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!

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

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. ​

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

Worksheet on Direct and Inverse Variation ICSE Class 8

 Certainly! Here's a worksheet on the topic of Direct and Inverse Variation for 8th-grade students following the ICSE board curriculum. The problems are categorized into easy, medium, and difficult levels.

**Direct and Inverse Variation 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. If y varies directly with x, and y = 12 when x = 4, find the constant of variation (k).

2. If y varies inversely with x, and y = 10 when x = 5, find the constant of variation (k).

3. If y varies directly with x, and y = 25 when x = 5, find y when x = 8.

4. If y varies inversely with x, and y = 6 when x = 9, find y when x = 12.

5. If y varies directly with x, and y = 15 when x = 3, find x when y = 30.

**Medium Problems:**

6. The cost (C) of printing flyers is directly proportional to the number of flyers (n). If it costs $40 to print 200 flyers, find the cost to print 600 flyers.

7. The time (t) it takes to complete a task is inversely proportional to the number of workers (w). If it takes 8 hours for 6 workers to complete the task, how long will it take for 12 workers to finish the same task?

8. A car travels at a constant speed. If it covers 60 miles in 2 hours, how long will it take to cover 150 miles at the same speed?

9. The force (F) of attraction between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (d) between them. If F = 12 when m1 = 4, m2 = 6, and d = 3, find F when m1 = 8, m2 = 9, and d = 5.

10. The pressure (P) in a closed container is inversely proportional to its volume (V). If P = 48 kPa when V = 4 liters, find the pressure when V = 10 liters.

**Difficult Problems:**

11. A car's fuel efficiency (miles per gallon) varies inversely with its speed (in miles per hour). If the car gets 30 miles per gallon at 60 mph, find the fuel efficiency at 70 mph.

12. The force (F) of gravity between two objects varies directly with the product of their masses (m1 and m2) and inversely with the square of the distance (d) between them. If F = 9.8 N when m1 = 5 kg, m2 = 10 kg, and d = 1 m, find F when m1 = 3 kg, m2 = 8 kg, and d = 2 m.

13. The time (t) it takes for a pendulum to complete one full swing varies directly with the square root of its length (L). If a pendulum takes 2 seconds to complete one swing when L = 9 meters, find the time it takes when L = 16 meters.

14. The resistance (R) in an electrical circuit is inversely proportional to the square of the current (I). If R = 25 ohms when I = 5 amperes, find R when I = 10 amperes.

15. The force (F) required to lift an object with a pulley system varies directly with the weight (W) of the object and inversely with the number (n) of supporting ropes. If F = 120 N when W = 600 N and n = 4, find F when W = 800 N and n = 6.

**Answers:**

**Easy Problems:**

1. k = 3

2. k = 50

3. y = 40

4. y = 4

5. x = 6

**Medium Problems:**

6. $150

7. 4 hours

8. 5 hours

9. F = 5.76

10. P = 19.2 kPa

**Difficult Problems:**

11. Fuel efficiency at 70 mph = 25 mpg

12. F = 4.35 N

13. Time = 3 seconds

14. R = 6.25 ohms

15. F = 160 N

Worksheet on Sets ICSE Class 8

 

**Sets Worksheet – ICSE – Class 8**

 

**Instructions:**

1. Answer all the questions.

2. Circle or underline your final answer.

3. Show your work or reasoning if required.

4. Answers to all the problems are given at the end. You should look at the solutions only after attempting all the problems.

 

 

**Questions:**

1. Define a "set" in your own words. Provide an example.

 

2. Classify the following into sets:

   a) The days of the week

   b) Even numbers less than 20

   c) Vowels in the English alphabet

 

3. List the elements of the set A = {2, 4, 6, 8, 10}. Also, find the cardinality of set A.

 

4. Create a set B with the first five prime numbers. Write it in the set-builder notation.

 

5. Determine whether the following statements are true or false:

   a) {1, 2, 3} ⊆ {1, 2, 3, 4, 5}

   b) {a, b, c} ⊇ {b, c, d}

   c) {2, 4, 6} ⊂ {1, 2, 3, 4, 5, 6}

 

6. Find the union of sets P = {1, 2, 3, 4, 5} and Q = {4, 5, 6, 7}. Write the result in roster form.

 

7. Calculate the intersection of sets X = {a, b, c} and Y = {b, c, d}. Write the result in set-builder notation.

 

8. Consider two sets: M = {1, 2, 3, 4} and N = {3, 4, 5, 6}. Find the difference M - N.

 

9. Solve the following set equation for set Z: Z ∩ {2, 3, 4} = {3, 4}. Write the result in roster form.

 

10. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, find the complement of the set V = {2, 4, 6, 8}.

