NMTC 2026 Comes to Kolkata: Cheenta Hosts the Preliminary Round at Mitra Institution

On 30th August 2026, young mathematics enthusiasts across Kolkata gathered at Mitra Institution (Main) for the preliminary round of the National Mathematics Talent Contest (NMTC) — one of India's longest-running mathematical olympiads. Cheenta took charge of conducting the exam in the city, with invigilation handled by Deepan Dutta, Raghunath J V, Shilpa Bera, Taniya Das and Sukdeb Mahato.

A Contest Rooted in History

NMTC is organised annually by the Association of Mathematics Teachers of India (AMTI), founded in 1965 to strengthen mathematics teaching and spot young talent nationwide. The contest itself has run since 1968, making 2026 roughly its 58th edition. What began as a regional initiative out of Tamil Nadu has grown into a national event, with well over a hundred thousand students from hundreds of schools sitting the screening test each year.

Over the decades, NMTC has become a recognised stepping stone in India's olympiad pipeline. Strong performers, especially at the Junior and Inter levels, often go on to represent their states at the Indian National Mathematics Olympiad (INMO), and from there to India's international olympiad teams.

Inside the Competition

NMTC is split into five age-based categories, each named after a mathematician: Primary (Gauss), Sub-Junior (Kaprekar), Junior (Bhaskara), Inter (Ramanujan), and Senior (Aryabhata), covering students from Standard V all the way through undergraduate level. The Preliminary Test — the round held this weekend — is a two-hour paper mixing multiple-choice and fill-in-the-blank questions, deliberately weighted toward difficulty: roughly a fifth of the questions are straightforward, while the rest demand real problem-solving. Only the top tenth of performers make it through to the Final Test, a longer three-hour paper that pushes further into advanced territory.

Kolkata's leg of this national exercise played out at Mitra Institution (Main), with Cheenta's invigilators ensuring the paper ran smoothly for the city's participating students.

How Cheenta Prepares Students

The exam day itself was only the visible part of a longer effort. In the weeks leading up to it, Cheenta ran a series of offline workshops at its Lake Place centre, giving students structured practice with the kind of unconventional, creative problems NMTC is built around — quite different from the routine drilling most school classrooms focus on.

This reflects Cheenta's broader approach to olympiad training: mathematical creativity is a skill that has to be practised through discussion, mistakes, and mentor feedback, not just picked up from a textbook. Through small-group problem-solving sessions at Lake Place, Cheenta has been helping students build the instincts they need — not just for NMTC, but for the longer olympiad journey that follows it, from RMO to INMO and beyond.

What the Curriculum Covers

NMTC draws its syllabus from pre-degree college mathematics, organised broadly around four areas: Algebra, Geometry, Number Theory, and Graph Theory & Combinatorics. Algebra questions typically involve polynomials, equations, inequalities, and complex numbers. Geometry centres on triangles and circles, and can often be approached through trigonometric, vector, complex-number, or transformation methods — giving students room to choose the technique that suits them best. Number theory and combinatorics round out the paper, rewarding pattern recognition and clean logical argument over formula recall.

Looking Ahead

With the preliminary round complete, Kolkata's NMTC aspirants now wait for results, while qualifiers begin preparing for the Final Test. For Cheenta, the Lake Place workshops and the exam day at Mitra Institution are two halves of the same effort — helping local students engage with mathematics as a creative pursuit, one contest at a time.

NMTC 2026 in Kolkata: Cheenta Runs a Month of Prep Workshops

Cheenta is hosting the National Mathematics Talent Contest (NMTC) 2026 Preliminary Round on 30th August 2026 at Mitra Institution (Main), Kolkata. In the weeks leading up to it, Cheenta ran an offline workshop at its Lake Place Road center, from 8th to 29th August to get students ready for the kind of thinking NMTC actually rewards.

A Talent Search Since 1968

NMTC is organised by the Association of Mathematics Teachers of India (AMTI), founded in 1965. The contest itself began in 1968, making 2026 its 58th edition, one of India's longest-running olympiad-style talent searches, predating even the Indian National Mathematical Olympiad. Its founding aim has stayed constant: to identify students capable of original, creative thinking on unfamiliar, non-routine problems, not just fast recall of textbook methods. Last year alone, over 82,000 students across India sat the screening round, with the top 10% at each level advancing to the final.

Inside NMTC 2026

NMTC runs in two stages, organised by grade:

Primary (Gauss, Classes 5–6), Sub-Junior (Kaprekar, 7–8), Junior (Bhaskara, 9–10), Inter (Ramanujan, 11–12), Senior (Aryabhata, first-year college).

Stage 1 — Preliminary Test: 30th August 2026, 10 AM–12 Noon. A 2-hour objective paper: 20 questions for Primary, 30 for the rest, with 80% of questions set at a distinctly Olympiad-like difficulty. Each school shortlists its own top 10% to advance.

Stage 2 — Final Test: 25th October 2026, a 3-hour subjective paper (at most 8 questions) pitched at IOQM/RMO/INMO level. Senior-level students skip the preliminary and go straight to this. Results and cash prizes for top scorers are announced in December.

How Cheenta Is Helping Students

Cheenta is hosting the preliminary exam itself at Mitra Institution (Main), giving Kolkata students a local, straightforward way to sit NMTC, continuing its long-standing role as a host center for olympiads like NMTC, the Australian Mathematics Competition, and the Sharygin Geometrical Olympiad.

Alongside that, Cheenta ran a dedicated prep workshop at its Lake Place Road center every weekend from 8th to 29th August. Rather than syllabus revision, sessions focused on working through non-routine, NMTC-style problems in person with mentors, building exactly the kind of reasoning the screening test is designed to reward.

What the Contest Tests

The syllabus is layered, each level building on the one below:

Primary: Fractions, ratios, divisibility, basic mensuration and geometry.
Sub-Junior: Adds surds, equations, 3D mensuration, and early number theory.
Junior: Adds circle theorems, modular arithmetic, and combinatorics (permutations, pigeonhole principle).
Inter: Adds inequalities, functional equations, and Diophantine equations.
Senior: A subjective paper at typical B.Sc. mathematics level.

Across every level, the emphasis stays on creative problem-solving over formula recall — exactly what Cheenta's Lake Place Road sessions were built to sharpen ahead of the 30th August exam.

NMTC 2023 Stage I - Gauss (Grade 5 & 6) - Problems

Question 1

The value of $\frac{999 \times 999 \times 999}{(999+999) \times 111 \times 111}$, when simplified, is

(a) $\frac{999}{2}$

(b) $\frac{111}{3}$

(d) 1000

Question 2

When $4 \frac{1}{2}$ is divided by $3 \frac{1}{4}$, the result is x. When $3 \frac{3}{4}$ is divided by $2 \frac{1}{8}$, the result is y. Then the numerical value of $13 x+17 y$ is

(a) 38

(b) 40

(d) 48

Question 3

There are 5 cards numbered as shown. In the figure. The number of ways in which one can choose 3 or less cards which contain only odd numbers is.

(a) 8

(b) 7

(d) 21

Question 4

ABC is a triangle. The bisector of $\angle \mathrm{C}$ meets AB at D. The bisectors of $\angle \mathrm{A}$ and $\angle \mathrm{BDC}$ meet at E. Then the measure of $\angle A E D$ is.

