CLASS-X
1. REAL NUMBERS
REAL NUMBER: Set of Rational and Irrational numbers are known as Real Number
LEMMA: A lemma is a proven statement used for proving another statement.
EUCLID’S DIVISION LEMMA
Given positive integers a and b, there exist unique whole number q and r satisfying a = bq + r, where
0 r<b .
Here we call ‘a’ as dividend, ‘b’ as divisor, ‘q’ as quotient and ‘r’ as remainder.
Dividend = (Divisor x Quotient) + Remainder
If in Euclid’s lemma r = 0 then b would be HCF of ‘a’ and ‘b’.
EUCLID’S DIVISION ALGORITHM
Euclid’s division algorithm is a technique to compute the Highest
Common Factor (HCF) of two given positive integers. Recall that the HCF
of two positive integers a andbis the largest positive integer d that
divides both a and b.
To obtain the HCF of two positive integers, say c and d, with c > d, follow
the steps below:
Step 1: Apply Euclid’s division lemma, to c and d. So, we find whole
numbers, q and r such that c = dq+ r, 0 0 r<d .
Step 2: If r = 0, d is the HCF of c and d. If r ≠0 apply the division lemma to
dandr.
Step 3: Continue the process till the remainder is zero. The divisor at this
stage will be the required HCF.
This algorithm works because HCF (c, d) = HCF (d, r) where the symbol
HCF (c, d) denotes theHCF of c and d, etc.
Fundamental Theorem of Arithmetic : Every composite number can be expressed ( factorised ) as a product of its primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Property of HCF and LCM of two positive integers ‘a’ and ‘b’:
HCF x LCM = First No. x Second No.
LCM =
HCF =
Let p be a prime number. If p divides a2, then p divides a, where a is positive integer.
√2, √5, 2+5√3 and 2-5√3 are Irrational Numbers
Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form where p and q are coprime, and the prime factorisation of q is of the form 2n5m , where n, m are non-negative integers.
Let x = be a rational number, such that the prime factorization of q is not of the form 2n5m , where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).
2.POLYNOMIAL
An algebraic expression of the form…………………+ is called polynomial of degree n, where…………………………………..are real numbers and n is non negative integer.
An expression of the form ax +b, ax2+bx+c and ax3 + bx2 + cx +d is called linear, quadratic and cubic polynomial respectively.
The highest power of variable in the polynomial is called degree of the polynomial.
A polynomial of form f(x)=c is called constant polynomial and its degree is 0
A polynomial of the form f(x) = 0 is called zero polynomial and its degree is not defined.
If α and β are the zeroes of the quadratic polynomial ax 2 + bx + c, then
α+β =− b/a,
If α, β, γ are the zeroes of the cubic polynomial ax 3 + bx2 + cx + d, then,
The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that
p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).
3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
1. Two linear equations in the same two variables are called a pair of linear equations in two
variables. The most general form of a pair of linear equations is
where , ,, , , are real numbers, such that
,
2. A pair of linear equations in two variables can be represented, and solved,
by the:
(i) Graphical method
(ii) Algebraic method
3. Graphical Method:
The graph of a pair of linear equations in two variables is represented by
two lines.
(i) If the lines intersect at a point, then that point gives the unique solution
of the twoequations. In this case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions — each
point on the line being a solution. In this case, the pair of equations is
dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In
this case, the pair of equations is inconsistent.
4. Algebraic Methods: We have discussed the following methods for finding
the solution(s) of a pair of linear equations :
Substitution Method;
Eg: x + y = 14 …………………. (1)
x - y = 4, ………………….....(2)
from (2) x = 4 + y, ………………(3)
put value of x in (1),
4 + y + y = 14,
4 + 2y = 14,
y = 5,
put y = 5 in (3)
x = 4 + 5,
x = 9
(ii) Elimination Method;
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
variablegetseliminated. 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.
