Mathematics and Science PYQs

Shortcut Trick

Product = N₁ × N₂ ⇒ −40/3 = (−5/2) × N₂

Other number (N₂) = (−40/3) × (−2/5) = 80/15 = 16/3

Value of x (reciprocal of N₂) = 3/16

Required value = 8 × (3/16) + 2 1/4 = 1.5 + 2.25 = 3.75

∴ The correct answer is 3 3/4.

Alternate Method

Given:

Product of two rational numbers = −40/3

One rational number = −5/2

Reciprocal of the other number = x

Formula Used:

Product = Number₁ × Number₂

Reciprocal of a/b = b/a

Calculations:

⇒ Let the other number be y.

⇒ (−5/2) × y = −40/3

⇒ y = (−40/3) ÷ (−5/2)

⇒ y = (−40/3) × (−2/5)

⇒ y = 80/15 = 16/3

⇒ Since x is the reciprocal of y, x = 1 ÷ (16/3) = 3/16

⇒ Now, substitute x in (8x + 2 1/4):

⇒ 2 1/4 = (2 × 4 + 1) / 4 = 9/4

⇒ Value = 8 × (3/16) + 9/4

⇒ Value = 3/2 + 9/4

⇒ Value = 1.5 + 2.25 = 3.75

⇒ 3.75 = 15/4 = 3 3/4

∴ The correct answer is 3 3/4.

Additional Information

Reciprocal Property

For any non-zero rational number a/b, its reciprocal is b/a such that (a/b) × (b/a) = 1.

Mixed Fraction Conversion

To convert a mixed fraction A b/c to an improper fraction, use the formula (A × c + b) ÷ c.

Rational Multiplication and Division

Division of fractions is equivalent to multiplying the first fraction by the reciprocal of the second: (a/b) ÷ (c/d) = (a/b) × (d/c).

Shortcut Trick

Observe that 4x² + 9y² + 12xy = (2x + 3y)².

Substitute u = 2x + 3y, simplifying the expression to u² − 5u − 14.

Factorize the quadratic u² − 5u − 14 as (u − 7)(u + 2).

Substitute back u = 2x + 3y to get the factors (2x + 3y − 7) and (2x + 3y + 2).

∴ The correct answer is 2x + 3y − 7.

Alternate Method

Given: 4x² + 9y² + 12xy − 5(2x + 3y) − 14

Formula Used: (a + b)² = a² + b² + 2ab

⇒ 4x² + 9y² + 12xy − 5(2x + 3y) − 14

⇒ [(2x)² + (3y)² + 2(2x)(3y)] − 5(2x + 3y) − 14

⇒ (2x + 3y)² − 5(2x + 3y) − 14

⇒ Let 2x + 3y = u

⇒ u² − 5u − 14

⇒ u² − 7u + 2u − 14

⇒ u(u − 7) + 2(u − 7)

⇒ (u − 7)(u + 2)

⇒ Substituting u: (2x + 3y − 7)(2x + 3y + 2)

∴ The correct answer is 2x + 3y − 7.

Additional Information

Perfect Square Trinomial

A trinomial that can be written as the square of a binomial, specifically a² + 2ab + b² = (a + b)².

Splitting the Middle Term

For a quadratic x² + Bx + C, find two numbers p and q such that p + q = B and p × q = C.

Factor Theorem

If f(k) = 0 for a polynomial f(x), then (x − k) is a factor of that polynomial.

The correct answer is Norm referenced tests

Key Points

• Norm-referenced tests are designed to compare and rank test takers in relation to one another. These tests report whether a student performed better or worse than a hypothetical average student, which is determined by comparing scores against the performance results of a statistically selected norm group.
• The primary objective of these tests is to determine the relative performance or standing of individuals within a specific group, rather than measuring mastery of specific content.
• Criterion-referenced tests, by contrast, measure a student's performance against a fixed set of predetermined criteria or learning standards. They determine what a student can or cannot do, regardless of how other students performed.
• Diagnostic tests are specifically used to identify a student's specific strengths and weaknesses in a subject area, typically conducted before or during instruction to guide remediation.
• Achievement tests are used to evaluate the knowledge or skills acquired by a student in a specific subject after a period of training or instruction. While achievement tests can be norm-referenced, the term itself refers to the content being tested rather than the method of comparison.

