Mathematics and Science PYQs

The correct answer is 52 m

Key Points

• To find the radius of the third circular field C₃, we first need to determine the radii of the two given circular fields C₁ and C₂.
• The formula for the perimeter (circumference) of a circle is 2πr, where r is the radius.
• For field C₁: 2πr₁ = 40π m. Dividing both sides by 2π, we get r₁ = 20 m.
• For field C₂: 2πr₂ = 96π m. Dividing both sides by 2π, we get r₂ = 48 m.
• The area of a circle is given by the formula πr².
• The area of the third field C₃ is the sum of the areas of C₁ and C₂:

\( Area(C_{3}) = Area(C_{1}) + Area(C_{2}) \)

\( \pi r_{3}^{2} = \pi r_{1}^{2} + \pi r_{2}^{2} \)

• Substitute the values of r₁ and r₂:

\( \pi r_{3}^{2} = \pi (20)^{2} + \pi (48)^{2} \)

r₃² = 400 + 2304

r₃² = 2704

• Calculate the radius r₃:

\( r_{3} = \sqrt{2704} = 52 \) m.

• Thus, the radius of the third circular field is 52 m.

Additional Information

• The relationship \( r_{3} = \sqrt{r_{1}^{2} + r_{2}^{2}} \) is derived directly from the principle of additive areas in circles. This is analogous to the Pythagorean theorem, where the radii 20, 48, and 52 form a Pythagorean triple (specifically, 4 × 5, 4 × 12, 4 × 13).
• If the problem had stated that the perimeter of C₃ is the sum of the perimeters of C₁ and C₂, then the radius would simply be the sum of the radii: r₃ = r₁ + r₂ = 20 + 48 = 68 m. However, since the areas are summed, we must use the squared relationship.

Shortcut Trick

Check if the given sides are multiples of a standard Pythagorean triplet.

Divide the numbers in option (1) by their common factor 2: 14/2, 48/2, 50/2 → 7, 24, 25.

Check the basic triplet: 7² + 24² = 49 + 576 = 625 = 25².

Since 7, 24, 25 is a triplet, its multiple 14, 48, 50 is also a right triangle triplet.

∴ The correct answer is 14, 50, 48 cm.

Alternate Method

Given sides: 14 cm, 50 cm, 48 cm.

Formula used: a² + b² = c² (Pythagoras Theorem, where c is the hypotenuse).

⇒ Let perpendicular (a) = 14 cm and base (b) = 48 cm.

⇒ a² + b² = 14² + 48²

⇒ 14² + 48² = 196 + 2304 = 2500

⇒ Let hypotenuse (c) be the longest side = 50 cm.

⇒ c² = 50² = 2500

⇒ Since a² + b² = c², the sides form a right-angled triangle.

∴ The correct answer is 14, 50, 48 cm.

Additional Information

Pythagorean Triplets

A set of three positive integers a, b, and c, such that a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

General Formula for Triplets

For any two positive integers m and n (where m > n), triplets can be generated using: a = m² − n², b = 2mn, and c = m² + n².

Nature of Triangles

If a² + b² > c², the triangle is acute-angled. If a² + b² < c², the triangle is obtuse-angled.

Shortcut Trick

Combine all terms under a single square root: √((5043 × 50) / (32 × 1323))

Simplify fractions inside the root: (5043 ÷ 1323 = 1681/441) and (50 ÷ 32 = 25/16)

New expression: √(1681/441 × 25/16) = (41/21) × (5/4)

Calculate final value: 205 / 84

∴ The correct answer is 205/84.

Alternate Method

Given:

Numerator = √5043 × √50

Denominator = √32 × √1323

Formula Used:

√(a × b) = √a × √b | √(a / b) = √a / √b

Calculations:

⇒ √5043 = √(1681 × 3) = 41√3

⇒ √50 = √(25 × 2) = 5√2

⇒ √32 = √(16 × 2) = 4√2

⇒ √1323 = √(441 × 3) = 21√3

⇒ Value = (41√3 × 5√2) / (4√2 × 21√3)

⇒ Value = (205 × √6) / (84 × √6)

⇒ Value = 205 / 84

∴ The correct answer is 205/84.

Additional Information

Product Rule for Surds

For any non-negative numbers a and b, √a × √b = √(ab).

Quotient Rule for Surds

For any non-negative number a and positive number b, √a / √b = √(a/b).

Common Square Roots

Remembering squares of numbers up to 50 helps in fast calculation: 21² = 441, 41² = 1681.

