, a2, b1, b2, c1, c2 are real numbers, such that 2 2 2 2
1 1 2 2a b 0,a b 0+ ≠ + ≠
4. A system of linear equations in two variables represents two lines in the plane.For two given lines there could be three possible cases:
(i) Intersecting lines, lines may intersect at a point. (ii) Parallel lins.
(iii) Overlapping or coincidental to each other,
5. If the lines intersect at a point, then that point gives the unique solution of the system of equations. In this case system of equations is said to be
consistent.
6. If the lines coincide (overlap), then the pair of equations will have infinitely many solutions. System of equations is said to be dependent and consistent.
7. If the lines are parallel, then the pair of equations has no solution. In this
case pair of equations is said to be inconsistent.
8. System of equations can be solved using Algebraic and graphical.
9. Graphical method can be used to obtain the solution of a system of
equations but it has its limitations in cases where the solution is non-integral.
10. Steps to be followed while using the method of substitution for solving
linear equations in 2 variables: Step1: Find the value of one variable, say y in terms of the other
variable. i.e. x from either equation, whichever is convenient.
Step2: Substitute this value of y in the other equation, and reduce
it to an equation in one variable, i.e. in terms of x, which
can be solved.
Step3: Substitute the value of x (or y) obtained in step2 in the equation used in step1 to obtain the value of the other variable.
Step 4: The values of x and y so obtained are the coordinates of the
solution of system of equations.
10. There could be three possibilities on substituting the variable in the other
equation: (i) Equation reduces to a linear equation in one variable x which
can be solved to get the value of x and then y. (ii) Equation reduces to a true equation involving no variable,
and then the given pair of equation has infinitely many solutions
(iii) Equation reduces to false equation involving no variable then
the given pair of equation has no solution.
11. Steps to be followed in Elimination Method of solving simultaneous linear equations:
Step 1: First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable
(either x or y) numerically equal. Step 2: Then add or subtract one equation from the other so that
one variable gets eliminated. If you get an equation in one variable, go to step 3.
If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.
If in Step 2, we obtain a false statement involving no variable, then
the original pair of equations has no solution, i.e. it is inconsistent. Step 3: Solve the equation in one variable (x or y) so obtained to
get its value.
Step 4: Substitute this value of x (or y) in either of the original equations to get the value of the other variable.
12. Steps to be followed in Cross Multiplication Method of solving
simultaneous linear equations:
Step1: Write the equations in the general form.
1 1 1
2 2 2
0
0
a x b y c
a x b y c
+ + =
+ + =
Step2: Arrange these in the following manner.
2 2
1 1
x
b c
b c
=
2 2 2 2
1 1 1 1
1y
c a a b
c a a b
=
Step3: Cross multiply.
1 2 2 1 2 1 1 2 1 2 2 1
1x y
b c b c a c a c a b a b
= =− − −
(1) (2) (3) (a) Comparing (1) and (3), we get the value of x
1 2 2 1
1 2 2 1
b c b c
x
a b a b
−=
−
(b) Comparing (2) and (3), we get the value of y
2 1 1 2
1 2 2 1
a c a c
y
a b a b
−=
−
13. Equations which are not linear but can be reduced to linear form by some
suitable substitutions are called equations reducible to linear form.
14. The speed of the boat downstream is the sum of speed of boat in still
water and speed of the stream.
15. The speed of the boat upstream is the difference of speed of boat in still water and speed of the stream.
16. Reduced equation can be solved by any of the algebraic method
(substitution, elimination or cross multiplication) of solving linear
equation.
Top Formulae
1. If 1 1 1a x b y c 0+ + = and 2 2 2a x b y c 0+ + = be the pair of linear equations,
then by cross multiplication method, the solution would be given by:
1 2 2 1
1 2 2 1
b c b c
x
a b a b
−=
− and 2 1 1 2
1 2 2 1
a c a c
y
a b a b
−=
−
2. A pair of linear equation is given by 1 1 1a x b y c 0+ + = & 2 2 2a x b y c 0+ + = ,
then we can have following conditions;
(i) Intersecting lines: 1 1
2 2
a b
a b≠ , pair of linear equation is
consistent.
(ii) Parallel lines: 1 1 1
2 2 2
a b c
a b c= ≠ , pair of linear equation is
inconsistent.
(iii) Coincident lines: 1 1 1
2 2 2
a b c
a b c= = , pair of linear equation is
dependent and consistent.
Top Diagrams
1. Intersecting line having unique solution.
2. Parallel lines having no solution.
4. Coincident lines having infinitely many solutions.
Chapter : Quadratic Equations
Top Definitions
1. A quadratic equation in the variable x is of the form ax2 + bx + c = 0,
where a, b, c are real numbers and a ≠ 0.
2. The value of x that satisfies an equation is called the solution or root of the equation.
3. Any quadratic equation can be converted to the form (x+a)2 - b2 =0 by adding and subtracting some term. This method of finding the root of
quadratic equation is called the method of completing the square.
4. For the equation ax2 + bx + c = 0, a ≠0 expression 24b ac− is known
as discriminant.
Top Concepts
1. A real number α is said to be a solution/root of the quadratic equation
ax2+bx+c=0 if aα2+bα+c=0.
