SSC CGL Tier 1 2019 General Awareness Questions with Solutions

Explanation

Total production of type B cars from 2013 to 2016 is calculated by summing the values for those years: 47 + 55 + 58 + 54 = 214 thousand.

Total production of all types of cars in 2017 is the sum of the production of cars A, B, C, D, and E in that year: 64 + 66 + 72 + 66 + 62 = 330 thousand.

To find the percentage by which the production of type B cars (2013-2016) is less than the total production in 2017, we use the formula:

Percentage less = (Total production in 2017 - Total B (2013-2016))/(Total production in 2017) × 100

Substituting the values:

Percentage less = (330 - 214)/(330) × 100 = ¹¹⁶⁄₃₃₀ × 100 ≈ 35.15%

The approximate value is 35%.

Explanation

When the train and the man move in the same direction, the relative speed is the difference between their speeds: Vᵣₑₗ₁ = 44 - 8 = 36 km/h. Converting this to meters per second gives 36 × ⁵⁄₁₈ = 10 m/s. The length of the train L is calculated by the formula L = Relative Speed × Time, so L = 10 m/s × 15 s = 150 meters.

When the train and the woman move in opposite directions, the relative speed is the sum of their speeds: Vᵣₑₗ₂ = Vₜᵣₐᵢₙ + V_wₒₘₐₙ. Using the train length and the time taken to cross the woman, the relative speed in m/s is Vᵣₑₗ₂ = (150 m)/(10 s) = 15 m/s. Converting this relative speed to km/h gives 15 × ¹⁸⁄₅ = 54 km/h. Setting up the equation 44 + V_wₒₘₐₙ = 54, we find that V_wₒₘₐₙ = 10 km/h.

Explanation

The sum of all eleven numbers is calculated as 11 × 54 = 594. The sum of the first four numbers is 4 × 48 = 192. The average of the next four numbers is 25% greater than 48, which is 48 × 1.25 = 60, so their sum is 4 × 60 = 240.

The combined sum of the 9th, 10th, and 11th numbers is the total sum minus the sum of the first eight numbers: 594 - (192 + 240) = 594 - 432 = 162. Let the 11th number be x. Based on the given conditions, the 9th number is x + 8 and the 10th number is x + 4.

Setting up the equation for the final three numbers: (x + 8) + (x + 4) + x = 162 3x + 12 = 162 3x = 150 x = 50

The 9th number is 50 + 8 = 58 and the 10th number is 50 + 4 = 54. The average of the 9th and 10th numbers is (58 + 54)/(2) = ¹¹²⁄₂ = 56.

Explanation

In a cyclic quadrilateral, the sum of each pair of opposite angles is 180 degrees. For the quadrilateral PQRS, the pairs of opposite angles are (P, R) and (Q, S). Thus, we have the relations ∠ P + ∠ R = 180° and ∠ Q + ∠ S = 180°.

Given that ∠ P = 3∠ R, substituting this into the first relation yields 3∠ R + ∠ R = 180°, which simplifies to 4∠ R = 180°, resulting in ∠ R = 45°.

Given that ∠ S = 5∠ Q, substituting this into the second relation yields ∠ Q + 5∠ Q = 180°, which simplifies to 6∠ Q = 180°, resulting in ∠ Q = 30°.

The sum of the angles Q and R is calculated as ∠ Q + ∠ R = 30° + 45° = 75°.

Explanation

Let the initial income be 100. Since Sudha saves 15% of her income, her initial savings S = 15 and her initial expenditure E = 100 - 15 = 85.

The increase in income Δ I is the sum of the increase in expenditure Δ E and the increase in savings Δ S. The increase in expenditure is 20% of 85: Δ E = ²⁰⁄₁₀₀ × 85 = 17

The increase in savings is 60% of 15: Δ S = ⁶⁰⁄₁₀₀ × 15 = 9

Total increase in income: Δ I = Δ E + Δ S = 17 + 9 = 26

The percentage increase in income is calculated as: Percentage increase = (Δ I)/(Initial Income) × 100 = ²⁶⁄₁₀₀ × 100 = 26%

Explanation

The production of type B cars in 2015 is 58 and type C cars in 2013 is 52. Their combined total is 58 + 52 = 110.

