CHAPTER TEST : Similarity, Ratio Proportion and Factorisation

 

Understanding Similarity, Ratio Proportion, and Factorisation for ICSE Class X

As students progress through their mathematics curriculum in ICSE Class X, they encounter crucial concepts that form the foundation of many advanced topics. Among these are similarity, ratio and proportion, and factorisation. This blog post aims to demystify these concepts, providing insights and tips to help students excel.

Similarity

What is Similarity?

In geometry, two figures are said to be similar if they have the same shape but not necessarily the same size. This means that corresponding angles are equal, and the lengths of corresponding sides are in proportion.

Key Properties of Similar Figures:

1. Angle-Angle (AA) Criterion: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
2. Side-Side-Side (SSS) Similarity: If the corresponding sides of two triangles are in proportion, then the triangles are similar.
3. Side-Angle-Side (SAS) Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, then the triangles are similar.

Applications of Similarity:

• Finding unknown lengths in geometric figures.
• Real-world applications like map scaling, architecture, and design.

Ratio and Proportion

Understanding Ratio:

A ratio is a way to compare two quantities by division. It tells us how many times one value contains or is contained within the other. Ratios can be expressed in several forms: as fractions, using the colon notation (a), or with the word "to" (a to b).

Applications of Ratios and Proportions:

• Solving problems involving mixtures, such as food recipes or chemical solutions.
• Scaling figures in similar triangles or maps.
• Financial calculations, like determining discounts or interest rates.

Factorisation

What is Factorisation?

Factorisation is the process of breaking down an expression into its constituent factors. It’s a crucial skill in algebra that helps simplify expressions and solve equations.

Applications of Factorisation:

• Solving quadratic equations.
• Simplifying algebraic fractions.
• Finding roots of polynomial equations.

Tips for Mastering These Concepts

1. Practice Regularly: Solve various problems related to similarity, ratio and proportion, and factorisation. This builds familiarity and confidence.
2. Visual Learning: Use diagrams for similarity and geometric ratios to enhance understanding.
3. Study in Groups: Explaining concepts to peers can reinforce your understanding and uncover new insights.
4. Use Online Resources: Leverage educational videos and interactive tools for visual and auditory learning.

Conclusion

Mastering the concepts of similarity, ratio and proportion, and factorisation is essential for success in ICSE Class X mathematics and beyond. These foundational skills not only enhance problem-solving abilities but also prepare students for more advanced studies in mathematics and related fields. With consistent practice and a positive attitude, students can excel in these topics and build a strong mathematical foundation. Happy studying!

Chapter Test : Quadratic Equation, AP and Probability for ICSE and CBSE

 As students prepare for their exams under the ICSE and CBSE curricula, mastering essential mathematical concepts is crucial for success. Among these concepts, quadratic equations, arithmetic progressions (AP), and probability hold significant importance. This blog post will explore these topics in detail, providing insights and sample questions to help students effectively prepare for chapter tests.

Understanding Quadratic Equations

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in the form:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$. The solutions to these equations can be found using various methods, including:

• Factoring
• Completing the square
• Quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Sample Questions

1. Solve the quadratic equation: $2x^2 - 4x - 6 = 0$
   
2. Factor the quadratic expression: $x^2 - 5x + 6$
   

Key Concepts

• The discriminant $D = b^2 - 4ac$ determines the nature of the roots:
  • $D > 0$: Two distinct real roots
  • $D = 0$: One real root (repeated)
  • $D < 0$: No real roots

Exploring Arithmetic Progressions (AP)

What is an Arithmetic Progression?

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference ($d$). The $n$-th term of an AP can be expressed as:

$$a_n = a + (n - 1)d$$

where $a$ is the first term and $n$ is the term number.

Sample Questions

1. Find the 10th term of the AP: $3, 7, 11, 15, \ldots$.
2. If the 5th term of an AP is 20 and the common difference is 4, find the first term.

Key Concepts

• The sum of the first $n$ terms ($S_n$) of an AP is given by:

$$S_n = \frac{n}{2} (2a + (n - 1)d)$$

or

$$S_n = \frac{n}{2} (a + l)$$

where $l$ is the last term.

Diving into Probability

What is Probability?

Probability is the measure of the likelihood of an event occurring, expressed as a number between 0 and 1. The basic formula for probability is:

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

Sample Questions

1. A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a red ball?
2. If two dice are rolled, what is the probability that the sum of the numbers is 8?

