Board Exam म" फोड़ना है - A4S Army
1
Quadratic Equations
Quadratic Polynomials: A polynomial of the form ax2 + bx + c is called a
quadratic polynomial in the variable x. This is a polynomial of the second
degree. In quadratic polynomial ax2 + bx + c, a ≠ 0 is the coefficient of x2, b
is the coefficient of x and c is the constant term (or coefficient of x0).
Quadratic Equation: An equation of the form ax2 + bx + c = 0, a ≠ 0, is
called a quadratic equation in one variable x, where a, b, c are constants.
For example, 4x2 – 3x + 1 = 0 and 3 – x – 7x2 = 0 are quadratic equation in
x.
!
#
#
But " ! + " − 5 = 0 is a quadratic equation in " , 𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 0.
Zeros of a quadratic polynomial or roots of a quadratic
equation
Let ax2 + bx + c be a quadratic polynomial if 𝑎𝛼 $ + 𝑏𝛼 + 𝑐 = 0, then 𝛼 is called
a zero of the quadratic polynomial ax2 + bx + c. if 𝛼 is a zero of ax2 + bx + c
i.e., if 𝑎𝛼 $ + 𝑏𝛼 + 𝑐 = 0, then we say x = 𝛼 satisfies the equation ax2 + bx + c =
0, and x = 𝛼 is a solution or root of equation ax2 + bx + c = 0.
Methods for solving a quadratic equation
By factorization
By completion of square
By solving quadratic formula
Quadratic Formula
If ax2 + bx + c = 0
𝐱=
−𝐛 ± √𝐛 𝟐 − 𝟒𝐚𝐜
−𝐛 ± √𝐃
𝐨𝐫 𝐱 =
𝟐𝐚
𝟐𝐚
Where 𝑫 = 𝒃𝟐 − 𝟒𝒂𝒄 is known as discriminant. This result is known as quadratic formula or
Sridharacharya formula
Nature of the roots
Case I: When D > 0, i.e., b2 – 4ac > 0. In this case, the roots are real and
distinct.
Case II: When D = 0, i.e., b2 – 4ac = 0. In this case, the roots are real and
equal.
Case III: When D < 0, i.e., b2 – 4ac < 0. In this case, the roots are not real.
Board Exam म" फोड़ना है - A4S Army
2
Arithmetic Progressions
Sequence :-Some numbers arranged in definite order, according to a definite rule are said to
form a sequence.
Progression: -Sequences which follow a definite pattern are called progressions.
Arithmetic Progression
An Arithmetic Progression (AP) is a list numbers in which each term is obtained by adding a fixed
number to the preceding term except the first term.
This fixed number is called the common difference (d) of the AP.
Note-It can be positive, negative or zero.
Calculation of ‘d’
a2 – a1 = d
a3 – a2 = d
a4 – a3 = d
an – an-1 = d and so on.
In General Terms in an A.P
a, a+d, a+2d, a+3d,……. Represent an arithmetic progression.
nth Term of an AP (General Term)
If the first term of an AP is ‘a’ and its common difference is ‘d’ then its nth term is given by the
formula
an = a+(n-1)d
NoteIn an AP, nth term is known as last term of an AP and it is denoted by I, which is given by the formula
Board Exam म" फोड़ना है - A4S Army
3
nth Term from the End of an AP
Let ‘a’ be the first term, ‘d’ be the commom difference and ‘l’ be the last term of an AP, then nth
term from the end can be found by the formula
Selection of Terms in an AP
Number Terms
of
terms
Common difference
3
𝒂 − 𝒅, 𝒂, 𝒂 + 𝒅
𝒅
4
𝒂 − 𝟑𝒅, 𝒂 − 𝒅, 𝒂 + 𝒅, 𝒂 + 𝟑𝒅
𝟐𝒅
5
𝒂 − 𝟐𝒅, 𝒂 − 𝒅, 𝒂, 𝒂 + 𝒅, 𝒂 + 𝟐𝒅
𝒅
Sum of First n Terms of an AP
If first term of an AP is ‘a’ and its common difference is ‘d’ , then the sum of its first n terms Sn , is
given by the formula
𝒏
Sn= [𝟐𝒂 + (𝒏 − 𝟏)𝒅]
𝟐
Or
𝒏
Sn = [𝒂 + 𝒍]
𝟐
Board Exam म" फोड़ना है - A4S Army
4
Calculation of nth term if sum of n terms is given
an = Sn- Sn-1
Arithmetic Mean
If a,b and c are in AP, then b is known as arithmetic mean of a and c, i.e. 𝐛 =
𝐚&𝐜
𝟐
.
Board Exam म" फोड़ना है - A4S Army
5
Some Applications of Trigonometry
Line of Sight
The line of sight is the line drawn from the eye of an observer to the point where the object is
viewed by the observer.
Horizontal Line
The line which goes parallel from eye to ground, is called horizontal line.
Angle of elevation
The angle of elevation of an object viewed, is the angle formed by the line of
sight with the horizontal when object viewed is above the horizontal level, i.e.
the case when we lower our head to look at the object.
Angle of Depression
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.
.