Shortcut Trick

CP of Article A = 382.50 ÷ 0.85 = ₹450

CP of Article B = 616 ÷ 1.12 = ₹550

Total CP of both articles = 450 + 550 = ₹1000

Total SP of both articles = ₹1180

Profit % = [(1180 − 1000) ÷ 1000] × 100 = 18%

∴ The correct answer is 18%.

Alternate Method

Given: SP_A = ₹382.50, Loss = 15%; SP_B = ₹616, Gain = 12%

Formula: CP = [SP ÷ (100 ± Gain/Loss%)] × 100

⇒ CP of Article A = [382.50 ÷ (100 − 15)] × 100 = (382.50 ÷ 85) × 100 = ₹450

⇒ CP of Article B = [616 ÷ (100 + 12)] × 100 = (616 ÷ 112) × 100 = ₹550

⇒ Total Cost Price = 450 + 550 = ₹1000

⇒ Total Selling Price = ₹1180

⇒ Total Profit = 1180 − 1000 = ₹180

⇒ Profit Percentage = (Total Profit ÷ Total CP) × 100

⇒ Profit Percentage = (180 ÷ 1000) × 100 = 18%

∴ The correct answer is 18%.

Additional Information

Cost Price Formula

When there is a gain: CP = [SP ÷ (100 + Gain%)] × 100. When there is a loss: CP = [SP ÷ (100 − Loss%)] × 100.

Overall Profit and Loss

Overall Profit or Loss depends on the total Cost Price and total Selling Price of all items combined.

Profit Percentage on CP

Gain% = (Profit ÷ Cost Price) × 100. Always calculate profit percentage on the Cost Price unless specified otherwise.

The correct answer is It would provide opportunity to the students to formally learn the mathematical concepts.

Key Points

• Planning a trip to a local market to teach Discount is a pedagogical strategy aimed at contextualizing mathematics. It allows students to move from abstract theories to concrete applications.
• The term formally learn in this context implies that the trip is a structured educational activity designed to meet specific curricular objectives. It is not just a casual visit but a teacher-led exploration to understand mathematical relations.
• Students get to observe the relationship between Marked Price (MP), Selling Price (SP), and the Discount offered. They can perform calculations like Discount = MP − SP using real-world data.
• While other options like seeing advertisements or enjoyment are valid outcomes, the primary pedagogical goal is to ensure students grasp the mathematical concepts through an experiential learning approach.
• Such trips help in breaking the monotony of classroom teaching and provide a social context to mathematical problem-solving, making the learning process more authentic and enduring.

Additional Information

• The National Curriculum Framework (NCF) 2005 strongly advocates for connecting school knowledge to life outside the school to make learning more meaningful.
• Discount Percentage is always calculated on the Marked Price (MP). The formula is: (Discount ÷ MP) × 100.
• Field trips are considered an active learning strategy that promotes critical thinking and observation skills among students.
• In real-world scenarios, students also encounter concepts like Successive Discounts and GST, which further enrich their Financial Literacy.

Shortcut Trick

x/y = (−1/3)⁻³ ÷ (2/3)⁻⁴ = (−3)³ ÷ (3/2)⁴ = −27 ÷ 81/16 = −27 × 16/81 = −16/3

Expression = (x/y + y/x)⁻¹ × (y/x)⁻¹ = [ (x² + y²) / xy ]⁻¹ × (x/y) = (xy / (x² + y²)) × (x/y) = x² / (x² + y²)

Substitute x = −16, y = 3: (−16)² / [(−16)² + 3²] = 256 / (256 + 9) = 256/265

Wait, let's re-evaluate the expression: ((x/y + y/x)⁻¹) × (y/x)⁻¹ = (xy / (x² + y²)) × (x/y) = x² / (x² + y²). Let's re-check the options.

Re-calculating: x/y = −16/3. Expression = ( (−16/3) + (−3/16) )⁻¹ × (−3/16)⁻¹ = ( (−256 − 9)/48 )⁻¹ × (−16/3) = (−48/265) × (−16/3) = (16 × 16) / 265 = 256/265.

∴ The correct answer is 256/265.

