CTET Paper 2 Maths & Science 10th Jan 2023 (English-Hindi-Sanskrit) (Previous Year Paper)

The affective domain of child development deals with the child's emotions, feelings, and values. It includes the child's ability to understand, express, and regulate their emotions, as well as their ability to form relationships with others.

Key Points

• The emotional domain of child development focuses on the affective component of a child's personality, which involves their emotions, feelings, and reactions to people, objects, and surroundings.
• It encompasses the understanding and expression of emotions, the development of empathy and compassion, and the regulation of emotional responses.
• During early childhood, children begin to develop a wide range of emotions, such as joy, sadness, anger, fear, and disgust. They also learn to recognize and interpret facial expressions and body language as cues to understand others' emotions.
• As children grow, they become more capable of understanding and expressing their own emotions, as well as recognizing and responding to the emotions of others.

Hint

• The physical domain of child development deals with the child's physical growth and development. It includes the child's development of motor skills, their health and nutrition, and their overall physical well-being.
• The spiritual domain of child development deals with the child's spiritual development. It includes the child's understanding of their own spirituality as well as their ability to connect with others on a spiritual level.
• The cognitive domain of child development deals with the child's intellectual development. It includes the child's ability to think, learn, and solve problems.

Hence, we can conclude that the emotional domain of child development deals with the affective component of a child's personality-her 'feelings' towards people, objects and surroundings.

The principle of individual differences states that each learner is unique and has their own set of characteristics, abilities, strengths, weaknesses, and learning styles.

Key Points

• The principle of individual difference variation states that there may be a difference in the speed of growth and development of children*.*
• This principle recognizes that children can vary in their rates of growth and development.
• Each child has their own unique pace and pattern of physical, cognitive, linguistic, and socio-emotional development.
• It acknowledges that not all children will reach developmental milestones at the same time or follow the same trajectory of growth.

Thus, it is concluded that option 3 is the correct answer.

Key Points

• ​Socialization refers to the process by which individuals learn the norms, values, and behaviors of their society or social group. It involves internalizing social expectations and acquiring the necessary skills to function effectively within a given social context.
• Acculturation  refers to the process of acquiring or adopting the cultural traits and practices of a different cultural group. This occurs when individuals or groups come into contact with a new culture and adjust their own behaviors and beliefs accordingly.

Thus, children learn their societal roles through the processes of socialization and acculturation.

Jean Piaget was a Swiss psychologist known for his theory of cognitive development. His theory proposes that children progress through distinct stages of cognitive development, characterized by different ways of thinking and understanding the world.

Key Points

• In the scenario described, when a child is shown a toy and it is then hidden, if the child actively searches for the toy, it demonstrates their developing ability to grasp the concept of object permanence.
• According to Jean Piaget's theory, object permanence develops during the sensory-motor stage of cognitive development, which typically occurs from birth to around 2 years of age.
• During this stage, children learn about the world through their senses and motor actions.
• Object permanence refers to the understanding that objects continue to exist even when they are not directly perceived or sensed.

Thus, it is concluded that, according to Jean Piaget, when a child is shown a toy, it is then hidden. Instead of forgetting about it, the child tries to find that toy. At the sensory-motor stage, this characteristic is found in the child, and it is called object permanence.

Key Points

• To facilitate cognitive development in children, Jean Piaget emphasized the role of activity, particularly hands-on exploration and interaction with the environment.
• Piaget believed that children construct their understanding of the world through active engagement in physical and mental activities, such as playing, experimenting, and problem-solving.
• Lev Vygotsky emphasized the importance of language in cognitive development.
• Vygotsky believed that language plays a crucial role in shaping children's thinking and cognitive processes.
• He emphasized the social and cultural aspects of learning, highlighting the influence of social interactions and language in scaffolding children's cognitive development.

Thus, it is concluded that To facilitate cognitive development in children, ​Jean Piaget gave importance to​ activity, while Lev Vygotsky emphasized language.

Lawrence Kohlberg's theory of moral development is a cognitive-developmental theory that describes how individuals develop their understanding of moral reasoning and decision-making.

Key Points

• In Lawrence Kohlberg's theory of moral development, the 'good boy-good girl' stage is also known as the interpersonal conformity or approval-seeking stage.
• At this stage, individuals base their moral judgements and actions on the desire for social approval and to maintain positive relationships with others.
• They seek to be seen as good and moral by those around them, and they strive to please others.
• The focus of moral reasoning in this stage is on meeting social expectations, following rules, and avoiding disapproval or conflict. The individual's sense of morality is largely shaped by the opinions and judgments of others.

Thus, it is concluded that According to 'Lawrence Kohlberg, the 'Good boy-good girl' stage of moral development is characterized by the individuals wanting to be respected by others and trying to do what will please others.

Vygotsky's theory emphasizes the importance of exploring what children can achieve with support and guidance rather than limiting them to their current capabilities.

Key Points

• Lev Vygotsky's sociocultural theory emphasizes the concept of the zone of proximal development (ZPD). According to Vygotsky, the ZPD represents the range of tasks or concepts that a child can perform or understand with the guidance and support of a more knowledgeable individual, such as a teacher or a peer. It is the area where learning and cognitive development occur.
• Vygotsky believed that effective instruction and learning should focus on tasks and concepts that are just beyond a child's current capabilities. By providing appropriate support, guidance, and scaffolding, the child can reach higher levels of understanding and performance.
• Vygotsky argued that it is through this process of guided learning within the ZPD that children can gradually internalize new knowledge and skills, eventually becoming capable of independent problem-solving and comprehension.

Thus, it is concluded that Lev Vygotsky believes in exploring what is currently beyond the children's capabilities but what they can do with support.

Lev Vygotsky, a prominent psychologist, emphasized the social and cultural aspects of cognitive development.

Key Points

• According to Vygotsky's sociocultural theory, cognitive development is deeply influenced by social interactions, language, and cultural practices.
• Vygotsky believed that cognitive processes are shaped through social interactions with more knowledgeable individuals, such as parents, teachers, and peers.
• Vygotsky proposed the concept of the "zone of proximal development" (ZPD), which refers to the gap between a learner's current level of independent functioning and their potential level of development with the guidance and support of a more skilled individual.
• In the ZPD, social interactions and scaffolding from more knowledgeable individuals play a crucial role in promoting cognitive growth and the acquisition of new skills.
• Language, according to Vygotsky, serves as a vital tool for cognitive development. Initially, children rely on social speech (communication with others) and later internalize it to form inner speech (self-directed thinking). Language allows individuals to organize their thoughts, regulate their behaviour, and solve problems.

Thus, it is concluded that Lev Vygotsky viewed cognition as inherently social and language-based.

In a child-centered classroom, the focus is on the needs, interests, and abilities of the individual child. It is an approach to education that emphasizes the importance of active learning, student autonomy, and personalized instruction.

Key Points

• The focus is on creating a supportive and inclusive environment where every child's individuality is respected and valued.
• The teacher aims to understand each student's strengths, challenges, and learning styles and adapt teaching methods and activities to meet their specific needs.
• Individual attention allows the teacher to provide personalized guidance, support, and feedback to help each learner reach their full potential.
• This approach promotes student engagement, active participation, and a sense of ownership in the learning process.

Thus, it is concluded that In a child-centered classroom, the teacher pays individual attention to each learner.

According to Gardner, there are multiple forms of intelligence that go beyond the traditional view of intelligence focused solely on cognitive abilities measured by IQ tests.

Key Points

• Howard Gardner proposed the theory of multiple intelligences, which suggests that individuals have different types of intelligence (such as linguistic, logical-mathematical, spatial, musical, bodily-kinesthetic, interpersonal, intrapersonal, and naturalistic) and that these intelligences can vary in degrees among individuals.
• This theory challenges the notion that intelligence is a single, fixed entity and highlights the diverse ways in which individuals can excel in different areas.

Thus, it is concluded that Howard Gardner believed that each individual possesses different types of intelligence to varying degrees.

Private speech refers to the act of talking to oneself either out loud or silently.

Key Points

• Vygotsky argued that when students engage in private speech, they are essentially using language to guide and regulate their own thoughts, actions, and problem-solving processes.
• By verbalizing their thoughts, students can better organize and control their thinking, leading to improved cognitive abilities and problem-solving skills.

Thus, it is concluded that, According to Lev Vygotsky, private speech helps students regulate their thinking.

