ITBP Head Constable (Telecommunication) 18 Sep 2023 Shift 1 Official Paper (Previous Year Paper)

The correct answer is Option 4.

 

Key Points

• The primary issue in the sentence is the incorrect form of the adverb "truly" being used.
• Adverbs modify verbs, adjectives, or other adverbs and typically end in "-ly".
• The correct phrase should be "released a truly masterful collection," ensuring grammatical correctness and clarity.

Therefore, the correct answer is Option 4.

 

Correct Sentence: "It took him his third try before he released a truly masterful collection."

 

Additional Information

• Option 1: "releases a truly" – This option is incorrect because the verb "releases" is in the wrong tense.
• Option 2: "No correction" – This option is incorrect because there is a grammatical error in the sentence.
• Option 3: "has truly released" – This option is incorrect because the present perfect tense is not appropriate in this context.

The correct answer is 'Option 4'.

Key Points

• The word "Knack" means a special skill, talent, or aptitude.
• "Knack" is the correct answer as it directly corresponds to the given description.

Hence, the correct answer is 'Option 4'.

Correct Usage: She has a knack for solving complex problems effortlessly.

Additional Information

• Option 1: "Lair" is specific to animals and does not indicate skill or aptitude.
• Option 2: "Ruddy" does not fit the context of a skill or talent.
• Option 3: "Chastise" is about discipline and criticism, not related to aptitude.

The correct answer is: Option 4 i.e. Queer someone’s pitch.

Key Points

• "Queer someone’s pitch" means to spoil someone’s plans or chances of doing something. (किसी के योजनाओं या अवसरों को बिगाड़ देना)
• Example: His constant interruptions queered my pitch during the presentation.

Therefore, the correct answer is: Option 4 i.e. Queer someone’s pitch.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Pull the plug (बंद करना): To stop something; to bring something to an end, especially by cutting off a source of support.
• Example: The company decided to pull the plug on the project due to lack of funds.
• Go bananas (बहुत उत्साहित या नाराज होना): To become very excited or angry.
• Example: The crowd went bananas when their team won the match.
• Hit the hay (सोने के लिए बिस्तर पर जाना): To go to bed in order to sleep.
• Example: I’m really tired, so I’m going to hit the hay early tonight.

The correct answer is Option 4.

 

Key Points

• The primary issue in the sentence is the preposition used with the word "resilience."
• The correct preposition to use with "resilience" in this context is "to."
• Thus, the phrase should be "improve resilience to heat," ensuring grammatical correctness and clarity.

Therefore, the correct answer is Option 4.

 

Correct Sentence: "How can regular physical activity improve resilience to heat?"

 

Additional Information

• Option 1: "No correction" – This option is incorrect because the original sentence contains a grammatical error.
• Option 2: "improving resilience on" – This option is incorrect because "on" is not the correct preposition to use with "resilience" in this context.
• Option 3: "improves resilience with" – This option is incorrect because "improves" should be in the base form "improve" to match the modal verb "can," and "with" is not the correct preposition in this context.

The correct answer is: Option 1.

Key Points

• The idiom "On the fence" means to be undecided or unsure about something. (दुविधा में होना)
• Example: Many students are on the fence about which course to choose for their future studies.

Therefore, the correct answer is: Option 1.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• As pale as a ghost (बहुत पीला): Extremely pale, usually due to fear or illness.
• Example: After hearing the shocking news, she looked as pale as a ghost.
• On cloud nine (बहुत खुश होना): Extremely happy or elated.
• Example: She was on cloud nine after winning the lottery.
• Green thumb (माली का हुनर): Having a natural talent for gardening.
• Example: My grandmother has a green thumb; her garden is always blooming beautifully.

The correct answer is option 4.

Key Points

The misspelled word is 'Maintanence'.

• The correct spelling would be 'Maintenance'.
• The word 'maintenance' is a noun which means 'the process of maintaining or preserving someone or something'. (रखरखाव)
• For Example -
• Regular maintenance of the machinery is crucial to ensure its smooth functioning.
• Therefore, the correct answer would be 'Maintanence'.

 

Additional Information

Let's understand the meanings of the other marked words - 

• Guidance - advice or information aimed at resolving a problem or difficulty, especially as given by someone in authority (मार्गदर्शन)
• Corpulent - (of a person) fat (मोटा)
• Blasphemy - the act or offense of speaking sacrilegiously about God or sacred things; profane talk (ईशनिंदा)

The correct answer is: Option 2 i.e. Pale.

Key Points

• The word "Dusky" means somewhat dark in color, typically used to describe a complexion, sky, or environment that is dim or shadowy. (धुंधला, सांवला)
• Example: The dusky sky indicated that it was almost nightfall.
• "Pale" refers to a light or washed-out color, often used to describe a complexion that lacks color or is very light. (पीला, फीका)
• Example: After staying indoors for months, her skin appeared pale and lacking in color.
• Hence, we can infer that the synonym of 'dusky' is 'pale'.

Therefore, the correct answer is: Option 2 i.e. Pale.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Lucent (उज्ज्वल): Shining, glowing with light.
• Example: The lucent moonlight reflected off the surface of the lake.
• Beamy (उज्ज्वल): Radiant; full of light.
• Example: The beamy sun filled the room with warmth and light.
• Mystical (रहस्यमय): Having a spiritual, mysterious, or otherworldly quality.
• Example: The old forest had a mystical aura that fascinated everyone who visited.

The correct answer is: Option 2 i.e., 'Solar panels may also reflect bats’ echolocation calls.'

Key Points

• The word "reflect" means to throw back (light, heat, sound, or an image) without absorbing it. (परावर्तित करना)
• Example: The mirror can reflect sunlight directly into the room.
• The word "echolocation" refers to the biological sonar used by animals like bats to navigate and locate objects by emitting sound waves. (गूंज स्थानिकरण)
• Example: Bats rely on echolocation to hunt for insects in the dark.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: "relfect" is a spelling error; the correct spelling is "reflect." (परावर्तित करना)
• Example: The water seemed to relfect the blue of the sky. (Incorrect spelling)
• Option 3: "ecolocation" is a spelling error; the correct spelling is "echolocation." (गूंज स्थानिकरण)
• Example: Dolphins use ecolocation to detect objects underwater. (Incorrect spelling)
• Option 4: "palens" is a spelling error; the correct spelling is "panels." (पैनल्स)
• Example: The solar palens are installed on the roof. (Incorrect spelling)

The correct answer is 'Option 1'.

Key Points

• The word "Osteology" refers to the study of the structure and function of the skeleton and bones. It involves the recovery and interpretation of human skeletons. (अस्थि विज्ञान)
• It helps in forensic investigations and understanding human evolution and anthropology.
• Example: Osteology has provided insights into the diet and health of ancient civilizations through the study of skeletal remains.

Hence, the correct answer is 'Option 1'.

Correct Sentence: Osteology is an essential branch of biological anthropology that aids in identifying human remains.

Additional Information

• Option 2: "Ophthalmology" deals with the diagnosis and treatment of eye disorders. (नेत्र विज्ञान)
• Example: She consulted an ophthalmologist for her vision problems.
• Option 3: "Dermatology" is the branch of medicine concerned with the skin and its diseases. (त्वचा विज्ञान)
• Example: Dermatology treatments are essential for severe acne cases.
• Option 4: "Cytology" is the study of cells, their structure, function, and chemistry. (कोशिका विज्ञान)
• Example: Cytology plays a vital role in diagnosing cancer through cell examination.

The correct answer is: Option 3 i.e. Exultation.

Key Points

• The word "lament" means to express sorrow, mourning, or regret for often demonstratively; it can also refer to a passionate expression of grief or sorrow. (विलाप करना)
• Example: She lamented the loss of her beloved pet.
• "Exultation" refers to a feeling of triumphant elation or jubilation; rejoicing. (उल्लास)
• Example: His exultation was evident when he received the news of his promotion.
• Hence, we can infer that the opposite of 'lament' is 'exultation'.

Therefore, the correct answer is: Option 3 i.e. Exultation.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Repent (पश्चाताप करना): Feel or express sincere regret or remorse about one's wrongdoing or sin.
• Example: He repented for his harsh words after realizing they hurt her feelings.
• Deplore (निंदा करना): Feel or express strong disapproval of (something).
• Example: Many people deplore the use of violence in any form.
• Bewail (शोक करना): Express great regret, disappointment, or bitterness over (something).
• Example: They bewailed the lack of progress in their community project.

सही उत्तर - परिश्रम करने पर भी लाभ कम प्राप्त होना।

Key Points

• लोकोक्ति - जब कोई पूरा कथन किसी प्रसंग विशेष में उद्धत किया जाता है, लोकोक्ति कहलाता है। इसी को कहावत कहते है।
• जैसे - 'एक आँख से देखना' का अर्थ - सबको बराबर समझना, वाक्य प्रयोग - विनोद चाचा सभी को एक आँख से देखते है।
• अन्य विकल्प -

Additional Information

सही उत्तर है- अभिषेक ने

Key Points

• अशुद्ध वाक्य - अभिषेक ने हंसमुख स्वभाव और बाल सुलभ बातें सुनकर कमल हँस पड़ा।
• शुद्ध वाक्य - अभिषेक का हंसमुख स्वभाव और बाल सुलभ बातें सुनकर कमल हँस पड़ा।
• वाक्य में कारक संबंधित अशुद्धि है।

Additional Informationवाक्य अशुद्ध- शुद्ध के कुछ उदाहरण-

सही उत्तर - लेकर।

Key Points

• अशुद्ध वाक्य - लोमड़ी और भेड़िए में किसी बात को पर आपस में लड़ाई हो गई।
• शुद्ध वाक्य - लोमड़ी और भेड़िए में किसी बात को लेकर आपस में लड़ाई हो गई।
• वाक्य में शब्द चयन संबंधित त्रुटि थी।

