Mathematics and Science PYQs

The teacher distributes different types of triangle-shaped paper cuttings and asks students to measure and tabulate the angles to find the sum of angles.

Key Points

• This encourages students to discover that the sum of the angles in a triangle is always 180 degrees through their own exploration and measurements.
• In the Discovery Method, learners are encouraged to explore and find out information for themselves rather than being directly taught.
• This approach is often used to enhance critical thinking and problem-solving skills.
• Student-Centered: The focus is on the students actively engaging with the material rather than passively receiving information.
• Exploratory Learning: Students are given opportunities to explore, experiment, and discover concepts on their own or in groups.
• Enhanced Retention: Since students discover the knowledge themselves, they are likely to retain the information better.
• Critical Thinking: This method fosters critical thinking and problem-solving abilities as students analyze and synthesize information.
• Teacher's Role: The teacher acts as a facilitator or guide, providing guidance and support as needed.

Hence, we can conclude that the method used by the teacher in the scenario is an excellent example of the Discovery Method as it allows students to learn through their own investigation and discovery.

Additional Information

• Deductive Method: This is a teacher-centered approach where the teacher provides a general rule or principle, and students apply it to specific cases. It is the opposite of the inductive or discovery method.
• Analytical Approach: This involves breaking down complex information into simpler parts to understand it better. While it can be student-centered, it does not necessarily involve discovery or exploration.
• Teacher-Centered Approach: In this approach, the teacher is the primary source of information, and students are passive recipients. This is contrary to the discovery method where students are active learners.

Correct Answer: 19

Concept:

To solve this problem, we need to use Euler's formula for polyhedra, which relates the number of faces (F), edges (E), and vertices (V) of a polyhedron. Euler's formula states:

V - E + F = 2

Calculation:

1. Given values: Faces (F) = 7, Edges (E) = 15, and Vertices (V) = x.
2. Substitute the given values into Euler's formula:

x - 15 + 7 = 2

x - 8 = 2

x = 10

3. Now that we know the value of x (vertices) is 10, we can calculate the value of the expression (2F + 3E - 4x):

2F + 3E - 4x = 2(7) + 3(15) - 4(10)

= 14 + 45 - 40

= 19

Additional Information:

By understanding Euler's formula and applying the given values, we can accurately determine the correct value of the expression (2F + 3E - 4x).

Thus, the correct answer is 19.

Concept:

Mean: The mean (average) of a set of numbers is calculated by dividing the sum of all the numbers by the count of numbers.

Range: The range of a set of numbers is the difference between the largest and smallest numbers in the set.

Explanation:

Mean = Total Sum / Number of Students

⇒ (36 + 37 + 31 + 42 + 23 + 38 + 17 + 18 + 35 + 25 + 29 + 35) / 12

⇒ 366 / 12 = 30.5

Range:

Maximum marks: 42

Minimum marks: 17

Range = Maximum - Minimum = 42 - 17 = 25

Difference between Mean and Range:

Difference = Mean - Range = 30.5 - 25 = 5.5

Hence, the correct answer is 5.5.

Solution:

Step 1: Simplify the Left-Hand Side (LHS):

Expand the LHS:

(2x + 5y)² - 5(2x + 5y) - 14

Let z = 2x + 5y

So, the equation becomes: z² - 5z - 14

Step 2: Factorize the Quadratic Expression:

We need to factorize z² - 5z - 14 into the form (z + p)(z + q)

To do this, we need to find two numbers whose product is -14 and whose sum is -5.

These numbers are -7 and 2.

Therefore, z² - 5z - 14 can be written as: (z - 7)(z + 2)

Step 3: Substitute Back:

Recall that z = 2x + 5y

So, (z - 7)(z + 2) becomes:

(2x + 5y - 7)(2x + 5y + 2)

Step 4: Identify p and q:

From the factorization, we have:

p = -7 and q = 2

Step 5: Calculate p + q:

p + q = -7 + 2 = -5

Shortcut Trick

Net profit percentage = Markup % − Discount % − (Markup × Discount)/100

Net profit percentage = 25 − 10 − (25 × 10)/100 = 15 − 2.5 = 12.5%

Overall Profit = 12.5% of Total Cost Price = 1/8 × 24,000

Overall Profit = ₹3,000

∴ The correct answer is ₹3000.

Alternate Method

Given:

Total Cost Price (CP) of 12 fans = ₹24,000

Markup Percentage = 25%

Discount Percentage = 10%

Formula Used:

Marked Price (MP) = CP × (100 + Markup%)/100

Selling Price (SP) = MP × (100 − Discount%)/100

Profit = SP − CP

Calculations:

⇒ Total Marked Price (MP) = 24,000 × (125/100) = ₹30,000

⇒ Total Selling Price (SP) = 30,000 × (90/100) = ₹27,000

⇒ Overall Profit = SP − CP = 27,000 − 24,000

⇒ Profit = ₹3,000

∴ The correct answer is ₹3000.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price is given by: CP/MP = (100 − Discount%)/(100 + Profit%).

