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LINEAR EQUATIONS IN TWO VARIABLES | Study
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Linear Equations
The equation of a straight line is the linear equation. It could be in one variable or two variables.
Linear Equation in One Variable
The equation with one variable in it is known as a Linear Equation in One Variable. (Scroll down to continue …)
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The general form for Linear Equation in One Variable is px + q = s, where p, q and s are real numbers and p ≠ 0.
Example:
x + 5 = 10
y – 3 = 19
These are called Linear Equations in One Variable because the highest degree of the variable is one.
Graph of the Linear Equation in One Variable
We can mark the point of the linear equation in one variable on the number line.
x = 2 can be marked on the number line as follows –
Graph of the Linear Equation in One Variable
Linear Equation in Two Variables
An equation with two variables is known as a Linear Equation in Two Variables. The general form of the linear equation in two variables is
ax + by + c = 0
where a and b are coefficients and c is the constant. a ≠ 0 and b ≠ 0.
Example
6x + 2y + 5 = 0, etc.
Slope Intercept form
Generally, the linear equation in two variables is written in the slope-intercept form as this is the easiest way to find the slope of the straight line while drawing the graph of it.
The slope-intercept form is y = mx+c
Where m represents the slope of the line.
and c tells the point of intersection of the line with the y-axis.
Remark: If b = 0 i.e. if the equation is y = mx then the line will pass through the origin as the y-intercept is zero.
Solution of a Linear Equation
There is only one solution in the linear equation in one variable but there are infinitely many solutions in the linear equation in two variables.
As there are two variables, the solution will be in the form of an ordered pair, i.e. (x, y).
The pair which satisfies the equation is the solution to that particular equation.
Example:
Find the solution for the equation 2x + y = 7.
Solution:
To calculate the solution of the given equation we will take x = 0
2(0) + y = 7
y = 7
Hence, one solution is (0, 7).
To find another solution we will take y = 0
2x + 0 = 7
x = 3.5
So another solution is (3.5, 0).
Graph of a Linear Equation in Two Variables
To draw the graph of a linear equation in two variables, we need to draw a table to write the solutions of the given equation, and then plot them on the Cartesian plane.
By joining these coordinates, we get the line of that equation.
The coordinates which satisfy the given Equation lie on the line of the equation.
Every point (x, y) on the line is the solution x = a, y = b of the given Equation.
Any point, which does not lie on the line AB, is not a solution of Equation.
Example:
Draw the graph of the equation 3x + 4y = 12.
Solution:
To draw the graph of the equation 3x + 4y = 12, we need to find the solutions of the equation.
Let x = 0
3(0) + 4y = 12
y = 3
Let y = 0
3x + 4(0) = 12
x = 4
Now draw a table to write the solutions.
x 0 4
y 3 0
Now we can draw the graph easily by plotting these points on the Cartesian plane.
Linear Equation in Two Variables
Equations of Lines Parallel to the x-axis and y-axis
When we draw the graph of the linear equation in one variable then it will be a point on the number line.
x – 5 = 0
x = 5
This shows that it has only one solution i.e. x = 5, so it can be plotted on the number line.
But if we treat this equation as the linear equation in two variables then it will have infinitely many solutions and the graph will be a straight line.
x – 5 = 0 or x + (0) y – 5 = 0
This shows that this is the linear equation in two variables where the value of y is always zero. So the line will not touch the y-axis at any point.
x = 5, x = number, then the graph will be the vertical line parallel to the y-axis.
All the points on the line will be the solution of the given equation.
Equations of Lines Parallel to the x-axis and y-axis
Similarly if y = – 3, y = number then the graph will be the horizontal line parallel to the x-axis.

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COORDINATE GEOMETRY | Study
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Coordinate Geometry: Cartesian Plane | Speed Notes
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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
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● The horizontal line is known as the x-axis and the vertical line is known
as the y-axis.
● The point where these two lines intersect each other is called the origin.
It is represented as ‘O’.
● OX and OY are the positive directions as the positive numbers lie in the
right and upward direction.
● Similarly, the left and the downward directions are the negative directions
as all the negative numbers lie there.
Quadrants of the Cartesian Plane The Cartesian plane is divided into four quadrants namely Quadrant I, Quadrant II, Quadrant III, and Quadrant IV anticlockwise from OX.
Coordinates of a Point To write the coordinates of a point we need to follow the following rules. ● Thex – coordinate of a point is marked by drawing perpendicular from the y-axis measured a length of the x-axis .It is also called the Abscissa.
They – coordinate of a point is marked by drawing a perpendicular from the x-axis measured a length of the y-axis .It is also called the Ordinate. ● While writing the coordinates of a point in the coordinate plane, the x – coordinate comes first, and then the y – coordinate. We write the coordinates in brackets. In figure, OB = CA = x coordinate (Abscissa), and CO = AB = y coordinate (Ordinate). We write the coordinate as (x, y).
Remarks:
As the origin, O has zero distance from the x-axis and the y-axis so its abscissa and ordinate are zero. Hence the coordinate of the origin is (0, 0). The relationship between the signs of the coordinates of a point and the quadrant of a point in which it lies.
Plotting a Point in the Plane if its Coordinates are Given
Steps to plot the point (2, 3) on the Cartesian plane:
● First of all, we need to draw the Cartesian plane by drawing the coordinate axes with 1 unit = 1 cm.
● To mark the x coordinates, starting from 0 moves towards the positive x-axis and counts to 2.
● To mark the y coordinate, starting from 2 moves upwards in the positive direction and counts to 3.
● Now this point is the coordinate (2, 3). Likewise, we can plot all the other points, like (-3, 1) and (-1.5,-2.5) in the figure.
Question: Are the coordinates (x, y) = (y, x)? Let x = (-4) and y = (-2) So (x, y) = (- 4, – 2) (y, x) = (- 2, – 4)
Let’s mark these coordinates on the Cartesian plane. You can see that the positions of both the points are different in the Cartesian plane. So, If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x), if x = y.
Example: Plot the points (6, 4), (- 6,- 4), (- 6, 4) and (6,- 4) on the Cartesian plane.
Solution: Since both numbers 6, 4 are positive the point (6, 4) lies in the first quadrant. For x coordinate, we will move towards the right and count to 6. Then from that point go upward and count to 4. Mark that point as the coordinate (6, 4). Similarly, we can plot all the other three points.

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POLYNOMIALS | Study

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Revision Notes on Polynomials

Polynomial

A polynomial is an algebraic expression that includes constants, variables, and exponents. It is the expression in which the variables have only positive integral powers.

Example

1. 4x³ + 3x² + x +3 is a polynomial in variable x.

2. 4x² + 3x^-1 – 4 is not a polynomial as it has negative power.

3. 3x^3/2+ 2x – 3 is not a polynomial. Polynomials

Polynomials are denoted by p (x), q (x), etc.
In the above polynomials, 2x², 3y, and 2 are the terms of the polynomial.
2 and 3 are the coefficients of x² and y, respectively.
x and y are the variables.
2 is the constant term, which has no variable.

Polynomials in One Variable

If there is only one variable in the expression, then this is called the polynomial in one variable.

Example

x³ + x – 4 is polynomial in variable x and is denoted by p(x).
r² + 2 is polynomial in variable r and is denoted by p(r).

Types of polynomials on the basis of the number of terms

Types of polynomials on the basis of the number of degrees

The highest value of the power of the variable in the polynomial is the degree of the polynomial.

Zeros of a Polynomial

If p(x) is a polynomial, then the number ‘a’ will be the zero of the polynomial with p(a) = 0. We can find the zero of the polynomial by equating it to zero.

Example: 1

The given polynomial is p(x) = x – 4

To find the zero of the polynomial, we will equate it to zero.

x – 4 = 0

x = 4

p(4) = x – 4 = 4 – 4 = 0

This shows that if we place 4 in place of x, we get the value of the polynomial as zero. So 4 is the zero of this polynomial. And also, we are getting the value 4 by equating the polynomial with 0.

So 4 is the zero of the polynomial or the root of the polynomial.

The root of the polynomial is basically the x-intercept of the polynomial.

If the polynomial has one root, it will intersect the x-axis at one point only, and if it has two roots, it will intersect at two points, and so on.

Example: 2

Find p (1) for the polynomial p (t) = t² – t + 1

p (1) = (1)² – 1 + 1

= 1 – 1 + 1

= 1

Remainder Theorem

We know the property of division which follows in the basic division, i.e.

Dividend = (Divisor × Quotient) + Remainder

This follows the division of polynomials.