Key Points 

• Assertion (A): This assertion emphasizes the importance of teachers holding the belief that every child, regardless of their gender, has the capacity to excel in logical-mathematical reasoning and possess linguistic skills. It promotes the idea that intellectual abilities are not determined by one's gender but rather by individual potential and opportunities for learning and development. By adopting this belief, teachers can create an inclusive and supportive learning environment that encourages all students to explore and develop their skills in these areas, fostering a sense of equality and fairness in education.
• Reason (R): The reason provided is that gender is a social construct. Gender refers to the roles, behaviours, and expectations that society assigns to individuals based on their perceived sex. It is distinct from biological sex, which is determined by physical and physiological characteristics. By understanding gender as a social construct, teachers can challenge traditional gender stereotypes and biases that may limit opportunities for certain students based on their perceived gender.

Thus, it is concluded that Both (A) and (R) are true and (R) is the correct explanation of (A).

A teacher should strive to include a wide range of diversity in their classrooms to create an inclusive and enriching learning environment.

Key Points

• Economic Diversity: This refers to the differences in the financial backgrounds and resources of students. It is crucial for a teacher to be aware of economic disparities among students and ensure that all students have equal access to educational resources, regardless of their economic circumstances.
• Regional Diversity: Regional diversity encompasses differences in geographical origins, cultural practices, and traditions. Students from different regions may bring unique perspectives and experiences to the classroom. Acknowledging and celebrating regional diversity helps create a culturally responsive classroom environment that values and respects various cultural backgrounds.
• Linguistic Diversity: Linguistic diversity involves recognizing and appreciating the various languages and dialects spoken by students. Teachers should acknowledge that students may have different levels of proficiency in the primary language of instruction, as well as varying abilities in other languages.
• Socio-cultural Diversity: Socio-cultural diversity encompasses differences in social, ethnic, and cultural backgrounds. It includes factors such as race, religion, ethnicity, and social customs. A teacher should foster an inclusive classroom where students from diverse socio-cultural backgrounds feel respected, valued, and represented.

Thus, it is concluded that (i) (ii) (iii) (iv) is the correct answer.

In a socio-constructivist classroom, assessment practices align with the principles of socio-constructivist learning theory. This approach emphasizes active learning, collaboration, and the construction of knowledge through social interactions.

Key Points

• Assessment includes a variety of tools, such as portfolios and classroom interactions.
• Portfolios allow students to showcase their learning progress, reflect on their learning experiences, and demonstrate their understanding through various artifacts.
• Classroom interactions, such as discussions, group projects, and presentations, provide opportunities for students to engage in collaborative learning and demonstrate their understanding through active participation.

Thus, it is concluded that Assessment includes a variety of tools including portfolios, classroom interactions  characterizes assessment in a socio-constructivist classroom.

The question is assessing the ability to think critically and develop solutions for creating a progressive classroom that respects the needs, interests, and capabilities of every child.

Key Points

• In this context, problem-solving skills would be necessary to analyze the challenges and complexities involved in creating an inclusive and respectful learning environment.
• Teachers would need to identify and address barriers, adapt instructional strategies to meet individual needs, and find innovative solutions to ensure that every child's needs, interests, and capabilities are respected and accommodated.

Thus, it is concluded that problem-solving skills are tested in the question.

Inclusion refers to the practice of ensuring that all individuals, regardless of their abilities, backgrounds, or characteristics, are fully integrated and provided with equal opportunities in educational settings. Inclusive education aims to create an environment where diverse learners can learn together, benefit from each other's strengths, and receive appropriate support to meet their individual needs.

Key PointsInclusion is promoted by: 

• Flexible grouping: This practice involves grouping students based on their learning needs and abilities, allowing for fluid and dynamic arrangements that can change based on the tasks or objectives.
• Reasonable accommodation: This practice involves providing necessary adjustments and supports to ensure that individuals with disabilities or special needs can fully participate in educational activities. Reasonable accommodations may include modifications to the physical environment, instructional materials, or teaching methods, enabling students to access learning opportunities on an equal basis with their peers.
• Differentiated instruction: This approach recognizes that students have diverse learning styles, interests, and readiness levels. Differentiated instruction involves tailoring teaching methods, content, and assessment to meet individual student needs and promote engagement and understanding. It promotes inclusion by addressing the unique requirements of each student and providing multiple pathways to learning.

Thus, it is concluded that Inclusion is not promoted by labelling and segregation.

Hint

• Labelling involves assigning fixed categorizations or stereotypes to individuals based on their characteristics or abilities.
• Segregation refers to the separation of students with disabilities or special needs from their peers in separate classrooms or educational settings.
• These practices create barriers, perpetuate stigma, and hinder the full participation and integration of individuals in the educational community, which goes against the principles of inclusive education.

When teaching children from diverse backgrounds in elementary classes**, the use of examples plays a crucial role** in enhancing their understanding and creating an inclusive learning environment.

Key PointsBy giving examples and also encouraging children to bring examples from their own lives, teachers can achieve several important objectives:

• Relatability and relevance: Providing examples that children can personally connect with helps make the content more relatable and relevant to their own experiences. This increases engagement and promotes deeper understanding.
• Inclusion and representation: Encouraging children to bring examples from their own lives allows for diverse perspectives and experiences to be shared and valued in the classroom. It promotes a sense of belonging and validates the diversity of backgrounds and cultures present among the students.
• Critical thinking and participation: Prompting children to bring examples encourages active participation and critical thinking skills.

Thus, it is concluded that option 2 is the correct answer.

Key Points 

• Assertion (A): This assertion is true. Students with ADHD often struggle with maintaining focus and attention and reducing distractions in the learning environment can help them concentrate better and engage in learning activities more effectively.
• Reason (R): This reason is also true and provides a correct explanation for the assertion. One of the key characteristics of ADHD is difficulty with sustained attention and a higher susceptibility to distractions. This reduced ability to maintain concentration can significantly impact a student's learning experience. Therefore, a well-designed approach that minimizes visual and auditory distractions can help create a more conducive learning environment for students with ADHD.

Thus, it is concluded that Both (A) and (R) are true and (R) is the correct explanation of (A).

Visual impairment refers to a condition in which an individual experiences a partial or complete loss of vision or has difficulty seeing.

Key Points

• The inclusion of students with visual impairments involves creating an accessible and inclusive learning environment that accommodates their specific needs.  The use of tactile and other materials that are embossed is the most appropriate and relevant choice for supporting students with visual impairments.
• Students with visual impairments often rely on their sense of touch to gather information and interact with their surroundings.
• The use of tactile materials, such as raised or embossed diagrams, maps, and models, allows students to access information through touch and promotes their understanding of concepts that may be difficult to grasp through visual means alone.

Thus, it is concluded that the Inclusion of students with visual impairments requires the use of tactile and other materials that are embossed.

Creativity is a cognitive and psychological phenomenon that involves the generation of new and original ideas, concepts, or solutions. It is characterized by the ability to think outside the box, break away from conventional thinking patterns, and produce novel and valuable outcomes.

Key Points

• Divergent thinking refers to the ability to generate multiple ideas, possibilities, or solutions in response to a given problem or stimulus. In divergent thinking, individuals explore various perspectives, think outside the box, and produce a wide range of diverse and unconventional ideas.
• Creativity involves breaking away from conventional or traditional ways of thinking and exploring new possibilities.
• Divergent thinking allows for the generation of multiple unique ideas and encourages originality and innovation.
• It is characterized by the ability to see connections between seemingly unrelated concepts, to think flexibly, and to generate alternative solutions or approaches.

Thus, it is concluded that Divergent thinking is an identifying feature of creativity.

Active engagement refers to the active participation and involvement of learners in the learning process. It involves students being mentally and physically engaged, taking an active role in their own learning, and being actively involved in activities, discussions, and interactions.

Key Points

• Enquiry and questioning are key aspects of active engagement.
• When students engage in inquiry and questioning, they are actively seeking knowledge, exploring concepts, and seeking deeper understanding.
• They are encouraged to ask thoughtful questions, investigate different perspectives, and actively participate in discussions or activities that promote higher-order thinking skills.

Thus, it is concluded that active engagement means enquiry and questioning.

Knowledge construction refers to the process by which individuals actively create, organise, and build new knowledge and understanding based on their prior knowledge, experiences, and interactions with the world.

