Mathematics and Science PYQs

Mathematics anxiety is a feeling of fear, apprehension, or dread associated with mathematics. 

Key Points

• Teachers' own anxiety about the subject can be communicated to students, making them feel anxious about the subject themselves.
• The perception that there are proper ways of doing mathematics which is characterized by unique answers can lead to students feeling like they are doing it wrong if they don't get the right answer. This can lead to anxiety and avoidance of the subject.
• Learning to reproduce procedures and algorithms which are away from the daily life of children can lead to students losing interest in the subject and developing anxiety about it.
• There is no evidence to suggest that mathematics is a more difficult subject for boys than it is for girls. In fact, studies have shown that girls are just as capable as boys in mathematics.

Hence, we can conclude that Mathematics is a difficult subject and is only for boys is the least contributing factor in developing mathematics anxiety in children.

Transposition in algebra refers to the process of moving or rearranging terms from one side of an equation to the other.

Key Points

• It's essentially a way to simplify or solve equations by changing the position or "transposing" terms.
• The fundamental principle behind this is the property of equality: if you do the same operation on both sides of an equation, the equation remains valid.
• Here's a step-by-step solution using transposition:

Given  5x - 8 = x + 4

• To get all x terms on one side and constants on the other, we will transpose terms:

5x - x = 4 + 8

4x = 12

x= 12/4

x= 3

Hence, we can conclude that Transposition is required to be developed in the child to solve the following problem.

An axiom, also known as a postulate, is a statement in mathematics and logic that is assumed to be true without proof.

Key Points

• Axioms are often basic and self-evident principles that form the basis for further reasoning and arguments in a particular mathematical system or theory.
• It's important to understand that axioms aren't "true" or "false" in an absolute sense.
• Instead, they are starting points for a system of thought. If you accept the axioms, then certain other statements will logically follow (those are called "theorems").
• A theorem is a statement that has been proven to be true, given the axioms of the system.
• A proof is a series of logical deductions from the axioms and previously proven theorems that end with the statement of the theorem being proven.
• A lemma is a "helper theorem," a result that is proven not for its own sake, but to assist in the proof of other theorems.

Hence, we can conclude that Axiom is taken to be true usually because it is self-evidently true.

Proportional thinking refers to the ability to understand and use ratios in varied situations and to recognize that a relationship or correspondence exists between equivalent ratios.

Key Points

• Out of the given problems, the one that does NOT necessarily require proportional thinking is "The height of a tree is 3 meters and the length of its shadow at 11:00 am is 4 meters. What would be the length of its shadow at 4 pm?".
• The reason is that the length of a shadow is influenced by the angle of the sun in the sky.
• This angle changes throughout the day, and the relationship between the height of an object and the length of its shadow at one time of day cannot be used to directly deduce the length of its shadow at another time of day without more information.
• Proportional reasoning alone can't determine the shadow's length at 4 p.m. based on its length at 11 a.m.

Hence, we can conclude that the question "The height of a tree is 3 meters and the length of its shadow at 11:00 am is 4 meters. What would be the length of its shadow a t4 pm?"  does not require proportional thinking.

The Van Hiele model of geometric thinking is a theory that describes the progression of geometric understanding in students. It comprises five levels, each building on the understanding of the previous one. 

Key Points

• Visualization (Level 0): At this level, students recognize geometric figures (like triangles, squares, and circles) based on their appearance but not based on their properties.
• Analysis (Level 1): Students at this level start to recognize properties of geometric figures but do not see relationships between them.
• Abstraction (Level 2): At this level, students begin to understand the relationships between properties of geometric figures. They can group shapes based on shared properties.
• Deduction (Level 3): Students can now understand and formulate proofs. They can deduce properties of a figure from given information, and they recognize the significance of definitions.
• Rigor or Axiomatic (Level 4): At this highest level, students can compare different axiomatic systems of geometry. They understand the relationships and differences between non-Euclidean geometries and Euclidean geometry.

Hence, we can conclude that Visualization is the starting level of Van Hieles' theory of geometric thinking.

The sum of the measures of the interior angles of a polygon is sum = (n-2)*180˚.

Key Points

Now, let's review the options:

• The sum of all interior angles of any polygon is 360 degrees. This statement is incorrect. Only a quadrilateral has interior angles that sum up to 360 degrees.
• The sum of all interior angles of a quadrilateral is three times the sum of all the interior angles of a triangle. The sum of a quadrilateral's angles is only two times that of a triangle. This statement is incorrect.
• When the number of sides increases the sum of the measures of interior angles of a polygon increases. As per the formula when the side of a polygon increases then the sum of its interior angles also increases. This statement is correct.
• When the number of sides increases the sum of the measures of exterior angles of a polygon decreases. The sum of the exterior angles of any polygon, regardless of the number of sides, is always 360 degrees. So, this statement is incorrect.