Additional Informationवाक्यों में निम्नलिखित अशुद्धियाँ  होती है - 

• संज्ञा संबंधित वाक्य शुद्धि
• अशुद्ध - मैं रविवार के दिन तुम्हारे घर आऊँगा।
• शुद्ध - मैं रविवार को तुम्हारे घर आऊँगा।
• लिंग संबंधित अशुद्धियाँ
• अशुद्ध - सविता ने जोर से हँस दिया।
• शुद्ध - सविता जोर से हँस दी।
• वचन संबंधित अशुद्धियाँ
• अशुद्ध -मेरे आँसू से रूमाल भींग गया।
• शुद्ध - मेरे आँसुओं से रूमाल भींग गया।
• कारक संबंधित अशुद्धियाँ
• अशुद्ध - सब से नमस्ते।
• शुद्ध - सब को नमस्ते।
• सर्वनाम संबंधित अशुद्धियाँ
• अशुद्ध - आप आपका काम करो।
• शुद्ध - आप अपना काम करो।

जो आज्ञा का पालन करता हो - आज्ञाकारी।

Key Points

• वाक्यांश के लिए उपयुक्त शब्द - अनेक शब्दों के लिए एक शब्द को प्रयुक्त करना ही वाक्यांश के लिए एक शब्द कहलाता है।
• अन्य विकल्प -

Additional Informationकुछ महत्वपूर्ण वाक्यांश के लिए एक शब्द -

अपने मुँह मियाँ मिडू बनना मुहावरे का सही अर्थ - अपनी प्रशंसा स्वंय करना।

Key Points

• मुहावरा - ऐसे वाक्यांश, जो सामान्य अर्थ का बोध न कराकर किसी विलक्षण अर्थ को प्रकट करते है, उसे मुहावरा कहते है।
• उदाहरण के लिए - कुत्ते की मौत मरना  - बुरी तरह मरना।
• मुहावरे के अन्य उदाहरण -

निर्धन का पर्यायवाची शब्द - गरीब।

Key Points

• पर्यायवाची - ऐसे शब्द जिनके अर्थ समान हों, पर्यायवाची शब्द कहलाते हैं।
• निर्धन के अन्य पर्यायवाची - दरिद्र, निर्धन, कंगाल, ग़रीबी, दीन आदि।
• अन्य विकल्प -

Additional Informationकुछ महत्वपूर्ण पर्यायवाची शब्द -

पराधीन का विलोम शब्द - स्वाधीन।

Key Points

• पराधीन का अर्थ - जो दूसरे के अधीन हो।
• स्वाधीन का अर्थ - जो अपने सिवा और किसी के अधीन न हो।
• विलोम - जब कोई एक शब्द किसी दूसरे शब्द का उल्टा या विपरीत अर्थ व्यक्त करे, तो उसे विलोम शब्द कहा जाता है।
• अन्य विकल्प -

Additional Informationकुछ अन्य महत्वपूर्ण विलोम शब्द -

सही उत्तर - तोड़कर।

Key Points

• अशुद्ध वाक्य - एक किसान खेत से सब्जियाँ तोड़ घर ले जाता है।
• शुद्ध वाक्य - एक किसान खेत से सब्जियाँ तोड़कर घर ले जाता है।
• वाक्य में पूर्वकालिक क्रिया का प्रयोग करना उचित होगा।

Additional Informationक्रिया व वचन संबंधी अशुद्धियाँ - 

सबसे आगे रहने वाला - अग्रणी।

Key Points

• वाक्यांश के लिए उपयुक्त शब्द - अनेक शब्दों के लिए एक शब्द को प्रयुक्त करना ही वाक्यांश के लिए एक शब्द कहलाता है।
• अन्य विकल्प -

Additional Informationकुछ महत्वपूर्ण वाक्यांश के लिए एक शब्द -

सही उत्तर है- जो बड़े को सम्मान

Key Points

• अशुद्ध वाक्य - आदर्श विद्यार्थी वही है जो बड़े को सम्मान करता हो तथा छोटों से स्नेह रखता हो ।
• शुद्ध वाक्य - आदर्श विद्यार्थी वही है जो बड़े का सम्मान करता हो तथा छोटों से स्नेह रखता हो ।
• वाक्य में कारक संबंधित त्रुटि थी।

Additional Information

 वचन, कारक, क्रिया विशेषण व विशेषण संबंधी अशुद्धियाँ - 

The correct answer is Kapil Dev.

Key Points

• Kapil Dev is a former Indian cricketer who captained the Indian national team to its first World Cup victory in 1983.
• He is widely regarded as one of the greatest all-rounders in the history of cricket.
• His autobiography, 'Straight from the Heart', provides an insight into his life, career, and the challenges he faced both on and off the field.
• Kapil Dev has received numerous accolades, including the Wisden Cricketer of the Year in 1983 and the Padma Bhushan in 1991.

Additional Information

• VVS Laxman:
• VVS Laxman is a former Indian cricketer known for his elegant batting style.
• He has authored an autobiography titled '281 and Beyond', which chronicles his cricket journey and memorable innings, especially his iconic 281-run knock against Australia in 2001.
• Yuvraj Singh:
• Yuvraj Singh is a former Indian cricketer famous for his explosive batting and all-round performances.
• His autobiography, 'The Test of My Life', details his battle with cancer and his journey back to cricket.
• Sachin Tendulkar:
• Sachin Tendulkar is one of the greatest cricketers of all time, often referred to as the 'Little Master' or 'Master Blaster'.
• His autobiography, 'Playing It My Way', offers a detailed account of his illustrious cricket career and personal life.

The correct answer is Article 21A.

Key Points

• Article 21A was inserted into the Indian Constitution by the 86th Amendment Act of 2002.
• It provides the right to free and compulsory education to all children in the age group of 6 to 14 years as a Fundamental Right.
• The State shall provide free and compulsory education to children of this age group in such manner as the State may, by law, determine.
• This amendment signifies the importance the Indian government places on education and its commitment to ensuring that every child receives basic education.

Additional Information

• Article 12:
• Article 12 of the Indian Constitution defines the term 'State' for the purpose of Part III (Fundamental Rights) of the Constitution.
• It includes the Government and Parliament of India, the Government and the Legislature of each of the States, and all local or other authorities within the territory of India or under the control of the Government of India.
• Article 21:
• Article 21 of the Indian Constitution provides for the protection of life and personal liberty.
• No person shall be deprived of his life or personal liberty except according to the procedure established by law.
• It has been interpreted by the Supreme Court to include a variety of rights such as the right to privacy, the right to a healthy environment, etc.
• Article 18:
• Article 18 of the Indian Constitution deals with the abolition of titles.
• No title, not being a military or academic distinction, shall be conferred by the State.
• This article aims to prevent the creation of a privileged class in the society.

The correct answer is Onam.

Key Points

• Onam is the most significant festival in the Indian state of Kerala, celebrated with great enthusiasm and joy.
• It is a harvest festival celebrated mainly by the Malayali community.
• Onam marks the homecoming of the legendary King Mahabali and is celebrated over a period of ten days.
• The festival includes various traditional activities such as Pookalam (floral decorations), Vallam Kali (boat races), Pulikali (tiger dances), and Onam Sadya (feast).

Additional Information

• Pongal
• Pongal is a harvest festival celebrated in the Indian state of Tamil Nadu.
• It is dedicated to the Sun God and marks the beginning of the sun’s six-month-long journey northwards (Uttarayana).
• Pongal is celebrated over four days and involves various rituals including the boiling of the first rice of the season.
• Bonalu
• Bonalu is a Hindu festival celebrated in the state of Telangana, especially in the twin cities of Hyderabad and Secunderabad.
• The festival is dedicated to Goddess Mahakali and involves offerings of food and rituals to seek her blessings.
• Ugadi
• Ugadi is the New Year’s Day for the states of Andhra Pradesh, Telangana, and Karnataka in India.
• The festival signifies the beginning of a new Hindu lunar calendar and is celebrated with family gatherings, special meals, and rituals.

The correct answer is Tipu Sultan's sword was made of wootz steel.

Key Points

• Wootz steel is a high-grade steel originating in South India and is known for its durability and ability to hold a sharp edge.
• Tipu Sultan, the ruler of Mysore, was known for his well-crafted swords made from wootz steel, which were a symbol of his power and military prowess.
• The wootz steel of Tipu Sultan's time was highly prized in Europe and the Middle East for its unique patterns and superior quality.
• This type of steel was used to make Damascus steel swords, which were renowned for their strength and distinctive patterns resembling flowing water.

Additional Information

• First Cotton Mill in India
• The first cotton mill in India was established in 1818 at Fort Gloster near Kolkata, not in Madras in 1864 as stated in the incorrect option.
• This marked the beginning of the modern textile industry in India.
• Chintz
• The word "chintz" comes from the Hindi word "chheent," meaning a printed or stained calico fabric, typically with a floral pattern. It does not come from the word "quartz."
• Chintz fabrics were highly popular in Europe during the 17th and 18th centuries.
• Decline of India's Textile Exports in the 18th Century
• India's textile exports did decline in the 18th century due to the imposition of colonial policies by the British, which favored British textile manufacturers.
• This led to a significant economic shift and the decline of India's traditional handloom industry.
• Wootz Steel
• Wootz steel is an ancient form of steel characterized by its high carbon content and distinctive patterns.
• It was produced using a unique process that involved the use of small crucibles to melt iron with charcoal.
• The resulting ingots were then forged into swords and other weapons, known for their sharpness and resilience.

The correct answer is 2030.

Key Points

• The Sustainable Development Goals (SDGs) were adopted by the United Nations in September 2015 as part of the 2030 Agenda for Sustainable Development.
• The SDGs are a set of 17 global goals designed to achieve a better and more sustainable future for all.
• The target year for achieving the SDGs is 2030.
• The goals cover a broad range of social, environmental, and economic development issues.

Additional Information

• Sustainable Development Goals (SDGs)
• They are a universal call to action to end poverty, protect the planet, and ensure that all people enjoy peace and prosperity by 2030.
• The 17 goals include no poverty, zero hunger, good health and well-being, quality education, gender equality, clean water and sanitation, affordable and clean energy, decent work and economic growth, industry, innovation, and infrastructure, reduced inequalities, sustainable cities and communities, responsible consumption and production, climate action, life below water, life on land, peace, justice and strong institutions, and partnerships for the goals.
• The SDGs build on the successes of the Millennium Development Goals (MDGs) while including new areas such as climate change, economic inequality, innovation, sustainable consumption, peace, and justice, among other priorities.
• They are integrated and indivisible, balancing the three dimensions of sustainable development: the economic, social, and environmental.
• 2030 Agenda for Sustainable Development
• It was adopted by all United Nations Member States in 2015.
• This agenda provides a shared blueprint for peace and prosperity for people and the planet, now and into the future.
• It includes 17 Sustainable Development Goals which are an urgent call for action by all countries in a global partnership.
• The agenda emphasizes the importance of tackling global challenges such as poverty, inequality, climate change, environmental degradation, peace, and justice.