Key Points

• Organising and reorganising previous knowledge: This process involves taking existing knowledge and actively organising and reorganising it to make sense of new information or experiences. By connecting new information with what is already known, learners can construct new knowledge and understanding.
• Exploration and discovery: The process of exploration and discovery involves actively engaging with new ideas, concepts, or experiences. It includes activities such as investigation, experimentation, problem-solving, and inquiry-based learning. Through exploration and discovery, learners can make connections, generate hypotheses, and develop new insights and understandings.
• Social interactions: Social interactions play a significant role in knowledge construction. Collaborative discussions, group work, and interactions with peers, teachers, and experts provide opportunities for sharing perspectives, debating ideas, and co-constructing knowledge. Through social interactions, learners can gain new insights, challenge their existing beliefs, and develop a deeper understanding of concepts and information.

Thus, it is concluded that (i), (ii), (iii) is the correct answer.

Key Points

• Assertion (A) This assertion is true. By providing students with a vocabulary of emotions, teachers help expand their emotional intelligence and understanding. Journal writing provides a safe space for students to express and reflect on their feelings, promoting self-awareness and emotional exploration.
• Reason (R)  This reason is also true and provides a valid explanation for the assertion. When students are equipped with a vocabulary of emotions and encouraged to express their feelings, they become better equipped to understand and manage their emotions effectively. Recognizing and expressing emotions is an essential component of emotional self-regulation, which supports students in developing healthy coping strategies and managing their emotional well-being.

Thus, it is concluded that Both (A) and (R) are true and (R) is the correct explanation of (A).

Means-end analysis is a problem-solving strategy that involves breaking down a complex problem into smaller, more manageable sub-goals or intermediate steps. 

Key Points

• In means-end analysis, the problem solver identifies the ultimate goal and then determines the differences between the current state (the initial problem) and the desired goal state.
• These differences are then broken down into sub-goals or intermediate steps, and the problem solver focuses on solving each sub-goal individually.
• By accomplishing the sub-goals, the problem solver moves closer to the ultimate goal.

Thus, it is concluded that in means-end analysis, the problem is divided into a number of intermediate goals and then the focus is on solving each sub-goal.

Mastery-oriented learners are individuals who exhibit specific characteristics and behaviours that reflect a focus on learning, growth, and mastery of the subject matter.

Key Points

• The primary characteristic of mastery-oriented learners is their focus on learning and their motivation to understand the concepts.
• Mastery-oriented learners are driven by the desire to acquire knowledge, develop skills, and deepen their understanding of the subject matter.
• Mastery-oriented learners are intrinsically motivated, meaning that they engage in learning for the inherent satisfaction of learning itself rather than solely for external rewards or recognition.
• They have a growth mindset, believing that their abilities and intelligence can be developed through effort, practice, and effective learning strategies.

Thus, it is concluded that option 4 is the correct answer.

Metacognition refers to the awareness and understanding of one's own thinking processes and the ability to regulate and control one's cognitive processes.

Key Points

• Metacognition involves reflecting on one's thoughts, experiences, and learning strategies.
• In the given scenario, Roshni encourages the students to reflect on their own learning experiences as readers, specifically asking them to consider what they have learned about themselves as readers.
• This reflection prompts students to engage in metacognitive thinking by evaluating their own reading abilities, strengths, challenges, and strategies.

Thus, it is concluded that she is promoting meta-cognition.

Key Points 

• Assertion (A): This assertion is true. Vygotsky's sociocultural theory emphasizes the importance of social interaction and collaboration in the learning process. According to Vygotsky, learning is enhanced when individuals engage in social interactions, participate in collaborative activities, and receive support and guidance from more knowledgeable others, such as teachers or peers.
• Reason (R): This reason is also true and provides a valid explanation for the assertion. Vygotsky's theory posits that learning is a social and cultural process. He emphasized the role of social interactions, language, and cultural tools in shaping cognitive development and learning. According to Vygotsky, learning occurs through interactions with others, through dialogue and collaborative problem-solving, and through the internalization of social knowledge and cultural tools.

Thus, it is concluded that Both (A) and (R) are true and (R) is the correct explanation of (A).

When students make errors or mistakes in the process of learning, it is important for a teacher to address them effectively.  Working out appropriate strategies to induce cognitive conflict aligns with best practices in education.

Key Points

• Cognitive conflict refers to the mental discomfort or discrepancy experienced by learners when their existing knowledge or beliefs are challenged or contradicted by new information or experiences.
• By intentionally designing learning experiences that induce cognitive conflict, teachers can encourage students to actively engage in critical thinking, reflection, and the revision of their existing understanding.

Thus, it is concluded that A teacher should address the errors and mistakes made by students in the process of learning by working out appropriate strategies to induce cognitive conflict.

Intrinsic motivation refers to engaging in an activity or behaviour for the inherent satisfaction, enjoyment, or interest it brings, rather than solely for external rewards or incentives.

Key Points

• In this case, Urmil's motivation for researching her project is driven by her personal interest in the topic.
• She finds the subject matter intriguing or enjoyable, which provides her with intrinsic motivation to engage in the research.
• Intrinsic motivation is often associated with increased engagement, perseverance, and creativity, as individuals are driven by their own interests and internal satisfaction rather than external factors.

Thus, it is concluded that Urmil researches her project because the topic interests her, which is an example that illustrates intrinsic motivation.

The teacher's pedagogy, as described in the scenario, aligns with the socio-constructivist perspective. The socio-constructivist perspective emphasizes the importance of social interactions, cultural contexts, and the active construction of knowledge through collaboration and meaningful learning experiences.

Key Points

• In this scenario, the teacher recognizes that students in the class have difficulty with Standard English and are more comfortable expressing themselves in their native language.
• By encouraging students to use their preferred language, the teacher acknowledges and values the students' linguistic diversity and cultural backgrounds.
• This approach promotes inclusivity and validates students' identities and experiences.

Thus, it is concluded that the Socio-constructivist perspective is what the teacher's pedagogy rooted in.

Calculation:

We know that Real numbers = rational numbers + irrational numbers

So, every rational number is a real number, but every real number is not a rational number.

∴ The correct answer is option 2

Concept used:

If the difference of alternate digits of a number is zero or multiple of 11, then the number will be divisible by 11

Calculation:

From option 3, (2 + 2 + 2 + 2) - (2 + 2 + 2) = 2 [ so it is NOT divisible by 11]

∴ The correct answer is 2222222.

Concept used:

If k be any number, then k⁰ = 1

a^-1 = 1/a

Calculation:

[(-1)0 + (-2)1 + (-3)2 + (-4)3] × (1)-3

⇒ [ 1 + (-2) + 9 + (- 64)] × (1)³

⇒ 1 - 2 + 9 - 64

⇒ - 56

∴ The correct answer is - 56

Calculation:

$1-\frac{1}{1-\frac{1}{2-\frac{1}{3}}}$

⇒  $1-\frac{1}{1-\frac{1}{\frac{5}{3}}}$ 

⇒  $1-\frac{1}{1-\frac{3}{5}}$ 

⇒ $1-\frac{1}{\frac{2}{5}}$

⇒  $1-\frac{5}{2}$ 

⇒ $-\frac{3}{2}$

∴ The correct answer is $-\frac{3}{2}$

Calculation:

$\sqrt{\frac{0.081 \times 2.5 \times 3.24}{0.0625 \times 7.29}}$

⇒ $\sqrt{\frac{\frac{81}{1000} \times \frac{25}{10}\times \frac{324}{100}}{\frac{625}{10000} \times \frac{729}{100}}}$ 

⇒ ${\frac{\frac{9 \times 5\times 18}{1000}}{\frac{25 \times27}{1000} }}$

⇒ $\frac{810}{675}$

⇒ 1.2

∴ The correct answer is 1.2

Concept used:

$\rm x+\frac{1}{x}=a$, then $\rm x^2+\frac{1}{x^2}=a^2 - 2$ and $\rm x^4+\frac{1}{x^4}=[x^2-\frac{1}{x^2}]^2 - 2$ 

Calculation:

$\rm x+\frac{1}{x}=4$ then $\rm x^2+\frac{1}{x^2}=(4^2 - 2)$ = 14 now, $\rm x^4+\frac{1}{x^4}= (14^2 - 2)$ = (196 - 2) = 194

∴ The correct answer is 194

Shortcut Trick

Half of 28 = 14.

