Mathematics and Science PYQs

Calculation:

For the equation y = 7x - 2

To find the number of solutions we need to calculate the 

Substituting x = 0

y = -2

Similarly,

for y = 0, x = 2/7

The given equation y = 7x - 2 is similar to equation of straight line y = mx + c,

As we know that, a straight line has an infinite number of solutions;

∴ y = 7x - 2 has an infinite number of solutions.

aⁿ - bⁿ is divisible by a - b for all n.

aⁿ - bⁿ is divisible by a + b if n is Even.

aⁿ + bⁿ is divisible by a + b if n is Odd.

(49)¹⁵ - (1)¹⁵, here n = 15 i.e. odd

So it will be divisible by 49 - 1 = 48

(49)¹⁵ can also be written as (7²)¹⁵ = (7)³⁰

∴ (49)¹⁵ - (1)¹⁵ = (7)³⁰ - (1)³⁰ 

So it will also be divisible by 7 + 1 = 8

Concept Used:

Calculation:

cos135° = cos(45 + 90) = - cos45° = - 1/√2

sin210° = sin(180 + 30) = - sin30° = - 1/2

cos150° = cos(90 + 60) = - sin60° = - √3/2

sin270° = sin(90 + 180) = - sin90 = - 1

cos315° = cos(360 - 45) = cos45° = 1/√2

Put the above value in equation:

{(-1/√2) × (-1/2) × (-√3/2)} / {(-1) × (1/√2)}

⇒ (- √3 / 4√2) / (- 1/ √2)

⇒ √3 / 4

∴ The required simplified value is √3/4.

Let the two odd numbers be x and x + 2

According to question:

x( x + 2) = 6723

x² + 2x = 6723

x² + 2x - 6723 = 0

So x = 81 and x + 2 = 83

Smaller number = 81

Square root of 81 = 9

GIVEN:

x2 + px - q      ----(1)

5x2 - 3px - 15q      ----(2)

(x - 3) is a factor 

CALCULATION:

As (x - 3) is a factor

On substituting x = 3, the equations (1) and (2) will be 0

Now, equation (1) becomes

9 + 3p - q = 0      ----(3)

Similarly, equation (2) becomes

45 - 9p -15q = 0      ----(4)

On solving (3) and (4) simultaneously, we get

∴ p = -5/3, q = 4

Given:

Hemisphere and a cone is having the same base and the same height

Formula used:

The curved surface area of hemisphere = 2πr²

Curved surface area of cone = πrl = πr(√r² + h²)

Calculation:

Given that base is also the same, so their radius will also be the same.

Also, the height of the hemisphere is equal to its Radius = r

Height of both hemisphere and cone = r or r = h

Curved surface area of cone = πrl = πr(√r2 + h2) = πr(√r² + r²) = πr√2 × r

∴ Ratio = 2πr² : πr√2 × r = √2 : 1

Calculation:

Let CP = Rs. 100

Profit = Cost Price + Profit on CP

15% gain = 100 + 15% of 100 = Rs. 115 

15% loss = 100 - 15% of 100 = Rs. 85

According to the question:

(115 - 85) unit = Rs. 450 (given)

∴  1 unit = Rs. 15

∴ Cost price = 100 × 15 = Rs. 1500

Calculation:

In isosceles triangle, 2 sides are equal,

 ∠ABC = 90°, then AB = BC  and AC will be hypotenuse.

By Pythagoras theorem

∴ AC² = AB² + BC² = 2AB²

(AC)² = 2(AB)²

Calculation:

Time taken to cover (4/5)th of the track = m minutes

Time taken to cover 1 revolution = m × (5/4) = 5m/4 minutes

4/5 part of the circular track is covered in m minutes.

Total Part = 5 unit

Part covered in m minutes = 4 unit

Efficiency is inversely proportional to the Time taken when the amount of work done is constant.

∴ Time taken to cover entire track is 5m/4 minutes

Alternate Method

Time taken to cover (4/5)th of the track = m minutes

Time taken to cover 1 revolution = m × (5/4) = 5m/4 minutes

∴ Time taken to cover entire track is 5m/4 minutes

Given:

x + y = 2z

Calculation:

x/(x - z) + z/(y - z)

⇒ (xy - xz + zx - z²)/(x - z)(y - z)     (Taking LCM of (x - z)(y - z))

⇒ (xy - z²)/xy - z(x + y) + z²

⇒ (xy - z²)/xy - z(2z) + z² (∵ x + y = 2z)

⇒ (xy - z²)/(xy - z²) = 1

The correct answer is Vitamin D.

