Mathematics and Science PYQs

Concept:

When two sides of a triangle are equal then, opposite angles are also equal

The Linear Pair Theorem states that two angles that form a linear pair are supplementary; that is, their measures add up to 180 degrees

Calculation:

AB = AC, hence ∠ABC = ∠ACB  (∵ Two sides of a triangle are equal then, opposite angles are also equal)

∠ACB + ∠ACD = 180° (Linear Pairs sum = 180°)

∠ACB = 180° - 105° = 75° (∵ ∠ACD = 105° ) 

As  ∠ABC = ∠ACB

∴ ∠ACB = 75° 

In triangle ABC:

∠ABC + ∠ACB + ∠BAC = 180° 

⇒ 75° + 75° + ∠BAC = 180° 

∴  ∠BAC = 180° - 150° = 30°

⇒ (30/100) × (B - A) = (18/100) × (A + B)

⇒ 5(B - A) = 3(A + B) ⇒ 5B - 5A = 3A + 3B

⇒ 2B = 8A ⇒ A/B = 1/4

∴ A: B = 1: 4

Given:

Ratio of Two numbers = 3 : 4

HCF × LCM = 2028

Concept:

HCF: The greatest common divisor of two or more positive integers happens to be

the largest positive integer that divides the numbers without leaving a remainder.

LCM: Least common multiple or lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b.

​Formula used:

Product of two numbers = Product of their HCF and LCM

Calculation:

Let the numbers be 3x, 4x respectively

HCF of 3x and 4x = x

LCM = 3 × 4 × x = 12x

Now, HCF × LCM = 2028

⇒ x × 12x = 2028

⇒ 12x² = 2028

⇒ x² = 169

⇒ x = 13

Sum of numbers = 3x + 4x = 7x = 7 × 13 = 91

Given:

a * b = (a × b) + b

Concept:

Now a = 5 and b = 7

Calculation:

 5 * 7 = (5 × 7) + 7 = 42

Mathematics is the study of numbers, shape, quantity, and patterns. Mathematics is the ‘queen of all sciences and its presence is there in all the subjects. It acts as the basis and structure of other subjects.

Teaching methods of mathematics include problem-solving, lecture, inductive, deductive, analytic, synthetic, and heuristic or Discovery methods. The teacher adopts any method according to the needs and interests of students.

Geometry is a branch of mathematics related to the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. 

• Geometry begins with the introduction of lines, line segments, and rays.
• The display method is the best method of teaching geometry.
• In this method, the teacher is used to explain the concept using concrete objects. So, it is easy to explain geometry using this method.

Additional Information

• Drill Method: It is a method that ensures the acquisition of knowledge by practicing and repeating the concept again and again.
• Play Method: Play method of teaching is a method that focuses on making children learn through enjoyable activities and play.

Hence, it becomes clear that display method is the best method of teaching geometry.

Given:

Initial profit = 20%

Decrease in Profit = 313%313%

Formula used:

Profit% = (Profit/CP) × 100

Calculation:

Let initial cost price be Rs. x

Profit of 20% = x + 20% of x = x/5

Selling price = x + x/5 = 6x/5

Given that cost price is increased by Rs. 50 and selling price is increased by Rs. 30

⇒ Cost price = x + 50

⇒ Selling Price = (6x/5) + 30

Profit = SP – CP = (6x/5 + 30) - (x + 50)

⇒ (x/5) – 20

Profit Percentage = [(x/5 - 20)/(x + 50)] × 100

Given that profit is decreased by (10/3)%

⇒ 20% - (10/3)% = (50/3)%

Profit percentage =

[(x/5 - 20)/(x + 50)] × 100 = 50/3

⇒ (x/5 - 20) / (x + 50) = 1/6

⇒ 6x/5 - 120 = x + 50

⇒ x/5 = 170

⇒ x = 850

∴ Original cost price is Rs. 850

Concept:

Minimum value = (4ac – b²)/4a

Calculation:

(x - 2)(x - 9)

⇒ x² - 11x + 18

We know ax² + bx + c = 0

Comparing this with above equation

⇒ a = 1, b = -11 and c = 18 

Minimum value = (4ac – b²)/4a

⇒ (4 × 1 × 18 - (-11)²)/(4 × 1)

⇒ (72 - 121)/4 

⇒ -49/4

∴ Minimum value of (x - 2)(x - 9) = -49/4

Additional Information

Proof of minimum value formula:

We have f(x) = ax² + bx + c

f(x) = ax² + bx + c, such that a > 0

Consider its first derivatives:

f′(x) = 2ax + b

and the second derivative

f′′(x) = 2a

Since, a > 0 

f″(x) is always positive.