Additional Information

• In Norm-referenced tests, results are often expressed as a percentile rank. For instance, a student in the 85^th percentile performed better than 85 percent of the students in the norm group.
• These tests are often constructed to produce a normal distribution of scores, commonly referred to as the Bell Curve, where the majority of students score in the middle range.
• Common examples of norm-referenced tests include standardized college entrance exams like the SAT, GRE, or IQ tests.
• Stanine (Standard Nine) scores are another common metric used in these tests to report results on a nine-point scale, providing a simplified way to compare relative standing.

Shortcut Trick

(a) False: For 0 < x < 1, x > x² (e.g., 0.5 > 0.25).

(b) True: For negative numbers, x > x³ (e.g., −0.5 > −0.125 and −2 > −8).

(c) False: For −1 < x < 0, x < x³ (e.g., −0.5 < −0.125) and for x > 1, x < x³.

(d) True: Square root of any prime number is always irrational.

∴ The correct answer is (b) and (d).

Alternate Method

Given: Four mathematical statements regarding properties of numbers.

Analysis of Statements:

Statement (a): A number never exceeds its square (x ≤ x²).

⇒ Let x = 0.5. Then x² = 0.25. Here 0.5 > 0.25. Statement (a) is False.

Statement (b): All negative numbers exceed their cubes (x > x³).

⇒ Case 1: x < −1 (e.g., −2). (−2)³ = −8. Since −2 > −8, it holds.

⇒ Case 2: −1 < x < 0 (e.g., −0.5). (−0.5)³ = −0.125. Since −0.5 < −0.125, it fails.

⇒ Note: Re-evaluating (b) based on standard options, if the question implies x < x³ for fractions, (b) is generally considered false for the interval (−1, 0). However, if only integers are considered, it differs. Given the options, let's re-verify (d).

Statement (d): The square root of a prime number is not a rational number.

⇒ By definition, for any prime p, √p cannot be expressed as a/b. It is always irrational. Statement (d) is True.

Conclusion: Since (d) is definitely true and (a), (c) are definitely false, and (b) is false for negative fractions, the only strictly true statement among the choices provided in the logic of the options is (d).

∴ The correct answer is (d) only.

Additional Information

Rational vs Irrational Numbers

A rational number can be written as p/q where q ≠ 0. Square roots of non-perfect squares (like all prime numbers) are irrational.

Powers of Fractions

For 0 < x < 1, increasing the power decreases the value (x > x² > x³). For x > 1, increasing the power increases the value.

Negative Number Comparisons

On a number line, the number further to the right is greater. −2 is greater than −8, but −0.5 is less than −0.125.

Shortcut Trick

Compare the side lengths of ΔABC and ΔDCB:

AB = 5 cm and CD = 5 cm ⇒ AB = CD

AC = 8 cm and BD = 8 cm ⇒ AC = BD

BC = 4 cm is a common side for both triangles.

Since all three corresponding sides are equal, the triangles are congruent by the SSS (Side-Side-Side) criterion.

∴ The correct answer is ΔABC ≅ ΔDCB, by SSS.

Alternate Method

Given:

AB = 5 cm, AC = 8 cm, BC = 4 cm

BD = 8 cm, CD = 5 cm

Method: SSS Congruence Theorem

Comparing ΔABC and ΔDCB:

⇒ Side AB = Side DC = 5 cm

⇒ Side AC = Side DB = 8 cm

⇒ Side BC = Side CB = 4 cm (Common side)

⇒ By SSS (Side-Side-Side) congruence rule:

⇒ ΔABC ≅ ΔDCB

∴ The correct answer is ΔABC ≅ ΔDCB, by SSS.

Additional Information

SSS Congruence Rule

Two triangles are congruent if all three sides of one triangle are equal to the corresponding three sides of the other triangle.

SAS Congruence Rule

Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding parts of the other triangle.

Vertex Correspondence

In congruence, the order of vertices is critical. If ΔABC ≅ ΔDCB, it implies A corresponds to D, B corresponds to C, and C corresponds to B.

CPCT Rule

Corresponding Parts of Congruent Triangles are equal. Once congruence is proved, all remaining angles and sides are automatically equal.

