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Question 1Subject-specific content (Post Graduate Teacher level)
In the context of the KVS PGT English syllabus, which literary period is characterised by the use of the heroic couplet as the dominant verse form in satire and didactic poetry?
- The Metaphysical period
- The Victorian period
- The Romantic period
- The Neoclassical (Augustan) period Correct answer.
Explanation
The Neoclassical or Augustan age, associated with Dryden and Pope, made the closed heroic couplet the standard vehicle for satire, epistle, and didactic verse, valuing wit, decorum, and reason over emotional spontaneity.
Question 2Subject-specific content (Post Graduate Teacher level)
A student writes 'The teacher gave the students their assignments'. Which grammatical description of 'the students' is accurate?
- Subject complement
- Direct object
- Object complement
- Indirect object Correct answer.
Explanation
In a ditransitive pattern (verb + recipient + thing), the first noun phrase after the verb is the indirect object and the second is the direct object. 'Their assignments' is what was given, so 'the students' receives it.
Question 3Subject-specific content (Post Graduate Teacher level)
In mathematics, which of the following is the derivative of sin(x²) with respect to x?
- −2x·cos(x²)
- 2x·sin(x²)
- 2x·cos(x²) Correct answer.
- cos(x²)
Explanation
Applying the chain rule, differentiate the outer function sine to get cosine of the inner expression, then multiply by the derivative of the inner function x², which is 2x. The result is 2x·cos(x²).
Question 4Subject-specific content (Post Graduate Teacher level)
Which quantity remains constant for an ideal gas undergoing an isothermal reversible expansion?
- Pressure
- Entropy of the gas
- Temperature Correct answer.
- Volume
Explanation
Isothermal means the temperature of the system is held constant throughout the process, so internal energy of an ideal gas also stays constant. Pressure falls and volume rises, and entropy increases.
Question 5Subject-specific content (Post Graduate Teacher level)
Which of the following best defines the term 'pedagogy' as used in teacher education?
- The administrative management of school finances
- The measurement of intelligence through standardised tests
- The historical study of examination boards
- The study of the theory and practice of teaching and learning Correct answer.
Explanation
Pedagogy refers to the principles, methods, and practice of instruction, encompassing how teachers plan, deliver, and assess learning. It is distinct from administration, psychometrics, and institutional history.
Question 6Subject-specific content (Post Graduate Teacher level)
Which Article of the Constitution of India abolishes untouchability?
- Article 21
- Article 17 Correct answer.
- Article 15
- Article 14
Explanation
Article 17 abolishes untouchability and forbids its practice in any form, making the enforcement of any disability arising from it an offence punishable by law.
Question 7Subject-specific content (Post Graduate Teacher level)
In a class, 60% of students passed Mathematics and 50% passed Science, while 30% passed both. What percentage of students passed at least one of the two subjects?
- 90%
- 110%
- 70%
- 80% Correct answer.
Explanation
By the inclusion–exclusion principle, the union equals the sum of the individual percentages minus the intersection: 60 + 50 − 30 = 80. The overlap is counted twice in the sum, so it must be subtracted once.
Question 8Subject-specific content (Post Graduate Teacher level)
Which of the following is the correct IUPAC name for CH₃–CH(CH₃)–CH₂–CH₃?
- 3-methylbutane
- pentane
- 2-methylbutane Correct answer.
- 2-methylpropane
Explanation
The longest chain has four carbon atoms (butane), and numbering from the end nearer the branch places the methyl group on carbon 2. The correct name is therefore 2-methylbutane.
Question 9Subject-specific content (Post Graduate Teacher level)
In the context of teaching-learning, which of the following is the best example of a formative assessment task?
- An annual standardised achievement survey
- A board examination at the end of the academic year
- A short ungraded quiz used to adjust the next lesson Correct answer.
- A final practical examination for certification
Explanation
Formative assessment is conducted during learning to gather evidence and inform immediate instructional decisions. A short ungraded quiz used to plan the next lesson serves exactly that purpose.
Question 10Subject-specific content (Post Graduate Teacher level)
In statistics, which measure of central tendency is most affected by extreme outliers in a data set?
