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Concept:

Each angle of n-sided polygon = ((n - 2) × 180)/n

Number of diagonals of n - sided polygon = n(n - 3)/2

Calculation:

Each interior angle = 150° 

150° = (n - 2) × 180)/n

⇒ 6n - 12 = 5n

n = 12 = Total side of the polygon

∴ Number of diagonals of n - sided polygon = n(n - 3)/2 = 108/2

∴ Number of diagonals of n - sided polygon = 54

Given:

Sum of ages of Mallika and Uttam is 40 years

Five years ago, the ratio of their ages = 7: 8

Calculation:

Let the ages of Mallika and Uttam 5 years ago be  7x and 8x

Present age ratio = (7x + 5): (8x + 5)

According to the question,

Sum of ages of Mallika and Uttam is 40 years  

⇒ 7x + 5 + 8x + 5 = 40

⇒ 15x = 30 or x = 2 years

⇒ Age of Mallika 5 years ago = 7x = 14

∴ Present age of Mallika = 14 + 5 = 19 years.

Short Trick:

Given that 5 years ago, Age of Mallika was 7x

∴ The present age of Mallika will be that option from which if we subtract 5, it will be multiple of 7 i.e. (19 - 5) = 14 which is multiple of 7.

Hence, 19 years is the correct answer.

Concept: 

(a + b)(a − b) = a² – b²

Calculation:

In the product 9547 × 9545 we find that the difference between 9547 & 9545 is 2.

So after 9545 and before 9547, there is one integer 9546.

So we can write 9547 = 9546 + 1 and 9545 = 9546 – 1.

So, 9547 × 9545 = (9546 + 1)(9546 – 1) = 9546² – 1².

∴ It is clearly found that if 1 is added to the product of 9547×9545 the result will be a perfect square.

∴ If we add 1 to 9547 × 9545, sum will become perfect square

Given:

Along the stream means downstream, Downstream speed = 8/1 = 8 km/hour

Against the stream means Upstream, Upstream speed = 2/1 = 2 km/hour

Formula Used:

Let the speed of boat be x km/hour and speed of stream be y km/hour

Downstream speed = (x + y)

Upstream speed =  (x - y)

Calculation:

According to question:

 (x + y) = 8       ---- (1)

 (x - y) = 2       ---- (2)

Solving above equation we get:

2x = 10 or x = 5 km/hour 

⇒ y = 8 - 5 = 3 km/hour  

∴ Speed of stream is 3 km/hr

Shortcut Trick

If the speed of the Man/Boat/Ship along the stream is u km/hr and the speed against the current is v km/hr, then

Speed of stream = \((u - v)\over2\) km/hr

Speed of Man/Boat/Ship = \((u + v)\over2\) km/hr

Speed of stream = \((u - v)\over2\) km/hr

⇒ \((8 - 2)\over2\) = 3 km/hr

∴ Speed of stream is 3 km/hr

The correct answer is option 2 i.e. transportation of worn-out sediments and deposition.

Igneous Rocks

• As igneous rocks form out of magma and lava from the interior of the earth, they are known as primary rocks.
• The igneous rocks (Ignis – in Latin means ‘Fire’) are formed when magma cools and solidifies.
• When magma in its upward movement cools and turns into solid form it is called igneous rock.
• The process of cooling and solidification can happen in the earth’s crust or on the surface of the earth.
• Igneous rocks are classified based on texture which depends upon size and arrangement of grains or other physical conditions of the materials.
• If molten material is cooled slowly at great depths, mineral grains may be very large; sudden cooling (at the surface) results in small and smooth grains; intermediate conditions of cooling would result in intermediate sizes of grains making up igneous rocks.
• Granite, gabbro, pegmatite, basalt, volcanic breccia and tuff are some of the examples of igneous rocks.

Sedimentary Rocks

• The word ‘sedimentary’ is derived from the Latin word sedimentum, which means settling.
• Rocks (igneous, sedimentary and metamorphic) of the earth’s surface are exposed to denudational agents, and are broken up into various sizes of fragments.
• Such fragments are transported by different exogenous agencies and deposited.
• These deposits through compaction turn into rocks. This process is called lithification.
• In many sedimentary rocks, the layers of deposits retain their characteristics even after lithification. Hence, we see a number of layers of varying thickness in sedimentary rocks like sandstone, shale etc.
• Depending upon the mode of formation, sedimentary rocks are classified into three major groups:

1. mechanically formed — sandstone, conglomerate, limestone, shale, loess etc. are examples;
2. organically formed — geyserite, chalk, limestone, coal etc. are some examples;
3. chemically formed — chert, limestone, halite, potash etc. are some examples.

Calculation:

Considering the given equation,

25x² - 20x + t²

The equation can be written as 

(5x)² - 2 × (5x) × 2 + t²

⇒ (5x)² - 2 (5x) × 2 + 2² + t² - 2²

⇒ (5x – 2²)² + t² - 2

If t² = 2, then above equation will be (5x – 2²)² which is perfect square:

∴ For t = ± 2

The correct answer is option 3 i.e. Typhoon.

