SSC CGL Tier 1 2019 General Awareness Questions with Solutions

Explanation

According to the property of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Given △ ABC ∼ △ EDF, the corresponding sides are determined by the order of the vertices: AB corresponds to ED, BC corresponds to DF, and CA corresponds to FE. The relationship is expressed as:

(ar(△ ABC))/(ar(△ DEF)) = ((BC)/(DF))²

Substituting the given area ratio 1:4 into the formula:

¼ = ((8)/(DF))²

Taking the square root of both sides gives the ratio of the sides:

½ = (8)/(DF)

Solving for DF:

DF = 8 × 2 = 16 cm

Explanation

For a period of 2 years, the difference between compound interest (CI) and simple interest (SI) is calculated using the formula D = P · (r/100)², where P is the principal amount, r is the annual interest rate, and D is the difference. Substituting the given values D = 60.50, r = 11, and P = x, the equation becomes 60.50 = x · (11/100)².

Expanding the square gives 60.50 = x · (121/10000). To find the value of x, rearrange the equation as x = (60.50 · 10000) / 121. This simplifies to x = 605000 / 121. Since 121 · 5 = 605, it follows that 605000 / 121 = 5000. Therefore, the value of x is 5000.

Explanation

To find the required ratio, we first calculate the sum of students playing cricket in schools A and B and the sum of students playing hockey in schools C and D based on the provided table.

Total number of students playing cricket in schools A and B = 125 + 250 = 375

Total number of students playing hockey in schools C and D = 200 + 150 = 350

The ratio of these totals is calculated as: Ratio = 375 : 350

By dividing both terms by their common factor of 25: ³⁷⁵⁄₂₅ : ³⁵⁰⁄₂₅ = 15 : 14

Explanation

The algebraic identity for the sum of cubes of three variables is given by the formula: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca))

This identity can be rewritten in terms of (a + b + c) and (ab + bc + ca) by using the relation (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Substituting a² + b² + c² = (a + b + c)² - 2(ab + bc + ca) into the original identity yields: a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca))

By substituting the given values a + b + c = 8 and ab + bc + ca = 20 into this equation: a³ + b³ + c³ - 3abc = 8 × (8² - 3 × 20) a³ + b³ + c³ - 3abc = 8 × (64 - 60) a³ + b³ + c³ - 3abc = 8 × 4 a³ + b³ + c³ - 3abc = 32

Explanation

The total number of students who play football across all schools is calculated by summing the values for football in the provided table: 175 + 200 + 250 + 125 = 750. The number of students playing football in school A is 175. To determine the percentage that school A represents relative to the total number of football players, the following formula is applied: Percentage = (Football players in School A)/(Total football players) × 100 Percentage = ¹⁷⁵⁄₇₅₀ × 100 Simplifying the fraction ¹⁷⁵⁄₇₅₀ by dividing both the numerator and the denominator by 25 yields ⁷⁄₃₀. The calculation becomes ⁷⁄₃₀ × 100 = ⁷⁰⁄₃, which results in 23.33... Rounding to one decimal place, the percentage is approximately 23.3.

Explanation

The relationship between speed and time for a constant distance is inversely proportional, expressed as S ∝ (1)/(T). If a person walks at ⁵⁄₇ of his usual speed, the time taken will be the reciprocal of this ratio, which is ⁷⁄₅ of his usual time.

Let the usual time be T minutes. The new time taken is ⁷⁄₅T. The difference between the new time and the usual time is given as 10 minutes. We can set up the equation:

⁷⁄₅T - T = 10

⅖T = 10

2T = 50

T = 25

Thus, the person's usual time to reach the office is 25 minutes. This can also be solved using the ratio method where the difference in time units is 7 - 5 = 2 units. Given that 2 units correspond to 10 minutes, 1 unit equals 5 minutes. The usual time, represented by the denominator of the speed fraction (5 units), is 5 × 5 = 25 minutes.