Hence, we can conclude that When the number of sides increases the sum of the measures of interior angles of a polygon increases is correct about the angles of the polygons.

As per the NCERT's expected learning outcomes for grade VI learners, students at this level are typically introduced to basic algebraic concepts, arithmetic operations with larger numbers, basic geometrical ideas, and initial problem-solving techniques.

Key Points

• In grade VI, students are expected to be competent in arithmetic operations with large numbers, including addition, subtraction, multiplication, and division.
• Instead of always dealing with specific numbers, students begin to use variables to represent unknown values. They learn to express general mathematical relationships and simple equations using variables.
• The unitary method is a fundamental problem-solving technique. This method is vital for solving a variety of problems in mathematics and everyday scenarios.
• Writing formal proofs of geometric theorems is a higher-level skill that is typically introduced in later grades, usually in grade IX and onwards. In grade VI, students are usually just getting introduced to basic geometric shapes, properties, and basic concepts like angles, lines, etc.

Hence, we can conclude that Writing formal proofs of geometric theorems is not one of the expected learning outcomes for grade VI learners.

Teaching methods refer to the principles and methods used for instruction. They are determined by the objectives, learning styles of students, content, and the context of the lesson.

Key Points

• Inductive method is one of many methods used to teach mathematics, and while it might work for some students and educators, it might not be the "best" method universally.
• "There is plenty of scope for project work in mathematics learning": This is generally true. Many educators use projects to help students understand complex math concepts, apply mathematical thinking, and link math to real-world situations.
• "Generalization is a characteristic of reasoning ability": This is true. Reasoning ability often involves taking specific instances or observations and making broader generalizations.
• Open-ended questions encourage learners to think deeply, consider various possibilities, and come up with solutions or answers based on their own reasoning, rather than simply recalling facts.

Hence, we can conclude that the best method to teach mathematics is the Inductive method incorrect statement from among the following.

Mathematical thinking is the process of using math to solve problems. This kind of thinking involves understanding mathematical concepts, not just memorizing them.

Key Points

• Recognizing patterns is fundamental to understanding mathematics. The ability to spot and understand patterns aids in prediction, assumption, and generalization.
• True mathematical thinking would involve understanding where a formula comes from, its application, and under what circumstances it can be used.
• The memorization of multiplication tables can make certain calculations faster and can be helpful in many areas of math, especially during early education, but mathematical thinking is about comprehension and application.
• Mathematical thinking involves not just applying the algorithm but understanding its basis and knowing when and how to use it.

Hence, we can conclude that Mathematical thinking is indicated by the identification of patterns.

Higher-order thinking Skills (HOTS) involve complex cognitive processes, including critical, logical, reflective, metacognitive, and creative thinking.

Key Points

• Identifying relationships among variables: This requires understanding and analyzing complex relationships, a hallmark of HOTS.
• Analyzing the data: Data analysis often requires synthesis, evaluation, and application of the information in new or novel contexts, all of which are components of HOTS.
• Making connections between mathematics and everyday life: Drawing connections between abstract concepts and real-world applications requires critical thinking and the ability to apply learned concepts in different scenarios.
• Solving a problem based on the given formula: This option primarily deals with applying a known procedure to a problem. If you're simply plugging numbers into a formula without much thought or understanding, then it's a lower-level skill.

Hence, we can conclude that Solving a problem based on the given formula is least related to Higher-Order thinking skills in mathematics.

Key Points

• An onion is a multicellular (consisting of many cells) plant organism.
• As in all plant cells, the cell of an onion peel consists of a cell wall, cell membrane, cytoplasm, nucleus and a large vacuole.
• The nucleus is present at the periphery of the cytoplasm.
• The vacuole is prominent and present at the centre of the cell.
• It is surrounded by cytoplasm.
• The presence of a cell wall and a large vacuole are indicators that help identify plant cells, such as seen in the onion peel.

Explanation:

The cells of an onion peel can be observed in a microscope. The Boundary of onion cell is called cell membrane(A)  which is covered by another thick covering called  cell wall (B). The central dense round body in the centre of cell is called nucleus (C) . The jelly-like substance between A and C is called cytoplasm(D)

Hence the correct answer is option 4

The Correct answer is Pressure.

