Mathematics and Science PYQs

Aryabhatta was one of the great mathematicians and astronomers of ancient India, born around 476 CE. 

Key Points

• Aryabhata wrote the "Aryabhatiya," a significant astronomical and mathematical text, around 499 CE, when he was only 23 years old.
• In the "Aryabhatiya," Aryabhata gave an approximate value for π (Pi), the ratio of the circumference of a circle to its diameter.
• In his approximation, Aryabhata considered the ratio of the circumference of a circle to its diameter which equals 3.1416, accurate to four decimal places.

Hence, we can conclude that Aryabhatta gave an approximate value of π correct to 4 decimal places as 3.1416.

The inductive method is a teaching and reasoning approach that begins with specific observations or examples and uses them to derive general principles or conclusions. 

Key Points

• It's essentially the process of moving from particular instances to broader generalizations.
• By showing class 8 students how interest is compounded over several periods through specific examples, they can begin to understand the general pattern and derive the formula themselves.
• This makes the concept more accessible and tangible.

Hence, we can conclude that the Inductive method is most suitable for establishing the formula for compound interest in class 8.

The Van Hiele model of geometric understanding describes a sequence of five levels of geometric understanding that students progress through.

Key Points

• Level 0: Visualization: Students recognize shapes by their overall appearance but don't necessarily understand the properties that define them.
• Level 1: Analysis: At this level, students can identify properties of shapes (like sides and angles) but don't fully see the relationships between different properties.
• Level 2: Abstraction or Relationship: Students can now understand and describe relationships between properties. They can see how changes to one property might affect others and can make logical deductions about geometric properties.
• Drawing a right-angled triangle and naming it a Right Scalene Triangle would require not just understanding the properties that define the shape but also the ability to see the relationships between these properties.
• Level 3: Deduction or Axiomatic: Students at this level can understand the role of undefined terms, definitions, axioms, and theorems in geometry. They can develop proofs, follow logical arguments, and understand the structure of geometric systems.
• Level 4: Rigor: This level is characterized by the ability to work within different axiomatic systems. Students can compare different geometries, understanding both the abstract structure and the specific content.

Hence, we can conclude that the correct answer is Relationship level.

Careless errors in mathematics are mistakes made not due to a lack of understanding of the material but rather through oversights, distractions, or lack of attention to detail.

Key PointsPreventive methods for rectifying careless errors typically involve:

• Teaching a concept in multiple ways - Helps students understand the concept more deeply, reducing the likelihood of errors.
• Encouraging learners to learn different steps in problem solving - Ensures that students understand and follow the correct sequence of steps, minimizing errors.
• Encouraging learners to practice a number of similar questions - Reinforces the concept and helps students become more proficient and attentive in their problem-solving approach.

Hint

• Encourage learners to take the help of calculator and pay attention during problem solving: Using calculators can help in performing accurate calculations, but it doesn't address the underlying issues causing careless errors. Calculators may prevent students from practicing and solidifying their computational skills, which is essential for avoiding careless errors.
• Encouraging the use of calculators does not inherently teach students how to pay closer attention to their work or understand the steps involved in problem-solving, both of which are critical for rectifying careless errors.

Hence, we can conclude that the correct answer is 'encourage learners to take the help of calculator and pay attention during problem solving'.

Errors in a mathematics class refer to mistakes or misconceptions that students may make while learning mathematical concepts or solving problems.

Key Points

• Errors are an essential part of the learning process. They provide insight into students' understanding and thought processes.
• Analyzing errors can give educators insight into how children are thinking about mathematical problems, which can be crucial for adapting instruction to their needs.
• Errors can then become opportunities for exploration, discussion, and clarification, fostering a deeper understanding of mathematical concepts.
• Expecting children to learn without making any mistakes sets an unrealistic expectation and overlooks the valuable insights and growth that can come from engaging with and learning from mistakes.

Hence, we can conclude that "Children learn only when there is no error in their responses for a given problem" is not true about errors in a mathematics class.

A theorem is a mathematical statement that has been proven to be true.

Key Points

• The proof is based on previously established statements, such as other theorems, and axioms.
• Since a theorem has been rigorously proven, it cannot be proven false.
• If a statement claimed to be a theorem is found to be false, it means there was a flaw in the purported proof, and the statement was never truly a theorem, to begin with.
• A conjecture is a mathematical statement that is suspected to be true but has not been proven.
• A hypothesis is a statement or assumption that is proposed for the sake of argument so that it can be used in a proof.
• A proposition is a statement that is either true or false, but unlike a theorem, it might not necessarily be a profound result.