Alternate Method

Given: x/y = (−1/3)⁻³ ÷ (2/3)⁻⁴

Formula Used: (a/b)⁻ⁿ = (b/a)ⁿ and a^m ÷ aⁿ = a^m−n

⇒ x/y = (−3/1)³ ÷ (3/2)⁴

⇒ x/y = −27 ÷ (81/16)

⇒ x/y = −27 × (16/81)

⇒ x/y = −16/3

⇒ y/x = −3/16

Now, evaluate (x/y + y/x)⁻¹ × (y/x)⁻¹:

⇒ (−16/3 − 3/16)⁻¹ × (−3/16)⁻¹

⇒ [ (−256 − 9) / 48 ]⁻¹ × (−16/3)

⇒ [ −265 / 48 ]⁻¹ × (−16/3)

⇒ (−48 / 265) × (−16 / 3)

⇒ (16 × 16) / 265

⇒ 256 / 265

∴ The correct answer is 256/265.

Additional Information

Negative Exponent Rule

For any non-zero rational number a/b, (a/b)⁻ⁿ = (b/a)ⁿ.

BODMAS in Fractions

Always solve the exponents first, then division (by multiplying with the reciprocal), and finally addition or subtraction within brackets.

Reciprocal Property

The expression (k)⁻¹ is simply 1/k. Thus, (x/y + y/x)⁻¹ = 1 / (x/y + y/x) = xy / (x² + y²).

Shortcut Trick

Complementary angles sum to 90°.

(2x − 10) + (3x − 50) = 90

5x − 60 = 90 ⇒ 5x = 150 ⇒ x = 30

Required value: 4x + 40 = 4(30) + 40 = 120 + 40 = 160

∴ The correct answer is 160°.

Alternate Method

Given:

Angle 1 = (2x − 10)°

Angle 2 = (3x − 50)°

Relationship: Complementary angles

Formula used:

Sum of complementary angles = 90°

Calculations:

⇒ (2x − 10) + (3x − 50) = 90

⇒ 5x − 60 = 90

⇒ 5x = 90 + 60

⇒ 5x = 150

⇒ x = 150 ÷ 5 = 30

Substitute x = 30 in the required expression (4x + 40):

⇒ 4 × 30 + 40

⇒ 120 + 40 = 160

∴ The correct answer is 160°.

Additional Information

Complementary Angles

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

Supplementary Angles

Two angles are said to be supplementary if the sum of their measures is exactly 180° (θ₁ + θ₂ = 180°).

Conjugate Angles

Two angles are said to be conjugate (or explementary) if the sum of their measures is exactly 360°.

Shortcut Trick

In a quadrilateral, if one pair of opposite sides is both parallel and equal (AB || DC and AB = CD), the quadrilateral is a parallelogram.

In a parallelogram, consecutive angles are supplementary (sum = 180°) and opposite angles are equal.

Given AB || DC ⇒ ∠B + ∠C = 180° ⇒ 70° + ∠C = 180° ⇒ ∠C = 110°.

Since ABCD is a parallelogram, opposite angles are equal: ∠A = ∠C = 110° and ∠D = ∠B = 70°.

∴ The correct answer is ∠A = 110°, ∠C = 110°.

Alternate Method

Given: Quadrilateral ABCD, AB || DC, AB = CD, and ∠B = 70°.

Formula: Properties of a parallelogram (Opposite sides parallel and equal, opposite angles equal, consecutive angles supplementary).

⇒ In quadrilateral ABCD, since AB || DC and AB = CD, ABCD is a parallelogram by definition.

⇒ In a parallelogram, consecutive interior angles along the transversal connecting parallel lines are supplementary.

⇒ ∠B + ∠C = 180° (Since AB || DC and BC is a transversal)

⇒ 70° + ∠C = 180° ⇒ ∠C = 110°

⇒ Since ABCD is a parallelogram, the other pair of sides is also parallel (AD || BC).

⇒ Therefore, ∠A + ∠B = 180° ⇒ ∠A + 70° = 180° ⇒ ∠A = 110°

⇒ Check opposite angles: ∠A = ∠C = 110° and ∠B = ∠D = 70°.

∴ The correct answer is ∠A = 110°, ∠C = 110°.

Additional Information

Sufficient Conditions for a Parallelogram

A quadrilateral is a parallelogram if: both pairs of opposite sides are parallel, both pairs of opposite sides are equal, OR one pair of opposite sides is both equal and parallel.

Interior Angles of a Parallelogram

Opposite angles of a parallelogram are always equal (∠A = ∠C and ∠B = ∠D), and consecutive angles are supplementary (∠A + ∠B = 180°).

Sum of Angles Property

The sum of all internal angles in any quadrilateral is always 360°.