Key Concepts

• Complementary Events: The probability of an event not occurring is $P(A') = 1 - P(A)$.

Conclusion

Mastering quadratic equations, arithmetic progressions, and probability is essential for students in ICSE and CBSE systems. Regular practice with chapter tests will enhance problem-solving skills and boost confidence. Incorporating a variety of question types, from basic to advanced, can further prepare students for their exams.

As you study these topics, remember to review key concepts, practice sample problems, and seek clarification on challenging areas. Good luck with your preparations, and may you achieve the results you strive for in your upcoming exams!

HS Mathematics Question Paper 2024 English Version Solved

HS Mathematics Question Paper 2024 English Version Solved

Detailed Solution of HS Math Paper 2024 English version for Higher Secondary Students of WBCHSE WBCHSE or the West Bengal Council of Higher Secondary Education. HS 2024 Math question paper or class 12 Mathematics question paper answer/solution. Topic 1) HS 2024 math question paper 2) class 12 mathematics question paper solution 3) Math question paper

Integrate (x^4 +1)/(x^6 + 1) Problem for #jeemains #WBJEE #CBSE #ISC #mathematics

Problem for #jeemains #WBJEE #CBSE #ISC #mathematics 

Algebra Problem on the Concept of Odd and Even Numbers

 Let m, n, p and q be four positive integers such that m+n+p+q = 200. If S = (-1)^m+(-1)^n+(-1)^p+(-1)^q, then what is the number of possible values of S? #algebra #ProblemSolving #schoolmathematics

Integration for JEE Mains

 Integrate sqrt ((cos x - cos^3 x)/(1 - cos^3 x) #jeemains #CBSE #wbjee #ISC #math

Geometry Problem for Secondary Students

ABC is an isosceles triangle whose ∠C is right angle. If D is any point on AB, then prove that, ABC is an isosceles triangle whose ∠C is right angle. If D is any point on AB, then let us prove that, AD² + DB² = 2CD². ​

Integration for JEE Mains

 Integration for JEE Mains

Height and Distance Worksheet - CBSE and ICSE Class 10

**Height and Distance Worksheet**

**Instructions:**

1. Solve all the problems.

2. Show your work and calculations where necessary.

3. Circle or underline your final answers.

**Easy Problems:**

1. A flagpole stands vertically on the ground. If the angle of elevation to the top of the flagpole is 45 degrees and you are standing 20 meters away from the flagpole, find the height of the flagpole.

2. From the top of a building 30 meters high, the angle of depression of an object on the ground is 60 degrees. Find the distance of the object from the base of the building.

3. If the length of a shadow of a 10-meter pole is 8 meters, find the angle of elevation of the sun.

4. A ladder leans against a wall. The angle of elevation of the ladder is 60 degrees, and the ladder reaches a height of 10 meters on the wall. Find the length of the ladder.

5. Two ships are sailing towards each other. They spot each other when they are 1,000 meters apart, and the angle of elevation is 30 degrees from one ship to the other. Find the altitude of each ship.

**Medium Problems:**

6. A 15-meter ladder is leaning against a wall. If the ladder makes a 45-degree angle with the ground, how far is the bottom of the ladder from the wall?

7. A person standing 50 meters away from a tree finds that the angle of elevation to the top of the tree is 30 degrees. Find the height of the tree.

8. An observer on a cliff sees a boat in the sea below at an angle of depression of 45 degrees. If the cliff is 60 meters high, find the distance between the boat and the observer.

9. A 12-meter ladder is placed against a wall. If the top of the ladder slides down the wall at a rate of 2 meters per second, how fast is the bottom of the ladder moving away from the wall when the top of the ladder is 9 meters above the ground?

10. Two buildings are 100 meters apart. From the top of one building, the angle of elevation to the top of the other is 30 degrees. If the height of the first building is 40 meters, find the height of the second building.

**Difficult Problems:**

11. From a point 15 meters above the water surface, the angle of elevation of a cliff is 60 degrees. If the cliff is 15 meters from the shore, find the depth of the water.

12. An airplane is flying at an altitude of 5,000 meters. An observer on the ground sees the airplane at an angle of elevation of 30 degrees. Find the horizontal distance between the observer and the airplane.

13. Two towers are 100 meters apart. From the top of the first tower, the angle of elevation to the top of the second tower is 45 degrees. If the height of the first tower is 60 meters, find the height of the second tower.