Hence, we can conclude that the Theorem can not be proven false.

The activity described by the teacher involves drawing rectangles that use the same number of squares on the graph sheet. 

Key Points

• This means that each rectangle drawn by the students must have the same area as the one drawn by the teacher.
• By keeping the area the same and allowing the lengths and widths to vary, students will see that rectangles with the same area can have different perimeters.
• They can create different shapes using the same number of squares, resulting in different lengths and widths and therefore different perimeters.
• The shapes can vary in length and width but must cover the same number of squares, meaning they have the same area.

Hence, we can conclude that rectangles with equal areas can have different perimeters describes the most appropriate objective of doing this activity.

The mathematics curriculum typically has a series of aims, including developing problem-solving skills, creating a positive attitude towards the subject, and highlighting the application of mathematics in real-world scenarios.

Key Points

• One of the objectives of primary-stage mathematics is indeed to foster a positive attitude towards the subject.
• In the primary stages, mathematics education typically focuses more on concrete, hands-on learning, and intuitive understanding rather than abstract concepts and formal algorithms.
• One of the goals of secondary mathematics education is to develop students' ability to reason logically, solve problems, and clearly articulate their mathematical thinking.
• A significant objective of the higher secondary mathematics curriculum is to show students how mathematics is applied in a wide range of fields and scenarios.

Hence, we can conclude that the primary-stage mathematics curriculum that focuses on teaching abstract concepts through a formal algorithm is not true for the mathematics curriculum.

The type of reasoning that typically proceeds from concrete to abstract is inductive reasoning.

Key Points

• Deductive Reasoning: Deductive reasoning starts with general principles or premises and derives specific conclusions from them. It moves from the abstract/general to the specific/concrete.
• Inductive Reasoning: Inductive reasoning is the process of making generalizations based on specific observations or instances. Inductive reasoning moves from the concrete/specific to the abstract/general.
• Reductio ad Absurdum: Reductio ad Absurdum is a method of proof in which a statement is proven true by assuming that it is false and then deriving a contradiction from that assumption.
• Analytical Reasoning: Analytical reasoning involves the ability to examine information or a complex situation and break it down into smaller, more manageable parts to solve a problem.

Hence, we can conclude that the correct answer is Inductive reasoning.

Experimental probability is determined by carrying out an experiment and recording the outcomes.

Key Points

• It is typically taught through hands-on activities where students can perform the experiment themselves and observe the results.
• Ask children to toss the coin 50 times and note the outcomes, by repeating the experiment 50 times, the children are more likely to observe a probability close to the theoretical probability.
• This exercise would allow them to better understand how experimental probability approaches theoretical probability with a larger number of trials.
• While examples from the textbook can be helpful in explaining concepts, they might not give the hands-on experience needed to grasp experimental probability.
• Tossing a coin only 5 times may not provide enough data to get a meaningful understanding of probability.
• While discussion is an essential part of teaching, for experimental probability, hands-on experience is often more effective.

Hence, we can conclude that the most appropriate method of teaching experimental probability, in this case, is to Ask children to toss the coin 50 times and note the outcomes.

The Correct answer is (A) and (C)

Concept:

 

Periscope:

• A periscope is an optical device that is used to see objects that are not in direct line of sight.
• A periscope works on the principle of laws of reflection of light.
• Any object in the line of sight reflects the light making it visible in the periscope.
• A simple periscope uses two mirrors placed in a Z shape, inclined at an angle of 45 degrees as shown in the figure.
• it is used by submarines to see ships over the surface of the water in the sea.
• Periscopes are used in tanks and also by soldiers in bunkers to see things outside.
• It can be used by one person at a time.