The correct answer is 59.3

Key Points

• The given observations are in ascending order: 28, 31, 47, 51, a, a + 3, 71, 74, 78, 90.
• The total number of observations (n) is 10, which is an even number.
• For an even number of observations, the median is the average of the (n/2)^th and (n/2 + 1)^th terms.

Here, n/2 = 5^th term and n/2 + 1 = 6^th term.

5^th term = a

6^th term = a + 3

• The median is given as 61.5. Therefore:

Median = (a + a + 3) ÷ 2 = 61.5

2a + 3 = 61.5 × 2

2a + 3 = 123

2a = 123 − 3

2a = 120

a = 60

• The 5^th and 6^th observations are 60 and 63 (60 + 3) respectively.
• The complete data set is: 28, 31, 47, 51, 60, 63, 71, 74, 78, 90.
• To find the mean of the data:

Mean = Sum of all observations ÷ Total number of observations

Sum = 28 + 31 + 47 + 51 + 60 + 63 + 71 + 74 + 78 + 90 = 593

Mean = 593 ÷ 10 = 59.3

Additional Information

• Mean: It is the arithmetic average calculated by dividing the sum of all values by the count of values.
• Median: It is the middle value of a data set when it is arranged in ascending or descending order. If n is odd, the median is the ((n + 1)/2)^th term.
• Mode: The value that appears most frequently in a data set.
• In a symmetrical distribution, the mean, median, and mode are all equal.

Shortcut Trick

Substitute k = 3x + 4 to simplify the terms in the equation.

Then 6x + 7 = 2(3x + 4) − 1 = 2k − 1 and 9x − 8 = 3(3x + 4) − 20 = 3k − 20.

Rewritten equation: 5k − 8(2k − 1) = 3k − 20

⇒ 5k − 16k + 8 = 3k − 20 ⇒ −11k + 8 = 3k − 20 ⇒ 14k = 28

⇒ k = 2

∴ The correct answer is 2.

Alternate Method

Given:

Equation: 5(3x + 4) − 8(6x + 7) = 9x − 8

Method Used:

Algebraic expansion and variable isolation.

Calculations:

⇒ 15x + 20 − (48x + 56) = 9x − 8

⇒ 15x + 20 − 48x − 56 = 9x − 8

⇒ −33x − 36 = 9x − 8

⇒ −33x − 9x = −8 + 36

⇒ −42x = 28

⇒ x = 28 ÷ (−42) = −2/3

⇒ Value of (3x + 4) = 3(−2/3) + 4

⇒ Value of (3x + 4) = −2 + 4 = 2

∴ The correct answer is 2.

Additional Information

Distributive Property

It states that a(b + c) = ab + ac, which is used to expand brackets in linear equations.

Variable Isolation

To solve for x, group all variable terms on one side and constant terms on the other, changing the sign during transposition.

Verification Rule

Always plug the solved value of x back into the original equation to ensure the Left Hand Side (LHS) equals the Right Hand Side (RHS).

CONCEPT:

Effect of Acids, Bases and Neutral Substances on Indicators

• Litmus is a natural indicator used to identify acidic and basic substances.
• Blue litmus turns red in acidic solutions.
• Red litmus turns blue in basic solutions.
• No colour change occurs in neutral solutions.
• Phenolphthalein is a synthetic indicator.
• It remains colourless in acidic and neutral solutions.
• It turns pink in basic solutions.

EXPLANATION:

• Test tube A contains glucose solution, which is a neutral substance.
• Blue litmus remains blue.
• Red litmus remains red.
• Phenolphthalein remains colourless.
• Test tube B contains turmeric solution. Turmeric itself is not acidic or basic and behaves as a neutral solution.
• Blue litmus remains blue.
• Red litmus remains red.
• Phenolphthalein remains colourless.
• Test tube C contains detergent solution, which is basic in nature.
• Blue litmus remains blue.
• Red litmus turns blue.
• Phenolphthalein turns pink.
• Test tube D contains pure water, which is neutral.
• Blue litmus remains blue.
• Red litmus remains red.
• Phenolphthalein remains colourless.
• Comparing these observations with the given statements shows that only the observations corresponding to the detergent solution are correct.

Therefore, detergent solution turns red litmus blue and phenolphthalein pink, while glucose solution, turmeric solution and pure water show no change in litmus colour and keep phenolphthalein colourless.

The correct answer is C < A < B < D.

