Mathematics and Science PYQs

Given:

The equation is: \( \text{ABA} \times 5 = 44B0 \)

Where A and B are digits, and we need to find the correct values for A and B.

Solution:

We can express the number ABA as: \( \text{ABA} = 100A + 10B + A = 101A + 10B \)

So, the equation becomes:

\( (101A + 10B) \times 5 = 4400 + 10B \)

Now, simplifying the equation:

\( 505A + 50B = 4400 + 10B \)

Subtract 10B from both sides:

\( 505A + 40B = 4400 \)

Now, let's test each option to find the correct values of A and B.

Option 1: A = 2, B = 4

Substituting A = 2 and B = 4 into the equation:

\( 505(2) + 40(4) = 1010 + 160 = 1170 \neq 4400 \)

So, this option is incorrect.

Option 2: A = 8, B = 9 

Substituting A = 8 and B = 9 into the equation:

\( 505(8) + 40(9) = 4040 + 360 = 4400 \)

This satisfies the equation, so this option is correct.

Option 3: A = 4, B = 2

Substituting A = 4 and B = 2 into the equation:

\( 505(4) + 40(2) = 2020 + 80 = 2100 \neq 4400 \)

So, this option is incorrect.

Option 4: A = 9, B = 8 

Substituting A = 9 and B = 8 into the equation:

\( 505(9) + 40(8) = 4545 + 320 = 4865 \neq 4400 \)

So, this option is incorrect.

Conclusion:

The correct values for A and B are 8 and 9, respectively. Therefore, the correct answer is:

Option 2: A = 8, B = 9

Given:

We are asked to find the smallest number \( x \) to be subtracted from 7751 to make it a perfect square. Then, we need to find the value of \( 3x - 5 \).

Concept:

To make a number a perfect square, we need to find the nearest perfect square less than or equal to the given number and subtract the difference.

Steps to solve the problem:

1. Find the square root of 7751:

First, we find the square root of 7751:

\( \sqrt{7751} \approx 88.07 \)

The nearest integer less than or equal to 88.07 is 88. 2. Find the perfect square less than 7751:

The square of 88 is:

\( 88^2 = 7744 \)

So, the nearest perfect square less than 7751 is 7744. 3. Calculate x:

Now, subtract the perfect square from 7751:

\( x = 7751 - 7744 = 7 \)

So, x = 7. 4. Find the value of 3x - 5:

Now, we calculate:

\( 3(7) - 5 = 21 - 5 = 16 \)

So, the value of \( 3x - 5 \) is 16.

Conclusion:

The value of \( 3x - 5 \) is 16.

Given:

The expression is: \( (5x - 3y)(6x + 5y) - (3x + 7y)(4x - 5y) \)

We need to find the coefficient of xy.

Concept:

To find the coefficient of  xy, we will expand the terms in the given expression and then collect the terms involving xy.

Step 1: Expanding the terms

1. Expand (5x - 3y)(6x + 5y):

Using the distributive property (FOIL method):

\( (5x - 3y)(6x + 5y) = 5x(6x) + 5x(5y) - 3y(6x) - 3y(5y) \)

\( = 30x^2 + 25xy - 18xy - 15y^2 \)

Simplifying:

\( = 30x^2 + (25xy - 18xy) - 15y^2 \)

\( = 30x^2 + 7xy - 15y^2 \)

2. Expand (3x + 7y)(4x - 5y):

Using the distributive property (FOIL method):

\( (3x + 7y)(4x - 5y) = 3x(4x) + 3x(-5y) + 7y(4x) + 7y(-5y) \)

\( = 12x^2 - 15xy + 28xy - 35y^2 \)

Simplifying:

\( = 12x^2 + (28xy - 15xy) - 35y^2 \)

\( = 12x^2 + 13xy - 35y^2 \)

Step 2: Subtract the second expansion from the first

Now, subtract the second expanded expression from the first expanded expression:

\( (30x^2 + 7xy - 15y^2) - (12x^2 + 13xy - 35y^2) \)

Simplifying the terms:

\( = (30x^2 - 12x^2) + (7xy - 13xy) + (-15y^2 + 35y^2) \)( = 18x^2 - 6xy + 20y^2 \)

Step 3: Coefficient of xy

From the simplified expression \( 18x^2 - 6xy + 20y^2 \), the coefficient of xy is -6.

Conclusion:

The coefficient of xy is -6.