Explanation:

 

• (A) Light from an object strikes the top mirror at 45° and bounces off at the same angle to the lower mirror so that a person can see the view outside: This is an accurate depiction of how a periscope functions. The periscope generally has two mirrors, one at the top and one at the bottom, both placed at an angle of 45 degrees. When a beam of light from an object (for example, a ship in the distance) falls on the top mirror, it gets reflected at an equal angle (45 degrees) into the periscope tube towards the lower mirror. The lower mirror then again reflects the light toward the eyepiece where someone inside the submarine can observe the object.
• (B) Light from the lower mirror enters the top mirror at 90° and bounces off at 90° to the top mirror to make the top view visible: This statement is incorrect because the angle of incidence in a periscope is 45 degrees, not 90 degrees. Also, in a periscope, the light doesn't go from the bottom mirror directly to the top mirror—it travels from the top mirror to the bottom mirror and then to the observer's eyepiece.
• (C) It is based on laws of reflection: This statement is correct. As explained above, periscopes operate on the law of reflection, where light reflects off at the same angle but in the opposite direction to the incident angle.
• (D) It is based on laws of refraction: This statement is incorrect because the light in a periscope is reflected, not refracted. Refraction is the bending of light as it passes from one medium to another (like from air into water), but this isn't what's happening in a periscope.

Hence the correct options are (A) and (C)

The Correct answer is Only (D)

CONCEPT:

• Density: The mass per unit volume is called density. It is denoted by ρ.
• The density of ice is less than that of water.

Density (ρ) = Mass (M)/Volume (V)

Explanation:

• (A) Being a solid, density of ice is greater than water: This statement is false. Although it is true that for most substances the solid state is denser than the liquid state, water is an exception due to its unique molecular structure. When water freezes and becomes ice, it actually expands and becomes less dense. That's why ice floats on water.
• (B) Ice combines with air which helps it float due to buoyancy: This statement is false. While air can influence buoyancy, the reason ice floats isn't because it combines with air. It floats because of its lower density compared to liquid water.
• (C) Ice undergoes a break in hydrogen bonding which makes it float: This statement is false. It's not the breaking of hydrogen bonds that makes ice float; in fact, hydrogen bonding plays an important role in why water expands when it freezes. When water turns into ice, the hydrogen bonds lead to a larger and more open structure – this results in more space between the water molecules, making the ice less dense than the liquid water, allowing it to float.
• (D) Molecular arrangement in ice is different than in water molecules: This statement is true. In liquid water, the individual water molecules are constantly moving and can be close together or further apart. But when water freezes into ice, the molecules arrange themselves into a hexagonal lattice structure. This structure takes up more space than the more random arrangement of liquid water molecules, making ice less dense and allowing it to float on water.

The Correct answer is assertion is true but reason is false.

Concept:

• A good conductor of electricity is a material that allows electric current to flow through it with minimal resistance.
• It is characterized by having a high electrical conductivity. In a good conductor, the outermost electrons of the atoms are loosely bound and are free to move within the material.
• These free electrons can easily carry electric charges, allowing the flow of electric current.
• Metals, such as copper, silver, aluminium, and gold, are known for their high electrical conductivity and are commonly used as conductors in electrical wiring, circuitry, and various electrical components.

Explanation:

• Assertion: Copper and aluminium wires are usually used for transmission of electricity. This assertion is true. Copper and aluminium wires are indeed frequently utilized in electricity transmission. Both metals are known for having high conductivity (meaning they easily allow the flow of electric current) and reliability, making them suitable for this purpose.
• Reason: Copper and aluminium are good conductors and have high resistivity. The reason is false due to its incorrect usage of the term 'resistivity.' Copper and aluminium do not have high resistivity; indeed, they have low resistivity, which is one of the reasons they are good conductors of electricity.
• Resistivity is a material's ability to resist electric current. A material with a high resistivity is a poor conductor because it opposes the passage of electric current. Conversely, a good conductor will have low resistivity because it allows electric current to flow readily.

Thus, copper and aluminium are good conductors because they have low resistivity, not high. Therefore, while the assertion is correct, the reason given is incorrect.

The Correct answer is There is decrease in atmospheric pressure at high altitudes which leads to lowering of boiling point of water.

Concept:

• The force per unit area is called pressure.
• As we go on higher altitudes the atmospheric pressure decreases.
• The heat required to change the phase of any substance without changing the temperature is called the latent heat of that substance.
• The temperature at which water starts boiling is called as the boiling temperature of water.