Key Points

The reactivity of metals can be determined using displacement reactions, where a more reactive metal displaces a less reactive metal from its salt solution. In this context, the reactivity hierarchy of the salts provided is Zinc (Zn) > Iron (Fe) > Copper (Cu).
Metal D is the most reactive in this set because it shows a white coating when added to ZnSO4 and a brown deposit with FeSO4. This indicates that it can displace Zinc and Iron from their respective solutions, placing it at the top of the reactivity order among the four metals. Hence, the statement regarding D's high reactivity is correct.
Metal B is less reactive than Metal D but more reactive than Iron. It produces a grey deposit with FeSO4 and a brown deposit with CuSO4, but shows no change with ZnSO4. This implies Metal B is between Zinc and Iron in the reactivity series. Hence, the positioning of B is correct.
Metal A shows no change with FeSO4 but creates a green deposit (or reaction evidence) with CuSO4. This proves that Metal A is more reactive than Copper but less reactive than Iron. Hence, the relative position of A is correct.
Metal C is the least reactive as it shows no change even with CuSO4, which is the salt of the least reactive metal in the tested group. This indicates Metal C cannot displace Copper. Hence, the statement regarding C's low reactivity is correct.
By combining these observations, the ascending order of reactivity is C < A < B < D. Hence, the correct answer is C < A < B < D.

Additional Information

Reactivity Series of Metals:

The Reactivity Series is an arrangement of metals in decreasing order of their chemical reactivity. It is essential in predicting the outcome of single-displacement reactions and understanding the extraction processes of metals.
Highly reactive metals like Potassium (K) and Sodium (Na) are at the top, while noble metals like Gold (Au) and Platinum (Pt) are at the bottom. The series often includes Hydrogen as a reference point for acids.

Chemical Displacement:

A Displacement Reaction occurs when a metal higher in the reactivity series reacts with a salt solution of a metal lower in the series. For example, Zinc displaces Copper from Copper (II) Sulfate, resulting in a color change from blue to colorless.
These reactions are exothermic and involve the transfer of electrons (Redox), where the more reactive metal acts as a reducing agent.

Metal Extraction context:

Metals at the top of the series are usually extracted via electrolysis of their molten ores because they form very stable compounds.
Metals in the middle (like Iron and Zinc) are often extracted by reduction with carbon, whereas metals at the bottom can be found in their native state or extracted by simple heating.

Important Points

Visual Indicators of Reactions:

Copper (II) Sulfate (CuSO4) is typically a blue solution. When displaced, the formation of a brown deposit indicates the presence of elemental Copper.
Iron (II) Sulfate (FeSO4) is a pale green solution. The displacement of Iron typically results in a greyish deposit.

Specific Metal Reactivity:

Zinc is more reactive than Iron because it sits higher in the electrochemical series. Any metal that can displace Zinc from Zinc Sulfate (ZnSO4), like Magnesium or Aluminum, is significantly more reactive than Iron.
Copper is one of the least reactive common transition metals. Metals that cannot displace Copper, such as Silver or Gold, are considered least reactive.

CONCEPT:

Professional Development of Science Teachers

• Professional development refers to activities that help teachers improve their subject knowledge, teaching skills, scientific attitude, and awareness of recent developments in science education.
• Effective professional development involves:
• Interaction and collaboration with other teachers.
• Exposure to scientific institutions, industries, and research environments.
• Learning new skills and technologies related to science.
• Keeping oneself updated beyond the prescribed curriculum.

EXPLANATION:

• Cultivating hobbies related to science such as robotics and electronics helps teachers acquire practical skills and stay connected with modern scientific developments.
• Regular meetings with teachers from neighbouring schools encourage exchange of ideas, teaching strategies, and professional experiences, thereby contributing to professional growth.
• Visits to planetariums, industries, and science museums provide exposure to real-life applications of science and enrich teachers' understanding of scientific concepts.
• Merely reading the science content given in school textbooks does not significantly contribute to professional development because it is limited to prescribed content and does not help teachers expand their knowledge, skills, or pedagogical practices beyond the curriculum.

Therefore, regular reading of science content given in science textbooks alone does not contribute significantly to the professional development of science teachers.

The correct answer is (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii).