• Vitamin D
• It is a hormone produced by the kidneys.
• The liver and kidneys convert vitamin D into the active hormone, which is called calcitriol.
• It primarily helps in controlling the concentration of calcium in the blood.
• It is also vital for the development of strong bones.

Additional Information

The correct answer is Monera.

Important Points

• The Monera Kingdom is classified into three sub-kingdoms- Archaebacteria, Eubacteria, and Cyanobacteria.
• Blue-green algae are cyanobacteria.
• They are found in the aquatic region.
• Characteristics of Monera-
• These are unicellular organisms.
• They reproduce asexually by binary fission or budding.
• It lacks organelles like mitochondria, lysosomes, plastids, Golgi bodies, endoplasmic reticulum, centrosome, etc.

Additional Information

• Fungi are eukaryotic organisms that include microorganisms such as yeasts, moulds, and mushrooms. 
• Fungi store their food in the form of starch.
• Kingdom Plantae includes all the plants. 
• They reproduce asexually by vegetative propagation or sexually.
• They make their own food hence are called autotrophs.
• Protists are eukaryotic organisms that are neither plants nor animals or fungi. 
• Most protists live in water, damp terrestrial environments, or even as parasites.

The correct answer is Blood.

Important Points

• Connective tissues are composed of a matrix consisting of living cells and a non-living substance, called the ground substance.
• It provides support, binds together, and protects tissues and organs of the body.
• Blood is a connective tissue because it has a matrix.
• ​The living cell types are red blood cells and white blood cells.
• The fluid portion of whole blood, its matrix, is commonly called plasma.

Additional Information

• The kidneys are bean-shaped organs in the renal system.
• It helps in regulating and filtering minerals from blood.
• It also helps in filtering waste materials from food and toxic substances.
• The lungs are the centre of the respiratory system.
• The primary function is to help oxygen enter the red cells in the blood.
• The ovaries produce and release eggs into the female reproductive tract at the mid-point of each menstrual cycle.
• They also produce the female hormones oestrogen and progesterone.

The correct answer is LH.

Key Points

• Ovulation is the release of a mature egg from one of the ovaries. 
• After the egg is released, it travels down the fallopian tube, where fertilization by a sperm cell may occur.
• LH-Luteinizing Hormone.
• When the level of estrogen is sufficiently high, it produces a sudden release of LH. It usually takes place around day thirteen of the cycle.
• LH at the peak during ovulation.
• Luteinizing hormone is produced and released in the anterior pituitary gland. It stimulates the ovaries to produce oestradiol. If fertilization occurs, luteinizing hormone will stimulate the corpus luteum, which produces progesterone to sustain the pregnancy.

Additional Information

• Estrogen
• This hormone is responsible for female physical features and reproduction.
• The ovaries, adrenal glands, and fat tissues produce estrogen.
• Both female and male bodies have this hormone, but females create more of it.
• It contributes to cognitive health, bone health, the function of the cardiovascular system, and other essential bodily processes.
• FSH
• The full form is Follicle-stimulating hormone.
• It is synthesized and secreted by the gonadotropic cells of the anterior pituitary gland.
• It regulates the development, growth, pubertal maturation, and reproductive processes of the body.
• Progesterone
• It is a female hormone that regulates the menstrual cycle.
• It is released by the corpus luteum in the ovary.
• It helps the uterus thicken in anticipation of receiving a fertilized egg.

The correct answer is Fish.

Key Points

• The genus Hippocampus contains more than 70 species found mostly in shallow tropical and temperate marine and estuarine waters throughout the world.
• Seahorse is the name given to 54 species of small marine fishes in the genus Hippocampus.
• Their habitats include coral reefs, mangroves, seagrass beds, and estuaries.

Additional Information

• Coelenterata
• Phylum Coelenterata is a group of aquatic, or marine organisms.
• They are diploblastic animals, in which, the body is made up of two layers of cells: Ectoderm and Endoderm.
• The nervous system and the circulatory system is absent.
• Coelenterates are classified into three different classes: Anthozoa, Hydrozoa, and Scyphozoa.
• Protochordate
• It is an informal category of organisms to describe the invertebrates that are closely related to vertebrates.
• Their body is bilaterally symmetrical, triploblastic, and coelomated.

The correct answer is Froth floatation process.

