SSC CGL Previous Paper (Held On: 5 March 2020 Shift 1) (Previous Year Paper)

The pattern followed here is

13 × 5 = 65, 13 × 9 = 117

Similarly,

15 × 5 = 75, 15 × 9 = 135

Hence, (15, 75, 135) is the correct answer.

When paper is unfolded it will appear like the following figure :

Hence, the correct answer is option 1.

According to the question :

If the positional value of a letter is "n", then the code of the letter will be "2n - 1".

H = 15, O = 29, N = 27, E = 9 and Y = 49

Therefore, the code for ‘HONEY’ is ‘152927949’

Hence, ‘152927949’ is the correct answer.

The least possible Venn diagram is as follows:

I. All parakeets are snakes. → True (All parakeets are cuckoos, All cuckoos are rabbits and, All rabbits are snakes.)

II. All snakes are cuckoos. → False(it is possible but not definite)

III. All rabbits are parakeets. → False (it is possible but not definite)

IV. All cuckoos are snakes. → True(All cuckoos are rabbits and All rabbits are snakes.)

Hence, Only conclusions I and IV follow.

The triangles are

MAF, NAG, AFG, LFD, FDH, FOH, OHG, GPE, GHE, DBK, DIB, DHR, DRI, HER, DRE, REJ, EJC, EQC, FHG, ADE, ABC, DRB, and ERC

Hence, there are total 23 triangles in the given figure.

1. 6, 3

6 + 5 × 4 – 24 ÷ 6 = 7 × 4 – 6 + 36 ÷ 3

6 + 5 × 4 – 4 = 7 × 4 – 6 + 12

6 + 20 - 4 = 28 – 6 + 12

22 ≠ 34

<br>

2. 4, 7

3 + 5 × 7 – 24 ÷ 3 = 4 × 7 – 3 + 36 ÷ 6

3 + 5 × 7 – 8 = 4 × 7 – 3 + 6

3 + 35 – 8 = 28 – 3 + 6

30 ≠ 31

<br>

3. 24, 36

3 + 5 × 4 – 36 ÷ 3 = 7 × 4 – 3 + 24 ÷ 6

3 + 5 × 4 – 12 = 7 × 4 – 3 + 4

3 + 20 - 12 = 28 – 3 + 4

11 ≠ 29

<br>

4.  5, 7

3 + 7 × 4 – 24 ÷ 3 = 5 × 4 – 3 + 36 ÷ 6

3 + 7 × 4 – 8 = 5 × 4 – 3 + 6

3 + 28 - 8 = 20 – 3 + 6

23 = 23

Hence, ‘5, 7’ is the correct answer.

1. h, f, g, h, f, h → f h h / g f f /h g g / f h h  / g f f /h f g → does not follow any pattern.

2. f, g, h, f, g, f → f f h / g g f / h h g / f f h / g g f / h f g → does not follow any pattern.

3. g, f, g, f, h, f → f g h / g f f / h g g / f f h / g h f / h f g → does not follow any pattern.

4. g, h, f, g, h, f → f g h / g h f / h f g / f g h / g h f / h f g → follows a pattern.

‘f g h / g h f / h f g’ repeats a cyclic pattern.

Hence, ‘g, h, f, g, h, f’ is the correct answer.

The pattern followed here is :

Hence, ‘31’ is the correct answer.

The pattern followed here is :

Hence, ‘uEx’ is the correct answer.

The mirror image is as follows:

Hence, the correct answer is option 3.

Given:

Total Amount is Rs. 5,500.

Product ‘z’ bought at Rs. 5000.

Therefore, the remaining amount is 500.

Product ‘z’ sell at 15% profit.

$SP = 5000 \times \frac{{115}}{{100}} = 5750$

Now total amount is = 500 + 5750

= 6250

Now he buys product ‘X’ at Rs. 6250 and then sell it at 25% profit then the total remaining amount

$= 6250 \times \frac{{125}}{{100}} = 7812.50$

Hence, ’Rs. 7812.50’ is the correct answer.

Given figure is hidden in option (3) as shown below.

Hence, option (3) is the correct answer.

The pattern followed here is :

Hence, ‘TVW‘ is odd among them.

Blunder (a careless mistake) is a type of Error (mistake).

Similarly,

Euphoria (intense happiness) is a state of Happiness.

Hence, Euphoria : Happiness is the correct answer.

Council is a collective noun used to denote a group of Ministers.

Similarly,

Crew is a collective noun used to denote a group of Sailors

Hence, ‘Crew’ is the correct answer.

Preparing the family tree using the following symbols,

Possible tree diagram will be,

Hence, ’L’ is the father of ‘K’.

Here, ‘7’ is common in only rectangle, circle and triangle.

So, 7 represent female students who play both table tennis and badminton.

Hence, ‘7’ is the correct answer.

The logic followed here is:

In a row,

4th number = (1st number × 3rd number) ÷ 2nd number 

In 1st row, 

4th number = (10 × 14) ÷ 4

= 140 ÷ 4 = 35

In 2nd row,

4th number = (15 × 5) ÷ 3

= 75 ÷ 3 = 25

In 3rd row,

4th number = (14 × 6) ÷ 7

= 84 ÷ 7 = 12

Similalry,

In 4th row,

4th number = (18 × 8) ÷ ? = 16

= 144 ÷ ? = 16 

? = 144 ÷ 16 = 9

Hence, "9" is the correct answer.

The correct logical sequence is,

Housefly → Lizard → Monkey → Crocodile → Whale

All are arranged in increasing order according to their physical size.

Hence, ‘4, 2, 5, 1, 3’ is the correct sequence.

Penury, indigence and destitution are synonyms for poverty but plenty is not.

Hence, ‘Plenty’ is odd among them.

U → C, K → H, L → U, N → E, S → L, E → K, and C → N

So, ’KNUCKLES’ is written as ‘HECNHUKL’.

Hence, ’HECNHUKL’ is the correct answer.

4 is adjacent to 2 and 3 in the first position of the dice.

3 is adjacent to 4 and 6 in the second position of the dice.

3 is adjacent to 6 and 1 in the third position of the dice.

3 is adjacent to 1 and 4 in the fourth position of the dice.

