Credit
Saturday, September 26, 2020
Polynomials
Quick Revision Polynomials
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
2. A quadratic polynomial in $x$ with real coefficients is of the form $a x^{2}+b x+c,$ where $a, b$, $c$ are real numbers with $a \neq 0$
3. The zeroes of a polynomial $p(x)$ are precisely the $x$ -coordinates of the points, where the graph of $y=p(x)$ intersects the $x$ -axis.
4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.
5. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $a x^{2}+b x+c,$ then
$\alpha+\beta=-\frac{b}{a}, \quad \alpha \beta=\frac{c}{a}$
6. If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $a x^{3}+b x^{2}+c x+d=0,$ then
$\alpha+\beta+\gamma=\frac{-b}{a} \alpha \beta+\beta \gamma+\gamma \alpha=\frac{c}{a}$
and $\alpha \beta \gamma=\frac{-d}{a}$
7. The division algorithm states that given any polynomial $p(x)$ and any non-zero polynomial $g(x)$, there are polynomials $q(x)$ and $r(x)$ such that
p(x)=g(x) q(x)+r(x)
where $\quad r(x)=0$ or degree $r(x)<$ degree $g(x)$
Real Number Theorems
Quick revision for theorems
Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b,
there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
Theorem 1.2 (Fundamental Theorem of Arithmetic) : Every composite number
can be expressed ( factorised) as a product of primes, and this factorisation is
unique, apart from the order in which the prime factors occur.
Theorem 1.3: Let $p$ be a prime number. If $p$ divides $a^{2},$ then $p$ divides a, where $a$ is a positive integer.
Theorem 1.4: $\sqrt{2}$ is irrational.
Theorem 1.5: Let $x$ be a rational number whose decimal expansion terminates.
Then $x$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are coprime, and the
prime factorisation of $q$ is of the form $2^{\circ} 5^{m},$ where $n, m$ are non-negative integers.
Theorem 1.6: Let $x=\frac{p}{q}$ be a rational mamber, such that the prime factorisation of $q$ is of the form $2^{n} 5^{\text {" }},$ where $n, m$ are non-negative integers. Then $x$ has a
decimal expansion which terminates.
Theorem 1.7: Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of $q$ is not of the form $2^{n} 5$ ", where $n, m$ are non-negative integers. Then, $x$ has a decimal expansion which is non-terminating repeating (recurring).
So we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.
Grade 10 Revision Questions Set I
Fast track revision
Q1 Find the largest four-digit number which when divided by 4,7 and 13 leaves a remainder 3 in each case.
Q2 Show that $\frac{2+3 \sqrt{2}}{7}$ is not a rational number, given that $\sqrt{2}$ is an irrational number.
Q3 Write the number of real roots of the equation $x^{2}+3|x|+2=0$
Q4 Solve for $\mathrm{x}$ $\frac{x+1}{x-1}+\frac{x-2}{x+2}=4-\frac{2 x+3}{x-2} ; x \neq 1,-2,2$
Q5 The longer side of a rectangular hall is $24 \mathrm{m}$ and the length of its diagonal is $26 \mathrm{m}$. Find the area of the hall.
Q6 Prove the trigonometric identity:
$\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta$
Q7 If $A=B=60^{\circ},$ verify that $\cos (A-B)=\cos A \cos B+\sin A \sin B$
Q8 From a point $\mathrm{P}$ on the ground the angle of elevation of the top of a $10 \mathrm{m}$ tall building is $30^{\circ} . \mathrm{A}$ flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from $\mathrm{P}$ is $45^{\circ} .$ Find the length of the flagstaff and distance of building from point $\mathrm{P}$. $\sqrt{3}=1.732]$
Q9 A pole casts a shadow of length $2 \sqrt{3} \mathrm{m}$ on the ground, when the Sun's elevation is $60^{\circ} .$ Find the height of the pole.
Q10 Find the values of $\mathrm{k}$ for which the given equation has real and equal roots:
$x^{2}-2 x(1+3 k)+7(3+2 k)=0$
Q11 If $\tan \theta=\frac{12}{13},$ evaluate $\frac{2 \sin \theta \cos \theta}{\cos ^{2} \theta-\sin ^{2} \theta}$
Q12 A solid metal cone with radius of base $12 \mathrm{cm}$ and height $24 \mathrm{cm}$ is melted to form solid spherical balls of diameter $6 \mathrm{cm}$ each. Find the number of balls formed.