The correct answer is Both I and II.

Key Points

• Simhastha, also known as Kumbh Mela, is a grand Hindu festival held in Ujjain, Madhya Pradesh.
• This festival is celebrated on the banks of the sacred river Kshipra in Ujjain.
• Simhastha signifies the celestial event when the planet Jupiter enters the zodiac sign of Leo (Simha Rashi).
• The festival occurs once every 12 years and is one of the four Kumbh Melas held in India, the other three being Haridwar, Nashik, and Prayagraj.

Additional Information

• Simhastha Kumbh Mela:
• It is one of the largest religious gatherings in the world, attracting millions of pilgrims and tourists.
• The festival is marked by ritual bathing in the Kshipra river, which is believed to cleanse devotees of their sins and grant spiritual merit.
• Various religious ceremonies, discourses, and cultural programs are conducted during the festival period.
• The event is managed by a combination of government and religious organizations to ensure the safety and well-being of the attendees.
• Kshipra River:
• The Kshipra river is considered sacred in Hinduism and is associated with several mythological stories.
• It originates from the Vindhya range in Madhya Pradesh and flows through the Malwa plateau.
• Ujjain, one of the seven holy cities in Hinduism, is situated on its banks, making it a significant pilgrimage site.
• During the Simhastha festival, the river is specially prepared to accommodate the large influx of pilgrims.
• Zodiac Sign of Leo (Simha Rashi):
• In Vedic astrology, the zodiac sign of Leo is known as Simha Rashi.
• It is ruled by the Sun and represents qualities such as leadership, courage, and confidence.
• The entry of Jupiter (Guru) into Leo is considered an auspicious event, influencing various aspects of life and spirituality.
• This celestial event serves as the basis for the timing of the Simhastha Kumbh Mela in Ujjain.

The correct answer is Himachal Pradesh.

Key Points

• As per Census 2011, Himachal Pradesh had the highest gap in rural-urban sex ratio among Indian states.
• The rural sex ratio in Himachal Pradesh was 974 females per 1000 males.
• The urban sex ratio in Himachal Pradesh was significantly lower at 877 females per 1000 males.
• This large disparity highlights the migration patterns and socio-economic differences between rural and urban areas in the state.

Additional Information

• Census of India 2011:
• The Census of India 2011 was the 15th National census since 1872.
• The census provides detailed information about the population, including demographic, social, and economic parameters.
• It was conducted by the Office of the Registrar General and Census Commissioner of India under the Ministry of Home Affairs, Government of India.
• The census data is used for policy-making, planning, and administration purposes.
• Sex Ratio:
• The sex ratio is the number of females per 1000 males in a given population.
• It is an important demographic indicator to understand the gender balance in the population.
• A low sex ratio can indicate gender imbalances due to various socio-cultural and economic factors.
• Efforts to improve the sex ratio include addressing gender discrimination, promoting female education, and empowering women.
• Rural-Urban Disparity:
• Rural-urban disparity refers to the differences in socio-economic indicators between rural and urban areas.
• These disparities can be in terms of income, education, healthcare, infrastructure, and gender ratios.
• Migration from rural to urban areas often leads to skewed sex ratios due to the movement of working-age males.
• Addressing these disparities requires targeted policies for rural development and urban planning.

The correct answer is ISRO.

Key Points

• The Indian Space Research Organisation (ISRO) successfully conducted the Mars Orbiter Mission (MOM), also known as Mangalyaan, which is the cheapest Mars mission ever.
• Mangalyaan was launched on 5 November 2013, and it successfully entered Mars' orbit on 24 September 2014.
• The mission's total cost was approximately INR 450 crore (around USD 74 million), making it the most economical Mars mission.
• ISRO became the fourth space agency in the world to reach Mars, and the first to do so in its maiden attempt.

Additional Information

• Indian Space Research Organisation (ISRO)
• ISRO is the space agency of the Government of India, headquartered in Bengaluru.
• It was founded in 1969 by Dr. Vikram Sarabhai, who is considered the father of the Indian space program.
• ISRO's vision is to "harness space technology for national development while pursuing space science research & planetary exploration."
• Some of the notable missions by ISRO include the Chandrayaan missions, the Mars Orbiter Mission (Mangalyaan), and the GSAT series of communication satellites.
• Mars Orbiter Mission (MOM) / Mangalyaan
• Mangalyaan is a space probe orbiting Mars since 24 September 2014.
• The primary objective of the mission is to develop the technologies required for interplanetary missions.
• It was designed to explore Mars' surface features, morphology, mineralogy, and the Martian atmosphere.
• The mission's success demonstrated India's capability in space technology and boosted its reputation globally.
• Cost-Effective Space Missions
• ISRO is known for its cost-effective and efficient space missions compared to other space agencies.
• For instance, the Mars Orbiter Mission (MOM) cost significantly less than NASA's MAVEN mission to Mars, which had a budget of USD 671 million.
• ISRO achieves cost efficiency through innovative engineering, frugal management, and leveraging its skilled workforce.
• This cost-effectiveness allows ISRO to undertake multiple missions and contribute to global space research and development.

The correct answer is Argentina.

Key Points

• Argentina won the FIFA World Cup 2022 held in Qatar.
• This victory marked Argentina's third World Cup title, with previous wins in 1978 and 1986.
• The final match took place on December 18, 2022, at the Lusail Iconic Stadium.
• Argentina defeated France in the final, with the match concluding in a penalty shootout after a thrilling 3-3 draw.

Additional Information

• FIFA World Cup
• The FIFA World Cup is an international football competition contested by the men's national teams of the member associations of FIFA.
• The championship has been awarded every four years since the inaugural tournament in 1930, except in 1942 and 1946 when it was not held because of World War II.
• The 2022 FIFA World Cup was the first to be held in the Arab world and the second to be held entirely in Asia after the 2002 tournament in South Korea and Japan.
• The tournament featured 32 teams and was played across eight venues in five cities in Qatar.
• Penalty Shootout
• A penalty shootout is a method used in football to determine the winner of a match that ends in a draw after the regular playing time and extra time.
• Each team takes turns to take penalty kicks, and the team that scores the most goals out of five attempts wins.
• If the score remains tied after the initial five rounds, the shootout proceeds to sudden death until one team has scored more goals than the other from the same number of attempts.
• Lusail Iconic Stadium
• The Lusail Iconic Stadium is a football stadium in Lusail, Qatar, which was constructed for the 2022 FIFA World Cup.
• It is the largest stadium in Qatar, with a capacity of 80,000 spectators.
• The stadium hosted the final match of the 2022 FIFA World Cup.

The correct answer is Only I.

Key Points

• Microeconomics focuses on the behavior of individual consumers and firms, rather than the economy as a whole.
• The study of different types of markets, such as perfect competition, monopoly, monopolistic competition, and oligopoly, falls under microeconomics.
• Microeconomics analyzes supply and demand in individual markets, the pricing of goods and services, and the allocation of resources.
• The behavior of aggregate measures of the economy, such as GDP, inflation, and unemployment, is studied in macroeconomics, not microeconomics.

Additional Information

• Microeconomics
• Microeconomics is a branch of economics that studies the behavior of individuals and firms in making decisions regarding the allocation of limited resources.
• It typically focuses on the patterns of supply and demand and the determination of price and output in individual markets (e.g., markets for goods and services).
• Key concepts in microeconomics include elasticity, marginal utility, consumer choice theory, and production theory.
• Microeconomics also analyzes market failures and the role of government intervention in addressing these failures.
• Macroeconomics
• Macroeconomics is the branch of economics that studies the behavior and performance of an economy as a whole.It focuses on aggregate changes in the economy such as growth rate, unemployment, and inflation.
• Key concepts in macroeconomics include national income accounting, aggregate demand and supply, monetary and fiscal policy, and international trade.
• Macroeconomists develop models that explain the relationship between factors like national income, output, consumption, unemployment, inflation, savings, investment, and international trade.

The correct answer is Only II.

Key Points

• Nanga Parbat is one of the peaks in the Himalayas that has a height of more than 8000 meters.
• It is the ninth highest mountain in the world at 8,126 meters (26,660 feet) above sea level.
• Nanga Parbat is located in the Gilgit-Baltistan region of Pakistan.
• The mountain is also known as the "Killer Mountain" due to its difficult climbing history.

Additional Information

• Nanda Devi
• Nanda Devi is the second highest mountain in India, with an altitude of 7,816 meters (25,643 feet).
• It is situated in the state of Uttarakhand, in the Garhwal Himalayas.
• Nanda Devi is a part of the Nanda Devi and Valley of Flowers National Parks, which is a UNESCO World Heritage Site.
• Namcha Barwa
• Namcha Barwa is a mountain in the Tibetan Himalayas, with a height of 7,782 meters (25,531 feet).
• It is located in the Nyingchi Prefecture of Tibet, China.
• Namcha Barwa forms the eastern anchor of the Himalayas and is the highest peak in its region.

The correct answer is Pradhan Mantri Shram Yogi Maan-Dhan Yojana.

Key Points

• The government of India has introduced a pension scheme for unorganized workers namely Pradhan Mantri Shram Yogi Maan-Dhan (PM-SYM) to ensure old age protection for Unorganized Workers.
• PM Narendra Modi, on 5 March 2019, launched the Pradhan Mantri Shram Yogi Maandhan (PM-SYM) Yojana at Vastral in Gujarat.
• The Pradhan Mantri Shram Yogi Maandhan Yojana was introduced in the Interim Budget 2019-20 presented by the Union Minister for Finance, Corporate Affairs, Railways & Coal, Shri Piyush Goyal in Parliament on 1 February 2019.
• PM-SYM is meant for old age protection and social security of Unorganized Workers (UW) who are mostly engaged as rickshaw pullers, street vendors, mid-day meal workers, head, brick kiln workers, cobblers, rag pickers, domestic workers, washermen, home-based, agricultural workers, construction workers, beedi workers, handloom workers, leather workers, audio-visual workers or in similar other occupations.
• There are an estimated 42 crore such unorganized workers in the country. The entry age for the beneficiary is 18-40 years and he/she should not be a member of ESIC/EPFO or an income taxpayer.
• The schemes are being implemented through the Life Insurance Corporation (LIC) of India and Common Service Centers. LIC is the Fund Manager and is responsible for pension pay-out.