(a) $\frac{\angle \mathrm{C}}{2}$

(b) $\frac{\angle \mathrm{C}}{3}$

(d) $\frac{\angle \mathrm{C}}{5}$

Question 5

Samrud secures 20% of the marks but fails by 30 marks. Saket gets 32% marks which is 42 marks more than the minimum pass marks. The maximum marks in the test would be

(a) 100

(b) 200

(d) 600

Question 6

The greatest number that divides 25, 73 and 97 to leave the same remainder is

(a) 19

(b) 22

(d) 37

Question 7

If $\frac{19}{7}=a+\frac{2}{a+\frac{b}{c}}$ where $a b c$, are natural numbers, then the numerical value of $(a+b+c)$ is

(a) 8

(b) 13

(d) 9

Question 8

In the adjoining figure, two equilateral triangles ABP, CDP are placed such that AB is parallel to CD. If $\mathrm{AB}=3 \mathrm{ cm}, \mathrm{CD}=1 \mathrm{ cm}$, the area (in $\mathrm{cm}^{2}$ ) of trapezium ABCD is.

(a) $\sqrt{3}$

(b) $\frac{2}{\sqrt{3}}$

(d) $4 \sqrt{3}$

Question 9

Siva found the average of 5 numbers. He got an answer 40 which is wrong because, while listing, instead of writing the number 43 he wrote 48. The correct average must be

(a) 38

(b) 39

(d) 31

Question 10

In the adjoining figure, ABCD is a square. Also $\mathrm{BE}=\mathrm{BF}$. Then the value of 2x (in degrees) is

(a) 210

(b) 220

(d) 225

Question 11

If the numerator and denominator of a fraction are increased by 20% and 30% respectively, then the fraction becomes $\frac{9}{13}$. If the original fraction is $\frac{p}{q}$, where $p$ and $q$ have no common factors, then $p+q$ is

(a) 3

(b) 6

(d) 9

Question 12

Gita divided 360 into 4 parts such that twice the first part, thrice the second part, five times the third part and six times the fourth part are all equal. Then the difference between the third and fourth parts is

(a) 20

(b) 15

(d) 7

Question 13

In the adjoining figure, the degree measure of $\angle \mathrm{FAB}$ is.

(a) 95°

(b) 105°

(d) 125°

Question 14

Consider the following sequence: 1, 3, 5, 7, 9, 7, 5, 3, 1, 3, 5, 7, 9, 7, 5, 3, 1, 3, 5, 7, 9, 7, 5, 3, 1,  The digit in the $2023{ }^{\text {rd }}$ place is

(a) 3

(b) 5

(d) 1

Question 15

In the two figures, there is a pattern of numbers which are same. Then the number in the head of the second figure,

(a) 6

(b) 13

(d) 10

Fill in the blanks

Question 16

There are two cars $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$. The speed of $\mathrm{C}_{1}$ is 20% less than that of $\mathrm{C}_{2}$. They travel a certain equal distance. The percentage of time does $\mathrm{C}_{1}$ need to travel than $\mathrm{C}_{2}$ is x%. Then x is = (____)

Question 17

Six equal unit squares are arranged in different shapes as shown in the diagram below:

[] In diagram (1), the perimeter is $A B C D$, which equals 10 . Similarly the perimeters of the other shapes also are found out. Let the perimeters be denoted by $P_{1}, P_{2}, P_{3}$ and $P_{4}$. Then the value of $\left(P_{1}+P_{4}\right)-\left(P_{2}+P_{3}\right)$ is_.

Question 18

In a two digit number, the digit in the tens place is twice the digit in the units place. If we swap the places of these two digits, a new two-digit number is formed. The sum of these two numbers is 132. The original number is. (____)

Question 19

An ant starts from A and wants to go to D. It isallowed to go along the lines and pass a line and a point only once. The number of different routes that it can take to go from A to D is (____).

Question 20

Three consecutive natural numbers are taken from 1 to 6 . With these three numbers, three digit numbers are formed. The total number of such 3-digit numbers is (____)

Question 21

In the adjoining figure, $\angle B A C=20^{\circ}, \angle B C A=10^{\circ}, \angle A C D=90^{\circ}$ and $\angle C D B=55^{\circ}$. If $\angle A B D=x^{\mathrm{o}}$, then $x=$ (____)

Question 22

The number of 5-digit numbers of the form $34 a 5 b$ (where $a, b$ are digits), each of which is divisible by 36 is (____)

Question 23

The units digit of the sum of all 2-digit numbers is (____)

Question 24

A natural number is taken. One sixth of this number is subtracted from it. From the resulting number, half of the number is taken and from this number one fifth is taken. If the resulting number is 3 , then the original number taken is (____).

Question 25

The least number that is added to 2716321 to make it exactly divisible by 3456 is (____)

Cheenta Hosts the Australian Mathematics Competition 2026 in Kolkata, Mumbai

Cheenta recently hosted the Australian Mathematics Competition (AMC) 2026 on 4th, 5th, and 6th August, starting at 2:30 PM, at its Kolkata center, and simultaneously at Innovation Story in Mumbai. Students from both cities sat for one of the world's most respected school-level mathematics contests, marking another year of Cheenta's continued involvement in bringing international olympiad opportunities to Indian students.

Participants on Day 1 at Innovation Story in Mumbai with Cheenta's representative Somnath Gayen.

Here's a closer look at the competition's history, what the contest actually involves, and how Cheenta supported students through the process.

A Contest Nearly 50 Years in the Making

The story of the AMC begins in 1976, in a much smaller and more local form. What later became the AMC started as a competition open only to students in the Australian Capital Territory, drawing around 1,200 entries, with the top performers awarded the Burroughs Medal. Two years later, in 1978, the contest went national — rebranded as the Australian Mathematics Competition for the Wales Awards — and roughly 60,000 students from Australia and New Zealand took part.

From there, the competition simply kept growing. It spread to Singapore, Fiji, Tonga, Taiwan, China, and Malaysia, with translated papers (including French and Chinese versions) making it accessible to non-English-speaking participants, and large-print and braille editions ensuring inclusivity for all students. Today, the AMC is sat by roughly 600,000 students a year, around 100,000 of them from outside Australia, across more than 30 countries — making it the largest mathematics competition in the world.

The Australian Maths Trust, the organisation behind the AMC, has its own interesting origin story. It was formed in 1992 through the merger of two older bodies: the Mathematics Olympiad (founded 1979) and the Australian Mathematics Foundation (founded 1987). Since then, AMT has grown into the umbrella organisation for a whole ecosystem of Australian maths and informatics competitions — including the Australian Mathematical Olympiad (run through the Australian Mathematical Olympiad Committee), which feeds talented students toward the International Mathematical Olympiad and the European Girls' Mathematical Olympiad. AMT also honours volunteer educators each year through the BH Neumann Award, named after one of Australian mathematics' most celebrated figures, and its flagship AMC itself is named in spirit after founding Executive Director Peter O'Halloran, whose legacy lives on through the competition's top individual honour, the Peter O'Halloran Award.

Inside AMC 2026

This year's competition ran from Tuesday 4th to Thursday 6th August 2026, with the official AMT paper offered in both online and printed formats depending on the host center — Cheenta's Kolkata center conducted it as an in-person, paper-based sitting starting at 2:30 PM across the three days.

A few key details of this year's edition:

Divisions by year level: Middle Primary (Years 3–4), Upper Primary (Years 5–6), Junior (Years 7–8), Intermediate (Years 9–10), and Senior (Years 11–12), or the equivalent grade for students outside Australia.
Duration: 60 minutes for the primary divisions, 75 minutes for the secondary divisions.
Question format: 30 questions in total — the first 25 multiple-choice, and the last 5 requiring an exact numerical answer — arranged in increasing order of difficulty.
Scoring: Questions 1–10 are worth 3 marks each, 11–20 are worth 4 marks, 21–25 are worth 5 marks, and the final five questions carry escalating weights up to 10 marks, for a maximum possible score of 135. There is no negative marking for wrong answers.
Awards: Students receive recognition on a sliding scale — Participation, Proficiency, Credit, Distinction, High Distinction, Prize, and Best in School — along with special honours like the Cheryl Praeger Medal (for the top-scoring female student nationally in each year level, secondary divisions) and AMC Medals for outstanding performers, awarded at AMT's discretion to roughly 1 in 5,000 students. A perfect score earns the Peter O'Halloran Award regardless of other honours.

