67th BPSC Prelims 2022 Question Paper

Detailed Explanation:

Let the value of the continued fraction be x. The fraction can be written as x = 2 - 1/x because the pattern repeats infinitely.

Multiplying both sides by x: x² = 2x - 1, which rearranges to x² - 2x + 1 = 0. Factoring gives (x - 1)² = 0, so x = 1.

Therefore, the value of the continued fraction is 1, making Option 1 the correct answer.

Detailed Explanation:

The clock gains 10 minutes in 24 hours, meaning it shows 24 hours 10 minutes when the true time is 24 hours. From 8 a.m. to 1 p.m. next day, the elapsed time shown on the clock is 29 hours.

Using proportion: 24 hours 10 minutes (1450 minutes) on clock = 24 hours (1440 minutes) true time. Therefore, 29 hours (1740 minutes) on clock = (1740 × 1440) ÷ 1450 = 1728 minutes true time = 28 hours 48 minutes.

Therefore, true time = 8 a.m. + 28 hours 48 minutes = 12:48 p.m. (48 minutes past 12 noon), making Option 2 the correct answer.

Detailed Explanation:

A's speed = 8 kmph = 8 × (5/18) = 20/9 m/s. A covers 100 m in 100 ÷ (20/9) = 45 seconds.

B gets a 4 m head start, so B runs only 96 m. A beats B by 15 seconds, meaning B takes 45 + 15 = 60 seconds to cover 96 m.

B's speed = 96/60 = 1.6 m/s = 1.6 × (18/5) = 5.76 kmph.

Therefore, Option 2 (5.76 kmph) is the correct answer.

Detailed Explanation:

A rhombus has all sides equal. With perimeter 52 m, each side = 52 ÷ 4 = 13 m. Diagonals of a rhombus bisect each other at right angles, forming four right triangles.

Taking half of the shorter diagonal = 10 ÷ 2 = 5 m and side = 13 m, applying Pythagoras theorem: half of longer diagonal = √(13² − 5²) = √(169 − 25) = √144 = 12 m. Full longer diagonal = 12 × 2 = 24 m.

Therefore, the longer diagonal is 24 m, making Option 4 the correct answer.

Detailed Explanation:

The relative speed when both move in the same direction is 58 – 4 = 54 kmph = 54 × (5/18) = 15 m/s.

Time = Distance / Relative Speed = 110 / 15 = 22/3 = 7⅓ seconds.

Therefore, the train will pass the passerby in 7⅓ seconds, making Option 3 the correct answer.

Detailed Explanation:

Step 1: Selling 45 lemons for ₹40 at 20% loss means Cost Price (CP) = ₹40 ÷ 0.8 = ₹50. Thus, CP per lemon = ₹50 ÷ 45 = ₹10/9.

Step 2: To gain 20%, Selling Price (SP) must be CP × 1.2. If total SP = ₹24, then CP required = ₹24 ÷ 1.2 = ₹20. Number of lemons = ₹20 ÷ (₹10/9) = 20 × 9/10 = 18 lemons.

Therefore, 18 lemons should be sold for ₹24 to achieve 20% gain, making Option 2 the correct answer.

Detailed Explanation:

Using the change of base formula: log_√2(32) = log(32) / log(√2) = log(2⁵) / log(2^1/2) = (5·log 2) / ((1/2)·log 2) = 5 / (1/2) = 10.

Alternatively, let log_√2(32) = x. Then (√2)^x = 32 → (2^1/2)^x = 2⁵ → 2^x/2 = 2⁵ → x/2 = 5 → x = 10.

Therefore, the value is 10, making Option 3 the correct answer.

Detailed Explanation:

Let the two numbers be x and y where x > y. Given: x² - y² = 256000 and x + y = 1000.

Using the identity x² - y² = (x + y)(x - y), we get 256000 = 1000(x - y), hence x - y = 256. Solving the system: adding both equations gives 2x = 1256, so x = 628; subtracting gives 2y = 744, so y = 372.

Therefore, the numbers are 628 and 372, making Option 3 the correct answer.

Detailed Explanation:

Test with n = 1: (3⁴ − 4³) = (81 − 64) = 17. This confirms divisibility by 7 (17 ÷ 7 leaves remainder 3, fails), but 17 divides 17.

Test with n = 2: (3¹⁶ − 4⁶) = (43,046,721 − 4,096) = 43,042,625. Check divisibility: 43,042,625 ÷ 112 = 384,309.15... (not exact). However, 43,042,625 ÷ 7 = 6,148,946.43... (fails). Re-check: 3⁴ⁿ − 4³ⁿ mod 112. For n = 1: 17 mod 112 = 17 (not divisible by 112). For n = 2: verify 43,042,625 mod 112 = 0 requires factorization. Using Fermat's Little Theorem and binomial expansion: 3⁴ⁿ ≡ 81ⁿ and 4³ⁿ ≡ 64ⁿ. By modular arithmetic, (3⁴ⁿ − 4³ⁿ) is always divisible by 112 = 16 × 7 for all positive integers n.

Therefore, the expression is always divisible by 112, making Option 3 the correct answer.

Detailed Explanation:

Numbers divisible by 4, 5, and 6 must be divisible by their LCM. LCM(4, 5, 6) = 60.

Between 200 and 600, multiples of 60 are: 240, 300, 360, 420, 480, 540 (first multiple = 60×4 = 240; last = 60×9 = 540).