The only number left is 5.

Hence, 3 is opposite to 5.

12 × 6 = 72, 12 × 9 = 108

Similarly,

14 × 6 = 84, 14 × 9 = 126

17 × 6 = 102, 17 × 9 = 153

Hence, ‘126’ is the correct answer.

The pattern followed here is

1. 11² = 121, 12² + 1 = 145

2. 17² = 289, 18² + 1 = 325

3. 13² = 169, 14²  + 1 = 197

4. 15² = 225, 16² + 1 = 257 ≠ 241

Hence, ‘225 - 241’ is the correct answer.

The elements move clockwise and in each step, one new element is added.

Hence, Image in option (2) is the next figure in the series.

The famous 11-day long ‘Dhanu Jatra’, considered as the largest open-air theatre in the world is celebrated in Odisha.

Key Points

• Spread over an 8 km radius area around the municipality of Bargarh, it is the largest open-air theatre in the world, one that is listed in the Guinness Book of World Records.
• It is based on Krishna's mythological tale (locally known as Krushna), and his demon-uncle Kansa.
• It is about the episode of the visit to Mathura by Krishna and Balaram to witness the ceremony of Dhanu, arranged by their (maternal) uncle Kansa.
• In November 2014, the Government of India's Department of Culture awarded Dhanu Yatra the status of the National Festival.

​In 1734, the Nahargarh Fort was built in Jaipur by Maharaja Sawai Jai Singh II.

• He was the ruler of the Amber kingdom.
• He founded the Jaipur city and made it capital.
• In 1699, the Mughal Emperor, Aurangzeb, who had summoned him to Delhi impressed by his intellect, gave him the title of Sawaii.
• The Mughal emperor Muhammad Shah awarded him the title of Saramad-i-Rajaha-i-Hind on 21 April 1721.

​As per the government rules, 15% of advance tax needs to be paid by 15th June by an individual who is liable to pay advance tax.

• Advance tax means earnings tax will be charged in advance, rather than lump-sum payment at year-end.
• The way you collect tax is also known as earn tax.
• Such payments are to be made in installments as given by the income tax department according to the due dates.
• When your gross tax liability in a financial year is Rs 10,000 or more you will pay the advance tax.

The 23rd National Youth Festival (NYF) 2020 was celebrated in Lucknow to commemorate the birth anniversary of Swami Vivekananda.

• It was organized by the Ministry of Youth Affairs and Sports and state government from 12^th to 16^th January 2020.
• The Government has been organizing National Youth Festival (NYF) since 1995.
• Cultural/musical events by local and renowned artists accompanied the inauguration.
• The festival's closing ceremony on 16^th January was graced by Uttar Pradesh Governor and Shri Kiren Rijiju.

​In 1833, the Currency Building was constructed in the BBD Bagh or Dalhousie area of Kolkata.

• It is also one of the country's oldest buildings that had functioned as a bank.
• It operated as Reserve Bank of India (RBI), as the name implies, after which it deteriorated till 2002.
• It has been designated a protected monument by the Archaeological Survey of India (ASI).
• The buildings are now being turned into lively cultural spaces even in the heart of the city for theater or music festivals.

Pallava dynasty built the Pancha rathas of Mahabalipuram.

• This was an Indian dynasty, which ruled a portion of southern India from 275 CE to 897 CE. Since the extinction of the Satavahana dynasty which the Pallavas served as feudatories, they gained prominence.
• During the reign of Mahendravarman I (571–630 CE), and Narasimhavarman I (630–668 CE) it became a major power.
• Pallavas built the Shore Temple in Mamallapuram, a UNESCO World Heritage Site.
• They developed the script for Pallava from which Grantha eventually descended.

​VISHWAS(Video Integration and State Wide Advanced Security), which is a major e-governance initiative launched by the government in January 2020.

• It was inaugurated by Union Home minister Amit Shah on 10^th January 2020.
• This project was launched for the Gujarat police and Indian Postal department in Gandhinagar.
• It improves law and order management, traffic management and control crime.
• With this project, the Union minister also launched these project: Netrang and AASHVAST.

Science and Technology: Rural Development was the theme of the 107th Indian Science Congress held in Bengaluru.

• India's biggest annual meeting of science luminaries from all over the world, the 107th Indian Science Congress held at the University of Agricultural Sciences, GKVK Campus, Bangalore, Karnataka, from 3 to 7 January 2020.
• It was inaugurated by Prime Minister Narendra Modi.
• To scientists, researchers and educators involved in different aspects of scientific developments and technology, Congress is a significant focal point.
• The five-day long gathering seeks to bring together the community of science around the world to address scientific creativity and research.

​Pongal festival is celebrated for four days in Tamil Nadu. The fourth day of Pongal is called Kaanum Pongal.

Key Points

• It is celebrated by south Indians during the Hindu Harvest festival.
• This is dedicated to the Hindu sun god, the Surya, and it corresponds to Makar Sankranti, the festival of harvests celebrated in India under many regional names.
• The first day of Pongal is called Bhogi Pongal.
• The second day of Pongal is called Surya Pongal.
• The third day of Pongal is called Mattu Pongal.

The white colour of the light emitted by the Sun.

• The actual Sun colour is white.
• The reason the Sun appears bright to us is that the Earth's atmosphere disperses higher wavelength colours less quickly, such as red, orange and violet. These wavelengths are, indeed, what we see, due to this the Sun appears yellow.
• The white colour is the composite of all visible light frequencies.
• The highest wavelength of light is Red and the lowest wavelength of light is violet.

​Reliance Industries is the first Indian company to hit the ₹10 lakh crore mark in market capitalization.

• The second most valued firm on the stock exchanges, Tata Consultancy Services has a market cap of Rs 7.81 lakh crore.
• Exxon Mobil is the world's most valuable 290 billion dollar energy firm.
• Only in August last year, RIL became the first corporation to cross the Rs 8 lakh crore mark.
• This was also the first company to cross the market cap mark of $100 billion in 2007.

​In January 2020, B. Sai Deepak set a Guinness World Record for 59 side lunges in 60 seconds. 