2. A quadratic equation can be solved by following algebraic methods.
i. Splitting the middle Term
ii. Completing Squares iii. Quadratic Formula
3. If ax2+bx+c, a≠0,can be reduced to the product of two linear factors, then the root of the quadratic equation ax2+bx+c = 0 can be found by
equating each factor to zero.
4. Method splitting the middle term of the equation ax2+bx+c=0 where
a ≠ 0.
i. Form the product “ac”
ii. Find a pair of numbers b1 and b2 whose product is “ac” and whose sum is “b” (if you can’t find such numbers, it can’t be factored).
iii. Split the middle term using b1 and b2 – that is express the
term bx as b1x + b2x .Now factor by grouping pairs of terms
5. Roots of the quadratic equation can be found by equating each linear factor to zero. Since product of two numbers is zero if either or both of
them are zero.
6. Method of completing the square for quadratic equation ax2+bx+c=0,
a ≠ 0.
i. Dividing through out by a we get 20
b cx x
a a+ + =
ii. Multiplying and dividing coefficient of x by 2
22 0
2
b cx x
a a+ + =
iii. Adding and subtracting 2
24
b
a
2 2
2
2 22. 0
2 4 4
b b b c
x x
a a a a
+ + − + =
2 2
2
4
2 4
− ⇒ + =
b b ac
x
a a
⇒
22 2
4
2 2
b b acx
a a
− + =
If 24 0b ac− ≥ then by taking square root
2
2
4
2 2
4
2
b b acx
a a
b b acx
a
− + = ±
− ± −⇒ =
7. Nature of the roots of a quadratic equation:
i. If b2 – 4ac > 0, the quadratic equation has two distinct real
roots
ii. If b2 – 4ac = 0, the quadratic equation has two equal real roots
iii. If b2 – 4ac < 0, the quadratic equation has no real roots
Top Formulae
1. Roots of 2
2 40 0 are
2
b b acax bx c ,a
a
− + −+ + = ≠ and
24
2
b b ac
a
− − − ,
where 24 0b ac− >
2. Roots of 20 0 are
2
bax bx c ,a
a
−+ + = ≠ and
2
b
a,where 2
4 0b ac− =
3. Quadratic identities:
i. (a + b)2 = a2 + 2ab + b2
ii. (a - b)2 = a2 - 2ab + b2 iii. a2 - b2 = (a + b) ( a – b)
4. Discriminant, D = b2-4ac
Chapter : Arithmetic Progressions
Top Definitions
1. An arithmetic progression is a list of numbers in which each term is
obtained by adding a fixed number d to the preceding term, except the first term.
2. The difference between the two successive term of an A.P is called the common difference.
3. Each of the number in the list of arithmetic progression is called a term
of an A.P
4. The arithmetic progression having finite number of terms is called a
finite arithmetic progression.
5. The arithmetic progression having infinite number of terms is called an infinite arithmetic progression.
Top Concepts
1. A list of numbers a1, a2, a3…… is an A.P, if the differences a2–a1, a3–a2, a4–a3 … give the same value i.e ak+1 – ak is same for all different values of k.
2. The general form of an A.P is a, a+ d, a+ 2d, a+3d…..
3. If the A.P a, a+d, a+ 2d……… � is reversed to � , � -d, � -2d………a, then
the common difference changes to negative of original sequence
common difference.
4. The nth term of an A.P is the difference of the sum to first n terms and the sum to first (n-1) terms of it.
i.e n n n 1a S S−
= −
Top Formulae
1. The general term of an A.P is given by: an = a + (n-1)d
where a is the first term and d is the common difference.
2. Sum of n terms of an A.P is given by:
n
nS 2a (n 1)d
2= + −
where a is the first term, d is the common difference and n is the total number of terms.
3. Sum of n terms of an A.P is also given by:
n
nS a
2= + �
Where a is the first term and � is the last term.
4. The general term of an A.P � , � -d, � -2d…….. is given by: a = �+ (n-1)(-d)
where � is the last term, d is the common difference and n is the number of terms.
Chapter: Triangles
Top Definitions
1. Two geometrical figures are called congruent if they superpose exactly on
each other that is they are of same shape and size.
2. Two figures are similar, if they are of the same shape but of different size.
3. Basic Proportionality Theorem (Thales Theorem): If a line is drawn
parallel to one side of a triangle to intersect other two sides in distinct
points, the other two sides are divided in the same ratio.
4. Converse of BPT: If a line divides any two sides of a triangle in the same
ratio then the line is parallel to the third side.
5. A triangle in which two sides are equal is called an isosceles triangle.
6. AAA (Angle-Angle-Angle) similarity criterion: If in two triangles,
corresponding angles are equal, then their corresponding sides are in the
same ratio (or proportion) and hence the two triangles are similar.
7. Converse of AAA similarity criterion: If two triangles are similar, then their
corresponding angles are equal.
8. SSS (Side- Side- Side) similarity criterion: If in two triangles, sides of one
triangle are proportional to (i.e., in the same ratio of) the sides of the
other triangle, then their corresponding angles are equal and hence the
two triangles are similar.
9. Converse of SSS similarity criterion: If two triangles are similar, then their
corresponding sides are in constant proportion.
10. SAS (Side-Angle-Side) similarity criterion: If one angle of a triangle is
equal to one angle of the other triangle and the sides including these
angles are proportional, then the two triangles are similar.