The production of type E cars in 2013 is 43 and in 2014 is 47. Their combined total is 43 + 47 = 90.

To find how much percent the first total is more than the second total, we use the formula for percentage increase:

Percentage More = (Sum 1 - Sum 2)/(Sum 2) × 100

Substituting the values:

Percentage More = (110 - 90)/(90) × 100 = ²⁰⁄₉₀ × 100 = ²⁰⁰⁄₉ ≈ 22.22%

Rounding to one decimal place, we get 22.2.

Explanation

For a number to be divisible by 88, it must be divisible by both 8 and 11. According to the divisibility rule for 8, the last three digits 3y2 must be divisible by 8. Testing possible values for y between 0 and 9: 312 / 8 = 39 352 / 8 = 44 392 / 8 = 49 The possible values for y are 1, 5, and 9.

According to the divisibility rule for 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11. Sum of digits at odd positions (2, 9, x, 4, y) = 2 + 9 + x + 4 + y = 15 + x + y Sum of digits at even positions (0, 4, 8, 3, 2) = 0 + 4 + 8 + 3 + 2 = 17 Difference = (15 + x + y) - 17 = x + y - 2

For the number to be divisible by 11, x + y - 2 must be 0 or 11. If x + y - 2 = 0 ⇒ x + y = 2. Using y = 1, we get x = 1. If x + y - 2 = 11 ⇒ x + y = 13. Using y = 5, we get x = 8. Using y = 9, we get x = 4. The largest possible value of x is 8, which occurs when y = 5. Substituting these values into the required expression: 5x - 7y = 5(8) - 7(5) = 40 - 35 = 5

Explanation

To determine the required ratio, we first identify the specific production figures from the table. The production of type A cars in 2017 is 64 thousand and the production of type D cars in 2015 is 56 thousand. The sum of these two values is 64 + 56 = 120 thousand.

Next, we identify the production of type B cars in 2013 as 47 thousand and the production of type E cars in 2013 as 43 thousand. The sum of these two values is 47 + 43 = 90 thousand.

The ratio of the first sum to the second sum is calculated as 120 : 90. By dividing both terms by their greatest common divisor, 30, the ratio simplifies to 4 : 3.

Explanation

In △ ABC, since DE is parallel to BC, the triangles ADE and ABC are similar by the Angle-Angle similarity criterion. A fundamental property of similar triangles states that the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Given the ratio AD:DB = 1:6, we can determine the length of the side AB relative to AD. Since AB = AD + DB, the ratio of AD to AB is 1 : (1 + 6), which simplifies to 1:7.

Using the area-side relationship for similar triangles: Area(ADE) / Area(ABC) = (AD / AB)² Area(ADE) / Area(ABC) = (1 / 7)² = 1 / 49

Thus, the ratio of the area of triangle ADE to the area of triangle ABC is 1:49.

Explanation

The expression is evaluated using the BODMAS rule, which dictates the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction. In this rule, the word of signifies a multiplication that must be performed before any division or standard multiplication in the sequence.

The first part of the expression, 16 ÷ 4 of 4, is simplified by first calculating the of part: 4 × 4 = 16. Then, performing the division, we get 16 ÷ 16 = 1.

Next, we evaluate the inner curly bracket {4 × 3 ÷ (3 + 3)}. Inside the parenthesis, 3 + 3 = 6. The curly bracket becomes {4 × 3 ÷ 6}. Following left-to-right order for multiplication and division, 12 ÷ 6 = 2.

The square bracket then becomes [3 ÷ 4 of 2]. Applying the of operator first, 4 × 2 = 8, so the bracket simplifies to 3 ÷ 8 or ⅜.

The final part of the expression is the parenthesis (2 ÷ 4 of 8). Applying the of operator first, 4 × 8 = 32. The result is 2 ÷ 32 = ¹⁄₁₆.

Combining these results back into the main expression, we have 1 × ⅜ ÷ ¹⁄₁₆. Changing the division by a fraction into multiplication by its reciprocal, we get 1 × ⅜ × 16. Calculating this gives 3 × 2 = 6.

Explanation

In △ ABD, since AB = BD, it is an isosceles triangle where the angles opposite to the equal sides are equal. Therefore, ∠ BDA = ∠ BAD = 70°. Using the angle sum property of a triangle, the third angle ∠ ABD = 180° - (70° + 70°) = 40°.