Explanation

For a number to be divisible by 72, it must be simultaneously divisible by both 8 and 9, as 72 is the product of these two coprime numbers. According to the divisibility rule for 9, the sum of the digits of the number 55350x2 must be a multiple of 9. Adding the digits together, we get 5 + 5 + 3 + 5 + 0 + x + 2 = 20 + x. For this sum to be divisible by 9, the value of 20 + x must be 27, which is the smallest multiple of 9 greater than 20. This gives x = 7.

To confirm the divisibility by 8, the last three digits of the number, 0x2, must form a three-digit number divisible by 8. Substituting x = 7, the last three digits are 072. Since 72 is exactly divisible by 8 (8 × 9 = 72), the condition for divisibility by 8 is satisfied. Thus, the value of x is 7.

Explanation

The total number of students who play hockey across all schools is the sum of the values in the hockey row: 75 + 125 + 200 + 150 = 550. To represent this data in a pie chart, the central angle of a sector is calculated using the formula θ = (Value of Component)/(Total Value) × 360°.

For school C, the number of students playing hockey is 200. Substituting the values into the formula gives the central angle for school C as ²⁰⁰⁄₅₅₀ × 360°. Simplifying the fraction ²⁰⁰⁄₅₅₀ by dividing both the numerator and denominator by 50 results in ⁴⁄₁₁. Therefore, the angle is ⁴⁄₁₁ × 360° = (1440°)/(11) ≈ 130.909°. Rounding to the nearest whole number, the central angle is 131°.

Explanation

When two articles are sold at the same price, one at a gain of g% and another at a loss of l%, the overall gain or loss percentage is determined by the formula: Overall % = (100(g - l) - 2gl)/(200 + g - l)

Substituting the given values g = 32 and l = 28: Overall % = (100(32 - 28) - 2(32 × 28))/(200 + 32 - 28) Overall % = (100(4) - 2(896))/(204) Overall % = (400 - 1792)/(204) = ⁻¹³⁹²⁄₂₀₄ ≈ -6.82%

The negative result signifies an overall loss, which, when rounded to one decimal place, is 6.8%.

Explanation

The triangles △ ABQ and △ ABC share the same vertex A and have their bases BQ and BC on the same straight line. This means they share a common altitude h dropped from vertex A to the line BC. The area of a triangle is given by the formula Area = ½ × base × height, so the ratio of the areas of these triangles is equal to the ratio of their bases.

Given the ratio BP : PC = 4 : 5, we can let BP = 4x and PC = 5x. The total length of the base BC is BP + PC = 4x + 5x = 9x. Since Q is the midpoint of BP, the length of BQ is ½ BP = ½ × 4x = 2x.

The ratio of ar(△ ABQ) to ar(△ ABC) is: (ar(△ ABQ))/(ar(△ ABC)) = (BQ)/(BC) = ²ˣ⁄₉ₓ = ²⁄₉

Explanation

The angle subtended by an arc at the centre of a circle is double the angle subtended by it at any point on the remaining part of the circle. Let D be a point on the major arc AC of the circle. Given that the arc ABC subtends ∠ AOC = 136° at the centre, the angle at the circumference is ∠ ADC = ½ × ∠ AOC = ½ × 136° = 68°.

Points A, B, C, and D lie on the circle, forming a cyclic quadrilateral ABCD. According to the property of cyclic quadrilaterals, the exterior angle formed by producing one side is equal to the interior opposite angle. When chord AB is produced to point P, the exterior angle ∠ CBP is equal to the interior opposite angle ∠ ADC. Therefore, ∠ CBP = 68°.

Explanation

Chepu is a significant mythological figure in the Newar culture of Nepal, traditionally depicted in religious and traditional architecture. He is most commonly found at the apex of a Torana, which is the ornamental archway placed over the entrances of temples and shrines in the Kathmandu Valley. According to Newar mythology, Chepu is one of three brothers, along with Garuda and Hitimanga. He is characterized by a representation consisting only of a head and hands; legend states that he was so bashful about his appearance that he hid his body from view when Manjushree, the deity who created the Kathmandu Valley, attempted to see him. As a cultural symbol, Chepu is revered as a powerful protective deity whose presence at temple gates is believed to ward off evil and protect the sanctity of the inner shrine.