How Cheenta Helped Students Get There

Participants on Day 1 at Cheenta's Lake Place Road center

For students in Kolkata, taking the AMC usually means either going through it via their school or finding an approved center that can register them and administer the paper under proper conditions. Cheenta Academy stepped into that role, organising registration and hosting the exam itself at its Kolkata center so that eligible students — whether or not their own school offered the AMC — had a straightforward way to participate.

Participants on Day 2 with Cheenta's representative Sayan Chakrabarty

This isn't new territory for Cheenta. The academy has a long-standing habit of opening its doors as a host and support center for international olympiads that Indian students might otherwise struggle to access locally — from conducting NMTC (National Mathematics Talent Contest) exams, to hosting the final round of the Sharygin Geometry Olympiad for Indian students in 2025, alongside its own mentorship programs that have helped students qualify for RMO, INMO, IOQM. Hosting the AMC fits squarely into that mission: lowering the logistical barrier between a motivated student and a world-class problem set, while giving them a taste of the kind of rigorous, non-routine problem-solving that Cheenta's own olympiad training emphasises.

Participants on Day 3 with Cheenta's representative Sayan Chakrabarty

What the Contest Actually Tests

Unlike syllabus-driven school exams, the AMC doesn't publish a fixed curriculum — its problems are designed fresh each year by a committee of educators and academics, deliberately built to reward clever thinking over rote formula recall. That said, the broad territory the questions are drawn from typically includes:

Basic arithmetic
Fractions and ratios
Algebra and pre-algebra
Geometry
Measurement
Statistics and probability
General problem-solving techniques, including enumeration and logical reasoning

No question on the paper requires calculus, and the difficulty is carefully graded — the early questions are meant to be accessible to nearly every student sitting the paper, while the last few are genuinely challenging even for strong mathematics students, which is part of why the AMC works equally well as a confidence-builder for beginners and a stretch target for advanced students preparing for further olympiad mathematics.

For Cheenta's students, the AMC often serves as exactly that: an early rung on a much longer ladder that, for some, eventually leads toward RMO, INMO, and beyond. Hosting it locally in Kolkata keeps that first rung within easy reach.

Sharygin Geometrical Olympiad 2026: Cheenta's Kolkata Center Hosts the Final Round

Cheenta recently hosted the final round of the Sharygin Geometrical Olympiad 2026 at its Kolkata center, spread across two days — 31st July and 1st August, from 12 PM to 4 PM each day. Students sat across the table from a panel of jurors and defended their geometric proofs the old-fashioned way: pen, paper, and conversation.

A Little About I.F. Sharygin and the Olympiad's History

The olympiad is named after Igor Fedorovich Sharygin (1937–2004), a Soviet and Russian mathematician best known for his lifelong devotion to elementary geometry. Sharygin wrote extensively for school-level students, and his books — including Problems in Geometry: Plane Geometry and Problems in Geometry: Solid Geometry — shaped how generations of young mathematicians in Russia (and later, well beyond it) learned to think about triangles, circles, and the quiet logic that connects them.

After his passing in 2004, several Russian mathematical and scientific organizations came together to launch an annual olympiad in his memory, starting in 2005. What began as a tribute has since grown into one of the most respected geometry-specific competitions in the world, run under the aegis of the Moscow Center for Continuous Mathematical Education, drawing correspondence-round entries from students across dozens of countries every year.

Details of the Competition

The Sharygin Geometrical Olympiad runs in two rounds:

Correspondence round — a qualifying round solved and submitted in writing, open to high-school students of the four senior-most grades (grades 8–11 in the Russian system, with equivalent grades used for international participants).
Final round — the round Cheenta's Kolkata center hosted this year, held as an oral examination. Participants don't submit a polished written paper. Instead, they work out their solutions on drafts and figures, and then walk a jury through the proof out loud, defending their reasoning in real time.

The scoring is refreshingly binary — a problem is marked either 1 (solved) or 0 (unsolved) — and each student gets up to three attempts to present a solution to a given problem before it's closed out. This oral format makes the final round quite different from most olympiads students are used to: it rewards not just finding the right idea, but being able to explain and defend it clearly under a jury's questioning.

The Jury

Presenting a geometry proof orally means facing genuine back-and-forth questioning, and Cheenta was fortunate to have an experienced panel of jurors (both at the center and online) across both days.

Dr. Sankhadip Chakraborty (INMO awardee, PhD in Mathematics)
Raghunath J V (B.Tech and M.Tech from IIT Chennai, Math Olympiad Coach at Cheenta, INMO and IMO Trainer)
Deepan Dutta (Bachelor of Science from the University of Calcutta)
Deepam Saha (BStat and MStat from Indian Statistical Institute, Kolkata)
Shayeef Murshid (B.Math and M.Math from ISI, INMO Merit List, Doctoral Scholar at Indian Statistical Institute)
Rishav Dutta (attended INMO camp)
Reyanksh Deb (attended INMO camp)

Broad Curriculum of the Contest

True to Sharygin's own body of work, the olympiad stays almost entirely within classical synthetic geometry rather than leaning on heavy algebraic or trigonometric machinery. Depending on grade level, problems typically draw from areas such as:

Triangle geometry — incenters, circumcenters, orthocenters, and the many special points and lines associated with a triangle
Circles — tangency, power of a point, radical axes, and cyclic quadrilaterals
Collinearity and concurrency problems, often solved through clever auxiliary constructions
Geometric transformations — reflections, rotations, spiral similarity, and inversion (more prominent in problems for senior grades)
Loci and constructions
Solid geometry, occasionally, for the more advanced problem sets

The problems are graded by difficulty and intended school grade, and students are free to attempt problems set for older grades — though solutions to problems meant for younger grades aren't considered for scoring if solved by senior students. The emphasis throughout is on elegant, insight-driven proof rather than computation.

How Cheenta Helps Students

Cheenta has hosted the Indian final round of the Sharygin Geometrical Olympiad at its Kolkata center for several years now, in coordination with the organizing committee, giving Indian students a genuine international competition experience without needing to travel abroad.

Beyond simply providing the venue, Cheenta's role is to prepare students for exactly the kind of thinking this olympiad demands. Because the final round is an oral defense rather than a written submission, Cheenta's olympiad training emphasizes not just solving problems but articulating and defending a proof clearly — a skill many students never get to practice until they're standing in front of a jury for real. Cheenta's broader olympiad programs (spanning RMO, INMO, IOQM, AMC, and other national and international contests) run problem-solving sessions through the week, and its geometry-focused resources and past-paper archives give students a way to build the specific pattern-recognition and synthetic-proof instincts that Sharygin-style problems reward.

For a student, the two-day final round is more than a test — it's a chance to sit with a real jury, defend an idea under scrutiny, and experience mathematics the way it was originally meant to be shared: out loud, in conversation, and with the person you're trying to convince sitting right across the table.

Mathematics in Non Routine Spaces for Elementary School

Mathematics lets us enjoy deep insights about the world around us. These insights may otherwise remain hidden from the eyes of a casual observer. The joy of this deeper understanding is comparable to literature or music. One may also build practical things using these insights. That engages the creative side of human nature.

Kids in elementary school can also enjoy this process of discovery and creation through mathematics. Designing lessons around this impetus can transform a child into a thinker and doer at a very early age. At this stage, the math should deliberately be easy. In particular, it should rarely involve complex formulae. Instead, math should be a vehicle of pattern discovery and elementary abstraction.

