Mathematics and Science PYQs

Given:

Volume = 4620 cm³

Length (Height) = 1.2 m = 120 cm

Concept used:

Volume(Cylinder) = πr²h

CSA = 2πrh

Calculation:

⇒ πr²h = 4620

⇒ πr²(120) = 4620

⇒ πr² = 38.5

⇒ r² = 12.25

⇒ r = 3.5 cm

CSA = 2πrh

⇒ 2 × π × 3.5 × 120 = 2640 cm²

∴ The curved surface area of the cylinder is 2640 cm².

Calculations:

A triangular prism has:

5 faces: 2 triangular faces and 3 rectangular faces.

6 vertices: 3 vertices on top and 3 beneath.

9 edges: 3 along the top, 3 along the bottom, and 3 vertical edges.

So we just need to substitute these numbers into each of the equation/options to find out which one is true.

4F + 2V - 3E = 6 ==> 4 x 5 + 2 x 6 - 3 × 9 = 20 + 12 - 27 = 5

2F + 3V - 2E = 10 ==> 2 x 5 + 3 x 6 - 2 × 9 = 10 + 18 - 18 = 10

2F + 4V - 3E = 8 ==> 2 x 5 + 4 x 6 - 3 × 9 = 10 + 24 - 27 = 7

3F + 2V - 2E = 11 ==> 3 x 5 + 2 x 6 - 2 × 9 = 15 + 12 - 18 = 9

 only the second option is correct, 2F + 3V - 2E = 10.

Shortcut Trick

Sum of angles on a straight line (Linear Pair) = 180°

Sum of ratio components = 7x + 5x = 12x

12x = 180° ⇒ x = 15°

Required value = (8 × 15°) − 10°

Required value = 120° − 10° = 110°

∴ The correct answer is 110°.

Alternate Method

Given: ∠QPA = 7x and ∠QPB = 5x

Formula Used: Sum of Linear Pair of angles = 180°

Since AB is a straight line and PQ is a ray starting from P on AB:

⇒ ∠QPA + ∠QPB = 180°

⇒ 7x + 5x = 180°

⇒ 12x = 180°

⇒ x = 180° ÷ 12

⇒ x = 15°

To find the value of (8x − 10°):

⇒ (8 × 15°) − 10°

⇒ 120° − 10°

⇒ 110°

∴ The correct answer is 110°.

Additional Information

Linear Pair Axiom

If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°. Such angles are called supplementary angles.

Straight Angle

An angle whose measure is exactly 180° is called a straight angle, which forms a perfectly straight line.

Adjacent Angles

Two angles are adjacent if they have a common vertex and a common arm, but no common interior points.

Shortcut Trick

Since point S lies on PQ and PS = QS, S is the midpoint of side PQ.

Given RS = PS = QS, S acts as the circumcenter of ΔPQR, making PQ the diameter of the circumcircle.

The angle subtended by a diameter at the circumference is a right angle, so ∠PRQ = 90°.

Using the angle sum property in ΔPQR: ∠PQR = 180° − 90° − 15° = 75°.

The required value is (2 ∠PQR − ∠PRQ) = 2(75°) − 90° = 150° − 90° = 60°.

∴ The correct answer is 60°.

Alternate Method

Given:

S is a point on side PQ.

RS = PS = QS

∠QPR = 15°

P
Q
S
R
15°
75°

Formula Used:

Angles opposite to equal sides of a triangle are equal.

Sum of interior angles of a triangle = 180°

Calculations:

⇒ In ΔPRS, since PS = RS, the angles opposite to them will be equal.

⇒ ∴ ∠SRP = ∠SPR = 15°

⇒ In ΔQRS, since QS = RS, the angles opposite to them will be equal.

⇒ Let ∠SQR = ∠SRQ = x

⇒ Now, considering the larger ΔPQR, the sum of all interior angles is 180°.