The correct answer is (a) and (c)

Key Points

• The Project Method is a modern teaching strategy in mathematics developed by William Heard Kilpatrick, which is deeply rooted in the philosophy of Pragmatism.
• Statement (a) is correct because the method is fundamentally based on the principle of Learning by Doing. It emphasizes that students should engage in purposeful activity and learn through practical experience in a natural environment.
• Statement (c) is correct because this method encourages students to investigate, explore, and solve problems independently, effectively turning the learner into a Discoverer. It fosters heuristic thinking where the student finds out facts for themselves.
• Statement (b) is incorrect because the Project Method is Student-centered. The role of the learner is Maximum, as they are responsible for planning and executing the project. The teacher only acts as a Facilitator or guide.
• Statement (d) is incorrect because the method is primarily Inductive. It starts with specific concrete activities and leads to the understanding of general mathematical concepts, rather than always starting from a general rule (deduction).

Additional Information

• The Project Method was inspired by the Instrumentalism of John Dewey, who believed that education is life itself, not just preparation for life.
• The six essential steps of the Project Method include: Creating a Situation, Selection and Purposing, Planning, Execution, Evaluation, and Recording.
• Kilpatrick classified projects into four types: Producer projects (construction), Consumer projects (appreciation), Problem projects (intellectual difficulties), and Drill projects (attainment of skills).
• This method helps in developing Social Skills, Cooperation, and Scientific Attitude among mathematics students by relating subject matter to real-life situations.

Key Points

• An inter-disciplinary approach in education refers to a method of teaching or problem-solving that integrates knowledge and skills from two or more academic disciplines to solve a problem or explain a concept.

Write your name and draw the lines of symmetry in each alphabet wherever possible integrates two distinct fields:

• Language Arts / Literacy: Identifying, writing, and analyzing alphabetic letters and word structures.
• Mathematics (Geometry): Applying spatial reasoning and identifying reflective symmetry within those textual symbols.

Thus,the correct answer is 3.

 Hint 

• Draw the mirror image of a given figure — involves geometry and reflection.
• How many lines of symmetry are there in a given figure? — involves identifying symmetry in geometry.
• Draw the lines of symmetry in a given geometrical figure — directly involves geometrical symmetry.

Shortcut Trick

A number is divisible by 72 if it is divisible by both 8 and 9 (since 8 × 9 = 72).

Shortcut for 9: Sum of digits must be divisible by 9.

Shortcut for 8: Last three digits must be divisible by 8.

Check 95142: Sum = 9 + 5 + 1 + 4 + 2 = 21 (not divisible by 9).

Check 95142: Last three digits 142 ÷ 8 = 17.75 (not divisible by 8).

∴ The correct answer is 95142.

Alternate Method

Given: Numbers are 95112, 95142, 88848, 88704.

Formula Used: Divisibility by 72 requires divisibility by co-prime factors 8 and 9.

⇒ Testing 95112: Sum of digits = 18 (Divisible by 9); Last three 112 ÷ 8 = 14 (Divisible by 8). ∴ Divisible by 72.

⇒ Testing 95142: Sum of digits = 21 (Not divisible by 9); Last three 142 ÷ 8 = 17.75 (Not divisible by 8). ∴ Not divisible by 72.

⇒ Testing 88848: Sum of digits = 36 (Divisible by 9); Last three 848 ÷ 8 = 106 (Divisible by 8). ∴ Divisible by 72.

⇒ Testing 88704: Sum of digits = 27 (Divisible by 9); Last three 704 ÷ 8 = 88 (Divisible by 8). ∴ Divisible by 72.

⇒ Only 95142 fails both divisibility tests for 8 and 9.

∴ The correct answer is 95142.

Additional Information

Divisibility by 8

A number is divisible by 8 if its last three digits are 000 or form a number divisible by 8.

Divisibility by 9

A number is divisible by 9 if the sum of all its digits is a multiple of 9.

Composite Divisibility Rule

To check divisibility by a composite number, check divisibility by its co-prime factors (e.g., 12 = 3 × 4, 15 = 3 × 5).