In this note we share a few open-ended questions that can be transformed into lesson plans in a non-routine mathematics program at the elementary school level (grades 1 to 6). These are not stand-alone lessons. In Cheenta, we combine non-routine problem solving sessions with math in non-routine spaces to produce a holistic experience for the children. Let us clarify this point before proceeding to the questions.

Non-routine problems in mathematics are found in books like Mathematics Can Be Fun by Yakov Perelman or Math Circles for Elementary School Students by Natasha Rozhkovskaya. There are several other books of this flavor. They contain problems that are deliberately non-repetitive. The motive is to help the child think rather than remember.

Mathematics in non-routine spaces, on the other hand, involves external objects such as gardens, road traffic, paintings, and the sky. These begin in spaces that are apparently non-mathematical in nature. A student and a mentor use elementary skills such as counting and pattern recognition to gain deeper insight about the space. That is part of the exercise. Additionally, they may want to create something with that insight. This layer may need more mathematical tools. The motive is to lead the child to think about mathematics in the context of the world around them, and to derive happiness and creativity in the process.

A word on method. Every problem below generates data: growth logs, sky maps, thread counts. The child maintains an observation journal throughout. The journal is the central tool of these activities. It gives the open-endedness a structure. It also shows parents what the child is actually doing, and over months it becomes a record of how the child's thinking matures.

Each of these problems is open-ended in nature. Kids may have different answers. Each problem has two key levers: discovery and creation. Discovery asks the child to observe, measure, and find patterns. Creation asks the child to build something with what was found. Both levers require some element of mathematical science. Each problem ends with an invitation to invent a definition, because definition-making is the deepest mathematical habit a young child can form.

Problem 1

Space: In-house or terrace garden.

Find a few saplings in a nearby nursery. Plant them in your home garden (this can be just one plant at the corner of a table). A plant may need soil, water, fertilizer, pesticide, and sunlight. Not every plant needs everything.

Discover.

Measure the growth of the plant over time and its relation to the inputs that you provide. Record both in your journal. Do you see any relation between the input and the growth? Observe the measurable aspects of the leaves of the plant (for example, the number of veins). Do you see any pattern? Is there a number such that most leaves have that many veins?

Create.

Can you design a more efficient input system next time to improve the growth of your plants? Can you draw a chart that predicts how tall your plant will be next week?

Invent a definition.

What does it mean for a plant to be "growing well"? Height alone? Number of leaves? Something else? Write down your own definition and test it against your plant.

Note for mentors and parents: the mathematical undercurrent here is data collection, tabulation, and elementary correlation. At higher levels the same problem supports rate of change and prediction.

Problem 2

Space: Sky

Look up. Do you see any bright objects in the night sky? Use a compass to understand directions and create a sky map in your journal.

Discover.

Observe the motion of your marked objects over a period of time. Is there a pattern? Is the moon close to certain other objects in the sky? Does it stay close over time?

Create.

Leave the compass at home. Just by looking at the night sky, can you identify north, south, east, and west? Are you able to make a map of your locality using that?

Invent a definition.

Can you find a way to define how far apart two celestial objects in your map are? You cannot use a measuring tape on the sky. What will you use instead?

Note for mentors and parents: the mathematical undercurrent here is coordinate systems, direction, and angular measure. The distance question quietly introduces the idea that a metric must be chosen, not assumed.

Problem 3

Space: Embroidery

Pick up a few embroidered clothes at your home. How many colors does a typical piece use?

Discover.

How many times does one color go beneath another color? Do you see any pattern? Do the patterns repeat? Does the design look the same if you turn the cloth around or view it in a mirror?

Create.

Can you design an embroidery style of your own that others find beautiful? Use graph paper to plan it before touching thread.

Invent a definition.

What makes two embroidery styles "different"? The colors? The repeating unit? The symmetry? Write down your own rule for telling styles apart, and test it on the pieces at home.

Note for mentors and parents: the mathematical undercurrent here is counting, repetition, and symmetry. At higher levels the same problem supports tessellations and transformation.

NSEJS 2017 Question Paper

Question 1

Two wave pulses I and II have the same wavelength. They are travelling in the directions as shown by the single headed arrows. The resultant sketch of the two wave pulses at some instant of time when $P$ coincides with $R$ is ____ .

Question 2

The equivalent resistance of two resistances in series is ' $S$ '. These resistances are now joined in parallel. The parallel equivalent resistance is ' $P$ '. If $S =n P$. Then the minimum possible value of $n$ is

(a) 2

(b) 4

(d) 5

Question 3

A copper disc of radius $a_{0}$ has a hole of radius $b_{0}$ at the centre, at $0^{\circ} \mathrm{C}$. The disc is now heated and maintained at $200^{\circ} \mathrm{C}$. The new radii of disc and hole are $a_{t}$ and $b_{t}$ respectively. For the heated disc it can be concluded that

(a) $a_{0}<a_{t}, b_{0}>b_{t}$ and density of disc increases

(b) $a_{0}<a_{t}, b_{0}>b_{t}$ and density of disc decreases

(d) $a_{0}<a_{t}, b_{0}<b_{t}$ and density of disc decreases

Question 4

A liquid, whose density doesn't change during the motion, is flowing steadily through a pipe of varying cross sectional area as shown in the given figure. If $a_{1}, a_{2}$ are the cross sectional areas, $v_{1}, v_{2}$ are the values of velocities (or speeds) at $L$ and $H$ respectively, then the correct relation between $a_{1}, a_{2}$ and $v_{1}, v_{2}$ is

(a) $a_{1} v_{1}=a_{2} v_{2}$

(b) $a_{1} v_{2}=a_{2} v_{1}$

(d) $a_{1} v_{1}^{2}=a_{2} v_{2}^{2}$

Question 5

A common hydrometer has a uniform scale and its stem is graduated downwards from 0 to 20 . While floating in water, it reads 0 and while floating in a liquid of density $1.40 \mathrm{ g} / \mathrm{cm}^{3}$, it reads 20 . Then the density of the liquid in which it will read 10 is ____ .

(a) $0.7 \mathrm{ g} / \mathrm{cm}^{3}$

(b) $0.85 \mathrm{ g} / \mathrm{cm}^{3}$

(d) $2.8 \mathrm{ g} / \mathrm{cm}^{3}$

Question 6

A boy throws a steel ball straight up. Consider the motion of the ball only after it has left the boy's hand but before it touches the ground and assume that forces exerted by the air are negligible. For these conditions, the force(s) acting on the ball is (are)

(a) A downward force of gravity along with a steadily decreasing upward force.

(b) A steadily decreasing upward force from the moment it leaves the boy's hand until it reaches its highest point; on the way down there is a steadily increasing downward force of gravity as the object gets closer to the earth.

(c) Constant downward force of gravity along with an upward force that steadily decreases until the ball reaches its highest point; on the way down there is only a constant downward force of gravity.

(d) Constant downward force of gravity only

Question 7

In bringing a $\alpha$-particle towards another $\alpha$-particle, the electrostatic potential energy of the system ____ .

(a) Increases

(b) Decreases

(d) Becomes zero

Question 8

An empty office chair is at rest on a floor. Consider the following forces

I. A downward force of gravity

II. An upward force exerted by the floor,

III. A net downward force exerted by the air

Then, which of the force(s) is (are) acting on the office chair?

(a) I only

(b) I and II

(d) None of the forces. (Since the chair is at rest there are no forces acting upon it.)

Question 9

If $x, v$ and $t$ represent displacement (m), velocity (m/s) and time (s) respectively for a certain particle then which pair of the following figures can be best correlated to each other?

(a) I and II

(b) I and III

(d) None

Question 10

In rural areas, an indigenous way of keeping kitchen materials cool is to put them in a box and wrap the box with wet blanket; the blanket is kept wet as tap is allowed to drip in to its corner Choose the correct statement:

(a) This method works because the water from the tap is cold. If one uses room temperature water, it will not work.