Key Points

• Adrenal Hormones (a - iv): The adrenal glands, located on top of the kidneys, secrete hormones like aldosterone and cortisol. Aldosterone is crucial for maintaining the balance of minerals, specifically sodium and potassium, in the body. A deficiency in these hormones can lead to a severe salt imbalance and conditions like Addison's disease. Hence, match (a)-(iv) is correct.
• Growth Hormone (b - i): This hormone is secreted by the anterior pituitary gland. It stimulates growth, cell reproduction, and cell regeneration. If there is a deficiency of growth hormone during childhood, it results in Dwarfism, a condition where the individual has a significantly short stature but usually maintains normal body proportions. Hence, match (b)-(i) is correct.
• Thyroxine (c - ii): Produced by the thyroid gland, thyroxine (T4) regulates the body's basal metabolic rate. The synthesis of thyroxine requires iodine. A deficiency of thyroxine, often due to a lack of dietary iodine, leads to Goitre, which is characterized by a visible swelling in the neck. Hence, match (c)-(ii) is correct.
• Insulin (d - iii): Insulin is a peptide hormone produced by the beta cells of the pancreas. It facilitates the uptake of glucose by the body's cells to lower blood sugar levels. A deficiency or inability to use insulin leads to Diabetes Mellitus, characterized by high blood glucose levels. Hence, match (d)-(iii) is correct.

Additional Information

• Endocrine System: This is a network of glands that produce and release hormones directly into the bloodstream to control many important body functions, including the body's ability to change calories into energy.
• Pituitary Gland: Often called the "Master Gland", it is located at the base of the brain. It produces several hormones that control other endocrine glands, such as the Thyroid-Stimulating Hormone (TSH) and Adrenocorticotropic Hormone (ACTH).
• The Pancreas: It is a unique organ as it functions as both an exocrine gland (secreting digestive enzymes) and an endocrine gland (secreting insulin and glucagon). It is located behind the stomach.
• Thyroid Health: Besides goitre, thyroxine imbalance can lead to Hyperthyroidism (excessive production) or Hypothyroidism (low production), affecting weight, heart rate, and body temperature.

Important Points

• Chemical Messengers: Hormones act as chemical messengers that travel through the blood to target organs. Even a tiny numerical change in their concentration can lead to significant physiological disorders.
• Hormonal Regulation: Most hormone levels are regulated by a feedback mechanism. For example, high blood glucose levels trigger the release of insulin, which then lowers the glucose, subsequently reducing further insulin secretion.
• Iodized Salt: The Government of India promotes the use of iodized salt to prevent iodine deficiency disorders (IDD) like goitre and cretinism in children.

CONCEPT:

Crystallization

• Crystallization is a method used to obtain a pure solid from its impure sample or from a solution.
• The technique is based on the difference in solubility of substances in a suitable solvent at different temperatures.
• It is commonly used for:
• Purification of salts and other solid compounds.
• Separation of a soluble solid from its impurities.
• Crystallization is not suitable for separating a liquid from a dissolved substance because the liquid does not form crystals.

EXPLANATION:

• Salt can be purified from an impure sample by crystallization because pure salt forms crystals while many impurities remain in the solution.
• Alum can also be separated from an impure sample by crystallization since it readily forms well-defined crystals on cooling its saturated solution.
• A mixture of zinc chloride and lead(II) chloride can be separated using differences in their solubilities, making crystallization a possible method.
• A solution of methyl alcohol and potassium permanganate cannot be separated by crystallization because methyl alcohol is a liquid and does not crystallize under ordinary conditions. The method is intended for obtaining crystalline solids, not for separating a liquid solvent from a dissolved substance.

Therefore, a solution of methyl alcohol and potassium permanganate cannot be separated by crystallization.

The correct answer is Hypothesizing.

Key Points

• Hypothesizing is a crucial integrated science process skill where a learner suggests a tentative explanation or an educated guess about an event or a relationship that can be tested through experimentation.
• In this activity, the teacher's question, "Why do you think the bowl did not sink?" prompts the students to move beyond simple observation and propose a logical reason or theory based on their prior knowledge. Hence, the primary skill being developed is hypothesizing.
• The follow-up question, "What will make it sink?" encourages students to identify variables and predict outcomes, which are essential components in forming a testable hypothesis.
• This process is fundamental to the Scientific Method, as it allows students to bridge the gap between gathering data and conducting experiments to verify their thoughts.
• By asking "why," the teacher is fostering higher-order thinking skills (HOTS), requiring students to conceptualize the relationship between mass, volume, and displacement before they even learn the formal laws of physics.