Explanation:

• The boiling point of a liquid is the temperature where the atmospheric pressure and the vapor pressure of the liquid are equal. At this temperature, liquid water starts to transform into water vapor (we see this as boiling).
• But when it comes to different altitudes, the atmospheric pressure changes - it decreases the higher you go. At lower altitudes, like at sea level, the atmospheric pressure is higher. This means you need a higher temperature to equalize the vapor pressure and atmospheric pressure and get the water to boil - typically this is 100 degrees Celsius.
• At higher altitudes, the atmospheric pressure is lower. Therefore, the temperature needed to make the vapor pressure of water equal to the atmospheric pressure is also lower, causing water to boil at a temperature below 100 degrees Celsius.

The Correct answer is Above 115 °C

Explanation:

• Sodium melts at 98 °C and boils at 882 °C, meaning it is in a liquid state anywhere between those two temperatures.
• Sulphur, on the other hand, melts at 115 °C and boils at 442 °C. Its liquid state is thus found between these two temperatures.
• To find a temperature where both elements would be in the liquid state, we need a temperature range that falls within the liquid state ranges of both elements. This is only achieved above the higher of the two melting points, which is 115 °C for sulphur, and below the lower of the two boiling points, which is 442 °C for sulphur.

So, both sodium and sulphur would be in a liquid state at temperatures above 115 °C but below 442 °C.

The Correct answer is (A) and (D).

Concept:

Force

• A push or a pull on an object is called a force.
• The SI unit of force is the newton which is represented by N.
• The forces in nature can be classified according to whether they act on the body by coming into physical contact with it or not.
• On the basis of this, the forces are of two types -contact forces and non-contact forces.

Contact force: 

The force exerted by one object on another by physical contact.

• There is no field required for contact force.
• Examples: friction, air resistance, etc.

Non-contact force:  

Force exerted by one object on another from a distance or without any physical contact.

• There is always a field required for non-contact force.
• ​Examples: Gravitational, magnetic, electrostatic, etc.

Explanation:

• (A) Electrostatic force is a non-contact force. This statement is correct. The electrostatic force, which is the force between electrically charged objects, can occur without the objects physically touching each other. For example, a charged balloon can attract small bits of paper even without touching them.
• (B) Ability to lift things with a magnet is an example of contact force. This statement is incorrect. Magnetic force is actually a non-contact force because it doesn't require physical contact to be effective. You can lift a paperclip off a table with a magnet without the magnet physically touching the paperclip.
• (C) Airplane flying through the air is an example of non-contact force. This statement is incorrect. It's primarily the contact force of air pressure difference on the wings (lift) that enables an airplane to fly, along with the force of thrust from the engines and the force of air resistance (drag).
• (D) Gravitational force is a non-contact force. This statement is correct. Gravity is a non-contact force; it acts at a distance, pulling objects toward each other without any requirement for physical contact. For example, gravity attracts an apple falling from a tree toward the Earth without any physical contact between the Earth and the apple.

Additional Information

• There are many types of force divided into two groups contact force and Non-contact (distance) force.

From the above table, we know that the force of friction is the contact force.

The force of friction:

• Friction is the force that resists motion when the surface of one object comes into contact with the surface of another.
• The frictional force also depends on the shape of the object and the nature of the fluid.​

The force of gravitational:

• The gravitational force is the force of attraction that exists between all objects in the universe that have mass.
• The force of gravity decreases with altitude.
• It also varies on the surface of the earth, decreasing from the poles to the equator.

Magnetic force:

• ​The magnetic force between two moving charges can be described as the effect exerted on either charge by a magnetic field created by the other.
• Magnetic force is another type of attractive or repulsive force that exists between two objects.
• Due to the magnetic force, the attraction or repulsion depends upon the polarity of the objects.

Electrostatic force: 

• The electrostatic force between particles is an attracting or repulsive force caused by the electric charges on their surfaces.
• The electrostatic force is the electric force that exists between two stationary charged bodies.

The Correct answer is Sound can travel through vacuum.