In △ BDC, it is given that BD = DC, making it an isosceles triangle where ∠ DBC = ∠ DCB. According to the exterior angle theorem, the exterior angle ∠ BDA of △ BDC is equal to the sum of the two interior opposite angles: ∠ BDA = ∠ DBC + ∠ DCB. Substituting the known values, 70° = 2 × ∠ DBC, which yields ∠ DBC = 35°.

The total measure of ∠ B (or ∠ ABC) is the sum of ∠ ABD and ∠ DBC: ∠ B = 40° + 35° = 75°

Explanation

Let the sum borrowed be P. Since the interest rate is 10% per annum compounded annually, the debt increases by a factor of 1.1 each year. The person clears the debt at the end of the third year by paying ₹13,200. The value of the debt at the start of the third year is calculated by discounting this final payment: Principal_Yₑₐᵣ₃ = (13200)/(1.1) = 12000

This ₹12,000 was the remaining balance after a payment of ₹6,634 was made at the end of the second year. Therefore, the total accumulated amount at the end of the second year, before the first repayment, was: Amount_Yₑₐᵣ₂ = 12000 + 6634 = 18634

The initial sum P grew to this amount over two years at a 10% compound interest rate. Using the formula A = P(1 + ʳ⁄₁₀₀)ᵗ: 18634 = P(1.1)² 18634 = 1.21P P = (18634)/(1.21) P = 15400

Explanation

To simplify calculations, assume the number of fruits bought at each rate is the least common multiple (LCM) of the given quantities 11 and 9, which is 99.

Cost price (CP) of the first batch of 99 fruits at the rate of 11 for ₹100 is ¹⁰⁰⁄₁₁ × 99 = ₹900. Cost price (CP) of the second batch of 99 fruits at the rate of 9 for ₹100 is ¹⁰⁰⁄₉ × 99 = ₹1100. Total CP for 99 + 99 = 198 fruits is 900 + 1100 = ₹2000.

The selling price (SP) of the 198 fruits at the rate of 10 for ₹100 is ¹⁰⁰⁄₁₀ × 198 = ₹1980.

Since the total cost price (₹2000) is greater than the total selling price (₹1980), the transaction results in a loss. Loss amount = 2000 - 1980 = ₹20. Loss percentage = (Loss)/(CP) × 100 = ²⁰⁄₂₀₀₀ × 100 = 1%.

Explanation

According to the President (Discharge of Functions) Act, 1969, enacted by the Parliament under the powers granted by Article 70 of the Indian Constitution, a specific hierarchy is established for the discharge of presidential duties during unforeseen contingencies. If the offices of both the President and the Vice President fall vacant simultaneously due to death, resignation, removal, or otherwise, the Chief Justice of India is mandated to act as the President. In the event that the Chief Justice is also unavailable, the senior-most judge of the Supreme Court of India available shall discharge these functions. This provision ensures that the constitutional head of the executive branch is always occupied to maintain governmental continuity. This protocol was notably invoked in 1969 when Justice M. Hidayatullah, then the Chief Justice of India, served as the Acting President.

Explanation

Tulsidas was a 16th-century saint and poet who became the most influential figure of the Rama-bhakti tradition within the Bhakti movement. He authored the epic poem Ramacharitamanasa, which retells the story of the Ramayana in the Awadhi dialect, making the life and virtues of Lord Rama accessible to the common people of North India. His literary contributions were instrumental in establishing Rama as a widely worshipped deity and a symbol of the ideal man (Maryada Purushottama). Tulsidas is also traditionally regarded as a contemporary of the Mughal Emperor Akbar and spent a significant portion of his life in the holy city of Varanasi.

Explanation

In March 2019, Padma Lakshmi, the Emmy-nominated television host and award-winning author, was named a Goodwill Ambassador for the United Nations Development Programme (UNDP). As a food expert and advocate for women's rights, her role involves mobilizing support for the Sustainable Development Goals, with a specific focus on fighting inequality, discrimination, and poverty globally. She is well-known for her work as the host and executive producer of the television series Top Chef, which aligns with the description of her as a food expert and television personality.