⇒ ∠QPR + ∠PQR + ∠PRQ = 180°

⇒ 15° + x + (∠SRP + ∠SRQ) = 180°

⇒ 15° + x + (15° + x) = 180°

⇒ 30° + 2x = 180°

⇒ 2x = 150°

⇒ x = 75°

⇒ Therefore, ∠PQR = 75°

⇒ And ∠PRQ = ∠SRP + ∠SRQ = 15° + 75° = 90°

⇒ Value to find: (2 ∠PQR − ∠PRQ)

⇒ (2 × 75°) − 90° = 150° − 90° = 60°

∴ The correct answer is 60°.

Additional Information

Circumcenter of a Right-Angled Triangle

The circumcenter of any right-angled triangle always lies exactly at the midpoint of its hypotenuse. The median to the hypotenuse is exactly half the length of the hypotenuse.

Angle in a Semicircle

By Thales's theorem, an angle subtended by a diameter of a circle at any point on its circumference is always a right angle (90°).

Isosceles Triangle Theorem

If two sides of a triangle are equal, then the angles opposite to those sides are also equal. Conversely, if two angles are equal, the sides opposite to them must be equal.

Calculations:

First, we need to find the cost price of item A and item B.

Given that the shopkeeper incurs a 20% loss when selling item A for Rs. 384

A_price = Rs. 384 / (1 - 0.20) = Rs. 480. For item B, given that the shopkeeper makes a 25% profit when selling it for Rs. 400

B_price = Rs. 400 / (1 + 0.25) = Rs. 320. Now we know the cost price of A and B, the total cost price for items A and B is:

Total_cost = A_price + B_price = Rs. 480 + Rs. 320 = Rs. 800. And, the shopkeeper sells both items for a total of Rs. 852. So, we can calculate the percent profit:

Profit_percent = [(Selling price - Cost price) / Cost price] ×  100.

Profit_percent = [(Rs. 852 - Rs. 800) / Rs. 800] × 100 = 6.5%.

So, the shopkeeper makes a profit of 6.5% when selling both items for a total of Rs. 852. Hence, option 1) "Profit, 6.5%" is the correct answer.

Shortcut Trick

Ratio of Amount to Principal = 15972 ÷ 12000 = 1331/1000 = (11/10)³.

Since time is 3 half-years, compounding factor = 1.1 ⇒ half-yearly rate = 10%.

Annual rate = 10% × 2 = 20%.

For 1.5 years compounded annually, effective multiplying factor = 1.20 (for 1^st year) × 1.10 (for 0.5 year) = 1.32.

New Amount = 12000 × 1.32 = 15840.

∴ The correct answer is Rs. 15,840.

Alternate Method

Given:

Principal (P) = Rs. 12,000

Amount (A) after 1.5 years = Rs. 15,972 (compounded half-yearly)

Time = 1.5 years = 3 half-years

Formula Used:

Compound Amount, A = P(1 + R/100)ⁿ

12,000
Principal

14,400
After 1 Year

15,840
After 1.5 Yrs

+20%

+10%

Annual Compounding Timeline

Calculations:

⇒ Let the annual rate of interest be R%.

⇒ For half-yearly compounding, rate = R/2 % and time = 3 periods.

⇒ 15972 = 12000 × (1 + R/200)³

⇒ (1 + R/200)³ = 15972 ÷ 12000

⇒ (1 + R/200)³ = 1331 ÷ 1000

⇒ 1 + R/200 = 11/10

⇒ R/200 = 1/10 ⇒ R = 20%

⇒ Now, the same sum is compounded annually for 1.5 years at 20% p.a.

⇒ Amount after 1^st year = 12000 × (1 + 20/100) = 14400

⇒ Interest for the remaining 0.5 year = 14400 × (20/2)/100 = 1440

⇒ Final Amount = 14400 + 1440 = 15840

∴ The correct answer is Rs. 15,840.

Additional Information

Half-Yearly Compounding

When interest is compounded half-yearly, the rate of interest is halved (R/2) and the time period is doubled (2T).

Fractional Year Compounding

If time is given in fractions like t = n + a/b years, the formula used is: Amount = P(1 + R/100)ⁿ × (1 + (a/b × R)/100).

Simple vs Compound Interest Difference

For the first year, both Simple Interest and Compound Interest are exactly the same. The difference arises from the second period onwards as interest is calculated on accumulated interest.