 

**Additional Challenging Questions:**

 

11. Let A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8, 9}. Find A ∪ B, A ∩ B, and A - B.

 

12. Consider a universal set U = {x | x is a positive integer less than 10}. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∩ B and A ∪ B.

 

13. Let U = {a, b, c, d, e, f, g} be the universal set. If A = {a, b, c, d} and B = {b, c, e, f}, find A ∪ B and A' (complement of A).

 

14. Define three sets A, B, and C as follows:

    A = {x | x is a multiple of 2 and 3}

    B = {x | x is a multiple of 2 and 5}

    C = {x | x is a multiple of 3 and 5}

   Find A ∩ B, A ∪ C, and B ∩ C.

 

15. Let U be the set of all students in a school, A be the set of students who play chess, and B be the set of students who play cricket. If there are 120 students in total, 60 play chess, and 80 play cricket, how many students play both chess and cricket?

 

16. Consider the set P = {x | x is a prime number less than 20} and the set Q = {x | x is an odd number less than 20}. Find P ∩ Q.

 

17. Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Find the symmetric difference of sets A and B.

 

18. Determine whether the following statement is true or false: For any two sets A and B, A ∪ B = B ∪ A.

**Answers:**

 

1. A set is a collection of distinct objects or elements. Example: Set of even numbers less than 10 = {2, 4, 6, 8}.

 

2.

   a) Set of days of the week = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

   b) Set of even numbers less than 20 = {2, 4, 6, 8, 10, 12, 14, 16, 18}

   c) Set of vowels in the English alphabet = {a, e, i, o, u}

 

3. Set A = {2, 4, 6, 8, 10}, Cardinality of A = 5.

 

4. Set B = {2, 3, 5, 7}. In set-builder notation: B = {x | x is a prime number and 1 < x < 10}.

 

5.

   a) True

   b) False

   c) True

 

6. P ∪ Q = {1, 2, 3, 4, 5, 6, 7}.

 

7. X ∩ Y = {b, c}. In set-builder notation: {x | x is an element of X and x is an element of Y}.

 

8. M - N = {1, 2}. (Elements in M but not in N)

 

9. Z = {3, 4}.

 

10. U - V = {1, 3, 5, 7, 9}.

 

**Answers to Additional Questions:**

 

11. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A ∩ B = {4, 5, 6}, A - B = {1, 2, 3}

 

12. A ∩ B = {3, 4}, A ∪ B = {1, 2, 3, 4, 5, 6}

 

13. A ∪ B = {a, b, c, d, e, f}, A' = {e, f, g}

 

14. A ∩ B = {x | x is a multiple of 2, 3, and 5}, A ∪ C = {x | x is a multiple of 2 or 3 or 5}, B ∩ C = {x | x is a multiple of 3 and 5}

 

15. Students who play both chess and cricket = 20 students.

 

16. P ∩ Q = {3, 5, 7, 11, 13, 17, 19}

 

17. Symmetric difference of sets A and B = {1, 2, 6, 7}

 

18. True. The union of two sets is commutative, so A ∪ B = B ∪ A.

 

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:

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

Geometry Solved Problems : Circles

Solved problems on circles for class X. Get solutions of problems shown in the post and many more. Visit the link Solved Problems: Circles

TRIGONOMETRY : TRIGONOMETRICAL RATIOS SOLVED PROBLEMS FOR 10TH GRADE CBSE ICSE OTHER STATE BOARDS

 

TRIGONOMETRY : TRIGONOMETRICAL RATIOS SOLVED PROBLEMS FOR 10TH GRADE ( CBSE, ICSE AND OTHER STATE BOARDS). SUBSCRIBE FOR MORE SOLVED PROBLEMS WHICH WILL BE UPLOADED SOON.

Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

• Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

${\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{c}}.}$

• Cosine function (cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.