Explanation

Kapurthala, a city in the state of Punjab, recorded an official population of 98,916 according to the Census of India 2011. This figure refers to the total number of inhabitants within the municipal council limits of the city. Based on the 2011 census data, this population count is the lowest in this specific set of urban centers. Kapurthala serves as the administrative headquarters of its district and is recognized for its unique architectural heritage, yet it maintains a smaller demographic footprint compared to the other cities listed.

Explanation

Ornithophily is the specialized form of pollination where birds serve as the biotic agents for transferring pollen grains from the anther to the stigma. Derived from the Greek roots ornithos for bird and phily for attraction or liking, this ecological interaction involves various nectar-feeding bird species such as hummingbirds and sunbirds. Plants that rely on this method have evolved specific floral characteristics to attract birds, including large, sturdy blossoms that can support the bird weight or allow for hovering. These flowers typically exhibit vivid colors like red, yellow, or orange and produce significant quantities of nectar to satisfy the high metabolic needs of the birds. Additionally, these flowers are often tubular in shape and lack a distinct fragrance, as birds rely more on visual cues than olfactory ones.

Explanation

According to the 2011 Census, Uttar Pradesh is the most populous state in India, accounting for approximately 16.5 percent of the country's total population. Maharashtra and Bihar follow it as the second and third most populous states respectively.

Explanation

Underwater volcanic activity is a recognized cause of tsunamis because it involves the sudden and violent displacement of water. When a volcano erupts beneath the ocean floor, the release of high-pressure gases and the movement of magma can push the water column above it vertically. Additionally, the collapse of a volcanic chamber or the rapid entry of large volumes of volcanic debris into the sea can generate significant wave energy. This energy propagates through the water in the form of long-wavelength waves that travel across the ocean at high speeds, eventually intensifying into a tsunami as they reach shallow coastal areas.

Explanation

Ramananda was a pioneering saint of the Bhakti movement in North India during the 14th and 15th centuries. He is credited with shifting the focus of devotion from the traditional worship of Vishnu and Lakshmi to the personalized worship of Lord Rama and his consort Sita. By preaching in the vernacular Hindi language rather than Sanskrit, he made the path of Rama-bhakti accessible to the common people regardless of their caste or social status. His teachings established Rama as the supreme reality and the ideal of dharma, significantly influencing the subsequent Rama-centric tradition in Indian spirituality.

Explanation

Babur founded the Mughal Empire in India after his victory in the First Battle of Panipat in 1526. He defeated Ibrahim Lodi, the last ruler of the Lodi dynasty, which effectively ended the Delhi Sultanate and established Mughal authority over northern India. A descendant of Timur and Genghis Khan, Babur ruled until 1530 and laid the foundational military and political structures for the empire that would dominate the Indian subcontinent for several centuries.

Explanation

According to Article 56(1)(a) of the Constitution of India, the President may resign from his office before the expiration of his five-year term by writing under his hand addressed to the Vice-President of India. Article 56(2) further mandates that any such resignation addressed to the Vice-President must be communicated by him immediately to the Speaker of the Lok Sabha to ensure the legislative branch is officially notified of the vacancy in the presidency.

Explanation

Superstats is a suite of advanced data metrics developed through a collaboration between researchers from the Indian Institute of Technology (IIT) Madras, Gyan Data (an IIT-M incubated company), and experts from the cricket news portal ESPNcricinfo. Launched in March 2019, these metrics utilize data science and machine learning algorithms to evaluate player performance and match situations in cricket more comprehensively than traditional statistics like batting averages or strike rates. The suite includes specialized tools such as the Luck Index, which quantifies the impact of fortuitous events like dropped catches or toss wins, and the Forecaster, which predicts match outcomes based on real-time data. By accounting for the specific context of every ball bowled, Superstats provides a deeper analytical framework for the sport of cricket.

Explanation

The Vivekananda Rock Memorial is a prominent monument situated on a small rocky island off the coast of Kanyakumari, Tamil Nadu. Built in 1970 to honor the visit of Swami Vivekananda to the site in December 1892, it marks the spot where the philosopher-monk is believed to have meditated for three days and attained enlightenment before embarking on his journey to the Parliament of the World Religions in Chicago. The memorial consists of two main structures, the Vivekananda Mandapam and the Shripada Mandapam, the latter being dedicated to the Goddess Kanya Kumari. Located at the confluence of the Arabian Sea, the Bay of Bengal, and the Indian Ocean, it stands as a symbol of national unity and spiritual significance at the southernmost tip of mainland India.

