Mathematics and Science PYQs

A transversal is a line that intersects two or more lines at distinct points. The distinct points are the points where the transversal crosses the other lines.  Parallel lines are lines that never intersect, no matter how far they are extended.

Key Points

• The railway line crossing several other lines can be used to teach students about transversals.
• The teacher can point out that the railway line intersects the other lines at distinct points.
• The road crossing two or more roads can also be used to teach students about transversals.
• The lines in the alphabet "H" and "Z" can be used to teach students about parallel lines.
• The teacher can point out that the lines in the letter "H" are parallel, but the lines in the letter "Z" are not parallel.

Thus, it is concluded that the teacher in her class discusses various examples from daily life situations like a railway line crossing several other lines, a road crossing two or more roads, and alphabets like 'H', 'Z', etc. Transversal and parallel lines can be introduced through these examples.

Additional Information Some of the properties of parallel lines: 

• Corresponding angles are equal.
• Alternate angles are equal.  ​
• Interior angles on the same side of the transversal are supplementary.

Mathematics is the systematic study of abstract structures, patterns, and relationships using logical reasoning and symbols. It provides a universal language to describe and understand the fundamental principles governing the world of numbers, shapes, and quantities.

Key Points Educators can create a stimulating and motivating learning environment that empowers students to harness their existing capabilities and develop a lifelong appreciation for mathematics. Some of the most appropriate ways to teach mathematics:

• Use concrete materials: Concrete materials, such as blocks, manipulatives, and real-world objects, can help students visualize mathematical concepts and make them more concrete.
• Teach problem-solving: Problem-solving is an essential skill in mathematics, and it is important to teach students how to solve problems in a variety of ways.
• Make connections to real-world applications: When students can see how mathematics is used in the real world, they are more likely to be engaged and motivated to learn.
• Encourage active learning: Students learn best when they are actively engaged in the learning process. This means giving them opportunities to ask questions, discuss ideas, and solve problems.
• Provide feedback: Feedback is essential for helping students learn. It should be timely, specific, and constructive.

Thus, it is concluded that the most appropriate way to teach mathematics is to motivate the learners to develop their existing capabilities.

Additional Information

Some additional tips for teaching mathematics: 

• Be positive and encouraging: Students are more likely to succeed if they feel supported and encouraged.
• Be creative: There is no right way to teach mathematics. Be creative and find ways to make the learning process engaging and interesting for students.
• Make learning fun: There are many ways to make learning mathematics fun, such as using games, puzzles, and activities.

Continuous and Comprehensive Evaluation (CCE) is an educational assessment method that involves assessing students' progress and learning consistently throughout the academic year, considering various aspects of their development including academic, co-curricular, and life skills.

Key Points The assessment process in Continuous and Comprehensive Evaluation (CCE) includes an assessment of previous knowledge, understanding level, and learning process:

• Diagnostic Assessment: CCE starts with an assessment of students' prior knowledge and understanding. This helps teachers understand what students already know and identify any knowledge gaps.
• Formative Assessment: Throughout the learning process, formative assessments are conducted to gauge students' understanding of ongoing lessons. These assessments focus on evaluating comprehension, critical thinking, and problem-solving skills.
• Varied Assessment Tools: CCE employs a diverse set of assessment tools such as quizzes, projects, group discussions, presentations, and practical demonstrations.
• Regular Feedback: Teachers provide continuous feedback to students, highlighting their strengths and suggesting areas for improvement. This feedback loop supports students in understanding their learning progress and making necessary adjustments.
• Learning Process Analysis: CCE emphasizes evaluating not only the final results but also the learning journey itself. Teachers assess the methods students use to approach problems, how they apply concepts, and their ability to adapt and learn from mistakes.

Thus, it is concluded that the assessment process includes an assessment of previous knowledge, understanding level, and learning process is true for Continuous and Comprehensive Evaluation (CCE).

A concept map in mathematics is a visual tool that connects ideas and concepts, showing their relationships and helping to organize and understand mathematical topics more clearly. It's like a roadmap that guides us through the interconnected world of mathematical ideas.