8 less than 14 = 6 (This is one-sixth of the number).

Original number = 6 × 6 = 36.

Required one-fourth = 36 ÷ 4 = 9.

∴ The correct answer is 9.

Alternate Method

Given:

Let the unknown number be x.

One-sixth of the number = (1/6)x

Half of 28 = 28 ÷ 2 = 14

Calculations:

According to the question:

⇒ (1/6) × x = (28 ÷ 2) − 8

⇒ (1/6)x = 14 − 8

⇒ (1/6)x = 6

⇒ x = 6 × 6 = 36

Now, calculate one-fourth of the number:

⇒ Required value = (1/4) × 36

⇒ Required value = 9

∴ The correct answer is 9.

Additional Information

Fraction of a Number

To find a fraction (a/b) of a number N, multiply the number by the numerator and divide by the denominator: (a × N) ÷ b.

Linear Equation Modeling

When solving word problems, translate is as = and less than as subtraction from the following term.

Reciprocal Rule

If 1/n of a number is k, then the full number is found by multiplying k by the reciprocal n.

Calculation:

Let's assume that the number is k

According to the question,

⇒ $\frac{2}{3}$k = 84

⇒ k = 84 × $\frac{3}{2}$ = 126

Then 45% of 126 is = 126 × 45/100 = 56.7

∴ The correct answer is 56.7

Calculation:

(2x - 3)(x + 4) + 2(x - 3)(x - 1) = 4x2 + 2x - 11

⇒ 2x² + 8x - 3x - 12 + 2[x² - x - 3x + 3] =  4x2 + 2x - 11

⇒ 2x2 + 5x - 12 + 2x² - 2x - 6x + 6 = 4x2 + 2x - 11

⇒ 4x² - 3x - 6 = 4x2 + 2x - 11

⇒ 5x = 5

⇒ x = 1

⇒ 2x + 3 = 5

∴ The correct answer is 5

Given:

A shopkeeper sold 10 toffees for Rs. 6 

Calculation:

Selling price of each toffee 6/10 = Rs. 0.6

The shopkeeper gain 20%, which means CP : SP = 5 : 6

Here, 6 unit → 0.6

Then, CP = 5 unit → $\frac{0.6}{6}$ × 5 = Rs. 0.5

With the cost price of Rs. 0.5, he get 1 toffee

Then, with the price of Rs. 4, he get = $\frac{1}{0.5}$ × 4 = 8

∴ The correct answer is 8

Concept used:

The line that divides a shape or an object into two equal and symmetrical parts is called the line of symmetry.

Calculation:

Rectangle and rhombus have exactly two lines of symmetry.

Important Points

• Square has four lines of symmetry

• Kite has one lines of symmetry

Calculation:

In the isosceles ΔABC, it is given that ∠ABC = 40° and BA = BC

Then ∠BAC = ∠BCA [ corresponding angles of equal sides ]

Now, ∠BAC = ∠BCA = (180° - 40°)/2 = 70° 

Now, ∠BCD = (180° - 70°) = 110° [ as ACD is a straight line ]

∴ The correct answer is 110°

Concept used:

The point of intersection of the joining of the angle bisector of a triangle is the incentre

Calculation:

According to the picture, in the ΔABC, BD and CD are angle bisectors

Then, D is the incentre, so ∠BDC = 90 + 1/2 ∠BAC 

⇒ ∠BDC =  90 + 1/2 × 110

⇒ ∠BDC = 145 °

In the ΔBDC ∠D = 145 °

∴ The correct answer is 145 °

Shortcut Trick

The difference between the supplementary angle and complementary angle of any angle is always a constant 90°.

Given ratio of Supplement : Complement = 3 : 1.

Difference in ratio parts = 3 − 1 = 2 units.

Equating to the constant difference: 2 units = 90° ⇒ 1 unit = 45°.

Supplementary angle corresponds to 3 units ⇒ 3 × 45° = 135°.

∴ The correct answer is 135°.

Alternate Method

Given: Ratio of Supplement to Complement = 3 : 1.

Formula Used: If an angle is θ, then its Complement = 90° − θ and its Supplement = 180° − θ.

 

Calculation:-

⇒ Let the required angle be θ.

⇒ According to the question: (180° − θ) / (90° − θ) = 3 / 1

⇒ 1 × (180° − θ) = 3 × (90° − θ)

⇒ 180° − θ = 270° − 3θ

⇒ 3θ − θ = 270° − 180°

⇒ 2θ = 90°

⇒ θ = 45°

⇒ Supplementary angle = 180° − 45° = 135°.

∴ The correct answer is 135°.

Additional Information

Complementary Angles

Two angles are said to be complementary if their sum is exactly 90°. Each angle is the complement of the other.

Supplementary Angles

Two angles are said to be supplementary if their sum is exactly 180°. They form a linear pair when placed adjacently on a straight line.

Reflex Angle

An angle which is greater than 180° but less than 360° is known as a reflex angle.

Angle at a Point

The sum of all angles around a single point is always 360°.

Calculation:

According to the picture, in the ΔABD, given that ∠ABD = 62° and ∠ADB = 90° then ∠DAB = (90 - 62) = 28° 

Now, ∠CAD = (90 - ∠DAB) = (90 - 28) = 62° 

∴ The correct answer is 62°

Concept used:

 In a rhombus, all sides are equal and diagonals bisect each other at right-angle

Calculation:

Each side of the rhombus = 60/4 = 15 cm

Area of Rhombus = 1/2 × Diagonal 1 × Diagonal 2

So, according to the picture, In the ΔBOC,

 ∠BOC = 90° [ as diagonals bisect each other at right-angle ], BO = 24/2 = 12 cm, BC = 15 cm, then CO = 9 [ as 9, 12, 15 Pythagorean triplet ]

Then the length of the other diagonal AC = 9 × 2 = 18 cm

Area of the rhombus = 1/2 × BD × AC = 1/2 × 24 × 18 = 216 cm²

∴ The correct answer is 216 cm²

Given:

The cost of painting four walls at the rate of Rs 25 per m2 is Rs 21000

Concept used:

Area of four walls = 2(L + B) × H

The perimeter of rectangular floor = 2(L + B)

Here, L = length, B = breadth and H = height

Calculation:

The perimeter of rectangular floor = 2(L + B) = 105 

Area of the four walls = 21000/25 = 840 m²

Then, according to the question,

⇒ 2(L + B) × H = 840

⇒ 105 × H= 840

⇒ H = 840/105 = 8 m

∴ The correct answer is 8 m

Shortcut Trick

Volume of milk remains constant when poured into the new container.

⇒ Volume of Cuboid = Volume of Cylinder

⇒ 22 × 22 × h = (22 / 7) × 10 × 10 × 77

⇒ 484 × h = 22 × 100 × 11 = 24200 ⇒ h = 50

∴ The correct answer is 50 cm.

Alternate Method

Milk in Cylinder Milk in Cuboid

10 cm

77 cm

22 cm 22 cm

h = ?

Poured Volume 1 = Volume 2

Given:

Radius of cylindrical can (r) = 10 cm

Height of milk in cylinder (h₁) = 77 cm

Base side of cuboidal can (a) = 22 cm

Let the new height of the milk be h₂.

Formula Used:

Volume of Cylinder = π × r² × h

Volume of Cuboid = Base Area × Height = a² × h

Calculation:

⇒ Since the milk is transferred, the volume remains exactly the same.

⇒ Volume of milk in cuboid = Volume of milk in cylinder

⇒ a² × h₂ = π × r² × h₁

⇒ 22² × h₂ = (22 / 7) × 10² × 77

⇒ 484 × h₂ = 22 × 100 × 11

⇒ 484 × h₂ = 24200

⇒ h₂ = 24200 ÷ 484

⇒ h₂ = 50 cm

∴ The correct answer is 50 cm.

Additional Information

Volume of a Cylinder

The capacity of a cylindrical shape is calculated as V = π × r² × h, where r is the base radius and h is the height.

Volume of a Cuboid

The capacity of a rectangular box is calculated as V = l × w × h. If the base is a square of side a, the formula simplifies to V = a² × h.

Conservation of Volume

When a liquid is transferred from one container to another, its total volume remains constant regardless of the change in shape. The liquid naturally takes the shape of the new container up to an equivalent volume level.

