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!

Madhyamik 2024 Math Paper English Version Solution

Get the detailed solution of Madhyamik Mathematics Paper 2024 in English Version. 

WB Board Class 10 Maths Question Paper PDF: Get here free PDF download of the West Bengal Board Class 10 Maths question paper for board exam 2024. Also, check the WBBSE Class 10 Maths answer key by experts here.

Surds Practice Problem Set

In Mathematics, surds are the values in square root that cannot be further simplified into whole numbers or integers. Surds are irrational numbers. The examples of surds are √2, √3, √5, etc., as these values cannot be further simplified. If we further simply them, we get decimal values, such as:

√2  = 1.4142135…

√3 = 1.7320508…

√5 = 2.2360679…

Surds Definition

Surds are the square roots  (√) of numbers that cannot be simplified into a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value. Consider an example, √2 ≈ 1.414213. It is more accurate if we leave it as a surd √2.

Surds Worksheet

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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.

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

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Geometry Inequality

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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