Key Points Essential features of a concept map in mathematics:

• Hierarchy and Structure: Concept maps in mathematics display a clear hierarchy of concepts, with the most general or foundational ideas at the top and more specific details branching out below. This structure helps to show how concepts relate to one another.
• Directional Arrows: These arrows indicate the relationships between concepts, showing which concepts serve as prerequisites for understanding others. This visual representation helps to grasp the logical progression and dependencies within a mathematical topic.
• Visual Representation: A concept map is a visual tool, often created using diagrams, shapes, and colors. This visual representation enhances understanding by presenting complex ideas in a more accessible and organized manner.
• Contributory Sub-Concepts: A concept map includes contributory sub-concepts, which are the fundamental building blocks that collectively form a complete understanding of the main concept. These sub-concepts demonstrate how different elements relate to one another.
• Hierarchical Structure: The map displays a hierarchical structure, with foundational or fundamental concepts at the top and more advanced or specific concepts below. This arrangement provides a visual representation of the logical progression and dependency of concepts.

Thus, it is concluded that the essential features of a concept map in mathematics represent a conceptual analysis of a concept using contributory sub-concepts, and directional, with arrows showing which concepts are prerequisites for which others.

Mathematics is the study of numbers, shapes, and patterns to solve problems and explain the way things work in our world. It's like a tool that helps us understand and make sense of different situations.

Key PointsNature of mathematics:

Statement III: Mathematics is based on deductive reasoning.

• This statement is true. Mathematics heavily relies on deductive reasoning, where conclusions are derived from general principles and axioms through logical steps.

Statement IV: Mathematics is much more abstract and hierarchic than most of the other subjects that children learn at the same age.

• This statement is true. Mathematics is often more abstract and has a hierarchical structure, where understanding more complex concepts depends on grasping simpler, foundational concepts first.

Therefore, statements III and IV accurately reflect the nature of mathematics.

Hint

Statement I: Mathematics at school level requires special aptitude in learners.

• This statement is not necessarily true. While some aspects of advanced mathematics may require a high level of aptitude, basic school-level mathematics is designed to be accessible to all students.

Statement II: Primary concepts in mathematics are abstract.

• This statement is not true. At the primary level, mathematical concepts are often introduced using concrete objects and visual aids to help students understand abstract ideas.

A mathematics classroom is a learning environment where students engage in mathematical activities, discussions, and problem-solving, guided by a teacher to enhance their understanding and application of mathematical concepts.

Key Points Argumentation in a mathematics classroom entails constructing a logical chain of reasoning to demonstrate mathematical truths. It aligns with the heuristic approach by fostering active engagement, deep understanding, and collaborative learning, ultimately leading to a more comprehensive grasp of mathematical concepts.

• Logical Reasoning: Argumentation in a mathematics classroom involves constructing a logical chain of reasoning to demonstrate the validity of a mathematical statement or result. It aims to provide clear and coherent steps that support why a particular mathematical conclusion is true.
• Explaining Mathematical Results: Argumentation serves as a vehicle to explain the rationale behind mathematical results.
• Heuristic Approach: Argumentation aligns with the heuristic approach in mathematics education, which emphasizes learning through discovery and exploration. Encouraging students to construct arguments allows them to actively engage with mathematical ideas, helping them discover the principles behind the results.
• Building Understanding: Through argumentation, students develop a solid grasp of mathematical concepts by piecing together interconnected ideas. Constructing arguments requires them to explore relationships, clarify assumptions, and recognize patterns, which collectively enhances comprehension.

Thus, it is concluded that it can be understood as a line of reasoning that intends to show or explain why a mathematical result is true and it can also be a part of the heuristic approach in mathematics education that is appropriate regarding "Argumentation" in a mathematics classroom.

A mathematics textbook is an educational resource that presents mathematical concepts, theories, and problem-solving techniques, designed to facilitate learning and understanding of mathematical principles. It serves as a structured guide for students to explore various mathematical topics.

Key Points Characteristics of a mathematics textbook:

• A mathematics textbook is organized in a systematic manner, presenting topics and concepts in a logical sequence.
• It follows a well-defined structure that helps students build a strong foundation by progressing from fundamental concepts to more advanced ones.
• A mathematics textbook should have varied exercises to give practice to students.
• A good mathematics textbook offers a diverse range of exercises, including practice problems, drill exercises, real-world applications, and challenging problems.
• Mathematics textbooks often include practical applications of mathematical concepts in real-world scenarios, demonstrating how mathematical principles are relevant in various fields, from science and engineering to economics and everyday life.
• A textbook of mathematics should not be very thick or heavy.
• Textbooks are designed to introduce concepts gradually, starting with basic principles and gradually increasing the complexity of problems and ideas.

Thus, it is concluded that the mathematics textbook should have solved examples only is not correct with regard to a mathematics textbook.