- Median
- Mode
- Interquartile range
- Arithmetic mean Correct answer.
Explanation
The arithmetic mean uses every observation in its calculation, so a single extreme value shifts it substantially. The median depends on position rather than magnitude, and the mode depends only on frequency.
Question 11Subject-specific content (Post Graduate Teacher level)
A teacher asks students to compare two historical accounts of the same event and identify differences in emphasis. Which cognitive level of Bloom's revised taxonomy does this task primarily address?
- Understanding
- Analysing Correct answer.
- Remembering
- Creating
Explanation
Analysing involves breaking material into parts and determining how they relate, including detecting bias and comparing perspectives. Comparing two accounts and identifying differing emphases is a clear analytical task.
Question 12Subject-specific content (Post Graduate Teacher level)
In physics, a body moves in a circle with constant speed. Which statement about its acceleration is correct?
- Acceleration is directed towards the centre and has magnitude v²/r Correct answer.
- Acceleration is directed away from the centre and has magnitude v²/r
- Acceleration is directed along the tangent to the circle
- Acceleration is zero because the speed is constant
Explanation
Uniform circular motion has constant speed but changing velocity direction, producing centripetal acceleration of magnitude v²/r directed towards the centre of the circle.
Question 13Subject-specific content (Post Graduate Teacher level)
Which of the following sentences contains a non-finite subordinate clause functioning as an adverbial?
- The students finished the experiment and recorded their observations.
- Because the students finished the experiment, they recorded their observations.
- The students who finished the experiment recorded their observations.
- Having finished the experiment, the students recorded their observations. Correct answer.
Explanation
'Having finished the experiment' is a participial (non-finite) clause modifying the main clause and expressing a prior circumstance, so it functions adverbially. The other options use finite verbs in their subordinate or coordinate clauses.
Question 14Subject-specific content (Post Graduate Teacher level)
In the KVS PGT context, which of the following is the primary purpose of a lesson plan?
- To document disciplinary actions taken against students
- To replace the syllabus prescribed by the examining body
- To record the teacher's salary and leave details
- To provide a structured sequence of objectives, activities, and assessment aligned to learning outcomes Correct answer.
Explanation
A lesson plan organises intended learning outcomes, instructional activities, resources, and assessment into a coherent sequence that guides classroom delivery. It supports the syllabus rather than replacing it.
Question 15Subject-specific content (Post Graduate Teacher level)
Which of the following Indian poets is best known for the collection 'Gitanjali', which won the Nobel Prize in Literature in 1913?
- Sarojini Naidu
- Sri Aurobindo
- Toru Dutt
- Rabindranath Tagore Correct answer.
Explanation
Rabindranath Tagore received the Nobel Prize in Literature in 1913, largely for the English translation of 'Gitanjali'. The collection is a landmark of Indian writing in English.
Question 16Subject-specific content (Post Graduate Teacher level)
In chemistry, which of the following explains why the first ionisation enthalpy of magnesium is higher than that of sodium?
- Magnesium has a larger atomic radius than sodium
- Magnesium has one more electron shell than sodium
- Magnesium has a half-filled outer subshell
- Magnesium has a higher effective nuclear charge acting on the outer electron Correct answer.
Explanation
Across a period, nuclear charge increases while electrons enter the same shell, so the outer electron of magnesium experiences a greater effective nuclear charge and is held more tightly than in sodium. Ionisation therefore requires more energy.
Question 17Subject-specific content (Post Graduate Teacher level)
A teacher wants to reduce social loafing during group projects. Which strategy is most directly supported by educational psychology?
- Assign individual accountability for specific sub-tasks within the group Correct answer.
- Let students choose their own groups without any defined roles
- Increase group size so that work is shared widely
- Remove grading so that students feel less pressure
Explanation
Social loafing decreases when individual contributions are identifiable and each member is accountable for a defined part of the task. Making sub-tasks individual and visible links effort to outcomes.
Question 18Subject-specific content (Post Graduate Teacher level)
Which of the following statements about the Fundamental Duties in the Constitution of India is correct?
- They were included by the 42nd Amendment on the recommendation of the Swaran Singh Committee Correct answer.