Calculations:

5x2 - 8y - 20x + 2xy = (x + a) (bx + 2y)

5x² - 20x + 2xy - 8y = ( x + a) (bx + 2y)

5x(x - 4) + 2y(x - 4) = (x + a) (bx + 2y)

(x - 4)(5x + 2y) = (x + a) (bx + 2y) 

comparing from both sides 

a = -4, b = 5 

∴ a - b = -4 -5 = -9 

Hence, The Correct option is 3.

Calculations:

A = x2 (1 - 3y2), B = x (xy2 -2x) and C = 3y (y - 4x2y)

A + B - C = x² - 3y²x² + x²y² - 2x² - (3y² - 12x²y²)

A + B - C = x² -  3y2x2 + x2y2 - 2x2 - 3y² + 12x²y²

A + B - C = -x² + 10x²y² - 3y²

Compare with px2 + qx2y2 + ry2. 

p = -1, q = 10, r = -3 

⇒ 5p + q - r = 5(-1) + 10 + 3 = 8 

Hence, The Correct option is 3.

Concept use:

A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 11 if the difference between the sum of digits occupying even and odd places is divisible by 11 (or is zero).

Calculations:

A) 8448:

Visibly, it's divisible by 2 (8 is an even number).

The sum of its digits is 8+4+4+8=24, which is divisible by 3.

The difference between the sum of digits at even places (4 + 8 = 12) and at odd places (8 + 4 = 12) is 0, which is divisible by 11. So, 8448 is divisible by 2, 3 and 11.

B) 9812:

Visibly, it's divisible by 2 (2 is an even number).

The sum of its digits is 9 + 8 + 1 + 2=20, which is NOT divisible by 3.

So, we don't need to check for divisibility by 11. 9812 is NOT divisible by 2, 3 and 11.

C) 9126:

Visibly, it's divisible by 2 (6 is an even number).

The sum of its digits is 9+1+2+6=18, which is divisible by 3.

The difference between the sum of digits at even places (1 + 6 = 7) and at odd places (9 + 2 = 11) is 4, which is NOT divisible by 11. So, 9126 is NOT divisible by 2, 3 and 11.

D) 9636:

Visibly, it's divisible by 2 (6 is an even number).

The sum of its digits is 9 + 6 + 3 + 6 = 24, which is divisible by 3.

The difference between the sum of digits at even places (6 + 6 = 12) and at odd places (9 + 3 = 12) is 0, which is divisible by 11. So, 9636 is divisible by 2, 3 and 11.

So, among the options 3 i.e A & D is Correct.

Calculations:

\(\rm \frac{(x+1)-(2 x+4)}{3-5 x}=\frac{1}{23}\)

(x + 1 - 2x - 4)/ (3 - 5x) = 1/23

(-x - 3)/(3 - 5x) = 1/23 

-23x - 69 = 3 - 5x 

-18x = 3 + 69

x = -4 

Take option 3. 

3(2x + 3) = 5(x + 1)

6x + 9 = 5x + 5 

x = -4 

Hence, The Correct Answer is option 3

The correct option is 2 that is Both (A) and (R) are true and (R) is the correct explanation of (A).

Concept :

• Any hydrogen-containing material that has the ability to give another substance a proton (hydrogen ion) is considered an acid.
• Any molecule or ion that can accept a hydrogen ion from an acid is known as a base.
• The human body makes numerous physiological adjustments to maintain homeostasis. One of them is preserving a proper acid-base balance.
• The pH of a healthy person's body ranges from 7.35 to 7.45, with an average of 7.40, when there are no pathological conditions present.

 

Explanation:

• Acids and bases are at work in your digestive system every time you digest food. Think about the pepsin enzyme, which aids in the stomach's breakdown of proteins.
• For pepsin to function, the environment must be acidic. Pepsin may function because the stomach secretes hydrochloric acid, a potent acid. So, Its concluded that the Assertion is True.
• Our stomachs contain hydrochloric acid, which dissolves food fragments and produces an acidic environment.
• Acidic digestive juices (such hydrochloric acid) are secreted by the stomach's inner walls. Juices like can help in breaking down food into simpler forms. 