11. Converse of SAS similarity criterion: If two triangles are similar, then one
of the angles of one triangle is equal to the corresponding angle of the
other triangle and the sides including these angles are in constant
proportion.
12. Pythagoras Theorem: In a right triangle, the square of the hypotenuse is
equal to the sum of the squares of the other two sides.
13. Converse of Pythagoras Theorem: If in a triangle, square of one side is
equal to the sum of the squares of the other two sides, then the angle
opposite the first side is a right angle.
Top Concepts
1. All congruent figures are similar but the similar figures need not be
congruent.
2. Two polygons are similar if
• Their corresponding angles are equal
• Their corresponding sides are in same ratio.
3. If the angles in two triangles are:
• Different, the triangles are neither similar nor congruent. • Same, the triangles are similar. • Same and the corresponding sides are the same size, the triangles are
congruent
4. A line segment drawn through the mid points of one side of a triangle
parallel to another side bisects the third side
5. The ratio of any two corresponding sides in two equiangular triangles is
always same.
6. All circles are similar.
7. All squares are similar.
8. All equilateral triangles are similar.
9. If two triangles ABC and PQR are similar under the corresponding A ↔ P,
B ↔Q and C ↔ R, then symbolically, it is expressed as ∆ ABC ∼∆ PQR.
10. If two angles of a triangle are respectively equal to two angles of another
triangle, then by the angle sum property of a triangle their third angles
will also be equal.
11. The ratio of the areas of two similar triangles is equal to the square of the
ratio of their corresponding sides.
12. The ratio of the areas of two similar triangles is equal to the ratio of the
squares of the corresponding medians.
13. Triangles on the same base and between the same parallel lines have equal
area.
14. In a rhombus sum of the squares of the sides is equal to the sum of
squares of the diagonals.
15. In an equilateral or an isosceles triangle, the altitude divides the base into
two equal parts.
16. The altitude of an equilateral triangle with side ‘a’ is3
a2
.
17. In a square and rhombus, the diagonals bisect each other at right angles
18. If a perpendicular is drawn from the vertex of the right triangle to the
hypotenuse then triangles on both sides of the perpendicular are similar
to the whole triangle and to each other.
Top Formulae 1. If ∆ ABC ∼ ∆ PQR, then
(1) ∠A = ∠P
(2) ∠B = ∠Q
(3) ∠C = ∠R
(4) AB BC AC
PQ QR PR= =
2. If ∆ ABC ∼ ∆ PQR, then
2 2 2ar(ABC) AB BC CA
ar(PQR) PQ QR RP
= = =
3. In triangle ABC right angled at B, AB2 + BC2 = AC2
Top Diagrams
1. ∆ ABC ∼ ∆ DEF
2. ∆ ABD ≅ ∆ DEF
Chapter : Coordinate Geometry
Top Definitions
1. Two perpendicular number lines intersecting at point zero are called coordinate axes. The horizontal number line is the x-axis (denoted by
X’OX) and the vertical one is the y-axis (denoted by Y’OY). 2. The point of intersection of x axis and y axis is called origin and
denoted by ‘O’.
3. Cartesian plane is a plane obtained by putting the coordinate axes perpendicular to each other in the plane. It is also called coordinate plane or xy plane.
4. The x-coordinate of a point is its perpendicular distance from y axis.
5. The y-coordinate of a point is its perpendicular distance from x axis.
6. The point where the x axis and the y axis intersect is coordinate points
(0, 0).
7. The abscissa of a point is the x-coordinate of the point.
8. The ordinate of a point is the y-coordinate of the point.
9. If the abscissa of a point is x and the ordinate of the point is y, then (x, y) are called the coordinates of the point.
Top Concepts
1. The axes divide the Cartesian plane into four parts called the quadrants (one fourth part), numbered I, II, III and IV anticlockwise
from OX.
2. The coordinate of a point on the x axis are of the form (x,0) and that
of the point on y axis are (0,y)
3. Sign of coordinates depicts the quadrant in which it lies. The coordinates of a point are of the form (+, +) in the first quadrant,
(-, +) in the second quadrant, (-,-) in the third quadrant and (+,-) in
the fourth quadrant.
4. Three points A, B and C are collinear if the distances AB, BC, CA are such that the sum of two distances is equal to the third.
5. Three points A, B and C are the vertices of an equilateral triangle if the distances AB = BC = CA.
6. The points A, B and C are the vertices of an isosceles triangle if the
distances AB = BC or BC = CA or CA = AB.
7. Three points A, B and C are the vertices of a right triangle if 2 2 2AB BC CA+ = .
8. For the given four points A, B, C and D
(i) AB = BC = CD = DA; AC = BD ⇒ ABCD is a square.
(ii) AB = BC = CD = DA; AC ≠ BD ⇒ ABCD is a rhombus.
(iii) AB = CD, BC = DA; AC = BD ⇒ ABCD is a rectangle.
(iv) AB = CD, BC = DA; AC ≠ BD ⇒ ABCD is a parallelogram.
9. Diagonals of a square, rhombus, rectangle and parallelogram always bisect each other.
10. Diagonals of rhombus and square bisect each other at right angle.
11. Four given points are collinear, if the area of quadrilateral is zero.
12. Centroid is the point of intersection of the three medians of a triangle.
13. Centroid divides the median in the ratio of 2:1.
14. The incentre is the point of intersection of internal bisector of the
angles. It is also the centre of the circle touching all the sides of a
triangle.