If p(x) and g(x) are two polynomials in which the degrees of p(x) ≥ degree of g(x) and g(x) ≠ 0 are given, then we can get the q(x) and r(x) so that:

P(x) = g(x) q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

It says that p(x) divided by g(x), gives q(x) as a quotient and r(x) as a remainder.

Let’s understand it with an example

Division of a Polynomial with a Monomial

We can see that ‘x’ is common in the above polynomial, so we can write it as

Hence, 3x²+ x + 1 and x the factors of 3x³ + x² + x.

Steps of the Division of a Polynomial with a Non –Zero Polynomial

Divide x² – 3x -10 by 2 + x

Step 1: Write the dividend and divisor in descending order, i.e., in the standard form. x² – 3x -10 and x + 2

Divide the first term of the dividend with the first term of the divisor.

x²/x = x this will be the first term of the quotient.

Step 2: Now multiply the divisor by this term of the quotient and subtract it from the dividend.

Step 3: Now the remainder is our new dividend, so we will repeat the process again by dividing the dividend by the divisor.

Step 4: – (5x/x) = – 5

Step 5:

The remainder is zero.

Hence x² – 3x – 10 = (x + 2)(x – 5) + 0

Dividend = (Divisor × Quotient) + Remainder

The Remainder Theorem says that if p(x) is any polynomial of degree greater than or equal to one and let ‘t’ be any real number and p (x) is divided by the linear polynomial x – t, then the remainder is p(t).

As we know,

P(x) = g(x) q(x) + r(x)

If p(x) is divided by (x-t) then

If x = t

P (t) = (t – t). q (t) + r = 0

To find the remainder or to check the multiple of the polynomial, we can use the remainder theorem.

Example:

What is the remainder if a⁴ + a³ – 2a² + a + 1 is divided by a – 1.

Solution:

P(x) = a⁴ + a³ – 2a² + a + 1

To find the zero of (a – 1), we need to equate it to zero.

a -1 = 0

a = 1

p (1) = (1)⁴ + (1)³ – 2(1)² + (1) + 1

= 1 + 1 – 2 + 1 + 1

= 2

So by using the remainder theorem, we can easily find the remainder after the division of the polynomial.

Factor Theorem

The factor theorem says that if p(y) is a polynomial with degree n≥1 and t is a real number, then

(y – t) is a factor of p(y), if p(t) = 0, and
P (t) = 0 if (y – t) is a factor of p (y).

Example: 1

Check whether g(x) = x – 3 is the factor of p(x) = x³– 4x²+ x + 6 using the factor theorem.

Solution:

According to the factor theorem, if x – 3 is the factor of p(x), then p(3) = 0, as the root of x – 3 is 3.

P (3) = (3)³– 4(3)² + (3) + 6

= 27 – 36 + 3 + 6 = 0

Hence, g (x) is the factor of p (x).

Example: 2

Find the value of k, if x – 1 is a factor of p(x) = kx²– √2x + 1

Solution:

As x -1 is the factor, p(1) = 0

Factorization of Polynomials

Factorization can be done by three methods

1. By taking out the common factor

If we have to factorise x² –x then we can do it by taking x common.

x(x – 1) so that x and x-1 are the factors of x² – x.

2. By grouping

ab + bc + ax + cx = (ab + bc) + (ax + cx)

= b(a + c) + x(a + c)

= (a + c)(b + x)

3. By splitting the middle term

x² + bx + c = x² + (p + q) + pq

= (x + p)(x + q)

This shows that we have to split the middle term in such a way that the sum of the two terms is equal to ‘b’ and the product is equal to ‘c’.

Example: 1

Factorize 6x² + 17x + 5 by splitting the middle term.

Solution:

If we can find two numbers p and q such that p + q = 17 and pq = 6 × 5 = 30, then we can get the factors.

Some of the factors of 30 are 1 and 30, 2 and 15, 3 and 10, 5 and 6, out of which 2 and 15 is the pair which gives p + q = 17.

6x² + 17x + 5 =6 x² + (2 + 15) x + 5

= 6 x² + 2x + 15x + 5

= 2 x (3x + 1) + 5(3x + 1)

= (3x + 1) (2x + 5)

Algebraic Identities

1. (x + y)²= x²+ 2xy + y²

2. (x – y)² = x²– 2xy + y²

3. (x + y) (x – y) = x² – y²

4. (x + a) (x + b) = x² + (a + b)x + ab

5. (x + y + z)²= x²+ y²+ z²+ 2xy + 2yz + 2zx

6. (x + y)³ = x³ + y³ + 3xy(x + y) = x³+ y³ + 3x²y + 3xy²

7. (x – y)³ = x³– y³ – 3xy(x – y) = x³– y³ – 3x²y + 3xy²

8. x³+ y³ = (x + y)(x² – xy + y²)

9. x³ – y³= (x – y)(x² + xy + y²)

10. x³+ y³ + z³– 3xyz = (x + y + z)(x² + y²+ z² – xy – yz – zx) x³ + y³ + z³ = 3xyz if x + y + z = 0

Example: 2

Factorize 8x³ + 27y³ + 36x²y + 54xy²

Solution:

The given expression can be written as

= (2x)³+ (3y)³ + 3(4x²) (3y) + 3(2x) (9y²)

= (2x)³ + (3y)³ + 3(2x)²(3y) + 3(2x)(3y)²

= (2x + 3y)³ (Using Identity VI)

= (2x + 3y) (2x + 3y) (2x + 3y) are the factors.

Example: 3

Factorize 4x² + y² + z² – 4xy – 2yz + 4xz.

Solution:

4x² + y² + z² – 4xy – 2yz + 4xz = (2x)²+ (–y)² + (z)² + 2(2x) (-y)+ 2(–y)(z) + 2(2x)(z)

= [2x + (- y) + z]² (Using Identity V)

= (2x – y + z)² = (2x – y + z) (2x – y + z)