So, Its concluded that the reason is also True and is the correct explanation of Assertion.

​

The correct option is option 4 that is Conversion of atmospheric nitrogen to a more usable form for living organisms.

Concepts:

• Nitrogen gas (N₂), which makes up almost 80% of the air we breathe, can only be directly utilized as a source of biological nitrogen by a certain class of bacteria.
• Lightning transforms atmospheric N₂ into nitrate (NO₃^-), a substance of biological significance.
• Biologically: Nitrogen gas diffuses into the soil from the atmosphere, where it is converted by different bacterial species which plants can utilise.
• Farmers frequently cultivate legumes because they have nodules on their roots that contain bacteria that fix nitrogen.

 

Explanation:

• Nitrogen is collected from the atmosphere's molecular form (N2) and transformed into nitrogen compounds useful for other biological processes through the process of nitrogen fixation.
• Fixation can happen as a result of industrial, biological, or atmospheric activities (lightning).
• By employing the nitrogenase protein complex, biological nitrogen fixation, also known as diazotrophy, transforms dinitrogen (N₂) gas into ammonia (NH₃).
• The production of all nitrogen-containing organic chemicals,  depends on fixed inorganic nitrogen compounds, which are necessary for life to exist.
• It's important for agriculture and the production of fertilizer because it's a part of the nitrogen cycle.

​

Hence, the correct option is 4.

The correct answer is The spring is neither compressed nor pulled out and is lying on the ground.​

Concept:

• The potential energy of a spring depends on its compression or extension from its equilibrium position, as well as its height above a reference point.
• The potential energy stored in a spring is given by the formula:
• Potential Energy = P.E = ½ k x²
• Where k is the spring constant (a measure of the stiffness of the spring)
• x is the displacement of the spring from its equilibrium position

Explanation:

• The spring is neither compressed nor extended and is lying on the ground:
• In this case, the displacement (x) is 0, which means there is no compression or extension of the spring.
• Therefore, the potential energy is also 0.
• The spring is compressed to the minimum and is lying on the ground:
• When the spring is compressed to the minimum, its displacement (x) is still not 0, and thus it has some potential energy.
• The spring is compressed to the minimum and is placed at a height:
• Here, the potential energy due to compression will exist as discussed in case 2.
• Additionally, there will be potential energy due to the height.
• The higher the spring is placed, the greater the potential energy due to the height.
• The spring is pulled out to the maximum and is placed at a height:
• When the spring is pulled out to the maximum, it is extended from its equilibrium position.
• This will lead to a positive displacement (x), resulting in potential energy due to extension.
• The potential energy due to the height will be greater.

Thus, the spring will have the least potential energy in case 1, where it is neither compressed nor pulled out and is lying on the ground.

The correct answer is Snake, Frog, Cockroach.

Concept:

Cold-blooded organisms

• Cold-blooded organisms are also known as ectothermic organisms.
• Cold-blooded organisms are incapable of controlling their body temperature.
• They rely on external sources of heat to regulate their body temperature.
• Snakes, frogs, lizards, earthworms, and cockroaches are cold-blooded organisms.

Warm-blooded organisms

• Warm-blooded organisms are also known as endothermic organisms.
• These organisms can internally regulate their body temperature, often maintaining a relatively constant temperature regardless of the external environment.
• Parrots, dolphins, pigeons, humans, and Whales are warm-blooded organisms.

Explanation:

Option 1: Lizard, Parrot, Dolphin

• Lizard: Lizards are cold-blooded reptiles that cannot internally regulate their body temperature and rely on external sources of heat.
• Parrot: Parrots are warm-blooded birds that can maintain a relatively constant body temperature through metabolic processes.
• Dolphin: Dolphins are warm-blooded marine mammals that have the ability to maintain a constant body temperature despite the surrounding water temperature.

Option 2: Pigeon, Humans, Lizard

• Pigeon: Pigeons are warm-blooded birds that can maintain a relatively constant body temperature through metabolic processes.
• Humans: Humans are warm-blooded mammals with the ability to regulate their body temperature internally.
• Lizard: As mentioned earlier, lizards are cold-blooded reptiles that rely on external sources of heat for temperature regulation.