Explanation

Metal oxides are generally basic in nature because they react with water to form metal hydroxides, which are alkaline. For example, when calcium oxide reacts with water, it forms calcium hydroxide as shown by the equation CaO + H₂O → Ca(OH)₂. Furthermore, metal oxides undergo neutralization reactions when they react with acids to produce salt and water, a defining characteristic of basic substances. A typical example is the reaction of magnesium oxide with hydrochloric acid: MgO + 2HCl → MgCl₂ + H₂O. This basic behavior is attributed to the presence of the oxide ion O²⁻, which acts as a Brønsted-Lowry base by accepting protons to form hydroxide ions.

Explanation

Bihar Diwas is celebrated every year on 22nd March to mark the formation of the state of Bihar. On this date in 1912, the British government carved out the province of Bihar from the Bengal Presidency. The separation followed the announcement made by King George V during the Delhi Durbar in 1911. The day is observed as a public holiday in the state to commemorate its distinct identity, history, and culture. The tradition of celebrating Bihar Diwas on a large scale was initiated by the state government in 2010 to restore the pride of the state and its people.

Explanation

While organic chemistry is broadly defined as the study of carbon-containing compounds, several specific groups of carbon compounds are traditionally classified as inorganic. These exceptions typically include simple carbon oxides like carbon monoxide (CO) and carbon dioxide (CO₂), carbonates, bicarbonates, carbides, and cyanides. Cyanides, which contain the cyano group (C ≡ N), are considered inorganic because they lack the carbon-hydrogen (C-H) bonds that characterize most organic molecules. Their chemical properties and reactions are more closely aligned with inorganic salts and minerals than with the complex structures of hydrocarbons and their derivatives. This classification is a historical convention in chemistry that separates simple carbon-bearing minerals and gases from the diverse array of molecules found in life sciences and synthetic organic chemistry.

Explanation

Sonam Kapoor was awarded the PETA India Person of the Year title for 2018 in recognition of her proactive advocacy for animal rights and her commitment to a meat-free lifestyle. She was specifically honored for her efforts to promote animal-free fashion through her vegan handbag line, Rheson, and for using her public platform to encourage fans to adopt compassionate choices. Her support for animal welfare causes, including speaking out against the use of animals in circuses and the exploitation of elephants, made her a prominent voice for the ethical treatment of animals.

Explanation

Carbon monoxide (CO) exhibits an extremely high affinity for hemoglobin, which is approximately 200 to 250 times greater than the affinity of oxygen (O₂). When inhaled, it competitively binds to the heme groups in hemoglobin to form a stable complex known as carboxyhemoglobin (COHb). Because carbon monoxide occupies the oxygen-binding sites, it prevents the attachment of oxygen molecules, thereby significantly decreasing the oxygen-carrying capacity of the blood. Furthermore, the formation of carboxyhemoglobin alters the structural conformation of the hemoglobin molecule, increasing its affinity for the remaining oxygen and making it harder for that oxygen to be released into the tissues, ultimately leading to cellular hypoxia.

Explanation

Marble cancer refers to the corrosion and degradation of marble structures caused by acid rain. This phenomenon occurs when atmospheric pollutants like sulfur dioxide (SO₂) and nitrogen oxides (NOₓ), primarily from industrial emissions and vehicular exhaust, react with water vapor and oxygen to form sulfuric acid (H₂SO₄) and nitric acid (HNO₃). When these acids fall as rain, they react chemically with the calcium carbonate (CaCO₃) present in marble. The chemical reaction is represented as CaCO₃ + H₂SO₄ → CaSO₄ + H₂O + CO₂. The resulting calcium sulfate (CaSO₄) is more soluble than the original carbonate and is easily washed away or leads to the formation of a powdery deposit. This process results in the yellowing, pitting, and gradual crumbling of the marble surface, a significant environmental concern for heritage monuments such as the Taj Mahal.