(iii) Cross-multiplication Method
Then
5. If a pair of linear equations is given by and , then the following situations can arise :
(i) : In this case, the pair of linear equations is consistent with
unique solution and lines will be intersecting.
(ii) : In this case, the pair of linear equations is inconsistent
with no solution and lines will be parallel.
(iii) : In this case, the pair of linear equation is dependent and
consistent with infinite many solutions and lines will be co-incident.
4. QUADRATIC EQUATIONS
Equation ax2+bx+c=0 where a≠0 is called quadratic equation.
Quadratic equations have at most two real solutions/roots.
Value of x which satisfies the equation is called the solution/roots of the equation.
There are two methods to solve quadratic equation:
(i) Splitting middle term (ii) completing square methods
(i)Splitting middle term:
ax2+bx+c=0.
Find a.c
If a.c>0 find two numbers whose sum is b and whose product is equal to a.c.
If a.c<0 find two numbers whose difference is b and whose product is equal to a.c.
(ii) Completing square methods:
ax2+bx+c=0. The solution is
5. The sum of the roots is-b/a.
6. Product of the roots is c/a.
7. Discriminants of the quadratic equation is D=)
8. Nature of roots.
(i)If D=0 roots are equal and real
(ii) If D>0 roots are distinct and real
(iii) If D<0 no real root.
5. ARITHMETIC PROGRESSION
8000, 8500, 9000………
10000, 12500, 15625, 19531.25………
In the examples above, we observe some patterns. In some, we find that the succeeding terms are obtained by adding a fixed number, in other by multiplying with a fixed number.
See these patterns in which succeeding terms are obtained by adding a fixed number to the preceding terms.
1, 2, 3, 4, . . .
100, 70, 40, 10, . . .
–3, –2, –1, 0, . . .
–1.0, –1.5, –2.0, –2.5 . . . Each of the numbers in the list is called a term.
Such list of numbers is said to form an Arithmetic Progression (AP).
So, an arithmetic progression is a list of 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 of the AP.
*Note that it can be positive, negative or zero.
nthTerm of an ARITHMETIC PROGRESSION ( AP )
nthterm of the AP with first term a and common difference d is given by
an = a + (n – 1) d.
nthterm from the end of an ARITHMETIC PROGRESSION ( AP )
Let the last term of an AP be ‘l’ and the common difference of an AP is ‘d’ then the nth term from the end of an AP is given by
ln = l – (n – 1) d.
Sum of First n Terms of an ARITHMETIC PROGRESSION (AP)
The sum of the first n terms of an AP is given by Sn = [2a + (n – 1) d]
Where a = first term, d = common difference and n = number of terms.
If we know the last term of the AP then we can write above formula in this manner also
Sn = [a + l]
where a = first term, l = last term and n = number of terms.
Note: let for an AP, a = first term, d = common difference
If you have to select 3 terms then the terms are a – d, a, a +d.
If you have to select 4 terms then the terms are a – 3d, a – d, a + d, a + 3d.
If you have to select 5 terms then the terms are a – 2d, a – d, a, a + d, a + 2d.
6. TRIANGLES
Any two figures with exactly same shapes is called similar figures and the event is called Similarity.
Two triangles are said to be similar if their
Corresponding angles are equal
Corresponding side are Proportional.
Basic Proportionality Theorem: In a triangle a line drawn parallel to one side to intersect other two sides at distinct points, divides the two sides in the same ratio.
Converse of BPT: If a line divides the two sides of a triangle in same ratio, the line must be parallel to the third side.
Criteria For Similarity: AAA, SSS, SAS
AAA: If in two triangles, corresponding angles are equal or the two corresponding angles are equal, then the triangles are similar.
SSS: If the corresponding sides of two triangles are proportional, then they are similar.
SAS: If in two triangles, one pair of corresponding sides are proportional and one included angle are equal, then the two triangles are similar.