Option 3: Snake, Frog, Cockroach

• Snake: Snakes are cold-blooded reptiles that rely on external heat sources to regulate their body temperature.
• Frog: Frogs are cold-blooded amphibians that have body temperatures influenced by the environment.
• Cockroaches: Cockroaches are cold-blooded insects that do not have the ability to regulate their body temperature internally.

Option 4: Whale, Frog, Earthworm

• Whale: Whales are warm-blooded marine mammals that can maintain a constant body temperature even in cold water.
• Frog: As mentioned earlier, frogs are cold-blooded amphibians that have body temperatures influenced by the environment.
• Earthworm: Earthworms are cold-blooded invertebrates that have body temperatures tied to their surroundings.

Thus, Snakes, frogs, and cockroaches are cold-blooded organisms.

The correct answer is kg m2/s2, Ws.

Explanation:

kg m²/s² and Ws:

• These units are equivalent and measure the same physical quantity. kg m²/s² is a unit of energy (joule, J), and Ws (watt-second) is also a unit of energy.
• They both measure energy.

kg m/s² and Nm:

• These units are not exactly the same, but they are related.
• kg m/s² represents force (Newton, N), and Nm represents torque, which is a type of rotational force.
• While they have a connection, they don't directly measure the same quantity.

kg m/s² and Ws:

• These units are not the same and measure different quantities.
• kg m/s² represents force (Newton, N), while Ws (watt-second) is a unit of energy.

kg m²/s² and J/s:

• These units are not the same and measure different quantities.
• kg m²/s² represents energy (joule, J), whereas J/s represents power (watt, W), which is the rate of energy transfer.

Thus, the pair of units that are used for measuring the same physical quantity are kg m²/s² and Ws.

The correct answer is the zygote, embryo, uterus, foetus.

Explanation:

• During fertilization, the sperm cells from the father and the egg cells of the mother fuse to form a zygote.
• This is the initial cell that carries the combined genetic material of both parents.
• After fertilization, the zygote begins to undergo multiple cell divisions through a process called cleavage.
• These divisions result in a ball of cells known as the embryo.
• The embryo is the early stage of development when the cells are beginning to differentiate and give rise to various tissues and organs.
• The developing embryo travels down the fallopian tube and eventually gets embedded in the wall of the uterus.
• The uterus provides the environment necessary for the embryo's further development and growth throughout pregnancy.
• As the embryo continues to develop, it eventually transitions into the fetal stage.
• The foetus is the stage of development where all major body parts are formed and can be identified.
• Hence, W, X, Y, Z respectively are -
• W: Zygote (fusion of sperm and egg)
• X: Embryo (ball of cells during early development)
• Y: Uterus (where embryo/foetus develops)
• Z: Foetus (stage with identifiable body parts)

Thus, in the above paragraph W, X, Y, and Z respectively are zygote, embryo, uterus, foetus.

The correct answer is Zn + CaSO4.

Concept:

• A metal can displace the other metal from its salt solution only when the other metal is placed above the first metal in the electrochemical series.
• To determine whether a reaction is feasible or not, we need to consider the reactivity of the elements and the solubility of the compounds involved.

Explanation:

The reactivity series of metals, also known as the activity series, is a list of metals ranked in order of decreasing reactivity. Here's one such list, starting with the most reactive:

Potassium (K)

Sodium (Na)

Calcium (Ca)

Magnesium (Mg)

Aluminum (Al)

Zinc (Zn)

Iron (Fe)

Lead (Pb)

Hydrogen (H) - Not a metal, but it's often included in the series for comparison

Copper (Cu)

Silver (Ag)

Gold (Au)

Platinum (Pt)

PbSO4 + Fe

• This reaction is feasible.
• Iron (Fe) is more reactive than lead (Pb), so it can displace lead from lead sulfate (PbSO4), resulting in the formation of iron sulfate and lead.

Al + CuSO4

• This reaction is feasible.
• Aluminum (Al) is more reactive than copper (Cu), so it can displace copper from copper sulfate (CuSO4), resulting in the formation of aluminum sulfate and copper.