(b) Method will work only if the box is a bad conductor of heat. If one uses tin box, it will not work.

(d) This method works because the latent heat necessary for evaporation of water in the blanket is taken from the box so the box and its content remain cool.

Question 11

An electron and $\alpha$-particle enter a region of uniform magnetic field (of induction $B$ ) with equal velocities. The direction of $B$ is perpendicular and into the plane of the paper. Then qualitatively identify the direction of paths of electron and the $\alpha$-particle.

(a) I for $\alpha$-particle, III for electron

(b) I for electron, II for $\alpha$-particle

(d) I for electron, III for $\alpha$-particle

Question 12

A concave mirror of radius of curvature 1 m is placed at the bottom of a water tank. The mirror forms an image of the sun when it is directly overhead. If the depth of water in the tank is 80 cm , then the distance of the image formed is ____ . (refractive index of water is 1.33 )

(a) 50 cm above mirror

(b) On surface of water

(d) Image cannot be formed

Question 13

As shown in adjacent figure, two plane mirrors $M_{1}$ and $M_{2}$ are inclined to each other at an angle $70^{\circ}$ (angle $M_{1} O M_{2}$ ). Incident ray $A B$ makes an angle of incidence $\theta$ on $M_{1}$. This ray after reflection at $B$ on $M_{1}$ and further at $C$ on $M_{2}$ travels along the direction $C D$, such that path $C D$ is parallel to $M_{1}$. Then angle $\theta$ is ____ -

(a) $45^{\circ}$

(b) $50^{\circ}$

(d) $60^{\circ}$

Question 14

For the same angle of incidence, the angle of refraction in three different media $A, B, C$ are $15^{\circ}$, $25^{\circ}$ and $35^{\circ}$ respectively. Then which statement is correct? $\left(\mu_{A}\right.$ is refractive index of $\left.A\right)$

(a) $\mu_{A}$ is maximum and velocity of light is minimum in medium $A$

(b) $\mu_{A}$ is minimum and velocity of light is maximum in medium $A$

(d) $\mu_{A}$ is minimum and velocity of light is minimum in medium $A$

Question 15

A large truck collides head-on with a small compact car. During the collision

(a) The truck exerts a greater force on the car than the car exerts on the truck

(b) The car exerts a greater force on the truck than the truck exerts on the car

(d) The truck exerts the same force on the car as the car exerts on the truck

Question 16

A magnet is placed between two coils $A B$ and $C D$ as shown. It is being moved in the direction as shown by the arrow, then which of the following statements is correct?

(a) Looking from end $A$, current in coil $A B$ will be anticlockwise and looking from end $D$ the direction of current in coil $C D$ will be clockwise

(b) Looking from end $A$, current in coil $A B$ will be clockwise and looking from end $D$, the direction of current in coil CD will be clockwise

(c) Looking from end $A$, current in coil $A B$ will be clockwise and looking from end $D$, the direction of current in coil CD will be anticlockwise

(d) Looking from end $A$, current in coil $A B$ will be anticlockwise and looking from end $D$, the direction of current in coil CD will be anticlockwise

Question 17

The ability of eye to focus both near and distant objects, by adjusting its focal length, is called

(a) Myopia

(b) Presbyopia

(d) Tyndall effect

Question 18

The take-off speed of Airbus A340 is $288 \mathrm{ km} / \mathrm{hr}$. From the taxi track it comes to the main runway and waits for a while for the final clearance from Air Traffic Control. The aircraft then achieves this speed within 50 seconds, Neglecting the effect of the wind direction and friction, what should be the minimum length of main runway decided by civil engineers for this aircraft for a take-off?

(a) 1800 m

(b) 2000 m

(d) 2400 m

Question 19

In the adjacent circuit what is the current flowing from $N$ to $K$ ?

(a) 3 A

(b) 2 A

(d) 0.5 A

Question 20

The positions of two blocks at successive 0.20 second time intervals are represented by the numbered squares in the figure below. The blocks are moving towards right.

The accelerations of the blocks are related as follows :

(a) Acceleration of 'a' is greater than acceleration of 'b'

(b) Acceleration of 'a' equals acceleration of 'b', both accelerations are greater than zero

(d) Acceleration of 'a' equals acceleration of 'b', both acceleration are zero

Question 21

If $x^{2}-3 x+2$ is a factor of $x^{4}-p x^{2}+q$, then $p, q$ are

(a) 0,0

(b) 2,3

(d) 5,4

Question 22

If the roots of the equation $\frac{x^{2}-b x}{a x-c}=\frac{m-1}{m+1}$ are equal and of opposite signs, then the value of $m$ is -.

(a) $\frac{a b}{a+b}$

(b) $\frac{a+b}{a b}$

(d) $\frac{a+b}{a-b}$

Question 23

If $p+q+r=2, p^{2}+q^{2}+r^{2}=30$ and $p q r=10$, the value of $(1-p)(1-q)(1-r)$ will be

(a) -18

(b) -24

(d) -35

Question 24

What is the radius of the circumcircle of a triangle whose sides are $30 \mathrm{ cm}, 36 \mathrm{ cm}$ and 30 cm ?

(a) 15 cm

(b) 16 cm

(d) 18 cm

Question 25

If $A B C D$ is a cyclic quadrilateral, $A B=204, B C=$ 104, $C D=195, D A=85$ and $B D=221$, then $A C=$

(a) 240

(b) 225

(d) 210

Question 26

By which smallest number we should divide 198396198 to get a perfect square?

(a) 14

(b) 18

(d) 28

Question 27

If $(a+b+c+d)=4$, then $\frac{1}{(1-a)(1-b)(1-c)}+\frac{1}{(1-b)(1-c)(1-d)}+$ $\frac{1}{(1-c)(1-d)(1-a)}+\frac{1}{(1-d)(1-a)(1-b)}=$ ____

(a) 0

(b) 0.25

(d) 4

Question 28

In triangle $A B C$, segment $A D$, segment $B E$ and segment $C F$ are altitudes. If $A B \times A C=172.8 \mathrm{ cm}^{2}$ and $B E \times C F=108.3 \mathrm{ cm}^{2}$ then $A D \times B C=$ ____

(a) $136.8 \mathrm{ cm}^{2}$

(b) $132.4 \mathrm{ cm}^{2}$

(d) $128.6 \mathrm{ cm}^{2}$

Question 29

$1 \frac{1}{2}+1 \frac{1}{6}+1 \frac{1}{12}+1 \frac{1}{20}+1 \frac{1}{30}+\ldots \ldots .+1 \frac{1}{380}=$ ____

(a) 19.85

(b) 19.95

(d) 20.25

Question 30

If $x=(\sqrt{21}-\sqrt{20})$ and $y=(\sqrt{18}-\sqrt{17})$, then

(a) $x=y$

(b) $x+y=0$

(d) $x<y$

Question 31

If $\left(x+\frac{1}{x}\right)=5$, then $\left(x^{3}+\frac{1}{x^{3}}\right)-5\left(x^{2}+\frac{1}{x^{2}}\right)+ \left(x+\frac{1}{x}\right)=$ ____ .

(a) 0

(b) 5

(d) 10

Question 32

The mean of the following frequency distribution is ____ .

__TABULAR__:W1siQ2xhc3MiXSxbImludGVydmFsIl1d	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
Frequency	4	6	8	10	12

(a) 25

(b) 28

(d) 32

Question 33

On seventy first 'Independence Day' there was Tuesday. After how many years there will be Tuesday on 'Independence Day'?

(a) 4 yrs .

(b) 5 yrs .

(d) 7 yrs .