Explanation:

• When a plastic bottle is filled with water and its mouth is closed with a cap, the water inside the bottle is in a relatively stable state, and its weight is evenly distributed throughout the volume of the water. This means that the pressure inside the bottle is equal on all sides.
• When a hole is made in the bottle using a push pin, the situation changes. The hole creates an opening through which the water can potentially flow out. But in the given scenario, it is observed that water doesn't come out of the hole. This phenomenon can be explained by the concept of pressure.
• By closing the mouth of the bottle, a sealed system is created that traps air and water inside it. The air above the water creates pressure acting downward, exerting force on the surface of the water. This downward pressure is greater than the upward pressure exerted by the same amount of water inside the bottle.
• Therefore, the pressure on the water surface is greater from the air side compared to the hole made by the push pin. This higher pressure pushes the water particles toward the hole. As a result, the water surface deforms and gets stretched around the hole, effectively blocking the flow of water out of the hole.
• This phenomenon demonstrates the concept of pressure exerted by a fluid. In this case, the pressure exerted by the air inside the bottle is greater than the pressure outside the hole, preventing the water from flowing out.

The observation that water doesn't come out of the hole in the plastic bottle in this activity illustrates the concept of pressure in fluids.

Additional Information

• Buoyant Force: This is the net upward force exerted on an object submerged in a fluid. This force is the reason objects float or are buoyant in a fluid.
• Density: Density is the measure of mass per unit volume of a material. It is one of the determinants of whether an object will sink or float in a fluid. If the object's density is less than the fluid's density, it will float, and vice versa.
• Pressure: In fluid dynamics, pressure increases with depth. This happens because fluid pressure is directly proportional to the height of the fluid above the point in question (P = h× ρ× g). Therefore, in a column of fluid, the pressure at the bottom is higher than at the top.
• Gravity: Gravity is an essential factor when discussing buoyancy as it pulls the objects downwards. The pressure difference between the top and the bottom of the object submerged in a fluid creates the buoyant force that opposes gravity.

The Correct answer is A - Refraction, B - Refraction.

Explanation:

• Prism: When light from a laser torch falls on a prism, the light undergoes refraction. Refraction is the bending or change in the direction of light as it passes from one medium to another with a different optical density.
• The prism refracts the light rays, causing them to bend or change direction based on their wavelength.
• Glass slab: When light from a laser torch falls on a glass slab, the light also undergoes refraction. The glass slab, being a transparent medium with a different optical density than air, causes the light to change speed and direction as it enters and exits the slab.
• The refraction of light at each surface of the glass slab results in the light rays bending or changing direction.

Therefore, the correct answer is A - Refraction, B - Refraction 

Additional Information

Dispersion of light:

• Dispersion of light is the phenomenon of splitting of a beam of white light into its constituent colours on passing through a prism.
• The band of seven colours so obtained is called the (visible) spectrum.
• The basic reason behind the dispersion of white light into seven different colours is that all the seven rays of light of different colours travel at different speeds through the glass prism.
• The degree of refraction also can be said as the bending of each light ray depends upon their individual speed while passing through the prism.
• All the different colours of light ray travel at different speeds in glass and so the angle of bending or angle of refraction of each light ray are different.

The Correct answer is  Combination reaction.

Explanation:

• In a combination reaction, two or more substances combine to form a new substance. In the case of photosynthesis, carbon dioxide, and water combine to form glucose, a sugar. The chemical equation for photosynthesis is:
• 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂
• In this equation, carbon dioxide (CO₂) and water (H₂O) are the reactants, and glucose (C6H12O6 ) and oxygen (O2) are the products. The sunlight provides the energy for the reaction to occur.
• Decomposition reactions, on the other hand, involve the breakdown of a single substance into two or more substances.
• Neutralization reactions involve the reaction of an acid and a base to form salt and water.
• Displacement reactions involve the replacement of one element by another in a compound.

So, the reaction carried out by plants to produce food using carbon dioxide and water in the presence of sunlight is a combination reaction.

Additional Information

• Decomposition reaction:
• The opposite of a combination reaction, a complex molecule breaks down to make simpler ones.
• Such reactions are represented by equations of the following form: AB → A + B.
• Displacement reaction:
• One element takes place with another element in the compound.
• Such reactions are represented by equations of the following form: A + BC → AC + B.
• Double displacement reaction: 
• It is a chemical reaction in which two compounds react and exchange their ions forming new products.
• This reaction often results in the formation of an insoluble compound called a precipitate.
• Such reactions are represented by equations of the following form: A+B- + C+D- → A+D- + B-C+

The Correct answer is By heating the solution.

Concept:

• A saturated solution is a solution that contains the maximum amount of solute that can dissolve in a solvent at a given temperature.
• If you try to dissolve more solute in a saturated solution, the excess solute will just precipitate out.