Explanation

During the third siege of Chittorgarh in 1567-1568, Emperor Akbar led a massive campaign against the Mewar capital. Jaimal Rathore and Patta Sisodia were entrusted with the defense of the fort after Maharana Udai Singh II moved to the hills. Jaimal was eventually wounded by a shot from Akbars musket, Sangram. In the final battle, Kalla Rathore carried the injured Jaimal on his shoulders so both could fight simultaneously, leading to the legend of the four-armed warrior. Both heroes achieved martyrdom at the gates of the fort while defending it against the Mughal forces. To commemorate their bravery, cenotaphs or chattris were erected at the spot where they fell, located specifically between the Hanuman Pol and Bhairon Pol within the Chittorgarh Fort.

Explanation

The 5th edition of the Economic Times Global Business Summit was held on February 22 and 23, 2019, at the Taj Palace Hotel in New Delhi. This flagship annual event is organized by the Times Network and The Economic Times to provide a platform for global thought leaders, policymakers, and corporate heads to discuss economic trends and business strategies. Prime Minister Narendra Modi delivered the keynote address during the summit, which focused on the theme India 1.5 trillion: Redefining the Indian Opportunity. New Delhi serves as the regular venue for this high-level international gathering.

Explanation

The Mansabdari system was the fundamental administrative and military framework of the Mughal Empire, introduced by Emperor Akbar to organize the nobility and military personnel. Every officer in the Mughal service was assigned a Mansab (rank), which was quantified through two designations: Jat and Sawar. The Jat rank determined the personal status of the officer within the imperial hierarchy and fixed their personal salary. The Sawar rank indicated the number of cavalrymen or horsemen the officer was required to maintain and provide for the service of the state. This dual ranking system ensured that the social standing and the military obligations of each member of the ruling elite were clearly defined and regulated by the central government.

Explanation

The motto "Let me win. But if I cannot win, let me be brave in the attempt" is the official athlete oath of the Special Olympics. This phrase was first introduced by Eunice Kennedy Shriver at the opening of the first International Special Olympics Summer Games held in Chicago in 1968. It embodies the movement's core philosophy of providing sports training and competition for individuals with intellectual disabilities, emphasizing the courage and effort of the athletes regardless of the final result. This motto served as the guiding spirit for the 2019 Special Olympics World Games hosted in Abu Dhabi, United Arab Emirates.

Explanation

Admiral Karambir Singh assumed command as the 24th Chief of the Naval Staff on May 31, 2019, succeeding Admiral Sunil Lanba. Prior to this appointment, he served as the Flag Officer Commanding-in-Chief of the Eastern Naval Command in Visakhapatnam. He was commissioned into the Executive Branch of the Indian Navy in July 1980 and earned his wings as a helicopter pilot in 1982. His tenure was notable as he was the first naval officer from the helicopter stream to lead the Indian Navy. He is a recipient of the Param Vishisht Seva Medal and the Ati Vishisht Seva Medal for his distinguished service.

Explanation

In March 2019, the Government of India extended a financial grant of 35.5 million Nepalese Rupees for the construction of the Siddheshwar Education Public Campus in the Ramechhap district of Nepal. This project was implemented under the India-Nepal Development Partnership Programme, which focuses on High Impact Community Development Projects. The grant was aimed at enhancing the educational infrastructure in Nepal, providing better facilities for students in the region and strengthening the bilateral ties through social sector development.

Explanation

The Asian Development Bank (ADB), in its Asian Development Outlook Supplement, maintained India's economic growth forecast at 7.3 percent for the fiscal year 2018-19. For the following financial year 2019-20, the bank projected a growth rate of 7.6 percent. This forecast was based on the expectation that the negative effects of the Goods and Services Tax (GST) and demonetization would fade, while a recovery in private investment and increased credit flow to the industry would support higher growth. The ADB highlighted that robust domestic demand remained the primary driver of the Indian economy, helping it withstand global trade tensions and volatile oil prices.