Question 34

If $x^{2}+x y+x z=135, y^{2}+y z+x y=351$ and $x^{2}+x z+y z=243$, then $x^{2}+y^{2}+z^{2}=$ ____

(a) 225

(b) 250

(d) 300

Question 35

What will be the remainder if the number $(7)^{2017}$ is divided by 25 ?

(a) 1

(b) 7

(d) 24

Question 36

The sum of two numbers is 13 and the sum of their cubes is 1066 . Find the product of those two numbers.

(a) 26

(b) 27

(d) 29

Question 37

Diagonals of a quadrilateral bisect each other. Therefore the quadrilateral must be a ____

(a) Parallelogram

(b) Rhombus

(d) Square

Question 38

A train is running at a speed of $54 \mathrm{ km} / \mathrm{hr}$. It is not stopping at a certain station. It crosses the person showing green flag in 20 seconds and crosses the platform in 36 seconds. What is the length of the train?

(a) 240 m

(b) 300 m

(d) 360 m

Question 39

What is the sum of all odd numbers between 500 and 600 ?

(a) 29500

(b) 27500

(d) 2600

Question 40

How many four digit numbers are there such that when they are divided by 101, they have 99 as remainder?

(a) 90

(b) 98

(d) 101

Question 41

An open vessel contains air at $27^{\circ} \mathrm{C}$. The vessel is heated till two-fifth of the air in it has been expelled. Assuming the volume of the vessel remains constant, find the temperature to which the vessel has to be heated?

(a) 500 K

(b) 550 K

(d) 750 K

Question 42

Rajiv, Nikhil, Shubha and Nilima wanted to establish a relationship between loss in weight of a solid with weight of water displaced by immersing it in tap water and sea water. After performing their experiment, they noted their observations for the same solid as follows: Rajiv : Loss of weight of solid is more in tap water. Nikhil : Loss of weight of solid is more in sea water. Shubha : Loss of weight of solid is equal in the tap water and the sea water. Nilima : Loss of weight of solid may be more in tap water or sea water, depending upon how deeply it is immersed. Identify the correct observation.

(a) Nikhil

(b) Nilima

(d) Rajiv

NSEJS 2016 Question Paper

Question 1

Rod AB of radius 2 r is joined with rod BC of radius r . They are of same material and are of same length. The combination carries a current I. Choose the correct statement.

(a) $V_{A B}=4 V_{B C}$

(b) Current per unit in AB and BC are equal

(d) $\mathrm{V}_{\mathrm{BC}}=4 \mathrm{ V}_{\mathrm{AB}}$ Ans. (d) Sol. $\frac{\mathrm{V}_{\mathrm{AB}}}{\mathrm{V}_{\mathrm{BC}}}=\frac{\mathrm{IR}_{\mathrm{AB}}}{\mathrm{IR}_{\mathrm{BC}}}=\frac{\mathrm{I} \rho \ell / \pi(2 \pi)^{2}}{\mathrm{I} \rho \ell / \pi \mathrm{r}^{2}}=\frac{1}{4}$ or $\mathrm{V}_{\mathrm{BC}}=4 \mathrm{ V}_{\mathrm{AB}}$

Question 2

The statement "a is not less than 4 " is correctly represented by

(a) $a<4$

(b) a $>4$

(d) $\mathrm{a} \leq 4$

Question 3

In the figure shown, the current carrying loop is fixed, where as current carrying straight conductor is free to move. Then straight wire will (ignore gravity)

(a) remain stationary

(b) move towards the loop

(d) rotate about the axis perpendicular to plane of paper

Question 4

Two friends A and B watched a car from the top of their buildings. Angle of depression for A was $10^{\circ}$ more than angle of depression for B , then

(a) A's apartment is taller than B's apartment

(b) B's apartment is taller than A's apartment

(d) We cannot compare the heights of the two apartments

Question 5

A convex mirror of focal length $f$ produces an image of size equal to $\frac{1}{n}$ times the size of the object. Then the object distance is

(a) nf

(b) $\frac{\mathrm{f}}{\mathrm{n}}$

(d) (n - 1)f

Question 6

Total surface area of a sphere $S$ with radius $\sqrt{2}+\sqrt{3} \mathrm{ cm}$ is

(a) $400 \pi(5+2 \sqrt{6}) \mathrm{sq} \mathrm{mm}$

(b) $\pi(\sqrt{2}+\sqrt{3})^{2}$ sq cm

(d) $40 \pi(5+2 \sqrt{6}) \mathrm{sq} \mathrm{mm}$

Question 7

The angle between the hour arm and the minute arm of a clock at 2:10 a.m. is

(a) Zero

(b) $4^{\circ}$

(d) $6^{\circ}$

Question 8

A craft teacher reshapes the wax from a cylinder of candle with section diameter 6 cm and the height 6 cm into a sphere. The radius of this sphere will be

(a) $r=6 \sqrt{3 / 2} \mathrm{ cm}$

(b) $r=6 \mathrm{ cm}$

(d) $\mathrm{r}=6 \sqrt[3]{2} \mathrm{ cm}$

Question 9

A point object O is kept at origin. When a concave mirror $\mathrm{M}_{1}$ placed at $\mathrm{x}=6 \mathrm{ cm}$, image is formed at infinity. When $\mathrm{M}_{1}$ is replaced by another concave mirror $\mathrm{M}_{2}$ at same position, image is formed at $\mathrm{x}=$ 30 cm , then ratio of the focal length of $\mathrm{M}_{1}$ to that of $\mathrm{M}_{2}$ is

(a) $\frac{3}{4}$

(b) $\frac{4}{3}$

(d) $\frac{1}{5}$

Question 10

The number $3^{8}\left(3^{10}+6^{5}\right)+2^{3}\left(2^{12}+6^{7}\right)$ is

(a) A perfect square and a perfect cube

(b) Neither a perfect squre nor a perfect cube

(d) A perfect square but not a perfect cube

Question 11

Melting point of a substance is $10^{\circ} \mathrm{C}$. What does this mean?

(a) The substance is a liquid at $10^{\circ} \mathrm{C}$.

(b) The substance is a solid at $10^{\circ} \mathrm{C}$.

(d) The substance is $50 %$ solid and $50 %$ liquid at $10^{\circ} \mathrm{C}$.

Question 12

Let the number of rectangles formed by 6 horizontal and 4 vertical lines be n and those formed by 5 vertical and 5 horizontal lines be m then we have

(a) $n=m$

(b) $\mathrm{n} \geq \mathrm{m}+1$

(d) $m>n+5$

Question 13

The effective resistance between $A$ and $D$ in the circuit shown in the adjacent figure is

(a) $5 \Omega$

(b) $10 \Omega$

(d) $20 \Omega$

Question 14

If ABCD is a rhombus and $\angle \mathrm{ABC}=60^{\circ}$ then

(a) The points $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are concyclic

(b) The quadrilateral has exactly half the area of the square with same sides as ABCD

(d) The diagonals of the quadrilateral ABCD are equal and bisect each other at right angle

Question 15

Three bulbs $\mathrm{B}_{1}, \mathrm{ B}_{2}$ and $\mathrm{B}_{3}$ having rated powers $100 \mathrm{ W}, 60 \mathrm{ W}$ and 60 W at 250 V are connected in a circuit as shown in the adjacent figure. If $\mathrm{W}_{1}, \mathrm{ W}_{2}$ and $\mathrm{W}_{3}$ are the output powers of the bulbs $\mathrm{B}_{1}, \mathrm{ B}_{2}$ and $\mathrm{B}_{3}$ respectively, then

(a) $\mathrm{W}_{1}>\mathrm{W}_{2}=\mathrm{W}_{3}$

(b) $\mathrm{W}_{1}>\mathrm{W}_{2}>\mathrm{W}_{3}$

(d) $\mathrm{W}_{1}<\mathrm{W}_{2}<\mathrm{W}_{3}$

Question 16

If $\mathrm{a}, \mathrm{b}>0$ then

(a) $a+b \leq \sqrt{a b}$

(b) a + b $>\sqrt{\text { ab }}$

(d) None of the above inequalities will hold

Question 17

Following diagram shows refraction of parallel beam of light through a spherical surface. Identify the correct ray diagram