• The previous record was created by Irfan Mehsood from Pakistan.
• A lunge can refer to any position of the human body where one leg is placed forward on the ground with knee bent and foot flat while the other leg is placed behind it.
• It is used by athletes in cross-training for sports, as a fitness exercise by weight-trainers and as part of an asana regime by yoga practitioners.
• These are good exercises to strengthen, shape and develop several muscle/muscle groups like the quadriceps, gluteus maximus, and hamstrings.

​The researchers of IIT Guwahati employed the nanoscale phenomenon called 'Electrokinetic streaming potential' to harvest energy from flowing water on a small scale like water flowing through household water taps.

• The different types of semiconducting materials "Contrasting Interfacial Activities" were used to generate power from stagnant water.
• The discovery of these electrokinetic phenomena and their potential use for the transfer of energy has been known for more than half a century, they have not been harnessed because of the low efficiency resulting from the inadequacy of fluid stream channels.
• The humble performance of potentially energy generation devices based on electrokinetic streaming is due to the trade-off between high flow rate and nanofluidic containment.
• Throughout household settings, new ways of generating energy can be used to help the idea of decentralization of energy sources.

​Jasprit Bumrah has been selected to receive the Polly Umrigar award for his performance in international cricket in the 2018-19 season.

• One of the BCCI Awards is the Polly Umrigar Award for International Cricket of the Year, given to Indian cricketers for outstanding performance in international cricket.
• The first recipient of this award was Sachin Tendulkar.
• Jasprit Bumrah is the first Asian bowler to take 5 wickets during the same calendar year at a test series in South Africa, England, and Australia.
• For a debut year with 48 wickets for 8 matches, he is also the third-highest wicket-taker in test matches.

Note: 

• Polly Umriga is one of the great players of cricket in India.
• Polly Umrigar was a right-handed middle-order batsman, who would roll his arms over for medium-pace or off-spin occasionally.

​Ishwar Sharma has been honoured with the Global Child Prodigy Award 2020. This award is associated with Yoga.

• He received his Certificate of Achievement at a special event in New Delhi in January 2020 attended by Nobel laureate Kailash Satyarthi and Lieutenant Governor Dr. Kiran Bedi of Puducherry.
• He was chosen from the UK for the awards that recognize children prodigies from 45 countries and 30 different categories, including riding, choreography, palaeontology, exercise, and martial arts.
• The award will now give him a chance to be a yoga ambassador and to spread the message.
• This award recognizes the kids below 15.

​The Indian Railways has integrated its helpline number into a single number 139.

• It converted into a single number on 1st January 2020.
• SMS facility also added into it and is available in 12 languages.
• Indian railway is the fourth largest railway network in the world.
• Ashwini Vaishnaw is the current minister of Railways.

The correct answer is Namdapha National Park.

Key Points

• Namdapha National Park is the largest protected area in the Eastern Himalayan sub-region.
• It is a wide protected area of 1,985 km² in Arunachal Pradesh of Northeast India.
• The national park harbours the northernmost lowland evergreen rainforests in the world at 27°N latitude.
• Even harbours vast dipterocarp forests, covering the northwestern parts of the eco-region rainforests Mizoram-Manipur-Kachin.
• It is the fourth largest national park in India.

Additional Information

G. Babita Rayudu took charge as an Executive Director for The Securities and Exchange Board of India(SEBI) in January 2020.

• It is the regulator of the Indian government-owned securities and commodity market.
• It was formed in 1988 and granted statutory powers through the 1992 SEBI Act, on 30 January 1992.
• Madhabi Puri Buch is the chairman of SEBI.
• The headquarters of SEBI in Mumbai.

Dibrugarh has been awarded the Plastic Waste Management Award -2020 for being the best district of India in the plastic waste management category during Swachhta Hi Seva 2019.

• Swachhta Hi Seva on Swachhta is a huge nationwide program of awareness and mobilization.
• The 2019 Swachhta hi Seva was held from 11 September till 2 October 2019.
• PM Modi had set the goal of making India ODF under the Swachh Bharat Mission (SBM) by October 2, the 150th anniversary of Mahatma Gandhi's birth.
• On October 2 the initiative also included a nationwide shramdaan for collecting and segregating plastic waste.

In terms of area, Madhya Pradesh has the largest forest cover in India.

• It is the second-largest state by area after Rajasthan in India, and the fifth-largest state by population.
• The capital of Madhya Pradesh is Bhopal.
• The largest city of Madhya Pradesh is Indore.
• As per the economy, it is the 10^th largest state economy in India.

Notes:

• Area-wise Madhya Pradesh has the largest forest cover in the country followed by Arunachal Pradesh, Chhattisgarh, Odisha, and Maharashtra.
• In terms of forest cover as a percentage of total geographical area, the top five States are Mizoram (85.41%), Arunachal Pradesh (79.63%), Meghalaya (76.33%), Manipur (75.46%) and Nagaland (75.31%).

In August Kranti Maidan, the Quit India Movement was launched by Mahatma Gandhi in 1942.

• It is also known as Gowalia Tank Maidan and located in Central Mumbai.
• On 8 August 1942, Mahatma Gandhi delivered the Quit India speech decreeing that the British must quit India immediately or else there would be mass agitations.
• He called for resolute yet passive resistance which meant the certainty that Gandhi foresaw the revolution, best represented by his call to do or die.
• Gandhi made the speech to help India become independent.

Chiropody is a branch of science related to Feet.

Key Points

• This is a field of medicine dedicated to researching, diagnosing, and treating foot, leg, and lower extremity conditions medicinally and surgically.
• Podopediatrics is a specialty of podiatric medicine which focuses on the treatment of lower limb medical problems in children.
• Nephrology is the study of kidney in the Human body.
• Pulmonology is the study of the lungs.
• Hepatology is the study of the liver.

Nandi river flows through Tiruttani a famous pilgrimage place of South India.

• Tiruttani is a town located in Tamil Nadu.
• This city is famous for the Temple of Tiruthani Murugan, one of the Arupadaiveedu, dedicated to Kartikeya.
• When Andhra state was established on 1 November 1953, Tirutani was part of Andhra Pradesh state's Chittoor district until 1960.
• Tirutani and Ramakrishnarajupet assembly areas of Andhra Pradesh are transferred to Madras state on 1 April 1960, with the Andhra Pradesh and Madras Alteration of Boundaries Act of 1959, in return for smaller territory.