15. Circum centre is the point of intersection of the perpendicular bisectors of the sides of the triangle.
16. Ortho centre is the point of intersection of perpendicular drawn from
the vertices on opposite sides (called altitudes) of a triangle and can be obtained by solving the equations of any two altitudes.
17. If the triangle is equilateral, the centroid, incentre, orthocentre, circum centre coincides.
18. If the triangle is right angled triangle, then orthocentre is the point
where right angle is formed.
19. If the triangle is right angled triangle, then circumcentre is the
midpoint of hypotenuse.
20. Orthocentre, centroid and circum centre are always collinear and centroid divides the line joining Orthocentre and circumcentre in the
ratio of 2:1.
21. In an isosceles triangle centroid, orthocentre, incentre, circumcentre lies on the same line.
22. Angle bisector divides the opposite sides in the ratio of remaining sides.
23. Three given points are collinear, if the area of triangle is zero.
Top Formulae
1. If x ≠ y, then (x,y)≠(y,x) and if (x,y) = (y,x), then x=y.
2. The distance between P(x1,y1) and Q(x2,y2) is 2 2
2 1 2 1(x x ) (y y ) .− + −
3. The distance of a point P(x,y) from origin is 2 2x y .+
4. Coordinates of point which divides the line segment joining the points
(x1,y1) and (x2,y2) in the ratio m : n internally are
2 1 2 1mx nx my nyx and y
m n m n
+ += =
+ +
5. Coordinates of mid-point which divides the line segment joining the
points (x1,y1) and (x2,y2) are 2 1 2 1x x y y
x and y2 2
+ += =
6. If A(x1, y1), B(x2,y2) and C(x3,y3) are vertices of a triangle, then the coordinates of centroid are
G = 1 2 3 1 2 3x x x y y y,
3 3
+ + + +
7. If A(x1, y1), B(x2,y2) and C(x3,y3) are vertices of a triangle, then the
coordinates of incentre are
I = 1 2 3 1 2 3ax bx cx ay ay ay,
a b c a b c
+ + + +
+ + + +
8. If A(x1, y1), B(x2,y2) and C(x3,y3) are vertices of a triangle, then the
area of triangle ABC is given by
Area of ABC∆ = 1 2 3 2 3 1 3 1 2
1x (y y ) x (y y ) x (y y )
2− + − + −
Top Diagrams
1. Sign of coordinates in various coordinates.
2. To plot a point P (3, 4) in the Cartesian plane. (i) A distance of 3 units along X axis.
(ii) A distance of 4 units along Y axis.
3. Area of quadrilateral ABCD = Area of ABC∆ + Area of ACD∆
A(x1, y1) B (x2, y2)
C (x3, y3) D (x4, y4)
4. Centroid (G) of a triangle.
5. Incentre (I) of a triangle.
6. Circumcentre (O) of a Triangle.
A
B C
F E
D
G
A
B C
F E
D
I
O
E
D
A
B C
F
7. Orthocentre (O) of a Triangle.
A
B C
F E
D
O
Chapter : Introduction to Trigonometry
Top Definitions
1. Trigonometry is the study of relationship between the sides and the angles of the triangle.
2. Ratio of the sides of the right triangle with respect to the acute angles
is called trigonometric ratios of the angle.
3. Pythagoras theorem: In a right triangle, square of the hypotenuse is
equal to the sum of the square of the other two sides. Top Concepts
1. Trigonometry is the combination of three Greek words ‘Tri (Three) +
gon (sides) + metron (measure)’.
2. When any two sides of a right triangle are given, its third side can be obtained by using Pythagoras theorem.
3. Angle measured in anticlockwise direction is taken as positive angle.
4. Angle measured in clockwise direction is taken as negative angle. 5. The values of the trigonometric ratios of an angle do not vary with the
length of the sides of the triangle, if the angles remain the same.
6. The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than 1 or equal to 1.
7. Each trigonometric ratio is a real number. It has no unit.
8. Only symbol cosine, sine, tangent, cotangent, sec and cosec has no meaning.
9. ( )n
sin θ is generally written as nsin θ , n being a positive integer.
Similarly other trigonometric ratios can also be written.
10. The value of sin θ increases from 0 to 1 when θ increases from 00 to
900.
11. The value of cos θ decreases from 1 to 0 when θ increases from 00 to
900.