Question 18

Tenth term in the sequence $12,18,20,28, \ldots$ is

(a) 336

(b) 63

(d) 68

Question 19

In the diagram $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ are two plane mirrors at right angles to each other. O is a luminous point object. Consider two images formed due to first reflection at $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$. The area of the triangle formed by the object and two images is

(a) $4 \mathrm{ cm}^{2}$

(b) $2 \mathrm{ cm}^{2}$

(d) $16 \mathrm{ cm}^{2}$

Question 20

The probability of a point within an equilateral triangle with side 1-unit lying outside its in-circle (inscribed circle) is

(a) $1-\frac{1}{2(\sqrt{3})}$

(b) $1-\frac{\pi}{3 \sqrt{3}}$

(d) $1-\frac{2 \pi}{3 \sqrt{3}}$

Question 21

A fisherman of height $h$ is standing on the bank of a lake. A fish in the water perceives his height as $h$ '. Then

(a) $\mathrm{h}^{\prime}>\mathrm{h}$

(b) $h^{\prime}<h$

(d) $h^{\prime}>h$ or $h^{\prime}<h$ depending on position of fish

Question 22

A triangle has perimeter 316 and its sides are of integer length. The maximum possible area for such a triangle is achieved for

(a) Single triangle

(b) Two triangles

(d) More than three triangle

Question 23

Object A is completely immersed in water. True weight of object A is $\mathrm{W}_{\mathrm{A}}$. Weight of water with beaker is $\mathrm{W}_{\mathrm{B}}$. Let B be the buoyant force. $\mathrm{W}_{1}$ and $\mathrm{W}_{2}$ are scale readings of spring balance and weighing scale respectively.

(a) $\mathrm{W}_{1}=\mathrm{W}_{\mathrm{A}}$

(b) $\mathrm{W}_{1}=\mathrm{W}_{\mathrm{A}}+\mathrm{B}$

(d) $\mathrm{W}_{2}=\mathrm{W}_{\mathrm{B}}+\mathrm{B}$

Question 24

Number of numbers less than 40 having exactly four divisors is

(a) 15

(b) 12

(d) 14

Question 25

Velocity of a particle moving along a straight line varies with time as shown in the figure. Net forces acting on the aprticle are $\mathrm{F}_{1}, \mathrm{ F}_{2}, \mathrm{ F}_{3}, \mathrm{ F}_{4}$ and $\mathrm{F}_{5}$ in the intervals $\mathrm{OA}, \mathrm{AB}, \mathrm{BC}, \mathrm{CD}$ and DE respectively. Indentify the correct statement

(a) $\mathrm{F}_{1}$ increases with time

(b) $\mathrm{F}_{5}$ is initially positive and then becomes negative

(d) $\mathrm{F}_{3}$ is negative

Question 26

If set $X$ consists of three elements then the number of elements in the power set of power set of $X$ is

(a) $3^{3}$

(b) $2^{3}$

(d) $2^{8}$

Question 27

A wooden block $(\mathrm{W})$ is suspended by using a cord from a heavy steel ball (B). The entire system is dropped from a height. Neglecting air resistance, the tension in the cord is

(a) Zero

(b) The difference in the masses of B and W

(d) The weights of $B$

Question 28

In a n - sided regular polygon, the radius of the circum-circle is equal in length to the shortest diagonal. the number of values of $\mathrm{n}<60$ for which this can happen is

(a) 0

(b) 1

(d) 2

Question 29

A circus performer of weight W is standing on a wire as shown in the figure. The tension in the wire is

(a) Approximately $\frac{\mathrm{W}}{4}$

(b) Approximately $\frac{\mathrm{W}}{2}$

(d) Much less than $\frac{W}{2}$

Question 30

Number of integers $n$ such that the number $1+n$ is a divisor of the number $1+n^{2}$ is :

(a) 0

(b) 1

(d) 2

Question 31

In the following diagrams O is point object and I is its image formed by a concave mirror. Identify the diagram. In which position of image I is nearly correct.

Question 32

If for $\mathrm{x}, \mathrm{y}>0$ we have $\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=2$ then the minimum value of xy is

(a) 2

(b) 1

(d) $\sqrt{2}$

Question 33

Two bodies $A$ and $B$ are charged with equal magnitude of charge but $A$ with positive charge and $B$ with negative. If $M_{A}$ and $M_{B}$ are masses before charging and $M^{I}{ }_{A}$ and $M^{I}{ }_{B}$ are the masses after charging the ( $\mathrm{m}_{0}$ is some constant mass)

(a) $\mathrm{M}_{\mathrm{A}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{A}}+\mathrm{m}_{0}$ and $\mathrm{M}_{\mathrm{B}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{B}}-\mathrm{m}_{0}$

(b) $\mathrm{M}_{\mathrm{A}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{A}}-\mathrm{m}_{0}$ and $\mathrm{M}_{\mathrm{B}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{B}}+\mathrm{m}_{0}$

(d) $M^{I}{ }_{A}=M_{A}-\frac{m_{0}}{2}$ and $M_{B}^{I}=M_{B}+m_{0}$

Question 34

The number of natural numbers $\mathrm{n} \leq 30$ for for which $\sqrt{\mathrm{n}+\sqrt{\mathrm{n}+\sqrt{\mathrm{n}+\ldots . .}}}$ is a natural number is

(a) 30

(b) Zero

(d) 5

Question 35

A conductor of length $L$ has a varying cross section with area $2 A$ at $P$ and $A$ at $Q$ as shown in the figure. If it carries a steady current I, then

(a) Net charge per unit volume near P is more than net charge per unit volume near Q .

(b) Net charge per unit volume near Q is less than net charge per unit volume.

(d) Current per unit area near P is less than current per unit area near Q .

Question 36

The number of natural numbers $n \leq 30$ for which $\sqrt{n+\sqrt{n+\sqrt{n+\ldots \ldots . .}}}$ is a prime number is

(a) Three

(b) Zero

(d) Two

Question 37

Vessels A and B are made of conducting material. Both contain water. Vessel A floats in B. Vessel B is now heated at a uniform rate, then

(a) Water in A boils first.

(b) Water in A boils some time after water in B starts boiling.

(d) Water in A does not boil.

Question 38

The number of squares formed by 5 vertical and 4 horizontal lines (all are equispaced) is

(a) 60

(b) 20

(d) 46

Question 39

Every major city in India has a pollution control board to monitor air and water pollution. The following data is from three different localities in Bangalore city from the year 2015.

ppb stands for parts per billion and ppm stands for parts per million. These are different units to express concentration. They are very similar to percentage (which is actually parts per hundred). Based on the above data, which place will you choose to live in?

(a) All localities are equally polluted, so I have no preference.

(b) P is more polluted than Y and Z , hence I will live in either Y or Z .

(d) Z and Y are more polluted than P , hence I will live in P .

Question 40

A body thrown vertically up reaches a maximum height and returns back. Its acceleration is

(a) Downward during both ascent and descent.

(b) Downward at all positions except at the highest point, where it is zero.

(d) Downward during ascent and upward during descent.

