Mathematics and Science PYQs

Concept used: 

To determine the least number to be added to 955 to make it a perfect square, we need to find the smallest integer x such that 955 + x is a perfect square. Once we find x, we can then calculate the value of 3x + 2.

• A perfect square is a number that can be expressed as the square of an integer.
• To find the smallest x, we first need to find the nearest perfect square greater than 955.

Calculation 

Find the nearest perfect square greater than 955:

The square root of 955 is approximately 30.90. The next integer is 31.

Calculate the next perfect square:

(31)² = 961

Determine the value of x:

⇒ x = 961 - 955 = 6

Now 3x + 2:

⇒ 3x + 2 = 3(6) + 2 = 18 + 2 = 20

∴ The Correct answer is Option (3): 20

Given :

• ∠A = 50º
• ∠B = 70º

Concept Used: 

In a triangle, the angle bisector of an angle divides the opposite side into segments that are proportional to the adjacent sides. 

Calculation: 

Triangle Sum Property: The sum of the interior angles of a triangle is always 180º. For triangle ΔABC, we have:

⇒∠A + ∠B + ∠C = 180º

⇒∠C = 180º - 50º  - 70º 

⇒ ∠C = 60º

Angle Bisector Property:

Since CD is the bisector of ∠ACB, it divides ∠C into two equal parts.

Thus, ∠ACD = ∠BCD = ∠C / 2 = 60º / 2 = 30º

Angle Calculation at Point D:

In ΔADC, we need to find ∠ADC.

The sum of angles in ΔADC is also 180º.

⇒∠A + ∠ACD + ∠ADC = 180º

Substitute the known values: 50º + 30º + ∠ADC = 180º

⇒∠ADC = 180º - 80º = 100º

∴ The Correct answer is Option (2): 100º

At the upper primary stage, the number system is a fundamental component of the mathematics curriculum designed to build a strong foundation in numerical understanding and operations. 

Key Points

• This stage typically includes various topics that help students grasp the concept of numbers and their applications in real-life scenarios.
• Complex numbers are numbers that have both a real and an imaginary part. They are generally represented in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.
• This concept is considered advanced and is typically introduced at the higher secondary or college level, not at the upper primary stage.
• Understanding complex numbers requires a prior understanding of real numbers, exponents, and other foundational mathematical concepts, which are covered in higher education levels.

Hence, we can conclude that as per NCERT, Complex Numbers have not been included under number system at upper primary stage.

Given:

BC = 48 cm, CD = 17 cm, AD = 40 cm

∠A = ∠B = 90°

Formula used:

Area of rectangle = Length × Breadth

Area of right-angled triangle = 1/2 × Base × Perpendicular

Hypotanesous² = Base² + Perpendicular²

Calculation:

In Δ DEC,

Perpendicular (DE) = √(DC² - EC²)

⇒ DE = √(172 - 82)

⇒ DE = √(289 - 64)

⇒ DE = √225 = 15 cm

Now,

Area of Δ DEC

⇒ 1/2 × EC × DE

⇒ 1/2 × 8 × 15

⇒ 1/2 × 120 = 60 cm²

Now, AB = DE = 15 cm

Area of rectangle ABED

⇒ AB × AD

⇒ 15 × 40 = 600 cm²

Thus,

Area of quadrilateral ABCD 

⇒ Area of Δ DEC + Area of rectangle ABED

⇒ 60 + 600

⇒ 660 cm²

∴ The correct answer is option (3).

Mathematical proofs are logical arguments that establish the truth of mathematical statements. They are a fundamental part of mathematics, serving multiple purposes such as validating results, understanding why they are true, and exploring deeper connections within mathematical structures.