Markup and Discount

Markup is always calculated on the Cost Price (CP), while Discount is always calculated on the Marked Price (MP).

Successive Percentage Change

When a value is increased by x% and then decreased by y%, the net change is (x − y − xy/100)%.

Shortcut Trick

Use the standard algebraic identity: (x + y)² − (x − y)² = 4xy.

Here, let x = (a − b) and y = c.

Numerator = 4 × (a − b) × c = 4c(a − b).

The denominator is (b − a), which is equal to −(a − b).

Expression = [4c(a − b)] ÷ [−(a − b)] = −4c.

∴ The correct answer is -4c.

Alternate Method

Given: ((a − b + c)² − (a − b − c)²) ÷ (b − a)

Formula Used: X² − Y² = (X + Y)(X − Y)

⇒ Let X = (a − b + c) and Y = (a − b − c).

⇒ X + Y = (a − b + c) + (a − b − c) = 2a − 2b = 2(a − b)

⇒ X − Y = (a − b + c) − (a − b − c) = a − b + c − a + b + c = 2c

⇒ Numerator = (X + Y)(X − Y) = 2(a − b) × 2c = 4c(a − b)

⇒ Expression = [4c(a − b)] ÷ (b − a)

⇒ Since (b − a) = −1(a − b), the term (a − b) cancels out.

⇒ Expression = 4c ÷ −1 = −4c

∴ The correct answer is -4c.

Additional Information

Difference of Squares

The identity a² − b² = (a − b)(a + b) is fundamental for simplifying quadratic expressions and rational functions.

Combined Identity

The expression (a + b)² − (a − b)² always simplifies to 4ab, while (a + b)² + (a − b)² simplifies to 2(a² + b²).

Sign Convention

In algebraic fractions, remember that (x − y) ÷ (y − x) = −1, provided that x ≠ y.

Explanation -

For Option 1: "The expression (x² + 10x + 25) cannot be simplified further" - This option is true because the expression is already in its simplest form, though it can be factored as (x + 5)².

For Option 2: "The value of the variable in an equation which satisfies the equation is called a solution of the equation" - This option is true because the solution of an equation is defined as the value(s) of the variable(s) that make the equation true.

For Option 3: The given expression 3 + 4 × 5 is a mathematical  expression. So this is not true.

Option 4: "Operations of multiplication can be done on mathematical expressions" - This option is true because multiplication is a valid operation that can be performed on mathematical expressions.

Hence Option (3) is true.

Bloom's Taxonomy is a framework that categorizes educational goals into different domains of learning. The taxonomy is divided into three primary domains: Cognitive, Affective, and Psychomotor.

Key Points

• The affective domain involves learning that is related to emotions, attitudes, appreciations, and values.
• It includes the manner in which we deal with things emotionally, such as feelings, values, appreciation, enthusiasms, motivations, and attitudes.
• Educational objectives in this domain are designed to foster emotional growth and the development of personal attitudes.
• The domain is structured into five levels: receiving, responding, valuing, organizing, and characterizing by a value or value complex.

Hence, we can conclude that the domain of learning that deals with attitudes and values is known as Affective domain.

Additional Information

• Cognitive Domain: Involves mental skills, knowledge, and intellectual abilities. This domain includes activities such as thinking, problem-solving, and understanding.
• Psychomotor Domain: Involves physical movement, coordination, and the use of motor skills.

The National Curriculum Framework (NCF) 2005 aims to provide a framework for creating a more effective and learner-friendly curriculum in India. 

Key Points

• Finding short-cuts in mathematics: While short-cuts can be useful for quick calculations, they often bypass a deeper understanding of mathematical concepts.
• The NCF prioritizes a solid grasp of fundamental principles over the ability to find quick solutions.
• Generalizing a mathematical formula: This process involves recognizing patterns and creating formulas that can be applied to a variety of problems.
• It is crucial for developing higher-order thinking skills and is highly valued in the curriculum.
• Solving day-to-day problems: Applying mathematical concepts to solve real-world problems helps students see the relevance of mathematics in their daily lives. This is a key objective of the NCF.
• Understanding when and how a mathematical technique is to be used: This involves not just knowing a technique but also understanding its appropriate application.
• This skill is essential for effective problem-solving and is emphasized in the curriculum.

Hence, we can conclude that National Curriculum Framework 2005 finding short-cuts in mathematics is least important in mathematics.

Explanation -

We have M is a point on side AB of a triangle ABC

such that AM = BM = CM.