Explanation:

• The solubility of a substance is the maximum amount of that substance that can dissolve in a given amount of solvent at a given temperature. The solubility of copper sulphate in water increases with temperature. This means that more copper sulphate can dissolve in water at higher temperatures than at lower temperatures.
• When a saturated solution of copper sulphate is heated, the solubility of the copper sulphate increases. This is because the heat provides energy to the copper sulphate molecules, allowing them to move around more freely and interact more easily with the water molecules. As a result, more copper sulphate can dissolve in the solution.
• If you heat a saturated solution of copper sulphate to a high enough temperature, all of the copper sulphate will dissolve. This is called a super-saturated solution. A super-saturated solution is unstable, and if you cool it slowly, the excess copper sulphate will precipitate out.
• So, to dissolve more copper sulphate in a saturated solution of copper sulphate, you can heat the solution. This will increase the solubility of the copper sulphate

Key Points Polio

• Polio is caused by Polio virus.
• This virus infects the central nervous system of the individual.
• It can be transmitted through direct contact with someone infected with the virus and through contaminated food and water.

Cholera

• Cholera is caused by a bacterial pathogen called Vibrio cholerae.
• It affects the alimentary system of the individual.
• It can be transmitted through contaminated food and water.

Hepatitis B 

• Hepatitis B is caused by hepatitis B virus (HBV).
• This virus affects the liver of the infected individuals.
• It is spread from the mother to child during birth, as well as through contact with blood or other body fluids during sex with an infected partner, unsafe injections or exposure to sharp instruments.

Measles

• Measles is caused by an extremely contagious virus called morbillivirus.
• In fact, if 10 people who weren't vaccinated were in a room with someone with measles, nine of those people would get measles.
• Measles is spread by: Contaminated droplets that are spread through the air when you cough, sneeze or talk.

Hence the correct answer is option 1

Key Points Movement in earthworm

• The body of earthworm is made up of many rings joined end to end.
• It has muscles which help to extend and shorten the body.
• Under its body it has large number of bristles called setae which are connected with muscles.
• These bristles help to get grip on the ground.
• During movement, the earthworm first extends the front part of the body, keeping the rear portion fixed to the ground.
• Then it fixes the front end and releases the rear end.
• It then shortens the body and pulls the rear end forward.
• This makes it move forward by small distances.
• Repeating such muscle contraction and relaxation the earthworm can move through soil.
• A slimy substance secreted by its body helps this movement.

Hence the correct answer is option 1.

Key Points

• Vitamin K is a fat-soluble vitamin that comes in two forms.
• The main type is called phylloquinone, found in green leafy vegetables like collard greens, kale, and spinach.
• The other type, menaquinones, are found in some animal foods and fermented foods.
• Menaquinones can also be produced by bacteria in the human body.
• Vitamin K helps to make various proteins that are needed for blood clotting and the building of bones.
• Prothrombin is a vitamin K-dependent protein directly involved with blood clotting.
• Osteocalcin is another protein that requires vitamin K to produce healthy bone tissue.
• Vitamin K is found throughout the body including the liver, brain, heart, pancreas, and bone.
• It is broken down very quickly and excreted in urine or stool. Because of this, it rarely reaches toxic levels in the body even with high intakes, as may sometimes occur with other fat-soluble vitamins.

Hence the correct answer is option 3.

Additional Information

The Correct answer is A-Sulphuric acid; B-Lime water; C-Calcium sulphate.

Explanation:

• When sulphuric acid (H2SO4) is added to lime water (Ca(OH)₂), calcium sulphate (CaSO₄) is produced along with a large amount of heat.
• This reaction is a neutralization reaction, which is a type of chemical reaction that occurs when an acid and a base react to form salt and water.
• The chemical equation for the reaction is:
• H₂SO₄ + Ca(OH)₂ → CaSO₄ + 2H₂O
• In this equation, sulphuric acid (H₂SO₄) is the acid, lime water (Ca(OH)₂ is the base, calcium sulphate (CaSO₄) is the salt, and water (H₂O) is the product.
• Calcium sulphate is used in the manufacture of plaster of Paris. When calcium sulphate is heated to a high temperature, it loses some of its water molecules and becomes a powder called plaster of Paris.
• When the plaster of Paris is mixed with water, it rehydrates and forms a hard solid. This property of the plaster of Paris makes it useful for making casts, moulds, and other objects.

The Correct answer is Sublimation, decantation, evaporation 

Explanation:.

Sublimation: Naphthalene is a substance that sublimates, that is, it directly changes from a solid to a gas state when heated. So by heating the mixture gently, you can separate naphthalene from the sand and salt.

Decantation: Now we are left with a mixture of sand and salt. By adding water to this mixture, the salt will dissolve. Then, you can pour off the saltwater (decantation), leaving the undissolved sand behind.

Evaporation: The saltwater solution now contains dissolved salt. By evaporating the water from this solution, you're left with the salt.

So by using the three methods in this combination (sublimation, decantation, evaporation), you can successfully separate a mixture of sand, naphthalene, and salt.