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CIRCLES | Study
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Introduction to Circles
There are many objects in our life which are round in shape. A few examples are the clock, dart board, cartwheel, ring, Vehicle wheel, Coins, etc. (Scroll down to continue …)
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Circles
Any closed shape with all points connected at equidistant from the centre forms a Circle.
Any point which is equidistant from anywhere from its boundary is known as the Centre of the Circle.
Radius is a Latin word which means ‘ray’ but in the circle it is the line segment from the centre of the circle to its edge. So any line starting or ending at the centre of the circle and joining anywhere on the border of the circle is known as the Radius of Circle.
Interior and Exterior of a Circle
In a flat surface, the interior of a circle is the line whose distance from the centre is less than the radius.
The exterior of a circle is the line in the plane whose distance from the centre is larger than the radius.
Terms related to circle
Chord: Any straight line segment that’s both endpoints falls on the boundary of the circle is known as Chord. In Latin, it means ‘bowstring’.
Diameter: Any straight line segment or Chord which passes through the centre of the Circle and its endpoints connects on the boundary of the Circle is known as the Diameter of Circle. So in a circle Diameter is the longest chord possible in a circle.
Arc: Any smooth curve joining two points is known as Arc. So in Circle, we can have two possible Arcs, the bigger one is known as Major Arc and the smaller one is known as Minor Arc.
Circumference: It is the length of the circle if we open and straighten it out to make a line segment.
Segment and Sector of the Circle
A segment of the circle is the region between either of its arcs and a chord. It could be a major or minor segment.
Sector of the circle is the area covered by an arc and two radii joining the centre of the circle. It could be the major or minor sector.
Angle Subtended by a Chord at a Point
If in a circle AB is the chord and is making ∠ACB at any point of the circle then this is the angle subtended by the chord AB at a point C.
Likewise, ∠AOB is the angle subtended by chord AB at point O i.e. at the centre and ∠ADB is also the angle subtended by AB at point D on the circle.
Theorem 1: Any two equal chords of a circle subtend equal angles at the centre.
Here in the circle, the two chords are given and PQ = RS with centre O.
So OP = OS = OQ = OR (all are radii of the circle)
∆POQ ≅ ∆SOR
∠POQ = ∠SOR
This shows that the angles subtended by equal chords to the centre are also equal.
Theorem 2: If the angles made by the chords of a circle at the centre are equal, then the chords must be equal.
This theorem is the reverse of the above Theorem 1.
Perpendicular from the Centre to a Chord
Theorem 3: If we draw a perpendicular from the centre of a circle to any chord then it bisects the chord.
If we draw a perpendicular from the centre to the chord of the circle then it will bisect the chord. And the bisector will make a 90° angle to the chord.
Theorem 4: The line which is drawn from the centre of a circle to bisect a chord must be perpendicular to the chord.
If we draw a line OB from the centre of the circle O to the midpoint of the chord AC i.e. B, then OB is the perpendicular to the chord AB.
If we join OA and OC, then
In ∆OBA and ∆OBC,
AB = BC (B is the midpoint of AC)
OA = OC (Both are the radii of the same circle)
OB = OB (same side)
Hence, ΔOBA ≅ ΔOBC (both are congruent by SSS congruence rule)
⇒ ∠OBA = ∠OBC (respective angles of congruent triangles)
∠OBA + ∠OBC = ∠ABC = 180° [Linear pair]
∠OBC + ∠OBC = 180° [Since ∠OBA = ∠OBC]
2 x ∠OBC = 180°
∠OBC = 90^o
∠OBC = ∠OBA = 90°
∴ OB ⊥ AC
Circle through Three Points
Theorem 5: There is one and only one circle passing through three given non-collinear points.
In this figure, we have three non-collinear points A, B and C. Let us join AB and BC and then make the perpendicular bisector of both so that RS and PQ the perpendicular bisector of AB and BC respectively meet each other at Point O.
Now take the O as centre and OA as the radius to draw the circle which passes through the three points A, B and C.
This circle is known as Circumcircle. Its centre and radius are known as the Circumcenter and Circumradius.
Equal Chords and Their Distances from the Centre
Theorem 6: Two equal chords of a circle are at equal distance from the centre.
AB and CD are the two equal chords in the circle. If we draw the perpendicular bisector of these chords then the line segment from the centre to the chord is the distance of the chord from the centre.
If the chords are of equal size then their distance from the centre will also be equal.
Theorem 7: Chords at equal distance from the centre of a circle are also equal in length. This is the reverse of the above theorem which says that if the distance between the centre and the chords are equal then they must be of equal length.
Angle Subtended by an Arc of a Circle
The angle made by two different equal arcs to the centre of the circle will also be equal.
There are two arcs in the circle AB and CD which are equal in length.
So ∠AOB = ∠COD.
Theorem 8: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at some other point on the remaining part of the circle.
In the above figure ∠POQ = 2∠PRQ.
Theorem 9: Angles from a common chord which are on the same segment of a circle are always equal.
If there are two angles subtended from a chord to any point on the circle which are on the same segment of the circle then they will be equal.
∠a = (1/2) ∠c (By theorem 8)
∠b = (1/2) ∠c
∠a = ∠b
Cyclic Quadrilaterals
If all the vertices of the quadrilateral come in a circle then it is said to be a cyclic quadrilateral.
Theorem 10: Any pair of opposite angles of a cyclic quadrilateral has the sum of 180º.
∠A + ∠B + ∠C + ∠D = 360º (angle sum property of a quadrilateral)
∠A + ∠C = 180°
∠B + ∠D = 180º
Theorem 11: If the pair of opposite angles of a quadrilateral has a sum of 180º, then the quadrilateral will be cyclic.
This is the reverse of the above theorem.
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TRIANGLES | Study
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Similar Geometric Figures:
Two geometric figures which are the same in shape, such that one is simply a copy of the other on a smaller scale or a larger scale, are called similar geometric figures.
Two geometric figures are said to be similar if and only if they have the same shape but not necessarily the same size. Two congruent geometric figures are always similar but converse may or may not be true. (Scroll down to read more …)
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Similar Polygons: Two polygons of the same number of sides are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in proportion or their corresponding sides are in the same ratio.
The same ratio of the corresponding sides is referred to as the representative fraction or the scale factor for the polygons.
Similar Triangles :
Two triangles are said to be similar,
if (i) their corresponding angles are equal and
ii) their corresponding sides are in proportion (are in the same ratio).
Basic Proportionality Theorem (or Thales Theorem) : 1
If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Or If a line is drawn parallel to one side of a triangle, intersecting the other two sides in distinct points, the other two sides are divided in the same ratio i .e.. If in ∆ABC, l∥ BC, intersecting in D and E. then
Converse of Basic Proportionality Theorem :
If a line divides any two sides of a triangle in the sameratio, the line is parallel to the third side i.e.
In ∆ABC, if l intersects AB in D and AC in E, such that:
Criteria for Similarity of Triangles:
Two triangles are said to be similar, if
(i) their corresponding angles are equal and (ii) their corresponding sides are in proportion (or are in the same ratio).
2 (i) AA or AAA Similarity Criterion : If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar. If two angles of one triangle are respectively equal to the two angles of another triangle, then the third angles of the two triangles are necessarily equal, because the sum of three angles of a triangle is always 180 ⁰ .
(ii) SAS Similarity Criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the two triangles are similar. Or If two sides of a triangle are proportional to two corresponding sides of another triangle and the angles included between them are equal, then the triangles are similar.
iii) SSS Similarity Criterion : If in two triangles, sides of one triangle are proportional (or are in the same ratio) to the sides of the other triangle, then the triangles are similar. If ∆ABC~ ∆PQR by any one similarity criterion, then ∠A=∠P, ∠B=∠Q, ∠C=∠R and
i.e., A and P, B and Q, C and R are the corresponding vertices, also AB and PQ. BC and QR. CA and RP are the corresponding sides. 3 Areas of Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. – The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. – The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes. – The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisectors. Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Converse of Pythagoras Theorem : In a triangle, if the 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. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and similar to each other i .e ..
If in ∆ABC, ∠B=90^0 and BD ⊥ AC, then (i) ∆ADB ~ ∆ABC (ii) ∆BDC ~ ∆ABC (iii) ∆ADB ~ ∆BDC