Key Points

• Thunderstorms
• Thunderstorms develop in hot, humid tropical areas like India very frequently.
• The rising temperatures produce strong upward rising winds. These winds carry water droplets upwards, where they freeze, and fall down again.
• The swift movement of the falling water droplets along with the rising air create lightning and sound. It is this event that we call a thunderstorm.
• Cyclones
• Before cloud formation, water takes up heat from the atmosphere to change into vapour. When water vapour changes back to liquid form as raindrops, this heat is released to the atmosphere.
• The heat released to the atmosphere warms the air around. The air tends to rise and causes a drop in pressure. More air rushes to the centre of the storm.
• This cycle is repeated. The chain of events ends with the formation of a very low-pressure system with very high-speed winds revolving around it.
• It is this weather condition that we call a cyclone. Factors like wind speed, wind direction, temperature and humidity contribute to the development of cyclones.
• The centre of a cyclone is a calm area. It is called the eye of the storm.
• A large cyclone is a violently rotating mass of air in the atmosphere, 10 to 15 km high.
• The diameter of the eye varies from 10 to 30 km. It is a region free of clouds and has light winds. Around this calm and clear eye, there is a cloud region of about 150 km in size. In this region there are high-speed winds (150–250 km/h) and thick clouds with heavy rain.
• Away from this region the wind speed gradually decreases. The formation of a cyclone is a very complex process.
• Tornadoes 
• A tornado is a dark funnel shaped cloud that reaches from the sky to the ground.
• Most of the tornadoes are weak. A violent tornado can travel at speeds of about 300 km/h.
• Tornadoes may form within cyclones.

A cyclone is known by different names in different parts of the world

The correct answer is Water pollution.

• The process in which nutrient-enriched water bodies support a dense plant population, which kills animal life by depriving it of oxygen and results in subsequent loss of biodiversity is known as Eutrophication.
• The organic chemicals are another group of substances that cause water pollution.

Key Points

• Petroleum products pollute many sources of water e.g., major oil spills in oceans; other organic substances with serious impacts are the pesticides that drift down from sprays or runoff from lands.
• Various industrial chemicals like polychlorinated biphenyls, (PCBs) which are used as cleansing solvent, detergents and fertilisers add to the list of water pollutants and are suspected to be carcinogenic.
• Nowadays most of the detergents available are biodegradable but the bacteria responsible for degrading biodegradable detergent feed on it and grow rapidly.
• While growing, they may use up all the oxygen dissolved in water and the lack of oxygen kills all other forms of aquatic life such as fish and plants.
• Fertilisers contain phosphates as additives and the addition of phosphates in water enhances algae growth.
• Such profuse growth of algae covers the water surface and reduces the oxygen concentration in water. This leads to anaerobic conditions, commonly with an accumulation of obnoxious decay and animal death.
• Thus, bloom-infested water inhibits the growth of other living organisms in the water body.

The correct answer is option 1 i.e. 20 Hz - 20,000 Hz.

• The audible sound for human beings extends from the frequency range of about 20 Hz to 20000 Hz (one Hz = one cycle/s).
• Children under the age of five and some animals, such as dogs can hear up to 25 kHz (1 kHz = 1000 Hz). The police use high frequency whistles which dogs can hear but humans cannot.
• Infrasonic sound or Infrasound
• Sounds of frequencies below 20 Hz are called infrasonic sound or infrasound.
• If we could hear infrasound we would hear the vibrations of a pendulum just as we hear the vibrations of the wings of a bee.
• Rhinoceroses communicate using infrasound of frequency as low as 5 Hz. Whales and elephants produce sound in the infrasound range. It is observed that some animals get disturbed before earthquakes. Earthquakes produce low-frequency infrasound before the main shock waves begin which possibly alert the animals.
• Ultrasonic sound or Ultrasound
• Frequencies higher than 20 kHz are called ultrasonic sound or ultrasound.
• Ultrasound is produced by dolphins, bats and porpoises. Moths of certain families have very sensitive hearing equipment. These moths can hear the high frequency squeaks of the bat and know when a bat is flying nearby, and are able to escape capture. Rats also play games by producing ultrasound.
• The ultrasound equipment, familiar to us for investigating and tracking many medical problems, works at frequencies higher than 20,000 Hz.
• Hearing Aid 
• As people grow older their ears become less sensitive to higher frequencies and people with hearing loss may need a hearing aid.
• A hearing aid is an electronic, battery operated device that receives sound through a microphone which converts the sound waves to electrical signals.
• These electrical signals are amplified by an amplifier and the amplified electrical signals are given to a speaker of the hearing aid.
• The speaker converts the amplified electrical signal to sound and sends to the ear for clear hearing.

Given:

Length of train = 300 m

Speed of walking man = 3 km/hour 

Train passed the man in time = 33 seconds

Formula used:

Speed = Distance/Time

Concept:

Length of man is neglected

Calculation:

Let the speed of train be x km/hr

Length of train = 300 meters

Relative speed in the same direction = (x - 3) km/hr

According to the question,

33 = 300/[(x - 3) × 5/18]

11x - 33 = 360

⇒ x = 393/11 km/hr = \(35\frac{8}{11}\) km/hr