Key Points

• Logical Reasoning: Proofs in mathematics are primarily based on logical reasoning rather than intuitive knowledge. While intuition can guide the formation of conjectures, the proof itself must be rigorously justified through logical steps.
• Validation: Proofs serve to validate the correctness of mathematical statements, ensuring that they hold true under all specified conditions.
• Explanatory Power: Proofs explain why a particular mathematical result must be true, providing insights into the underlying principles that govern the result.
• Connections and Insights: Proofs help to reveal connections between different areas of mathematics and provide deeper insights into the structure and relationships within the field.

Hence, we can conclude that Proofs are built on intuitive knowledge and not reasoning is least appropriate regarding the proofs in mathematics.

Creativity in mathematics involves the ability to think outside the box, approach problems from multiple angles, and generate innovative solutions. 

Key Points 

• Convergent thinking is about finding a single, correct solution to a problem.
• It is more focused and linear, often used in situations where there is a clear, definitive answer.
• While convergent thinking is important in mathematics, it is not typically associated with creativity.
• Students use multiple and alternate problem-solving strategies: This indicates a high level of creativity as students are not confined to a single method. They explore various approaches to find the best possible solution.
• Students can pose questions: The ability to ask questions is a crucial aspect of creativity. It shows that students are curious and engaged, seeking to understand the subject deeply rather than just accepting information passively.
• Students can think flexibly: Flexible thinking is a hallmark of creativity. It involves adapting to new situations, seeing connections between different concepts, and being open to changing one's approach as needed.

Hence, we can conclude that Students using convergent thinking in different contexts is not an indicator of creativity in Mathematics.

Given: 

Arun is three years older than Varun

Eight years ago \(\frac{5}{6}th\) of Arun's age exceeded \(\frac{3}{5}th\) of Varun's age by 6 years.

The present age of Varun x years

Calculation: 

Since Arun is 3 years older than Varun: 

The present age of Arun x +3 

Eight years ago:

Arun's age was (x + 3) - 8 = x - 5 years

Varun's age was x - 8 years

According to the problem, Eight years ago \(\frac{5}{6}th\) of Arun's age exceeded \(\frac{3}{5}th\) of Varun's age by 6 years.

⇒ \(\frac{5}{6}\times (x - 5) - \frac{3}{5}\times (x - 8) =6\)

This matches the third option.

∴ The correct answer is Option (3): \(\frac{5}{6}\times (x - 5) - \frac{3}{5}\times (x - 8) =6\)

Rubrics are a scoring tool that lists the criteria for a piece of work or "what counts." They help in clarifying expectations and providing detailed feedback to students. 

Key Points

• Rubrics consist of a clear performance target for students. This means that rubrics specify what is expected in terms of performance, making the objectives transparent.
• Contrary to the incorrect statement, rubrics make evaluation more objective, not subjective.
• By providing specific criteria and performance levels, rubrics reduce ambiguity and help ensure consistency in grading.
• Rubrics have a fixed scale that categorizes different levels of performance.
• For example, a scale might range from "Excellent" to "Needs Improvement," with each level describing specific characteristics of the work.
• Rubrics include a list of characteristics describing performance for each point on the scale.
• This helps students understand what is required to achieve each performance level and provides detailed feedback on their strengths and areas for improvement.

Hence, we can conclude that rubrics make evaluation more subjective is not correct with respect to Rubrics as a tool of assessment.

Given

The formula for the volume of a right circular cylinder is: V = πr²h, where r is the base radius and h is the height.

Original radius (r) = 6 cm

Original height (h) = 9 cm

Calculation:

Original Volume (V_original) = π × (6)² × 9 = π × 36 × 9 = 324π cm³

New Dimensions of the Cylinder:

New radius = 6 cm / 2 = 3 cm

New height = 9 cm + 3 cm = 12 cm

New Volume of the Cylinder:

New Volume (V_new) = π × (3)² × 12 = π × 9 × 12 = 108π cm³

Now, we need to compare the new volume with the original volume:

Comparing Volumes:

Original Volume (V_original) = 324π cm³

New Volume (V_new) = 108π cm³

New Volume / Original Volume = 108π / 324π = 1/3

Thus, the new volume is one-third of the original volume.