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STATISTICS | Study
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Statistics
Statistics is the study of collection, organization, analysis and interpretation of data.
Data
Data is a distinct piece of information in the form of fact or figures collected or represented for any specific purpose. The word data is derived from the Latin word Datum.
Collection of Data
In general, data is of two types. They are:
Primary Data
Secondary Data
Primary Data
Primary data is the data collected from any firsthand experience for an explicit use or purpose.
Secondary data
Secondary data is the data collected by any third party for a different purpose other than the user.
Presentation of Data
After collecting data it is important to present it meaningfully. There are many ways to present data.
1. Raw Data or Ungrouped Data
a. Raw Data or Ungrouped Data is the collected data without any change in its form.
Example
The marks obtained by 10 students in a Mathematics test are:
55 36 95 73 60 42 25 78 75 62
Range
Range is the difference between the highest and the lowest values of data.
For Above data: Range = 95 – 36 = 59
b. Frequency Distribution Table – Frequency Distribution Table is the data of a large number of items converted into tabular form.
Frequency is the number of times the item comes to the table.
2. Grouped Data
To present the very large number of items in the data we use a grouped distribution table.
Grouped Distribution Table
a. Class Interval – The group used to classify the data is called the class interval i.e. 20 – 30, 30 – 40.
b. Upper Limit – In each class interval, the greatest number is the upper-class limit.
c. Lower Limit – In each class interval, the smallest number is the lower class limit.
d. Class Size – It is the difference between the upper limit and the lower limit i.e. 10.
Class Size = Upper Limit – Lower Limit
e. Class Mark – The midpoint of each class interval is the class mark.
Grouped data could be of two types as below:-
Inclusive or discontinuous Frequency Distribution.
Exclusive or continuous Frequency Distribution
Inclusive or discontinuous Frequency Distribution – If the upper limit of a class is different from the lower limit of its next class then it is said to be an Inclusive or discontinuous Frequency Distribution.
Example
Draw the histogram of the following frequency distribution.
Daily earnings (in Rs) 700 – 749 750 – 799 800 – 849 850 – 899 900 – 949 950 – 999
No. of stores 6 9 2 7 11 5
Exclusive or continuous Frequency Distribution – If the upper limit of a class is the same as the lower limit of its next class then it is said to be exclusive or continuous Frequency Distribution
Example
Draw the histogram of the following frequency distribution.
Daily earnings (in Rs) 700 – 750 750 – 800 800 – 850 850 – 900 900 – 950 950 – 1000
No. of stores 6 9 2 7 11 5
Graphical Representation of Data
Since a picture represents better than a thousand words, The data is presented graphically. Some of the methods of representing the data graphically are:
1. Bar Graph
2. Histogram
3. Frequency Polygon
1. Bar Graph
It is the easiest way to represent the data in the form of rectangular bars so it is called Bar graph.
The thickness of each bar should be the same.
The space between the bars should also be the same.
The height of the bar should be according to the numerical data to be represented.
Example
Represent the average monthly rainfall of Nepal for the first six months in the year 2014.
Month Jan Feb Mar Apr May Jun
Average rainfall 45 65 40 60 75 30
Solution
On the x-axis mark the name of the months.
On the y-axis mark the class interval which we have chosen.
Then mark the average rainfall respective to the name of the month by the vertical bars.
The bars could be of any width but should be the same.
This is the required bar graph.
2. Histogram
It is similar to Bar graph, but it is used in case of a continuous class interval.
The class intervals are to be taken along an x-axis.
The height represents the frequencies of the respective class intervals.
Example
Draw the histogram of the following frequency distribution.
Daily earnings (in Rs) 700 – 750 750 – 800 800 – 850 850 – 900 900 – 950 950 – 1000
No. of stores 6 9 2 7 11 5
Solution:
Mark the daily earnings on the x-axis.
Mark the no. of stores on the y-axis.
As the scale is starting from 700 so we will mark the zigzag to show the break.
Mark the daily earnings through the vertical bars.
3. Frequency Polygon
Procedure to draw the frequency polygon
First, we need to draw a histogram.
Then join the midpoint of the top of the bars to a line segment and the figure so obtained is the required frequency polygon.
The midpoint of the first bar is to be joined with the midpoint of the imaginary interval of the x-axis
The midpoint of the last bar is to be joined with the midpoint of the next interval of the x-axis.
If we need to draw the frequency polygon without drawing the histogram then first we need to calculate the class mark of each interval and these points will make the frequency polygon.
Example
Draw the frequency polygon of a city in which the following weekly observations were made in a study on the cost of living index without histogram.
Step 1: First of all we need to calculate the class mark of each class interval.
Step 2: Take the suitable scale and represent the class marks on the x-axis.
Step 3: Take the suitable scale and represent the frequency distribution on the y-axis.
Step 4: To complete the frequency polygon we will join it with the x-axis before the first class and after the last interval.
Step 5: Now plot the respective points and join to make the frequency polygon.
Measures of Central Tendency
To make all the study of data useful, we need to use measures of central tendencies. Some of the tendencies are
1. Mean
2. Median
3. Mode
1. Mean (Average)
The mean is the average of the number of observations. It is calculated by dividing the sum of the values of the observations by the total number of observations.
It is represented by x bar or.
The meanof n values x1, x2, x3, …… xn is given by
Mean of Grouped Data (Without Class Interval)
If the data is organized in such a way that the frequency is given but there is no class interval then we can calculate the mean by
where, x1, x2, x3,…… xn are the observations
f1, f2, f3, …… fn are the respective frequencies of the given observations.
Example
Here, x1, x2, x3, x4, and x5 are 20, 40, 60, 80,100 respectively.
and f1 , f2 , f3 , f4, f5 are 40, 60, 30, 50, 20 respectively.
2. Median
The median is the middle value of the given number of the observation which divides into exactly two parts.
For median of ungrouped data, we arrange it in ascending order and then calculated as follows
If the number of the observations is odd then the median will be Term
As in the above figure the no. of observations is 7 i.e. odd, so the median will beterm.
= 4^th term.
The fourth term is 44.
If the number of observations is even then the median is the average of n/2 and (n/2) +1 term.
Example
Find the median of the following data.
6, 7, 10, 12, 13, 4, 8, 12
1. First, we need to arrange it in ascending order.
4, 6, 7,8,10,12,12,13
2. The no. of observation is 8. As the no. of observation is even the median is the average of n/2 and (n/2)+1.
3.
4. 4^th term is 8 and the 5^th term is 10.
5. So the median
3. Mode
The mode is the value of the observation which shows the number that occurs frequently in data i.e. the number of observations which has the maximum frequency is known as the Mode.
Example
Find the Mode of the following data:
15, 20, 22, 25, 30, 20,15, 20,12, 20
Solution
Here the number 20 appears the maximum number of times so
Mode = 20.
Remark: The empirical relation between the three measures of central tendency is: 3 Median = Mode + 2 Mean
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Plane figure

The figures which we can be drawn on a flat surface or that lie on a plane are called Plane Figure.

Example – Circle, Square, Rectangle etc.

Solid figures

The 3D shapes which occupy some space are called Solid Figures.

Example – Cube, Cuboid, Sphere etc. (Scroll down to continue …)

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Volume

Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.

Surface Area

The area of all the faces of the solid shape is its total surface area. The unit of surface area is a square unit.

Lateral or Curved Surface Area

The surface area of the solid shape after leaving the top and bottom face of the figure is called the lateral surface of the shape. The unit of lateral surface area is a square unit.

Surface Area and Volume of a Cube

Cube is a solid shape having 6 equal square faces.

Lateral surface area of a cube	4s²
Total surface area of a cube	6s²
The volume of a cube	s³
Diagonal	√3 s, s = edge of the cube = side length of face of cube

Surface Area and Volume of a Cube

Example

What is the capacity of a cubical vessel having each side of 8 cm?

Solution

Given side = 8 cm So, Volume of the cubical vessel = l³ = (8)³ = 256 cm³.

Surface Area and volume of a Cuboid

Cuboid is a solid shape having 6 rectangular faces at a right angle.

Lateral surface area of a cuboid	2h(l + b)
Total surface area of a cuboid	2(lb + bh + lh)
Volume of a cuboid	lbh
Diagonal	l = length, b = breadth, h = height

Surface Area and volume of a Cuboid

Example

What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?

Solution

Given, length = 20 cm, breadth = 8 cm, Height = 30 cm

Total surface area of the cereal box = 2(lb + bh + lh)

= 2(20 × 8 + 8 × 30 + 20 × 30)

= 2(160 + 240 + 600)

= 2(1000) = 2000 cm².

Surface Area and Volume of a Right Circular Cylinder

If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder.

Curved surface area of a Right circular cylinder	2πrh
Total surface area of a Right circular cylinder	2πr² + 2πrh = 2πr(r + h)
The volume of a Right circular cylinder	πr²h
	r = radius, h = height

Surface Area and Volume of a Right Circular Cylinder

Surface Area and Volume of a Hollow Right Circular Cylinder

If a right circular cylinder is hollow from inside then it has different curved surface and volume.

Curved surface area of a Right circular cylinder	2πh (R + r)
Total surface area of a Right circular cylinder	2πh (R + r) + 2π(R² – r²)
	R = outer radius, r = inner radius, h = height

Surface Area and Volume of a Hollow Right Circular Cylinder

Example

Find the Total surface area of a hollow cylinder whose length is 22 cm and the external radius is 7 cm with 1 cm thickness. (π = 22/7)

Solution

Given, h = 22 cm, R = 7 cm, r = 6 cm (thickness of the wall is 1 cm).

Total surface area of a hollow cylinder = 2πh(R + r) + 2π(R²– r²)

= 2(π) (22) (7+6) + 2(π)(7²– 6²)

= 572 π + 26 π = 598 π

= 1878.67 cm²

Surface Area and Volume of a Right Circular Cone

If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.

Curved surface area of a Right Circular Cone	πrl = πr[√(h2 + r2)]
Total surface area of a Right Circular Cone	πr²+ πrl = πr(r + l)
The volume of Right Circular Cone	(1/3) πr²h
	r = radius, h = height, l = slant height

Surface Area and Volume of a Right Circular Cone

Surface Area and Volume of a Sphere

A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.

Curved or Lateral surface area of a Sphere	4πr²
Total surface area of a Sphere	4πr²
Volume of a Sphere	(4/3) πr³
	R = radius

Surface Area and Volume of a Sphere

Surface Area and Volume of a Hemisphere

If we cut the sphere in two parts then is said to be a hemisphere.

Curved or Lateral surface area of a Sphere	2πr²
Total surface area of a Sphere	3πr²
Volume of a Sphere	(2/3) πr³
	r = radius

Surface Area and Volume of a Hemisphere

Example

If we have a metal piece of cone shape with volume 523.33 cm³ and we mould it in a sphere then what will be the surface area of that sphere?

Solution

Given, volume of cone = 523.33 cm³

Volume of cone = Volume of Sphere

Volume of sphere = 100 π cm³

125 = r³

r = 5

Surface area of a sphere = 4πr²

= 314.28 cm².

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Introduction:
Euclid’s Division Lemma/Euclid’s Division Algorithm :
Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.
This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder. (Scroll down to continue …).
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Introduction:
Euclid’s Division Lemma/Euclid’s Division Algorithm:
Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.
This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder.
NOTE:
1. Lemma is a proven statement used for proving another statement.
2. Euclid’s Division Algorithm can be extended for all integers, except zero i.e., b 0.
HCF of two positive integers :
HCF of two positive integers a and b is the largest integer (say d ) that divides both a and b(a>b) and is obtained by the following method :
Step 1. Obtain two integers r and q, such that a=bq+r, 0r<b.
Step 2. If r=0, then b is the required HCF.
Step 3. If r0, then again obtain two integers using Euclid’s Division Lemma and continue till the remainder becomes zero. The divisor when remainder becomes zero, is the required HCF.
The Fundamental Theorem of Arithmetic :
Every composite number can be factorised as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.
Irrational Number :
A number is an irrational if and only if, its decimal representation is non-terminating and non-repeating (non-recurring).
OR
A number which cannot be expressed in the form of pq , q 0 and p, qI, will be an irrational number. The set of irrational numbers is generally denoted by Q.
NOTE:
1. The rational number pq will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2ⁿ5^m, where n, m are some non-negative integers.