CONCEPT:

• Sound wave: The longitudinal wave in an elastic medium that produces an audible sensation is called a sound wave.
• As sound waves are longitudinal waves, the air particles vibrate to and fro in the direction of propagation of sound.
• It is a wave of compression and rarefaction.
• Compressions and rarefactions are part of a sound wave.
• Compression: A region in a longitudinal wave where the particles are closest together is called compression. Compression has high density and high pressure

Rarefaction: A region in a longitudinal wave where the particles are furthest apart is called rarefaction. The rarefaction has a low density and low pressure

Explanation:

• Vibration produces sound in humans: This statement is True. In humans, our vocal cords vibrate to create sound waves that we perceive as speech, singing, etc.
• Sound can travel through a vacuum: This statement is False. Sound requires a medium to travel, whether that be a solid, liquid, or gas. In a vacuum, where there are no particles to vibrate and transmit the sound waves, sound cannot travel.
• If the amplitude becomes twice, the loudness will be four times: This statement is True (with some qualifications). The loudness of sound is generally proportional to the square of the amplitude, so if the amplitude doubles, the loudness will be about four times greater. However, the perception of loudness by human ears is not exactly linear with the intensity, due to the logarithmic nature of human hearing.
• Higher the frequency of vibration, the shriller the sound: This statement is True. The frequency of a sound wave determines its pitch. As frequency increases, the pitch of the sound becomes higher, which we perceive as a shrill sound.

The Correct answer is B > D > C > A.

Concept:

• The pH scale is the scale used to express the acidity or alkalinity of a substance based on the concentration of hydrogen ions in its solution.
• pH scale was discovered by Soren Sorensen.
• The range of the pH scale is 0 to 14.
• If pH < 7 then the solution is acidic.
• If pH > 7 then the solution is basic.
• If pH = 7 then the solution is neutral.

Explanation:

Given pH values of solutions A, B, C, and D are respectively 9, 2, 7, 4 

pH of A = 9, so it is alkaline in nature (base)

pH of B = 2, so it is acidic in nature

pH of C = 7, so it is neutral in nature

pH of D= 4, so it is acidic in nature

Hence the correct order of acidic strength. B > D > C > A.

The Correct answer is Use separating funnel to remove water and oil, followed by evaporating water from salt.

Concept: 

Different techniques are used to separate mixtures based on the physical and chemical properties of the components.

1. Sublimation: This process is used to separate a mixture of substances where one of them can turn directly from the solid state to the gaseous state without becoming liquid. An example of such a substance is naphthalene.
2. Crystallization: This method is used to purify substances. It's usually employed to separate solids that have dissolved in a liquid and need to be recovered.
3. Evaporation: This process is used to separate a soluble substance from a solution. It is usually used to separate a solid substance that has dissolved in a liquid and needs to be retrieved.
4. Separating Funnel: This method is typically used to separate two immiscible liquids. Immiscible liquids are those that do not mix to form a homogeneous mixture. An example is oil and water.

Explanation:

 

• Use a separating funnel: The mixture is first placed in a separating funnel. Since oil and water are immiscible (do not mix), they form two separate layers, with oil being less dense floating on top of the denser water layer. The stopcock of the funnel can be opened to allow the lower layer (water containing the dissolved salt) to be removed while keeping the oil in the funnel.
• Evaporate the water from the salt: After separating the water-salt solution from the oil using the separating funnel, the water can be evaporated using heat, leaving behind the salt.

Therefore, option 2) describes the correct process.

The Correct answer is Reactivity of Mg is more than Cu but less than Na.

Explanation:

Mg + CuO → MgO + Cu

• This reaction is feasible. Magnesium (Mg) is more reactive than copper (Cu) and is higher in the reactivity series. When magnesium reacts with copper oxide (CuO), the more reactive magnesium will displace copper from the compound, resulting in the formation of magnesium oxide (MgO) and copper metal (Cu).
• Mg + CuO→  MgO + Cu.

2 Na + MgO:

• This reaction is not feasible. Sodium (Na) is more reactive than magnesium (Mg) and is higher in the reactivity series. When sodium reacts with magnesium oxide (MgO), the more reactive sodium will displace magnesium from the compound resulting in the formation of sodium oxide (Na2O).
• 2 Na + MgO →  Na2O + Mg.

 Zn + CuO → ZnO +Cu

• This reaction is feasible. Zinc (Zn) is more reactive than copper (Cu) and is higher in the reactivity series. When zinc reacts with copper oxide (CuO), the more reactive zinc will displace copper from the compound, resulting in the formation of zinc oxide (ZnO) and copper metal (Cu). 
• Zn + CuO →  ZnO + Cu.

Additional Information 

• Sodium is the most reactive of the three elements (Mg, Cu, and Na).
• Magnesium is less reactive than sodium but more reactive than copper.
• Copper is the least reactive.