Key Points

• Froth floatation
• It is a process that selectively separates materials based upon whether they are water-repelling (hydrophobic) or have an affinity for water (hydrophilic).
• If the minerals are of Hydrophobic nature then only can get attracted toward froth and not with water.
• Ores containing multiple metals such as lead, copper and zinc can be selectively extracted by using froth floatation.
• Froth flotation can primarily be done under two common techniques:
• Direct floatation technique- The mineral sticks to the air bubble and the remaining gangue settles down at the bottom.
• Reverse floatation technique- The gangue sticks to the air bubble and ore particle settle down in the bottom.

Important Points

• Copper is extracted from its principal ore Copper pyrites (CuFeS₂).
• The ore is concentrated by froth flotation process.
• Then, the concentrated ore is roasted in a furnace where reactions remove sulphur and Arsenic gases:
• S+O₂ ​ →SO₂​ and 4As+3O₂ ​ ⟶2As₂ ​O₃ ​
• The copper pyrite gets converted to a mixture of Cu₂S and FeS by the reactions given below:
• 2CuFeS₂ ​ +O₂ ​ ⟶ Cu₂ ​S+2FeS+SO₂ ​
• These Sulphides are then oxidized and smelted. Hence, copper is extracted.

The correct answer is 1.

Key Points

• 2-methyl propane is  H₃C−CH(CH₃)−CH₃.
• A methyl group substitutes the chain at C₂.
• There is one "three-carbon" longest chain in 2-methyl propane.
• 2-methyl propane is also known as Isobutane.
• It is a colourless, odourless gas.
• It is obtained by isomerization of butane.
• It is also the principal feedstock in alkylation units of refineries.

The correct answer is 2-methoxy propane.

Key Points

• The International Union of Pure and Applied Chemistry (IUPAC) is an international federation of National Adhering Organizations that represents chemists in individual countries.
• It is best known for its works standardizing nomenclature in chemistry.
• The IUPAC nomenclature of organic chemistry is a method of naming organic chemical compounds as recommended by IUPAC.
• It is published in the Nomenclature of Organic Chemistry.
• 2-methoxy propane is an ether compound having methyl and isopropyl as the two alkyl groups. The molecular formula is C₄H₁₀O.

Additional Information

• The molecular formula of 2-methoxy-2-methyl ethane is C₈H₁₈O.

• The molecular formula of 1-methoxy-1-methyl ethane is C₅H₁₂O₂.

• The molecular formula of Isopropyl methyl ether is C4H10O. But, it cannot be the answer as it is not the IUPAC name of the compound given.

The correct answer is Si.

Important Points

• Catenation is the binding of an element to itself through covalent bonds to form chain or ring molecules.
• Carbon is the most common element that exhibits catenation.
• It can form long hydrocarbon chains and rings like benzene.
• Please note that the smaller the atom, the stronger the tendency of catenation.
• All the elements of the carbon family or group 4 family exhibit catenation property.
• The carbon family includes carbon (C), silicon (Si), germanium (Ge), tin (Sn), lead (Pb), and flerovium (Fl).
• The ability of an element to catenate is primarily based on the bond energy of the element to itself, which decreases with more diffuse orbitals.
• The bond length decreases across a row in the periodic table and increases down a group.
• The bond energy of S-S is 266 Kj/mol.
• The bond energy of Si-Si is 340 Kj/mol.
• The bond energy of P-P is 54 Kj/mol.
• The bond energy of N-N is 160 Kj/mol.

Additional Information

• Catenation tendency decreases down the group.
• This happens because the atomic size increases down the group and the strength of the covalent bond decreases.

The correct answer is A, B and C.

Key Points

• Mole fraction is the total number of moles of a solute divided by the total number of moles in a solution.
• It is given that nx mole of substance x (solute) was dissolved in ny mole of substance y (solvent).
• Mole fraction of x = Total number of moles of substance x /(Total number of moles of substance x + Total number of moles of substance y)
• Mole fraction of x = n_x/(n_x + n_y)
• Mole fraction of y = Total number of moles of substance y /(Total number of moles of substance x + Total number of moles of substance y)
• Mole fraction of y = n_y/(nx + ny)
• The total mole fractions = nx/(nx + ny) + ny/(nx + ny) = (nx + ny) /(nx + ny)  = 1
• Hence, the total of mole fractions of substance x and y remain 1.

 

• The statement in option D is incorrect since the total number of moles in a solution is not considered in denominator.