Mathematics is the study of patterns, quantities, and relationships using logical reasoning and symbolic language, helping us understand the fundamental structures governing the world around us.

Key Points A conjecture in mathematics is a preliminary statement that suggests a possible mathematical truth based on patterns or observations, awaiting further investigation and mathematical scrutiny to either confirm its validity or disprove it.

• **Hypothetical

Statement:

** A conjecture in mathematics is a proposition or conclusion that is put forth based on observation, pattern recognition, or intuition, without formal proof. It's a hypothesis about a mathematical relationship or property that appears to hold true.

• Proposed Patterns: Conjectures often arise when mathematicians notice consistent patterns or relationships among numbers, shapes, or structures. These patterns spark curiosity and lead to the formulation of a conjecture to describe the observed trend.
• Lack of Rigorous Proof: Unlike theorems, which are proven statements, conjectures lack formal mathematical proof. They represent educated guesses or ideas awaiting confirmation through mathematical reasoning and evidence.
• Exploration and Investigation: Conjectures play a crucial role in mathematical exploration. They guide mathematicians towards uncharted territories, encouraging them to delve deeper into the subject to uncover the underlying principles and relationships.

Thus, it is concluded that a conjecture in mathematics is a proposition or conclusion that is assumed to be true on a tentative basis without proof.

Prime numbers are positive integers greater than 1 that have no divisors other than 1 and themselves, making them indivisible within the set of natural numbers.

Key Points The statement about prime numbers and multiples of 6 exemplifies a conjecture a hypothesis based on observation that requires further investigation and rigorous mathematical analysis to either confirm its truth or uncover its limitations.

• Hypothesis Based on Observation: A conjecture is a statement that is based on initial observations or patterns but hasn't been proven true or false yet. In this case, the statement suggests that specific numbers (1 less or 1 more than multiples of 6) have a prime number property, which is derived from observing some instances.
• Potential Patterns: The statement hints at a pattern involving prime numbers and multiples of 6. However, patterns observed in a limited set of cases might not necessarily hold true for all cases, as demonstrated by numerous counterexamples in the history of mathematics.
• Need for Verification: To confirm the validity of the conjecture, mathematical proof or counterexamples are required. The statement implies a specific relationship between prime numbers and multiples of 6, which needs to be rigorously examined to determine if it's universally true.
• Role in Mathematical Exploration: Conjectures play a vital role in driving mathematical research and exploration. They inspire mathematicians to investigate patterns, search for counterexamples, and either establish them as true theorems or disprove them, contributing to the advancement of mathematical knowledge.

**​Thus, it is concluded that the statement "1 less than or 1 more than a multiple of 6 are prime numbers" is an example of Conjecture.**​

Additional Information

A mathematics teacher is an educator who imparts knowledge of mathematical concepts, problem-solving skills, and critical thinking to students, fostering their understanding and proficiency in the subject. 

Key Points The unitary method simplifies complex scenarios by establishing a clear relationship between known and unknown values and allows for efficient calculations based on the concept of one unit's value.

• Proportional Relationships: The unitary method is a mathematical technique used to solve problems involving proportional relationships. It helps find an unknown quantity by considering the relationship between the known quantity and its corresponding value.
• Direct and Inverse Proportions: The unitary method is applicable to both direct and inverse proportions. In direct proportion, as one quantity increases, the other also increases, and vice versa. In inverse proportion, as one quantity increases, the other decreases, and vice versa.
• Real-World Applications: The unitary method is commonly used to solve everyday problems involving rates, ratios, prices, and quantities. It finds applications in areas such as finance, commerce, science, and engineering, where proportional relationships play a vital role.
• Cross-Multiplication: Once the proportion is set up, you cross-multiply to find the unknown value. This involves multiplying the numerator of one fraction by the denominator of the other fraction. This step simplifies the proportion to a simple equation.

45. So, the cost of 1 banana is Rs. 45/36 = Rs. 1.25. The number of bananas that can be purchased for Rs. 12.50 is 12.50/1.25 = 10. | | C | The third problem, C, is not based on the unitary method. It is a problem of relative speed. |

Thus, it is concluded that statements A and B are daily life problems based on the unitary method is correct that the mathematics teacher posed the problems to her students.