Avogadro’s Law states that equal volume of all gases under the same conditions of temperature and pressure contain an equal number of molecules.

• It is an experimental gas law, which relates the volume of a gas to the amount of gas present.
• For example equivalent quantities of molecular hydrogen and nitrogen, when they are at the same temperature and pressure, contain the same number of molecules and exhibit ideal gas behavior.
• Ohm's law: Relation between the voltage and current at a constant temperature.
• Charle's law: a law stating that the volume of an ideal gas at constant pressure is directly proportional to the absolute temperature.
• Boyle's Law: the relation between volume and pressure of a gas at a constant temperature.

The Gujarat Police was honoured with the President's Colours award in December 2019.

• Gujarat became India's seventh state police force to be given the President's Colors Honour.
• The Vice President, Shri M. Venkaiah Naidu, addressed this.
• He made homage to Gujarat Police martyrs.
• The request for President's Colors was introduced by the office of Director-General (DG) and Police Inspector General (IGP), the Gujarat Police Chief.

Given:

D is the midpoint of side BC of ΔABC 

CE = AC/3 

BE and AD intersect at G 

Calculations:

Let DP is parallel to BE 

In ΔBCE and ΔDCP 

∠CEB = ∠CPD (The corresponding angles between two parallel lines are equal)

∠CBE = ∠CDP ( The corresponding angles between two parallel lines are equal)

By AA (Rule of similarity), ΔBCE ∼ ΔDCP 

∴ (BC/DC) = (CE/CP) 

⇒ (BD + DC)/DC = (CP + PE)/CP

⇒ BD/DC + 1 = 1 + PE/CP

⇒ 1/1 = PE/CP

⇒ PE = CP = CE/2 

⇒ PE/CE = 1/2

In ΔADP and ΔAGE 

∠ADP = ∠AGE 

∠APD = ∠AEG  

By AA (Rule of similarity),ΔADP ∼ ΔAGE  

∴ (AD/AG) = (AP/AE)

⇒ (AG + GD)/AG = (AE + PE)/AE 

⇒ 1 + GD/AG = 1 + PE/AE 

⇒ GD/AG = PE/AE 

⇒ AG/GD = AE/PE

⇒ AG/GD = AE/(CE/2)

⇒ AG/GD = 2 × AE/CE 

⇒ AG/GD = 2 × 2/1 

⇒ AG/GD = 4 ∶ 1 

∴ AG/GD = 4 ∶ 1

Alternate Method

Given,

D is the midpoint of side BC

So, BD : DC = 1 : 1

Point E lies on AC such that CE = AC/3

So, AC : CE = 3 : 1

So, CE is 1 and AE = AC - CE

So, AE = 2

AE : CE = 2 : 1

From the following figure balancing sides of the triangle, then

LCM of 2, 1, 1 and 1 is 2

∴ AG : GD = 4 : 1

Shortcut Trick

The product 16a⁴ + 36a²b² + 81b⁴ is the product of factors (4a² + 9b² + 6ab) and (4a² + 9b² − 6ab).

Given: 13 × (4a² + 9b² + 6ab) = 91 ⇒ (4a² + 9b² + 6ab) = 7.

To find 3ab, use the difference of factors: 12ab = 7 − 13 = −6.

3ab = −6 ÷ 4 = −3/2.

∴ The correct answer is −3/2.

Alternate Method

Given:

16a⁴ + 36a²b² + 81b⁴ = 91

4a² + 9b² − 6ab = 13

Formula Used:

(x² + xy + y²)(x² − xy + y²) = x⁴ + x²y² + y⁴

Calculations:

⇒ (4a² + 9b² + 6ab)(4a² + 9b² − 6ab) = 16a⁴ + 36a²b² + 81b⁴

⇒ (4a² + 9b² + 6ab) × 13 = 91

⇒ 4a² + 9b² + 6ab = 7 ----(Equation 1)

⇒ 4a² + 9b² − 6ab = 13 ----(Equation 2)

Subtracting Equation 2 from Equation 1:

⇒ (4a² + 9b² + 6ab) − (4a² + 9b² − 6ab) = 7 − 13

⇒ 12ab = −6

⇒ 3ab = −6 ÷ 4

⇒ 3ab = −3/2

∴ The correct answer is −3/2.

Additional Information

Algebraic Identity for Quartic Expressions

The identity (x⁴ + x²y² + y⁴) = (x² + xy + y²)(x² − xy + y²) is frequently used in competitive exams to solve for specific terms like xy or (x² + y²).

Sum of Squares Calculation

If the question asked for 4a² + 9b², you would add the two factors and divide by 2: (7 + 13) ÷ 2 = 10.

Standard Format

For any equation a⁴ + a²b² + b⁴ = K, if (a² + b² + ab) = x, then (a² + b² − ab) = K/x.

Total work = 40

Time taken to fill the tank A and B together = 40/(1 + 4) = 8 hrs

Let the efficiency of C be x.

If all three pipe A, B and outlet pipe C open, time taken to fill the tank = 8 hr + 80 min = 9 + 1/3 = 28/3 hrs

Efficiency of A,  B and C = 40/(28/3) = 30/7

According to the question

4 + 1 + x = 30/7

⇒ x = 30/7 – 5

⇒ x = (–5/7)

Work done by A and B in 7 hrs = (4 + 1) × 7 = 35

∴ Pipe C empty the tank filled by A and B in 7 hrs = 35/(5/7) = 49 hrs

Average of 24 numbers = 56

Sum of 24 numbers = 56 × 24 = 1344

Average of first 10 numbers = 71.7

Sum of first 10 numbers = 71.7 × 10 = 717

Average of next 11 numbers = 42

Sum of next 11 numbers = 42 × 11 = 462

Sum of 22^nd, 23^rd and 24^th = 1344 – 717 – 462 = 165

Ratio of 22^nd, 23^rd and 24^th = 1/2 : 1/3 : 5/12 = 12/2 : 12/3 : (12 × 5)/12 = 6 : 4 : 5