Top Formulae
1. ( )1 1sec cos cos
− −θ = θ ≠ θ
2. Trigonometric ratio’s
(i) sidopposite to A p
sin Ahypotenuse h
∠= =
(ii) side adjacent to A b
cos Ahypotenuse h
∠= =
(iii) side adjacent to A p
tan Aside opposite to A b
∠= =
∠
(iv) hypotenuse h
cos ecAside opposite to A p
= =∠
(v) hypotenuse h
s ecAside adjacent to A b
= =∠
(vi) side adjacent to A b
co t Aside opposite to A p
∠= =
∠
3. Relation between trigonometry ratios
(vii) sin
tancos
θθ =
θ
(viii) 1
cos ecsin
θ =θ
(ix) 1
secco s
θ =θ
(x) 1 co s
cottan sin
θθ = =
θ θ
4. Trigonometric ratios of complementary angles
(i) sin (90 – A) = cos A (ii) cos (90 – A) = sin A
(iii) tan (90 – A) = cot A (iv) cot (90 – A) = tan A (v) sec (90 – A) = cosec A
(vi) cosec (90 – A) = sec A
5. Trigonometric Identities
(i) 2 2sin co s 1θ + θ =
(ii) 2 21 tan sec+ θ = θ
(iii) 2 21 cot sec+ θ = θ
Top Diagrams
1. Sides of right triangle
2. Values of Trigonometric ratios:
A∠ 0o 30
o 45
o 60
o 90
o
sin A 0 1
2
1
2
3
2
1
cos A
1 3
2
1
2
1
2
0
tan A
0 1
3
1 3 Not defined
cosec A
Not defined 2
2 2
3
1
sec A
1 2
3
2 2 Not defined
cot A Not defined 3 1 1
3
0
Chapter : Some Application of Trigonometry
Top Definitions
1. The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.
2. The angle of elevation of an object viewed is the angle formed by the line of sight with the horizontal when it is above the horizontal level,
i.e., the case when we raise our head to look at the object. 3. The angle of depression of an object viewed is the angle formed by the
line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object.
4. Pythagoras theorem: In a right triangle, square of the hypotenuse is
equal to the sum of the square of the other two sides.
5. Ratio of the sides of the right triangle with respect to the acute angles
is called trigonometric ratios of the angle. Top Concepts
1. The height or length of an object or the distance between two distant
objects can be determined by the help of trigonometric ratios. 2. When any two sides of a right triangle are given, its third side can be
obtained by using Pythagoras theorem.
3. Angle measured in anticlockwise direction is taken as positive angle. 4. Angle measured in clockwise direction is taken as negative angle.
5. The values of the trigonometric ratios of an angle do not vary with the
length of the sides of the triangle, if the angles remain the same.
6. Each trigonometric ratio is a real number. It has no unit. 7. Only symbol cosine, sine, tangent, cotangent, sec and cosec has no
meaning.
8. The two heights above and below the ground level in case of reflection from the water surface are equal.
Top Formulae
1. ( )1 1sec cos cos
− −θ = θ ≠ θ
2. Trigonometric ratio’s
(i) sidopposite to A p
sinAhypotenuse h
∠= =
(ii) side adjacent to A b
cosAhypotenuse h
∠= =
(iii) side adjacent to A p
tanAside opposite to A b
∠= =
∠
(iv) hypotenuse h
cosecAsideopposite to A p
= =∠
(v) hypotenuse h
secAsideadjacent to A b
= =∠
(vi) sideadjacent to A b
co t Aside opposite to A p
∠= =
∠
3. Relation between trigonometry ratios
(vii) sin
tancos
θθ =
θ
(viii) 1
cosecsin
θ =θ
(ix) 1
seccos
θ =θ
(x) 1 cos
cottan sin
θθ = =
θ θ
4. Trigonometric ratios of complementary angles
(i) sin (90 – A) = cos A
(ii) cos (90 – A) = sin A (iii) tan (90 – A) = cot A (iv) cot (90 – A) = tan A (v) sec (90 – A) = cosec A (vi) cosec (90 – A) = sec A
5. Trigonometric Identities
(i) 2 2sin cos 1θ + θ =
(ii) 2 21 tan sec+ θ = θ
(iii) 2 21 cot sec+ θ = θ
Top Diagrams
1. Angle of elevation.
2. Angle of depression.
3. Values of Trigonometric ratios.
A∠ 0o 30
o 45
o 60
o 90
o
sin A 0 1
2
1
2
3
2
1
cos A
1 3
2
1
2
1
2
0
tan A
0 1
3
1 3 Not defined
cosec A
Not defined 2
2 2
3
1
sec A
1 2
3
2 2 Not defined
cot A Not defined 3 1 1
3
0
Chapter : Circles
Top Definitions 1. A tangent to a circle is a line that intersects the circle only at one
point.
2. The common point of the circle and the tangent is called point of contact.
3. The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the
tangent. 4. Pythagoras theorem: In a right triangle, square of the hypotenuse is
equal to the sum of the square of the other two sides.
Top Concepts
1. A tangent to a circle is a special case of the secant when the two end points of the corresponding chord coincide.
2. There is no tangent to a circle passing through a point lying inside the circle.
3. There are exactly two tangents to a circle through a point outside the
circle.
4. At any point on the circle there can be one and only one tangent.
5. The tangent at any point of a circle is perpendicular to the radius
through the point of contact.
6. The lengths of the tangents drawn from an external point to a circle
are equal. 7. The centre lies on the bisector of the angle between the two tangents.
8. There can be infinite number of chords passing through a point which
is inside the circle.
Top Diagrams
1. Various lines on a circle.
2. Two equal tangents (PA = PB) from an external point P.
3. Radius ⊥ Tangent.
Chapter: Constructions
Top Definitions
1. The ratio of the side of the triangle to be constructed with the
corresponding sides of the given triangle is known as their scale factor.
2. Reduced scale factor figures are the geometric figure to be constructed is smaller in size.
3. Enlarged scale factor figures are the geometric figure to be constructed is larger in size.
Top Concepts
1. To divide a line segment internally in a given ratio m: n, where both m and n are positive integers, we follow the following steps:
Step 1: Draw a line segment AB of given length by using a ruler.