Additional Information

• Ayushman Bharat - Pradhan Mantri Jan Arogya Yojana (AB-PMJAY)
• It is a national health insurance scheme spearheaded by the Ministry of Health and Family Welfare
• It is considered to be the world's largest government-financed health assurance initiative.
• It aims to provide free access to health insurance coverage for India's low-income earners.
• AB-PMJAY provides a cover of Rs. 5 lakhs per family per year, catering specifically to secondary and tertiary care hospitalization across public and private impanelled hospitals throughout the country.
• This significant scheme is fully financed by the Indian government, serving as an important stepping stone towards achieving universal healthcare in India.
• Pradhan Mantri Suraksha Bima Yojana (PMSBY)
• ​It is a government-backed accident insurance scheme in India launched on May 9, 2015, by PM Narendra Modi.
• PMSBY aims to provide a highly affordable insurance scheme for the poor and underprivileged people.
• It is available to individuals in the age group of 18 to 70 years who own a bank account.
• At a nominal annual premium of Rs.12, beneficiaries receive a risk coverage of Rs.2 lakh for accidental death and full disability, and Rs.1 lakh for partial disability.
• This prime ministerial scheme plays a pivotal role in uplifting the vulnerable strata of society and is a step towards protection against unforeseen risks.
• Pradhan Mantri Jeevan Jyoti Bima Yojana (PMJJBY)
• It was launched on 9th May 2015 by PM Narendra Modi
• It is a government-backed life insurance scheme in India aimed at increasing the penetration of life insurance coverage among Indians.
• It is available to individuals aged 18 to 50 years with a bank account.
• This scheme offers a life cover of Rs. 2 lakhs for a one-year period, which extends from June 1st to May 31st of the next year.
• The annual premium for this policy is ₹ 436 (US$5.50).
• The amount will be automatically debited from the subscriber's account annually.
• The PMJJBY scheme is an important initiative taken by the Indian government to provide affordable life insurance to its citizens.

The correct answer is $\rm M^{d}_{T} = k.T$.

Key Points

• The transaction demand for money, is a key concept in monetary economics that reflects the amount of money people in an economy want to hold for the purpose of making transactions within a specific time period.
• The transaction demand is influenced by the velocity of money—the rate at which money circulates in the economy.
• Faster circulation (lower k) means each unit of currency is used more frequently, reducing the need for holding more of it.
• Changes in T or k can signal shifts in economic conditions or policy impacts, influencing monetary policy decisions.
• For instance, an increase in T due to economic growth or an increase in k due to decreased confidence can lead to higher demand for cash holdings.
• The formula thus provides a straightforward way to estimate the amount of money that needs to be in circulation to accommodate economic activities, forming a basis for setting monetary policy and predicting economic trends.

The correct answer is Kaveri.

Key Points

• Shivasamudram is a famous waterfall located in the Mandya district of Karnataka, India.
• It is formed by the river Kaveri (Cauvery) as it drops downstream into the gorge.
• The Kaveri River splits into two branches that flow through rocky terrain before plunging into the gorge, creating the beautiful Shivasamudram Falls.
• The waterfall is a popular tourist destination and is known for its scenic beauty and the hydroelectric power station which is one of the oldest in Asia.

Additional Information

• Kaveri River
• The Kaveri River, also spelled Cauvery, is one of the major rivers in southern India.
• It originates from the Brahmagiri Hill in the Western Ghats in Karnataka and flows through the states of Karnataka and Tamil Nadu before emptying into the Bay of Bengal.
• The river is considered sacred in Hinduism and is often referred to as the "Ganga of the South".
• It plays a crucial role in the irrigation and water supply for the regions it flows through, supporting agriculture and drinking water needs.
• Hydroelectric Power Station
• The Shivasamudram Hydroelectric Power Station, established in 1902, is one of the oldest in Asia.
• It was set up by the Diwan of Mysore, Sir K. Seshadri Iyer, and is still in operation today.
• The power station uses the gravitational force of falling water from the Shivasamudram Falls to generate electricity.
• This initiative marked the beginning of the harnessing of hydroelectric power in India.
• Tourism
• The Shivasamudram Falls attract a large number of tourists every year due to their stunning natural beauty.
• Visitors often enjoy the lush green surroundings, the mist from the falls, and the serene atmosphere.
• There are also several viewing points and facilities for tourists to enhance their experience.
• During the monsoon season, the falls are particularly spectacular with increased water flow.

The correct answer is Both I and II.

Key Points

• The National Mission for Sustaining the Himalayan Ecosystem and the National Mission on Strategic Knowledge for Climate Change are both part of the National Action Plan on Climate Change (NAPCC).
• The NAPCC was launched by the Government of India in 2008 to address climate change and promote sustainable development.
• The Ministry of Science and Technology is responsible for the National Mission on Strategic Knowledge for Climate Change.
• The National Mission for Sustaining the Himalayan Ecosystem is also associated with the Ministry of Science and Technology, focusing on the sustainable development of the Himalayan region.

Additional Information

• National Action Plan on Climate Change (NAPCC)
• NAPCC is an initiative by the Government of India to outline its strategy to meet the challenges of climate change.
• It consists of eight national missions, including the National Mission for Sustaining the Himalayan Ecosystem and the National Mission on Strategic Knowledge for Climate Change.
• These missions focus on various aspects such as energy efficiency, renewable energy, sustainable agriculture, and water conservation.
• National Mission for Sustaining the Himalayan Ecosystem
• This mission aims to understand the complex interactions between the Himalayan glaciers, hydrological systems, and the climate.
• It focuses on the conservation of biodiversity, forest resources, and wildlife in the Himalayan region.
• The mission also addresses the concerns of the local communities dependent on the Himalayan ecosystem for their livelihoods.
• National Mission on Strategic Knowledge for Climate Change
• This mission aims to develop a better understanding of climate science, impacts, and adaptation measures.
• It focuses on building a knowledge system that integrates scientific, technical, and traditional knowledge related to climate change.
• The mission also promotes research and innovation in climate science and technology.

The logic follows here is:

So, the next number in the series is 369.

Hence, the correct answer is "Option 2".

Given:

A is the sister of B, B is the brother of C, 

C is the mother of D, D is the father of E, E is the wife of F

So, C is the grandmother of E. 

Hence, the correct answer is "Option 2".

Given:

Eight students A, B, C, D, E, F, G and H are sitting around a circular table facing towards the centre.

F is to the immediate right of D, A is third to the right of F, G is third to the left of D

B is the immediate neighbour of G,  D is not the immediate neighbour of E and C is not the immediate neighbour of D or A

So, C is sitting third to right of H.

Hence, the correct answer is "Option 2".

The least possible Venn diagram is as follows:

Conclusions:

I. Some water are white → Follow(As, Some white are water, So some water are white is definite)

II. Some white are jug → Does not Follow(As, their is no direct relation so, it can be possible but definite)

III. Some water are jug → Does not Follow(As, No jug is water so, it cannot be possible)

∴ Here, Only conclusion I follows.

Hence, the correct answer is "Option 3".

The logic follows here:

All shapes are rotating in a clockwise direction.

Hence, the correct answer is "Option (1)".

The logic follows here is:

MF : LS

Similarly,

LY : ?

So, LY : ER

Hence, the correct answer is "Option 2".

Given:

Five persons A, B, C, D and E are sitting around a circular table facing towards the centre.

B is sitting second to the left of D, A is sitting to the immediate right of B and Only one person is sitting between C and D.

Thus, D is sitting second to the right of B.

Hence, the correct answer is "Option 2".

The pattern in the given series is:

So, JN will complete the series.

Hence, the correct answer is "Option 4".

 Decoding the information, 

 

Given:

1. 30 □ 15 ○ 40 à 100 D 10 = 460

After replacing the signs: 30 × 15 + 40 - 100 ÷ 10 = 460 

⇒ 30 × 15 + 40 - 100 ÷ 10 = 460 

⇒ 30 × 15 + 40 - 10 = 460 

⇒ 450 + 40 - 10 = 460 

⇒ 490 - 10 = 460 

⇒ 480 ≠ 460  

II. 80 D 20 □ 30 ○ 40 à 50 = 110 

After replacing the signs: 80 ÷ 20 × 30 + 40 - 50 = 110 

⇒ 4 × 30 + 40 - 50 = 110 

⇒ 120 + 40 - 50 = 110 

⇒ 160 - 50 = 110 

⇒ 110 = 110 

Hence, 'Only II' are the correct answer.

The logic follows here is:

Logic : 1^st letter + 6 = 3^rd letter

2^nd letter + 6 = 4^th letter

Option 1) BA – HG 

Follows the logic.

Option 2) SQ – YW 

Follows the logic.

Option 3) MD – SK

Does not follows the logic.

Option 4) JY – PE

Follows the logic.

So, the Option 3) does not follows the logic.

Hence, the correct answer is "Option 3".

Given equation: 5 + 7 × 4 – 8 ÷ 2 = 17

BODMAS Table: 

Let's check each option one by one.

Option 1) 5 and 4

⇒ 4 + 7 × 5 – 8 ÷ 2 = 17

⇒ 4 + 7 × 5 – 4 = 17

⇒ 4 + 35 – 4 = 17

⇒ 39 – 4 = 17

⇒ 35  ≠ 17 ⇒ LHS ≠ RHS

Option 2) 4 and 2

⇒ 5 + 7 × 2 – 8 ÷ 4 = 17

⇒ 5 + 7 × 2 – 2 = 17

⇒ 5 + 14 – 2 = 17

⇒ 19 – 2 = 17

⇒ 17 = 17 ⇒ LHS = RHS

Option 3) 7 and 5

⇒ 7 + 5 × 4 – 8 ÷ 2 = 17

⇒ 7 + 5 × 4 – 4 = 17

⇒ 7 + 20 – 4 = 17

⇒ 27 – 4 = 17

⇒ 23 ≠ 17 ⇒ LHS ≠ RHS

Option 4) 5 and 8

⇒ 8 + 7 × 4 – 5 ÷ 2 = 17

⇒ 8 + 7 × 4 – 2.5 = 17

⇒ 8 + 28 – 2.5 = 17

⇒ 36 – 2.5 = 17

⇒ 33.5 ≠ 17⇒ LHS ≠ RHS

Hence, the correct answer is "Option 2".