Ratio of 22^nd, 23^rd and 24^th = 6x : 4x : 5x

6x + 4x + 5x = 165

⇒ 15x = 165

⇒ x = 11

22^nd number = 6x = 6 × 11 = 66

24^th number = 5x = 5 × 11 = 55

∴ Average of 22^nd and 24^th number = (66 + 55)/2 = 121/2 = 60.5

Short Trick:

Average of 24 numbers = 56

Deviation on first 10 numbers = (56 – 71.7) × 10 = –157

Deviation on next 11 numbers = (56 – 42) 11 = 154

Sum of 22^nd, 23^rd and 24^th = 56 × 3 – 157 + 154 = 168 – 3 = 165

Ratio of 22^nd, 23^rd and 24^th = 1/2 : 1/3 : 5/12 = 12/2 : 12/3 : (12 × 5)/12 = 6 : 4 : 5

Average of 22^nd and 24^th number = (6 + 5)/2 = 5.5

6 + 4 + 5 = 15 unit

⇒ 15 unit = 165

⇒ 1 unit = 11

⇒ 5.5 unit = 60.5

Shortcut Trick

Multiply 8 years ago ratio (2 : 3) by 2 → 4 : 6 to equalize age difference with 4 years ago ratio (5 : 7).

Change in parts = 5 − 4 = 1 unit, which corresponds to the 4 years passed.

Time gap from 4 years ago to 8 years from now is 12 years ⇒ 12 ÷ 4 = 3 units.

New ratio 8 years from now = (5 + 3) : (7 + 3) = 8 : 10 = 4 : 5.

∴ The correct answer is 4 : 5.

Alternate Method

-8 Yrs -4 Yrs Present +8 Yrs

2 : 3

5 : 7

3 : 4

4 : 5

→ 4 : 6 → 5 : 7 → 6 : 8 → 8 : 10

4 yrs = 1 unit

12 yrs = 3 units

Given:

Ratio of A's age to B's age 8 years ago = 2 : 3

Ratio of their ages 4 years ago = 5 : 7

Calculations:

⇒ Let the ages of A and B 8 years ago be 2x and 3x respectively.

⇒ 4 years ago, A's age = 2x + 4 and B's age = 3x + 4

⇒ According to the question, (2x + 4) / (3x + 4) = 5 / 7

⇒ 7 × (2x + 4) = 5 × (3x + 4)

⇒ 14x + 28 = 15x + 20

⇒ 15x − 14x = 28 − 20

⇒ x = 8

⇒ Present age of A = 2x + 8 = 2 × 8 + 8 = 24 years

⇒ Present age of B = 3x + 8 = 3 × 8 + 8 = 32 years

⇒ A's age 8 years from now = 24 + 8 = 32 years

⇒ B's age 8 years from now = 32 + 8 = 40 years

⇒ Required ratio = 32 : 40 = 4 : 5

∴ The correct answer is 4 : 5.

Additional Information

Constant Age Difference

The difference between the ages of two individuals always remains constant. Equalizing the horizontal difference between ratio parts simplifies age problems without requiring algebraic equations.

Cross-Multiplication Method

If initial ratio a : b changes to final ratio c : d after adding time periods x and y, you can apply the cross-product rule: (a × d) − (b × c) = (c × y) − (d × x) to solve directly.

Ratio Properties

Multiplying or dividing both terms of a ratio by the same non-zero number does not change its value. A duplicate ratio of a : b is given as a² : b².

x² – 2√5 x + 1 = 0

Divide by x

x – 2√5 + 1/x = 0

x + 1/x = 2 √5      ------ (1)

Squaring on both sides equation (1)

(x + 1/x)² = (2 √5)²

⇒ x² + 1/x² + 2 = 20

⇒ x² + 1/x² = 20 – 2

(x² + 1/x²) = 18      ------ (2)

Cube of equation (1)

(x + 1/x)³ = (2 √5)³

⇒ x³ + 1/x³ + 3 (x + 1/x) = 40 √5

⇒ x³ + 1/x³ + 3 (2√5) = 40√5

⇒ x³ + 1/x³ = 40√5 – 6√5 = 34√5

⇒ x³ + 1/x³ = 34√5      ------ (3)

Multiply by equation (2) and equation (3)

(x² + 1/x²) (x³ + 1/x³) = 18 × 34√5

⇒ x⁵ + 1/x + x + 1/x⁵ = 612√5

⇒ x⁵ + 1/x⁵ = 612√5 – (x + 1/x)

⇒ x⁵ + 1/x⁵ = 612 √5 – 2 √5

⇒ x⁵ + 1/x⁵ = 610 √5

Short trick:

As we know,

If x + 1/x = a, then

⇒ x⁵ + 1/x⁵ = (a² – 2)(a³ – 3a) – a

If x + 1/x = 2√5, then

⇒ x⁵ + 1/x⁵ = [(2√5)² – 2)][(2√5)³ – 3 × 2√5)] – 2√5

⇒ x⁵ + 1/x⁵ = (20 – 2) (40√5 – 6√5) – 2√5 = 18 × 34√5 – 2√5 = 610√5

Shortcut Trick

The angle between two intersecting chords is half the sum of the central angles of the intercepted arcs.

⇒ ∠APD = 180° − ∠APC = 180° − 95° = 85°

⇒ ∠APD = (∠AOD + ∠BOC) ÷ 2

⇒ 85° = (110° + ∠BOC) ÷ 2

⇒ 170° = 110° + ∠BOC

⇒ ∠BOC = 60°

∴ The correct answer is 60°.

Alternate Method

Given:

∠APC = 95° (Angle between chords AB and CD)

∠AOD = 110° (Central angle subtended by arc AD)

Formula Used:

The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs (central angles).

Calculations:

⇒ Chords AB and CD intersect at P.

⇒ ∠APC and ∠APD are linear pair angles.

⇒ ∠APD = 180° − ∠APC

⇒ ∠APD = 180° − 95° = 85°

⇒ By the Intersecting Chords Angle Theorem:

⇒ ∠APD = (Measure of arc AD + Measure of arc BC) ÷ 2

⇒ ∠APD = (∠AOD + ∠BOC) ÷ 2

⇒ 85° = (110° + ∠BOC) ÷ 2

⇒ 170° = 110° + ∠BOC

⇒ ∠BOC = 170° − 110°

⇒ ∠BOC = 60°

∴ The correct answer is 60°.