Area of two similar triangles:- The ratio of area of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Pythagoras Theorem: -In a Right angled triangle the square of hypotenuse is equal to the sum of the squares of other two sides.
Converse of Pythagoras Theorem: -In a triangle if the square of one side is equal to sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.
7. COORDINATE GEOMETRY
Location of a point in a plane
To locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the y-axis is called its x-coordinate, or abscissa. The distance of a point from the x-axis is called its y-coordinate, or ordinate. The coordinates of a point on the x-axis are of the form (x, 0), and of a point on the y-axis are of the form (0, y).Thus point is represented by (x,y)
To locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the y-axis is called its x-coordinate, or abscissa. The distance of a point from the x-axis is called its y-coordinate, or ordinate. The coordinates of a point on the x-axis are of the form (x, 0), and of a point on the y-axis are of the form (0, y).Thus point is represented by (x,y)
To locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the y-axis is called its x-coordinate, orabscissa. The distance of a point from the x-axis is called its y-coordinate, or ordinate. The coordinates of a point on the x-axis are of the form (x, 0), and of a point on the y-axis are of the form (0, y).Thus point is represented by (x,y)
Uses of coordinate geometry
linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a straight line.
Further, we have seen the graph of y = ax2 + bx+ c (a ≠ 0), is a parabola.
In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures.
It helps us to study geometry using algebra, and understand algebra with the help of geometry.
Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art!
Gist of the lesson
1. How to find the distance between the two points whose coordinates are given,
How find the area of the triangle formed by three given points.
How to find the coordinates of the point which divides a line segment joining two given points in a given ratio.
1.Distance formula
D=
2.Section Formula
x = y=
3.Formula for Area of Triangle
(½)[x1 ( y2 – y3) + x2 ( y3 – y1) + x3 ( y1 – y2)]
Uses of distance formula
1.distance between two points
2.Colinearity of points
3.To prove properties of different figures.
Uses of Section formula
To find the point of division of a line by a point.
Mid point
Point of trisections
Centroid of a Triangle
Uses of Area of Triangle Formula
1.area of triangles and Quadrilaterals
2.Colinearity of points.
8. INTRODUCTION TO TRIGONOMETRY
1.Definition- Ratio of any two sides of a triangle is called trigonometric ratio.
i) (Tricks to remember T.Ratios) Sin cos tan
ii) C
B A
SinA= CosA= TanA=
CosecA= SecA= CotA=
2. Relation between different trigonometric ratios-
SinA=SinA .CosecA=1
CosA=CosA.SecA=1,
TanA=TanA.CotA=1
TanA= , CotA=
3.Trigonometric Ratio of some specific angles-
00
300
450
600
900
Sin
0
1
1
0
0
1
Not Defined
Not Defined
1
0
Sec
1
2
Not Defined
Cosec
Not Defined
2
1
4.Trigonometric ratios of complimentary angles-
Sin(900-)=Cos
Cos(900-)=Sin
Tan(900-)=Cot
Cot(900-)=Tan
Sec(900-)=Cosec
Cosec(900-)=Sec
5.Trigonometric Identities-
An equation involving trigonometric ratios of an angle is called trigonometric identities if it is true for all values of angles.
Sin2+ Cos2=1, 1-Sin2Cos2 1-Cos2=Sin2
1+Tan2=Sec2,Sec2-1=Tan2,Sec2-Tan2=1
1+Cot2=Cosec2,Cosec2-1=Cot2Cosec2-Cot2=1
9. SOME APPLICATIONS TO TRIGONOMETRY
1 Line of sight
Line segment joining the object to the eye of the observer is called the line of sight.
2 Angle of Elevation
When an observer sees an object situated in upward direction, the angle formed by line of sight with horizontal line is called angle of elevation.