14. A man stands on the top of a hill and sees a car coming towards him. If the angle of depression of the car is 30 degrees when it is 300 meters away, find the height of the hill.

15. A 20-meter ladder leans against a wall. If the ladder makes an angle of 75 degrees with the ground, find the height at which the ladder touches the wall.

**Answers:**

**Easy Problems:**

1. 20 meters

2. 30 meters

3. 60 degrees

4. 20 meters

5. Altitude of each ship is 500 meters.

**Medium Problems:**

6. 15 meters

7. 25 meters

8. 60 meters

9. 2 m/s

10. 20 meters

**Difficult Problems:**

11. 15 meters

12. 10,000 meters (10 km)

13. 60 meters

14. 150 meters

15. 5√3 meters

Bayes' Theorem Problem from ISC 2023 Maths Paper

In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. as per the annual report of the company, 80% of the graduate employees and 10% of the non-graduate employees are in the administrative positions. find the probability that an employee selected at random from those working in administrative positions will be a graduate. 

Answer:

Step-by-step explanation:

Let G be the event that the selected employee is a graduate, and NG be the event that the selected employee is non-graduate.

Clearly, G and NG forms a mutually exclusive and exhaustive set of events.

Further, let A be the event that the selected employee works in administrative office.

According to the problem, we have to find, the selected employee is a graduate given he/she works in the administrative position P(G/A).

By Bayes' theorem,

Now, Probability of an employee to be graduate =

Probability of an employee to be non-graduate =

Probability of an employee working in administrative office given he is a graduate =

( as 80% of the graduate employee works in the administrative positions)

Probability of an employee working in administrative office given he is a non-graduate =

( as 10% of the non-graduate employee works in the administrative positions)

Substituting the values in

, we get,

Combinations SN Dey Solved

1. An executive committee of 6 is to be formed from 4 ladies and 7 gentlemen. In how many ways can this be formed when the committee contains (i) only 2 lady members, (ii) at least 2 lady members?

2. Find the number of committees of 5 members that can be formed from 6 gentlemen and 4 ladies if each committee has at least one lady and two gentlemen.

 

3. A committee of 5 is to be formed from six ladies and four gentlemen. In how many ways this can be done so that the committee contains (i) exactly two ladies, (ii) at least two ladies, (iii) at most two ladies?

 

4. In a cricket team of 14 players 6 are bowlers. How many different teams of 11 players can be selected keeping at least 4 bowlers in the team?

 

5. A box contains 12 lamps of which 5 are defective. In how many ways can a sample of 6 be selected at random from the box so as to include at most 3 defective lamps?

 

6. An examinee is required to answer 6 questions out of 12 questions which are divided into two groups each containing 6 questions, and he is not permitted to answer more than 4 questions from any group. In how many ways can he answer 6 questions?

 

7. A question paper contains 10 questions, which are divided into two groups each containing 5 questions. A candidate is asked to answer 6 questions only, and to choose at least 2 questions from each group. In how many different ways can the candidate make up his choice?

 

8. In how many ways can a team of 11 cricketers be chosen from 9 batsmen and 6 bowlers to give a majority of batsman if at least 3 bowlers are to be included?

 

9. The Indian Cricket Eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that the team contains at least three bowlers?

10. How many combinations can be formed of eight counters marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one odd and one even counter in each combination?

 

11. Find the number of permutations of the letters of the words FORECAST and MILKY taking 5 at a time of which 3 letters from the first word and 2 from the second.

 

12. In how many ways can the crew of an eight-oared boat be arranged if 2 of the crew can row only on the stroke side and 1 can row only on the bow side?

 

13. Of the 17 articles, 12 are alike and the remaining 5 are different. Find the number of combinations, if 13 articles are taken at a time.

 

14. Out of 3n given things 2n are alike and the rest are different. Show that a selection of 2n things can be made from these 3n things in 2" different ways.

 

15. Show that there are 136 ways of selecting 4 letters from the word EXAMINATION.

 

16. Find the total number of ways of selecting 5 letters from the letters of the word INDEPENDENT.

 

17. (i) Find the number of combinations in the letters of the word STATISTICS taken 4 at a time.

      (ii) Find the number of permutations in the letters of the word PROPORTION taken 4 at a time.

 

18. How many different numbers of 4 digits can be formed with the digits 1, 1, 2, 2, 3, 4?

 

19. (i) From 4 apples, 5 oranges mangoes, how many selections of fruits can be made, taking at least one of each kind if the fruits of the same kind are of different shapes?

