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Mind Map Overal Idea Content Speed Notes Quick Coverage 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,… readmore
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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.
The 3D shapes which occupy some space are called Solid Figures.
Example – Cube, Cuboid, Sphere etc. (Scroll down the till the end of the page)
Volume
Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.
The area of all the faces of the solid shape is its total surface area. The unit of surface area is a square unit.
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.
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 |
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.
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 |
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
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.
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 |
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 |
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)
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
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 |
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 |
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 |
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?
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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Mind Map Overal Idea Content Speed Notes Quick Coverage Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Linear Equations in One Variable… readmore
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Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down till end of the page)
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Linear Equations in One Variable
An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.
A linear equation may have for its solution any rational number.
An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.
Just as numbers, variables can, also, be transposed from one side of the equation to the other.
Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.
The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved
using linear equations.
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Mind Map Overal Idea Content Speed Notes Quick Coverage Any expression of the form a0xn+a1xn-1+a2xn-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 …)… readmore
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Any expression of the form a0xn+a1xn-1+a2xn-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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Mind Map Overal Idea Content Speed Notes Quick Coverage ntegers are a bigger collection of numbers which is formed by whole numbers and their negatives. You have studied inthe earlier class, about the representation of integers onthe number lineand their addition and subtraction. (Scroll down till end of the page) Study Tools Audio, Visual &… readmore
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ntegers are a bigger collection of numbers which is formed by whole numbers and their negatives. You have studied inthe earlier class, about the representation of integers onthe number lineand their addition and subtraction. (Scroll down till end of the page)
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We now study theproperties satisfied by addition andsubtraction.
(a) Integers are closed for addition and subtraction both. That is, a + b and a – b are again integers, where a andb are anyintegers.
(b) Addition is commutative forintegers, i.e., a + b = b + a for allintegers a andb.
(c) Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.
(d) Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a. We studied, how integers could be multiplied, andfound that product of a positive and a negative integer is a negative integer, whereas the product of two negative integers isa positive integer. For example, –2 × 7 = –14 and –3 × – 8 =24.
Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative. Integers showsome properties under multiplication.
(a) Integers are closed under multiplication. Thatis, a × b isan integer forany two integers a and b.
(b) Multiplication is commutative for integers. Thatis, a × b = b × a forany integers a and b.
(c) The integer 1 is theidentity under multiplication, i.e., 1 × a = a × 1 = a forany integer a.
(d) Multiplication is associative for integers, i.e.,(a × b) × c = a × (b × c) for anythree integers a,b and c.
Under addition and multiplication, integers show a property called distributive property.
That is, a× (b +c) = a × b+ a × c forany three integers a, b andc.
The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier. We alsolearn how to divide integers. We found that,
(a) When a positive integer is divided by a negative integer, the quotient obtained is a negative integer and vice-versa. (b) Division of a negative integer by another negative integer gives a positive integer as quotient. For any integer a,we have
1) The numbers. . . , —4,—3, —1, 0, 1, 2,3, 4, etc.are integers.
2) 1, 2, 3, 4, 5. . . . are positive integers and —1,-2, —3,.. are negative integers.
3) 0 isan integer which is neither positive nornegative.
4). On an integer number line, all numbers to the right of 0 arepositive integers andall numbers tothe left of0 are negative integers.
5) 0 is less than everypositive integer and greater than everynegative integer.
6) Every positive integer is greater than every negative integer.
7) Two integers thatare at thesame distance from 0, but onopposite sides of it are called opposite numbers.
8. The greater the number, the lesser is its opposite.
9. The sumof an integer and its opposite is zero.
10. The absolute valueof an integer is the numerical value of theinteger without regard to its sign.
The absolute value of an integer a isdenoted by |a| and is given by a,if a is positive or 0 a = -a,if a is negative
11. The sum oftwo integers of the same sign is an integer of the same sign whose absolute value is equal to the sum of the absolute values of the given integers.
12. The sum of two integers of opposite signs is an integer whose absolute value is the difference of the absolute values of addend and whose sign isthe sign ofthe addend having greater absolute value.
13. To subtract an integer b from another integer a, we change the sign ofb and addit to a. Thus, a − b = a + (−b)
14. All properties of operations onwhole numbers aresatisfied by theseoperations on integers.
15. If aand b are two integers, then(a − b) is alsoan integer.
16. −a and aare negative oradditive inverses of each other.
17. To find theproduct of twointegers, we multiply theirabsolute values andgive the result a plus signif both thenumbers have the same sign or a minussign otherwise.
18. To find thequotient of oneinteger divided by another non-zero integer, we divide their absolute values and give the result a plus sign if both the numbers have the same sign or a minus signotherwise.
19. All the properties applicable to wholenumbers are applicable to integers in addition, the subtraction operation has the closure property.
20. Any integer whenmultiplied or divided by 1 gives itself and whenmultiplied or divided by-1 gives its opposite.
21. When expression hasdifferent types ofoperations, some operations haveto be performed before the others. That is, each operation has its own precedence. The order in which operations are performed is division, multiplication, addition and finally subtraction (DMAS).
22. Brackets are usedin an expression when we wanta set of operations to be performed before the others.
23. While simplifying anexpression containing brackets, the operations within the innermost set of brackets are performed first and then those brackets are removed followed by the ones immediately after them tillall the brackets are removed.
24. While simplifying arithmetic expressions involving various brackets and operations, we use BODMAS rule.
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Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Content … Key Terms Topic Terminology Term Important Tables Table: . Assessments Test Your Learning readmore
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