Calculation:

The mean of 14, 13, 18, 16, k, (k + 3) = 13 which means

⇒ [14 + 13 + 18 + 16 + k + (k + 3)]/6 = 13

⇒ [ 2k + 64] = 78

⇒ k = 7

Then, the mean of k, 8, 9, 11, 5, 10, 6 is

⇒ [k + 8 + 9 + 11 + 5 + 10 + 6]/7 

⇒ [7 + 8 + 9 + 11 + 5 + 10 + 6]/7 

⇒ 56/7

⇒ 8

∴ The correct answer is 8

Concept used:

Probability of an event P(A) = (favorable outcomes)/(total number of outcomes)

Calculation:

There are 150 marbles of three colours

One marble is drawn, and if the probability of drawing a white marble is $\frac{7}{10}$ , then making the denominator equal to 150 we get

⇒ $\frac{7 \times 15}{10 \times 15}$ = 105/150, so there are 105 white marbles

Similarly, $\frac{3 \times 6}{25 \times 6}$ = 18/150, So total number of black marble = 18

Now, total number of green marble is 150 - (105 + 18) = 27

∴ The correct answer is 27

The characteristics of reasoning in mathematics include "Looking for patterns" and "Making and testing conjectures," which highlight key aspects of mathematical thinking.

Key Points

• "Looking for patterns" involves recognizing regularities in data or structures, which can lead to the formulation of conjectures or hypotheses.
• "Making and testing conjectures" involves creating educated guesses about mathematical properties and then rigorously examining their validity through logical reasoning or empirical evidence.
• These two aspects are essential in problem-solving, theorem proving, and advancing mathematical knowledge.
• They allow mathematicians to explore, understand, and communicate complex concepts, ultimately contributing to the continuous development and enrichment of the field.

Hence, it can be concluded that option 2 is the correct answer.

Mathematisation refers to the process of identifying, formulating, and solving problems using mathematical methods and principles.

Key Points

• The framework emphasizes that mathematics education should go beyond merely teaching computational skills and memorization of formulas.
• It aims to enable students to apply mathematical thinking and reasoning to real-world situations, fostering their problem-solving skills and analytical abilities.
• By developing mathematisation skills, students can better understand the world around them and appreciate the relevance and significance of mathematics in various fields beyond the classroom.

Hence, it can be concluded that option 3 is the correct answer.

By physically conducting experiments and recording outcomes, students actively participate in the learning process, making it more meaningful and memorable.

Key Points

• Through this hands-on approach, students develop a concrete understanding of probability concepts like sample space, equally likely outcomes, and relative frequency.
• They observe patterns and begin to grasp the concept of probability intuitively.
• This activity also encourages critical thinking as students analyze the data they collect and draw conclusions about the likelihood of different outcomes.
• Moreover, the activity promotes teamwork and communication skills as students discuss their findings and observations with their peers.
• It creates an interactive and dynamic learning environment, fostering curiosity and a deeper appreciation for mathematics.

Hence, it can be concluded that Asking students to throw a dice or flip a coin and note down the number of times each possibility occurs is the best approach.

When teaching students how to write a geometrical proof, it is essential for the teacher to guide them by providing logical arguments and justifications for each step taken. This involves explaining the importance of reasoning and providing evidence to support each statement made in the proof.

Key Points

• The teacher can demonstrate the process of constructing a proof by breaking it down into smaller steps.
• They can explain the significance of using axioms, definitions, and previously proven theorems as the foundation for the proof.
• By emphasizing the use of deductive reasoning, students can learn to derive new statements based on the given information.
• Through explicit instruction and examples, the teacher can show students how to make logical connections between statements, use appropriate mathematical language, and follow a clear and organized structure in their proofs.
• Additionally, the teacher can encourage students to ask questions and challenge their own reasoning to strengthen their proof-writing skills.

Hence, it can be concluded that option 3 is the correct answer.

A clinometer is a device used to measure angles of inclination or elevation.

Key Points

• It can be constructed using simple materials such as a protractor, a straw or a straw-like object, and a weight or a plumb bob.
• By constructing a clinometer, students can explore the concepts of trigonometry, specifically trigonometric ratios like sine, cosine, and tangent.
• They can measure the angle of elevation or inclination of various objects or locations and use trigonometric ratios to calculate the corresponding side lengths or heights.

Hence, it can be concluded that a clinometer is the correct answer.

Alternative interpretations refer to students' understanding of mathematical concepts that deviate from conventional or expected interpretations.

Key Points

• These alternative interpretations can lead to errors because they may not align with established mathematical principles and problem-solving strategies.
• To address this, it is important for educators to actively identify and address students' misconceptions or alternative interpretations.
• This can be achieved through targeted instruction, including clear explanations, concrete examples, and guided practice.
• By providing multiple representations of mathematical concepts, such as visualizations, real-world applications, or manipulatives, educators can help students develop a more accurate and comprehensive understanding.

Hence, it can be concluded that option 3 is the correct answer.

Asking open-ended questions in mathematics allows students to demonstrate their depth of understanding and problem-solving abilities.

Key Points

• These questions do not have a single correct answer and instead encourage students to think critically, explore multiple approaches, and justify their reasoning.
• By engaging with open-ended questions, students analyze mathematical problems, make connections between concepts, and apply their knowledge creatively.
• This assessment method fosters independent thinking, encourages students to explore different strategies, and develops their conceptual understanding beyond rote memorization.
• Through open-ended questions, teachers gain insights into students' ability to think analytically, evaluate information, and communicate mathematical ideas effectively.

Hence, it can be concluded that option 1 is the correct answer.

Mathematical discussion in the classroom plays a crucial role in enhancing students' mathematical understanding.

Key Points

• By engaging in discussions, students can share their ideas, explore different perspectives, and deepen their comprehension of mathematical concepts.
• Through collaborative problem-solving and explanation of their reasoning, students develop a deeper understanding of mathematical principles and strategies.
• Discussions also allow students to learn from each other, challenge their thinking, and refine their mathematical reasoning skills.
• Encouraging mathematical discussion fosters a more active and participatory learning environment, promoting the development of students' mathematical understanding.

Hence, it can be concluded that option 1 is the correct answer.

Creative thinking in mathematics involves solving real-life problems using mathematical abstractions and finding alternate ways to solve problems.

Key Points

• When students engage in solving real-life problems using mathematical abstractions, they go beyond the surface level and apply mathematical concepts in practical contexts.
• They creatively analyze the problem, identify relevant mathematical principles, and devise innovative approaches to arrive at solutions.
• This requires them to think critically, make connections between abstract concepts and real-world scenarios, and adapt mathematical tools to address the complexities of the problem at hand.
• Finding alternate ways to solve problems demonstrates creative thinking by encouraging students to explore different strategies and approaches beyond conventional methods.
• It challenges students to think flexibly, consider multiple perspectives, and come up with innovative solutions.
• By encouraging students to find alternate solutions, the teacher fosters their creativity, problem-solving skills, and the ability to think outside the box.

Hence, it can be concluded that statements A and B are correct.

Emphasising the significance of zero in various contexts" correctly describes the teacher's approach to conducting the debate on whether zero is a significant number.

Key Points

• By encouraging every child to express their views on the topic, the teacher is fostering critical thinking and exploring the different contexts in which zero plays a significant role in mathematics.
• This approach allows students to understand the concept of zero beyond its numerical value and recognize its importance in different mathematical operations, place value, algebra, and other mathematical concepts.
• The teacher's emphasis on zero's significance promotes a deeper understanding of mathematical principles and encourages students to think analytically and creatively.

Hence, it can be concluded that the teacher is Emphasising on the significance of zero in various contexts.

The correct answer is Both (A) and (R) are true and (R) is the correct explanation of (A).

Explanation:

• Nylon, Polyester, and Terylene are man-made threads; they are made of things like toothbrushes, rope, curtains, etc.
• Polyester is a synthetic fabric made of Adipic acid and Hexamethylene Diamene.
• Polyester and Terylene fabrics tend to dry quicker than cotton fabrics.
• The synthesized fabric soaks little water causing it to dry quickly.
• Synthesized fabric is a type of "Polymer".
• Synthetic fabric is more shiny and soft. The clothes made of these fabrics do not appear to have a worm.
• Cotton fabrics are natural fibers made from cotton.
• The cotton cloth absorbs moisture quickly. That's why towels and cotton clothes are mostly used for heat.