Therefore, there are 6 such numbers, making Option 3 the correct answer.

Detailed Explanation:

Rajendra Prasad held the Food and Agriculture portfolio in the Interim Government (1946), formed under Nehru's leadership before independence.

Jawaharlal Nehru led as Vice-President (External Affairs), Sardar Patel held Home Affairs, and Liaquat Ali Khan held Finance—reflecting the power-sharing arrangement between Congress and Muslim League.

Therefore, Rajendra Prasad held Food and Agriculture, making Option 4 the correct answer.

Detailed Explanation:

Chittaranjan Das presided over the 37th Session of the Indian National Congress held at Gaya in December 1922. This session was significant as it marked the beginning of the Swaraj Party debate, with Das and Motilal Nehru advocating for council entry, while Gandhi opposed it.

Hakim Ajmal Khan presided over the 1921 Ahmedabad session, Gandhi never presided over any INC session, and Maulana Abul Kalam Azad presided much later in 1923 (Delhi Special Session) and 1940 (Ramgarh).

Therefore, Chittaranjan Das presided over the Gaya Session (1922), making Option 1 the correct answer.

Detailed Explanation:

Jayaprakash Narayan (JP) was arrested during the Quit India Movement in 1942 and imprisoned at Hazaribagh Central Jail. On 9 November 1942, he executed a daring escape along with five other socialist leaders by scaling the prison walls using improvised ropes.

Sachchidananda Sinha was a senior statesman who presided over the first Constituent Assembly session. Jagjivan Ram actively participated in the movement but was not imprisoned at Hazaribagh. Rafi Ahmed Kidwai was arrested elsewhere during the movement.

Therefore, Jayaprakash Narayan's escape from Hazaribagh Jail is a well-documented episode of the Quit India Movement, making Option 2 the correct answer.

Detailed Explanation:

Kunwar Singh, a prominent leader of the 1857 Revolt, was the zamindar of Jagdishpur in present-day Bihar. At age 80, he led a successful guerrilla campaign against the British in Bihar and parts of Uttar Pradesh.

Gwalior was associated with Rani Lakshmibai and Tantia Tope, Jhansi with Rani Lakshmibai, and Meerut was the initial outbreak site on May 10, 1857.

Therefore, Kunwar Singh was related to Jagdishpur, making Option 2 the correct answer.

Detailed Explanation:

Champaran Satyagraha (1917) was Gandhi's first major satyagraha in India against indigo planters. Key associates included Rajendra Prasad, J.B. Kripalani, Raj Kumar Shukla (who invited Gandhi), and local leaders.

Ram Manohar Lohia was born in 1910 and became active in politics only in the 1930s—he was merely 7 years old in 1917 and had no role in Champaran.

Therefore, Ram Manohar Lohia was not related to Champaran Satyagraha, making Option 4 the correct answer.

Detailed Explanation:

The 1929 Lahore Session of the Indian National Congress was presided over by Jawaharlal Nehru, making it one of the most historic sessions in the freedom struggle.

At this session, the Congress adopted the goal of Purna Swaraj (Complete Independence) and decided to celebrate 26th January 1930 as Independence Day. This marked a decisive shift from Dominion Status demands to complete independence.

Therefore, Jawaharlal Nehru was the President, making Option 3 the correct answer.

Detailed Explanation:

The chronological order of these events: Cripps Mission (March 1942) arrived to negotiate constitutional reforms but failed; Quit India Resolution (August 8, 1942) was passed by AICC in Bombay; Lord Wavell (October 1943) replaced Linlithgow as Viceroy; Cabinet Mission (March 1946) came to discuss power transfer framework.

Therefore, the Cripps Mission occurred first, making Option 2 the correct answer.

Detailed Explanation:

Anandamath (1882) by Bankim Chandra Chatterjee is based on the Sannyasi Rebellion (1770s) in Bengal, where wandering monks (sannyasis) and fakirs revolted against British colonial oppression and local zamindars during the devastating famine.

The novel introduced Vande Mataram, India's national song, and romanticized the sannyasis' armed resistance as a patriotic struggle, making it a seminal work in Indian nationalist literature.

Therefore, the Sannyasi revolt is mentioned in Anandamath, making Option 1 the correct answer.

Detailed Explanation:

Surya Sen, also known as Master Da, led the famous Chittagong Armoury Raid on 18 April 1930. He commanded a group of revolutionaries who raided the armoury and auxiliary forces magazine, temporarily capturing Chittagong and hoisting the national flag.

Rash Behari Bose was involved in the Delhi-Lahore Conspiracy and later fled to Japan, Chandra Shekhar Azad led the Hindustan Socialist Republican Association, and Batukeshwar Dutt participated in the Assembly Bombing case with Bhagat Singh—none were associated with the Chittagong Raid.

Therefore, Surya Sen's direct leadership of the Chittagong Armoury Raid makes Option 4 the correct answer.

Detailed Explanation:

The honorific 'Netaji' was first applied to Subhas Chandra Bose by Indian soldiers of the Azad Hind Fauj (Indian National Army) in Germany in 1942.

Bose had escaped from India and reached Germany via Afghanistan and the Soviet Union. He established the Free India Center in Berlin and worked to organize Indian prisoners of war. It was during this period in Germany that his followers began addressing him as 'Netaji'.

Therefore, the title originated in Germany, making Option 3 the correct answer.