- They apply only to citizens holding public office
- They were part of the original Constitution of 1950
- They are enforceable directly by writs under Article 32
Explanation
Fundamental Duties were inserted by the 42nd Amendment Act, 1976, following the recommendations of the Swaran Singh Committee. They are moral obligations and are not directly enforceable by courts.
Question 19Subject-specific content (Post Graduate Teacher level)
In computer science, what is the time complexity of binary search on a sorted array of n elements?
- O(1)
- O(n)
- O(n log n)
- O(log n) Correct answer.
Explanation
Binary search halves the search interval at each step, so the number of comparisons grows logarithmically with the input size. The worst-case complexity is therefore O(log n).
Question 20Subject-specific content (Post Graduate Teacher level)
In a normal distribution, approximately what percentage of observations lie within one standard deviation of the mean?
- 68% Correct answer.
- 50%
- 99.7%
- 95%
Explanation
The empirical rule states that about 68% of observations fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.
Question 21Subject-specific content (Post Graduate Teacher level)
Which of the following is the correct order of stages in language acquisition according to widely accepted developmental accounts?
- One-word stage, cooing, babbling, telegraphic stage
- Babbling, cooing, telegraphic stage, one-word stage
- Cooing, babbling, one-word stage, telegraphic stage Correct answer.
- Telegraphic stage, one-word stage, babbling, cooing
Explanation
Infants typically produce vowel-like cooing sounds first, then repetitive babbling, followed by single-word utterances, and later two-word telegraphic speech. This sequence reflects increasing phonological and syntactic control.
Question 22Subject-specific content (Post Graduate Teacher level)
A body is projected with speed u at an angle θ to the horizontal on level ground, neglecting air resistance. Which expression gives its maximum height?
- u² cos²θ / (2g)
- u² sin2θ / g
- u² sin²θ / (2g) Correct answer.
- u² sin²θ / g
Explanation
The vertical component of velocity is u sinθ. Using v² = u² − 2as with final vertical velocity zero at the top gives h = (u sinθ)² / (2g), which is the maximum height.
Question 23Subject-specific content (Post Graduate Teacher level)
In literary theory, which concept, associated with Mikhail Bakhtin, describes the coexistence of multiple voices and ideological positions within a single text?
- Defamiliarisation
- Intentional fallacy
- Heteroglossia Correct answer.
- Negative capability
Explanation
Bakhtin's heteroglossia refers to the stratification of language into multiple social voices and viewpoints that interact within a text, especially in the novel. It is central to his dialogic theory.
Question 24Subject-specific content (Post Graduate Teacher level)
In organic chemistry, which product predominates when propene undergoes hydroboration followed by oxidation with alkaline hydrogen peroxide?
- Propanal
- Propan-2-ol
- Propane
- Propan-1-ol Correct answer.
Explanation
Hydroboration–oxidation adds water across the double bond in an anti-Markovnikov fashion, placing the hydroxyl group on the less substituted carbon. Propene therefore gives propan-1-ol.
Question 25Subject-specific content (Post Graduate Teacher level)
A teacher wants to design a test item that measures higher-order thinking rather than recall. Which item best serves this purpose?
- Define the term 'photosynthesis'.
- Given data on two cities, evaluate which has the greater range of temperature and justify your conclusion. Correct answer.
- State the year in which the Constitution of India came into force.
- List the four fundamental forces of nature.
Explanation
Evaluating and justifying a conclusion from given data requires analysis and judgement, which are higher-order processes. Items that ask for dates, lists, or definitions test retrieval of remembered information.
Question 26Subject-specific content (Post Graduate Teacher level)
In the context of the KVS PGT Biology syllabus, which of the following is the primary function of the Golgi apparatus in a eukaryotic cell?
- Synthesis of ATP through oxidative phosphorylation
- Modification, sorting, and packaging of proteins and lipids for secretion or delivery to other organelles Correct answer.
- Replication of chromosomal DNA during the S phase
- Translation of mRNA into polypeptide chains
Explanation
The Golgi apparatus consists of flattened membranous sacs (cisternae) that receive transport vesicles from the endoplasmic reticulum. It modifies proteins and lipids by glycosylation and other processes, then sorts and packages them into vesicles for secretion or transport to other organelles.