The logic follows here is:

CP : GT

Option 1) TS

 DQ : ?

Does not follows logic.

Option 2) HU 

DQ : ?

Follows logic.

Option 3) WX

DQ : ?

Does not follows logic.

Option 4) YZ

DQ : ?

Does not follows logic.

So, the HU is correct answer.

Hence, the correct answer is "Option 2".

The logic follows here is:

So, the next number in the series is -25.

Hence, the correct answer is "Option 4".

Given: G X S N F L T N G Z F B J M J Z B F Q C X S P M J

Condition: immediately followed by letter B or F.

G X S N F L T N G Z F B J M J Z B F Q C X S P M J

Therefore, 5 letters are letters are immediately followed by letter B or F.

Hence, the correct answer is "Option 1".

The logic followed here is:

Option 1) OPS – QMR 

Option 2) JZS – LWT 

Option 3) TJC – VGD

Option 4) DTM – FQN

Hence, the correct answer is "Option 1".

The correct answer is Only I.

Key Points

• Coulomb's Law agrees with Newton's Third Law.
• Newton's Third Law states that for every action, there is an equal and opposite reaction. Similarly, Coulomb's Law states that the force exerted by one charge on another is equal in magnitude and opposite in direction to the force exerted by the second charge on the first.
• The SI unit of permittivity of free space, also known as the electric constant (ε₀), is actually C²/N·m² and not c²/N·m. This unit can be derived from Coulomb's Law, where the force between two point charges is inversely proportional to the permittivity of free space.

 Additional Information

• Permittivity of Free Space
• The permittivity of free space (ε₀) is a fundamental physical constant that describes how electric fields interact with the vacuum. It is crucial in the equations of electrostatics and electromagnetism.
• The correct SI unit of ε₀ is F/m (farads per meter), but in the context of Coulomb's Law, it is often represented as C²/N·m².

The correct answer is option 3.

Key Points

• Potential energy is a type of energy associated with the position or configuration of an object in a conservative force field.
• The SI unit of potential energy is the Joule (J), which is equivalent to kg·m²/s².
• The dimension of potential energy can be derived from its formula: Potential Energy (U) = mgh, where:
• m = mass (kg)
• g = acceleration due to gravity (m/s²)
• h = height (m)
• By analyzing the formula U = mgh, the dimensional formula is:
• m (mass) = M
• g (acceleration due to gravity) = LT⁻²
• h (height) = L
• So, the dimensional formula of potential energy is:
• U = M × (LT⁻²) × L = ML²T⁻²
• Therefore, the dimension of potential energy is correctly represented by ML²T⁻².

 Additional Information

• Option 1 (ML⁻²T⁻²)
• This dimensional formula does not align with the physical quantities involved in potential energy, as it incorrectly represents the dimensions of mass, length, and time.
• Option 2 (MLT⁻²)
• This represents the dimension of force, not potential energy, as force is defined by the product of mass and acceleration (F = ma).
• Option 4 (ML²T⁻¹)
• This dimensional formula is incorrect for potential energy. It does not correctly account for the dimensions of time in the context of energy.

The Correct answer is Gold-leaf electroscope.

Key Points

• The Gold-leaf electroscope is an instrument used for detecting and measuring electric charge.
• It consists of a vertical metal rod with two thin gold leaves attached at the bottom.
• When a charged object comes into contact with the metal rod, the charge gets transferred to the gold leaves.
• Due to like charges repelling, the gold leaves diverge, indicating the presence of electric charge.
• This apparatus is highly sensitive and can detect even small amounts of charge.
• It is widely used in physics experiments and educational demonstrations to show the presence of static electricity.

Additional Information

• Silver-leaf electroscope
• No common apparatus named Silver-leaf electroscope exists in scientific literature or common usage.
• Platinum-leaf electroscope
• Similar to the silver-leaf electroscope, no common apparatus named Platinum-leaf electroscope is known.
• Silver-rod electroscope
• This term is not typically used to describe an electroscope. The most commonly used type is the gold-leaf electroscope due to its sensitivity.

The correct answer is accelerate.

Key Points

• The law of inertia, also known as Newton's first law of motion, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
• When an object is moving down an inclined plane, it experiences a component of the gravitational force acting along the plane.
• This component of gravitational force causes the object to accelerate down the slope.
• The acceleration of the object on the inclined plane is given by the equation a = g sin(θ), where g is the acceleration due to gravity and θ is the angle of the incline.
• The absence of other forces (like friction) would mean the object continuously accelerates as it moves down the plane.

 Additional Information

• Option 1: "is an intermediate situation"
• This option is too vague and does not accurately describe the behavior of objects on an inclined plane according to the law of inertia.
• Option 3: "retard"
• "Retard" means to slow down. However, objects on an inclined plane do not decelerate unless there is a significant opposing force like friction.
• Option 4: "has constant velocity"
• Constant velocity would imply no net force and no acceleration, which contradicts the effect of the gravitational component along the inclined plane.

The correct answer is option 3.

Key Points

• Kepler's laws describe the motion of planets around the Sun.
• There are three laws:
• Law of Orbits: States that planets move in elliptical orbits with the Sun at one focus.
• Law of Areas: States that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
• Law of Periods: States that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
• The Law of Mass is not one of Kepler’s laws. It does not describe the motion of planets but is related to the mass of objects.

Additional Information

• Law of Orbits
• Each planet's orbit around the Sun is an ellipse with the Sun at one focus.
• Law of Areas
• A line segment between a planet and the Sun sweeps out equal areas in equal times.
• Law of Periods
• The square of a planet's orbital period is proportional to the cube of the average distance from the Sun.

The correct answer is Option 1.

Key Points

• Power of a Lens:
• The lens's power is its ability to converge or diverge light rays.
• It is calculated using the formula: $P = \frac{1}{f}$, where P is the power in diopters and f is the focal length in meters.
• Positive Power:
• Convex lenses have positive power as they converge light rays.
• Used in magnifying glasses and corrective lenses for hyperopia (farsightedness).
• Negative Power:
• Concave lenses have negative power as they diverge light rays.
• Commonly used in corrective lenses for myopia (nearsightedness).
• Zero Power:
• A lens with no converging or diverging effect has zero power.
• This occurs when the lens is flat, having an infinite focal length: $P = \frac{1}{\infty} = 0$.
• Measurement Unit:
• Lens power is measured in diopters (D), where: $1 , \text{D} = 1 , \text{m}^{-1}$.

Additional Information

• Option 2:
• This option is incorrect because it excludes positive power. Convex lenses, which are widely used in magnifying glasses and corrective lenses for hyperopia, have positive power.
• Option 3:
• This option is incorrect because it excludes negative power. Concave lenses, used in corrective lenses for myopia, have negative power.
• Option 4:
• This option is incorrect because it excludes zero power. Lenses with zero power do not alter the path of light rays, such as flat glass.

The correct answer is option 3 (errors).

Key Points

• In any measurement, there is always some degree of uncertainty associated with the process.
• This uncertainty is commonly referred to as errors in the context of measurements and instrumentation.
• Errors can arise due to various factors, including limitations of the measuring instrument, environmental conditions, and human factors.
• Errors can be categorized into different types such as systematic errors and random errors.
• Systematic errors are predictable and typically consistent in their nature, while random errors are unpredictable and vary in their impact on measurement results.
• Understanding and minimizing errors is crucial for improving the accuracy and precision of measurements.

Additional Information

• Precision
• Precision refers to the closeness of repeated measurements to each other. It indicates the consistency or repeatability of measurements.
• Accuracy
• Accuracy refers to the closeness of a measured value to the true or accepted value. It indicates how close a measurement is to the actual value.
• Random
• Random errors are unpredictable variations that affect measurement results. They can be caused by unpredictable fluctuations in the measuring process.

The correct answer is option 3.

Key Points

• Eddy currents are loops of electric current induced within conductors by a changing magnetic field in the conductor.
• These currents flow in closed loops within the plane of the conductor, perpendicular to the magnetic field.
• Magnetic braking in trains: Eddy currents are used in magnetic braking systems where they create a magnetic field that opposes the motion of the train, thereby slowing it down.
• Electromagnetic damping: Eddy currents are utilized in electromagnetic damping systems, where they provide resistance to motion by converting kinetic energy into heat, thus dampening oscillations.
• Electric power meters: Eddy currents are employed in electric power meters (such as induction-type meters) to measure the power consumption by generating a magnetic field that interacts with a rotating disk, causing it to rotate proportional to the power consumption.

 Additional Information

• Explanation for Option 1:
• Option 1 includes "Only I and III", which is incorrect because it excludes electromagnetic damping, a key application of eddy currents.
• Explanation for Option 2:
• Option 2 includes "Only I and II", which is incorrect because it excludes electric power meters, another significant application of eddy currents.
• Explanation for Option 4:
• Option 4 includes "Only II and III", which is incorrect because it excludes magnetic braking in trains, a crucial application of eddy currents.

The correct answer is option 2.

Key Points

• Magnetic field is a vector quantity:
• A vector quantity has both magnitude and direction.
• The magnetic field is represented by the symbol B and is measured in teslas (T).
• The direction of the magnetic field is the direction in which the north end of a compass needle points.
• Magnetic field-lines never cross each other:
• Magnetic field lines represent the direction of the magnetic field at various points in space.
• If two magnetic field lines were to cross, it would imply that the magnetic field has two different directions at the point of intersection, which is impossible.
• This is because at any given point, the magnetic field has a unique direction.

 Additional Information

• Option 1: Only II
• This option is incorrect because it ignores the fact that the magnetic field is a vector quantity (statement I).
• Option 3: Only I
• This option is incorrect because it does not acknowledge that magnetic field lines never cross each other (statement II).
• Option 4: Neither I nor II
• This option is incorrect as both statements I and II are correct.
• Option 5:
• This option is invalid as it does not provide any specific answer.

The correct answer is Option 3.

