Mathematics and Science PYQs

Explanation:

1) A rational number includes integers 

 A number which of the p/q where p , q are integers and q≠ 0

All integers can be written in the form of p/q 

example -1, -2, 3, 4 and many more can be witten as -1/1, -2/1, -3/1, 4/1 and so on 

Hence,  A rational number includes integers is Correct.

2) 0 (zero) is not a rational number is incorrect 

because 0 can also be written in the form p/q = 0/1.

3) All fractions are rational numbers is Correct 

because they can be written in the form p/q where q≠ 0

Hence, (a) and (c) are Correct 

Hence, Option 3 is Correct.

Concept use:

The angle-angle-side (AAS) theorem is used to determine when two triangles are congruent.

By the theorem, two triangles are congruent if they have two angles of equal measure and one side adjacent to only one of the angles that are equal in length.

Explanation:

In ΔABC & ΔDEF 

∠F = ∠A (40°)

∠B = ∠E (90°)

BC = DE (Equal in Length)

∴ ∠ABC ≅ ∠FED (by Angle - Angle - Side Property) 

Hence, Option "1" is the correct answer.

Key Points

• It provides an opportunity for students to engage in proofs and examining patterns: Mathematics involves logical reasoning, which is evident in the process of proving theorems and examining patterns. By engaging in proofs and pattern recognition, students develop critical thinking skills and deepen their understanding of mathematical concepts.
• Students use appropriate strategies for solving various new problems: Mathematics encourages students to employ problem-solving strategies to tackle unfamiliar or challenging problems. By applying appropriate techniques and methods, students develop problem-solving skills and become adept at addressing a wide range of mathematical problems.

Hence, the correct answer is option 4.

Explanation:

Let's denote the three numbers as 2�2�2�2�2�2�

    2x, 3�3�3�3�3�3�

    3x, and 4�4�4�4�4�4�

    4x, since they are in the ratio 2: 3: 4.

(2x)³ + (3x)³ + (4x)³  = 33957 

8x³ + 27x³ + 64x³ = 33957

99x³ = 33957

x³ = 33957/99 = 343

x = (343)^1/3 = 7

First Number 2x = 14, Second Number = 3x = 21, Third Number = 4x = 28 

Sum of the Number = 14 + 21 + 28 = 63.

Hence, The Correct Answer is 63.

Explanation:

Let's denote the three consecutive integers as ����

x, �+1�+1�+1�+1

x+1, and �+2�+2�+2�+2

x+2.

According to the given conditions:

• 3�3�3�3�

  3x is the result when the first integer is multiplied by 3.

• 5(�+1)5(�+1)5(�+1)5(�+1)

  5(x+1) is the result when the second integer is multiplied by 5.

• 2(�+2)2(�+2)2(�+2)2(�+2)

  2(x+2) is the result when the third integer is multiplied by 2.

The sum of these results is 99:

3�+5(�+1)+2(�+2)=993�+5(�+1)+2(�+2)=993�+5(�+1)+2(�+2)=993�+5(�+1)+2(�+2)=99

3 × x + 5 × (x+1) + 2 × (x+2) = 99

Now, let's solve this equation:

3�+5�+5+2�+4=993�+5�+5+2�+4=993�+5�+5+2�+4=993�+5�+5+2�+4=99

3x + 5x + 5 + 2x + 4 = 99

10�+9=9910�+9=9910�+9=9910�+9=99

10x + 9 = 99

10�=9010�=9010�=9010�=90

10x = 90

�=9�=9�=9�=9

x = 9

So, the three consecutive integers are 9, 10, and 11.

Now, the sum of the original first and third integers is 9+11=209+11=209+11=209+11=20

9 + 11=20.

∴ the correct answer is Option (4) 20

Concept use:

Proportion of a Particular Item = (Number of Particular Items) / (Total number of Items) 

Angle of sector representing Particular Item  = Proportion of Item × 360°

Explanation:

Given:

Number of mangoes = 44

Number of apples = 56

Number of oranges = 42

Number of guavas = 30

Number of pomegranates = 38

Total number of fruits = 44 + 56 + 42 + 30 + 38 = 210

Now, to find the proportion of apples:

Proportion of apples = (Number of apples) / (Total number of fruits)

= 56 / 210

To find the angle of the sector representing apples, we'll multiply this proportion by 360 degrees (the total degrees in a circle):

Angle of sector representing apples = Proportion of apples × 360 degrees

= 56 /210 × 360° 

= 96 ° 

So, the angle of the sector representing apples in the pie chart will be 96 degrees.