The correct answer is Geoboard

Key Points

• A Geoboard is a mathematical manipulative consisting of a board with pegs arranged in a grid (lattice). It is the most effective tool among the options for introducing geometric concepts because students can stretch rubber bands around the pegs to create various polygons.
• Shapes: It allows students to physically construct and visualize triangles, quadrilaterals, and other polygons, helping them identify vertices and edges.
• Perimeter: Students can calculate perimeter by counting the unit segments between the pegs along the boundary of the shape they have created.
• Area: It is an excellent tool for teaching area through Pick's Theorem or by simply counting the unit squares enclosed within the rubber band.
• Characteristics of Polygons: It helps in understanding symmetry, congruency, and angles (right, acute, obtuse) in a hands-on manner.
• Why other options are incorrect:
• Hanoi Tower: This is a mathematical puzzle used to teach logical reasoning, problem-solving, and recursion, rather than geometry.
• Cuisenaire rods: These are colored rods of different lengths used primarily to teach arithmetic operations (addition, subtraction, multiplication, division), fractions, and ratio.
• Tangrams: While these are dissection puzzles consisting of seven flat shapes used to teach spatial visualization and shape composition, they are not as effective as a Geoboard for the systematic measurement of area and perimeter on a grid.

Additional Information

• The Geoboard was invented by the Egyptian mathematician and pedagogue Caleb Gattegno in the 1950s.
• There are three main types of Geoboards: Square Lattice (for general polygons), Isometric Lattice (specifically for equilateral triangles and hexagons), and Circular Geoboards (for studying circles and angles).
• Manipulative materials like the Geoboard are essential in the Concrete-Pictorial-Abstract (CPA) approach to mathematics, helping transition students from physical experience to conceptual understanding.

Shortcut Trick

Identify given values: a + b = 17/6 and ab = 2.

Use the identity: (a − b)² = (a + b)² − 4ab.

(a − b)² = (17/6)² − 4 × 2 = 289/36 − 8.

(a − b)² = (289 − 288) / 36 = 1/36.

Taking the square root (since a > b): a − b = 1/6.

∴ The correct answer is 1/6.

Alternate Method

Given: a + b = 17/6 and ab = 2

Formula Used: (a − b) = √[(a + b)² − 4ab]

⇒ (a − b)² = (a + b)² − 4ab

⇒ (a − b)² = (17/6)² − 4 × 2

⇒ (a − b)² = 289/36 − 8

⇒ (a − b)² = (289 − 288) / 36

⇒ (a − b)² = 1/36

⇒ a − b = √(1/36) = 1/6

∴ The correct answer is 1/6.

Additional Information

Algebraic Identity for Sum and Difference

The relationship between the square of a sum and the square of a difference is given by: (a + b)² − (a − b)² = 4ab.

Sum and Product of Roots

For a quadratic equation x² − Sx + P = 0, the sum of roots is S = a + b and the product is P = ab.

Difference of Squares

The formula for the difference of two squares is: a² − b² = (a + b)(a − b).

The correct answer is (a) and (c)

Key Points

• Formative Assessment, also known as Assessment for Learning, is an ongoing process used by teachers and students during instruction to provide feedback and adjust ongoing teaching and learning to improve students’ achievement of intended instructional outcomes.
• Making assessment a part of teaching (a): This is a core purpose of formative assessment. It is not a separate, final event but is integrated into the daily teaching-learning process. This helps in identifying learning gaps as they occur.
• Providing feedback to the children (c): This is perhaps the most critical component of formative assessment. Constructive feedback helps students understand their current performance levels and guides them on how to reach their learning goals.
• Ranking and placement of children (b): This is not the purpose of formative assessment. Ranking and placement are characteristics of Summative Assessment (Assessment of Learning), which occurs at the end of a course or unit to evaluate student learning against a standard or benchmark.

Additional Information

• Summative Assessment: Focuses on the final outcome, grading, ranking, and certification. It summarizes the development of learners at a particular time.
• Continuous and Comprehensive Evaluation (CCE): Formative assessment is a key pillar of CCE, ensuring that student development is monitored regularly rather than just through annual examinations.
• Tools of Formative Assessment: Common tools include anecdotal records, portfolios, class observations, quizzes, and self-assessment.
• Diagnostic Nature: Formative assessment is diagnostic and remedial, helping teachers provide tailored support to students based on their individual needs.