 Key Points

• Work is defined in physics as the product of force and displacement in the direction of the force.
• Mathematically, work (W) is given by the equation: W = F × d × cos(θ), where:
• F is the force applied.
• d is the displacement of the object.
• θ is the angle between the force and the displacement vectors.
• Force (F) and displacement (d) are the primary quantities upon which work depends.
• Distance can be a factor in calculating displacement, but work is specifically the product of force and displacement.
• Thus, the correct quantities are Force and Displacement, making the correct answer Option 3: Only I and II.

 Additional Information

• Option 1: Only I and III
• This option is incorrect because it includes distance instead of displacement. Work is calculated based on displacement, not just distance.
• Option 2: Only II and III
• This option is incorrect because it omits the force, which is a crucial component in the calculation of work.
• Option 4: I, II, and III
• This option is incorrect because it includes distance. While distance can affect displacement, work is specifically the product of force and displacement.

The correct answer is normal reaction.

Key Points

• The normal reaction force is the component of contact force that is perpendicular to the surface of contact.
• It occurs as a response to other forces acting on the body, such as the force due to gravity.
• In a situation where an object is resting on a horizontal surface, the normal reaction force is equal in magnitude and opposite in direction to the weight of the object.
• The normal reaction force ensures that objects do not pass through the surface they are in contact with, maintaining equilibrium.
• This force acts perpendicular to the contact surface and is critical in calculating frictional forces.

Additional Information

• Friction
• Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. It acts parallel to the surface of contact.
• Equilibrium
• Equilibrium refers to the state where the sum of forces and the sum of torques acting on a body are zero, resulting in no net motion.
• Linear
• This term generally refers to objects or forces in a straight line, not specifically related to the concept of forces normal to surfaces.

The Correct answer is 10 Hz to 1 MHz.

Key Points

• Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light.
• The frequency range of radio waves is typically from about 10 Hz to 1 MHz.
• Radio waves are used for various forms of wireless communication, including AM and FM radio broadcasting, television, cell phones, and satellite communications.
• Lower frequency radio waves (<10 Hz) are typically used for communication with submarines because they can penetrate seawater.
• Higher frequency radio waves (up to 1 MHz) are used in applications such as medium-wave (MW) broadcasting and long-wave (LW) broadcasting.

 Additional Information

• 30 MHz to 40 MHz
• This range is typically used for VHF (Very High Frequency) applications such as television broadcasting and FM radio.
• 20 MHz to 30 MHz
• This frequency range falls into the HF (High Frequency) band, often used for shortwave radio broadcasting and amateur radio.
• 2 MHz to 10 MHz
• This range is part of the HF (High Frequency) band, used for marine and aviation communications as well as international shortwave broadcasts.

The correct answer is option 3.

Key Points

• Coulomb's Law describes the force between two point charges. The formula is given by F=K⋅|q1⋅q2|r2F=K⋅|q1⋅q2|r2F=K⋅|q1⋅q2|r2

  , where:

• F is the magnitude of the force between the charges.

• q₁ and q₂ are the amounts of the charges.

• r is the distance between the charges.

• K is the Coulomb's constant.

• The SI unit of Coulomb's constant (K) is derived from the formula. Since:

• Force (F) is measured in Newtons (N).

• Charge (q) is measured in Coulombs (C).

• Distance (r) is measured in meters (m).

• Rewriting the formula K=F⋅r2|q1⋅q2|K=F⋅r2|q1⋅q2|K=F⋅r2|q1⋅q2|

  , we can deduce that the SI unit of KKK

  is N⋅m2C2[Math Processing Error]N⋅m2C2

  .

 Additional Information

• Option 1: (N-m)/(C²)
• This unit does not correctly represent Coulomb’s constant, as it should include the square of the distance in the numerator.
• Option 2: (N-m²)/(C)
• This unit includes an extra Coulomb term in the denominator and is not the correct unit for Coulomb’s constant.
• Option 4: (N-m)/(C)
• This unit does not include the necessary square terms for both distance and charge, making it incorrect for Coulomb’s constant.

The correct answer is option 4.

Key Points

• Total internal reflection occurs when a light ray travelling from a denser medium to a rarer medium hits the boundary at an angle greater than the critical angle, causing the light to be reflected entirely within the denser medium.
• A mirage is an optical phenomenon where light rays are bent to produce a displaced image of distant objects or the sky. This occurs due to total internal reflection in layers of air of different temperatures.
• Diamonds sparkle brilliantly due to total internal reflection. The high refractive index of diamond causes light to be internally reflected multiple times within the gem before it exits, enhancing the sparkle and brilliance.
• Optical fibres use total internal reflection to transmit light signals over long distances with minimal loss. The light signals are confined within the core of the fibre by repeated total internal reflection.

Additional Information

• Explanation of Incorrect Options
• Option 1 (Only I and II): This option excludes optical fibres, which is incorrect as optical fibres are a well-known application of total internal reflection.
• Option 2 (Only II and III): This option excludes mirages, which is incorrect as mirages are also caused by total internal reflection.
• Option 3 (Only I and III): This option excludes diamonds, which is incorrect as diamonds exhibit total internal reflection to enhance their brilliance.

The correct answer is Only I.

Key Points

• The work done by the spring force in a cyclic process is zero.

• This is because in a cyclic process, the system returns to its initial state, meaning the net change in potential energy of the spring is zero.

• When a spring is compressed or stretched, it stores potential energy given by 12kx212kx212kx212kx2

  , where kkkk

  is the spring constant and xxxx

  is the displacement from the equilibrium position.

• During a complete cycle, the spring goes through equal amounts of compression and extension, leading to equal and opposite work done by the spring force.

• As a result, the total work done by the spring force over one complete cycle is zero, confirming that the correct answer is only I.

 Additional Information

• Positive Work (Option II)
• Positive work occurs when the force applied and the displacement are in the same direction. However, in a cyclic process, this is not the case over the entire cycle.
• Negative Work (Option III)
• Negative work occurs when the force applied and the displacement are in opposite directions. Similar to positive work, this is not the case over the entire cycle.
• Combination of I, II, and III (Option IV)
• While different segments of the cycle may involve positive or negative work, the net work done over a complete cycle is zero, ruling out the possibility of all three being correct simultaneously.

The correct answer is Neither I nor II.

Key Points

• The magnetic field inside a solenoid is indeed in the form of parallel straight lines, indicating a uniform magnetic field.
• The statement that the magnetic field is the same at all points inside the solenoid is correct. This is because the field lines are uniformly distributed and parallel to each other.
• Since both the statements I and II are correct, the statement asked in the question is about which of the statements are NOT correct. Therefore, the correct answer is that neither I nor II are incorrect statements.

 Additional Information

• Solenoid
• A solenoid is a coil of wire that generates a magnetic field when an electric current passes through it.
• The magnetic field inside a solenoid is strong and uniform, making it useful in various applications such as electromagnets, inductors, and transformers.
• Uniform Magnetic Field
• A uniform magnetic field is one where the magnetic field lines are parallel and equally spaced, indicating that the field strength is the same at all points.
• Inside a solenoid, the uniform magnetic field is a result of the tightly wound coils and consistent current flow.

The correct answer is option 4.

Key Points

• The parallax method is a technique used to measure the distance of celestial objects such as stars and planets from the Earth.
• This method involves observing the apparent shift in the position of the object when viewed from two different points along Earth's orbit around the Sun.
• The parallax angle (p) is the angle subtended by the radius of Earth's orbit at the distant star.
• Using the parallax angle and the baseline (distance between the two observation points), the distance (d) to the star can be calculated using the formula:
  

d = 1/p (where p is measured in arcseconds and d is in parsecs).

• The parallax method is highly accurate for measuring distances to nearby stars within a few hundred light-years from Earth.

 Additional Information

• Option 1: Base
• The term "base" is not specific enough to describe a method for measuring astronomical distances.
• In the context of the parallax method, the "base" refers to the baseline or the distance between two observation points, but this alone is not sufficient to measure distance.
• Option 2: Light Rays
• Light rays are involved in the parallax method as they are the means by which we observe the position of stars.
• However, stating "light rays" does not describe a specific method for measuring distance.
• Option 3: Metre Scale
• A metre scale is a tool for measuring short distances and is not suitable for measuring astronomical distances.
• It is impractical to use a metre scale for distances on the scale of light-years or parsecs.

The correct answer is Option 1

Key Points

• Momentum is a vector quantity. This means that it has both magnitude and direction. It is calculated as the product of mass and velocity (p = m × v).
• Force is also a vector quantity, not a scalar quantity. Force has both magnitude and direction, and it is defined as mass times acceleration (F = m × a).

 Additional Information

• Scalar Quantities
• Scalar quantities have only magnitude and no direction. Examples include mass, temperature, and time.
• Vector Quantities
• Vector quantities have both magnitude and direction. Examples include velocity, acceleration, and displacement.

The correct answer is cd.

Key Points

• The symbol cd stands for candela, which is the SI unit of luminous intensity.
• Luminous intensity refers to the amount of light power emitted by a source in a particular direction, per unit solid angle.
• The definition of candela is based on the light emitted by a monochromatic source with a frequency of 540 × 10¹² Hz and a radiant intensity of 1/683 watt per steradian.
• The candela is one of the seven base units in the International System of Units (SI).
• This unit is crucial in fields such as lighting design, photography, and various scientific research areas that involve light measurement.

 Additional Information

• cm
• cm stands for centimeter, which is a unit of length in the metric system. It is not related to luminous intensity.
• LI
• LI is not a standard unit or symbol for any physical quantity in the International System of Units (SI).
• Lm
• Lm stands for lumen, which is the SI unit of luminous flux. It represents the total quantity of light emitted by a source in all directions, not just in a specific direction as luminous intensity does.

The Correct answer is Nm²/kg².

Key Points

• The gravitational constant (G) is a key quantity in Newton's law of universal gravitation.

• Newton's law of universal gravitation states that every point mass attracts every other point mass by a force acting along the line intersecting both points.

• This force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

• The formula for the gravitational force (F) is given by: F = G * (m₁ × m₂ / r²) Where:

• F = gravitational force between two masses

• G = gravitational constant

• m₁ and m₂ = masses of the objects

• r = distance between the centers of the two masses

• The SI unit of the gravitational constant (G) is Nm²/kg².

• This unit can be derived from the force equation: G = F × r² / (m₁ × m₂) Where:

• F is in Newtons (N)

• r is in meters (m)

• m₁ and m₂ are in kilograms (kg)

Thus, the unit of G becomes Nm²/kg².