Question 27Subject-specific content (Post Graduate Teacher level)
A KVS PGT Mathematics question asks: If the roots of the quadratic equation x² – 5x + k = 0 are equal, what is the value of k?
- 25
- 25/4 Correct answer.
- 10
- 5
Explanation
For a quadratic equation ax² + bx + c = 0, the roots are equal when the discriminant b² – 4ac = 0. Here a = 1, b = –5, c = k, so (–5)² – 4(1)(k) = 0 ⇒ 25 – 4k = 0 ⇒ k = 25/4.
Question 28Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Physics, which of the following statements correctly describes the photoelectric effect?
- The photoelectric current is directly proportional to the frequency of incident light.
- The number of emitted electrons is independent of the frequency of incident light.
- The emission of electrons occurs only if the frequency of incident light exceeds a threshold frequency. Correct answer.
- The kinetic energy of emitted electrons increases with the intensity of incident light.
Explanation
The photoelectric effect is the emission of electrons from a metal surface when light of sufficient frequency strikes it. A minimum threshold frequency is required, regardless of intensity, because energy is transferred in quanta (photons) and must exceed the work function.
Question 29Subject-specific content (Post Graduate Teacher level)
In the KVS PGT English syllabus, which of the following works is a tragedy written by William Shakespeare?
- A Midsummer Night's Dream
- The Tempest
- Twelfth Night
- King Lear Correct answer.
Explanation
King Lear is one of Shakespeare's major tragedies, depicting the downfall of the title character and the tragic consequences of his decisions. It is grouped with Hamlet, Othello, and Macbeth as one of the four great tragedies.
Question 30Subject-specific content (Post Graduate Teacher level)
A KVS PGT History question asks: Who was the founder of the Maurya Empire in ancient India?
- Chanakya
- Ashoka
- Bindusara
- Chandragupta Maurya Correct answer.
Explanation
Chandragupta Maurya founded the Maurya Empire around 321 BCE after overthrowing the Nanda dynasty. He was aided by his mentor Chanakya (Kautilya).
Question 31Subject-specific content (Post Graduate Teacher level)
In the KVS PGT Geography context, which of the following is the correct order of the layers of the atmosphere from the Earth's surface upward?
- Mesosphere, Troposphere, Stratosphere, Thermosphere
- Stratosphere, Troposphere, Mesosphere, Thermosphere
- Troposphere, Mesosphere, Stratosphere, Thermosphere
- Troposphere, Stratosphere, Mesosphere, Thermosphere Correct answer.
Explanation
The atmosphere is divided into layers based on temperature variations. Starting from the surface, the layers are the troposphere, stratosphere, mesosphere, and thermosphere, each separated by a pause (tropopause, stratopause, mesopause).
Question 32Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Computer Science, what is the primary function of the Domain Name System (DNS) in networking?
- To route packets between different networks based on IP addresses
- To encrypt data transmitted over the internet
- To assign dynamic IP addresses to devices on a network
- To translate human-readable domain names into IP addresses Correct answer.
Explanation
DNS is a hierarchical naming system that resolves domain names (like www.example.com) into numerical IP addresses, enabling browsers and other applications to locate resources on the internet.
Question 33Subject-specific content (Post Graduate Teacher level)
A KVS PGT Mathematics question on probability asks: If a fair die is rolled once, what is the probability of getting a number greater than 4?
- 1/2
- 2/3
- 1/3 Correct answer.
- 1/6
Explanation
A fair die has six equally likely outcomes: 1, 2, 3, 4, 5, 6. The numbers greater than 4 are 5 and 6, so there are 2 favorable outcomes. The probability is 2/6 = 1/3.
Question 34Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Biology, which of the following is the correct sequence of stages in the process of photosynthesis?
- Electron transport chain followed by chemiosmosis
- Glycolysis followed by the Krebs cycle
- Light-dependent reactions followed by the Calvin cycle Correct answer.