Given Expression:

We need to factorize the quadratic expression:  81x2 - 126xy + 49y2

Solution:

Step 1: Check for a perfect square trinomial

This expression looks like a quadratic trinomial of the form:

\( a^2 - 2ab + b^2 = (a - b)^2 \)

Let's check if this is a perfect square trinomial. We have:

• First term:  81x² is a perfect square, i.e., 9x²
• Last term: 49y²  is a perfect square, i.e., 7y²
• Middle term:-126xy is equal to  -2 × 9x × 7y , which matches the pattern of a perfect square trinomial.

Therefore, the factorization is:81x2 - 126xy + 49y² = (9x-7y)²  

Step 2: Identify one of the factors

The factor of the expression is \( (9x - 7y) \).

Conclusion:

The factor of \( 81x^2 - 126xy + 49y^2 \) is (9x - 7y).

Given:

We are given the equation: \( \frac{7x - 1}{4} - \frac{1}{7} \left( 2x - \frac{1 - x}{2} \right) = 4 \)

We need to find the value of \( \frac{5x - 3}{3x - 1} \).

Solution:

Step 1: Simplify the expression

First, simplify the expression inside the parentheses:

\( 2x - \frac{1 - x}{2} = \frac{4x}{2} - \frac{1 - x}{2} = \frac{4x - (1 - x)}{2} = \frac{5x - 1}{2} \)

Now, substitute this back into the equation:

\( \frac{7x - 1}{4} - \frac{1}{7} \left( \frac{5x - 1}{2} \right) = 4 \)

Step 2: Multiply the second term

Multiply the second term by \( \frac{1}{7} \):

\( \frac{1}{7} \times \frac{5x - 1}{2} = \frac{5x - 1}{14} \)

Now, the equation becomes:

\( \frac{7x - 1}{4} - \frac{5x - 1}{14} = 4 \)

Step 3: Find a common denominator

The common denominator of 4 and 14 is 28. So, we rewrite both fractions with a denominator of 28:

\( \frac{7x - 1}{4} = \frac{49x - 7}{28} \)

\( \frac{5x - 1}{14} = \frac{10x - 2}{28} \)

Now, the equation is:

\( \frac{49x - 7}{28} - \frac{10x - 2}{28} = 4 \)

Step 4: Simplify and solve for x 

Now, combine the fractions:

\( \frac{49x - 7 - (10x - 2)}{28} = 4 \)

Simplifying the numerator: \( 49x - 7 - 10x + 2 = 39x - 5 \) So, the equation becomes: \( \frac{39x - 5}{28} = 4 \) Multiply both sides by 28: \( 39x - 5 = 4 \times 28 = 112 \) Now solve for x: \( 39x = 112 + 5 = 117 \)( x = \frac{117}{39} = 3 \)

Step 5: Find the value of \( \frac{5x - 3}{3x - 1} \)

Substitute x = 3 into the expression \( \frac{5x - 3}{3x - 1} \):

\( \frac{5(3) - 3}{3(3) - 1} = \frac{15 - 3}{9 - 1} = \frac{12}{8} = \frac{3}{2} \)

Conclusion:

The value of \( \frac{5x - 3}{3x - 1} \) is \( \frac{3}{2}\) .

Given:

A person sold two articles, A and  B, for ₹348 each. He earned a profit of 20% on article A and suffered a loss of 13% on article B. We need to find his profit in the entire transaction.

Concept:

• The selling price (SP) of an article is related to the cost price (CP) by the following formulas:
• For profit: \( \text{SP} = \text{CP} \times \left( 1 + \frac{\text{Profit%}}{100} \right) \)
• For loss: \( \text{SP} = \text{CP} \times \left( 1 - \frac{\text{Loss%}}{100} \right) \)

We can use these formulas to calculate the cost prices of both articles and then determine the total profit.

Step 1: Calculate the cost price of article A

For article A, the selling price is ₹348, and the profit percentage is 20%. Using the profit formula:

\( 348 = \text{CP}_A \times 1.2 \)

Now solve for CP_A:

\( \text{CP}_A = \frac{348}{1.2} = 290 \)

So, the cost price of article A is ₹290.

Step 2: Calculate the cost price of article B

For article B, the selling price is ₹348, and the loss percentage is 13%. Using the loss formula:

\( 348 = \text{CP}_B \times 0.87 \)

Now solve for CP_B:

\( \text{CP}_B = \frac{348}{0.87} \approx 400 \)

So, the cost price of article B is ₹400.