f(x) will achieve minima when f′(x) = 0

2ax + b = 0

x = −b/2a

Putting x = −b/2a in the original equation we get:

f(x) = ax² + bx + c

⇒ a(−b/2a)² + b(b/2a) + c

⇒ b²/4a – b²/2a + c

⇒ (4ac – b²)/4a

Concept:

a2 - b2 = (a + b) (a - b)

Calculation:

a² - b² = (a + b) (a - b)

so, (272)² - (128)² = (272 + 128) (272 - 128)

⇒ (400 × 144) = 57600

√57600 = 240 (∵ 242 = 576)

∴ Square root of (272)² - (128)² = 240

Explanation:

Surface tension: 

• Surface tension is the property by virtue of which liquid tries to minimize its free surface area.
• In a spherical shape, the surface area is minimum and for this reason, the raindrops are spherical.
• Surface tension is measured as the force acting per unit length of an imaginary line drawn on the liquid surface.

\(Surface;tension = \frac{{Force}}{{length}}\)

Two factors can affect the surface tension of a liquid. The factors are-

 

• Temperature: If temperature increases then the surface tension of a liquid decreases.
• Soluble Impurities: In the case of less soluble impurities, the surface tension decreases. But, for highly soluble impurities in the liquid the surface tension increases.

Additional Information

• Surface tension is the result of the unbalanced cohesive forces among the liquid molecules, which always tries to make the surface area of the liquid droplet minimum.
• Hence due to exertion of surface tension by the molecules, water droplets always take a round shape whenever they are allowed to move freely, and thus because of this we see a nearly spherical drop of rain.

Raindrops are spherical in shape because each drop tends to acquire minimum surface area due to surface tension, and for a given volume, the surface area of the sphere is minimum.

The correct answer is Carbon dioxide.

Explanation:

Greenhouse Effect: The solar energy trapped by the earth atmosphere and radiate it slowly so to cover our earth with a warm blanket. This is the natural process of the greenhouse effect on earth to maintain its temperature and makes the earth perfect for life.

Greenhouse gases are carbon dioxide, other greenhouse gases are methane, water vapour, nitrous oxide, CFCs and ozone. 

Values of radiation absorbed by the greenhouse gases are as follow:

 

• From the above table it is clear that carbon dioxide absorbs the maximum radiation in the earth’s atmosphere

Important Points

• Global warming: If the amount of greenhouse gases crosses by delicate proportion, the natural greenhouse balance may get disturbed and there is a continuous rise in temperature of the earth by trapping the solar heat continuously. This creates the overall effect of Global warming.

Hence, in the above options, the correct answer is Carbon dioxide.

Explanation:

Mineral: A mineral is an element or chemical compound that is normally crystalline and that has been formed as a result of geological processes.

Some common minerals/ores are described below:

 

Gypsum:

• Plaster of Paris changes into gypsum on mixing with water and is used for decoration.
• Plaster of Paris, commonly called as P.O.P is Calcium sulphate hemihydrate.
• The chemical formula of Plaster of Paris is CaSO4.1/2 H2O.
• Crystals of gypsum are produced when Plaster of Paris is mixed with water.

 

CaSO4.1/2H2O + H2O → CaSO4.2H2O (Gypsum).

Here,

Bauxite → Aluminium ore

Hematite → Iron ore

Limonite → Iron ore

Gypsum → Plaster or pairs + water

So, Gypsum is not a mineral.

Explanation:

• Bromine the only non-metallic element that is in liquid state at room temperature.
• It is a member of the halogen elements (Group 17) of the periodic table.
• Atomic number of Bromine is 35.
• Bromine is a highly volatile reddish brown liquid at room temperature.
• Bromine is found in nature dispersed throughout Earth’s crust only in compounds as soluble and insoluble bromides.

Bromine: 

• Atomic number - 35
• Atomic weight - 79.909
• Melting point - − 7.2 °C.
• Boiling point - 59 °C.

Explanation:

Henri Becquerel (1852-1908) observed that there are certain elements which emit radiation on their own and named this phenomenon as radioactivity and the elements known as radioactive elements. 

• This field was developed by Marie Curie, Piere Curie, Rutherford, and Fredrick Soddy.
• It was observed that three kinds of rays i.e., α, β- and γ-rays are emitted.
• For this property, spontaneous radiation happens in an unstable high atomic number element; for example:- Uranium, Polonium, etc.
• The SI unit of Radioactivity is Becquerel(Bq) and another unit of it is Curie(Ci).
• 1 Becquerel is equal to 1 decay per second.
• ​E.g. Radium, Uranium, Polonium, etc.

The correct answer is Acid rain.

• Stone cancer is a phenomenon in which the discoloration of stones/marbles occurs due to acid rain.
• It is also known as Marble cancer.
• The pollutants present in acid rain reacts with the stone to change their colour.