- Calvin cycle followed by light-dependent reactions
Explanation
Photosynthesis consists of two main stages: the light-dependent reactions, which occur in the thylakoid membranes and produce ATP and NADPH, and the Calvin cycle (light-independent reactions), which uses these products to fix carbon dioxide into sugars.
Question 35Subject-specific content (Post Graduate Teacher level)
A KVS PGT Physics question asks: A body is moving with uniform acceleration. Which of the following graphs between velocity and time is a straight line inclined to the time axis?
- Position-time graph
- Velocity-time graph Correct answer.
- Acceleration-time graph
- Force-time graph
Explanation
For uniform acceleration, velocity changes at a constant rate, so the velocity-time graph is a straight line with a slope equal to the acceleration. The position-time graph would be a parabola.
Question 36Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Chemistry, which of the following statements about the periodic table is correct?
- Ionisation energy generally decreases across a period from left to right.
- Atomic radius generally decreases across a period from left to right. Correct answer.
- Electronegativity generally decreases across a period from left to right.
- Metallic character generally increases across a period from left to right.
Explanation
Across a period, the nuclear charge increases while electrons are added to the same shell, leading to a stronger attraction that pulls electrons closer. As a result, atomic radius generally decreases from left to right.
Question 37Subject-specific content (Post Graduate Teacher level)
A KVS PGT English question asks: In the poem 'The Road Not Taken' by Robert Frost, what does the speaker's choice of the less travelled road symbolise?
- The regret of not taking the easier path
- The inevitability of fate and lack of free will
- The importance of following the crowd
- The making of a significant, individual choice that shapes one's life Correct answer.
Explanation
The poem uses the fork in the road as a metaphor for life's choices. The speaker's decision to take the less travelled road symbolises making a nonconformist, individual choice that has lasting consequences, even though both paths were actually similar.
Question 38Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Mathematics, what is the value of the integral ∫₀¹ x eˣ dx?
- e – 1
- e
- 1 Correct answer.
- e + 1
Explanation
Using integration by parts with u = x and dv = eˣ dx, we get du = dx and v = eˣ. Then ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C. Evaluating from 0 to 1 gives (1·e – e) – (0 – 1) = 1.
Question 39Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Physics, which of the following statements about the Bohr model of the hydrogen atom is correct?
- The energy of the electron increases as the principal quantum number decreases.
- The angular momentum of the electron is quantised in integer multiples of h/2π. Correct answer.
- The radius of the orbit is inversely proportional to the square of the principal quantum number.
- The electron can revolve in any orbit without radiating energy.
Explanation
Bohr postulated that the electron can only occupy certain stationary orbits where its angular momentum is an integer multiple of h/2π (L = n h/2π). This quantisation condition explains the discrete energy levels of the hydrogen atom.
Question 40Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Chemistry, which of the following is the correct IUPAC name for the compound CH₃–CH₂–CO–CH₂–CH₃?
- Pentanal
- 3-Pentanone Correct answer.
- 2-Pentanone
- Diethyl ketone
Explanation
The compound is a ketone with a carbonyl group at the centre of a five-carbon chain. The longest chain containing the carbonyl carbon has five carbons, and numbering from the end nearer the carbonyl gives the carbonyl carbon position 3. Hence the IUPAC name is pentan-3-one (3-pentanone).
Question 41Subject-specific content (Post Graduate Teacher level)
A KVS PGT Biology question asks: Which of the following is the correct sequence of events during the process of translation in protein synthesis?
- Initiation, termination, elongation
- Elongation, initiation, termination
- Termination, elongation, initiation
- Initiation, elongation, termination Correct answer.
Explanation
Translation proceeds in three stages: initiation, where the ribosome assembles on the mRNA and the first tRNA binds; elongation, where amino acids are added one by one; and termination, where a stop codon is reached and the polypeptide is released.
Question 42Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT History, which of the following was a major consequence of the Revolt of 1857 in India?
- The Mughal Empire was restored to full power.
- The British withdrew all their troops from India.
- The British granted immediate independence to India.
- The East India Company was abolished and India came under direct rule of the British Crown. Correct answer.