Explanation

Let the three numbers be x, y, and z. Based on the given information, we can set up the following equations: (x + y)/(2) + z = 164 ⟹ x + y + 2z = 328 (y + z)/(2) + x = 158 ⟹ 2x + y + z = 316 (z + x)/(2) + y = 132 ⟹ x + 2y + z = 264

By summing these three equations, we combine the terms for each variable: (x + y + 2z) + (2x + y + z) + (x + 2y + z) = 328 + 316 + 264 4x + 4y + 4z = 908

Dividing the entire equation by 4 gives the sum of the three original numbers: x + y + z = ⁹⁰⁸⁄₄ = 227

The average of the original three numbers is calculated by dividing their sum by 3: Average = (x + y + z)/(3) = ²²⁷⁄₃ = 75 ⅔

Explanation

The given trigonometric equation is tan²θ - 3sec θ + 3 = 0. Using the fundamental identity tan²θ = sec²θ - 1, the equation can be rewritten in terms of sec θ as (sec²θ - 1) - 3sec θ + 3 = 0. This simplifies to the quadratic equation sec²θ - 3sec θ + 2 = 0. Factoring this quadratic expression gives (sec θ - 1)(sec θ - 2) = 0, yielding potential solutions sec θ = 1 or sec θ = 2.

Since the problem specifies the range 0° < θ < 90°, the value sec θ = 1 is excluded because it corresponds to θ = 0°. Therefore, we take sec θ = 2, which implies cos θ = 1/2 and θ = 60°. Substituting this angle into the required expression sin θ + cot θ, we calculate sin 60° + cot 60° = √3{2} + (1)/(√(3)). Combining these terms with a common denominator gives (3 + 2)/(2√(3)) = (5)/(2√(3)). Rationalizing the expression by multiplying the numerator and denominator by √3 results in 5√3{6}.

Explanation

Let D and U be the speed of the boat downstream and upstream respectively in km/h. According to the formula Time = Distance / Speed, the given conditions are:

7.2/D + 3.2/U = 2 ... (1) 1.5/D + 0.6/U = 24/60 = 0.4 ... (2)

To eliminate U and find D, multiply equation (1) by 3 and equation (2) by 16 to equate the coefficients of 1/U to 9.6:

3 × (7.2/D + 3.2/U) = 3 × 2 ⇒ 21.6/D + 9.6/U = 6 16 × (1.5/D + 0.6/U) = 16 × 0.4 ⇒ 24/D + 9.6/U = 6.4

Subtracting the first result from the second:

(24/D - 21.6/D) + (9.6/U - 9.6/U) = 6.4 - 6 2.4/D = 0.4 D = 2.4 / 0.4 = 6

The speed of the boat downstream is 6 km/h.

Explanation

The central angle θ for a specific category in a pie chart is determined by the formula: θ = (Value of the category)/(Total value) × 360°

First, find the total production of type E cars for all given years: Total production = 48 + 55 + 64 + 65 + 68 = 300

The production of type E cars in 2013 is 55. Substituting these values into the formula to find the central angle for 2013: θ = ⁵⁵⁄₃₀₀ × 360° θ = ¹¹⁄₆₀ × 360° θ = 11 × 6° = 66°

Explanation

Let the original list price of the article be 100. The dealer buys the article at a 20% discount, so the cost price is CP = 100 - 20 = 80. The dealer then marks the article at 25% above the original list price, making the new list price 100 + 25 = 125. A discount of 20% is then allowed on this new list price to determine the selling price: SP = 125 - (0.20 × 125) = 125 - 25 = 100.

The profit percentage is calculated using the formula Profit % = (SP - CP)/(CP) × 100. Substituting the values into the formula, we get (100 - 80)/(80) × 100 = ²⁰⁄₈₀ × 100 = ¼ × 100 = 25%.

Explanation

The average production of type C cars from 2012 to 2016 is calculated by summing the production for those years and dividing by 5. The values are 44, 45, 67, 63, and 76. Sum of type C production = 44 + 45 + 67 + 63 + 76 = 295. Average of type C production = 295 / 5 = 59.

The total production of type D cars in 2012 and type E cars in 2014 is: Total = 46 (Type D, 2012) + 64 (Type E, 2014) = 110.

To find how much percent the average of type C is less than this total, we use the formula: Percentage Less = [(Total - Average) / Total] × 100 Percentage Less = [(110 - 59) / 110] × 100 Percentage Less = (51 / 110) × 100 = 510 / 11 = 46.3636...%

Rounding to one decimal place, we get approximately 46.4%.