Explanation

Padmaja Naidu served as the Governor of West Bengal for a continuous period of over 10 years, from November 3, 1956, to June 1, 1967. Her tenure of approximately 10 years and 210 days established her as the longest-serving governor of any Indian state since independence. She was a prominent freedom fighter and the daughter of Sarojini Naidu, and she was honored with the Padma Vibhushan in 1962 for her distinguished service to the nation.

Explanation

Blood production, scientifically known as hematopoiesis, occurs primarily within the bone marrow. This soft, spongy tissue is found in the central cavities of certain bones, such as the pelvis, ribs, sternum, and vertebrae. The bone marrow contains multipotent hematopoietic stem cells that divide and differentiate to create all types of blood cells. These include red blood cells (erythrocytes), which transport oxygen; white blood cells (leukocytes), which are essential for the immune system; and platelets (thrombocytes), which are necessary for blood clotting. In adults, the red bone marrow is the highly vascularized site responsible for the continuous generation of these cellular components to replace those that have reached the end of their lifespan, ensuring a constant and healthy blood supply throughout the circulatory system.

Explanation

The Battle of Saragarhi was fought on 12 September 1897. It is recognized as one of historys greatest last stands, involving 21 soldiers of the 36th Sikhs of the British Indian Army who defended a small signal post against an estimated 10,000 to 12,000 Afghan tribesmen. The battle occurred in the Tirah region of the North-West Frontier Province, now in modern-day Pakistan. Led by Havildar Ishar Singh, the soldiers maintained their position to ensure communication between Fort Lockhart and Fort Gulistan remained intact. All 21 defenders were posthumously awarded the Indian Order of Merit, the highest gallantry award available to Indian soldiers at that time. To honor their bravery, the British Parliament halted its proceedings to pay tribute to them, and the date is still commemorated annually as Saragarhi Day.

Explanation

Pawan Kumar Chamling held the record for the longest-serving Chief Minister of an Indian state as of April 2019. Serving as the Chief Minister of Sikkim, he held the office from December 12, 1994, until May 27, 2019, spanning five consecutive terms. He officially surpassed the previous record of 23 years and 137 days, held by Jyoti Basu of West Bengal, on April 29, 2018. By April 2019, Chamling had completed over 24 years in the position, representing the Sikkim Democratic Front.

Explanation

The Byopa is a traditional headgear worn by the men of the Nyishi tribe, who primarily inhabit the state of Arunachal Pradesh. It is a distinctive helmet woven from cane or bamboo. Traditionally, the Byopa is decorated with the beak of a Great Indian Hornbill, which holds significant cultural and symbolic value for the tribe. In recent years, as part of conservation efforts to protect the hornbill, many tribesmen have shifted to using replicas made from wood or other synthetic materials. This headgear is a vital part of the Nyishi traditional attire and is prominently worn during major cultural festivals such as Nyokum.

Explanation

The cerebellum is located at the back of the skull and is the primary center for motor control in the brain. Its main function is to coordinate voluntary muscle movements and maintain the body's posture, balance, and equilibrium. It achieves this by integrating sensory information from the vestibular system of the inner ear, which detects head position and movement, along with proprioceptive data from muscles and joints throughout the body. By processing this information, the cerebellum ensures that movements are smooth and precise, and it makes constant, subconscious adjustments to muscle tone to keep the body stable and upright during both static and dynamic activities.