3 Angle of depression
When an observer sees an object situated in downward direction the angle formed by line of sight with horizontal line is called angle of depression
4. Trigonometric Ratio.
C
B A
SinA= CosA= TanA=
CosecA= SecA= CotA=
4. Trigonometric Ratio of some specific angles-
00
300
450
600
900
Sin
0
1
1
0
0
1
Not Defined
Not Defined
1
0
Sec
1
2
Not Defined
Cosec
Not Defined
2
1
Chapter- 10 (Circles)
Definitions, statements and summary
A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.
The perpendicular from the centre of a circle to a chord bisects the chord.
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Angles in the same segment of a circle are equal.
Angle in a semicircle is a right angle.
If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.
If sum of a pair of opposite angles of a quadrilateral is 1800, the quadrilateral is cyclic.
A tangent to a circle is a line that intersects the circle at only one point.
The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Chapter- 11 (Construction)
Summary and key points
1. To bisect a given angle.
2. To draw the perpendicular bisector of a given line segment.
3. To construct an angle of 60° etc.
4. To construct a triangle given its base, a base angle and the sum of the other two sides.
5. To construct a triangle given its base, a base angle and the difference of the other two sides.
6. To construct a triangle given its perimeter and its two base angles.
To divide a line segment in a given ratio. 2. To construct a triangle similar to a given triangle as per a given scale factor which may be less than 1 or greater than 1. 3. To construct the pair of tangents from an external point to a circle.
Chapter- 12 (Area related to circles)
Summary and key points
1-Circumference of a circle = 2 π r.
2. Area of a circle = π r2.
3. Length of an arc of a sector of a circle with radius r and angle with degree measure
4. Area of a sector of a circle with radius r and angle with degrees measure θ is =
5. Area of segment of a circle = Area of the corresponding sector – Area of the corresponding triangle.
13. SURFACE AREA AND VOLUMES
The figures that can be drawn on notebooks or blackboards easily are called plane figure.
Solid regular figures can be obtained by piling up plane congruent shapes on each other vertically. Some solids obtained in this way are cube, cuboid and cylinder.
If we leave top and bottom faces of cuboid in this case we find four faces, the area of these four faces is called lateral surface area.
Now, Lateral Surface area of cuboid = 2(L + B) × H
And total surface area of cuboid = 2(LB + BH + HL).
Where L,B and H are the lengths of three edges of the cuboid
The Unit of area is taken as the square unit. ( cm2 , m2 etc )
A cuboid whose length, breadth and height are equal, is called a cube.
Hence surface area of a cube = 6 a2
Lateral surface area = 4 a2.
If we take a number of circular sheets of paper and stack them up vertically as we stacked up rectangular sheets earlier what we get is called a right circular cylinder.
And if we take a number of circular sheets of paper and stack them up not vertically, then what kind of cylinder we find is not a right circular cylinder. In this chapter we have to deal with only right circular cylinder, so the word cylinder would mean right circular cylinder.
If a cylinder is to be covered with coloured paper, then we need a rectangular sheet of paper whose length is just enough to go round the cylinder and whose breadth is equal to the height of the cylinder.
In this way curved surface area of the cylinder = area of the rectangular sheet.
Curved surface area of cylinder = 2╥rh
Total surface area of a cylinder = 2╥r (h + r)
Where r is the radius of the base of the cylinder and h is the height.
Cut out a right angled triangle ABC right angled at B paste a long thick
string along AB .hold the string and rotate the triangle about the string.
This is called a right circular cone.
Curved surface area of cone = ╥rl
Total Curved surface area of cone = ╥r(r + l)
Where r is the radius of the base and l is the slant height.
Also l2= r2 + h2. Here his vertical height of the cone.
12. If we rotate a circular plate rapidly then we see a new solid it is
Called sphere.Therefore a sphere is a three dimensional figure,
which is made up of all points in the space, which lie at a constant
distance called the radius, from a fixed point called the centre of the
sphere.