Sunday, April 26, 2020
Polynomials Questions
Q1 Number of zeroes in $X^3-1$. are ...... Give reason to support your answer.
Sol Number of zeroes in $X^3 -1$ are 3. As the equation is cubic or has degree three.
Q2 The sum and product of the zeroes of the quadratic polynomial are 2 and 15 then what will be the polynomial is
Sol
Given $\alpha + \beta=2$
$\alpha\beta=15$
Quadratic polynomial is given by
$x^2-(\alpha+\beta)x + \alpha\beta$
Hence the equation is
$x^2-2x+15$
Sol Number of zeroes in $X^3 -1$ are 3. As the equation is cubic or has degree three.
Q2 The sum and product of the zeroes of the quadratic polynomial are 2 and 15 then what will be the polynomial is
Sol
Given $\alpha + \beta=2$
$\alpha\beta=15$
Quadratic polynomial is given by
$x^2-(\alpha+\beta)x + \alpha\beta$
Hence the equation is
$x^2-2x+15$
Monday, March 23, 2020
POEM
Trust
Trust is like a glass
Once broken would never last
No matter how much you try to repair,
So it's better to be beware
Never-ever try to break
it because it's must
it because it's must
And once broken every relation will then turn into dust
Trust is like the glass
Once broken would never last
Once you are trustworthy,
you are loved, valued, respected.
you are loved, valued, respected.
And once the trust is broken you are left dissociated
Trust is important,
It must be supported,
But never try to break it,
Just create it and keep it
because..
Trust is like a glass
Once broken would never last....
Written by Rhea
unedited intentionally :)
Sunday, March 22, 2020
Number System
Infate
Academy
Number
System
Natural Numbers (N)
Counting numbers are known as natural
numbers N= {1,2,3,4,5…………………}
Whole Numbers (W)
Whole Numbers together with 0 form whole numbers
W = {0,1,2,3,4,5……….}
Integers (I or Z)
All natural numbers, 0 and the negatives of
natural numbers form the collection of integers
Z= {-$\infty$ ……….-5,-4,-3,-2,-1,0,1,2,3,4,5,…………….$\infty$}
Rational Numbers (Q)
The number of the form $\frac{P}{Q}$ where P and Q are integers and Q$\ne$0 is in the lowest form i.e P and Q have no common factor, is called a rational
numbers.
All integers and fractions are a rational
number. The set of rational numbers is denoted by Q. A rational numbers is said
to be positive if its numerator and denominator are either both positive or
both negative
Irrational Numbers
$\frac{5}{2}$ = 2.5
is rational number
Ï€ =3.14159… this number cannot be
expressed in the form of ratio. So any number that cannot be represented in the
form of ratio is called irrational number
Real numbers include whole numbers,
rational numbers and irrational numbers. Real numbers can also be positive,
negative and zero.
$\sqrt {-2}$ ie of any negative number is called imaginary
numbers and such numbers are not real numbers
Also infinity is not a real number
Real Numbers
The collection of rational and irrational
numbers are called real numbers and is denoted by R
Tuesday, February 25, 2020
Grade 7 Science worksheet
Science worksheet
Science formulae of acid, bases and salts
Note:This is a mere support extended towards students. It does not mean to replace classroom solutions.