Shortcut Trick

Area of Circle = Area of Rectangle

⇒ (22/7) × r² = 176 × 56

⇒ r² = (176 × 56 × 7) ÷ 22

⇒ r² = 8 × 56 × 7 = 3136 ⇒ r = 56 cm

⇒ Circumference = 2 × (22/7) × 56 = 352 cm

∴ The correct answer is 352 cm.

Alternate Method

Given:

Length of rectangle (L) = 176 cm

Breadth of rectangle (B) = 56 cm

Area of circle = Area of rectangle

Formula Used:

Area of rectangle = L × B

Area of circle = πr²

Circumference of circle = 2πr

Calculations:

⇒ Area of circle = Area of rectangle

⇒ πr² = 176 × 56

⇒ (22/7) × r² = 9856

⇒ r² = (9856 × 7) ÷ 22

⇒ r² = 448 × 7

⇒ r² = 3136

⇒ r = √3136 = 56 cm

⇒ Circumference = 2 × (22/7) × 56

⇒ Circumference = 2 × 22 × 8

⇒ Circumference = 352 cm

∴ The correct answer is 352 cm.

Additional Information

Area and Circumference of Circle

The area represents the space enclosed by the shape where A = πr². The circumference refers to the total boundary length, expressed as C = 2πr.

Area of Rectangle

It is the amount of flat space occupied within the boundaries of a rectangle, calculated as Area = Length × Breadth.

Shortcut Trick

Total Surface Area (TSA) = (Total Cost ÷ Rate) × 100

TSA = (1032 ÷ 20) × 100 = 51.6 × 100 = 5160 cm²

2(lb + bh + hl) = 5160 ⇒ (40 × 36) + X(40 + 36) = 2580

1440 + 76X = 2580 ⇒ 76X = 1140 ⇒ X = 15

∴ The correct answer is 15.

Alternate Method

Given:

Length (l) = 40 cm, Breadth (b) = 36 cm, Height (h) = X cm

Total Cost of painting = ₹1,032

Rate of painting = ₹20 per 100 cm²

Formula Used:

Total Surface Area (TSA) of a cuboid = 2(lb + bh + hl)

TSA = Total Cost ÷ (Rate per cm²)

Calculations:

⇒ Rate per cm² = 20 ÷ 100 = ₹0.2

⇒ Total Surface Area (TSA) = 1032 ÷ 0.2 = 5160 cm²

⇒ 2(40 × 36 + 36 × X + X × 40) = 5160

⇒ 2(1440 + 36X + 40X) = 5160

⇒ 1440 + 76X = 5160 ÷ 2

⇒ 1440 + 76X = 2580

⇒ 76X = 2580 − 1440

⇒ 76X = 1140

⇒ X = 1140 ÷ 76

⇒ X = 15

∴ The correct answer is 15.

Additional Information

Volume of Cuboid

The space occupied by the cuboid is calculated as Volume = Length × Breadth × Height (l × b × h).

Lateral Surface Area (LSA)

The area of the four vertical faces (excluding top and bottom) is LSA = 2 × Height × (Length + Breadth).

Diagonal of a Cuboid

The longest rod that can be placed inside a cuboid is the diagonal, given by d = √(l² + b² + h²).

Explanations:

PQRS is a Quadrilateral in which PQ = PS & RQ = RS is Kite.

Kite is Quadrilateral whose Adjacent sides are equal

Hence, By the Properties of Kite 

The diagonals are perpendicular to each other.

Hence, option 2 is Correct.

Shortcut Trick

In a rhombus, adjacent sides are equal (AD = AB), making ΔABD an isosceles triangle.

⇒ ∠ABD = ∠ADB = 25°

⇒ ∠BAD = 180° − (25° + 25°) = 130°

⇒ ∠ABC = 2 × ∠ABD = 50° (Since the diagonal bisects the angle)

⇒ 2 × ∠BAD − ∠ABC = 2 × 130° − 50° = 210°

∴ The correct answer is 210°.

Alternate Method

Given:

ABCD is a rhombus.

∠ADB = 25°

Calculations:

In a rhombus, all four sides are of equal length, so AD = AB.

⇒ In ΔABD, sides AD and AB are equal, which means the angles opposite to them are also equal.

⇒ ∠ABD = ∠ADB = 25°

The sum of all interior angles in a triangle is 180°.

⇒ ∠BAD + ∠ABD + ∠ADB = 180°

⇒ ∠BAD + 25° + 25° = 180°

⇒ ∠BAD = 180° − 50° = 130°

We know that the diagonal of a rhombus acts as an angle bisector for the interior angles.

