Mathematics and Science PYQs

Given:

Pentagon ABCDE with angles:

• \(\angle A = (2x + 9^\circ)\)
• \(\angle B = (2x + 1^\circ)\)
• \(\angle C = (2x - 1^\circ)\)
• \(\angle D = (2x + 5^\circ)\)
• \(\angle E = (2x - 4^\circ)\)

Concept:

The sum of interior angles of a pentagon is \(540^\circ\).

Step 1: Sum all angles and set equal to 540°

(2x + 9° ) + (2x + 1° ) + (2x - 1° ) + (2x + 5° ) + (2x - 4° ) = 540° 

Step 2: Simplify the equation

\(10x + 10^\circ = 540^\circ\)

Step 3: Solve for x

\(10x = 530^\circ\)

x = 53° 

Step 4: Calculate (2x+10∘)(2x+10∘)(2x+10∘)(2x+10∘)

2x + 10°  = 2(53° ) + 10°  = 106°  + 10°  = 116°

Given:

Polynomial expressions:

• \(A = 3x^2 - 8x + 11\)
• \(B = -2x^2 + 12x\)
• \(C = -4x^2 + 7\)
• \(D = x^2 - x - 11\)

Step 1: Compute (A + B + C) 

\( \begin{align*} A + B + C &= (3x^2 - 8x + 11) + (-2x^2 + 12x) + (-4x^2 + 7) \ &= (3x^2 - 2x^2 - 4x^2) + (-8x + 12x) + (11 + 7) \ &= -3x^2 + 4x + 18 \end{align*} \)

Step 2: Subtract D from the sum

\( \begin{align*} (A + B + C) - D &= (-3x^2 + 4x + 18) - (x^2 - x - 11) \ &= -3x^2 - x^2 + 4x + x + 18 + 11 \ &= -4x^2 + 5x + 29 \end{align*} \)

Step 3: Compare with px² + qx + r

From the equation:

\(-4x^2 + 5x + 29 = px^2 + qx + r\)

We get:

• \(p = -4\)
• \(q = 5\)
• \(r = 29\)

Step 4: Calculate (p + q + r)

\( p + q + r = -4 + 5 + 29 = 30 \)

Assessing learners in mathematics involves more than checking recall of facts, it requires evaluating their ability to think with numbers, recognize patterns, and apply concepts to real situations.

Key Points

• Effective assessment targets both abstract reasoning and practical problem‑solving, ensuring students can connect mathematical ideas to everyday life.
• The ability to use abstractions enables learners to see relationships between quantities and operations, such as understanding how changing one value affects another.
• This skill is foundational for algebraic thinking and for generalizing patterns beyond specific examples.
• Applying concepts to daily‑life problems shows that students can transfer classroom learning into meaningful contexts, whether estimating costs at a store or calculating time intervals.
• Together, these capabilities demonstrate both deep conceptual understanding and real‑world relevance, which are central goals of mathematics education.

Hint

• Memorizing every formula does not guarantee understanding of when or why to use it, and may lead to rote application without insight.
• Focusing on achieving full marks emphasizes performance over genuine comprehension and fails to reveal how well students can think mathematically.

Hence, the correct answer is ‘(a) and (c)’.

Given:

• Principal (P) = ₹8000
• Amount (A) = ₹9500
• Simple Interest Rate (r) = 7.5% p.a.
• Time = x years

Part 1: Find Time (x) using Simple Interest

Simple Interest Formula:

\( A = P(1 + \frac{r \times x}{100}) \)

Substitute values:

\( 9500 = 8000(1 + \frac{7.5 \times x}{100}) \)

Simplify:

\( \frac{9500}{8000} = 1 + 0.075x \)

\( 1.1875 = 1 + 0.075x \)

\( 0.1875 = 0.075x \)

\( x = \frac{0.1875}{0.075} = 2.5 \) years

Part 2: Calculate Compound Amount at 20% p.a.