${\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{c}}.}$

• Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

${\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}}={\frac {a/c}{b/c}}={\frac {\sin A}{\cos A}}.}$

If you have any query feel free to connect us at any of the following platforms. Free help for your queries. Helping Students! Feel free to send your problems. Follow us on the most popular platforms. Whatsapp: +91-9038126497

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Interesting Property of a Trapezium

Prove that the straight line that passes through the point of intersection of the diagonals of a trapezium and through the point of intersection of its non-parallel sides, bisects each of the parallel sides of the trapezium.

Class IX West Bengal Board Ganit Prakash Chapter 20 Solutions- Co-ordinate Geometry

Class IX West Bengal Board Ganit Prakash Chapter 20 Solutions - Coordinate Geometry

In this pdf you will get all the solved problem from the exercise 20. Before going through the solutions you are advised to first try the problem yourself. If you get stuck, seek help from the solutions. Don't copy blindly. If you have any query, comment below.

Solved Trigonometry Problems: 10th Grade

Here is a list of few good problems at the 10th grade. Get the full solution below and if you want
more comment or mail us at maths.programming@gmail.com!


Download here

Circle: Geometry

Given below is a fine problem based on simple properties of circles ( tangent and chord ). Try yourself before looking at the solution!

Here is the solution!

Solved Problems : Logarithm

A collection of solved problems on logarithm meant for secondary students ( 9th and 10th grade). The collection covers almost all types of problems at the level mentioned.



Download the file here: Solved Logarithm Problems 

Geometry Inequality

Post by Maths.

Prove that in any quadrilateral, the sum of the diagonals is greater than the half of its perimeter.

Consider the quad. In the above diagram. Let E be the point of the intersection.

Now, AE+EB > AB

EB+EC > BC

AE+ED > AD

EC+ED > DC (Using Triangle Inequality)

Adding the above four inequalities we get

2(AE+EC+EB+ED) > AB+BC+AD+DC

=> AC + BD > ½(AB+BC+AD+DC)

Thus sum of the diagonals is greater than the half of its perimeter Q.E.D

In any triangle four times the sum of its medians is greater than 3 times its perimeter.

We know that difference of any two sides of a triangle is less than the third side (prove it)

In triangle ABE,

AE > AB-BE

In triangle ACE,

AE > AC-CE

Adding above two inequalities we get,

2AE > AB + AC -(BE+CE)

=> AE > ½(AB+AC-BC)

=> 4AE > 2(AB+AC-BC).........(1)

Similarly,

4BD > 2(AB+BC-AC).............(2) and 4CF > 2(AC+BC-AB)........(3)

Adding (1),(2) and (3) we have,

4(AE+BD+CF) > 2(AB+AC-BC+AB+BC-AC+AC+BC-AB)

=> 4(AE+BD+CF) > 2(AC+AB+BC)

=> sum of the lengths of the medians is greater than half the perimeter

We can strengthen the inequality by using the fact that the point 'O' divides the medians AE,BD,CF internally in the ration 2:1

Therefore, OD:OB = 1:2

=> (OB+OD):OB = (1+2):2

=> BD:OB=3:2

=> OB = 2/3 BD........(a)

Similarly, OC = 2/3 CF.........(b) and OA = 2/3 AE.......(c)

Now in triangle OBC, OB+OC> BC

=>2/3(BD+CF)>BC [using (a) and (b)]

=> 2(BD+CF)>3BC

Similarly, 2(CF+AE)>3AC and 2(BD+AE)>3AB

Adding the last three inequalities we get 4(AE+BD+CF) > 3(AB+BC+CA)

In the triangle ABC, AE,BD and CF are the medians where O is the point of there intersection

Solved Problems on Circles (Tangent Properties)

In these material we explore the properties of circles, tangents and common tangents. Using congruency and similarity of triangles many of the desired property has been deduced.

Post by Maths.

Coordinate Geometry

Solved Problems on Coordinate Geometry for 10th grade. Problems based on distance formula, section ratio formula   

 

GEOMETRY Thales Theorem or Basic Proportionality Theorem Solved Problems

Solved Problems on Thales Theorem useful for students of 10th grade of CBSE and ICSE and other state board. This material is also useful for the preparation of Regional Mathematics Olympiad. Basic Geometry is a pre-requisite  for any Olympiad.