Step 3: Calculate the total cost price and total selling price

The total cost price of both articles is:

\( \text{Total CP} = \text{CP}_A + \text{CP}_B = 290 + 400 = 690 \)

The total selling price of both articles is:

\( \text{Total SP} = 348 + 348 = 696 \)

Step 4: Calculate the total profit

The total profit is the difference between the total selling price and the total cost price:

\( \text{Profit} = \text{Total SP} - \text{Total CP} = 696 - 690 = 6 \)

Conclusion:

The profit in the entire transaction is ₹6.

Given:

• Principal (P) = ₹16,000
• Time (t) = 7/3 years
• Rate (R) = 15% p.a. (compounded yearly)

Calculation:

1. For 2 years compound interest:

\(A_1 = 16,!000 \times (1.15)^2 = ₹21,!160\)
2. For remaining 1/3 year simple interest:

\(\text{Interest} = 21,!160 \times 0.15 \times \frac{1}{3} = ₹1,!058\)

\(A = 21,!160 + 1,!058 = ₹22,!218\)

Final Amount = 22,128

Given:

The difference between two complementary angles is 40°. The smaller of these angles is y. We need to find the value of 2y + 15⁰).

Concept:

• Complementary angles are two angles whose sum is 90°. If the smaller angle is y, then the larger angle is 90⁰ - y.

Step 1: Set up the equation

The difference between the two angles is 40°. Thus, the difference between the larger and smaller angle is:

\( (90^\circ - y) - y = 40^\circ \)

Step 2: Solve for y

Simplify the equation:

\( 90^\circ - 2y = 40^\circ \)

Subtract 40⁰ from both sides:

\( 90^\circ - 40^\circ = 2y \)

\( 50^\circ = 2y \)

Now, divide both sides by 2:

\( y = \frac{50^\circ}{2} = 25^\circ \)

So, the smaller angle y is 25°.

Step 3: Calculate 2y + 15⁰

Now, substitute y = 25⁰ into the expression 2y + 15⁰:

\( 2(25^\circ) + 15^\circ = 50^\circ + 15^\circ = 65^\circ \)

Conclusion:

The value of 2y + 15⁰ is 65°.

Lines of symmetry refer to imaginary lines that divide a shape into two identical halves. A figure can have one or more lines of symmetry depending on its geometric properties.

 Key Points

• Understanding symmetry helps students in visual reasoning and recognizing patterns, which are important aspects of geometry.
• Kite and parallelogram do not have an equal number of lines of symmetry.
• A kite has one line of symmetry — along the line that joins its unequal sides — while a general parallelogram has no lines of symmetry because neither its opposite sides nor angles are necessarily symmetric in all directions.

Hint

• Both rectangle and rhombus have two lines of symmetry.
• Kite and isosceles triangle both have one line of symmetry.
• Kite and isosceles trapezium also have one line of symmetry each.

Hence, the correct answer is kite and parallelogram.

Given:

PS ⊥ QR and PS is the bisector of ∠QPR

Calculation:

In Δ PQS and Δ PRS,

∠ QPS = ∠ RPS (As PS is the angle bisector of ∠QPR)

PS = PS (common in both)

∠ PSQ = ∠PSR = 90°

So, ΔPQS and ΔPRS are congruent by ASA.

Hence, the correct answer is "ΔPQS ≅ ΔPRS, by ASA"

Calcualtion:

Let's analyze the materials provided to both groups:

Group A: Three rectangular glass strips, a circular plane glass plate, some pieces of bangles, and a cardboard tube and disk.

Group B: Two plane mirrors and a hard cardboard/box.

Gadget 1: Kaleidoscope

A Kaleidoscope can be made using the following materials:

• Group A: Glass strips, cardboard tube, and disk (for making the tube and the reflective surfaces inside).
• Group B: Plane mirrors (for reflecting the images inside the kaleidoscope).

Thus, a Kaleidoscope can be made using the materials from both groups.

Gadget 2: Periscope

A Periscope requires mirrors to reflect light, typically used for looking over obstacles:

• Group A: Glass plate or pieces of bangles can be used for the reflective surface.
• Group B: Plane mirrors (for reflecting the light). The cardboard box could be used as the casing.

Therefore, a Periscope can also be made using the materials from both groups.

Correct Answer: Option 1: Kaleidoscope and Periscope can be made using the given materials.