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Introduction :
Polynomial :
Any expression of the form a₀xⁿ+a₁x^n-1+a₂x^n-2+….an is called a polynomial of degree n in variable x ; a₀≠0, where n is a non-negative integer and a₀, a₁, a₂, ….., and are real numbers, called the coefficients of the terms of the polynomial. (Scroll down to continue …)
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A polynomial in x can be denoted by the symbols p(x), q(x), f(x), g(x), etc.
Degree Of Polynomial: The highest power of x in p(x) is called the degree of the polynomial p(x).
Linear Polynomial : A polynomial of degree one is called a linear polynomial.
Quadratic Polynomial :
A polynomial of degree two is called a Quadratic Polynomial.
Generally, any quadratic polynomial in x is of the form ax²+bx+c, a ≠ 0 and a, b, c are real numbers.
Cubic Polynomial :
A polynomial of degree three is called a Cubic Polynomial.
Generally, any cubic polynomial in x is of the form ax³+bx²+cx+d, a≠0 and a, b, c, d are real numbers.
Value of a Polynomial :
If we replace x by ‘ -2’ in the polynomial p(x) = 3x³-2x²+x-1
we have p(-2) =3(-2)³-2(-2)²+(-2)-1
= -24-8-2-1 =-35
Thus, on replacing x by ‘ -2 ‘ in the polynomial p(x), we get -35, which is called the value of the polynomial.
Hence, if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of the polynomial p(x) at x=k, and generally, denoted by p(k).
Zeros of a Polynomial :
A real constant, k is said to be a zero of a polynomial p(x) in x, if p(k)=0
For example, the polynomial p(x) = x²+x-12 gives p(3)=3²+3-12=0 and p(-4)=(-4)²+(-4)-12=0.
Thus, 3 and -4 are two zeroes of the polynomial p(x).
A linear polynomial (degree one) has one and only one zero, given by;
Zero of the linear polynomial = -(constant term )coefficient of x
Geometrical Representation of the Zeroes of a Polynomial :
Let us consider a linear polynomial p(x)=3x-6.
We know that, graph of a linear polynomial is a straight line.
Therefore, graph of p(x)=3x-6 is a straight line passing through the points (1,-3),(3,3),(2,0).
Table for p(x)=3x – 6
From the graph of p(x)=3x-6, we observe that it intersects the x-axis at the point (2,0).
Zero of the polynomial [p(x)=3x-6] = -(-6)3 = 63 = 2.
Thus, we conclude that the zero of the polynomial p(x) = 3x – 6 is the x-coordinate of the point where the graph of p(x) = 3x – 6 intersects the x-axis.
Similarly, the zeroes of a quadratic polynomial, p(x) = ax²+bx+c, a≠0, are the x-coordinates of the points where the graph (parabola) of p(x)=ax²+bx+c, a≠0, intersects the x-axis.
Graph of p(x) = ax²+bx+c, a≠0 intersects the x-axis at the most in two points and hence the quadratic polynomial can have at most two distinct real zeros.
A cubic polynomial can have at most three distinct real zeros.
Relation between Zeroes and Coefficients of a Polynomial :
Let the quadratic polynomial be p(x) = ax²+bx+c, a≠0 and having zeroes as α and β, then
Sum of the zeroes = α + β
= -(coefficient of x) /(coefficient of x²) = -b/a
Product of the zeroes = αβ
Let the cubic polynomial be p(x) = ax³+bx²+cx+d, a≠0 and having zeroes as α , β and γ, then Sum of the zeroes = α + β + γ
α + β + γ = -(coefficient of x² )/(coefficient of x³)= -b/a
αβ = (constant term) /(coefficient of x²) = c/a
Sum of the products of zeroes taken two at a time αβ+βγ+γα
αβ+βγ+γα = (coefficient of x) /(coefficient of x³)= c/a
and
Product of the zeroes = αβγ
αβγ = (constant term) /(coefficient of x³)= -d/a
Division Algorithm for Polynomials :
For any two polynomials p(x) and g(x) ; g(x) ≠0, we can find two polynomials q(x) and r(x), such that p(x)=g(x) × q(x)+r(x).
Where r(x)=0 or degree of r(x) is less than the degree of g(x). Here, q(x) is called quotient, r(x) is called remainder, p(x) is called dividend and g(x) is called divisor. This result is known as a division algorithm for polynomials.

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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
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Coordinate Axes:
The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.
The horizontal number line is called x-axis and vertical number line is called y axis.
A point that lies on X Axis is (x,0)
A point that lies on Y Axis is (0,y)
Equation of Y Axis is x = 0
Equation of X Axis is y = 0
Equation of a lne parallel to Y Axis is x = a
Equation of a lne parallel to X Axis is y = a
Equation of a lne perpendicular to X Axis is X = a
Equation of a lne perpendicular to X Axis is X = a
The point of intersection of the two lines is called origin.
is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.
Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.
● In the first quadrant, x is + ve and y is + ve
● In the second quadrant, x is – ve and y is + ve
● In the third quadrant, x is – ve and y is – ve
● In the fourth quadrant, x is + ve and y is -ve.
Distance Formula:
Example:
Example:
Collinearity of three points:
Three points P, Q and R are said to be collinear, if they lie in the same straight line.
i.e., PR = PQ + QR
i.e., PQ = PR + RQ
i.e., QR = QP + PR
If three points are not collinear, they always form a triangle.
Special Polygons:
(i) In Case of Triangle
(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.
(b) an equilateral triangle, if all the three sides are equal.
(c) an isosceles triangle, if any two sides are equal.
(ii) In Case of Quadrilateral
(a) parallelogram, if opposite sides are equal and diagonals are not equal.
(b) rectangle, if opposite sides are equal and diagonals are equal.
(c) square, if all the four sides are equal and diagonals are equal.
(d) rhombus, if all the four sides are equal and diagonals are not equal.
Section Formula (Internal division only)
Midpoint Formula:
Point Dividing Two points in K : 1 Ratio:
Note:
If k is positive, the point divides the given points internally.
If k is Negative, the point divides the given points externally
Coordinates of the centroid of a triangle:
Points of Trisection:
If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.
In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.
Area of a Triangle:
The area of the triangle formed by the points
is calculated by the following expression.
Area of ∆PQR =
Area of Quadrilateral:
Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.
Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS
Condition for collinearity of three points :
Three given points will be collinear, if the area of the triangle formed by these points is zero.
Rule to prove that three given points are collinear:
Step 1. Find the area of the triangle formed by the given points.
Step 2. Show that the area of the triangle formed by the given points is zero.
* The coordinates of the origin are O(0,0)
* The coordinates of any point on x-axis are (x, 0)
i.e., y=0 or ordinate is zero.
* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.