 Additional Information

• N/m²kg²
• This unit incorrectly combines the units of force and area in a manner not suitable for the gravitational constant.
• Nm/kg²
• This unit is missing the square of the distance term (m²), thus it is incorrect.
• N/m²kg
• This unit is also incorrect as it does not properly represent the relationship between force, mass, and distance in Newton's law of gravitation.

CONCEPT:

Reduction Reaction

• A reduction reaction involves the gain of electrons by a molecule, atom, or ion.
• It is the opposite of an oxidation reaction, which involves the loss of electrons.
• In a reduction reaction, the oxidation state of the species decreases.

EXPLANATION:

• In the given reactions:

I. 2Ag⁺(aq) + 2e^- → 2Ag(s)

II. Cu(s) → Cu²⁺(aq) + 2e^-

• Reaction I: 2Ag⁺(aq) + 2e^- → 2Ag(s)
• Silver ions (Ag⁺) are gaining electrons to form silver metal (Ag).
• This is a reduction reaction because there is a gain of electrons.
• Reaction II: Cu(s) → Cu²⁺(aq) + 2e^-
• Copper metal (Cu) is losing electrons to form copper ions (Cu²⁺).
• This is an oxidation reaction because there is a loss of electrons.
• Therefore:
• Only Reaction I is a reduction reaction.
• Reaction II is an oxidation reaction.

The correct answer is option 4: Only I.

CONCEPT:

IUPAC Naming of Alcohols

• The IUPAC naming system is a standardized method for naming chemical compounds as recommended by the International Union of Pure and Applied Chemistry (IUPAC).
• For alcohols, the base name is derived from the longest carbon chain that includes the carbon atom bonded to the hydroxyl group (-OH).
• The position of the hydroxyl group is indicated by a number, and the suffix -ol is added to the base name of the hydrocarbon.
• If there are multiple hydroxyl groups, the suffixes -diol, -triol, etc., are used along with the appropriate numbers to indicate their positions.

EXPLANATION:

• Glycerol is a simple polyol compound. It is a colorless, odorless, viscous liquid that is sweet-tasting and non-toxic.
• The structure of glycerol includes a three-carbon chain with each carbon bonded to a hydroxyl group (-OH).
• According to IUPAC naming rules:
• The base name is propane, as the longest carbon chain has three carbon atoms.
• The three hydroxyl groups are located on the first, second, and third carbon atoms.
• Therefore, the IUPAC name of glycerol is Propane-1,2,3-triol.

Therefore, the IUPAC name of Glycerol is Propane-1,2,3-triol.

CONCEPT:

Drawback of Octet Theory

• The octet theory, proposed by Gilbert Lewis and Walther Kossel, primarily focuses on the tendency of atoms to prefer having eight electrons in their valence shell.
• Although it has been widely accepted and used to explain the formation of chemical bonds, it has certain limitations.

EXPLANATION:

• Statement I: This theory does not account for the shape of molecules.
• The octet theory focuses on the stability provided by having eight electrons in the valence shell but does not provide any insight into the three-dimensional arrangement of atoms in a molecule.
• It does not explain the geometric shapes of molecules, such as the linear, trigonal planar, tetrahedral, or other complex shapes observed in molecules.
• Statement II: It does not explain the relative stability.
• The octet theory also falls short in explaining why some molecules are more stable than others even if they have complete octets.
• Stability is influenced by factors such as bond energy, resonance, hybridization, and molecular interactions, which are not considered by the octet theory.
• Both statements highlight significant limitations of the octet theory, making both statements correct.

Therefore, the correct answer is Both I and II.

CONCEPT:

Boyle's Law

• Boyle's Law states that the pressure (p) of a given mass of gas is inversely proportional to its volume (V) at a constant temperature.
• The mathematical expression of Boyle's Law is:

p ∝ 1/V or pV = k (where k is a constant)

EXPLANATION:

• In Boyle's Law, when plotting the graph of pressure (p) against the reciprocal of volume (1/V), it shows a direct relationship.
• This means that as the volume decreases (and thus 1/V increases), the pressure increases proportionally.
• Hence, the graph of (p) versus (1/V) is a straight line passing through the origin.

Therefore, the graph between (p) and (1/V) in Boyle's law is a straight line.

CONCEPT:

Butanedioic Acid

• Butanedioic acid is a dicarboxylic acid with the formula HOOC-CH2-CH2-COOH.
• It is a naturally occurring substance found in various organisms and is involved in the citric acid cycle.
• In its common form, butanedioic acid is known by a different name.

EXPLANATION:

• The common name of butanedioic acid is derived from its structure and origin.

Butanedioic acid has two carboxyl groups (-COOH) attached to a four-carbon chain.

• In the given options:
• Option 1: Glutaric acid – This is a different dicarboxylic acid with five carbon atoms (HOOC-(CH2)3-COOH).
• Option 2: Adipic acid – This is another dicarboxylic acid with six carbon atoms (HOOC-(CH2)4-COOH).
• Option 3: Succinic acid – This is the common name for butanedioic acid (HOOC-CH2-CH2-COOH).
• Option 4: Malonic acid – This is a dicarboxylic acid with three carbon atoms (HOOC-CH2-COOH).
• The correct match for butanedioic acid is Succinic acid

Therefore, the common name of butanedioic acid is succinic acid.

CONCEPT:

Oxidation of Alcohols

• Oxidation of alcohols is a process where the hydroxyl group (-OH) is converted into a carbonyl group (C=O).
• Primary alcohols can be oxidized to aldehydes, and secondary alcohols can be oxidized to ketones.
• Cyclohexanol is a secondary alcohol, and its oxidation will result in the formation of cyclohexanone.

EXPLANATION:

• The transformation of cyclohexanol to cyclohexanone involves the oxidation of the secondary alcohol to a ketone.
• Several reagents can be used for this oxidation:
• Anhydrous CrO₃ (Chromium trioxide) - A strong oxidizing agent that can oxidize secondary alcohols to ketones.
• PCC (Pyridinium chlorochromate) - A mild oxidizing agent suitable for oxidizing secondary alcohols to ketones without over-oxidation.
• Diisobutylaluminum hydride (DIBAL-H) - Typically used for the reduction of esters and nitriles, not for the oxidation of alcohols.
• O₃/H₂O - Zn dust (Ozonolysis) - Used for oxidative cleavage of alkenes, not for the oxidation of alcohols.
• Among the given options, Anhydrous CrO₃ (Option 1) is the most appropriate reagent for the oxidation of cyclohexanol to cyclohexanone.

Therefore, the correct reagent to bring about the transformation of cyclohexanol to cyclohexanone is Anhydrous CrO₃ (Option 1).

CONCEPT:

Charles's Law

• Charles's law states that the volume of an ideal gas is directly proportional to its absolute temperature, provided the pressure and the amount of gas are held constant.
• The mathematical expression for Charles's law is:

V ∝ T (at constant pressure and amount of gas)

• This means that if the temperature of a gas increases, its volume increases proportionally, and vice versa.

EXPLANATION:

• In a volume vs temperature graph for Charles's law, each line is called an isobar.
• An isochore represents a constant volume process, where the volume of the gas does not change.
• Other terms such as isotopes, isotones, and isobars are not related to this concept:
• Isotopes are atoms of the same element with different numbers of neutrons.
• Isotones are nuclides having the same number of neutrons but different numbers of protons.

Therefore, the correct answer is isobars.

CONCEPT:

Electronic Configuration

• The electronic configuration of an element describes the distribution of electrons in the atomic orbitals.
• Copper (Cu) has an atomic number of 29, which means it has 29 electrons.
• According to the Aufbau principle, electrons are filled in the order of increasing energy levels (1s, 2s, 2p, 3s, etc.).
• However, due to increased stability associated with a fully filled or half-filled d-subshell, copper exhibits an exception to the typical filling order.

EXPLANATION:

• The typical filling order for elements leads to the configuration 3d⁹ 4s² for copper.
• However, copper's actual electronic configuration is 3d¹⁰ 4s¹ because a completely filled d-subshell (3d10) is more stable than a partially filled one (3d9).
• This stability arises due to the exchange energy and symmetric distribution of electrons in the subshell.
• Therefore, one electron from the 4s orbital is promoted to the 3d orbital to achieve a more stable configuration.

Therefore, the electronic configuration of copper (Cu) is 3d¹⁰4s¹.

CONCEPT:

Preparation of Alcohols

• Alcohols can be prepared by various methods, including:
• Acid Catalysed Hydration:
• In this method, alkenes react with water in the presence of an acid catalyst to form alcohols.
• For example, ethene reacts with water in the presence of sulfuric acid to form ethanol.

C₂H₄ + H₂O → C₂H₅OH

• Reduction of Carboxylic Acids:
• Carboxylic acids can be reduced to primary alcohols using strong reducing agents like lithium aluminium hydride (LiAlH₄).
• For example, ethanoic acid can be reduced to ethanol.

CH₃COOH + 4[H] → CH₃CH₂OH + H₂O

EXPLANATION:

• In the given question, two methods of preparation of alcohols are mentioned:
• Statement I: By acid catalysed hydration
• This statement is correct as alcohols can be prepared by the hydration of alkenes in the presence of an acid catalyst.
• 

• Statement II: By reduction of carboxylic acids
• This statement is also correct as carboxylic acids can be reduced to primary alcohols using strong reducing agents like LiAlH₄.
• 

Since both methods mentioned are correct, the correct answer is Both I and II

CONCEPT:

Rate Law (Rate Equation or Rate Expression)

• The rate law (also known as the rate equation or rate expression) expresses the relationship between the rate of a chemical reaction and the concentration of its reactants.
• The general form of the rate law for a reaction aA + bB → cC + dD is:

Rate = k[A]]m[B]]n

• Where:
• Rate is the reaction rate.
• k is the rate constant.
• [A] and [B] are the concentrations of the reactants A and B.
• m and n are the reaction orders with respect to reactants A and B.

EXPLANATION:

• The rate law is also known as the rate equation or rate expression.
• Both terms, rate equation and rate expression, are used interchangeably to describe the mathematical relationship that defines the rate of a reaction.
• Therefore, the correct answer is that another name for the rate law is both the rate equation and the rate expression.

Therefore, the correct answer is Both I and II.