Mg + ZnSO4

• This reaction is feasible.
• Magnesium (Mg) is more reactive than zinc (Zn), so it can displace zinc from zinc sulfate (ZnSO4), resulting in the formation of magnesium sulfate and zinc.

Zn + CaSO4

• This reaction is not feasible.
• Zinc (Zn) is less reactive than calcium (Ca), so it can not displace calcium from calcium sulfate (CaSO₄).

The correct answer is A, B and D.

Explanation:

A. Distance - time graph of a body at rest:

• This graph would be a straight horizontal line at the distance axis, showing that the body is not changing its position over time.

B. Velocity - time graph of a body moving with uniform acceleration:

• This graph would be a straight line inclined upwards or downwards, indicating a constant change in velocity over time due to uniform acceleration.

C. Distance - time graph of a body moving with uniform acceleration:

• This graph would be a curve, not a straight line. The slope of the curve would change, indicating the changing velocity due to uniform acceleration.

D. Velocity - time graph of a body at rest:

• This graph would be a straight horizontal line at the velocity axis, showing that the body's velocity remains constant (which is zero in this case) over time.

Hence, the correct options are A, B, and D.

Explanation:

Based on the given information about the densities of liquids A, B, C, and D:

Liquid A has a density of 785 kg/m³.

Liquid B has a density of 820 kg/m³.

Liquid C has a density of 924 kg/m³.

Liquid D has a density of 1020 kg/m³.

 1) An object that floats in liquid C may or may not float in liquid A.

This statement is correct. Whether an object floats in liquid A or not depends on its density compared to liquid A, and the density of liquid C doesn't provide enough information to make a definite conclusion about its floatation in liquid A.

2) An object that floats in liquid A may or may not float in liquid B.

 This statement is correct. Whether an object floats in liquid B or not depends on its density compared to liquid B, and the density of liquid A doesn't provide enough information to make a definite conclusion about its floatation in liquid B.

3) An object that floats in liquid B surely floats in liquid A.

This statement is incorrect. Liquid A has a lower density (785 kg/m³) than liquid B (820 kg/m³), so an object that floats in liquid B does not necessarily float in liquid A. It may sink in liquid A.

4) An object that floats in liquid C surely floats in liquid B.

This statement is incorrect. Liquid B has a higher density (820 kg/m³) than liquid C (924 kg/m³), so an object that floats in liquid C does not necessarily float in liquid B. It may sink in liquid B.

Therefore, the correct statements are 1) and 2).

Note:

But here we can choose option 1 as it is more accurate

The correct answer is (A) is false, but (R) is true.

Concept:

Scalar quantity

• A scalar quantity is a physical quantity that only has magnitude.
• It has no direction.
• A scalar quantity is represented by a number only.
• For example – Energy, volume, mass, electric current, distance, speed, temperature, area.

Vector quantity

• A vector quantity is a physical quantity that has both a magnitude and a direction.
• For example – Torque, weight, acceleration, displacement, force, momentum, angular velocity.

Explanation:

• Electric current does have a direction: It flows from a point of high potential to a point of low potential. However, it is still considered a scalar quantity and not a vector quantity. Because vector quantities behave and are added together.
• The addition of vector quantities follows specific rules called vector addition.
• When two vector quantities are added, the result depends not just on their magnitudes but also on their directions.
• For instance, if you're traveling north and then east, your resulting direction is different than if you traveled east and then north.
• In contrast, the magnitude of the current at any point in a circuit is the sum of the currents through that point, regardless of their directions.
• This is the main rule for adding scalar quantities, not vectors.
• Thus, despite having a direction, electric current does not abide by the laws of vector addition and is therefore considered a scalar quantity.

This is why statements, (A) Electric current is a vector quantity is false, and (R) Electric current is a quantity having magnitude as well as direction is true.

 Mistake PointsCurrent is a vector because it has a magnitude and a direction. But the thing is a vector always obeys the law of addition of vectors. Since the current doesn't obey it and it follows algebraic addition, currents are scalar.