Compound Interest Formula:

\( A = P(1 + \frac{r}{100})^x \)

Given:

• P = ₹8000
• r = 20%
• x = 2.5 years

 

Since compounding is yearly but time is 2.5 years, we calculate:

\( A = 8000(1 + 0.20)^2 \times (1 + 0.20 \times 0.5) \)

Breakdown:

1. First 2 years: \( 8000 \times (1.20)^2 = 8000 \times 1.44 = ₹11520 \)
2. Next 0.5 year (simple interest on ₹11520): \( 11520 \times (1 + 0.20 \times 0.5) = 11520 \times 1.10 = ₹12672 \)

Given:

We are asked to find the smallest number x which must be subtracted from 2605 to make it a perfect square. Then, we need to find the value of 4x + 7.

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.

Step 1: Find the square root of 2605

First, we find the square root of 2605:

\( \sqrt{2605} \approx 51.06 \)

The nearest integer less than or equal to 51.06 is 51.

Step 2: Find the perfect square less than 2605

The square of 51 is:

\( 51^2 = 2601 \)

So, the nearest perfect square less than 2605 is 2601.

Step 3: Calculate x

Now, subtract the perfect square from 2605:

\( x = 2605 - 2601 = 4 \)

So, x = 4.

Step 4: Calculate 4x + 7

Now, calculate 4x + 7:

\( 4(4) + 7 = 16 + 7 = 23 \)

Conclusion:

The value of 4x + 7 is 23.

Given Range:

We need to find rational numbers between \(\frac{-7}{10}\) and \(\frac{-1}{4}\).

Step 1: Convert fractions to decimal for easier comparison

• \(\frac{-7}{10} = -0.7\)
• \(\frac{-1}{4} = -0.25\)

So we're looking for numbers between -0.7 and -0.25.

Step 2: Evaluate each option

Option 1: \(\frac{-3}{5}, 0, \frac{-1}{5}\)

• \(\frac{-3}{5} = -0.6\) → Valid (-0.7 < -0.6 < -0.25)
• \(0 = 0\) → Not valid (0 > -0.25)
• \(\frac{-1}{5} = -0.2\) → Not valid (-0.2 > -0.25)

Only \(\frac{-3}{5}\) qualifies in this option.

Option 2: \(\frac{-2}{3}, \frac{-3}{5}, \frac{-3}{4}\)

• \(\frac{-2}{3} ≈ -0.666\) → Valid
• \(\frac{-3}{5} = -0.6\) → Valid
• \(\frac{-3}{4} = -0.75\) → Not valid (-0.75 < -0.7)

\(\frac{-2}{3}\) and \(\frac{-3}{5}\) qualify in this option.

Option 3: \(\frac{5}{-8}, \frac{2}{3}, \frac{-3}{5}\)

• \(\frac{5}{-8} = -0.625\) → Valid
• \(\frac{2}{3} ≈ 0.666\) → Not valid (positive)
• \(\frac{-3}{5} = -0.6\) → Valid

\(\frac{5}{-8}\) and \(\frac{-3}{5}\) qualify in this option.

Option 4: \(\frac{5}{-8}, \frac{-2}{3}, \frac{-3}{5}\)

• \(\frac{5}{-8} = -0.625\) → Valid
• \(\frac{-2}{3} ≈ -0.666\) → Valid
• \(\frac{-3}{5} = -0.6\) → Valid

All three numbers in this option qualify.

Conclusion:

The option where all given numbers lie between \(\frac{-7}{10}\) and \(\frac{-1}{4}\) is:

Final Answer:

\( \frac{5}{-8}, \frac{-2}{3}, \frac{-3}{5} \)

(Option 4 is the complete correct answer where all three numbers satisfy the condition)

Given:

• Radius of the wheel = 25 cm
• Number of rounds = 350
• Use \( \pi = \frac{22}{7} \)

Concept:

To find the distance covered by the wheel, we need to calculate the distance covered in one round (i.e., the circumference of the wheel), and then multiply it by the number of rounds.

Step 1: Calculate the circumference of the wheel

The formula for the circumference C of a circle is:

\( C = 2\pi r \)

Where:

-\( \pi = \frac{22}{7} \)

- r is the radius of the wheel, which is 25 cm.