Additional Information

Intersecting Chords Theorem (Internal)

If two chords AB and CD intersect at a point P inside a circle, then PA × PB = PC × PD.

Central Angle and Arc Measure

The measure of a central angle is equal to the measure of its intercepted arc. The total measure of a circle is 360°.

Angle Subtended at Circumference

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Average export of items B during six years = (128 + 134 + 138 + 169 + 182 + 209)/6 = 960/6 = 160

⇒ 1.4 of 160 = 224

In 2013, the exports of item D were 1.4 times the average exports of item B during six years.

∠ADC = 75° and ∠BEC = 52°

As we know, In a cyclic quadrilateral, the sum of opposite angles are 180°.

∠ADC + ∠ABC = 180°

⇒ ∠ABC = 180° – 75° = 105°

⇒ ∠ABC + ∠CBE = 180° Δ [straight line]

⇒ ∠CBE = 180° – 105° = 75°

In ΔBEC

∠CBE + ∠BEC + ∠ECB = 180°

⇒ ∠ECB = 180° °– 75° – 52° °= 53°

⇒ ∠ECB + ∠BCD = 180°  [straight line]

⇒ ∠BCD = 180° – 53° = 127°

⇒ ∠BAD + ∠BCD = 180°

⇒ ∠BAD = 180° – 127° = 53°

In ΔAFB

∠BAF + ∠ABF + ∠AFB = 180°

⇒ ∠AFB = 180° – 53° – 105° = 22°

∴ Difference between ∠BAD and ∠AFB = 53° – 22° = 31°

Shortcut Trick

Let the Marked Price (MP) of 1 kg rice be 1 unit.

Total rice received = 80 + (80 ÷ 20) × 1 = 80 + 4 = 84 kg.

Total Marked Price of 84 kg rice = 84 units.

Cost Price (CP) for 80 kg at 10% discount = 80 × 0.9 = 72 units.

Profit % = (84 − 72) ÷ 72 × 100 = 12 ÷ 72 × 100 = 1/6 × 100 = 16(2/3)%.

∴ The correct answer is 16(2/3)%.

Alternate Method

Given: MP of 1 kg rice = Rs. 10; Discount = 10%; Free rice = 1 kg for every 20 kg.

Formulas: CP = MP × (100 − Discount%) ÷ 100 | Profit % = (Total SP − Total CP) ÷ Total CP × 100

⇒ Total quantity received = 80 kg + (80/20) kg free = 84 kg.

⇒ Cost Price (CP) for shopkeeper = (80 kg × Rs. 10) × 0.90 = Rs. 720.

⇒ Selling Price (SP) at Marked Price = 84 kg × Rs. 10 = Rs. 840.

⇒ Profit = 840 − 720 = Rs. 120.

⇒ Profit % = (120 ÷ 720) × 100 = 1/6 × 100 = 16(2/3)%.

∴ The correct answer is 16(2/3)%.

Additional Information

Free Items Concept

When items are offered free, the effective cost price per unit decreases. Net discount % = [Free items ÷ (Paid items + Free items)] × 100.

Successive Discounts

The shopkeeper receives a double benefit: a direct cash discount and a quantity discount. Total Profit % = [(Total SP − Total CP) ÷ Total CP] × 100.

Shortcut Trick$\frac{{{x^3} + {y^3}}}{{{{\left( {x - y} \right)}^2} + 3xy}} \div \frac{{{{\left( {x + y} \right)}^2} - 3xy}}{{{x^3} - {y^3}}} \times \frac{{{{\left( {x + y} \right)}^2} + {{\left( {x - y} \right)}^2}}}{{{x^2} - {y^2}}}$

Put x = 1 and y = 0, then

$\Rightarrow \frac{{1 + 0}}{{1 + 0}} \div \frac{1}{1} \times \frac{{1 + 1}}{1}$

⇒ 1 ÷ 1×2

⇒ 1 × 2

⇒ 2

From option 1.

⇒ 2 (x² + y²)

⇒ 2 × (1 + 0)

⇒ 2 × 1

⇒ 2 (satisfied)

Alternate Method $\frac{{{x^3} + {y^3}}}{{{{\left( {x - y} \right)}^2} + 3xy}} \div \frac{{{{\left( {x + y} \right)}^2} - 3xy}}{{{x^3} - {y^3}}} \times \frac{{{{\left( {x + y} \right)}^2} + {{\left( {x - y} \right)}^2}}}{{{x^2} - {y^2}}}$

$\Rightarrow \frac{{{x^3} + {y^3}}}{{{x^2} + {y^2} - 2xy + 3xy}} \times \frac{{{x^3} - {y^3}}}{{{x^2} + {y^2} + 2xy - 3xy}} \times \frac{{2({x^2} + {y^2})}}{{\left( {x + y} \right)\left( {x - y} \right)}}$

$\Rightarrow \frac{{{x^3} + {y^3}}}{{\left( {x - y} \right)({x^2} + {y^2} + xy)}} \times \frac{{{x^3} - {y^3}}}{{\left( {x + y} \right)({x^2} + {y^2} - xy)}} \times \frac{{2({x^2} + {y^2})}}{1}$

$\Rightarrow \frac{{{x^3} + {y^3}}}{{{x^3} - {y^3}}} \times \frac{{{x^3} - {y^3}}}{{{x^3} + {y^3}}} \times \frac{{2({x^2} + {y^2})}}{1}$

⇒ 2(x² + y²)