⇒ ∠ABC = 2 × ∠ABD

⇒ ∠ABC = 2 × 25° = 50°

Now, substitute these values into the required expression (2 × ∠BAD − ∠ABC):

⇒ 2 × 130° − 50°

⇒ 260° − 50° = 210°

∴ The correct answer is 210°.

Additional Information

Properties of a Rhombus

All four sides are of equal length. Opposite angles are equal, and any two adjacent angles are supplementary (their sum is 180°).

Diagonals of a Rhombus

The diagonals bisect each other at right angles (90°) and completely bisect the interior angles at the vertices they connect.

Area of a Rhombus

The area can be calculated using the lengths of the diagonals: Area = (d₁ × d₂) ÷ 2, or by using the base and height: Area = Base × Height.

Given:

The data elements provided are: 17, 8, 11, 13, 9, 14, 10, 11, 16, 23, 15, 7

 let's arrange this data in ascending order: 7, 8, 9, 10, 11, 11, 13, 14, 15, 16, 17, 23

Concept used:

Mode is the number which appears most frequently in a data set. A data set may have one mode, more than one mode, or no mode at all.

Median is the middle number of a sorted data set; if the data set has an even number of observations

Range is the difference between the highest and lowest values in a data set.

Calculations:

The mode for this dataset is 11.

The median is the average of the two middle numbers. Here, it is (11+13)/2 = 12.

In this case, range = 23 - 7 = 16.

Now that we have the mode (11), median (12), and range (16), we can compute the mean of these three values:

Mean = (Mode + Median + Range) / 3 Mean = (11 + 12 + 16) / 3 = 39 / 3 = 13

Hence, the mean of the mode, median, and range of the given data is 13.

\(\mathrm{x}=√{198} × √{550}\)

x = √(2 × 3 × 3 × 11) × √ (5 × 5 × 2 × 11)

x = 3 × 5 × √22 × √22 = 330 

\(\mathrm{y}=\sqrt[3]{99} × \sqrt[3]{363}\)

y = (3 × 3 × 11)^1/3 × (3 × 11 × 11)^1/3

y = (11)^{2/3 + 1/3} × (3)^{2/3 + (1/3)} = 33

1/x + 1/y = 1/330 + 1/33 = 11/330 = \(0 .0 \overline{3}\)

Hence, Option 2 is correct.

Calculations:

The given expression is:

p = (12.34 × 10¹⁰) - (5.67 × 10⁹)

we need to express these numbers in the same power of ten.

We can rewrite 5.67 × 10⁹ as 0.567 × 10¹⁰.

p = (12.34 × 10¹⁰) - (0.567 × 10¹⁰) = 11.773 × 10¹⁰

To express this in standard scientific notation, we would have it as:

p = 1.1773 × 10¹¹

Hence, The Correct Option is 1.

Calculations:

(-1/4)^-1 = -4

(-7/4)^-1 = (-4/7)

The Reciprocal of \(\left(-\frac{7}{4}\right)^{-1} \) = (-7/4)

Let the Number be x that should be multiplied to \(\left(-\frac{1}{4}\right)^{-1} \) so that product is reciprocal of (-7/4)^-1

(-1/4)^-1 × x = (-7/4)

 x = (-7/4 × -4)

⇒ x = 7/16 

Hence, option 1 is correct.

Concept used: 

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. 

It is divisible by only 1 and itself. For example, 2, 3, 5, 7, 11, 13 are prime numbers.

Calculation:

The last digit of 313632, is 2, means one of the prime number is even, i.e, 2.

Factors (313632) = 2⁵ × 3⁴ × 11²

⇒ p = 11

⇒ q = 2

⇒ r = 3

⇒ p + q - 2r = 11 + 2 - 6 = 7

∴ The value of (p + q - 2r) is 7.

Concept use:

In a pie chart, the total degrees is 360°, which represents the total marks obtained in all subjects.

The angle of each sector representing each subject is proportional to the marks obtained in that subject.

Calculations:

First we need to find the total marks Sunita obtained:

Total Marks = 72 (English) + 84 (Hindi) + 96 (Mathematics) + 80 (Science) + 68 (Social Science) = 400 marks

Angle for Mathematics = (Marks in Mathematics / Total Marks) ×  Total degrees = (96 / 400) × 360 = 0.24 ×  360 = 86.4°

So, option 3) 86.4° represents the angle of the sector for Mathematics marks in the pie chart.