∴ The Correct answer is Option (2): The new volume will be one-third the original volume.

3Concept used: 

BODMAS

Step 1 : Bracket 

Step 2 : OF

Step 3 : Divide

Step 4 : Multiplication

Step 5 : Addition

Step 6 : Substraction

Calculation: 

Evaluate the square root: √(441) = 21.

Substitute the value back into the expression: 52 × 3 ÷ 4 + 21 ÷ 7 × 3 + 5 - 32 ÷ 8 × 12.

Perform the multiplication and division from left to right:

52 × 3 ÷ 4 + 21 ÷ 7 × 3 + 5 - 32 ÷ 8 × 12.

52 × 3 ÷ 4 + 3 × 3 + 5 - 4 × 12.

52 × 3/4 + 3 × 3 + 5 - 4 × 12.

 52 × 3/4 + 9 + 5 - 48 

13 × 3 + 9 + 5 - 48

39 + 14 - 48

53 - 48 = 5

 

∴ The Correct answer is Option (4): 5

Socio-scientific issues are complex, real-world problems that involve both scientific and societal aspects. These issues are grounded in scientific principles, research, and understanding. They often require scientific knowledge to address and solve.

Key PointsDiscussion on socio-scientific issues in class helps in:

• (a) Developing argumentative skills in learners: Engaging in discussions about socio-scientific issues helps students learn how to argue and present their viewpoints effectively.
• (b) Developing multiple perspectives on an issue: These discussions expose students to different viewpoints and help them understand various aspects of a topic.
• (c) Encouraging citizen science activities: Discussing these issues can motivate students to participate in science-related activities that benefit the community.

Hint

• (d) Appreciating the role of society in development of science is also a benefit, it is more about understanding the relationship between society and science, which is broader and not always the direct focus of such discussions.

Hence, the correct answer is (a), (b) and (c).

The correct answer is Option 1.

Concept

In this problem, we need to distinguish between observations (O) and inferences (I) based on given statements about chemical properties and reactions. Observations are direct sensory experiences or measurements, while inferences are logical conclusions drawn from those observations.

 Key Points

• Observation (O): A direct measurement or sensory experience, such as seeing lime water turn milky when carbon dioxide is passed through it.
• Inference (I): A logical conclusion based on observations, such as deducing that air expands when heated.

Explanation:

• O - C (Lime water turns milky on passing carbon dioxide) is a direct observation based on the reaction between lime water and carbon dioxide, forming calcium carbonate.
• I - A (Metal oxides are basic in nature) is an inference drawn from a different experiment involving metal oxides, not directly from observation C.
• I - B (Air expands on heating) can be inferred based on heating experiments, though it may not be directly related to a specific observation here.
• I - D (Sodium hydroxide and hydrochloric acid neutralize each other) is inferred from the reaction of an acid and a base.

Thus, the correct answer is Option 1.

CONCEPT:

Buoyancy and Density:

• When an object floats in a liquid, the buoyant force is equal to the weight of the liquid displaced by the submerged part of the object.
• The fraction of the volume of the object that remains submerged is inversely proportional to the ratio of the density of the liquid to the density of the object.
• If an object floats with a smaller fraction of its volume outside the liquid, it is in a denser liquid.

Explanation:

• Let V be the total volume of the object and \( \rho_0\) be the density of the object.
• For liquid A:1/4 V is outside the liquid, so 3/4 V is inside the liquid.
• Using Archimedes' principle:
• \(\rho_A \cdot \frac{3}{4}V = \rho_0 \cdot V \ \ implying \ \ \rho_A = \frac{4}{3} \rho_0\)
• For liquid B: 3/4 V is outside the liquid, so1/4 V is inside the liquid.
• Using Archimedes' principle: \(\rho_B \cdot \frac{1}{4}V = \rho_0 \cdot V \ \ implying\ \ \rho_B = 4\rho_0\)
• Comparing densities: \(\rho_A = \frac{4}{3} \rho_0 \ \ and \ \ \rho_B = 4\rho_0\)

Since 4 (density of liquid B) is greater than 4/3 (density of liquid A), the correct statement is:

• 1. Density of liquid B is greater than density of liquid A.