Explanation:

Thickness of Wood = 3cm 

Length = 1.48 m = 148 cm

Breadth = 1.16 m = 116 cm

Height = 83 cm

 

Inner length = Outer length - twice width = 148 - 6 = 142 cm

Breadth = 116 - 6 = 110 cm

Height = 80 cm [ since it is open box]

 

Area of inner surface = 2(lb+bh+hl)−lb = 2[142×110+110×80+80×142]−[142×110] = 55940cm² =  5.5940 m²

 

The cost of painting its inner surface at ₹150 per m2 

So, The Cost of Painting its inner surface = 150 × 5.5940 m² = ₹839.10 

Hence, The Correct Answer is Option 3.

Concept use:

Median of the data when number of observation is even =  (n/2^th + n/2^th + 1)/2 term 

Median of the data when number of observation is odd = n/2^th term

Explanation:

To find the median of the original data set and the modified data set,

let's first arrange the numbers in ascending order:

Original data set: 3, 7, 8, 15, 17, 21, 23, 24, 29, 30

Modified data set: 7, 18, 15, 17, 21, 23, 24, 29, 30, 33

Now, let's find the median for each set:

For the original data set:

Since the number of observations is even (10), the median is the average of the 5^th and 6^th numbers when arranged in ascending order.

Median (x) = (17 + 21) / 2 = 19

For the modified data set:

Since the number of observations is still even (10), the median is the average of the 5^th and 6^th numbers when arranged in ascending order.

Median (y) = (21 + 23) / 2 = 22

Now, let's find the difference between y and x:

Difference = y - x

= 22 - 19

= 3

So, the difference between y and x is 3.

• Least Common Multiple (LCM): The concept of LCM helps find this smallest distance. LCM is the smallest multiple that is common to all the given numbers. In this case, it represents the minimum distance each person should walk to synchronize their steps.

Key Points

• Understanding the Problem: The problem involves determining the minimum distance each person should walk so that all can cover the same distance in complete steps.
• Step Lengths: Three persons with step lengths of 70 cm, 85 cm, and 95 cm respectively start walking together.
• Common Distance: To ensure that all three persons cover the same distance in complete steps, we need to find the smallest distance that is divisible evenly by all three step lengths.
• Solution: By calculating the LCM of the given step lengths (70 cm, 85 cm, and 95 cm), we can determine the minimum distance each person should walk so that all can cover the same distance in complete steps.

Hence, the correct answer is concept of LCM.

In the context of mathematics education, Learning-Teaching Resources (LTRS) refer to any tools, materials, or strategies that can be used by both teachers and students to facilitate the learning and teaching of mathematical concepts. 

Key PointsMost appropriate Learning-Teaching Resources (LTRS) for visually challenged students in mathematics classroom:

• (b) Taylor's abacus: This manipulative tool allows students to perform calculations using touch and movement, making it accessible and useful for developing mathematical concepts.
• (c) Computer: With the help of screen readers and specialized software, computers can be incredibly valuable for visually challenged students to access educational materials, practice calculations, and learn mathematical concepts through audio descriptions and tactile interactions.

Hint

• (a) Geogebra: While a powerful tool for mathematics exploration and visualization, Geogebra primarily relies on visual elements, making it unsuitable for students with visual impairments.
• (d) Spreadsheets can potentially be adapted for accessibility with the use of screen readers and specialized features, they are not specifically designed for teaching mathematics and may not be as universally effective as Taylor's abacus and computers in this context.

Therefore, (b) and (c) provides the most appropriate combination of resources for supporting the learning of visually challenged students in a mathematics classroom.

The correct answer is Only (b)

Explanation:-

(a) Rust of wheat: Rust diseases are caused by various species of fungi belonging to the genera Puccinia and Uromyces.

• These fungi infect the leaves of wheat and other cereal crops, leading to the formation of characteristic rust-colored pustules on the leaf surface.
• These pustules contain spores that can spread the infection to other plants. The rust of wheat is not caused by bacteria.

(b) Citrus canker: Citrus canker is a bacterial disease caused by the bacterium Xanthomonas citri .

• It affects citrus trees, including oranges, lemons, limes, and grapefruits.
• The disease manifests as raised lesions or pustules on the leaves, fruits, and stems of infected trees.
• These lesions eventually lead to defoliation, premature fruit drop, and reduced fruit quality, impacting citrus production.

(c) Yellow vein mosaic of bhindi (okra): Yellow vein mosaic disease of bhindi is caused by a group of viruses known as begomoviruses, specifically transmitted by whiteflies.