(ii) In how many ways can one or more fruits be selected from 4 apples, 5 oranges and 3 mangoes, if the fruits of the same kind be of the same shape?

20. Find the total number of combinations taking at least one green ball and one blue ball, from 5 different green balls, 4 different blue balls and 3 different red balls.

 

21. How many different algebraic quantities can be formed by combining a, b, c, d, e with + and - signs, all the letters taken together?

 

22. There are n points in space, no four of which are in the in the same place with the exception of m points, all of which are in the same plane. How many planes can be formed by joining them?

 23. n1, n2 and n3, points are given on the sides BC, CA and AB respectively of the triangle ABC. Find the number of triangles formed by taking these given points as vertices of a triangle.

 

24. A man has 7 relatives, 4 of them are ladies and 3 gentlemen; his wife has also 7 relatives, 3 of them are ladies and 4 are gentlemen. In how many ways can they invite dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

 

25. Eighteen guests have to be seated, half on each side of long table. Four particular guests desire to sit on one particular side and three others on the other side Determine the number of ways in which the arrangements can be made. 

Permutations SN Dey Solved XI Maths

Solved problems from the book of SN Dey Class XI WBCHSE, Permutations [ 4 and 5 marks only]

Problem from Geometry - Circles

In the figure given below, ABD is a right-angled triangle at B. Taking AB as diameter, a circle has been drawn intersecting AD at F. Prove that the tangent drawn at point F bisects BD.

Solution:

Trigonometry SN Dey Solved Problems - HS - Class XI

1. Trigonometric functions of Standard Angles [ 4 and 5 marks ]

2. Trigonometric functions of Associate Angles [ 4 and 5 marks ]

3. Trigonometric transformations of sums and products [ 4 and 5 marks]

4. Trigonometric functions of Compound Angles [ 4 and 5 marks]

5. Trigonometric functions of Multiple Angles [4 and 5 marks]

6. Trigonometric functions of Sub-multiple Angles [ 4 and 5 marks]

7. Trigonometric Equations  [ 4 and 5 marks]

Vectors SN Dey Solved Problems

 Introduction of Vectors Solved Problems SN Dey - Long Answer Type Questions 5 Marks

Product of Vectors Solved Problems SN Dey - Long Answer Type Questions 5 Marks

Set Theory Solved Problems : Class XI

The concept of set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets. The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”. In this Chapter, we discuss some basic definitions and operations involving sets.

Empty Sets

The set with no elements or null elements is called an empty set. This is also called a Null set or Void set. It is denoted by {}.

For example: Let, Set X = {x:x is the number of students studying in Class 6th and Class 7th}

Since we know a student cannot learn simultaneously on two classes, therefore set X is an empty set.

Singleton Set

The set which has only one element is called a singleton set.

For example, Set X = { 2 } is a singleton set.

Finite and Infinite Sets

Finite sets are the one which has a finite number of elements, and infinite sets are those whose number of elements cannot be estimated, but it has some figure or number, which is very large to express in a set.

For example, Set X = {1, 2, 3, 4, 5} is a finite set, as it has a finite number of elements in it.

Set Y = {Number of Animals in India} is an infinite set, as there is an approximate number of Animals in India, but the actual value cannot be expressed, as the numbers could be very large.

Equal Sets

Two sets X and Y are said to be equal if every element of set X is also the elements of set Y and if every element of set Y is also the elements of set X. It means set X and set Y have the same elements, and we can denote it as;

X = Y

For example, Let X = { 1, 2, 3, 4} and Y = {4, 3, 2, 1}, then X = Y

And if X = {set of even numbers} and Y = { set of natural numbers} the X ≠ Y, because natural numbers consist of all the positive integers starting from 1, 2, 3, 4, 5 to infinity, but even numbers starts with 2, 4, 6, 8, and so on.

Subsets

A set X is said to be a subset of set Y if the elements of set X belongs to set Y, or you can say each element of set X is present in set Y. It is denoted with the symbol as X ⊂ Y.

We can also write the subset notation as;

X ⊂ Y if a ∊ X

a ∊ Y

Thus, from the above equation, “X is a subset of Y if a is an element of X implies that a is also an element of Y”.

Each set is a subset of its own set, and a null set or empty set is a subset of all sets.

Power Sets

The power set is nothing but the set of all subsets. Let us explain how.