If angles A and B are respectively x and 70°

Now 

By isosceles triangle -

∠ACM = x and ∠BCM = 70

∠ACB = ∠ACM + ∠MCB

∠ACM = ∠CAM = x    [As AM = AC]

∠MCB = ∠CBM = 70^∘   [As BM = MC] 

∠ACB = x + 70^∘ 

Now use the triangle property -

∠CAM + ∠CBM + ∠ACB = 180

⇒ x + 70 + x + 70 = 180

⇒ 2x + 140 = 180

⇒ 2x = 180 - 140

⇒ 2x = 40

⇒ x = 20

Now 3x + 25° = 3 × 20 + 25

= 60 + 25 = 85° 

Hence Option (4) is correct.

Explanation -

Given fractions are -

6/7 = 0.857

\(2\frac{3}{5} = \frac{13}{5}\) = 2.6

3/8 = 0.375

\( \ 2\frac{8}{15} = \frac{38}{15} = 2.533\)

Clearly,  smallest fraction is 3/8 and largest fraction is \(2\frac{3}{5} = \frac{13}{5}\)

Sum = 3/8 + \(2\frac{3}{5} \)

= 3/8 + 13/5

= (15+ 104)/40

= 119/40 = \(2\frac{39}{40} \)

Hence Option (2) is correct.

Shortcut Trick

Ratio of numbers = 3 : 7

HCF = LCM ÷ (Product of ratio terms) = 630 ÷ (3 × 7)

HCF = 630 ÷ 21 = 30

Sum of LCM and HCF = 630 + 30 = 660

∴ The correct answer is 660.

Alternate Method

Given:

Ratio of two numbers = 3 : 7

LCM of the numbers = 630

Formula Used:

If the ratio of two numbers is a : b and their HCF is x, then the numbers are ax and bx. Their LCM is given by (a × b × x).

Calculations:

⇒ Let the numbers be 3x and 7x, where x is the HCF.

⇒ LCM (3x, 7x) = 21x

⇒ 21x = 630

⇒ x = 630 ÷ 21

⇒ x = 30

⇒ HCF = 30

⇒ Sum = LCM + HCF

⇒ Sum = 630 + 30 = 660

∴ The correct answer is 660.

Additional Information

Product Rule of HCF and LCM

The product of two numbers is always equal to the product of their HCF and LCM (i.e., n₁ × n₂ = HCF × LCM).

Co-prime Numbers

For two co-prime numbers (ratio in simplest form), the HCF is always 1 and the LCM is the product of the numbers.

LCM and HCF Relationship

The LCM of any set of numbers is always a multiple of their HCF.

Concept: 

A right triangle is a type of triangle that has one angle measuring 90 degrees. The sides of a right triangle are referred to as the base, the height, and the hypotenuse (the longest side opposite the right angle).

Calculation: 

The area  A of a right triangle can be calculated using the formula:

A = 1/2 × Base × Height 

⇒(1/2)× 30× h = 240​

⇒ h = 16 cm 

Using the Pythagorean Theorem to calculate hypotenuse

c² = 30² + 16²

⇒c² = 900 + 256

⇒c² = 1156

⇒c = 34 

Perimeter of the Triangle: The perimeter is the sum of all sides: Perimeter = base + height + hypotenuse.

⇒Perimeter = 30 + 16 + 34

⇒Perimeter = 80 cm

∴ The Correct answer is Option (3): 80 cm

Given: 

A is the amount after 2 years = 3267 

r is the rate of interest = 10% 

n is  the number of years = 2 years 

Concept Used: 

A = P (1 + \(\frac{r}{100}\) )2

S.I = \(\frac{p× r × t}{100}\)

Calculation: 

Determine the principal amount 

A = P (1 + \(\frac{r}{100}\) )²

⇒ 3267 = P (1 + 10/100)²

⇒ P = 3267 ×  100/ 110  ×  100/ 110   = 2700

Now Simple Interest

S.I = \(\frac{p× r × t}{100}\)

S.I = 2700 × 10× 2.5 /100 = 675

∴ The Correct answer is Option (1): Rs. 675

Concept used: 

To determine how many natural numbers between 1 and 500 are divisible by multiple numbers (in this case, 3, 5, and 7), we need to find the least common multiple (LCM) of these numbers. Once we have the LCM, we can count how many multiples of this LCM exist within the given range.

Calculation: 

LCM Calculation: 

LCM of 3, 5, and 7: Since 3, 5, and 7 are all prime numbers, their LCM is simply their product: 3 × 5 × 7 = 105.

Counting Multiples:

To find how many multiples of 105 are there between 1 and 500, we divide the upper limit (500) by the LCM (105).

Number of multiples = 500 / 105 ≈ 4.76

Since we're looking for whole numbers, we take the integer part of the division: 4.