Scientific investigations refer to the systematic and methodical process of exploring natural phenomena, acquiring knowledge, and answering questions through observation, experimentation, and analysis.

Key PointsFeatures of Scientific Investigations:

• (b) Social context may influence conclusions in any investigation: This statement is true because the context in which scientific research is conducted, including social, cultural, and economic factors, can influence how conclusions are drawn.
• (d) There is no universal scientific method: This is true as well because while there are common elements in scientific methods, the approach and methodology can vary depending on the field of study and the specific nature of the investigation.

Hint

• (a) Results are not influenced by procedure: This is not true because the procedure or methodology of an investigation can significantly affect the results. A change in procedure might lead to different results.
• (c) Data and evidence are same: This is incorrect because while data refers to the raw information collected during an investigation, evidence involves the interpretation of this data to support or refute a hypothesis.

Hence, the correct answer is (b) and (d).

The correct answer is (a) is false and (b) is true.

Concept:

Self-pollination and cross-pollination are two types of pollination in plants. 

• Self-pollination: This occurs when pollen from the anther (male part) of a flower transfers to the stigma (female part) of the same flower or another flower on the same plant. This generally happens in bisexual (hermaphroditic) flowers that contain both male and female reproductive organs.
• Cross-pollination: This involves the transfer of pollen from the anther of one flower to the stigma of a flower on a different plant of the same species. Bisexual flowers can participate in both self and cross-pollination.

**Explanation of Given

Statements:

**

• (a) Self-pollination takes place only in bisexual flowers: This statement is false. Self-pollination can occur in both bisexual flowers (containing both male and female reproductive organs) and unisexual flowers in monoecious plants, where male and female flowers are on the same plant (e.g., maize). Therefore, it is incorrect to say that self-pollination takes place only in bisexual flowers.
• (b) Bisexual flowers can pollinate both by self and cross pollination: This statement is true. Bisexual flowers have both male and female parts, which allow them to participate in both self-pollination and cross-pollination.

Therefore, the correct option is (a) is false and (b) is true.

The correct answer is (a), (b), (d)

Concept:

Rainwater harvesting involves collecting and storing rainwater for future use instead of allowing it to run off. This practice has numerous benefits including, but not limited to, water conservation, reducing the risk of flooding, and providing water for irrigation.

Explanation:

• (a) It conserves ground water: This is true because rainwater harvesting helps recharge groundwater levels by allowing rainwater to percolate into the ground.
• (b) It reduces local flooding: This is true because rainwater harvesting systems capture and store rainwater, thereby reducing the volume of runoff and the risk of local flooding.
• (c) It creates drainage problems: This statement is false. Rainwater harvesting reduces drainage problems by reducing the volume of runoff.
• (d) It can be used for irrigation of crops: This is true as the stored rainwater can be used for various purposes including irrigation, reducing the dependency on other water sources.

Therefore, the statements that are true regarding rainwater harvesting are (a), (b), and (d).

Concept:-

The correct factors responsible for the generation and flow of winds on the Earth are

• (a) Uneven heating of the Earth and
• (b) Uneven heating of the oceans.

Winds are primarily driven by differences in atmospheric pressure, which are in turn caused by the uneven heating of the Earth's surface by the sun. This differential heating creates areas of high and low pressure, and the movement of air from high to low pressure areas generates wind. Understanding the factors that contribute to this uneven heating is crucial for comprehending the mechanisms behind wind formation.