Total exports of item A from 2012 to 2014 = 225 + 370 + 425 =  1020

Total exports of all the four items in 2015 = 400 + 209 + 306 + 275 = 1190

∴ Required percentage = (1190 – 1020)/1190 × 100 = 14.3%

In right triangle ABC

AC² = AB² + BC²

⇒ AC² = 5² + 12² = 25 + 144 = 169

⇒ AC = √169 = 13 cm

If AD is bisector of ∠BAC, then

AB/AC = BD/DC

⇒ 5/13 = BD/DC

⇒ DC = 13BD/5

⇒ BD + DC = BC

⇒BD + 13BD/5 = 12

⇒18BD/5 = 12

⇒ BD = 12 × (5/18) = 10/3

In right triangle ABD

AD² = AB² + BD²

⇒ AD² = 5² + (10/3)² = 25 + 100/9 = 325/9

⇒ AD = √325/9 = 5 √13/3 cm

As we know,

Diagonal of square = √2 a

Area of square = a²

Area of square = (diagonal)²/2

Diagonal of square A = (a + b)

Area of square A = (a + b)²/2

Area of square B = 2 × Area of square A

Area of square B = 2 × (a + b)²/2 = (a + b)²

Side of square B = √(a + b)² = (a + b)

Diagonal of square B = √2 (a + b)

Given,

Side of other square = diagonal of square B = √2 (a + b)

Area of other square = √2 (a + b) × √2 (a + b) = 2 (a + b)²

Other Method:

Let a = b = 1, then

Diagonal of square A = 2

Area of square A = 2²/2 = 2

Area of square B = 2 × 2 = 4

Sides of square B = √4 = 2

Diagonal of square B = 2 √2

Given,

Side of other square  = diagonal of square B = 2 √2

Area of other square = 2 √2 × 2 √2 = 8

From option 3

2 (a + b)²

⇒ 2 (1 + 1)²

⇒ 2 × 4

⇒ 8 (satisfied)

Maximum marks of the examination = 500

A scored = 500 × (80/100) = 400

B scored = 400 × (100/125) = 320

C scored = 320 × (100/160) = 200

D scored = 200 × (100/80) = 250

Percentage of D = 250/500 × 100 = 50%

Other method:

Ratio of A to B = 5 : 4

Ratio of B to C = 8 : 5

Ratio of C to D = 4 : 5

Taking only ratio A to D = 160 : 100

Divide by 2 only, then A : D = 80 : 50

If A got 80%, then D got 50%

Given:

5 sin θ = 4

∴  sin θ = 4/5

We have to calculate the value of $\frac{{sec\theta + 4\cot \theta }}{{4tan\theta - 5cos\theta }}$

Calculation:

Perpendicular = P, Hypotenuse = H and Base = B.

sin θ = P/H = 4/5

P = 4 and H = 5

As we know,

H² = P² + B²

⇒ 5² = 4² + B²

⇒ B² = 25 – 16 = 9

⇒ B = √9 = 3

⇒ cos θ = B/H = 3/5

⇒ sec θ = 1/cos θ = 5/3

tan θ = P/B = 4/3

cot θ = 1/tan θ = 3/4

Now,

$\frac{{sec\theta + 4\cot \theta }}{{4tan\theta - 5cos\theta }}$

$\Rightarrow \frac{{\frac{5}{3} + 4 \times \frac{3}{4}}}{{4 \times \frac{4}{3} - 5 \times \frac{3}{5}}}$

$\Rightarrow \frac{{\frac{5}{3} + 3}}{{\frac{{16}}{3} - 3}}$

$\Rightarrow \frac{{\frac{{14}}{3}}}{{\frac{7}{3}}}$

⇒ 2

So, the value is 2.

Sudha sold an article to Renu for = Rs. 576

Cost price of the article for Sudha = Rs. 576 × (100/80) = Rs. 720

Final cost price of the article for Renu = 576 + 224 = Rs. 800

Profit gain on the article for Renu = Rs. 720 × (124/100) = Rs. 892.8

Profit percentage on the article for Renu = (92.8/800) × 100 = 11.6%

Shortcut Trick

$\frac{{{{\sec }^6}\theta - {{\tan }^6}{\rm{\theta }} - 3{{\sec }^2}\theta {{\tan }^2 \theta} + 1}}{{{{\cos }^4}\theta - {{\sin }^4}\theta + 2si{n^2}\theta + 2;}}$

Concept Used:

Put θ = 0

⇒ (1 – 0 – 0 + 1)/(1 + 0 + 0 + 2)

 

⇒ 2/3 (satisfied)

Alternate Method

$\frac{{{{\sec }^6}\theta - {{\tan }^6}{\rm{\theta }} - 3{{\sec }^2}\theta {{\tan }^2 \theta} + 1}}{{{{\cos }^4}\theta - {{\sin }^4}\theta + 2si{n^2}\theta + 2;}}$

$\Rightarrow \frac{{{{({{\sec }^2}\theta )}^3} - {{({{\tan }^2}{\rm{\theta }})}^3} - 3{{\sec }^2}\theta {{\tan }^2 \theta} + 1}}{{\left( {{{\cos }^2}{\rm{\theta }} + {{\sin }^2}{\rm{\theta }}} \right)({{\cos }^2}\theta - {{\sin }^2}\theta ) + 2si{n^2}\theta + 2;}}$

$\Rightarrow \frac{{({{\sec }^2}\theta - {\rm{ta}}{{\rm{n}}^2}\theta )\left( {{\rm{se}}{{\rm{c}}^4}\theta + {\rm{ta}}{{\rm{n}}^4}\theta + {\rm{se}}{{\rm{c}}^2}\theta {\rm{ta}}{{\rm{n}}^2}\theta } \right) - 3{{\sec }^2}\theta {{\tan }^2 \theta} + 1}}{{{{\cos }^2}\theta + {\rm{si}}{{\rm{n}}^2}\theta + ;2}}$

$\Rightarrow \frac{{{{\sec }^4}\theta + {{\tan }^4}\theta + {{\sec }^2}\theta {{\tan }^2}\theta - 3{{\sec }^2}\theta {{\tan }^2 \theta} + 1}}{{1 + 2}}$

$\Rightarrow \frac{{{{\sec }^4}\theta + {{\tan }^4}\theta - 2{{\sec }^2}\theta {{\tan }^2}\theta + 1}}{3}$

$\Rightarrow \frac{{{{({{\sec }^2}\theta - {{\tan }^2}\theta )}^2} + 1}}{3}$

⇒ (1 + 1)/3

⇒ 2/3

The total exports of item D in 2010, 2012 and 2014 = 214 + 247 + 309 = 770

Total exports of all the four items in 2011 and 2012 = 250 + 134 + 244 + 282 + 225 + 138 + 230 + 247 = 1750

∴ Required percentage = 770/1750 × 100 = 44%

Given:

A train takes $2\frac{1}{2}$hours less for a journey of 300 km, if its speed is increased by 20 km/h from its usual speed.