| S# | Chemical Name | Formula | Uses | Special Point |
| 1 | Acetic Acid | CH3COOH | Used in vinegar | Ethanoic acid |
| 2 | Hydrochloric acid | HCL | Clean washrooms | Muriatic acid |
| 3 | Nitric acid | HNO3 | Used in making fertilizers and plastic Purification of gold |
Spirit of niter or aqua fortis Highly corrosive |
| 4 | Sulphuric acid | H2SO4 | Used in chemical industries | Oil of vitriol |
| 5 | Carbonic acid | H2CO3 | Used in cold drinks | Carbonated water, acid of air, carbon di oxide solution |
| 6 | Phosphoric acid | H3PO4 | Toothpaste, food industry | Weak acid also called orthophosphoric acid |
| 7 | Calcium hydroxide | Ca(OH)2 | White wash walls | Slaked lime |
| 8 | Magnesium hydroxide | Mg(OH)2 | Antacid, also used as food additive | Milk of magnesia |
| 9 | Sodium hydroxide | NaOH | Chemical pulping, tissue digestion, used in soaps | Caustic soda |
| 10 | Potassium hydroxide | KOH | Used as an electrolyte Used in making soft soaps. Used as ph control agent and food stabilizer Used in manicure as chemical cuticle removing agent |
Caustic potash |
| 11 | Ammonium hydroxide | NH4OH | Used in window cleaners Used in making of Nylon |
Aqua ammonia or ammonia water |
| 12 | Sodium bicarbonate | NaHCO3 | Used in baking cakes, biscuits etc | Baking soda |
| 13 | Sodium Chloride | Nacl | Food preservation, taste agent in food, fire extinguisher, cleaning agent | Common salt |
| 14 | Ammonium chloride | NH4Cl | Used in metal work, medicine, food etc | Nushadir salt, Sal ammoniac |
| 15 | Magnesium sulphate | MgSO4 | Food preparation, medicine, agriculture etc | Epsom salt |
| 16 | Copper sulphate | CuSO4 | Used in pesticides,soil sterilization, disinfecting agent. It is toxic. | Cuprous sulfate or Dicopper sulfate |
| 17 | Calcium Carbonate | CaCO3 | Used to make cement, health and dietary applications | Limestone, marble, chalk |
| 18 | Potassium nitrate | KNO3 | Use in production of nitric acid, meat processing, pharmacological use, fertilizer etc | Salt peter, nitrate of potash |
| 19 | Potassium chloride | KCl | Used in fertilizers, medical uses and culinary | Sylvite, muriate of potash |
| 20 | Iron oxide | Fe2O3.10H2O | Used in iron ores, pigments, catalysts and in thermite. Used as pigment in color concrete, paints and durables |
Rust |
| 21 | Sodium carbonate | Na2CO3 | Used in glass manufacturing, washing powders. It is also used as water softner | Soda ash, washing soda |
Thursday, February 13, 2020
Data Structures
Grade BCA
Data structures
Question Bank
- Write advantages of Hashing.
- Explain the difference between B- Tree and B+ Tree.
- Explain traversal of graph.
- What is m-way Search tree?
- What is Data structure?
- List out the areas in which data structures are applied extensively?
- Explain variable length record.
- What is topological sorting?
- What is the basic difference between direct and access and index sequential files?
- Describe two applications of general trees?
- What is B+ tree?
- What are the data structures used to perform recursion?
- What is the difference between graph and tree?
- Discuss advantages of graphs.
- Discuss major features of B-trees.
- Explain AVL Tree.
- Describe the complexity of heap sort.
- Discuss the complexity of Binary search.
- Draw all binary search trees with 5 nodes A,B,C,D and E that
satisfy the balance requirement of AVL trees.
- Explain Sorting.
- What is spanning tree?
- Explain File.
- Write the use and advantages of files.
- What is file organization?
- Discuss few important applications of search trees.
- What is binary search tree? Explain in detail the insertion and deletion in a binary search tree.
- What are threaded binary trees? What are the applications?
OR
What is meant by threading? Discuss its relevance.
OR
What are threads? What are its advantages and
disadvantages? How threads are implemented in trees?
Explain through an example.
- Explain the role of threads in binary search tree with suitable Example.
- What do you mean by AVL tree? How to insert node into AVL tree? Explain.
- Discuss uses and advantages of AVL search tree with suitable
- What uses and advantages of AVL search tree with suitable
examples.
- What is B-Tree? write an algorithm to insert a key into a B-
Tree and delete a key from B-Tree?
- Write short note on B-Tree?
- Explain B+ trees and its advantages with suitable
examples.
OR
Write a short note on B+ Tree?
- Give a brief Idea about general trees?
OR
What are general Trees? discuss the steps for conversion of
general Trees to binary Trees?
- What are huffman‘s algorithm ?how it is useful and used?
Discuss with Examples?
OR
Describe Huffman ‘s algorithm?