Given:

Formula Used:

Total value = Sum of computers sold in all months

Central angle of a sector = (Value of specific category ÷ Total value) × 360°

Difference between central angles = [(Value₁ − Value₂) ÷ Total value] × 360°

Calculation:

According to the question,

First, we calculate the total number of computers sold:

⇒ Total = 500 + 700 + 840 + 750 + 890 + 820

⇒ Total = 4500

Now, find the values for the months mentioned:

⇒ Number of computers sold in March = 840

⇒ Number of computers sold in January = 500

The difference between the number of computers sold in March and January:

⇒ Difference in values = 840 − 500 = 340

Now, calculate the difference between the central angles:

⇒ Difference in central angles = (340 ÷ 4500) × 360°

⇒ Difference in central angles = (34 ÷ 450) × 360°

⇒ Difference in central angles = 34 × (360 ÷ 450)°

⇒ Difference in central angles = 34 × (4 ÷ 5)°

⇒ Difference in central angles = 34 × 0.8°

⇒ Difference in central angles = 27.2°

∴ The correct answer is 27.2°.

Shortcut Trick

Total Area to be plastered = Area of floor + Area of 4 walls

Area = (20 × 15) + 2 × 4 × (20 + 15) = 300 + 280 = 580 m²

Total Cost = Total Area × Rate = 580 × 42

Cost = ₹ 24,360

∴ The correct answer is ₹ 24,360.

Alternate Method

Given: Length (L) = 20 m, Breadth (B) = 15 m, Depth (H) = 4 m

Plastering Rate = ₹ 42 per m²

Area of the floor = Length × Breadth

⇒ Floor Area = 20 × 15 = 300 m²

Area of the four walls = Lateral Surface Area = 2 × Height × (Length + Breadth)

⇒ Walls Area = 2 × 4 × (20 + 15)

⇒ Walls Area = 8 × 35 = 280 m²

Total Area to be plastered = Floor Area + Walls Area

⇒ Total Area = 300 + 280 = 580 m²

Cost of plastering = Total Area × Rate

⇒ Cost = 580 × 42

⇒ Cost = ₹ 24,360

∴ The correct answer is ₹ 24,360.

Additional Information

Lateral Surface Area of Cuboid

The area of the four vertical walls of a cuboid is given by the formula 2H(L + B).

Total Surface Area of an Open Box

For a box open at the top (like a swimming pool), the total surface area is LB + 2BH + 2HL.

Volume of a Cuboid

The amount of space occupied by the cuboid is calculated as Length × Breadth × Height (L × B × H).

Algebra is fundamentally the study of generalized relationships and structures.

Key Points

• Role of Generalization: When students analyze a pattern (e.g., 2, 4, 6, 8, ......), they observe specific numerical or visual instances and determine the underlying rule.
• Transition to Algebra: Expressing this general rule algebraically (e.g., 2n) transitions learners from concrete observations to abstract variables and expressions.
• Because identifying and formulating rules from patterns builds the exact logical reasoning required to express general algebraic relationships, pattern recognition is considered a core foundation for teaching algebra.

Thus, the correct answer is 3.

Hint

• Ease of learning is subjective and not the primary cognitive/pedagogical reason for teaching algebra.
• Algebra itself has extensive real-life applications, so stating patterns apply to real life "unlike algebra" is incorrect.
• While exploring patterns can foster creativity, it is not the main reason pattern study is tied to algebraic thinking.

Shortcut Trick

Rajiv: R = 20%/2 = 10%, n = 3 half-years ⇒ Effective CI% = 33.1%

Interest for Rajiv = 33.1% of 7,000 = 0.331 × 7,000 = ₹2,317

Amit: R = 10%, n = 2 years ⇒ Effective CI% = 10 + 10 + (10×10)/100 = 21%

Interest for Amit = 21% of 8,000 = 0.21 × 8,000 = ₹1,680

Required Difference = ₹2,317 − ₹1,680 = ₹637

∴ The correct answer is ₹637.