• This viral disease affects the leaves of okra plants, causing yellowing, mottling, and vein clearing symptoms.
• It can significantly reduce yield and quality in okra cultivation.

Conclusion:-

• Rust of wheat (option a) is caused by fungi, not bacteria.
• Citrus canker (option b) is caused by the bacterium Xanthomonas citri .
• Yellow vein mosaic of bhindi (option c) is caused by viruses, specifically begomoviruses transmitted by whiteflies.

Key Points

• A fact is an observation that has repeatedly been found to be true over time.

This statement is correct. A fact is a statement that can be proven to be true based on evidence and repeated observations.

• A law is a theory that has repeatedly been found to be true over time.

This statement is incorrect. A law is a statement that describes a consistent and universal relationship observed in nature. It is more definitive than a theory but is not derived from a theory.

Therefore, only statement (a) is correct

The correct answer is a - v, b - i, c - ii, d - iii

Explanation:-

(a) Rose - Roses commonly reproduce asexually through methods such as cutting, layering, or grafting, all of which are forms of vegetative propagation.

(b) Yeast - Yeast reproduces asexually primarily through budding, where a small outgrowth (bud) forms on the parent cell and eventually separates to become a new individual.

(c) Ferns - Ferns reproduce asexually through the formation and dispersal of spores, which are produced in structures called sporangia on the underside of fern fronds.

(d) Spirogyra - Spirogyra is a filamentous green algae that reproduces asexually through fragmentation, where a filament breaks apart into smaller fragments, each of which can grow into a new individual.

Therefore, the correct set is:

(a) Rose - (v) Vegetative propagation

(b) Yeast - (i) Budding

(c) Ferns - (ii) Spore formation

(d) Spirogyra - (iii) Fragmentation

Using a real-life artifact like a doctor's prescription as a teaching tool offers a unique and memorable way to introduce the complex topic of deficiency diseases.

Key Points

This approach can have several benefits and unique advantages in an educational setting:

• Real-world Connection: Bringing a real doctor's prescription into the classroom immediately grabs students’ attention by providing a tangible connection to the real world. It emphasizes the practical relevance of what they are learning.
• Engagement through Curiosity: The specificity of a real medical document related to a deficiency disease can spark curiosity among students. They may be more inclined to ask questions and engage in the discussion, eager to understand the details and reasons behind the prescription.
• Starting with a Case Study: This approach effectively uses a case study method, starting with a specific example (the prescription and the deficiency disease it addresses) before generalizing about deficiency diseases. This can help students understand the direct impact of nutritional deficiencies on health.
• Interactive Learning: Discussing the prescription encourages interactive learning. It gives students a platform to express their thoughts, ask questions, and learn through dialogue, which can be more effective than passive learning methods.
• Introduction to Critical Thinking: Analyzing a prescription and discussing it can also introduce students to critical thinking skills. They can learn to interpret information, connect it to their existing knowledge about nutrition and health, and consider the broader implications of deficiency diseases.

Hence, we can conclude that "Arrange for a doctor's prescription about a deficiency disease and hold a discussion around it"  would be the most suitable strategy.

Key Points

• Discuss chemical reactions observed in their lives: This strategy engages students by connecting the concept of chemical reactions to their everyday experiences, making the learning more relatable and contextual.

Hint

• Explain the concept of valence electrons and electronic configuration: While the concept of valence electrons is important, it may be too abstract and detailed for an initial introduction to chemical reactions at the upper primary level.
• Write chemical equations in symbolic form and ask students to observe: Introducing symbolic chemical equations may be overwhelming for students without a prior understanding of the concept of chemical reactions. Observing equations may not effectively convey the fundamental concept.
• Define a chemical reaction and give examples: Providing a definition and examples is a straightforward approach, but it may not engage students as actively as a discussion based on their observations and experiences.

The correct answer is  Discuss chemical reactions that students are likely to have observed in their lives.

The correct answer is "Force of gravity" 

Explanation:

• Work done by gravity: The work done by gravity varies during the upward and downward motion. It is negative when the ball moves up and positive when it moves down.
• Acceleration: While the magnitude of the acceleration due to gravity is constant (approximately \((9.8 , \text{m/s}^2)\)), the direction of the acceleration due to gravity is always downward. This means it remains the same in magnitude but not the same in direction relative to the motion of the ball, impacting its status as equal in strict comparison.
• Force of gravity: The force of gravity is also constant in magnitude ((F = mg)) but, like acceleration, it always acts downward. Thus, its direction remains constant regardless of whether the ball is moving up or down.
• Displacement: Displacement as a vector quantity depends on direction as well, but if we look at the overall round trip (upward and downward movement), the displacements can be equal in magnitude. If considering specific points (e.g., halfway up and halfway down), the displacement from the starting point will have identical numeric magnitudes.
• For example, if the ball is ( h ) meters above the ground on the way up, it is also ( h ) meters above the ground on the way down before it reaches the thrower.