Concept use:

Angle in Radians = Angle in Degree × π/180°

Calculation:

Given 1 Degree 

Angle in Radian =  Angle in Degree × π/180° = 1 × 3.14 / 180° 

Angle in Radian = 0.01746

Hence, The Correct Answer is Option 3.

Concept use:

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by g ∘ f, is defined as the function g ∘ f : A → C given by g ∘ f (x) = g(f (x)), ∀ x ∈ A.

It is similar like putting one Function to another 

Calculation:

Given f(x) = 16x² and g(x) = x^1/2

gof(x) = g(16x²) = (16x²)^1/2 = 4x 

Hence, The Correct Answer is Option 3.

To determine the magnitude of a vector with given endpoints:

$\text{Magnitude} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.$

Calculation:

$(x_1, y_1, z_1) = (5, -5, 0)$ and $(x_2, y_2, z_2) = (2, 3, 0) .$

Step 1: Subtract the coordinates

$x_2 - x_1 = 2 - 5 = -3, \quad y_2 - y_1 = 3 - (-5) = 8, \quad z_2 - z_1 = 0 - 0 = 0.$

Step 2: Square the differencesStep 3: Sum the squares

$(-3)^2 = 9, \quad 8^2 = 64, \quad 0^2 = 0.$

⇒ 9 + 64 + 0 = 73

Step 4: Take the square rootFinal Answer:

$\text{Magnitude} = \sqrt{73}.$

Hence, The Correct Answer is Option 4.

Concept:

Dot Product: it is also called the inner product or scalar product

Let the two vectors be $\rm \vec{a}$ and $\rm \vec{b}$ then dot Product of two vectors:  ${\bf{\vec a}}.;{\bf{\vec b}} = ;;\left| {\bf{a}} \right|\left| {\bf{b}} \right|;{\bf{cos}};{\bf{θ }}$ 

Where, |a| = Magnitude of vectors a and |b| = Magnitude of vectors b and θ is the angle between a and b

Note:  

If $\rm \vec{a}$ and $\rm \vec{b}$ are two vectors perpendicular to each other then $\rm \vec{a}\cdot \vec{b} =0$

If \(\rm \vec{i},\vec{j}, \vec{k}\) are mutually perpendicular vectors then ${\rm{\vec i}}.{\rm{\vec i;}} = {\rm{;\vec j}}.{\rm{\vec j;}} = {\rm{;\vec k}}.{\rm{\vec k;}} = {\rm{;}}1{\rm{;and}}\overrightarrow {{\rm{;i}}} .{\rm{\vec j;}} = {\rm{;\vec j}}.\overrightarrow {{\rm{i;}}} = {\rm{;\vec i}}.{\rm{\vec k;}} = \overrightarrow {{\rm{;k}}} .\overrightarrow {{\rm{i;}}} = {\rm{;\vec j}}.{\rm{\vec k;}} = {\rm{;\vec k}}.\overrightarrow {{\rm{j;}}} = {\rm{;}}0$

If two vectors are perpendicular to each other, their dot product is zero.

Calculation

Given:

Vectors are perpendicular i.e. θ is 90,  cos 90 = 0

⇒ a⃗ .b⃗ cos θ = 0

\((ai ̂+9j ̂).(2i ̂+2j ̂)\)

⇒ 2a + 18 = 0

⇒ a *= -*9 

Hence, The Correct Answer is option 2.

Concept use:

The Cartesian product of two sets is the set of all ordered pairs where the first element is from one set and the second element is from the other set

Calculation:

A = [2, 5] and B = [3, 8] 

A × B = [(2, 3), (2, 8), (5, 3), (5, 8)]

Hence, The Correct Answer is Option 2.

Calculation:

R ∩ 3  = [3, 6] ∩ 3 = 3 

R ∩ 3 × P = 3× [4, 5]  = {(3, 4), (3, 5)} 

Hence, The Correct Answer is option 3.

Concept use:

Concept -

The Length of AB where coordinates of point A and B =

The distance between two points (x1, y1) and (x2, y2), you can use the distance formula:

 $d = √{(x_2 - x_1)^2 + (y_2 - y_1)^2 }$

Explanation -

Given the points A(-2, 0) and B(2, 3), you can substitute these values into the formula:

$d = √{(2 - -2)^2 + (3 - 0)^2}$

$d = √{4^2 + (3)^2} = √ {9+16} = √25 = 5$

Therefore, Length of AB where the points A(-2, 0) and B(2, 3) is 5 units

Option 2:

 Let's take two real functions: f(x) = x² g(x) = sin(x)

Point wise multiplication (f×g)(x) = x²×sin(x)

At x = π/2: f(π/2) × g(π/2) = (π/2)² × 1

At x = π: f(π) × g(π) = π² × 0

At x = 3π/2: f(3π/2) × g(3π/2) = (3π/2)² × (-1)

Here, we multiply the values at each corresponding point, making it truly point wise.

Option 1:

If we take a vector v = [1, 2, 3] and try to multiply it with our function f(x) = x², this operation isn't defined as point wise because:

The vector has discrete elements

The function is continuous

They don't have corresponding points to multiply

Option 3:

If we tried to multiply f(x) = x² with a purely imaginary function like h(x) = i×sin(x), the result would be:

(f × h)(x) = i × x² × sin(x) This involves complex numbers and isn't a real function multiplication

Option 4:

If we multiply f(x) = x² by a scalar k = 2:

(k × f)(x) = 2x² This is just scaling the function, not point wise multiplication as we're not multiplying two functions at each point.

Hence, Option 2 is Correct.

Concept use:

cot(180° – θ)  = −cotθ 

sin(180° – θ) = sinθ

cos(180° – θ) = −cosθ

tan(180° – θ) = −tanθ

cosec (180° – θ) = cosec θ

sec(180° – θ) = −secθ

 

Calculation:

$\cos \left(\frac{5 π}{6}\right)$ = cos(π° – π/6) = −cos(π/6) 

Now, cos(π/6) = √3/2 

So, -cos(π/6) = -√3/2 

Hence, The Correct Answer is option 1.

Concept use:

$\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)$ = 1

Calculation:

$\lim _{x \rightarrow 0}\left(\frac{\sin 3x}{x}\right)$

Now, Multiplying and Divide the Expression by 3 

$\lim _{x \rightarrow 0}3\left(\frac{\sin 3x}{3x}\right)$ By the Given Property 

= 3 × 1 

Hence, The Correct Answer is option 4.

Concept use:

Domain of the Functions:

The set of all possible values that qualify as inputs to a function is known as the domain of the function, or it can also be defined as the entire set of values possible for independent variables.

The domain can be found in – the denominator of the fraction is not equal to zero and the digit under the square root bracket is positive. (In the case of a function with fraction values).

Calculation:

The Given Function is in the form of Fractions

$f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$

= $\frac{x^{2}+3 x+5}{(x - 4)(x- 1)}$

The numerator of the Following functions is not affected by the value of x 

The denominator of the Fraction gets the Value of 0 for x = 4 & 1 which will make the whole function value to ∞ 

Hence, The Domain of Function is R - {1, 4}

Hence, The Correct option is 1.

Concept use:

The co-ordinate system is used to locate the position of a point in a plane using two perpendicular lines. Points are represented in the form of coordinates (x, y) in two-dimension with respect to x- and y- axes. In this article, we will learn about the Cartesian Coordinate system.

Calculation:

as Second Quadrant Contains Negative x axis and Positive y axis 

Hence, (-5, 2) Lies inside the Second Quadrant.

Concept use:

For Finding the Range of f(x): Convert the Function in the Form of Y then find out the domain of that function 

Calculation:

$√{9-x^{2}}$ = f(x) (given)

$√{9-x^{2}}$ = y 

Squaring both sides 

y² = 9 - x²

x² = 9 - y²

x = √9 - y²

We can put y from -3 to 3 

From the Given options [0, 3] Satisfied the Value of y

Concept use:

Quadrantal angles are angles that lie on the axes in a coordinate system, which means they form angles of 90°, 180°, 270°, or 360° with the positive x-axis. These angles are integral multiples of π/2 radians.

Calculations:

• π/3 ≈ 60° (not a quadrantal angle)
• π ≈ 180° (this is a quadrantal angle, but it's 2 times π/2)
• π/6 ≈ 30° (not a quadrantal angle)
• π/2 = 90° (this is the basic unit for quadrantal angles)

Therefore, when we say "integral multiples of π/2", we get:

• 1 × π/2 = 90° (first quadrantal angle)
• 2 × π/2 = π = 180° (second quadrantal angle)
• 3 × π/2 = 270° (third quadrantal angle)
• 4 × π/2 = 2π = 360° (fourth quadrantal angle)

Hence, The Correct Answer is 4.

We are tasked with evaluating:

$\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right) + cosec^{-1}(2)$

Step 1: Evaluate $\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right)$

We know that: $\cos^{-1}(x) \text{ gives the angle } \theta \text{ such that } \cos(\theta) = x \text{ and } \theta \in [0, π].$

For $\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right)$

$\cos(\theta) = -\frac{1}{\sqrt{2}}.$

The angle corresponding to this cosine value is: $\theta = \frac{3π}{4}.$

Thus:

$\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right) = \frac{3π}{4}$

Step 2: Evaluate $cosec^{-1}(2)$

We know that: $cosec^{-1}(x) \text{ gives the angle } \theta \text{ such that } cosec(\theta) = x \text{ and } \theta \in \left[-\frac{π}{2}, \frac{π}{2}\right] \setminus {0}.$

For $cosec^{-1}(2)$The angle corresponding to this sine value is $cosec(\theta) = 2 \implies \sin(\theta) = \frac{1}{2}.$

 $cosec^{-1}(2) = \frac{π}{6}.$

Final Answer:

3π/4 + π/ 6 = 11π/12

Hence, The Correct Answer is 2.

Concept use:

For a set with n elements, the number of possible subsets is 2n.

To find the total number of relations from A into B

We need to find all possible subsets of the Cartesian product A × B.

Calculation:

 let us find A × B:

A × B = {(7,6), (7,8), (5,6), (5,8)}

Since a relation is any subset of A × B

The Cartesian product A × B has 4 elements.

For a set with n elements, the number of possible subsets is 2ⁿ.

This is because for each element, we have 2 choices: include it or not.

∴ The total number of relations from A into B is: 2⁴ = 16

Hence, The Correct answer is 16 relations.