Explanation

Prime Minister Narendra Modi announced the renaming of Havelock Island as Swaraj Dweep on December 30, 2018, during his visit to Port Blair. This decision was taken to honor the memory of Netaji Subhash Chandra Bose and mark the 75th anniversary of his hoisting of the national flag on the islands. Netaji had originally proposed the names Shahid and Swaraj for the Andaman and Nicobar Islands in 1943. The renaming serves as a tribute to the freedom struggle and replaces the colonial name associated with Sir Henry Havelock with a title that signifies self-rule and national pride.

Explanation

Kabaddi was officially designated as the national sport of Bangladesh in 1972, shortly after the country gained independence. The sport has deep historical and cultural roots in the region, where it is traditionally played in rural areas and often referred to locally as Ha-du-du. To formalize the sport and oversee its development, the Bangladesh Kabaddi Federation was established in 1973. While other sports like cricket and football are highly popular, Kabaddi maintains its status as the national sport due to its indigenous heritage and the fact that it requires minimal equipment, making it accessible to the entire population.

Explanation

Wind is the movement of air molecules caused by differences in atmospheric pressure and temperature. In physics, any mass in motion possesses kinetic energy, which is defined by the formula Eₖ = ½mv², where m represents the mass of the air and v represents its velocity. As the wind flows across the specially designed blades of a turbine, it creates a lift and drag effect that forces the rotor to spin. This physical interaction captures the kinetic energy inherent in the moving air stream and transforms it into rotational mechanical power through the turbine's shaft.

Explanation

Jawaharlal Nehru Port, also known as Nhava Sheva Port, is the premier container-handling port in India. Located in Navi Mumbai, Maharashtra, it was commissioned in 1989 to decongest the Mumbai Port and has since grown to become the country's primary hub for international trade. As of 2019, it handles more than half of the total containerized cargo traffic across all major Indian ports. The port features state-of-the-art infrastructure, including multiple dedicated container terminals operated by both the port trust and private global operators. Its strategic location provides excellent connectivity to the industrial and commercial hinterlands of North, West, and Central India through an extensive network of rail and road systems, ensuring its position as the largest and most technologically advanced container gateway in the nation.

Explanation

The Heckscher-Ohlin Model, also known as the factor proportions theory, posits that international trade patterns are determined by a country's relative factor endowments. It suggests that a nation will export products that utilize its abundant and cheap factors of production, such as labor, capital, or land, and import products that require factors it lacks. By specializing in goods that align with its most plentiful resources, a country can produce and export those goods more efficiently than other nations. This model extends the concept of comparative advantage by identifying the specific resource differences that allow countries to trade effectively in the global market.

Explanation

Dr. B.R. Ambedkar authored The Buddha and His Dhamma, which was published posthumously in 1957. This monumental work serves as the primary scripture for the Navayana Buddhist movement, often referred to as Neo-Buddhism. In this text, Ambedkar provides a critical reinterpretation of the Buddha's life, his enlightenment, and the fundamental principles of the Dhamma. He emphasizes the social and ethical dimensions of Buddhism, focusing on equality, fraternity, and the rejection of the caste system. The book was completed just before his death in December 1956 and remains one of the most significant intellectual contributions to modern Buddhist literature in India.

Explanation

SAMPANN, which stands for System for Accounting and Management of Pension, is a Comprehensive Pension Management System (CPMS) launched by the Department of Telecommunications in December 2018. It was designed to facilitate the seamless processing, sanctioning, and disbursement of pensions directly into the bank accounts of telecom pensioners. The system provides an end-to-end online solution that allows retirees to track their pension status, view payment history, and register grievances, thereby ensuring transparency and efficiency in the delivery of pensionary benefits.

Explanation

In the sport of boxing, a bleeder is a professional term used to describe a fighter who has a physical predisposition to cutting easily during a bout. This susceptibility often results from the presence of significant scar tissue around the facial area, particularly near the eyebrows and nose, built up over many matches. When a boxer is labeled a bleeder, it signifies that even relatively light or glancing blows can cause lacerations that may obstruct their vision or lead to a technical knockout if the ringside physician deems the injury too severe to continue the fight.