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Introduction To Trigonometry | Study
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Introduction To Trigonometry | Speed Notes
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An angle is positive if its rotation is in the anticlockwise and negative if its rotation is in the clockwise direction.
(Scroll down to continue …)
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Trigonometric Ratios
If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ration of the angle can be determined.
Two angles are said to be complementary, if their sum is 90⁰ and each one of them is called the complement of the other.
sin (90⁰ – θ) = Cos θ
Cos (90⁰– θ)= Sin θ
tan (90⁰– θ) = Cot θ
Cot(90⁰– θ) = tan θ
sec (90⁰– θ)= cosec θ
cosec (90⁰– θ) = sec θ
Trigonometric Identities
An equation with trigonometric ratios of an angle θ, which is true for all values of ‘ θ ‘, for which the given trigonometric ratios are defined, is called an identity.
The three fundamental trigonometric identities are
sin² θ +cos² θ = 1
⇒ sin² θ =1-cos² θ
⇒ sin² θ =(1-cos θ)(1+cos θ)
⇒ (1- cos θ) = (sin² θ) /(1+ cos θ)
⇒ (1+ cos θ) = (sin² θ) /(1- cos θ)
⇒ cos² θ + sin² θ = 1
cos² θ =1- sin² θ
⇒ cos² θ =(1- sin θ)(1+ sin θ)
⇒ (1+ sin θ) = (cos² θ) /(1- sin θ)
⇒ (1- sin θ) = cos² θ /(1+sin θ)
(b) sec² θ = 1 + tan² θ
⇒ sec² θ – tan² θ =1
⇒ (sec θ – tan θ)(sec θ + tan θ) = 1
⇒ (sec θ – tan θ) = 1/ (sec θ + tan θ)
⇒ (sec θ + tan θ) = 1/ (sec θ – tan θ)
⇒ sec² θ – 1 = tan² θ
⇒ (sec θ – 1)( sec θ – 1) = tan² θ
(c) cosec² θ = 1+cot²θ
⇒ cosec² θ – cot² θ = 1
⇒ (cosec θ – cot θ)(cosec θ + cot θ)=1
(cosec θ+ cot θ) =1cosec θ – cot θ
(cosec θ- cot θ) = 1cosec θ + cot θ
⇒ Cosec² θ – 1 = cot² θ
⇒ (Cosec θ – 1)( Cosec θ – 1) = Cot² θ
Supportive Formulae:
(a+b)²=+a²+b²+2ab
(a-b)² = a²+b²-2ab
(a+b)²+(a-b)²= 2 (a²+b²)
(a+b)²– (a-b)²= 4ab
(a-b)²– (a+b)²= – 4ab
(a+b)²= (a-b)²+ 4ab
(a-b)²= (a+b)²– 4ab
(a²-b²)=(a+b)(a-b)
a+b=(a²-b²) /(a-b)
a-b=(a²-b²) /(a-b)
(a+b)²= (a-b)²+ 4ab
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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
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Coordinate Axes:
The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.
The horizontal number line is called x-axis and vertical number line is called y axis.
A point that lies on X Axis is (x,0)
A point that lies on Y Axis is (0,y)
Equation of Y Axis is x = 0
Equation of X Axis is y = 0
Equation of a lne parallel to Y Axis is x = a
Equation of a lne parallel to X Axis is y = a
Equation of a lne perpendicular to X Axis is X = a
Equation of a lne perpendicular to X Axis is X = a
The point of intersection of the two lines is called origin.
is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.
Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.
● In the first quadrant, x is + ve and y is + ve
● In the second quadrant, x is – ve and y is + ve
● In the third quadrant, x is – ve and y is – ve
● In the fourth quadrant, x is + ve and y is -ve.
Distance Formula:
Example:
Example:
Collinearity of three points:
Three points P, Q and R are said to be collinear, if they lie in the same straight line.
i.e., PR = PQ + QR
i.e., PQ = PR + RQ
i.e., QR = QP + PR
If three points are not collinear, they always form a triangle.
Special Polygons:
(i) In Case of Triangle
(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.
(b) an equilateral triangle, if all the three sides are equal.
(c) an isosceles triangle, if any two sides are equal.
(ii) In Case of Quadrilateral
(a) parallelogram, if opposite sides are equal and diagonals are not equal.
(b) rectangle, if opposite sides are equal and diagonals are equal.
(c) square, if all the four sides are equal and diagonals are equal.
(d) rhombus, if all the four sides are equal and diagonals are not equal.
Section Formula (Internal division only)
Midpoint Formula:
Point Dividing Two points in K : 1 Ratio:
Note:
If k is positive, the point divides the given points internally.
If k is Negative, the point divides the given points externally
Coordinates of the centroid of a triangle:
Points of Trisection:
If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.
In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.
Area of a Triangle:
The area of the triangle formed by the points
is calculated by the following expression.
Area of ∆PQR =
Area of Quadrilateral:
Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.
Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS
Condition for collinearity of three points :
Three given points will be collinear, if the area of the triangle formed by these points is zero.
Rule to prove that three given points are collinear:
Step 1. Find the area of the triangle formed by the given points.
Step 2. Show that the area of the triangle formed by the given points is zero.
* The coordinates of the origin are O(0,0)
* The coordinates of any point on x-axis are (x, 0)
i.e., y=0 or ordinate is zero.
* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.

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Some Applications Of Trigonometry| Study
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Some Applications Of Trigonometry | Speed Notes
Notes For Quick Recap
Introduction :
One of the main applications of trigonometry is to find the distance between two or more than two places or to find the height of the object or the angle subtended by any object at a given point without actually measuring the distance or heights or angles.
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Trigonometry is useful to astronomers, navigators, architects and surveyors etc. in solving problems related to heights and distances.
The directions of the objects can be described by measuring :
(i) angle of elevation and (ii) angle of depression
Angles of elevation or angles of depression of the objects are measured by an instrument called The odolite.
The odolite is based on the principles of trigonometry, which is used for measuring angles with a rotating telescope.
In 1856, Sir George Everest first used the giant theodolite, which is now on display in the Museum of the survey of India in Dehradun.
Angle of Elevation:
Let P be the position of the object above the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of elevation. It is called the angle of elevation, because the observer has to elevate (raise) his line of sight from the horizontal OA to see the object P. [ When the eye turns upwards above the horizontal line.]
Angle of Depression:
Let P be the position of the object below the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of depression.
It is called the angle of depression because the observer has to depress (lower) his line of sight from the horizontal OA to see the object P.
[When the eye turns downwards below the horizontal line].

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Speed Notes
Notes For Quick Recap
Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
Scroll Down To Continue …
Study Tools
Audio, Visual & Digital Content
Coordinate Axes:
The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.
The horizontal number line is called x-axis and vertical number line is called y axis.
A point that lies on X Axis is (x,0)
A point that lies on Y Axis is (0,y)
Equation of Y Axis is x = 0
Equation of X Axis is y = 0
Equation of a lne parallel to Y Axis is x = a
Equation of a lne parallel to X Axis is y = a
Equation of a lne perpendicular to X Axis is X = a
Equation of a lne perpendicular to X Axis is X = a
The point of intersection of the two lines is called origin.
is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.
Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.
● In the first quadrant, x is + ve and y is + ve
● In the second quadrant, x is – ve and y is + ve
● In the third quadrant, x is – ve and y is – ve
● In the fourth quadrant, x is + ve and y is -ve.
Distance Formula:
Example:
Example:
Collinearity of three points:
Three points P, Q and R are said to be collinear, if they lie in the same straight line.
i.e., PR = PQ + QR
i.e., PQ = PR + RQ
i.e., QR = QP + PR
If three points are not collinear, they always form a triangle.
Special Polygons:
(i) In Case of Triangle
(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.
(b) an equilateral triangle, if all the three sides are equal.
(c) an isosceles triangle, if any two sides are equal.
(ii) In Case of Quadrilateral
(a) parallelogram, if opposite sides are equal and diagonals are not equal.
(b) rectangle, if opposite sides are equal and diagonals are equal.
(c) square, if all the four sides are equal and diagonals are equal.
(d) rhombus, if all the four sides are equal and diagonals are not equal.
Section Formula (Internal division only)
Midpoint Formula:
Point Dividing Two points in K : 1 Ratio:
Note:
If k is positive, the point divides the given points internally.
If k is Negative, the point divides the given points externally
Coordinates of the centroid of a triangle:
Points of Trisection:
If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.
In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.
Area of a Triangle:
The area of the triangle formed by the points
is calculated by the following expression.
Area of ∆PQR =
Area of Quadrilateral:
Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.
Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS
Condition for collinearity of three points :
Three given points will be collinear, if the area of the triangle formed by these points is zero.
Rule to prove that three given points are collinear:
Step 1. Find the area of the triangle formed by the given points.
Step 2. Show that the area of the triangle formed by the given points is zero.
* The coordinates of the origin are O(0,0)
* The coordinates of any point on x-axis are (x, 0)
i.e., y=0 or ordinate is zero.
* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.