So, the circumference of the wheel is:

\( C = 2 \times \frac{22}{7} \times 25 = \frac{1100}{7} , \text{cm} \)

Step 2: Convert the circumference to meters

Since the distance is in centimeters, we convert it to meters by dividing by 100 (since 1 meter = 100 cm):

\( C = \frac{1100}{7} , \text{cm} = \frac{1100}{7 \times 100} = \frac{11}{7} , \text{meters} \)

Step 3: Calculate the total distance covered in 350 rounds

Now, multiply the distance covered in one round by the number of rounds (350):

\( \text{Total distance} = 350 \times \frac{11}{7} = 50 \times 11 = 550 , \text{meters} \)

Conclusion:

The distance covered by the wheel in 350 rounds is 550 meters.

Given:

• Point O lies on line AB
• Rays OC and OD are on the same side of AB
• \(OC \perp OD\)  (they are perpendicular)
• \(\angle AOC = x\)
• \(\angle BOD = 3x\)

Step 1: Understand the Diagram

Since O is on AB, angles AOC and BOD form a linear pair with COD:

\(\angle AOC + \angle COD + \angle BOD = 180^\circ \)

Step 2: Use the Perpendicular Condition

Given \(OC \perp OD\) :

\( \angle COD = 90^\circ \)

Step 3: Substitute Known Values

\(x + 90^\circ + 3x = 180^\circ \)

\(4x + 90^\circ = 180^\circ \)

\( 4x = 90^\circ \)

 \(x = 22.5^\circ \)

Step 4: Find  \(\angle BOD \)

 \(\angle BOD = 3x = 3 \times 22.5^\circ = 67.5^\circ\) 

Final Answer:

\( \boxed{67\frac{1}{2}^\circ} \)

Given:

• Cost price (CP) = ₹750
• Selling price (SP) = ₹608
• Discount = 24% on the marked price (MP)

Concept:

To find the profit percent if the article had been sold at the marked price, we need to follow these steps:

Step 1: Calculate the marked price (MP)

We know that a 24% discount was given, so the selling price is 76% of the marked price (because 100%−24%=76%100%−24%=76%100%−24%=76%100%−24%=76%100%−24%=76%

  ).

\( \text{SP} = 76% \times \text{MP} \)

Substituting the known values:

\( 608 = 0.76 \times \text{MP} \)

Solving for MPMPMPMPMP

  :

\( \text{MP} = \frac{608}{0.76} = 800 \)

So, the marked price of the article is ₹800.

Step 2: Calculate the profit if the article was sold at the marked price:

The profit is the difference between the selling price (if the article were sold at the marked price) and the cost price (CP):

\( \text{Profit} = \text{MP} - \text{CP} = 800 - 750 = 50 \)

Step 3: Calculate the profit percent:

The profit percent is calculated using the formula:

\( \text{Profit Percent} = \frac{\text{Profit}}{\text{CP}} \times 100 \)

Substituting the values:

\( \text{Profit Percent} = \frac{50}{750} \times 100 = \frac{5000}{750} = 6.67% \)

Conclusion:

If the article had been sold at its marked price, the profit percent would have been 6.67%.

In geometry, transformations describe how a shape changes its position, orientation, or size. For upper primary learners, associating each transformation with a simple, intuitive term can aid in understanding.

Key Points To help Nitu grasp these transformations, her teacher assigned each one a simple term:

• Reflection: "Flip" – This term captures the idea of flipping a shape over a line, creating a mirror image.
• Rotation: "Turn" – This term describes turning a shape around a fixed point, akin to spinning.
• Translation: "Slide" – This term conveys sliding a shape from one position to another without altering its orientation.
• Enlargement: "Bigger Figure" – This term indicates increasing the size of a shape.

Thus, the correct sequence of terms corresponding to the transformations is Flip, Turn, Slide and Bigger Figure.​

Scientific Laws are concise statements that describe patterns in nature. They are often expressed mathematically and represent consistent relationships observed under specific conditions (e.g., Newton’s Law of Gravitation). They describe what happens.

Key Points

• Statement 1 correctly distinguishes laws and theories: laws generalize observations, and theories explain them.
• Statement 2 is incorrect because the age of a law or theory does not define it—some laws and theories are recent discoveries.
• Statement 3 is false because scientific laws can be refined or reinterpreted as new evidence emerges; theories are also constantly tested and updated, so this distinction is not accurate.