Step 2: Draw any ray AX making an acute angle with AB.
Step 3: Along AX mark off (m + n) points A1, A2,………AM,
Am+1,………,Am+n, such that AA1 = A1A2 = Am+n-1 Am+n.
Step 4: Join B Am+n
Step 5: Through the point Am draw a line parallel to Am+n B by making
an angle equal to ∠AAm+n B at Am.
Suppose this line meets AB at point P.
The point P so obtained is the required point which divides AB
internally in the ratio m: n.
2. Constructions of triangles similar to a given triangle:
(a) Steps of constructions when m < n:
Step 1: Construct the given triangle ABC by using the given data.
Step 2: Take any one of the three side of the given triangle as base.
Let AB be the base of the given triangle.
Step 3: At one end, say A, of base AB. Construct an acute angle ∠BAX
below the base AB.
Step 4: Along AX mark off n points A1, A2, A3,………, An such that
AA1 = A1A2 = ……… = An-1 An
Step 5: Join AnB
Step 6: Draw AmB’ parallel to AnB which meets AB at B’.
Step 7: From B’ draw B’C’||CB meeting AC at C’.
Triangle AB’C’ is the required triangle each of whose sides is
thm
n
of
the corresponding side of ∆ABC.
(b) Steps of construction when m > n:
Step 1: Construct the given triangle by using the given data.
Step 2: Take any one of the three sides of the given triangle and
consider it as the base. Let AB be the base of the given triangle.
Step 3: At one end, say A, of base AB. Construct an acute angle ∠BAX
below base AB i.e., con the opposite side of the vertex C.
Step 4: Along AX mark off m (large of m and n) points A1, A2,
A3,………An of AX such that AA1 = A1A2 = ………= Am-1Am.
Step 5: Join AnB to B and draw a line through Am parallel to AnB,
intersecting the extended line segment AB at B’.
Step 6: Draw a line through B’ parallel to BC intersecting by the extended line segment AC at C’.
Step 7: ∆AB’C’ so obtained is the required triangle.
3. Constructions of tangent to a circle:
a. To draw the tangent to a circle at a given point on it, when the
centre of the circle is known.
Given : A circle with centre O and a point P on it.
Required : To draw the tangent to the circle at P.
Steps of construction:
i. Join OP
ii. Draw a line AB perpendicular to OP at the point P. APB is the
required tangent at P.
4. To draw the tangent to a circle at a given point on it, when the centre
of the circle is not known.
Given : A circle and point on it.
Required : To draw the tangent to the circle at P.
Steps of construction:
i. Draw any chord PQ and join P and Q to a point R in major arc �PQ (or minor arc PQ).
ii. Draw ∠QPB equal to ∠PRQ and on opposite side of chord PQ.
The line BPA will be a tangent to the circle at P.
5. To draw the tangent to a circle from a point outside it (external point) when its center is known.
Given : A circle with center O and a point P outside it.
Required : to construct the tangents to the circle from P.
Steps of construction:
i. Join OP and bisect it. Let M be the mid point of OP.
ii. Taking M as centre and MO as radius, draw a circle to intersect
C (O,r) in two points, say A and B.
iii. Join PA and PB. These are the required tangents from P to C
(O,r).
6. To draw tangents to a circle from a point outside it (when its centre is
not known)
Given : P is a point outside the circle.
Required : To draw tangents from a point P outside the circle.
Steps of construction:
i. Draw a secant PAB to intersect the circle at A and B.
ii. Produce AP to a point C, such that PA = PC.
iii. With BC as a diameter, draw a semicircle.
iv. Draw PO ⊥ CB, intersecting the semicircle at O.
v. Taking PO as radius and P as centre, draw arcs to intersect the circle at T and T’.
vi. Join PT and PT’. Then PT and PT’ are the required tangents.
7. Two tangents can be drawn to a circle through a point outside the circle
and pair of these tangents are always equal in length.
Chapter : Area Related to Circles
Top Definitions
1. A circle is a collection of all points in a plane which are at a constant
distance from a fixed point in the same plane.
2. A line segment joining the centre of the circle to a point on the circle is
called its radius.
3. A line segment joining any two points of a circle is called a chord. A
chord passing through the centre of circle is called its diameter.
4. A part of a circle is called an arc.
5. A diameter of circle divides a circle into two equal arcs, each known as
a semicircle.
6. The region bounded by an arc of a circle and two radii at its end points
is called a sector.
7. A chord divides the interior of a circle into two parts, each called a
segment.
8. An arc of a circle whose length is less than that or a semicircle of the same circle is called a minor arc.
9. An arc of a circle whose length is greater than that of a semicircle of
the same circle is called a major arc.
10. Circles having the same centre but different radii are called concentric
circles.
11. Two circles (or arc) are said to be congruent if we can superpose
(place) one over the other such that they cover each other completely.
12. The distance around the circle or the length of a circle is called its
circumference or perimeter.
Top Concepts
1. The mid – point of the hypotenuse of a right triangle is equidistant
from the vertices of the triangle.
2. Angle subtended at the circumference by a diameter is always a right angle.
3. Angle described by minute hand in 60 minutes is 360°.
4. Angel described by hour hand in 12 hours is 360°
Top Formulae
1. Circumference (perimeter) or a circle = πd or 2πr, where r is the radius
of the circle and 22
7π = .