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Circles | Study
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Pre-Requisites
Circle:
Circle is a round shaped figure has no corners or edges.
A circle is the locus of all points in a plane which are at constant distance
(called radius) from a fixed point (called centre). A circle with centre O and radius r is denoted by C (O, r).
Radius:
A line segement that joins the centre and circumference or boundary of the circles is called the radius of the circles.
A line segement that divides the circle into two halves is called he diameter of the circle.
Diameter = 2x radius
Radius = Diameter/2
Chord:
A line segment joining any two points on a circle. The largest chord of a circle is a diameter.
Position of A point With Respect To a Circle:
In a plane a point P can lie either inside, or on the circle or outside the given circle.
Position of A Line With Respect to A Circle
If a circle C(O, r) and a straight line ‘l’ are in the same plane, then only three possibilities are there. These are :
Outside The Circle:
(i) The line ‘l’ does not intersect the circle at all. The line ‘l’ is called a non-intersecting line with respect to the circle.
Inside of the Circle – Secant To A Circle:
The line ‘ l ‘ intersects the circle in two distinct points say A and B. The line which intersects the circle in two distinct points is called a secant line.
Touching The Circle – Tangent To Circle:
A tangent to a circle is a special case of the secant when the two end points of the corresponding chord are coincide.
That is the line ‘ l ‘ touches the circle in only one point. Such a line which touches the circle only in one point is called a tangent line.
Tangent To Circle :
Etimology of Tangent:
The word ‘tangent’ comes from the Latin word ‘tangere’, which means to touch and was introduced by the Danish mathematician Thomas Fincke in 1583.
Tangent is a line that intersects the circle in exactly one point.
A tangent to a circle is the limiting position of a secant when its two points of intersection with the circle coincide.
The common point of the circle and the tangent is called the point of contact.
In other words the point, at which the tangent touches the circle is called
the point of contact.
Number of Tangents From A Point To Circle:
Number of tangents to a circle from a point (say P) depends upon the position of the point P.
(a)
When point ‘P’ lies outside the circle: There are only two lines, which touch the circle in one point only, all the remaining lines either intersect in two points or do not intersect the, circle. Hence, there are only two tangents from point P to the circle.
(b)
When point ‘ P ‘ lies on the circle : There is only one line which touches the circle in one point, all other lines meet the circle in more than one point. Hence, there is one and only one tangent to the circle through the point P lies on the circle.
(c)
When point ‘ P ‘ lies inside the circle: Every line passing through the point P (lies inside the circle) intersect the circle in two points. Hence, there is no tangent through the point P lies inside the circle
There is only one tangent at a point on the circumference of the circle.
Point of contact is the common point of the tangent and the circle.
The tangent at any point of a circle is perpendicular to the radius through the point of
contact.
Theorems :
(i) Tangent-Radius Theorem
The line perpendicular to the tangent and passing through the point of contact, is known as the normal.
Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
The converse of above theorem is also true.
Theorem :
The tangents at any point of a circle is perpendicular to the radius through the point of contact. Or At the point of contact the angle between radius and tangents to a circle is 90^0 .
Theorem :
The length of tangents drawn from an external point to a circle are equal.
Important Results:
If two circles touch internally or externally, the point of contact lies on the straight line through the two centres.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
The length of the tangents drawn from an external point to a circle are equal.
Length of the tangent from a point P’ lies outside the circle is given by
PT =PT’ =
The distances between two parallel tangents drawn to a circle is equal to the diameter of the circle.
Facts:
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
More Points To Remember !
There is no tangent to a circle passing through a point lying inside the circle.
At any point on the circle there can be one and only one tangent.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
There are exactly two tangents to a circle through a point outside the circle.
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.
The lengths of the tangents drawn from an external point to a circle are equal.
The line containing the radius through the point of contact of tangent is called the normal to the circle at the point.
There is no tangent to the circle passing through a point lying inside the cirele.
There are exactly two tangents to a cirele through a point lying outside the circle
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.
The length of tangents drawn from an external point to a circle are equal.
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Areas Related to Circles | Study
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Areas Related To Circles | Speed Notes
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Introduction :
Circumference of a circle Or Perimeter of a circle :
The distance around the circle or the length of a circle is called its circumference or perimeter. (Scroll down to continue …)
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AREAS AND PERI METER RELATED TO CIRCLES
Introduction :
Circumference of a circle Or Perimeter of a circle :
The distance around the circle or the length of a circle is called its circumference or perimeter.
Circumference (perimeter) of a circle = d or 2r, where d is a diameter and r is a radius of the circle and = 227 or 3.14
Area of a circle:
Area of a circle = πr²
Area of a semicircle :
Area of semicircle = 12 πr²
Area of a quadrant :
Area of a quadrant (quarter circle) = r24
Perimeter of a semicircle:
Perimeter of a semicircle or protractor =πr+2r
Area of the ring :
Area of the ring or an annulus
= πR² – πr²
= π(R² – r²)
= π(R- r)(R+r)
Length of the arc AB
= 2πrθ3600 or πrθ1800
Area of a sector:
Area of sector OACBO =πrθ3600
OR
Area of sector OACBO = 12(r×l)
Perimeter of a sector :
Perimeter of sector OACBO
= Length of arc AB+2r
= πrθ1800+2r
Other important formulae :
(i) Distance moved by a wheel in 1 revolution = Circumference of the wheel.
(ii) Number of revolutions in one minute = Distance moved in 1 minuteCircumference
(iii) Angle described by minute hand in 60 minutes =360⁰
(iv) Angle described by hour hand in 12 hours = 360⁰
The midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.
Angle subtended at the circumference by a diameter is always a right angle.
Area of a segment:
(i) Area of minor segment ACBA
= Area of sector OACBO- Area of ∆OAB
= πrθ3600- 12r2sinθ
(ii) Area of major segment BDAB
= Area of the circle – Area of minor segment ACBA
=πr²– Area of minor segment ACBA
If a chord subtends a right angle at the center, then
Area of the corresponding segment = 4-12r2
If a chord subtends an angle of 60⁰ at the center, then
Area of the corresponding segment = 6-32r2
If a chord subtends an angle of 120⁰ at the center, then Area of the corresponding segment = 3-34r2

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Basics Of Geometry
Lines And Angles
Triangles
Basics Of Quadrilaterals
Quadrilaterals

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Cuboid
What is a cuboid?
A cuboid is a three-dimensional geometric shape that resembles a rectangular box or a rectangular prism. A cuboid has 3 Pairs of opposite, congruent and parallel rectangular faces, 12 edges, and 8 vertices.
Note 1: All squares are rectangles.
Note 2: Cuboid may have one, or Three equal pairs of squares. (Square is a special type of Rectangle.
Note 3: If All three pairs of faces of a cuboid are squares then it it becomes a Cube.
Note 4: A cube is a special case of cuboid.
Parts And Their Alignment Of A Cuboid
Faces
The flat surfaces of a cuboid are known as its faces.
A cuboid has six faces, and each face is a rectangle.
These faces are arranged such that three pairs of opposite faces are parallel to each other.
The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.
Note 1: All squares are rectangles.
Note 2: Rectangle may have one or two pairs of squares.
Note 3: If All three pairs of faces of a rectangle are squares then it it becomes a Cube.
Note 4: A cube is a special case of cuboid.
Edges
An edge is a line segment where the two surfaces of a cuboid meet.
There are 12 edges in a cuboid, where three edges meet at each vertex.
All edges form right angles with the adjacent edges and faces.
Vertices
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cuboid has eight vertices.
Diagonals
Diagonal of a cuboid is a line segment that joins two opposite vertices.
The cuboid has four space diagonals.
Length of the diagonal of cuboid = √(length² + breadth²+ height²) units.
Symmetry
Cuboids exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
Features of a Cuboid
It is a three-dimensional, Rectangular figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are rectangles.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cuboid are right angles.

Mensuration of Cuboid
Surface Area of a Cuboid
The total surface area of a cuboid is defined as the area of its surface (Appearing face).
The Lateral Surface Area of a Cube.
Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)
Derivation of Total Surface Area of a Cuboid
Since the total surface area of a cuboid (TSA) is the area of its surface.
Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]
Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]
TSA = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height) + (Length ✕ Breadth) + (Length ✕ Breadth)
TSA = 2(Length ✕ Height) + 2(Breadth ✕ Height) + 2(Length ✕ Breadth)
TSA = 2[(Length ✕ Height) + (Breadth ✕ Height) + (Length ✕ Breadth)]
The Volume of a Cube
Volume

The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.
The volume of a cuboid can be calculated using the formula, V = lbh, where,
l = length, b = breadth or width, h = height
This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.
The volume is calculated by multiplying the object’s length, breadth, and height.
Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height
Cuboids in Our Daily Life
Cuboids are commonly used in everyday objects, such as boxes, books, and building blocks.
They are used in architectural and engineering designs for modeling rooms, buildings, and structures.
In mathematics and geometry, cuboids serve as fundamental examples for teaching and understanding concepts related to three-dimensional shapes.
Similar Shapes:
A cube is a special type of cuboid where all sides are equal in length, making it a regular hexahedron.
Real-world Examples:
A shoebox is an example of a cuboid.
Most refrigerators, ovens, and TV screens have cuboidal shapes.
Buildings and houses often have cuboidal rooms.
Fun Fact:
Cuboids are among the simplest and most familiar three-dimensional shapes, making them a fundamental concept in geometry.
Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.
What is a cube?
A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.
Parts And Their Alignment Of In A Cube
Faces
The flat surfaces of a cube are known as its faces.
A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.
The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.
All the edges have the same length.
A cube also has 8 vertices and 12 edges.
Edges
An edge is a line segment where the two surfaces of a cube meet.
There are twelve edges in a cube, where three edges meet at each vertex.
All edges have equal length and form right angles with the adjacent edges and faces.
Vertices
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cube has eight vertices.
Diagonals
The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.
Symmetry
Cubes exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
Features of a Cube
It is a three-dimensional, square-shaped figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are squares with equal area.
All sides have the same length.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cube are right angles.
Mensuration of Cube

Surface Area of a Cube
The total surface area of a cube is defined as the area of its outer surface.
Derivation of Total Surface Area of a Cube
Since the total surface area of a cube is the area of its outer surface.
total surface area of a cube = 6 ✕ area of one face.
We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.
Let the length of each edge is “s”.
Area of one square face = length of edge ✕ length of edge
Area of one square face == s ✕ s = s²
Therefore, the total surface area of the cube = 6s²
The total surface area of the cube will be equal to the sum of all six faces of the cube.
The Lateral Surface Area of a Cube.

Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = 4a²
The Volume of a Cube
Volume

The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.
The volume of a cube can be calculated using the formula V = s³, where “s” represents the length of one side of the cube.
This formula shows that the volume of a cube is directly proportional to the cube of its side length.
The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.
Hence the volume of the cube is s ✕ s ✕ s = s³
Cubes in Our Daily Life
We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.
Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.
Also, Cubes have many applications in mathematics, engineering, architecture and art etc.
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Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.
In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.
Let’s start with the basic definition of a right circular cylinder.
A cylinder is a solid shape that has two parallel circular bases of equal size and shape.
The curved surface that connects the two bases is called the lateral surface.
The axis of the cylinder is a line passing through the center of both bases.
Types Of Cylinders
(i). Solid Cylinder
(ii). Hollow Cylinder
Area Of A Solid Cylinder
Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.
Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)
Or
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)
Therefore the formula to calculate the surface area of a cylinder is expressed as the following:
$$SA = 2\pi r h + 2\pi r^2$$
where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.
Here, Curved Surface Area, $$CSA = 2\pi r h$$ and
Cicular Base Area = Top Surface Area = $$\pi r^2$$
Volume Of A Solid Cylinder
The formula to calculate the volume of a cylinder is given by:
$$V = \pi r^2 h$$
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:
$$V = \pi (4)^2 (10) = 160\pi$$
Therefore, the volume of the cylinder is 160π cubic cm.
To calculate its surface area, we can use the formula:
$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$
Therefore, the surface area of the cylinder is 120π square cm.
In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.
Hollow Cylinder
A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.
Volume Of A Hollow Cylinder
The volume of a hollow cylinder can be calculated using the formula V = πh(R²-r²), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Surface Area Of A Hollow Cylinder
The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.
In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.
Cone
A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.
Surface Area of a Cone
The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:
$$A = \pi r (r + l)$$
Where:
( A ) is the surface area of the cone
$$\pi = 3.14159$$
( r ) is the radius of the base of the cone
( l ) is the slant height of the cone
Volume of a Cone
The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
( V ) is the volume of the cone
( \pi ) is a mathematical constant approximately equal to 3.14159
( r ) is the radius of the base of the cone
( h ) is the height of the cone
Example Equations
Here are a few example equations related to cones:
Equation for calculating the slant height of a cone: $$l = \sqrt{r^2 + h^2}$$
Equation for calculating the radius of a cone given its slant height and height: $$r = \sqrt{l^2 – h^2}$$
Equation for calculating the height of a cone given its volume and radius: $$h = \frac{3V}{\pi r^2}$$
Sphere
Solid Sphere
Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.
Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.
Volume of Solid Sphere The formula to calculate the volume ($V_s$) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: $V_s$ = Volume of the solid sphere, $\pi$ ($\pi$) ≈ 3.14159, $r$ = Radius of the sphere.
Surface Area of Solid Sphere The formula to calculate the surface area ($A_s$) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]
Where: $A_s$ = Surface area of the solid sphere, $\pi$ ($\pi$) ≈ 3.14159, $r$ = Radius of the sphere
Mass of Solid Sphere The mass ($m_s$) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: $m_s$ = Mass of the solid sphere, $\text{Density}$ = Density of the material making up the sphere (usually in $kg/m^3$), $V_s$ = Volume of the solid sphere (calculated using the previous formula)
Hollow Sphere
Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.
Characteristics – The hollow sphere has a well-defined outer radius ($R$), inner radius ($r$), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.
Volume of Hollow Sphere The formula to calculate the volume ($V_h$) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – $V_h$ = Volume of the hollow sphere, $\pi$ ($\pi$) ≈ 3.14159, $R$ = Outer radius of the sphere, $r$ = Inner radius of the sphere
Surface Area of Hollow Sphere The formula to calculate the surface area ($A_h$) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: $A_h$ = Surface area of the hollow sphere, $\pi$ ($\pi$) ≈ 3.14159, $R$ = Outer radius of the sphere, $r$ = Inner radius of the sphere
Mass of Hollow Sphere The mass ($m_h$) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: $m_h$ = Mass of the hollow sphere, $\text{Density}$ = Density of the material making up the sphere (usually in $kg/m^3$), $V_h$ = Volume of the hollow sphere (calculated using the previous formula)
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Probability:
Probability is a measure of uncertainity (an event may happen or may not happen).
Probability is classified into the following two types.
(i) Empirical probability
(ii) theoretical probability.
Empirical Probability Or Experimental Probability
Empirical Probability is the probability of events on the basis of the results obtained through actual experiments.
Learn More
Theoretical probability Or Classical Probability:
Theoretical probability Or Classical Probability is the probability of events based on the results obtained from theoretical approach.
Theoretical Approach
Here, we try to predict the outcomes without performing an actual experiment.
We assume that the outcomes of an experiment are equally likely.
We find that the experimental probability of an event approaches its theoretical probability if the number of trials of an experiment is very large.
Key Terms
Experiment:
Experiment is An activity that causes some well defined outcomes.
Random Experiment
Random experiment is an experiment that
may not give the same result on repitition
under identical conditions,
Trial: Each repition of the activity is called A Trial.
Outcome
Each possible result of a random experiment or a Trial is called its outcome.
Sample Space
The collection of all possible outcomes in a random experiment is called Sample space.
Or
The total number of possible outcomes of a random experiment.
As the number of trials in an experiment increases we may expect the empirical and theoretical probabilities to be nearly the same.
Event:
Any subset of smaple space is called an Event.
Types of Events
Compound Event An event connected to a random experiment is a compound event if it is obtained by combining two or more elementary events connected to the random experiment.
Occurrence of an event An event corresponding to a random experiment is said to occur if any one of the elementary events corresponding to the event is the outcome.
Impossible Events The event which never occurs is an impossible event.
So the probability ofan impossible eventis always zero.
Sure Event The event which certainly occurs is a sure event.In general, it is truethat for anevent E, ( ̅) ( ) Here theevent E is representing “not E”. This is called the compound ofthe event ‘E’.So ‘E’ and E are complementary events.
Theoretical Probability of An Event:
If there are n events associated with a random experiment and m of them are favorable to an event E, then the probability of the event E is denoted by P(E) and is calculated using the follwoing formula.
i.e.,
Probability of an event E lies between 0 and 1.
If P (E) =1, then event E is called a certain event or sure event.
If P (E) =0, then E is called an impossible event.
Number of non-occurrence of event E = n – m
Where the event
representing non-occurrence of event E, is called the complement of the event E.
Hence, E and
are complementary events.
Sum of the probabilities of all the elementary events of an experiment is
i.e.,
P (E1) + P (E2) + P (E3) … + P (En) = 1
Playing Cards:
A pack of cards consists of four suits. They are
Each suit consists of 13 cards, nine cards numbered 2, 3, 4 ….10, an Ace (B) a Jack (J), a Queen (Q), and a King (K).
Spades and Clubs are black in colour.
Hearts and Diamonds are red in colour.
So there are 26 black cards and 26 red cards.
King, queen and jack are called face cards.
There are totally 12 (4 x3) face cardsin a packof 52 cards.
I.e. in eachsuit we have3 face cards.
Coins:
A coin has two sides namely head and tail.
In the experiment of tossing a coin for once, there are 2 possible outcomes. They are 1 head, 1 tail.
P (Head) = 1/2 = P (Tail)
Dice:
A die is a well-balanced cubewith six facesnumbered from 1 to 6.
Dice is the pluralform. There are six equally likely outcomes 1, 2,3, 4, 5,6 in a single throw. Geometric Probability:
If the total number of outcomes of a trial in a random experiment is infinite, then the above definition is not sufficient to find the probability of an event.
In such cases, the definition of probability is modified and probability so obtained is called Geometric Probability.
The geometric probability p of an event is given by
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