Hence, S1 is true; S2 and S3 are false.

CONCEPT:

Physical and Chemical Changes

• Physical changes involve changes in the state or appearance of a substance without altering its composition.
• Chemical changes involve changes that alter the chemical composition of a substance, resulting in the formation of new substances.

EXPLANATION:

• A Crystallization is a physical change because it involves the transition of a substance from a liquid to a solid state without changing its chemical composition.
• B Photosynthesis is a chemical change because it converts carbon dioxide and water into glucose and oxygen, which involves a change in the chemical composition of the reactants.
• C Ripening of fruit is a chemical change because it involves the conversion of starches to sugars and other chemical changes within the fruit.
• D Formation of curd from milk is a chemical change because it involves the conversion of lactose to lactic acid and alters the chemical composition of milk.

Therefore, the correct answer is 2 B and D.

CONCEPT:

Phase Change and Latent Heat

• During a phase change, such as melting or boiling, the temperature of a substance remains constant.
• This occurs because the energy added to the system is used to break the intermolecular bonds rather than increasing the kinetic energy (temperature) of the molecules.
• The specific amount of heat required to convert a unit mass of a solid into a liquid at its melting point is called the latent heat of fusion.

EXPLANATION:

• When ice melts at 0°C, it undergoes a phase change from solid to liquid.
• During this melting process, the temperature of the ice-water mixture remains constant at 0°C until all the ice has melted.
• The heat energy supplied is used to break the bonds between the ice molecules, converting them into liquid water.
• Thus, there is no change in temperature; it remains the same.

Therefore, the correct answer is 3) remains same.

The correct answer is: Option 3) 

Explanation:

Assertion (A): Nichrome is used as a heating element in devices like electric toaster and electric geyser.

• This is true. Nichrome, an alloy of nickel and chromium, is commonly used in heating elements due to its ability to withstand high temperatures and its resistance to oxidation. It is used in devices like toasters, electric geysers, and other heating appliances.

Reason (R): Nichrome has high melting point and low resistivity.

• This statement is true but incorrect as an explanation for assertion (A). Nichrome does have a high melting point, which is why it is suitable for high-temperature applications like heating elements. However, it does not have low resistivity; rather, it has a moderate resistivity. The high resistivity of Nichrome is actually one of the key factors that allows it to generate heat when an electric current flows through it.

CONCEPT:

Buoyant Force

• Buoyant force is the force exerted by a fluid that opposes the weight of an immersed object.
• It is based on Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object.
• This force always acts in the opposite direction to gravity.

EXPLANATION:

• When an object is immersed in a liquid, the liquid exerts an upward force on the object.
• This upward force, known as the buoyant force, helps the object stay afloat or reduces its effective weight.
• The direction of the buoyant force is always upward, opposing the downward force of gravity.

Therefore, the correct answer is 1) Upward direction.

Inquiry-based learning in science emphasizes active student engagement through exploration, questioning, and problem-solving. It shifts the classroom focus from rote memorization or direct teaching to hands-on investigation and student-led discovery. 

Key Points

• Engaging students in floating and sinking activities fosters inquiry-oriented learning as it encourages them to investigate phenomena directly.
• Through experimentation with different objects, students can observe patterns, predict outcomes, and develop an understanding of buoyancy.
• This process allows them to construct the principle behind why some objects float while others sink, gradually guiding them toward understanding concepts like density and Archimedes' principle.

Hint

• Defining and explaining Archimedes' principle is a teacher-led strategy where students receive information passively, which limits exploration and personal understanding.
• Demonstrating loss in weight of an object in water is slightly more engaging, but students are still observers rather than investigators.
• Explaining the difference between real and apparent weight also remains a theoretical and lecture-based approach, which does not support hands-on learning or student inquiry.

Hence, the correct answer is engaging students in floating and sinking activities.

Divergent questions are those that have multiple possible answers or invite creative, open-ended thinking. These types of questions encourage students to explore various possibilities, apply imagination, and think critically beyond a fixed response. 