Additional Information

• Observing:
• This is a basic science process skill where students use their five senses to notice characteristics of objects or phenomena.
• In the given scenario, students used observation when they saw that the steel spoon sank while the steel bowl remained afloat. Statement 2 is not the skill being prompted by the teacher's specific questions.
• Experimenting:
• This involves the active testing of a hypothesis under controlled conditions to see if the results match the prediction.
• While the teacher's questions set the stage for experimentation, the act of "thinking why" occurs before the actual test is performed. Statement 3 is the step that follows the development of the hypothesis.
• Inferring:
• Inference is an explanation or interpretation of an observation based on prior experience.
• While similar to hypothesizing, inferring usually explains a past or present event ("The spoon sank because it is heavy"), whereas hypothesizing creates a general statement that can be tested against new evidence. Statement 4 is closely related but less accurate in the context of predicting what will make it sink.

Important Points

• Density and Buoyancy:
• The activity relates to Archimedes' Principle, which states that any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
• The steel bowl floats because its hollow shape increases its volume, allowing it to displace a weight of water greater than or equal to its own weight.
• The steel spoon, having a much smaller volume relative to its mass, cannot displace enough water to create an upward buoyant force that counteracts its weight.
• Teacher's Role in Inquiry:
• By using open-ended questions, the teacher facilitates Inquiry-based learning, which is a student-centered approach recommended by the National Curriculum Framework (NCF) 2005.
• This method encourages critical thinking and helps students build a conceptual understanding of density (mass per unit volume) rather than just memorizing a formula.

The correct answer is Scientific discovery can be serendipitous..

Key Points

• Serendipity in science refers to the phenomenon of making unexpected or accidental discoveries while looking for something entirely different.
• The provided excerpt describes how French scientist Henri Becquerel discovered radioactivity in 1896 because of a stormy day that prevented him from carrying out his planned experiment with sunlight.
• Becquerel’s observation that the photographic plate was fogged despite the lack of external light led to the realization that the uranium salts emitted radiation spontaneously. Hence, the statement that scientific discovery can be serendipitous is correct.
• This discovery illustrates that while scientific methods are structured, chance events and keen observation of anomalies often lead to major breakthroughs.
• The excerpt contradicts the idea that science is purely procedural; it shows that the creative interpretation of an accidental result is vital to progress. Hence, statement 4 is incorrect.

Additional Information

• Henri Becquerel: He was a physicist who shared the 1903 Nobel Prize in Physics with Marie Curie and Pierre Curie for the discovery of spontaneous radioactivity.
• Radioactivity: This is the process by which an unstable atomic nucleus loses energy by radiation. Becquerel's work proved that certain elements emit energy without any external stimulation.
• The Becquerel (Bq): In honor of his contribution, the SI unit of radioactivity is named the Becquerel, which represents one nuclear decay per second.
• Phosphorescence vs. Radioactivity: Becquerel was originally studying phosphorescence (materials that glow after light exposure), but his "accident" helped differentiate it from the internal, constant atomic process of radioactivity.
• Other Serendipitous Discoveries: History is full of such instances, such as Alexander Fleming's discovery of Penicillin in 1928 and Wilhelm Röntgen's discovery of X-rays in 1895.

Important Points

• Scientific Knowledge is Tentative: This means that scientific theories are subject to change or revision when new evidence arises. While true for science in general, the excerpt specifically highlights the mode of discovery (luck/chance) rather than the revision of a theory. Hence, statement 3 is not the best fit for this specific context.
• Science as a Pursuit: Although the excerpt shows Becquerel working in his lab, science is rarely a solitary pursuit in the long run. It involves peer review, collaboration, and building on the work of others. Hence, statement 2 is incorrect.
• Nature of Science (NOS): This concept emphasizes that science involves human error, creativity, and curiosity. Becquerel’s decision to check the plate even though he "knew" there was no sunlight is a hallmark of the scientific temper.
• Date and Location: The discovery took place in Paris in February 1896, marking a turning point in nuclear physics.

The correct answer is Body temperature of Priya is more than that of Sana.

Key Points

• To compare the body temperatures of Sana and Priya, both must be converted to the same unit, either Celsius (°C) or Fahrenheit (°F).
• Sana's temperature is recorded as 98.6°F. Using the conversion formula C = (F - 32) × 5/9, we calculate: (98.6 - 32) × 5/9 = 66.6 × 5/9 = 37°C. This is considered the average normal human body temperature.
• Priya's temperature is recorded as 38.4°C. Converting this to Fahrenheit using F = (C × 9/5) + 32, we get: (38.4 × 1.8) + 32 = 69.12 + 32 = 101.12°F.
• Comparing the two, 38.4°C (Priya) is higher than 37°C (Sana). Alternatively, 101.12°F (Priya) is higher than 98.6°F (Sana).
• Statement 1: "Body temperature of Sana is more than that of Priya." Since 37°C is less than 38.4°C, this statement is incorrect.
• Statement 2: "Body temperature of Priya is more than that of Sana." Since 38.4°C is greater than 37°C, this statement is correct.
• Statement 3: "Both Sana and Priya have the same body temperature." Since the values are 37°C and 38.4°C respectively, this statement is incorrect.
• Statement 4: "Both Sana and Priya are suffering from fever." Sana's temperature is perfectly normal (37°C), while Priya's temperature is elevated (38.4°C), indicating only Priya has a fever. Hence, this statement is incorrect.