Explanation
After the Revolt of 1857, the British government passed the Government of India Act 1858, which abolished the East India Company and transferred administration to the Crown, marking the beginning of the British Raj.
Question 43Subject-specific content (Post Graduate Teacher level)
A KVS PGT Geography question asks: Which of the following is the primary cause of the Coriolis effect on Earth?
- The tilt of the Earth's axis
- The gravitational pull of the Moon
- The uneven heating of the Earth's surface
- The rotation of the Earth on its axis Correct answer.
Explanation
The Coriolis effect is an apparent deflection of moving objects (like air masses) caused by the Earth's rotation. As the Earth spins, points at different latitudes move at different speeds, leading to a deflection to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.
Question 44Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Computer Science, which of the following is the correct order of the phases in the Software Development Life Cycle (SDLC)?
- Requirement analysis, design, implementation, testing, deployment, maintenance Correct answer.
- Requirement analysis, implementation, design, testing, deployment, maintenance
- Implementation, design, requirement analysis, testing, deployment, maintenance
- Design, requirement analysis, implementation, testing, deployment, maintenance
Explanation
The SDLC typically follows a sequential order: requirements gathering and analysis, system design, implementation (coding), testing, deployment, and maintenance. This structured approach ensures that each phase builds on the previous one.
Question 45Subject-specific content (Post Graduate Teacher level)
A KVS PGT Mathematics question asks: What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...?
- 1
- 4
- 2 Correct answer.
- 3/2
Explanation
The series is geometric with first term a = 1 and common ratio r = 1/2. Since |r| < 1, the sum to infinity is S = a/(1 – r) = 1/(1 – 1/2) = 2.
Question 46Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Physics, which of the following statements about the refraction of light through a prism is correct?
- The angle of deviation is minimum when the angle of incidence equals the angle of emergence. Correct answer.
- The angle of deviation is always zero for any angle of incidence.
- The angle of deviation is maximum when the angle of incidence equals the angle of emergence.
- The angle of deviation is independent of the angle of incidence.
Explanation
For a prism, the angle of deviation varies with the angle of incidence. It reaches a minimum (angle of minimum deviation) when the ray passes symmetrically through the prism, i.e., when the angle of incidence equals the angle of emergence.
Question 47Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Chemistry, which of the following statements about the SN1 reaction mechanism is correct?
- The rate of the reaction depends only on the concentration of the substrate. Correct answer.
- The reaction proceeds with complete inversion of configuration.
- The rate of the reaction depends on both the substrate and the nucleophile concentrations.
- The reaction is favoured by primary alkyl halides.
Explanation
SN1 is a two-step mechanism: first, the leaving group departs to form a carbocation (rate-determining step), and then the nucleophile attacks. Since the nucleophile is not involved in the rate-determining step, the rate law is first order, depending only on the substrate concentration.
Question 48Subject-specific content (Post Graduate Teacher level)
A KVS PGT Biology question asks: Which of the following is the correct sequence of the stages of mitosis?
- Telophase, anaphase, metaphase, prophase
- Prophase, anaphase, metaphase, telophase
- Metaphase, prophase, anaphase, telophase
- Prophase, metaphase, anaphase, telophase Correct answer.
Explanation
Mitosis consists of four main stages: prophase (chromosomes condense), metaphase (chromosomes align at the equator), anaphase (sister chromatids separate), and telophase (nuclear envelopes reform). This order ensures the proper distribution of genetic material.
Question 49Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT English, which literary device is used in the phrase 'The pen is mightier than the sword'?
- Simile
- Hyperbole
- Personification
- Metaphor Correct answer.
Explanation
The phrase uses 'pen' to represent writing or diplomacy and 'sword' to represent military force, making an implicit comparison without using 'like' or 'as'. This is a metaphor.
Question 50Subject-specific content (Post Graduate Teacher level)
In the context of KVS PGT Mathematics, which of the following is the derivative of f(x) = ln(sin x) with respect to x?
- sec x
- tan x
- cot x Correct answer.
- csc x
Explanation
Using the chain rule, d/dx ln(sin x) = (1/sin x) · cos x = cot x. The derivative of the inner function sin x is cos x, and the derivative of ln u is 1/u.