Board Exam म" फोड़ना है - A4S Army
6
Value of trigonometric ratios of standard angles
sin 𝜃
𝟎𝟎
0
cos 𝜃
1
tan 𝜃
0
cot 𝜃
Not
defined
1
sec 𝜃
cosec 𝜃
Not
defined
𝟑𝟎𝟎
1
2
√3
2
1
𝟒𝟓𝟎
1
√2
1
𝟔𝟎𝟎
√3
2
1
2
√3
1
1
2
√2
√3
2
√3
2
√2
2
√2
1
√3
√3
𝟗𝟎𝟎
1
0
Not
defined
0
Not
defined
1
√3
Board Exam म" फोड़ना है - A4S Army
7
Chapter–10 Circles
Circles-A circle is a collection of all those points in a plane which are at a constant
distance (radius) from a fixed point of that plane. Constant distance is length of radius and
fixed point is centre.
Note- Two or more circles having the same centre are called concentric circles.
Secant: A line which intersects circle in two distinct points is called a secant of the circle.
Tangent: The tangent to a circle is a line that meets the circle at exactly one point.
Length of the Tangent
•
The length of the segment of the tangent from the external point and the point of
contact with the circle is called the length of the tangent from the external point to the
circle.
In the above figure, AB is called the length of tangent.
Board Exam म" फोड़ना है - A4S Army
8
Number of Tangent from a Point on a Circle
(i) There are exactly two tangents to a circle through a point lying outside the circle.
circle ,i.e. PT1 and PT2
(ii) There is one and only one tangent to a circle passing through a point lying on the circle.
(iii) There is no tangent to a circle passing through a point lying inside the circle.
Board Exam म" फोड़ना है - A4S Army
9
Some Important Terms Related to Chapter 12 Area Related to Circles
Chord :-A line segment joining any two points on the circumference of the circle is called a chord
of the circle. If this chord passes through the centre, then the chord (or diameter) is the longest
chord of the circle
Semi – Circle
A diameter of a circle divides it into two equals parts or in two equal arcs. Each of these two arcs is
called a semi-circles
Circumference
The length of the complete circle is called the circumference of the circle.
Arc (Minor and Major)
A continuous piece of a circle is called an arc. In adjoining figure, P and Q are two points on a circle
which divide it into two parts, called the arcs. The larger part is called the major arc QRP and the
smaller part is called the minor arc PMQ
Sector
The region between an arc and the two radii, joining the ends of the arc to the centre, is called a
sector.
The sector formed by minor arc, is called minor sector and
The sector formed by major arc, is called major sector
Board Exam म" फोड़ना है - A4S Army
10
Segment
The region between a chord and either of its arc is called a segment of the circular region or simply a
segment of the circle. The segment formed by minor arc along with chord, is called minor segment
and the segment formed by major arc, is called the major segment.
Important Results based on Class 9th
(i) The perpendicular drawn from the centre of a circle to a chord bisects it and vice-versa.
(ii) Equal chords of a circle are equidistant from the centre
(iii) The angle subtended by an arc (or corresponding chord ) at the centre of the circle is twice the
angle subtended by the same arc at any point on the remaining part of the circle
(iv) Equals chords of a circle subtend equal angles at the centre.
(v) The angle in a semi-circle is a right angle.
(vi) Angles in the same segment of a circle are equal.
(vii) The sum of any pair of opposite angle of a cyclic quadrilateral is 180o.
(viii) If two circles intersect at two points, then the line through the centres in the perpendicular
bisector of the common chord.
Theorem Related to Tangent of circle (Class 10th Theorem on Latest Syllabus)
Theorem 1 The Tangent at any point of a circle is perpendicular to the radius through the point of
contact.
Here, O is centre of circle and AB is tangent of circle at P and it is point of contact and OP is radius.
∴ 𝐎𝐏 ⊥ 𝐀𝐁.
Board Exam म" फोड़ना है - A4S Army
11
Theorem 2 The lengths of two tangents drawn from an external point to a circle are equal.
Here,P is exterior point and PA and PB are tangents PA=PB
Important Results
(i) If two circles touch internally or externally, then point of contact lies on the straight line through
the two centres .
(ii) The opposite sides of a quadrilateral circumscribing acircle subtend supplementary angles at the
centre of the circle.
Board Exam म" फोड़ना है - A4S Army
12
Surface Areas and Volumes
Cuboid:
TSA(Total Surface Area) = 2(lb + bh + hl)
Lateral Surface Area (LSA) = 2h (l + b)
Volume = lbh.