So, quantities have the same values during the upward and downward motion of the ball is Force of gravity .

Explanation:

• The most suitable reason behind Newton's theory of gravitation being regarded as an excellent theory is: It explains a larger number of observations by making fewer assumptions.
• Its ability to make accurate predictions with minimal assumptions about the nature of the gravitational interaction between objects is what has made it an enduring and fundamental part of physics. This principle of parsimony, often referred to as Occam's razor, suggests that among competing hypotheses that predict equally well, the one with the fewest assumptions should be selected.
• Newton's law of gravitation essentially states that every point mass attracts every other point mass by a force acting along the line intersecting both points.
• The force is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• This simple yet powerful law enables the prediction and explanation of the elliptical orbits of planets, the tides, and various other phenomena with remarkable precision.
• The other options, while they may have elements of truth or contribute to the theory's development and acceptance, do not capture the fundamental reason Newton's theory is considered excellent:
• The development time of a theory does not inherently contribute to its quality.
• While experimentation and empirical evidence are crucial for the acceptance and refinement of scientific theories, the core reason for Newton's theory's excellence is its explanatory and predictive power, not just the method of its development.
• The use of precise mathematical language is instrumental in formulating and expressing the theory clearly, but the most compelling reason for its excellence is its explanatory and predictive capacity with minimal assumptions.

The correct answer is (a) is an observation while (b) and (c) are inferences.

Explanation:

• (a) "the electroscope leaves move further apart" -
• This is an observation that can be directly made when a negatively charged rod is brought near an uncharged electroscope.
• The leaves of the electroscope, being initially uncharged, repel each other when they acquire the same charge due to the presence of the negatively charged rod.
• (b) "both the electroscope leaves acquire similar charge" -
• This statement is an inference drawn from the observation.
• While it is true that the leaves of the electroscope move further apart, indicating that they acquire the same charge (negative), it's an inference based on the observed behavior.
• (c) "the negative charges pass on to the electroscope causing its leaves to move further apart" -
• Similarly, this statement is an inference based on the observed behavior.
• It explains why the leaves move further apart but isn't directly observable.

Thus, Statement (a) describes the observation, while statements (b) and (c) are inferences drawn from the observed behavior.

The correct statement regarding the forces acting on the box is: A pair of balanced forces act on it

Explanation:

• A box of mass 500 gm (or 0.5 kg) lying on a horizontal table experiences two main forces: the gravitational force acting downward and the normal force acting upward from the table.
• The gravitational force (weight) acting on the box can be calculated using the equation F = m × g, where m is the mass and g is the acceleration due to gravity.
• Therefore, the gravitational force acting downward is:

F = m × g = 0.5 kg ×  10 m/s² = 5 N

• This means there is a downward force of 5 N due to gravity.
• To prevent the box from moving through the table, the table exerts an upward force known as the normal force. For the box to remain in equilibrium (not accelerating), this normal force must be equal in magnitude and opposite in direction to the gravitational force acting on the box.
• Thus, there's also an upward normal force of 5 N acting on the box.
• Since these two forces (the gravitational force downward and the normal force upward) are equal in magnitude and opposite in direction, they balance each other out.
• This means that there are indeed a pair of balanced forces acting on the box, which keeps it in a state of static equilibrium (not moving) on the table.

Key Points

• Using the senses to gather information: This is not indicative of hypothesizing skill. Using the senses to gather information is a fundamental aspect of observation and data collection, but it does not directly involve formulating hypotheses.

Hint

• Identifying variables for an experiment: Indicative of hypothesising skill. It involves recognizing the factors that may influence the outcome of an experiment and formulating hypotheses about their relationships.
• Formulating questions which lead to inquiry: Indicative of hypothesizing skill. It involves generating questions that stimulate inquiry and hypothesis formulation.
• Attempting to explain observations or relationships in terms of some principle: Indicative of hypothesizing skill. It involves making connections between observations or relationships and proposing explanations based on scientific principles.

The correct answer is Using the senses to gather information.