Explanation

Let the present ages of A and B be 8x and 15x respectively. According to the problem, eight years ago, the ratio of their ages was 6:13. This relationship is expressed by the equation:

(8x - 8)/(15x - 8) = ⁶⁄₁₃

Cross-multiplying the terms results in 13(8x - 8) = 6(15x - 8). Expanding both sides gives 104x - 104 = 90x - 48. Solving for x, we subtract 90x from both sides and add 104 to both sides to get 14x = 56, which yields x = 4.

The present age of A is 8 × 4 = 32 years and the present age of B is 15 × 4 = 60 years. Eight years from now, A will be 32 + 8 = 40 years old and B will be 60 + 8 = 68 years old. The ratio of their ages after 8 years is 40 : 68, which simplifies to 10 : 17 when divided by 4.

Using the ratio method, the difference between the present ratio (8:15) and the ratio eight years ago (6:13) is 2 units for both A and B. This 2-unit difference corresponds to the 8-year interval, meaning 1 unit represents 4 years. To find the ratio 8 years from now, we add 8 years (or 2 units) to the present ratio units, resulting in (8 + 2) : (15 + 2) = 10 : 17.

Explanation

In compound interest, if the time intervals are equal, the amounts form a geometric progression. Let the principal be P, the amount after 3 years be A₃ = 18600, and the amount after 6 years be A₆ = 27900. The constant ratio k for the 3-year interval is given by k = A₆ / A₃ = 27900 / 18600 = 3 / 2. Since the growth factor remains constant over equal time periods, we have A₃ = P × k. Substituting the values, we get 18600 = P × (3 / 2). Solving for P, we find P = (18600 × 2) / 3 = 6200 × 2 = 12400.

Explanation

Based on the given percentages, we can establish the following mathematical relationships: A = B × (1 - 0.20) = 0.8B D = A × (1 - 0.25) = 0.75A C = D × (1 + 0.30) = 1.3D

To determine the relationship between B and C, we substitute the equation for A into the equation for D, and then substitute that result into the equation for C: D = 0.75 × (0.8B) = 0.6B C = 1.3 × (0.6B) = 0.78B

Multiplying the coefficients 1.3 × 0.75 × 0.8 yields 0.78, which confirms that C is 0.78 times B.

Explanation

The area of a sector of a circle with radius r and central angle θ is given by the formula: Area = (θ)/(360°) × π r²

Given the diameter of the circle is 42 cm, the radius r is ⁴²⁄₂ = 21 cm. The central angle θ is 150°. Substituting the values and using π = ²²⁄₇: Area = ¹⁵⁰⁄₃₆₀ × ²²⁄₇ × 21 × 21

Simplifying the expression: Area = ⁵⁄₁₂ × 22 × 3 × 21 Area = ⁵⁄₄ × 22 × 21 Area = (5 × 11 × 21)/(2) Area = ¹¹⁵⁵⁄₂ Area = 577.5 cm²

Explanation

The overall profit percentage on the total investment is calculated as (Total Gain)/(Total Cost) × 100 = ³⁰⁰⁄₁₂₀₀₀ × 100 = 2.5%.

By applying the rule of allegation, the ratio of the cost price of tables to the cost price of chairs is determined using their respective profit/loss percentages and the mean profit percentage. The profit on tables is +15%, the loss on chairs is -10%, and the mean profit is +2.5%.

The difference between the mean profit and the chair loss is 2.5 - (-10) = 12.5. The difference between the table profit and the mean profit is 15 - 2.5 = 12.5.

The ratio of the cost price of tables to chairs is 12.5 : 12.5, which simplifies to 1 : 1. Since the total cost of ₹12,000 is divided into two equal parts based on this ratio, the total cost of the tables is ¹⁄₁₊₁ × 12000 = ½ × 12000 = 6000.

Explanation

The total production of type A cars in 2014 and 2016 along with type C cars in 2013 is calculated by summing the values from the table: 56 + 64 + 45 = 165. The total production of type B and type D cars in the year 2014 is the sum of their respective production figures: 63 + 57 = 120. To find the required ratio, we compare these two totals: 165 : 120. Dividing both sides of the ratio by their greatest common divisor, 15, yields 165 ÷ 15 = 11 and 120 ÷ 15 = 8, which simplifies to 11 : 8.