2. Area of a circle = πr2
3. Area of a semi circle = 21r
2π
4. Perimeter of a semi circle or protractor = πr + 2r
5. Area of a ring or an annulus = π (R + r) (R-r)
6. Length of arc AB =
22 r r
or360 180
π θ π
° °
7. Area of a sector =
2r
360
π θ
° Or Area of sector = ( )
1r
2�
8. Area of minor segment =
22r 1r sin
360 2
π θ− θ
°
9. Area of major segment = Area of the circle – Area of minor segment
πr² - Area of minor segment.
10. If a chord subtend a right angle at the centre, then
Area of the corresponding segment = 21r
4 2
π −
11. If a chord subtend an angle of 60° at the centre, then
Area of the corresponding segment = 23r
6 2
π−
12. If a chord subtend an angle of 120° at the centre, then
Area of the corresponding segment = 23r
3 4
π−
13. Distance moved by a wheel in 1 revolution = Circumference of the
wheel.
14. Number of revolutions in one minute =
Dis tance moved in 1 minute
Circumference
15. Perimeter of sector = r
2r180
π θ+
°
Chapter : Surface Areas and Volumes
Top Definitions
1. A Cube is a special type of cuboids in which length = breadth = height.
Also called an edge of a cube.
2. A sphere is a perfectly round geometrical object in three-dimensional
space, such as the shape of a round ball.
3. A cylinder is a solid or a hollow object that has a circular base and a
circular top of the same size.
4. A hemisphere is half of a sphere.
5. If a right circular is cut off by a plane parallel to its base, then the
portion of the cone between the plane and the base of the cone is called a frustum of the cone.
Top Concepts
1. The total surface area of the solid formed by the combination of solids
is the sum of the curved surface area of each of the individual parts.
2. A solid is melted and converted to another, volume of both the solids remains the same, assuming there is no wastage in the conversions.
The surface area of the two solids may or may not be the same.
3. A frustum can be obtained by cutting a cone by a plane, parallel to the
base of the cone.
4. The solids having the same curved surface do not necessarily occupy
the same volume.
Top Formulae
1. Cuboids:
Lateral surface area Or Area of four walls = 2(ℓ + b) h
Total surface area = 2(ℓb + bh + hℓ)
Volume = ℓ x b x h
Diagonal of a cuboids = 2 2 2
b h+ +�
2. Cube
Lateral surface area Or Area of four walls = 4 x (edge)2
Total surface area = 6 x (edge)²
Volume = (edge)²
Diagonal of a cube = 3 x edge.
3. Right circular cylinder:
Area of each end or Base area = πr²
Area of curved surface or lateral surface area
= perimeter of the base x height = 2π r (h + r)
Total surface area (including both ends)
= 2 πrh + 2πr² = 2πr (h + r)
Volume = (Area of the base0 x height = πr²h
4. Right circular hollow cylinder:
Area of curved surface
= (External surface) + (Internal surface)
= (2πRh + 2πrh) = 2 (πR² - πr²)
= [2πh(R+ r) + 2π (R² - r²)]
= [2π(R + r) (h + R – r)]
Volume of the material
= (External volume) – (Internal volume)
= (πR²h - πr²h) = πh (R² - r²)
5. Right circular cone:
Slant height (ℓ) = 2 2
h r+
Area of curved surface = πrℓ = πr 2 2
h r+
Total surface area = Area of curved surface + Area of base
= πrℓ + πr² = πr (ℓ + r)
Volume = 21r h
3π
6. Sphere:
Surface area = 4 πr²
Volume = 4
r²3
π
7. Spherical shell:
Surface area (outer) = 4πR²
Volume of material = 4 4
r³ r³3 3
π − π
= ( )4
R³ r³3
π −
8. Hemisphere:
Area of curved surface = 2 πr²h
Total surface Area = Area of curved surface + Area of base
= 2 πr² + πr²
= 3πr²
Volume = 2
r²3
π
9. Frustum of a cone:
Total surface area = π[R² + r² + ℓ (R + r)]
Volume of the material = 1
h R² r² Rr3
π + +
Top Diagrams
1. Cuboid
2. Cube
3. Right circular cylinder:
4. Right circular hollow cylinder:
5. Right circular cone:
6. Sphere:
7. Spherical shell:
8. Hemisphere:
9. Frustum of a cone:
Chapter : Statistics
Top Definitions
1. When a frequency distribution is obtained by dividing an ungrouped
data in a number of strata, according to the value of variety under study, such information is called grouped data or classified data.
2. The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding the given class to
the frequency of the class.
3. In a less than ogive the upper limit of a class is plotted against its
cumulative frequency as a point on the ogive.
4. In a ‘more than ogive’ the upper limit of a class is plotted against its
cumulative frequency as a point on the ogive
5. The mode for ungrouped data is the value that occurs most often.
6. In the case of a grouped frequency distribution a class with maximum
frequency is called as the modal class.
7. A distribution in which the values of mean, median and mode coincide
(i.e. mean = median = mode) is known as a symmetrical distribution.
8. Distribution for which values of mean, median and mode are not equal is known as asymmetrical or skewed distribution.
Top Concepts
1. A cumulative frequency distribution can be represented graphically by
means of an ‘ogive’.
2. The ogives can be drawn only when the given class intervals are continuous and if this is not the case then you don’t need to worry. All
you need to do is simply make the class intervals continuous.