- Describe m-way search Tree?
- Describe the various operation on graph?
OR
Explain the different operations performed on graph ?
- Describe Warshall algorithm for shortest path with
Example?
OR
What do you mean by shortest path? explain Warshall
algorithm for shortest path?
- Describe Dijkstra algorithm for shortest path?
- What do you mean by shortest path?
- Describe Dijkstra algorithm for shortest path with example?
- Discuss topological sorting and in the advantage with suitable example?
- Explain topological sorting?
- What is traversal of graph? How is it useful and use Explain with suitable example?
- Explain Traversal of graph?
- What do you mean by graph traversal? Explain depth first and breadth first traversal of graph.
- Explain various types of graph?
- What is the quick sort? how it is use and useful? Explain its complexity with suitable examples?
- Explain quick sort algorithm?
- Explain merge sort and its complexity with examples?
- Explain merge sort algorithm?
- Explain linear search and its advantages, disadvantages and complexity?
- Describe binary search?
- Write an algorithm to search an aliment using binary search?
- Differentiate between linear search and binary search?
- What are the differentiate between internal and external sorting?
- Which sorting algorithm is the best on the basis of complexity and why?
- Compare various sorting algorithm on basis of complexity?
- Describing merging?
- Define File. Describe constituent of a file?
- Write a short note on file operations?
- Describe various kind of operations required to maintain files briefly?
- Explain various types of file? According to function they perform?
- Discuss serial file organisation its advantages and disadvantages?
- Write a short note on sequential file organisation?
- What are the steps to access a record of a file? Also draw flow chart of this.
- What are the steps for updating a sequential file?
- Discuss the advantages and disadvantages of sequential file processing?
- Explain direct access file organisation there use and advantages?
- What do you understand by indexed by sequential file? Discuss the different technique for handling overflows in indexed sequential file?
- What is multi list file organisation? Explain by an example.
- Explain inverted list file organisation by taking examples.
- Differential between inverted list and multi list file organisation?
- Describe fixed and variable length records with examples?
- Describe primary and secondary key with Examples?
- Differentiate between fixed and variable length record?
- Explain hashing function and their relatives and merits and demerits?
- Write a short note on hashing?
- What do you understand by hashing? Explain various hashing function using suitable examples?
- What do you mean by hashing function? Describe various types of hashing function. Give suitable examples.
- What are the collisions? How are these harmful and resolved? Discuss with Examples?
Wednesday, January 29, 2020
Grade 4 Revision
Grade 4
Mock Test
Q2 Convert 18 minutes to seconds.
Q3 Convert 540 minutes to hours.
Q4 Convert 875 seconds to minutes.
Q5 Is “Wednesday at 24:00” and “ Thursday at 00:00 mean exactly the same time?
Q6 Convert the following in 24-hour clock time
- 02:15 am
- 02:15 pm
- 10:10 am
- 10:20 pm
Q8 Time 12 noon can be written as 12 pm. State the statement as true or false.
Q9 Convert to 12 hour clock time.
- 23:24 hours
- 10:00 hours
Q11 Find the difference between 6 hours 20 min and 12 hours 25 min.
Q12 Find 3 hours 20 minutes before 10:00 pm.
Q13 My father’s office timing are 10:30 am to 6:30 pm. How many hours does he spend in office?
Q14 Convert the following
- 8 km 200 m into m
- 9 m 82 mm into mm
- 50 mm into cm
- 625 cm into m
- 12 kg into g
- 19 g to mg
- 2385 gm to kg
- 4200 mg to g
- 15kl to l
- 29 l to ml
- 15 m 25 cm + 8 m 65 cm
- 75 kg 250g + 62 kg 100 g
Q16 A pack of juice contains 2l 200 ml of juice. Rahul drank 750 ml of juice. How much juice is left in the pack?
Q17 Subtract
- 11 km 375m - 8 km 215 m
- 96 l 545 ml - 47 l 400ml
Q19 The perimeter of an equilateral triangle is 36 cm. Find the side of the equilateral triangle.
Q20 Reduce 85/100 in lowest form.