13. Surface area of a sphere = 4╥r2
Curved surface area of a hemisphere = 2╥r2
Total surface area of a hemisphere = 3╥r2
14. Measure of space occupied by an object is called the Volume of the
object. If the object is hollow then air or some liquid can be filled in
the interior of the object. In such way the volume of substance filled
in interior of object is called the capacity of the container or hollow
object. Hence unit of both is cubic unit.
15. Cuboid is a regular solid so its volume = base area × height
= L × B × H
16. Volume of cube = side × side × side
17. If we pile up congruent circles on each other vertically, then the
shape formed is called right circular cylinder.
Volume of cylinder = base area × height =╥r2 × h,
Where r is radius of the base of cylinder and h is its height.
18. If we rotate a right angled triangle with one of its leg vertical and
Other horizontal. We will get a solid shape this is called right circular
cone.
19. If we take a cone and a right circular cylinder with equal radius of
Base and equal height. If the cone filled with sand is emptied in a
cylinder then the cylinder is filled in three times. Therefore
Volume of cone = volume of cylinder
= ╥r2 h
20. Volume of sphere = ╥r3
Volume of hemisphere = ╥r3.
14. STATISTICS
KEY POINTS
The three measures of central tendency are :
(i) Mean
(ii) Median
(iii) Mode
Mean Of grouped frequency distribution can be calculated by the following methods.
Direct Method
Mean = = Where Xi is the class mark of the ith class interval and fi frequency of that class
Assumed Mean method or Shortcut method
Mean = = a +Where a = assumed mean
And di= Xi - a
Step deviation method.
Mean = = a +Where a = assumed mean
h = class size
And ui= (Xi – a)/h
Median- When raw data is written in order, the middlemost observation is called Median.
Median class- The class which has the cumulative frequency just greater than n/2.
Median of a grouped frequency distribution can be calculated by
Median = l +
Where
l = lower limit of median class
n = number of observations
cf = cumulative frequency of class preceding the median class
f = frequency of median class
h = class size of the median class.
Mode- The observation having highest frequency.
Modal Class- The class having highest frequency.
Mode of grouped data can be calculated by the following formula.
Mode = l +
Where
l = lower limit of modal class
h = size of class interval
f1 = Frequency of the modal class
fo = frequency of class preceding the modal class
f2= frequency of class succeeding the modal class
Empirical relationship between the three measures of central tendency.
3 Median = Mode + 2 Mean
Ogive- is the free hand curve of the cumulative frequency distribution. It is of two types:
Less than type ogive.
More than type ogive
Median by graphical method
The x-coordinated of the point of intersection of ‘less than ogive’ and ‘more than ogive’ gives the median.
15. PROBABLITY
KEY POINTS
Probability:- Probability is the numerical measurement of uncertainty.
The theoretical probability of an event E, written as P(E) is defined as.
P(E) =
Where we assume that the outcomes of the experiment are equally likely
Equally Likely- The experiment in which each outcome has equal chance to happen.
The probability of a sure event (or certain event) is 1.
The probability of an impossible event is 0.
The probability of an Event E is number P (E) such that 0 ≤ P(E) ≤ 1.
Elementary events:- An event having single outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.
For any event E, P(E)+P()=1, where stands for not E, E and are called complementary event.
Sample space:-The set of all possible outcomes in an experiment is called sample space.
Probability:- Probability is the numerical measurement of uncertainty.
The theoretical probability of an event E, written as P(E) is defined as.
P(E) =
Where we assume that the outcomes of the experiment are equally likely.
Equally Likely- The experiment in which each outcome has equal chance to happen.
The probability of a sure event (or certain event) is 1.
The probability of an impossible event is 0.
The probability of an Event E is number P (E) such that 0 ≤ P(E) ≤ 1.
Elementary events:- An event having single outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.
For any event E, P(E)+P()=1, where stands for not E, E and are called complementary event.
Sample space:-The set of all possible outcomes in an experiment is called sample space.
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