 Key Points

• Uneven Heating of the Earth:
• The Earth receives different amounts of solar radiation at different latitudes due to its spherical shape and axial tilt.
• Equatorial regions receive more direct sunlight year-round, leading to higher temperatures compared to the poles.
• This uneven heating creates pressure differences, as warm air rises at the equator and cold air sinks at the poles, setting up a circulation pattern known as atmospheric circulation.
• Uneven Heating of the Oceans:
• Oceans and landmasses heat up and cool down at different rates due to their specific heat capacities.
• During the day, land heats up faster than water, creating low-pressure areas over land and high-pressure areas over the ocean, resulting in sea breezes.
• At night, the reverse occurs, and land cools down more quickly than the ocean, leading to land breezes.
• This differential heating and cooling contribute to local wind patterns and can influence larger-scale atmospheric circulation.

Additional Information:

• Distance between Earth and Sun:
• The distance between the Earth and the Sun does not significantly change over short periods and thus does not directly influence daily wind patterns.
• While the Earth's elliptical orbit does cause slight variations in solar radiation received throughout the year, this is not a primary driver of wind formation.
• Gravitational Force on the Earth:
• Gravitational force is responsible for keeping the atmosphere in place but does not directly cause wind.
• Wind is generated by the movement of air due to pressure differences, not by gravitational forces.

Thus, the correct factors responsible for the generation and flow of winds on the Earth are Uneven heating of the Earth and Uneven heating of the oceans.

CONCEPT:

Properties of Tungsten:

• Tungsten is chosen for the filament of incandescent bulbs because of its unique properties.
• Tungsten has a very high melting point (about 3422°C or 6192°F).
• It also has high resistivity, which means that it can withstand high temperatures before melting and can emit light efficiently when a current passes through it.

Explanation:-

• Assertion (A) states that the filament of an incandescent bulb is made up of tungsten. This is correct because tungsten is the material commonly used for bulb filaments due to its high melting point.
• Reason (R) states that tungsten has low resistance and high melting point. While the high melting point part is correct, tungsten actually has high resistance, not low resistance. This part of the reason is incorrect.

**On analyzing both the

Statements:

**

• 1. (A) is true and (R) is false.

Gender responsive pedagogy in science involves practices that address and promote gender equality.

Key PointsA gender responsive pedagogy in science entails:

• (a) Use of gender sensitive language: Employing language that avoids reinforcing gender stereotypes and ensures inclusivity.
• (b) Engaging students in examining stereotypes: Encouraging students to critically analyze and challenge gender stereotypes and biases within science and beyond.

Hint

• (c) Making gender based groups is not typically a component of gender responsive pedagogy as it might reinforce stereotypes rather than address them.
• (d) Using differentiated instruction and assessment is valuable for accommodating diverse learning needs but is not specifically focused on gender responsiveness.

Hence, a gender responsive pedagogy in science entails (a) and (b).

CONCEPT:

Conduction of Heat:

• Conduction is the process by which heat energy is transmitted through collisions between neighboring atoms or molecules.
• Conduction occurs in solids, liquids, and gases but is most efficient in solids where the particles are closely packed.
• Different materials have different thermal conductivities, which means they allow heat to flow through them at different rates.
• For conduction to occur, there must be a temperature gradient and direct contact between the objects.

Explanation:-

• 1. Conduction does not take place in gases - This statement is not true. Conduction does occur in gases, though it is less efficient compared to solids and liquids.
• 2. All materials do not allow heat to flow through them in the same way - This statement is true. Different materials have varying thermal conductivities.
• 3. Conduction is faster in solids than liquids - This statement is true. In solids, particles are more closely packed, allowing heat to transfer more quickly.
• 4. Two objects should be in contact with each other for heat flow - This statement is true for conduction, as direct contact is required for heat to transfer via conduction.

 Additional Information

• In gases, conduction is relatively inefficient because the particles are widely spaced, reducing the frequency of collisions that transfer energy.
• In liquids, conduction is more effective than in gases but less efficient than in solids due to the intermediate spacing of particles.
• In practical applications, materials with high thermal conductivity, such as copper and aluminum, are used for heat sinks and other thermal management solutions.

The incorrect statement is 1) Conduction does not take place in gases.