The correct answer is Preservation of food items in homes

Explanation:

• Microorganisms play a significant role in various beneficial processes, making them essential in multiple industries and natural ecosystems.
• They contribute to soil fertility, the preparation of medicines, and the breakdown of organic wastes, among other functions.
• Preservation of food items in homes: Microorganisms are not typically responsible for the preservation of food items in homes. While certain microorganisms are used in industrial food preservation (such as the use of bacteria in fermentation), preservation in homes is usually achieved through methods like refrigeration, freezing, and drying, which do not rely on microorganisms.
• Preparation of medicines: Microorganisms are used extensively in the pharmaceutical industry. They are involved in the production of antibiotics (e.g., Penicillin produced by the mold Penicillium), vaccines, vitamins, and enzymes, which are essential for medical treatments and maintaining health.
• Breaking down of organic wastes: Microorganisms such as bacteria and fungi decompose organic matter in the environment. This process is vital for recycling nutrients back into the ecosystem, thus maintaining soil health and fertility and reducing the accumulation of organic waste.
• Increasing soil fertility: Microorganisms contribute to soil fertility through processes like nitrogen fixation (e.g., Rhizobium bacteria in leguminous plants' root nodules), decomposition of organic matter, and the formation of humus. These activities enhance the nutrient content and structure of the soil, supporting plant growth.

The correct answer is Maintain minimum distance between the crops while sowing

Concept:

• Soil fertility refers to the ability of soil to provide essential nutrients to plants in adequate amounts for their growth and development.
• Several agricultural practices can significantly enhance soil fertility, including crop rotation, organic manure use, and allowing fields to rest.

Explanation:

• Grow different crops alternately every season: This practice, known as crop rotation, helps prevent soil depletion of specific nutrients, reduces pest and disease cycles, and improves soil structure and organic matter content. It contributes significantly to maintaining and enhancing soil fertility.
• Maintain minimum distance between the crops while sowing: Adequate spacing between crops can enhance plant growth by reducing competition for sunlight, water, and nutrients. However, this practice primarily focuses on optimizing plant growth rather than directly increasing soil fertility.
• Substitute fertilizers by organic manure: Using organic manure such as compost or green manure enriches the soil with essential nutrients, improves soil structure, increases water retention, and promotes beneficial microbial activity. This practice is highly effective in enhancing soil fertility.
• Leave the field uncultivated between two crops: Also known as fallowing, this practice allows the soil to recover its nutrient levels naturally, improves its structure, and reduces the buildup of pests and diseases. This rest period can lead to increased fertility and better crop yields in subsequent planting seasons.

The correct answer is Thyroxine

Concept:

• Metamorphosis in amphibians, such as frogs, is the biological process through which the organism undergoes a significant transformation from its larval stage to adult form.
• This process involves drastic changes in physiology, morphology, and behavior, allowing the frog to adapt to its changing environment.

Explanation:

• Thyroxine: Thyroxine is a thyroid hormone that is essential for the process of metamorphosis in frogs. It regulates the transformation of the tadpole into an adult frog by controlling the growth and differentiation of cells, tissues, and organs. Thyroxine triggers the resorption of the tail, the development of limbs, and the changes in the respiratory and digestive systems necessary for terrestrial life.
• Insulin: Insulin is a hormone produced by the pancreas that regulates blood glucose levels by facilitating the uptake of glucose into cells.
• Adrenaline: Adrenaline (or epinephrine) is a hormone produced by the adrenal glands that prepares the body for a 'fight or flight' response during stressful situations.
• Growth hormone: Growth hormone is produced by the pituitary gland and stimulates growth, cell reproduction, and regeneration in the body.

Alternative conceptions, or misconceptions, are incorrect ideas that students hold about scientific concepts. In chemistry, learners often form these based on superficial observations or incomplete understanding, which teachers need to identify and correct.

Key Points

• Among the statements given, “Allotropes of carbon have same atomic number” is scientifically correct.
• Carbon exists in different allotropes like graphite, diamond, and graphene, but all have the same atomic number 6.
• This is a factual statement and does not reflect a learner’s misconception.

Hint

• Adding oxygen in a reaction is oxidation is an outdated or incomplete idea; oxidation involves electron transfer, not just addition of oxygen.
• Carbon atom is black in color  is incorrect because atoms are not colored; color refers to bulk material.
• A chemical bond is a line between two atoms is a symbolic representation, not a literal fact about bonding, so it is also a misconception.