Formula Used:

Speed = Distance/Time

Calculation:

$2\frac{1}{2} = \frac{5}{2}$

Let speed of train be x km/hr

According to the question

[300/x] – [300/(x + 20)] = 5/2

⇒ 300 [x + 20 – x]/[x (x + 20) = 5/2

⇒ 6000/(x² + 20x) = 5/2

⇒ 12000 = 5x² + 100x

⇒ x² + 20x – 2400 = 0

⇒ x² + 60x – 40x – 2400 = 0

⇒ x (x + 60) – 40 (x + 60) = 0

⇒ (x + 60) (x – 40) = 0

Taking,

⇒ (x – 40) = 0

⇒ x = 40 km/hr

Speed of the train = 40 km/hr

∴ Time taken to cover 192 km by train = 192/40 = 4.8 hrs

Shortcut Trick Let the usual speed = x

Increased  speed is (x + 20)

[300/x] – [300/(x + 20)] = 5/2

Using different values x = 40, - 60

As speed is 40.

Time taken to cover 192/40 = 4.8hrs

12cos² θ – 2sin²θ + 3cosθ = 3

⇒ 12 cos² θ – 2 (1 – cos² θ) + 3 cos θ = 3

⇒ 12 cos² θ – 2 + 2 cos² θ + 3 cos θ = 3

⇒ 14 cos² θ + 3 cos θ – 5 = 0

⇒ 14 cos² θ + 10 cos θ – 7 cos θ – 5 = 0

⇒ 2 cos θ (7 cos θ + 5) – 1 (7cos θ + 5) = 0

⇒ (2 cos θ – 1) (7 cos θ + 5) = 0

Taking,

(2 cos θ – 1) = 0

⇒ cos θ = 1/2

⇒ cos θ = cos 60

⇒ θ = 60

$\Rightarrow \frac{{cosec\theta + sec\theta }}{{tan\theta + cot;\theta }}$

$\Rightarrow \frac{{cosec60 + sec60}}{{tan60 + cot;60}}$

$\Rightarrow \frac{{\frac{2}{{\sqrt 3 }} + 2}}{{\sqrt 3 + \frac{1}{{\sqrt 3 }}}}$

$\Rightarrow \frac{{\frac{{2\sqrt 3 + 2}}{{\sqrt 3 }}}}{{\frac{4}{{\sqrt 3 }}}}$

$\Rightarrow \frac{{\sqrt 3 + 1}}{2}$

Detailed Method:

Rate = $16\frac{2}{3} = \frac{{50}}{3}{\rm{\% }}$

Time = 3 years

CI = 6350

As we know,

CI = P [(1 + r/100)³ – 1]

6350 = P [{1 + (50/3 × 100)³} – 1]

⇒ 6350 = P [(1 + 1/6)³ – 1]

⇒ 6350 = P [(7/6)³ – 1]

⇒ 6350 = P [343/216 – 1]

⇒ 6350 = P [127/216]

⇒ P = 6350 × (216/127)

⇒ P = 10800

Now,

P = 10800, $r = 16\frac{2}{3}\% = \frac{{50}}{3}\% ;and;T = 5\frac{2}{3} = \frac{{17}}{3}year;$ 

As we know,

SI = Prt/100

SI = (10800 × 50 × 17)/(3 × 3 × 100)

SI = 10200

Short trick:

Rate = $16\frac{2}{3} = \frac{1}{6}$ and time = 3 years and CI = 6350

Ratio of P to A = 6³ : 7³ = 216 : 343

⇒ 343 – 216 = 127

⇒ 127 unit = 6350

⇒ 1 unit = 50

⇒ 216 unit = 10,800

Now,

P = 10800, $r = 16\frac{2}{3}\% = \frac{{50}}{3}\% ;and;T = 5\frac{2}{3} = \frac{{17}}{3}year;$ 

Interest rate for 1 year = 50/3 %

Interest rate for 17/3 year = 850/9 %

Simple interest = 10800 × (850/900) = 10200

Hint:

As given, time = 17/3 years

So, we can say answer divisible by 17. Only option 2 is divisible by 17.

Total exports of items A in 2014 and 2015 = 425 + 400 = 825

Total export of item C in 2011 and 2015 = 244 + 306 = 550

∴ Required ratio = 825 : 550 = 3 : 2

$\frac{{3\frac{2}{3} \div \frac{{11}}{{30}}of\frac{2}{3} - \frac{1}{4}of;2\frac{1}{2} \div \frac{3}{5} \times 4\frac{4}{5}}}{{\frac{2}{5}of;7\frac{1}{2} \div \frac{3}{4} - \frac{3}{4} \times 1\frac{1}{2} \div 2\frac{1}{4}}}$

$\Rightarrow \frac{{\frac{{11}}{3} \div \frac{{11}}{{30}} \times \frac{2}{3} - \frac{1}{4} \times \frac{5}{2} \div \frac{3}{5} \times \frac{{24}}{5}}}{{\frac{2}{5} \times \frac{{15}}{2} \div \frac{3}{4} - \frac{3}{4} \times \frac{3}{2} \div \frac{9}{4}}}$

$\Rightarrow \frac{{\frac{{11}}{3} \div \frac{{11}}{{45}} - \frac{5}{8} \div \frac{3}{5} \times \frac{{24}}{5}}}{{3 \div \frac{3}{4} - \frac{3}{4} \times \frac{3}{2} \times \frac{4}{9}}}$

$\Rightarrow \frac{{\frac{{11}}{3} \times \frac{{45}}{{11}} - \frac{5}{8} \times \frac{5}{3} \times \frac{{24}}{5}}}{{3 \times \frac{4}{3} - \frac{1}{2}}}$

$\Rightarrow \frac{{15 - 5}}{{4 - \frac{1}{2}}}$

⇒ 10/(7/2)

⇒ 20/7