Alternate Method

Given:

Rajiv: P₁ = ₹7,000, R₁ = 20% p.a., T₁ = 1.5 years (Half-yearly compounding)

Amit: P₂ = ₹8,000, R₂ = 10% p.a., T₂ = 2 years (Annual compounding)

Formula Used:

Amount (A) = P(1 + R/100)ⁿ | CI = A − P

Calculations:

For Rajiv:

⇒ New Rate (r) = 20% ÷ 2 = 10% per half-year

⇒ Total periods (n) = 1.5 × 2 = 3 half-years

⇒ Amount = 7000(1 + 10/100)³ = 7000 × (1.1)³

⇒ Amount = 7000 × 1.331 = ₹9,317

⇒ Interest (CI_R) = 9317 − 7000 = ₹2,317

For Amit:

⇒ Rate (R) = 10% p.a., Time (n) = 2 years

⇒ Amount = 8000(1 + 10/100)² = 8000 × (1.1)²

⇒ Amount = 8000 × 1.21 = ₹9,680

⇒ Interest (CI_A) = 9680 − 8000 = ₹1,680

Difference:

⇒ Difference = CI_R − CI_A = 2317 − 1680 = ₹637

∴ The correct answer is ₹637.

Additional Information

Compounding Frequency Rule

When compounded half-yearly, the rate is divided by 2 (R/2) and the time is multiplied by 2 (2n). Similarly, for quarterly, divide rate by 4 and multiply time by 4.

Effective Rate for 3 years

For a rate of r%, the effective compound interest rate for 3 periods is 3r + 3r²/100 + r³/10000. For 10%, this equals 30 + 3 + 0.1 = 33.1%.

Effective Rate for 2 years

The effective interest rate for 2 years can be calculated using the successive percentage formula: (x + y + xy/100)%.

The correct answer is 2x + 3y − 2z = 8

Key Points

• A triangular pyramid, also known as a tetrahedron, is a solid figure with a triangular base and three other triangular faces that meet at a single vertex (apex).
• Let us determine the values for the given variables:
• Number of faces (x): There is 1 base and 3 lateral faces, so x = 4.
• Number of vertices (y): There are 3 vertices at the base and 1 vertex at the top, so y = 4.
• Number of edges (z): There are 3 edges at the base and 3 edges connecting the base vertices to the top vertex, so z = 6.
• Now, we test the given options using x = 4, y = 4, and z = 6:
• Option 1 (2x + 3y − 2z = 8): 2(4) + 3(4) − 2(6) = 8 + 12 − 12 = 8. This matches the equation.
• Option 2 (3x + 2y − 3z = 4): 3(4) + 2(4) − 3(6) = 12 + 8 − 18 = 2 (not 4).
• Option 3 (2x + 2z − 3y = 6): 2(4) + 2(6) − 3(4) = 8 + 12 − 12 = 8 (not 6).
• Option 4 (3x + 3y − 2z = 6): 3(4) + 3(4) − 2(6) = 12 + 12 − 12 = 12 (not 6).
• Therefore, the core logic confirms that the relationship 2x + 3y − 2z = 8 is the only correct one.

Additional Information

• All convex polyhedra follow Euler's formula, which states that F + V − E = 2, where F is faces, V is vertices, and E is edges.
• Applying Euler's formula to a triangular pyramid: 4 + 4 − 6 = 2.
• A Regular Tetrahedron is a special triangular pyramid where all four faces are equilateral triangles.
• A pyramid is named after the shape of its base; hence, a pyramid with a square base is called a square pyramid.

The correct answer is Contradiction

Key Points

• The most appropriate method for proving the irrationality of √2 is Proof by Contradiction (also known as reductio ad absurdum).
• In this method, we begin by assuming the negation of the statement we want to prove. To prove √2 is irrational, we first assume that √2 is rational.
• By definition, a rational number can be expressed as a fraction p/q, where p and q are coprime integers (integers with no common factors other than 1) and q ≠ 0.
• If √2 = p/q, then squaring both sides results in 2 = p²/q², which simplifies to p² = 2q².
• This equation shows that p² is an even number, which implies that p itself must be even (since the square of an odd number is always odd).
• We can then let p = 2k for some integer k. Substituting this into the equation gives (2k)² = 2q², which simplifies to 4k² = 2q², or q² = 2k².
• This shows that q² is also even, meaning q must be even.
• Since both p and q are even, they share a common factor of 2. This directly contradicts the initial assumption that p and q are coprime.
• Because the assumption leads to a logical impossibility, the assumption must be false. Therefore, √2 cannot be rational and must be irrational.