Diagonal of cuboid = √𝑙 $ + 𝑏 $ + ℎ$
Cube: LSA = 4a2
TSA = 6a2
Volume = a3,
Diagonal of cube = √3𝑎
Board Exam म" फोड़ना है - A4S Army
13
Right circular cylinder
CSA(Curve Surface Area) = 2𝜋𝑟ℎ
TSA = 2𝜋𝑟ℎ + 2𝜋𝑟 $ = 2𝜋𝑟(𝑟 + ℎ)
Volume = 𝜋𝑟 $ ℎ
Hollow cylinder
Thickness of cylinder = R – r
External CSA = 2𝜋𝑅𝐻
Internal CSA = 2𝜋𝑟ℎ
TSA = External curved area + internal curved area + area of two ends
= 2𝜋𝑅ℎ + 2𝜋𝑟ℎ + 2𝜋(𝑅$ − 𝑟 $ )
= 2𝜋(𝑅ℎ + 𝑟ℎ + 𝑅$ − 𝑟 $ )
Volume of material = 𝜋𝑅$ ℎ − 𝜋𝑟 $ ℎ = 𝜋(𝑅$ − 𝑟 $ )ℎ
Board Exam म" फोड़ना है - A4S Army
14
Cone
CSA = 𝜋𝑟𝑙 = 𝜋𝑟 √𝑟 $ + ℎ$
TSA = 𝜋𝑟𝑙 + 𝜋𝑟 $ = 𝜋𝑟(𝑙 + 𝑟)
#
Volume = ! 𝜋𝑟 $ ℎ
Slant height = 𝑙 = <(𝑟 $ + ℎ$ )
Sphere
CSA = 𝟒𝝅𝒓𝟐
TSA = 𝟒𝝅𝒓𝟐
)
Volume = ! 𝜋𝑟 ! ,
Board Exam म" फोड़ना है - A4S Army
15
Spherical Shell
Thickness = R – r
)
Volume = ! 𝜋 (𝑅! − 𝑟 ! ).
Hemisphere
CSA = 2𝜋𝑟 $
TSA = 3𝜋𝑟 $
$
Volume = ! 𝜋𝑟 ! ,
Hemispherical Shell
External CSA = 2𝜋𝑅$
Internal CSA = 2𝜋𝑟 $
TSA = 2𝜋𝑅$ + 2𝜋𝑟 $ + 𝜋(𝑅$ − 𝑟 $ ) = 𝜋(3𝑅$ + 𝑟 $ )
$
Volume of material = ! 𝜋(𝑅! − 𝑟 ! )
Board Exam म" फोड़ना है - A4S Army
16
Statistics
Statistics measures of central tendency
Mean – the arithmetic mean (or, simply mean) is the sum of the values of all
the observations divided by the total number of observations.
Mean of ungrouped data
The mean of n numbers x1, x2, x3, … xn denoted by 𝑋 (read as X bar) is
defined as:
𝑥# + 𝑥$ + 𝑥! + … + 𝑥* ∑ 𝑥
𝑋=
=
𝑛
𝑛
Where Σ is a Greek alphabet called sigma. Thus, ∑ 𝑥 means sum of all x.
Mean of grouped data
Direct method: if the variates observations x1, x2, x3, … xn have frequencies
f1, f2, f3, …. Fn respectively, then the mean is given by:
Mean
𝑓# 𝑥# + 𝑓$ 𝑥$ + … + 𝑓* 𝑥* ∑ 𝑓+ 𝑥+
𝑋=
=
∑ 𝑓+
𝑓# + 𝑓$ + … + 𝑓*
Short cut method: in some problems, where the number of variates is large
or the values of xi or fi are larger, then the calculations become tedious. To
overcome this difficulty, we use short cut or deviation method.
Assumed mean method
Find the class mark or mid-value of each class, as:
,-./0 ,+2+3&455/0 ,+2+3
Xi = class mark = G
H
$
In this method, an approximate mean, called assumed mean or provisional
mean is taken. This assumed mean is taken preferably near the middle, say
A and the deviation di = xi – A for each variate xi.
The mean is given by the formula:
Mean 𝑋 = 𝐴 +
∑ 7" 8"
∑ 7"
Board Exam म" फोड़ना है - A4S Army
17
Mode – The mode of a distribution is the value of observation with highest
frequency. In a continuous frequency distribution with equal class interval,
mode is obtained by locating a class with the maximum frequency.
𝑓# − 𝑓9
𝑚𝑜𝑑𝑒 = 𝑙 +
×ℎ
2𝑓# − 𝑓9 − 𝑓$
Where,
l = lower limit of the modal class
f1 = frequency of the modal class
f0 = frequency of the class preceding the modal class
f2 = frequency of the class succeeding the modal class.
h = size of the modal class.
Median
The median gives the value of the middle – most observation in the data.
Median of ungrouped data
For finding median of ungrouped data, we first arrange the data in
ascending order.
*&#
• If n is odd, median = G $ H 𝑡ℎ observation
•
# *
*
If n is even, median = $ O$ 𝑡ℎ + G$ + 1H 𝑡ℎ Q observation.
•
Median of grouped data
To find median class, we locate the class whose cumulative frequency is
greater than (nearest to) /2 median of a grouped or continuous frequency
distribution
Median= l + R
#
:;7
!
7
S×ℎ
Where, l = lower limit of the median class
n = number of observations
f = frequency of the median class
h = size of the median class (assuming class size to be equal)
cf = cumulative frequency of the class preceding the median class
Empirical relationship between the three measure of central tendency.
3 median = mode + 2 mean
Board Exam म" फोड़ना है - A4S Army
18
Board Exam म" फोड़ना है - A4S Army
19