Explanation

World AIDS Day is observed annually on 1 December to raise awareness about the HIV/AIDS pandemic and mourn those who have died from the disease. For the year 2018, which marked the 30th anniversary of the first World AIDS Day, the global theme was Know your Status. This theme was chosen by UNAIDS to highlight the critical importance of HIV testing as a gateway to all HIV treatment and prevention services. It focused on the fact that despite significant advancements in medical science, millions of people living with HIV worldwide were still unaware of their infection, making the expansion of testing services a key priority in the effort to end the AIDS epidemic.

Explanation

The Purchasing Power Parity (PPP) theory, popularized by Gustav Cassel, is the primary economic framework used for forecasting long-term exchange rate movements in a flexible system. It is based on the Law of One Price, which suggests that in the absence of transaction costs and trade barriers, identical goods should have the same price when expressed in a common currency. In the long run, the exchange rate between two countries adjusts to reflect changes in their respective price levels. If one country experiences higher inflation than another, its currency will depreciate to maintain parity in purchasing power. The relationship is often expressed as E = Pₕ / P_f, where E is the nominal exchange rate and Pₕ and P_f represent the price levels in the home and foreign countries, respectively. This theory focuses on the relative valuation of currencies based on their actual command over goods and services over extended periods.

Explanation

Entomology is the branch of zoology that focuses on the scientific study of insects. The term is derived from the Greek word entomon, which means notched or segmented, referring to the segmented body plan characteristic of insects. This field encompasses the investigation of insect taxonomy, morphology, physiology, ecology, and behavior. Given that insects comprise the largest and most diverse group of animals on Earth, entomology plays a critical role in understanding biodiversity, agricultural pest management, and the transmission of diseases.

Explanation

Glaucoma refers to a group of eye conditions that lead to progressive damage of the optic nerve, which is the critical link that transmits visual information from the retina to the brain. This damage is typically caused by an abnormally high intraocular pressure resulting from the improper drainage of aqueous humor, the fluid inside the eye. As the nerve fibers gradually deteriorate, the patient's peripheral vision begins to fade, and without medical intervention, the condition worsens over time. This progression can eventually lead to total and irreversible blindness. Because the symptoms often develop slowly and painlessly, the damage to the optic nerve may reach an advanced stage before a person notices a change in their vision.

Explanation

Nikolai Bulganin visited India in November 1955 while serving as the Chairman of the Council of Ministers of the Soviet Union, a position equivalent to that of Prime Minister. Accompanied by Nikita Khrushchev, then the First Secretary of the Communist Party, Bulganin's visit marked the first time a head of the Soviet government traveled to independent India. This landmark three-week tour across various Indian cities followed Prime Minister Jawaharlal Nehru's visit to the USSR earlier that year and was instrumental in formalizing the strategic, economic, and diplomatic ties between the two nations during the early Cold War period.

Explanation

The Parliament of India normally convenes for three sessions in a calendar year. The first is the Budget Session, which is the most significant and longest, typically held from February to May. This is followed by the Monsoon Session, which usually takes place from July to September. The third is the Winter Session, the shortest of the three, held from November to December. According to Article 85 of the Indian Constitution, the maximum interval between two consecutive sessions cannot exceed six months, which means the Parliament must meet at least twice a year. However, the convention of holding three sessions is consistently followed to manage the legislative business effectively.

Explanation

The Ahom dynasty was established by Chaolung Sukaphaa, a Tai prince who entered the Brahmaputra valley in 1228. This dynasty ruled the Ahom kingdom in present-day Assam for nearly 600 years, successfully resisting numerous external invasions, most notably the Mughal expansion during the 17th century. The Ahom administration was known for its unique Paik system, a labor-based social and military organization. The dynasty's long period of sovereignty ended in 1826 after the Burmese invasions and the subsequent signing of the Treaty of Yandabo, which led to the annexation of the region by the British East India Company.