We know the empty set is a subset of all sets and every set is a subset of itself. Taking an example of set X = {2, 3}. From the above given statements we can write,

{} is a subset of {2, 3}

{2} is a subset of {2, 3}

{3} is a subset of {2, 3}

{2, 3} is also a subset of {2, 3}

Therefore, power set of X = {2, 3},

P(X) = {{},{2},{3},{2,3}}

Universal Sets

A universal set is a set which contains all the elements of other sets. Generally, it is represented as ‘U’.

For example; set X = {1, 2, 3}, set Y = {3, 4, 5, 6} and Z = {5, 6, 7, 8, 9}

Then, we can write universal set as, U = {1, 2, 3, 4, 5, 6, 7, 8, 9,}

Note: From the definition of the universal set, we can say, all the sets are subsets of the universal set. Therefore,

X ⊂ U

Y ⊂ U

And Z ⊂ U

Union of sets

A union of two sets has all their elements. It is denoted by ⋃.

For example, set X = {2, 3, 7} and set Y = { 4, 5, 8}

Then union of set X and set Y will be;

X ⋃ Y = {2, 3, 7, 4, 5, 8}

Properties of Union of Sets:

X ⋃ Y = Y ⋃ X ; Commutative law

(X ⋃ Y) ⋃ Z = X ⋃ (Y ⋃ Z)

X ⋃ {} = X ; {} is the identity of ⋃

X ⋃ X = X

U ⋃ X = U

Intersection of Sets

Set of all elements, which are common to all the given sets, gives intersection of sets. It is denoted by the symbol ⋂.

For example, set X = {2, 3, 7} and set Y = {2, 4, 9}

So, X ⋂ Y = {2}

Difference of Sets

The difference of set X and set Y is such that, it has only those elements which are in the set X and not in the set Y.

i.e. X – Y = {a: a ∊ X and a ∉ Y}

In the same manner, Y – X = {a: a ∊ Y and a ∉ X}

For example, if set X = {a, b, c, d} and Y = {b, c, e, f} then,

X – Y = {a, d} and Y – X = {e, f}

Disjoint Sets

If two sets X and Y have no common elements, and their intersection results in zero(0), then set X and Y are called disjoint sets.

It can be represented as; X ∩ Y = 0

ICSE Class IX - Practice Set

Practice Set 32 Marks - Real Numbers and Compound Interest - Click Here

Practice Set 25 Marks -7 Questions- Real Numbers, Exponents, Compound Interest and Expansions - Set I

Practice Set 28 Marks-8 Questions- Real Numbers, Exponents, Compound Interest and Expansions - Set II

Practice Set 15 Questions - Real Numbers - Click Here

Harder Problems 22 Questions - Algebraic Identities - Click Here

Practice Set 39 Questions - Factorisations - Click Here

Practice Set 23 Questions - Expansion and Indices - Click Here

Practice Set 23 Questions - Real Numbers and Compound Interest - Click Here

WBJEE MATHEMATICS PAPER 2021 SOLVED

 The West Bengal Joint Entrance Examinations Board

The West Bengal Joint Entrance Examinations Board (WBJEEB) was established in 1962 by Government of West Bengal in exercise of the powers conferred under article 162 of the Constitution of India in pursuant to No. 828-Edn(T), dated 02.03.1962.

Subsequently in 2014, the Government of West Bengal enacted the West Bengal Act XIV of 2014 to form The West Bengal Joint Entrance Examinations Board and empowered it to conduct Common Entrance Examinations for selection of candidates for admission to undergraduate and postgraduate Professional, Vocational and General Degree Courses in the State of West Bengal and to conduct on-line counselling process or otherwise adopting a single-window approach.

WBJEEB has been instrumental in the admission process based on online application and allotment through e-Counselling since 2012. It advocates fairness and transparency, ensures no-error, and adopts state-of-the-art technology.

WBJEE 2022 Mathematics Syllabus

S.No.	Topics
1	Algebra Arithmetic Progression G.P., H.P Sets, Relations and Mappings Logarithms Complex Numbers Permutation and combination Polynomial equation Principle of mathematical induction Matrices Binomial theorem (positive integral index) Statistics and Probability
2	Trigonometry
3	Coordinate geometry of two dimensions
4	Coordinate geometry of three dimensions
5	Differential calculus Calculus Integral calculus Application of Calculus Differential Equations Vectors