Hence**, the statement about carbon allotropes having the same atomic number is not an alternative conception.**

CONCEPT:

Mitochondria as the powerhouse of the cell

• The mitochondrion is often referred to as the powerhouse of the cell because it generates the cell's supply of adenosine triphosphate (ATP), which is used as a source of chemical energy.
• Mitochondria are crucial for energy production:

ATP synthesis occurs through processes such as oxidative phosphorylation and the citric acid cycle (Krebs cycle).

EXPLANATION:

• Mitochondria contain their own DNA, and they reproduce independently within the cell.
• The inner membrane of mitochondria has folds called cristae that increase surface area for ATP production.
• Energy conversion process:
• Nutrients such as glucose are broken down.
• The breakdown of nutrients results in the production of ATP.
• The ATP produced is then used by the cell for various functions including muscle contraction, cell division, and synthesis of molecules.

Therefore, mitochondria are rightly considered the powerhouse of the cell due to their role in energy production.

A debate on a topic like nuclear energy can serve multiple educational objectives, as it touches upon aspects of science, technology, societal impact, and responsible decision-making.  

Key PointsThe objectives of the debate could include:

• (A) Making students understand the connection between science, technology, and society, as nuclear energy involves both scientific principles and its societal implications.
• (B) Familiarizing students with global issues like energy sustainability and the ethical implications of nuclear energy, which are of significant concern worldwide.
• (D) Helping students grasp the responsible use of science for the benefit of society, ensuring that they understand how scientific advancements should be used for the greater good.

While conceptual understanding about nuclear reactors (C) is important, a debate may not focus deeply on the technical details of nuclear reactors, making it less central to the objectives of a debate.

Hence**, the correct answer is (A), (B), and (D).**

The contemporary understanding of science emphasizes flexibility, critical thinking, and the application of the scientific method in a dynamic way. The scientific method is not a rigid, step-by-step procedure but rather a flexible, iterative process that allows for modification and refinement as new data is gathered.

Key Points

• The statement "Following one scientific method is surely and the only way to scientific discovery" is not consistent with the contemporary understanding of science.
• This view is outdated, as modern science recognizes that the process of discovery can take many paths.
• The scientific method is flexible, and scientists often revisit and revise steps based on the outcomes of their investigations.
• There is no single "only way" to discover scientific knowledge.

The other statements are consistent with contemporary science:

• Investigations can involve repeated actions on various steps of the scientific method.
• The scientific method does indeed offer a logical approach to problem solving.
• The scientific method is about explaining and predicting phenomena, not about making aesthetic or value judgments.

Hence, the correct answer is following one scientific method is surely and the only way to scientific discovery.

The nature of science involves understanding that scientific knowledge evolves over time, often as a result of new theories or experimental findings that challenge existing concepts.

 Key Points

• The statement about Bohr's theory illustrates a fundamental aspect of science, its ability to evolve and refine based on new ideas.
• The excerpt illustrates that science is tentative. Bohr's theory marked a revolution in science, and it was an advancement that refined the earlier models of the atom.
• However, as scientific knowledge progresses, newer models and theories (like quantum mechanics) replaced or modified Bohr’s ideas.
• This shows that scientific theories are open to change and improvement, reflecting the tentative nature of science.

Hint

• Science is socially and culturally embedded refers to how scientific practice is influenced by the society and culture in which it is conducted, which is not directly illustrated in the excerpt.
• Science is subjective suggests that science is influenced by personal opinions or biases, which is not the case in this context.
• Science is value-neutral refers to science being unaffected by moral or ethical considerations, which isn't directly relevant to the nature of Bohr’s theory as described.

Hence, the correct answer is science is tentative.

In the philosophy of science, the concept of paradigm shifts refers to significant changes in the fundamental concepts and experimental practices of a scientific discipline. 

Key Points

• The assertion (A) "Science undergoes paradigm shifts" is true, as scientific fields often experience dramatic changes when new theories or discoveries challenge the existing understanding. For example, the shift from Newtonian mechanics to Einstein's theory of relativity represented a paradigm shift.
• The reasoning (R) "Existing scientific theories and laws may be discarded in light of new evidence" is also true because, in the process of paradigm shifts, older theories or models are often revised or replaced when new evidence emerges that they cannot explain. For instance, classical physics was challenged by quantum mechanics and relativity due to new experimental data.

Both (A) and (R) are true, and (R) provides a correct explanation for (A). The reasoning aligns with the idea that science progresses through paradigm shifts when new evidence leads to the replacement of existing theories.

Hence, the correct answer is both (A) and (R) are true and (R) is the correct reasoning for (A).