Explanation

Mahabaleshwar, located in the Sahyadri range of the Western Ghats in Maharashtra, was recorded as the wettest place in India during the 2018 monsoon season by the India Meteorological Department (IMD). Between June 1 and September 30, 2018, the station received a cumulative rainfall of 5,619 mm. This exceptional volume of rain was driven by a vigorous Southwest Monsoon, characterized by a persistent offshore trough along the Arabian Sea and frequent low-pressure systems that funneled massive amounts of moisture toward the Western Ghats. The orographic lifting of these moisture-laden winds resulted in sustained heavy precipitation, allowing Mahabaleshwar to surpass the rainfall totals typically recorded in the traditional high-rainfall regions of Northeast India for that specific year.

Explanation

Mount Pedro, also known as Pidurutalagala, is the highest mountain in Sri Lanka with an elevation of 2,524 meters (8,281 feet) above sea level. It is located in the Central Province near the city of Nuwara Eliya. The peak is a prominent geographical feature of the central highlands and serves as a critical site for the nation's telecommunications and military radar systems. Because of its strategic importance, the summit is maintained as a high-security zone under the control of the Sri Lankan security forces.

Explanation

Wajid Ali Shah was the eleventh and final Nawab of Awadh, reigning from 1847 to 1856. His kingdom was annexed by the British East India Company on February 7, 1856, under the orders of Governor-General Lord Dalhousie. The British justified the annexation by citing widespread maladministration and misgovernance within the state, which was a violation of the Treaty of 1801. Following his deposition, Wajid Ali Shah was exiled to Garden Reach in Calcutta, where he spent the remainder of his life. He is widely remembered as a great patron of the arts, particularly for his contributions to Kathak dance and Hindustani classical music. The annexation of his kingdom was one of the primary grievances that led to the Indian Rebellion of 1857.

Explanation

In △ ABC, since vertex A is common to both △ ABQ and △ ABC, and their bases BQ and BC lie on the same straight line, the ratio of their areas is equal to the ratio of the lengths of their bases. Given the ratio BP : PC = 4 : 11, let BP = 4x and PC = 11x. The total length of the base BC is BP + PC = 4x + 11x = 15x. Since Q is the midpoint of BP, the length of segment BQ = ½ BP = ½(4x) = 2x. The ratio of the area of △ ABQ to the area of △ ABC is given by (ar(△ ABQ))/(ar(△ ABC)) = (BQ)/(BC) = ²ˣ⁄₁₅ₓ = ²⁄₁₅.

Explanation

The selling price after successive discounts is determined by the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)). Given the marked price MP = 480 and the first discount d₁ = 25%, the price after the first discount is 480 × (1 - ²⁵⁄₁₀₀) = 480 × ¾ = 360.

This intermediate price of 360 is then subjected to a second discount of x% to reach the final selling price of 288. The discount amount is 360 - 288 = 72. The value of x is calculated as the percentage of this discount relative to the intermediate price: x = ⁷²⁄₃₆₀ × 100. Simplifying the fraction, ⁷²⁄₃₆₀ = ⅕, so x = ⅕ × 100 = 20.

Explanation

In a class of 50 students, the number of girls is 40% of 50, which is 50 × 0.40 = 20, and the number of boys is 50 - 20 = 30. The total marks of the class are calculated by multiplying the total number of students by the class average: 50 × 64.4 = 3220. The total marks of the boys are calculated by multiplying the number of boys by their average: 30 × 62 = 1860. The total marks of the girls are the difference between the class total and the boys' total: 3220 - 1860 = 1360. Dividing the girls' total marks by the number of girls gives the average marks of the girls: 1360 / 20 = 68.

Alternatively, using the deviation method based on the ratio of girls to boys (40:60 or 2:3), the boys' average of 62 is 2.4 units below the class average of 64.4. The total deviation for the boys is 3 × (-2.4) = -7.2. Since the net deviation from the mean must be zero, the total deviation for the 2 parts of girls must be +7.2. The deviation per girl part is 7.2 / 2 = 3.6. Adding this to the class average gives the girls' average marks: 64.4 + 3.6 = 68.