Additional Information

• Temperature Scales:
• Celsius Scale: Designed by Anders Celsius in 1742. It is based on 0°C for the freezing point of water and 100°C for the boiling point at standard atmospheric pressure.
• Fahrenheit Scale: Proposed by Daniel Gabriel Fahrenheit in 1724. In this scale, the freezing point of water is 32°F and the boiling point is 212°F.
• Normal Body Temperature: While 98.6°F (37°C) is the standard average, a normal range is generally between 97°F (36.1°C) and 99°F (37.2°C). Temperatures can vary based on age, time of day, and physical activity level.
• Fever (Pyrexia): A medical condition where the body temperature rises above the normal range. In adults, a temperature of 100.4°F (38°C) or higher is typically considered a fever. Priya’s temperature of 38.4°C (101.12°F) clearly falls into this category.
• Measurement Tools: Temperatures are measured using a thermometer. A clinical thermometer is specifically designed to measure human body temperature and typically has a range from 35°C to 42°C (or 94°F to 108°F).

Important Points

• Conversion Formulas:
• To convert Celsius to Fahrenheit: F = (C × 9/5) + 32.
• To convert Fahrenheit to Celsius: C = (F - 32) × 5/9.
• Mercury vs. Digital Thermometers: Traditional mercury thermometers use the expansion of liquid mercury, but they are being replaced by digital thermometers due to the toxic nature of mercury and the higher accuracy and safety of electronic sensors.
• Kink in Clinical Thermometers: A clinical thermometer has a kink (constriction) near the bulb that prevents the mercury level from falling on its own, allowing the user to read the temperature accurately after removing it from the body.

CONCEPT:

Scientific Temper

• Scientific temper refers to an attitude of logical thinking, curiosity, objectivity, and a willingness to examine evidence before accepting any claim.
• It involves:
• Questioning and investigating phenomena systematically.
• Being creative in identifying and solving problems.
• Maintaining a critical and sceptical outlook towards unsupported beliefs and claims.
• Relying on evidence, observation, and reasoning rather than tradition or authority alone.

EXPLANATION:

• Perseverance in carrying out scientific investigations is an important characteristic of scientific temper because scientific inquiries often require patience, repeated observations, and continuous effort.
• Creativity in identifying important problems and breaking them into smaller, manageable parts is a key aspect of scientific thinking and problem-solving.
• Scepticism towards non-scientific beliefs is also associated with scientific temper, as scientific conclusions must be based on evidence and rational analysis rather than unverified claims.
• Commitment towards cultural preservation is a valuable social and cultural attribute, but it is not a defining component of scientific temper. Scientific temper is concerned primarily with inquiry, evidence, and rational thinking.

Therefore, scientific temper is associated with perseverance in scientific investigations, creativity in problem-solving, and scepticism towards non-scientific beliefs.

CONCEPT:

Momentum, Force and Work

• Momentum of a particle is given by:

p = mv

• According to Newton's Second Law, the rate of change of momentum is equal to the net force acting on the particle.

F = dp/dt

• Therefore, a change in momentum requires a non-zero net force.
• Work done is related to the displacement of the particle in the direction of force.

W = F.s

• A force can change the direction of momentum without doing any work if it acts perpendicular to the displacement.

EXPLANATION:

• The first statement claims that non-zero work must be done on a moving particle to change its momentum.
• This statement is not always correct. Consider a particle moving in a circular path with constant speed. The centripetal force continuously changes the direction of momentum, yet it remains perpendicular to the displacement.
• Since the force is perpendicular to the displacement, the work done is zero, although the momentum changes continuously.
• Hence, a change in momentum does not necessarily require non-zero work.
• The second statement states that a non-zero net force should act on a particle to change its momentum.
• According to Newton's Second Law, momentum can change only when a net external force acts on the particle.
• Therefore, the second statement is correct.

Therefore, a particle's momentum can change without any work being done on it, but a non-zero net force is necessary for the change in momentum.