The correct answer is Liver, bile, gall bladder, and fats.

Explanation:

• The liver is a reddish-brown gland situated in the upper part of the abdomen on the right side.
• It is the largest gland in the body.
• It secretes bile juice, which is stored in a sac called the gall bladder.
• Bile juice plays an important role in the digestion of fats.
• Bile does not contain any enzyme but is very essential in the process of digestion
• Bile juice helps with digestion and performs two main functions:
• It converts fats into fatty acids, which are then absorbed by the digestive tract.
• The food from the stomach is acidic and cannot be digested directly by pancreatic fluids since they require an alkaline medium to work properly. Bile alkalinizes the food and environment, allowing pancreatic enzymes to function.

The correct answer is Rabi, wheat, mustard.

Explanation:

• Three types of crops are grown.

1. ​​​Rabi Crops
2. Kharif Crops
3. Zaid crops

• ​​​Rabi Crops: The crops which are sown around mid-November and harvested in April-May
• Under this crop, wheat, barley, gram, peas, lentils, linseed, etc.
• Kharif Crops: These types of crops need more water and temperature. So, it is sown from May to July.
• Under this crop, Rice, corn, millet, jute, etc.
• Zaid crops: These types of crops need more water. So, it is sown in the March–April months and reaped until June–July.
• Under this crop, cucumber, cucumber, watermelon, melon gourd, etc.

Based on these considerations, it appears that the city that best matches the data is Shimla.

Explanation:

Let's consider each city one by one.

1. Kochi is a coastal city located in the state of Kerala, South India. It generally has a tropical monsoon climate with high humidity throughout the year. But the temperatures seem too low to be in Kochi, as the maximum temperature there rarely falls below 30 °C, even in the winter months.

2. Shimla is located in the hilly state of Himachal Pradesh in North India. It has a subtropical highland climate. The city experiences mild summers and cold, snowy winters. The given data could possibly belong to Shimla, as the temperatures for January, February, and March align quite well with Shimla's climate.

3. Karnal is in the state of Haryana in North India and has a hot-summer humid subtropical climate. The summer months (April to June) can get quite hot, with temperatures often exceeding 40°C, which is higher than the given data.

4. Chennai, a coastal city in South India, has a tropical wet and dry climate. The temperature here does not fluctuate much throughout the year, and it generally remains hot and humid. Again, the temperatures in the data are lower than what you would typically see in Chennai.
   

Note: Here in the official answer key, option 3 is correct, but according to the question and options, option 2 seems to be right.

The correct answer is (A) is false but (R) is true.

Explanation:

• Phosphorus was discovered by Hennig Brand in 1669.
• Phosphorus has the smell of garlic.
• Elemental phosphorus exists in two different forms, namely white phosphorus and red phosphorus.
• Phosphorus is never found as a free element on earth due to its high reactivity.
• Phosphorus is stored in water to prevent contact with atmospheric oxygen.
• The element kept in water is white phosphorus.
• White phosphorus emits green light upon exposure to the atmosphere.
• White phosphorus burns in the atmosphere at 30º C.
• Red phosphorus is used for making safety matches.
• Phosphorus is used in the fertilizer industry.
• Sodium (Na) and potassium (K) are kept in Kerosene to avoid moisture and oxidation.
• White phosphorous is kept underwater to avoid oxidation and it is also insoluble in water.

The Correct answer is (B) or (C)

Concept:

Force:  

• It is an influence that tends to set a stationary body in motion or stop a moving body, or that tends to change the speed and direction of a moving body, or that tends to change the shape (and size) of a body.

Balanced forces:

• The resultant of all the forces acting on a body is called the balanced force.
• A body under the action of balanced forces does not change its position of rest( or of uniform motion) and it appears as if no force is acting on it.
• If several balanced forces act on a stationary body, the body continues to remain in its stationary position.
• If several balanced forces act on a body in uniform motion, the body continues to be in its state of uniform motion.
• Balanced forces cannot produce motion in a stationary body or stop a moving body; they can change the shape of a body.
• An example of a balanced force is the squeezing of a rubber ball or a balloon.
• In this case, we apply equal and opposite force on our hands.
• When the resultant of all the forces is not equal to zero, then it is called unbalanced forces

Explanation:

• No force is acting on it: This statement is incorrect because even when an object is at rest, forces do act upon it. The absence of observable motion does not mean an absence of forces.
• Forces are in place to ensure the object stays at rest, but they are balanced, meaning their net sum is zero, ensuring the object does not move.
• A pair of balanced forces are acting on it: This statement is correct. The pair of forces here is the downward force of gravity and the normal (upward) force of the reaction from the surface.
• When an object lies at rest on a surface, the force of gravity pulling the object down is equal to the normal force pushing up from the surface, thereby remaining balanced and ensuring the object's state of equilibrium.
• Multiple pairs of balanced forces acting on it: could also be correct, depending on the circumstances. For instance, besides the downward force of gravity and the upward normal force, there can be other pairs of forces at play, such as static friction and an applied force.
• If you try to push the object, but it doesn't move, it means the static frictional force is equal and opposite to the applied push. Thus, there are multiple pairs of balanced forces keeping the object at rest.

Based on the principles outlined by Newton's third law of motion and the concepts of balanced forces (B) and (C) are answers.

The correct answer is that both (A) and (R) are true and (R) is the correct explanation of (A).

Concept:

Digestion:

• The process of converting complex food substances to simple absorbable forms is called digestion and is carried out by our digestive system through mechanical and biochemical methods
• The digestion is performed in the following manner:
• Mouth – Oesophagus – Stomach – Small intestine – Large intestine.

Explanation:

• The food from the stomach enters the small intestine. This is the longest part of the alimentary canal, which is fitted into a compact space because of extensive coiling.
• The length of the small intestine differs in various animals depending on the food they eat.
• Herbivores eating grass need a longer small intestine to allow the cellulose to be digested, as it is harder to digest
• Meat is easier to digest; hence, carnivores like tigers have a shorter small intestine.

The correct answer is Quick lime.

Explanation:

• The chemical formula for quick lime calcium oxide is CaO.
• It reacts with water to form calcium hydroxide.
• The product was left immediately after the calcination of pure limestone.
• It is commonly used in making porcelain and glass and also in preparing bleaching powder
• Limestone forms from the accumulation of shell, coral, algal, and fecal debris that have calcium carbonate in them.
• The chemical composition of chalk is calcium carbonate, with minor amounts of silt and clay.
• Marble (Calcium carbonate) is a biogenic rock and is more compact than chalk.
• As with marble, calcium carbonate is a coarse-crystalline, metamorphic rock that is formed when chalk or limestone is recrystallized under conditions of high temperature and pressure.

Concept:

• The liquefaction process involves applying conditions such as decreased temperature and increased pressure to bring the gas particles closer together, allowing attractive forces to overcome their kinetic energy and form a more ordered, condensed liquid state.

Explanation;

• The correct option for the liquefaction of gases is: 3) Decrease in temperature and increase in pressure
• Liquification of gases is achieved by subjecting the gas to conditions of low temperature and high pressure. This combination causes the gas particles to come closer together and lose energy, eventually leading to the formation of a liquid.
• Option 1) An increase in temperature and a decrease in pressure are not helpful in liquefication. Raising the temperature generally makes gases less likely to liquefy, as the gas particles gain more kinetic energy and move further apart. Decreasing the pressure may allow the gas to expand and become less dense, making it even harder to liquefy.
• Option 2) An increase in both temperature and pressure is also not helpful for liquefication. Increasing the temperature, as explained earlier, makes gases less likely to liquefy. Raising the pressure alone might not be sufficient to cause liquefaction unless the temperature is lowered as well.
• Option 4) A decrease in both temperature and pressure is not effective for liquefication either. Lowering both temperature and pressure can lead to the formation of a solid or directly cause sublimation (from gas to solid) without going through a liquid phase.
• In summary, for successful gas liquefaction, you need to decrease the temperature and increase the pressure, as stated in option 3.