3. The ‘less than ogive’ is a rising curve.
4. The ‘more than ogive’ is a falling curve.
5. Direct Method of finding Mean Step 1: First we find the mid values (also called class marks) of the
intervals, denoted by ‘x’ or ‘m’
LowerLimit UpperLimit
x2
+=
Step 2: Multiply frequency with corresponding mid values obtained in step1.
Step 3: Mean is calculated by using the following formula
Arithmetic mean = f x
xf
====∑∑∑∑∑∑∑∑
6. Short Cut Method/ Assumed Mean Method Step1: Find the class marks
Step2: Find the assumed mean (A) from the mid values Step 3: Calculate deviation (d), d = x – A
Step 4: find the product of frequency with the corresponding deviations Step 5 : Calculate mean by using the following formula
A +fd
xf
====∑∑∑∑∑∑∑∑
7. Step Deviation Method Step1: Find the class marks
Step2: Find the assumed mean (A) from the mid values Step 3: Calculate deviation (d). d = x – A Step 4: After calculating deviations (d), we make one more column of
values by dividing ‘d’ by ‘h’ Step 5: This is new value called step deviation (d’ or u) is multiplied
with corresponding frequencies. Step 6: Calculate mean by using the formula
'
A +fd
x hf
= ×= ×= ×= ×∑∑∑∑∑∑∑∑
8. The mode may be greater than, less than or even equal to the mean.
9. For finding the median we must arrange the given information i.e. the
given data in increasing or decreasing order.
10. The last of the cumulative frequencies will be always equal to the total
of all frequencies.
11. If the number of observations, n is even, so the median is the average
of the (n/2)th observation and the (n/2+1)th observation.
12. The step deviation method will be convenient to apply if all the
deviations (d’s) have a common factor
13. If class marks so obtained are in decimal form, then step deviation
method is preferred to calculate mean.
14. The median of a grouped data can be obtained graphically as the x coordinate of the point of intersection of two ogives for the data.
15. The most commonly used method of central tendency is the mean. The biggest problem with mean is that it is effected by the extreme
values one large or small number can distort the average. In that case the median is a better measure of central tendency while when the
most repeated value or the most wanted one is required, and then mode is used.
16. The most frequently used measure of central tendency is the mean,
because the mean is calculated by taking into account all the
observations of a given data. And it lies between the smallest and the largest value of the data.
17. In general, Mean median and mode could be connected as follows
• Mean<=Median<=Mode
• Mean>=Median>=Mode • Mode<Median and Mean<Median • Mode>Median and Mean>Median
Top Formulae
1. Direct Method
Mean = f x
f
∑
∑
2. Assumed Mean Method/ Short Cut Method
fd
x A +f
=∑∑
3. Step Deviation Method
fd'
x A + hf
= ×∑∑
4. Mode for a grouped data is given by
Mode= l + 1 0
1 0 2
f -f×h
2f -f -f
l = lower limit of the modal class
h = size o f the class interval
1f = frequency of the modal class
0f =frequency of the class preceding the modal class
2f = frequency of the class succeeding the modal class
5. Formula for median of a grouped data
Median= l +
n-cf
2× h
f
Where, l= the lower limit of median class. cf = the cumulative frequency of the class preceding the median class.
f = the frequency of the median class. h =the class size
6. 3 Median = Mode + Median
Top Diagrams
1. Less than Ogive
2. More than Ogive
3. Median calculated graphically.
4. Symmetric Distribution
5. Asymmetrical or skewed distribution
Chapter : Probability
Top Definitions
1. Probability is a quantitative measure of certainty.
2. Any activity associated to certain outcome is called a random
experiment. e.g. (i) tossing a coin (ii) throwing a dice (ii) selecting a card.
3. Outcome associated with an experiment is called an event. E.g (i)
Getting a head on tossing a coin (ii) getting a face card when a card is
drawn from a pack of 52 cards.
4. The event whose probability is one are called sure events/ certain
event.
5. The event whose probability is zero are called impossible events.
6. An event with only one possible outcome is called an elementary
event.
7. In a given experiment, if two or more events are equally likely to occur
or have equal probabilities, then they are called equally likely events.
Top Concepts
1. Probability of an event lies between 0 and 1.
2. Probability can never be negative.
3. A pack of playing cards consist of 52 cards which are divided into 4
suits of 13 cards each. Each suit consists of one ace, one king, one
queen, one jack and 9 other cards numbered from 2 to 10. Four suits named spades, hearts, diamonds and clubs.
4. King, queen and jack are face cards.
5. The sum of the probabilities of all elementary events of an experiment
is 1.
6. Two events A and B are said to be complements of each other if the
sum of their probabilities is 1.
Top Formulae
1. Probability of an event E denoted as P(E) is given by:
Number of outcomes favourable toE
P(E)TotalNumber of Outcomes
=
2. For an event E, P(E) 1 P(E)= − , where the event E representing ‘not E”
is the complement of event E.
3. For A and B two possible outcomes of an event.
(i) If P(A) > P(B) then event A is more likely to occur than event B. (ii) If P(A) = P(B) then events A and B are equally likely to occur.
4. a
Top Diagrams
1. Suits of Playing Card
Heart Spades Diamond Club
2. Face Cards
A King of diamond A Queen of club A Jack of Clubs