Additional Information

• The best example of stone cancer can be seen on the Taj Mahal.
• An illustrative picture shows the stone/marble cancer effect on the Taj Mahal.
• The left side of the picture shows the image of the Taj before the stone cancer, and the right side shows the image after the effect of the stone cancer.

Explanation:

• The Brine is the solution of Sodium chloride (NaCl) and water.
• They occur naturally on the Earth’s surface in salt lakes, or even the benthic zone of the oceans.
• They occur due to the evaporation of ground saline leaving behind little water with high concentration of Sodium Chloride.
• Applications include meat packing, pickling, steel quenching, heat-transfer medium.

Important Points

The chemical name of bleaching powder is

• Calcium hypochlorite is commonly known as Bleaching powder.
• Its formula is Ca(ClO)2.
• It is a pale yellow coloured powder with a strong smell of chlorine.

Explanation:

Acetylene gas (C₂H₂) is commonly used in some countries to artificially ripen fruits.

When calcium carbide comes in contact with moisture, it produces acetylene gas (C₂H₂) which is similar in its effects to the natural ripening agent, ethylene (C2H₄). Acetylene accelerates the ripening process.

• Fruit ripening agents can be used to ripen fruits artificially and to provide fruits with the desired colour and taste within a short time.
• Calcium carbide, ethephon, and ethylene have been used as artificial ripening agents.
• Calcium carbide is alkaline in nature.

Explanation:

Nitrogen and Oxygen are the main components in the Earth's atmosphere by volume as these two gases together account for 99% of the atmosphere.

• Nitrogen(N₂): A colourless and tasteless gas, accounts for approx 78% of the atmosphere. It is a major component of chlorophyll and amino acids, the building blocks of proteins.
• Oxygen(O₂): A colourless, odourless and tasteless gas, accounts for approx 21% of the atmosphere and makes life possible.

 

While other gases like-

The atmospheric composition of gases are as follow

Explanation:

Ozone: A layer of ozone is present in the stratosphere of Earth and prevents the UV rays from reaching the Earth's surface. It is an allotrope of Oxygen.

• Ozone, or tri-oxygen, is an inorganic molecule with the chemical formula O₃.
• It is a pale blue gas with a distinctively pungent smell.
• Ozone is an excellent oxidizing agent as it breaks down into oxygen gas and nascent oxygen. { O₃ → O2 + [O] }
• It is an allotrope of oxygen that is much less stable than the diatomic allotrope O₂, breaking down in the lower atmosphere to O₂ or di-oxygen

Concept:

• Kinetic energy (K.E): The energy possessed by a body by the virtue of its motion is called kinetic energy.

The expression for kinetic energy is given by:

\(KE = \frac{1}{2}m{v^2}\)

Where m = mass of the body and v = velocity of the body

• Momentum (p): The product of mass and velocity is called momentum.

Momentum (p) = mass (m) × velocity (v)

The relationship between the kinetic energy and Linear momentum is given by:

As we know,

 \(KE = \frac{1}{2}m{v^2}\)

Divide numerator and denominator by m, we get

\(KE = \frac{1}{2}\frac{{{m^2}{v^2}}}{m} = \frac{1}{2}\frac{{;{{\left( {mv} \right)}^2}}}{m} = \frac{1}{2}\frac{{{p^2}}}{m};\) [p = mv]

\(\therefore KE = \frac{1}{2}\frac{{{p^2}}}{m};\)

\(p = \sqrt {2mKE} \)

 

Explanation:

So, from the given condition if kinetic energy is denoted as KE and linear momentum is denoted as p by using the above relation, we can say that:

⇒ \(KE = \frac{{{p^2}}}{{2m}}\) 

This is the correct option.

Concept:

• Resistance: The obstruction offered to the flow of current is known as the resistance. It is denoted by R.
• When two or more resistances are connected one after another such that the same current flows through them then it is called resistances in series.
• Resistance can be placed with two combination:

 

 

Effective resistance: The overall resistance or combination of series and parallel gives effective resistance.

 

Calculation:

Given:

R₁ = 4 Ω, R₂ = 10 Ω and R₃ = 20 Ω 

\(⇒ \text{ }!!~!!\text{ }\frac{1}{{{R}_{eq}}}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}}\)

\(⇒ \frac{1}{{R}_{eq}}=\frac{1}{4}+\frac{1}{{{10}}}+\frac{1}{20}⇒ {{R}_{eq}}=~2.5~\text{ }!!Ω!!\text{ }\)

Now, 

Ohm's Law says V = I × R

V = 15 V, R = 2.5 Ω then

\(\Rightarrow I = \frac{V}{R} = \frac{{15;V}}{{2.5;{\rm{\Omega }}}} = 6;A\)

The correct option is 6 Ampere.

 

Important Points

• When two resistances are given we can find the effective resistance on the parallel circuit by using \(⇒ {{R}_{ef}}=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}}\)