Explanation

The combined efficiency of A, B, and C is 5 + 3 + 2 = 10 units per hour. Since the three together complete the task in 21 hours, the total work is calculated as 10 × 21 = 210 units.

The specific portion of the task B needs to complete is 40% of the total work, which is 0.40 × 210 = 84 units. Given that the efficiency of B is 3 units per hour, the time taken by B alone to complete this portion is ⁸⁴⁄₃ = 28 hours.

Explanation

The value of the expression a³ + b³ + c³ - 3abc is determined using the standard algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca))

To simplify the calculation, the term (a² + b² + c² - (ab + bc + ca)) can be rewritten using the square of the sum (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). By subtracting 3(ab + bc + ca) from both sides of the squared identity, we get the modified identity: a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca))

Substituting the given values a + b + c = 10 and ab + bc + ca = 32 into this formula: a³ + b³ + c³ - 3abc = 10 × (10² - 3 × 32) a³ + b³ + c³ - 3abc = 10 × (100 - 96) a³ + b³ + c³ - 3abc = 10 × 4 = 40

Explanation

The angle subtended by an arc at the centre of a circle is double the angle subtended by it at any point on the remaining part of the circle. Given that arc ABC subtends ∠ AOC = 110° at the centre O, let us consider a point D on the major arc of the circle. The angle subtended by the same arc at point D is ∠ ADC = ½ × ∠ AOC = (110°)/(2) = 55°.

In the cyclic quadrilateral ABCD, the side AB is produced to point P. According to the exterior angle property of a cyclic quadrilateral, the exterior angle formed at a vertex is equal to the interior opposite angle. Therefore, the exterior angle ∠ CBP is equal to the interior opposite angle ∠ ADC. By substituting the value calculated above, we get ∠ CBP = 55°.

Explanation

In a circle, the perpendicular from the center to a chord bisects the chord. This geometry creates a right-angled triangle where the radius r serves as the hypotenuse, the distance from the center d is one leg, and half the length of the chord x is the other leg.

According to the Pythagorean theorem: r² = d² + x²

Substituting the given values r = 17 cm and d = 8 cm: 17² = 8² + x² 289 = 64 + x² x² = 225 x = √(225) = 15 cm

The total length of the chord is 2x, which equals 2 × 15 = 30 cm.

Explanation

The central angle θ for any component in a pie chart is calculated using the formula θ = (Component Value)/(Total Value) × 360°.

First, calculate the total number of cars sold in each year: Total in 2011 = 500 + 450 + 400 = 1350 Total in 2012 = 480 + 420 + 450 = 1350 Total in 2013 = 520 + 530 + 460 = 1510 Total in 2014 = 620 + 480 + 520 = 1620 Total in 2015 = 650 + 520 + 540 = 1710 Total in 2016 = 630 + 400 + 430 = 1460

The grand total of cars sold over all six years is the sum of these yearly totals: Grand Total = 1350 + 1350 + 1510 + 1620 + 1710 + 1460 = 9000

To find the central angle for the year 2013, substitute the total sales for 2013 (1510) and the grand total (9000) into the formula: θ = ¹⁵¹⁰⁄₉₀₀₀ × 360° θ = ¹⁵¹⁄₉₀₀ × 360° θ = ¹⁵¹⁄₅ × 2° θ = ³⁰²⁄₅ = 60.4°

Rounding to the nearest whole number, the central angle is 60°.

Explanation

The relationship between expenditure, price, and consumption is defined by the formula E = P × C, where E is expenditure, P is price, and C is consumption. Let the initial price be 100 and the initial consumption be 100, resulting in an initial expenditure of 100 × 100 = 10,000.