The Correct answer is (A) - (iii), (B) - (iv), (C) - (i), (D) - (ii)

Explanation:

• Sawdust and water: Sawdust is insoluble in water, so it can be separated by filtration. The mixture can be passed through a filter paper, and the sawdust will be retained on the filter paper while the water passes through.
• Iodine and sand: Iodine is a solid that can undergo sublimation, which means it can change directly from a solid to a gas without passing through the liquid phase. By heating the mixture, the iodine will sublime, leaving behind the sand.
• Salt and water: Salt is soluble in water, so the mixture can be separated by evaporation. By heating the mixture, the water will evaporate, leaving the salt behind.
• Cream from curd: Cream can be separated from curd by centrifugation. Centrifugation involves spinning the mixture at high speed, causing the denser cream to separate from the lighter curd.

The Correct answer is (BF)lead > (BF)zinc > (BF)alum

Concept:

• To compare the buoyant forces on the lead, aluminium, and zinc blocks submerged in water, we need to consider Archimedes' principle, which states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object

Calculation:

The buoyant force (BF) ⇒  BF = ρ ×V × g 

where:

ρ is the density of the fluid (in this case, water).

V is the volume of the fluid displaced by the object.

g is the acceleration due to gravity.

Let's calculate the buoyant forces for each block:

Lead block:

Density of lead (ρlead) = 11.29 g/cm³

The buoyant force on the lead block (BF_lead) = ρ_water ×  V × g

Aluminium block:

Density of aluminium (ρalum) = 2.7 g/cm³

The buoyant force on the aluminium block (BF_alum) = ρ_water * V * g

Zinc block:

Density of zinc (ρzinc) = 7.13 g/cm³

The buoyant force on zinc block (BF_zinc) = ρ_water ×  V × g

Density of lead (ρ_lead) = 11.29 g/cm³

Density of aluminium (ρ_alum) = 2.7 g/cm³

Density of zinc (ρ_zinc) = 7.13 g/cm³

 (BF)lead > (BF)zinc > (BF)alum

Note: In the official answer key, option 4 is correct, but according to the data given, option 3 is correct.

The correct statement is: 3) A concave mirror can form a real as well as a virtual image.

Explanation:

1. A concave lens can form only virtual images. It diverges light and cannot bring it to a focus to form real images.

2. A plane mirror can only form virtual images. The image formed in a plane mirror appears behind the mirror and is virtual (not real).

3. A concave mirror can form both real and virtual images. When the object is placed beyond the focal point, a real and inverted image is formed. When the object is placed between the focal point and the mirror, a virtual and erect image is formed.

4. A convex mirror can only form virtual images. It diverges light and cannot bring it to a focus to form real images.

So, the correct statement is that a concave mirror can form a real as well as a virtual image.

The correct answer is Both (A) and (R) are true and (R) is the correct explanation of (A).

Concept:

• Good conductors of electricity are materials that allow an electric current to pass through them.
• Materials that do not easily allow an electric current to pass through them are poor conductors of electricity.
• Common salt (NaCl) conducts electricity in an aqueous solution or molten state.

NaCl (aq) → Na+ + Cl-

• Due to the mobility of ions, common salt conducts electricity.

Explanation:

• ​Electricity cannot be conducted by all liquids. However, some of them are considered good conductors of electricity, while others are considered poor conductors of electricity.
• Tap Water with dissolved salts and minerals always conducts electricity. For example, water obtained from taps, hand pumps, wells, and ponds.
• Electricity cannot be conducted by distilled water that does not contain any salts.
• Any acid or base solution can conduct electricity.

The correct answer is Turnip, Carrot.

Concept:

• Roots are the underground structures that help in the absorption of water and minerals, provide a proper anchorage to the plant parts, store reserve food material (Radish), and synthesise plant growth regulators.
• Sometimes roots get modified to perform different functions like Storage of food, respiration, climbing, etc. These types of roots are called modified roots.

Explanation:

• Prop roots, or pillar roots, are modified adventitious roots.
• These roots arise from the branches of the plant and grow downward towards the soil
• These roots provide support for the trees. E.g., Banyan (Ficus bengalensis)

Additional Information 

The correct answer is cholera, typhoid.

Concept:

• The organisms that cannot be seen with the naked eye are called microorganisms.
• Microorganisms are classified into four major groups. These groups are bacteria, fungi, protozoa, and some algae.
• Viruses are also microscopic.
• Microorganisms are both beneficial as well as harmful.

Explanation:

• Microorganisms are harmful in many ways. Some microorganisms cause diseases in human beings, plants, and animals.
• Such disease-causing microorganisms are called pathogens.
• On the basis of these pathogens, diseases can be categorized as - Viral diseases, Bacterial diseases, Nematode diseases, etc.
• Some of the viral diseases include- Measles, Chickenpox, Polio, Common cold, AIDS, etc.

Additional InformationSome common human diseases are caused by Microorganisms.

The correct answer is The pole strength of each new piece will be one-fourth of the original.

Concept:

Magnet:

• An object that is capable of producing a magnetic field, attracting Ni, Co, Fe, etc., and aligning in the geographical north-south direction when suspended freely is called a magnet.
• Pole strength:
• The strength of a magnetic pole to attract magnetic material towards itself is known as pole strength.
• It is a scalar quantity.
• Its SI unit is amp×m.
• Poles of a magnet:
• The point on the magnet that has the maximum attraction property is called the pole of the magnet.
• The poles of the magnet lie slightly inside from the end.
• If a magnet is cut perpendicular to its axis, then its pole strength does not change.
• If a magnet is cut along its axis, then its pole strength decreases.
• If a magnet is cut into two equal parts along its length, then the pole strength of each part becomes half of the original magnet.

Explanation:

• The pole strength of each new piece will be one-fourth of the original. When a magnet is cut into parts along its length, the pole strength of each part becomes a fraction of the original magnet's pole strength.
• In this case, since the magnet is divided into four parts, the pole strength of each piece would be one-fourth of the original's.

The Correct answer is 14.5 km/h

Concept:

• Average speed is a measure of the total distance traveled divided by the total time taken. Unlike average velocity, which takes into account the net displacement and total time, average speed accounts for the total ground covered.

The formula for average speed is:

Average Speed = Total Distance / Total Time

Calculation:

⇒ Time taken while going = Distance / Speed = 10 km / 12 km/h = 5/6 hours

⇒ Time taken while returning = Distance / Speed = 10 km / 18 km/h = 5/9 hours

The total time for the trip is the sum of the time taken while going and returning:

Total Time = (5/6) hours + (5/9) hours = 25/18 hours

The average speed:

Average Speed = Total Distance / Total Time

= 20 km / (25/18) hours

= 20 km ×  (18/25) hours = 14.4 km\h

The correct answer is Spinach.

Explanation:

• To enhance the intake of iron, which is important for haemoglobin production, the recommended food item would be spinach
• Spinach is a good source of dietary iron and other nutrients. Iron is a crucial component of haemoglobin, the protein responsible for carrying oxygen in red blood cells.
• Increasing the intake of iron-rich foods, like spinach, can help improve haemoglobin levels and alleviate the symptoms of anaemia.
• Nuts can provide some nutrients, but they are not typically high in iron.
• Rice and butter do not significantly contribute to increasing haemoglobin levels. Rice is a carbohydrate-rich food, while butter is primarily a source of fat.

The Correct answer is (B) and (C)

Explanation:

• The statement (A) that water is transparent is not a valid reason for not using it as a thermometric liquid.
• Transparency, or the ability to see through a substance, does not directly impact its suitability for use as a thermometric liquid.
• Transparency is not a determining factor in the selection of a thermometric liquid.
• Statement (B): Water has a high coefficient of expansion. The coefficient of expansion is a measure of how much a substance expands or contracts with changes in temperature.
• Water has a relatively high coefficient of expansion compared to many other liquids commonly used as thermometric liquids.
• Statement (c): Water has a higher freezing point and a lower boiling point as compared to other liquids.
• This limits the temperature range over which water can be effectively used as a thermometric liquid.
• Additionally, water's boiling point is 100 degrees Celsius at sea level, which is lower compared to other liquids like mercury.
• This further restricts the range of temperatures where water can be utilized as a thermometric liquid.

Note: In the official answer key, option 2 is correct, but according to the data given, option 3 is correct.

The Correct answer is (A) - (ii), (B) - (iii), (C) - (iv), (D) - (i)

Explanation:

Pseudoscientific theories are formulated in such a manner that they cannot be tested for incorrectness," correctly describes one of the key differences between science and pseudoscience.