Thus, 4 natural numbers between 1 and 500 are divisible by 3, 5, and 7.

∴ The Correct answer is Option (4): 4

Given:

The two angles given are 2x + 15 and 3x - 20

Concept: 

When two parallel lines are intersected by a transversal, several types of angles are formed.

One of the key properties of these angles is that the sum of the two interior angles on the same side of the transversal is always supplementary, meaning they add up to 180º.

Calculation: ​

Set up the equation for the angles 

⇒ 2x + 15 + 3x - 20 = 180... ( angles are supplementary) 

⇒ 5x - 5 =180

⇒ x = 37 

Now substituting the value of x in 4x + 6

⇒ 4(37) + 6 = 154

∴ The Correct answer is Option (3): 154

Concept Used: 

In a pie chart, different sectors represent different categories with their central angles proportional to the data values.

The sum of all central angles in a pie chart is 360º. If the total marks obtained by a student were 540, we can use the central angles to find the marks obtained in each subject.

Calculation: 

Determine marks per degree

The total central angle of a pie chart is 360°. The total marks obtained by the student are 540.

So, marks per degree can be calculated as

Marks Per degree = 540/360 = 1.5 Marks per degree

Calculate the marks for Mathematics and English

Mathematics: The central angle is 90°. so the marks obtained in Mathematics are:

⇒ 90 × 1.5 = 135  marks

English: The central angle is 50°. so the marks obtained in English are: 

⇒ 50 × 1.5 = 75 marks 

The difference in marks between Mathematics and English is

⇒ 135- 75 = 60 Marks

∴ The Correct answer is option (3): 60

Assessment is a broad term used in education to refer to the process of gathering, interpreting, and using information about students' learning and performance.

Key Points 

• Diagnose Individual Differences: Assessment helps in identifying the unique strengths and weaknesses of each student.
• This diagnostic function enables teachers to tailor their instructional strategies to meet the diverse needs of their students.
• Gather Information on Student's Performance: Assessments are used to collect data on how well students are understanding the material and performing academically.
• This information is crucial for monitoring progress and making informed decisions about instruction.
• Evaluation: While assessment itself is a broader term, it often includes evaluation, which is the process of analyzing assessment data to make judgments about student performance.
• Evaluation helps in determining the effectiveness of teaching methods and the achievement of learning objectives.
• **Labeling students (**e.g., calling them "high achievers" or "slow learners") can have negative consequences, such as reinforcing stereotypes or limiting students' potential.
• True assessment aims to provide constructive feedback and promote growth, rather than reducing students to fixed labels.

Hence, we can conclude that It helps to label the students does not imply the meaning of assessment.

Van Hiele's theory is a hierarchical model that describes how students learn geometry through various levels of understanding. The theory consists of five levels, but for this question, we focus on the relevant levels to the given statements.

Key Points

• Level 0 - Visualization: At this level, students recognize shapes based on their appearance without considering their properties. They can identify and name shapes as they are.
• This corresponds to statement (d) "Identification of geometrical shapes as they are."
• Level 1 - Analysis: At this level, students start to recognize the properties of shapes. They can identify specific characteristics such as the number of sides or angles.
• This corresponds to the statement (a) "Identification of properties of geometrical shapes."
• Level 2 - Informal Deduction (Abstraction): At this level, students can classify shapes based on their properties. They understand the relationships between different properties and can group shapes accordingly.
• This corresponds to a statement (b) "Classification of different shapes."
• Level 3 - Deduction: At this level, students can logically deduce properties and relationships between shapes. They can understand and form proofs.
• This corresponds to statement (c) "Finding relationships between the properties of geometrical shapes."

Hence, we can conclude that the correct sequence according to Van Hiele's theory of geometrical learning is (d), (b), (a), (c).

Given :

• OP = (x + 7) cm
• OQ = (x + y) cm
• OR = 20 cm
• OS = 16 cm

Concept used: 

Properties of a Parallelogram:

• In a parallelogram, the diagonals bisect each other. This means that the diagonals PR and QS intersect at point O such that PO = OR and QO = OS.

Calculation: 

Since the diagonals bisect each other at O, we have:

OP = OR, thus (x + 7) = 20

OQ = OS, thus (x + y) = 16

From the equation (x + 7) = 20:

We solve for x: x = 20 - 7 = 13

Next, we use the value of x to solve for y from the equation (x + y) = 16:

Substitute x = 13 into (x + y) = 16:

13 + y = 16

Solve for y: y = 16 - 13 = 3

Finally, we calculate the value of (7y + 5):

Substitute y = 3 into (7y + 5):

(7 ×  3) + 5 = 21 + 5 = 26

Thus, the value of (7y + 5) is 26.

∴ The Correct answer is Option (1): 26