Key Points

• A question like "What if there was no gravity on the Earth?" is a divergent question because it asks students to imagine a hypothetical scenario.
• It invites multiple lines of thought about human movement, atmosphere, ocean behavior, structure of buildings, and much more.
• It has no single correct answer and requires synthesis of knowledge, prediction, and creative reasoning, making it ideal for stimulating open-ended discussion.

Hint

• Asking "Why does a ball thrown up come down?" is a convergent question, leading to a factual answer (due to gravity).
• "How does gravity vary with altitude?" and "What would happen to gravity if the size of Earth is reduced to half?" are also convergent, though slightly exploratory, they aim for specific, scientifically supported answers.

Hence, the correct answer is what if there was no gravity on the earth.

The correct answer is (A), (C), (D)

Explanation:

Whales and dolphins are marine mammals, which means they have specific respiratory adaptations for living in the water. Unlike fish, which have gills to extract oxygen from water, whales and dolphins have lungs and need to breathe air. Their respiratory system is adapted to allow them to take in large amounts of oxygen quickly when they surface.

• (A) They cannot use dissolved oxygen: This is true. Whales and dolphins cannot extract oxygen from water through gills like fish do. They must come to the surface to breathe air through their lungs.
• (C) They have large nostrils: This is true. Whales and dolphins have blowholes on top of their heads, which are their nostrils. These blowholes allow them to quickly exchange air at the surface.
• (D) They often come up to the water surface to breathe: This is true. Whales and dolphins must regularly surface to breathe since they rely on atmospheric oxygen. They come up to the surface, exhale forcefully to clear the blowhole, and then inhale fresh air.

Other Options:

• (B) They have gills: This is incorrect. Whales and dolphins do not have gills. They have lungs and breathe air just like humans.

CONCEPT:

Chemical Reactions of Sodium Carbonate and Hydrochloric Acid

• When sodium carbonate (Na2CO3) reacts with dilute hydrochloric acid (HCl), it produces sodium chloride (NaCl), water (H2O), and carbon dioxide (CO2).
• The produced carbon dioxide gas can be tested by passing it through lime water (calcium hydroxide solution, Ca(OH)2), which results in the formation of calcium carbonate (CaCO3) and water.

EXPLANATION:

A Na2CO3 + 2HCl → 2NaCl + H2O + CO2

• Sodium carbonate (Na2CO3) reacts with hydrochloric acid (HCl) to produce sodium chloride (NaCl), water (H2O), and carbon dioxide (CO2) gas.
• The produced carbon dioxide gas is then passed through lime water:

B Ca(OH)2 + CO2 → CaCO3 + H2O

• Calcium hydroxide (Ca(OH)2) reacts with carbon dioxide (CO2) to form calcium carbonate (CaCO3) and water (H2O).
• This reaction causes the lime water to turn milky, indicating the presence of carbon dioxide.

Therefore, the correct representation of the chemical reactions for A and B is option 2.

CONCEPT:

Atomic Number, Mass Number, Protons, Electrons, and Neutrons

• The atomic number of an element is equal to the number of protons in the nucleus of an atom. Since atoms are electrically neutral, the number of electrons is also equal to the atomic number in a neutral atom.
• The mass number is the sum of the protons and neutrons in the nucleus of an atom.
• The number of neutrons can be calculated by subtracting the atomic number (number of protons) from the mass number.

EXPLANATION:

• For the first row (Atomic number = 9):
• Element is Fluorine (F), with atomic number 9, and mass number A is 19 (Fluorine's isotope).
• Since it is neutral, the number of electrons B is also 9 (equal to atomic number).
• Number of protons C is 9 (equal to atomic number).
• To find the number of neutrons F, subtract atomic number (9) from mass number (19), which gives 10 neutrons.
• For the second row (Atomic number = D):
• Element is Sodium (Na), with atomic number 11, and mass number A is 23 (Sodium's isotope).
• Number of electrons E is also 11 (equal to atomic number in neutral state).
• Number of protons C is 11 (equal to atomic number).
• To find the number of neutrons F, subtract atomic number (11) from mass number (23), which gives 12 neutrons.

Therefore, the correct values for A, B, C, D, E, and F are: 19, 9, 9, 12, 12, 12