Key Points

• In science, theories are formulated in such a way that they can be tested through experimentation, observation, and empirical evidence.
• The scientific method relies on the ability to test hypotheses and theories and potentially prove them incorrect through rigorous experimentation and observation.
• Pseudoscientific theories, on the other hand, often lack the ability to be tested or verified through empirical evidence.
• They may rely on unverifiable claims, subjective experiences, or flawed methodologies that make them immune to falsification.
• This lack of testability and falsifiability is a distinguishing characteristic of pseudoscience.

Hence, it can be concluded that option 2 is the correct answer.

The strategy described in the scenario aligns well with inquiry-oriented pedagogy. It encourages students to engage in observation and critical thinking, which are key aspects of inquiry-based learning.

Key Points

• By explaining the meaning of floatation, the teacher provides students with a solid foundation for their subsequent observations and analysis.
• Understanding the concept from the start enables students to make meaningful connections between the objects they observe and the underlying principles of floatation.
• It sets the stage for inquiry-based learning by engaging students' curiosity and encouraging them to explore further.
• By establishing a strong conceptual framework, the strategy effectively supports students' inquiry-oriented learning and promotes a deeper understanding of the laws of flotation.

Hence, it can be concluded that option 1 is the correct answer.

While this indicator is important for assessing students' understanding of specific subject matter, it does not directly measure their ability to think rationally.

Key Points

• Memorizing and recalling formulae and laws primarily involve rote learning and retention of information.
• Rational thinking, on the other hand, involves critical analysis, logical reasoning, and the ability to evaluate and solve complex problems.
• While understanding and applying formulae and laws are essential components of rational thinking in certain contexts, they do not fully capture the breadth of skills required for rational thinking.

Thus, option 2 is not a suitable indicator for assessing students' ability to think rationally.

Cognitive validity refers to the appropriateness and alignment of the content with the cognitive development and abilities of the students.

 Key Points

• When teaching a topic, it is essential to consider the student's prior knowledge, age, and cognitive abilities to ensure effective learning.
• By introducing the concepts of electrostatic charge, electrostatic field, and electrostatic potential before explaining electrostatic force to Class VI students, the teacher may be violating cognitive validity.
• These concepts are more complex and typically introduced at higher grade levels when students have developed a foundational understanding of basic forces and have the cognitive ability to comprehend and apply abstract concepts effectively.

Hence, it can be concluded that cognitive validity is the correct answer.

In the experiment conducted by the students to investigate the effect of temperature on the resistance of a given length of metallic wire, the correct statement is that the resistance of the wire would be a dependent variable.

 

Key Points

• As the students vary the temperature of the wire, the resistance will change accordingly.
• This is because temperature influences the resistance of a wire; as temperature increases, the resistance typically increases.
• By measuring and analyzing the changes in resistance at different temperatures, the students can determine the relationship between temperature and resistance.
• The resistance of the wire is thus the dependent variable, as it is the outcome that is being observed and measured in response to the independent variable, which is the temperature.

Hence, it can be concluded that option 2 is the correct answer.

Encouraging students to freely describe their erroneous understandings of a given phenomenon promotes an environment of open communication and active learning.

Key Points

• It allows students to express their ideas, misconceptions, and questions without fear of judgement.
• By doing so, the teacher can identify and address any misconceptions or gaps in understanding, which is crucial for the learning process.
• This practice fosters critical thinking, engagement, and a deeper understanding of the subject matter.
• It also promotes a classroom culture where students feel comfortable exploring and questioning scientific concepts, leading to a more collaborative and effective learning environment.
• It is important for a science teacher to create an inclusive and supportive atmosphere where students are encouraged to participate, ask questions, and share their ideas.
• By doing so, students can actively engage in the learning process, develop their scientific thinking skills, and construct a deeper understanding of scientific concepts.

Hence, it can be concluded that option 3 is the correct answer.

The pedagogic device that the teacher is employing in this scenario is an analogy.

Key Points

• An analogy is a teaching technique that involves drawing similarities between two different concepts or situations to enhance understanding.
• In this case, the teacher is using the analogy of water flowing from higher to lower pressure to explain the flow of electric current in a circuit.
• By comparing the movement of electrons in a circuit to the movement of water in a system, the teacher is making the concept more relatable and easier to comprehend for the students.
• This pedagogic device helps bridge the gap between unfamiliar or abstract concepts and familiar or concrete ones, facilitating better understanding and retention of the subject matter.

Hence, it can be concluded that option 1 is the correct answer.

In scientific research and experimentation, various concepts and methods are used to understand the natural world.

Key Points

• A theory in science refers to a well-substantiated explanation of a natural phenomenon. It is formed based on a comprehensive body of evidence and has withstood rigorous testing and scrutiny.
• A hypothesis is a proposed explanation for a phenomenon or a scientific question. It is an educated guess that serves as a starting point for further investigation.
• A hypothesis is tentative and requires testing through experimentation or further research to determine its validity.
• A scientific law is a concise statement that describes a fundamental relationship or pattern observed in nature
• A scientific experiment is a carefully controlled procedure designed to test a hypothesis or investigate a specific scientific question. It involves manipulating variables under controlled conditions to observe and measure the resulting outcomes.

Hence, it can be concluded that option 4 is the correct answer.

Field trips to places of scientific interest, such as science museums, research labs, or natural habitats, can enhance students' understanding by linking theoretical knowledge learned in textbooks with practical, real-world examples.

Key Points

• This connection can make learning more meaningful, engaging, and memorable for students.
• Visiting places of scientific interest can bring textbook knowledge to life.
• When students see science in action, they can better understand abstract concepts. For instance, seeing a fossil in a museum can deepen our understanding of prehistoric life far more effectively than a textbook image.
• Students might see the practical applications of scientific principles, which can spark curiosity and deepen engagement.
• For example, a visit to a local river can demonstrate the effects of pollution and the importance of conservation efforts, giving context to environmental science lessons.
• Such visits often involve aspects of other disciplines as well, like history, sociology, or art, encouraging holistic learning.

Hence, it can be concluded that option 1 is the correct answer.

"Convince students that the science subject is not meant for all" is an erroneous notion that undermines a core principle of education: that learning is accessible to everyone.

Key Points

• By suggesting that science is not for everyone, a teacher might inadvertently discourage students who are struggling or who may not yet have discovered their potential.
• This can lead to reduced confidence and a lack of motivation to try harder, which in turn could result in poorer performance not just in science but in other subjects as well.
• Even students who don't pursue careers in science benefit from scientific literacy.
• They gain critical thinking and problem-solving skills and a better understanding of the world around them.
• They're also better equipped to make informed decisions on important issues that involve scientific knowledge, like climate change, health-related matters, and technology usage.
• Modern educational psychology emphasizes the importance of a "growth mindset" — the belief that abilities and intelligence can be developed through dedication and hard work.

​Hence, it can be concluded that option 1 is the correct answer.

The correct answer is 'We have choice of action'.

Key Points

• In the passage, the author identifies human beings as a privileged lot because we have the ability to make choices.
• This is highlighted specifically in the first line: "We are a privileged lot, gifted with choice of action. No other species has this ability."
• None of the other options — sages making choices for us, value-based decisions making us happy, or choices sculpting our future — properly answer the question regarding why human beings are particularly privileged according to the passage.

Therefore, the correct answer is Option 3.

The correct answer is 'Adjective'

Key Points

• The clause 'Which holds the intellect hostage' describes a characteristic of the 'strong desire,' qualifying it.
• In English grammar, a clause that modifies or describes a noun (in this case, 'strong desire') is termed as an adjective clause.

Hence, the correct answer is Option 3, 'Adjective.'

The correct answer is 'Every action has an impact; choose wisely the impact you want to have. - Mindy Hall'

Key Points

• This quote encapsulates the message of the passage accurately. The passage discusses the importance of choice, consequences, and the wisdom to choose wisely.
• It underlines the significance of mindful actions and their subsequent impact, which aligns with the theme of the passage.
• The other quotes, although valuable, do not encompass the essence of the passage as holistically as the chosen quote does.

Therefore, the correct answer is Option 1.

The correct answer is 'Choices made in haste are mostly benign.'

Key Points

• The passage repeatedly emphasizes the importance of taking time while making choices and warns against hasty decisions.
• It implies that choices made impulsively, without adequate thought, often lead to undesirable outcomes.
• Thus, the statement that choices made in haste are 'mostly benign' contradicts the passage's primary message.

Therefore, the correct answer is Option 4.