Wednesday, January 15, 2020
Annual exam practice
Annual Exam 2020 Practice
Grade 4
Practice Dodging with 12 and 13
https://quizizz.com/admin/quiz/5e1ebc897ef318001f93ff31
Fractions 1
https://quizizz.com/admin/quiz/5e2a8f8a703e48001b5d7cb8
Friday, November 29, 2019
English Grammar
Join Infate English club on Quiziz. Click link below to join English class
Grammar Practice
a) Conjunctions
b)Tenses
c) Comprehension
d) Advertisement
Monday, November 25, 2019
Relation and Functions
Equivalence and partial order relations with examples
Equivalence relations:
A relation ‘R’ on set ‘A’ is said to be equivalent if R is
- Reflexive. It needs to be diagonal pair such as (a,a),(1,1),(8,8)
- Symmetric: it should have element such as (a,b) and (b,a) for example (1,4) and (4,1) in the relation set for symmetric
- Transitive: It needs to be in the form of (a,b), (b,c) so that we have (a,c)
Let us understand with an example
A={1,2,3} elements =2^3
|
A(1,1) |
A(1,2) |
A(1,3) |
|
A(2,1) |
A(2,2) |
A(2,3) |
|
A(3,1) |
A(3,2) |
A(3,3) |
R1={NULL} it is not reflexive and therefore equivalence does not holds.
R2={(1,1),(2,2) (3,3)} it is reflexive as it has diagonal pairs such as (a,a)
It is symmetric as all diagonal pair makes the set symmetric
Also if also transitive if it has only diagonal elements.
Therefore the relation is equivalence
R3={(1,1),(2,2) (3,3)(2,1)} It is reflexive as it has all diagonal elements
It is not symmetric as there exist only (2,1) and not (1,2) in R3. There the relation is not in
equivalence
R4={(1,1),(1,3),(2,1)(3,1)} It does not has all diagonal elements ref to table above, hence it does not have equivalence relation
R5={(1,1),(2,2),(3,3),(1,2)(1,3)(2,1)(3,1)} It is reflexive as it has all diagonal elements ref table
It is symmetric as we have (1,2) and (2,1). Also we have (1,3) and (3,1)
It is transitive as if we take (1,1) (3,3) then we have (3,1) as in (a,b) and (b,c) we have (a,c) for it to become transitive. Hence equivalence hold here.
R6={(1,1),(1,2),(2,1),(2,3),(3,1)(3,2),(3,3)} It is not reflexive as we do not have all diagonal elements ref table above and you will find (2,2) is missing.
Partial order relations
- Reflexive. It needs to be diagonal pair such as (a,a),(1,1),(8,8)
- Anti Symmetric: it should not have element such as (a,b) and (b,a) for example (1,4) and (4,1) in the relation set for symmetricity. If it has (a,b) and not (b,a) then it is said to be in anti symmetric form
- Transitive: It needs to be in the form of (a,b), (b,c) so that we have (a,c)
A=(1,2,3}
R1={NULL} it is not reflexive and therefore partial order relation does not holds.
R2={(1,1),(2,2) (3,3)} it is reflexive as it has diagonal pairs such as (a,a)
It is antii-symmetric. Also it is transitive if it has only diagonal elements.
Therefore the relation is in partial order
R3={(1,1),(2,2) (3,3)(2,1)} It is reflexive as it has all diagonal elements
It is anti symmetric as there exist only (2,1) and not (1,2) in R3. There the relation is in
Partial order
R4={(1,1),(1,3),(2,1)(3,1)} It does not has all diagonal elements ref to table above, hence it does not have reflexive property and therefore partial order does not holds for this relation
R5={(1,1),(2,2),(3,3),(1,2)(1,3)(2,1)(3,1)} It is reflexive as it has all diagonal elements ref table
It is symmetric as we have (1,2) and (2,1). Also we have (1,3) and (3,1). Therefore it is not anti symmetric
It is transitive as if we take (1,1) (3,3) then we have (3,1) as in (a,b) and (b,c) we have (a,c) for it to become transitive. Hence the relation is not in partial order
R6={(1,1),(1,2),(2,1),(2,3),(3,1)(3,2),(3,3)} It is not reflexive as we do not have all diagonal elements ref table above and you will find (2,2) is missing.Therefore relation is not set to be in partial order.
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