With a 17% increase in price, the new price becomes 117. The person intends to increase the expenditure by only 5%, making the new target expenditure 10,000 × 1.05 = 10,500. To find the required percentage decrease in consumption, we compare the difference between the updated price and the allowed expenditure level against the updated price:

Percentage Decrease = (New Price - Target Expenditure)/(New Price) × 100

Substituting the values: Percentage Decrease = (117 - 105)/(117) × 100 Percentage Decrease = ¹²⁄₁₁₇ × 100 = ¹²⁰⁰⁄₁₁₇

Performing the division: 1200 ÷ 117 ≈ 10.256...

Rounding to the nearest decimal place gives approximately 10.3%.

Explanation

Total cars sold by showroom B during the years 2014 and 2016 is calculated by adding the values from the provided table: 480 + 400 = 880.

Total cars sold by showroom C during the years 2015 and 2016 is calculated by summing the respective values: 540 + 430 = 970.

The required ratio is the relationship between the total for showroom B and the total for showroom C: Ratio = 880 : 970

Simplifying this ratio by dividing both components by 10 yields 88 : 97.

Explanation

Total number of cars sold by all three showrooms in the year 2015 is calculated by adding the sales of showroom A, B, and C for that year: 650 + 520 + 540 = 1710.

Total number of cars sold by all three showrooms in the year 2016 is calculated by adding the sales of showroom A, B, and C for that year: 630 + 400 + 430 = 1460.

The decrease in the number of cars sold in 2016 as compared to 2015 is 1710 - 1460 = 250.

The percentage decrease is calculated using the formula Percentage Decrease = (Decrease)/(Original Value) × 100.

Substituting the values, we get: Percentage Decrease = ²⁵⁰⁄₁₇₁₀ × 100 = ²⁵⁰⁰⁄₁₇₁ ≈ 14.6198%

Rounding to one decimal place, the percentage decrease is 14.6%.

Explanation

The given ratio can be expressed as a linear equation: (5a - 3b)/(4a - 2b) = ⅔

By cross-multiplying the denominators with the numerators on the opposite sides, we get: 3(5a - 3b) = 2(4a - 2b) 15a - 9b = 8a - 4b

Next, group the terms containing a on one side and the terms containing b on the other side of the equation: 15a - 8a = 9b - 4b 7a = 5b

To find the ratio a : b, divide both sides by b and by 7: (a)/(b) = ⁵⁄₇

The ratio a : b is equal to 5 : 7.

Explanation

The cost price (CP) of each article is determined from the selling price (SP) and the respective gain or loss percentage using the formula CP = (SP × 100)/(100 ± R). For the first article sold at a 30% gain, CP₁ = (962 × 100)/(130) = 740. For the second article sold at a 26% loss, CP₂ = (962 × 100)/(74) = 1300.

The total cost price for both articles is 740 + 1300 = 2040. The total selling price is 962 × 2 = 1924. Since the total cost price is higher than the total selling price, the seller incurs an overall loss. The net loss is 2040 - 1924 = 116. The overall loss percentage is calculated as (Net Loss)/(Total CP) × 100 = ¹¹⁶⁄₂₀₄₀ × 100 ≈ 5.686%. Rounding to the nearest one decimal place results in a 5.7% loss.

Explanation

For a number to be divisible by 99, it must be divisible by both 9 and 11 because 99 = 9 × 11.

Applying the divisibility rule for 9, the sum of the digits of 8439x53 must be divisible by 9. The sum is 8 + 4 + 3 + 9 + x + 5 + 3 = 32 + x. For 32 + x to be a multiple of 9, the smallest possible value for x is 4, as 32 + 4 = 36, which is divisible by 9.

Applying the divisibility rule for 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be 0 or a multiple of 11. Sum of digits at odd positions: 8 + 3 + x + 3 = 14 + x. Sum of digits at even positions: 4 + 9 + 5 = 18. The difference is (14 + x) - 18 = x - 4. For this difference to be 0, x - 4 = 0, which gives x = 4.

Since x = 4 satisfies the divisibility criteria for both 9 and 11, the number 8439453 is divisible by 99.