Additional Information

• Direct Proof: This method involves starting from established axioms, definitions, and previously proven theorems to reach a conclusion through a series of logical steps. It is commonly used for proving that the sum of two even numbers is even.
• Contrapositive Proof: This is used to prove a conditional statement of the form 'If P, then Q' by proving the logically equivalent statement 'If not Q, then not P'.
• Mathematical Induction: This method is specifically used to prove statements or formulas that are asserted for all natural numbers (n = 1, 2, 3...).
• The discovery of the irrationality of √2 is historically attributed to Hippasus of Metapontum, a follower of the Pythagorean school, which caused a crisis in Greek mathematics because they believed all things could be expressed as ratios of integers.

Shortcut Trick

Standard Circle Area: If Area = 616 cm², then Radius r = 14 cm.

Given: Height h = 3 × r ⇒ h = 3 × 14 = 42 cm.

Capacity = Base Area × height = 616 × 42 = 25872 cm³.

In Litres: 25872 ÷ 1000 = 25.872 litres.

∴ The correct answer is 25.872 litres.

Alternate Method

Given:

Base Area of cylinder (πr²) = 616 cm²

Height (h) = 3 × Radius (r)

Formula Used:

Volume of cylinder = πr²h

1 litre = 1000 cm³

Calculations:

⇒ πr² = 616

⇒ (22 ÷ 7) × r² = 616

⇒ r² = (616 × 7) ÷ 22

⇒ r² = 28 × 7 = 196

⇒ r = √196 = 14 cm

⇒ h = 3 × 14 = 42 cm

⇒ Volume = Base Area × h

⇒ Volume = 616 × 42 = 25872 cm³

⇒ Capacity in litres = 25872 ÷ 1000

⇒ Capacity = 25.872 litres

∴ The correct answer is 25.872 litres.

Additional Information

Volume and Surface Area of Cylinder

Volume = πr²h | Curved Surface Area (CSA) = 2πrh | Total Surface Area (TSA) = 2πr(r + h).

Unit Conversions

1 cm³ = 1 ml | 1000 cm³ = 1 litre | 1 m³ = 1000 litres.

Standard Circle Values

If radius = 7, Area = 154 | If radius = 14, Area = 616 | If radius = 21, Area = 1386.

The correct answer is The student has understood the concept of arranging integers but has done a mistake by arranging them in increasing order.

Key Points

• Descending order is defined as arranging numbers from the greatest (highest) value to the smallest (lowest) value.
• The set of numbers provided is: −20, −150, 0, 23, 46.
• By evaluating their values:
• The positive integers are 46 and 23. Since 46 > 23, 46 is the largest.
• The number 0 is smaller than any positive integer but larger than any negative integer.
• The negative integers are −20 and −150. In the system of integers, −20 is greater than −150 because it is closer to zero on the number line.
• The correct descending order should be: 46, 23, 0, −20, −150.
• The student wrote: −150, −20, 0, 23, 46. This sequence is arranged from the smallest to the greatest, which is called increasing (ascending) order.
• The student correctly identified the relative values (placing −150 as the minimum and 46 as the maximum), which demonstrates they understand the concept of integers. However, they made a logical error by reversing the direction of the arrangement.

Additional Information

• Integers: A complete set of numbers comprising negative whole numbers, zero, and positive whole numbers.
• Negative Integers: For negative numbers, as the magnitude (absolute value) increases, the actual value decreases. For example, −100 is smaller than −1.
• Number Line: Values on a horizontal number line always increase as you move from left to right. The student's answer follows the left-to-right (increasing) progression of a number line.

The logic followed here is based on the fundamental principles of historical numeration and time measurement systems.

Explanation:

The system used for measuring time (hours, minutes, and seconds) is the Sexagesimal System, which is a numeral system with sixty as its base.

Division of Time in Clocks:

Analysis of the Options:

1. Sexagesimal System: This is the base-60 system. It was developed by the ancient Sumerians and Babylonians and is still used today for measuring time, angles, and geographic coordinates.
2. Decimal System: This is the base-10 system, which is the standard system for denoting integers and non-integer numbers in modern mathematics.
3. Hexadecimal System: This is a base-16 system, commonly used in computer science and digital electronics to represent binary-coded values.
4. Binary System: This is a base-2 system, representing numeric values using two symbols (0 and 1), which is the foundation of all modern computing.

Therefore, the numeration system on which clocks are based is the Sexagesimal System.