Question 41

The number of integers $a, b, c$ for which $a^{2}+b^{2}-8 c=3$ is

(a) 2

(b) Infinite

(d) 4

NSEJS 2020 Question Paper

Question 1

Gravitational collapse is the contraction of an astronomical object under its own gravity. This draws the matter inwards towards the centre of gravity. A neutron star is an example of the collapsed core of a giant star. A certain neutron star of radius 10 km is of mass $1.5 M_{\odot}$. The acceleration due to gravity on the surface of the neutron star is nearly

(a) $2.0 \times 10^{8} \mathrm{ m} / \mathrm{s}^{2}$

(b) $2.0 \times 10^{12} \mathrm{ m} / \mathrm{s}^{2}$

(d) $2.6 \times 10^{20} \mathrm{ m} / \mathrm{s}^{2}$

Question 2

Two illuminated point objects $\mathrm{O}_{1}$ and $\mathrm{O}_{2}$ are placed at a distance 24 cm from each other along the principal axis of a thin convex lens of focal length 9 cm such that images of both the objects are formed at the same position. Then the respective distances of the lens from $\mathrm{O}_{1}$ and $\mathrm{O}_{2}$ (in cm ) are

(a) 12 and 12

(b) 18 and 6

(d) 16 and 8

Question 3

A nuclear reactor is working at $30 %$ efficiency (i.e. conversion of nuclear energy to electrical energy). In this reactor ${ }_{92}^{235} \mathrm{U}$ nucleus undergoes fission and releases 200 MeV energy per atom. If 1000 kW of electrical power is obtained in this reactor, then the number of atoms disintegrated (undergone fission) per second in the reactor is

(a) $1.04 \times 10^{17}$

(b) $6.5 \times 10^{12}$

(d) $3.25 \times 10^{32}$

Question 4

Two blocks $A$ and $B$ are in contact with each other and are placed on a frictionless horizontal surface. A force of 90 N is applied horizontally on block A (situation I ) and the same force is applied horizontally on block B (situation II). Mass of A is 20 kg and B is 10 kg . Then the correct statement is

(a) Since both the blocks are in contact, magnitude of force by block A on B will be 90 N (situation I) and magnitude of force by block B on A will also be 90 N (situation II).

(b) Magnitude of force by block A on B is 30 N (situation I) and magnitude of force by block B on $A$ is 60 N (situation II).

(c) Magnitude of force by block A on B is 60 N (situation I) and magnitude of force by block B on A is 30 N (situation II).

(d) The 90 N force will produce acceleration of different magnitudes in A and B .

Question 5

In the adjoining circuit, $R=5 \Omega$. It is desired that the voltage across $R_{x}$ should be 6 V , then the value of $R_{x}$ should be

(a) $4 \Omega$

(b) $12 \Omega$

(d) $20 \Omega$

Question 6

If $x^{2}+a x+b=0$ and $x^{2}+b x+a=0$ have one common root, then

(a) $a+b=0$

(b) $a+b=1$

(d) $a^{2}+b^{2}=1$

Question 7

Six circles each of radius 3 cm are inscribed in an equilateral triangle ABC such that they touch each other and also touch the sides of the triangle as shown in the adjacent figure. Then height of triangle $A B C$ is

(a) $6(2 \sqrt{3}+3)$

(b) $3(2 \sqrt{3}+6)$

(d) $6(2+\sqrt{3})$

Question 8

Find the remainder when $x^{51}$ is divided by $x^{2}-3 x+2$

(a) $X$

(b) $\left(2^{51}-2\right) x+2-2^{51}$

(d) 0

Question 9

If $\frac{3}{x-2}<1$, where $x$ is a real number, then

(a) $2<x<5$

(b) $x<2$ or $5<x$

(d) None of these

Question 10

If $100^{25}-25$ is written in decimal notations, then the sum of its digits is

(a) 444

(b) 442

(d) 422

Question 11

$A B C$ is a triangle, the bisector of angle $A$ meets $B C$ in $D$. The relation between $A D, A B$ and $A C$ is

(a) $A D>\sqrt{A B \cdot A C}$

(b) $A D>A B \cdot A C$

(d) $A D<\sqrt{A B \cdot A C}$ PART: A - 2 MORE THAN ONE CORRECT OPTIONS. BUBBLE ALL CORRECT OPTIONS ONLY.

Question 12

An infinitely long conductor when carrying current $I$, produces a magnetic field $B$ around it. If such a conductor is placed along the X-axis, then the magnitude of $B$ at a distance $r$ is given by the relation $B=\frac{\mu_{0}}{4 \pi} \frac{2 I}{r}$, (where $\frac{\mu_{0}}{4 \pi}=10^{-7} \mathrm{NA}^{-2}$ is a constant). The following figure shows such an infinitely long conductor placed along X -axis carrying current $I$ and $B$ at $S$ is $2 \times 10^{-4} \mathrm{ T}$, directed into the plane of the paper at S . Given $r=1 \mathrm{ cm}$. Then, the correct statements are

(a) $I=10 \mathrm{ A}$

(b) The number of electrons transported across the cross section of the conductor during time 1s is $6.25 \times 10^{19}$

(d) The electrons will flow in the direction $X_{2}$ to $X_{1}$.

Question 13

The ratio of the charge of an ion or subatomic particle to its mass $(q / m)$ is called specific charge. Then the correct options are

(a) SI unit of specific charge can be written as $\mathrm{A} \cdot \mathrm{s} / \mathrm{kg}$.

(b) If all the isotopes of hydrogen are ionized then tritium will have least specific charge among them.

(d) specific charge ratio of an electron is $1.75 \times 10^{11} \mathrm{C} / \mathrm{kg}$.

Question 14

If $0 \leq x \leq \pi$ and $81^{\sin ^{2} x}+81^{\cos ^{2} x}=30$, then $x=$

(a) $\frac{\pi}{6}$

(b) $\frac{\pi}{3}$

(d) $\frac{2 \pi}{3}$ [0pt] [Useful information: $\pi^{c}=180^{\circ}, \sin (180-\theta)=\sin \theta, \sin \theta \geq 0$ when $0 \leq \theta \leq 180^{\circ}$ ]

Question 15

Given $(a-b)^{2}+(a-c)^{2}=(b-c)^{2}$, then which of the following statements are true?

(a) equation is valid when $b=c$ and $a \neq c$

(b) equation is valid when $a=b$

(d) Given equation is not valid when $a, b$ and $c$ are distinct.

NSEJS 2021 Question Paper

Question 1

The axes of a coordinate system $\mathrm{S}_{2}$ are inclined at an angle $\theta$ to those of another coordinate system $\mathrm{S}_{1}$. The origins of both the systems are coinciding. A particle $\mathrm{P}_{1}$ at rest in system $\mathrm{S}_{1}$, starts from point ($-2,0$) and travels along positive direction of $\mathrm{X}_{1}$ axis with uniform acceleration of $1.25 \mathrm{ m} / \mathrm{s}^{2}$ for 4 s and stops. In system $\mathrm{S}_{2}$, particle $\mathrm{P}_{2}$, starts from rest from the origin and travels for 2 s along positive direction of $\mathrm{X}_{2}$ axis with uniform acceleration $5 \mathrm{ m} / \mathrm{s}^{2}$ and stops. If the final distance between $P_{1}$ and $P_{2}$ is 6 m , then the angle between $+\mathrm{Y}_{1}$ axis and $+\mathrm{X}_{2}$ axis is

(a) $36.8^{\circ}$

(b) $53.2^{\circ}$

(d) $126.8^{\circ}$

Question 2

The variation of a certain physical parameter $Z$ with variable $u$ is given by the relation $\mathrm{Z}=A\left(\frac{R}{R+u}\right)^{3}$, where $R$ and $A$ are constants and the maximum value of $u \ll R$. Then to find $R$, a student plots a graph of variation of $Z$ ( Y axis) against $u$ ( X axis). The graph is a

(a) straight line passing through origin and slope $=\frac{R}{3}$

(b) straight line with intercept $\frac{3 A}{2}$ and slope $=-\frac{R}{3 A}$

(d) straight line with intercept $-\frac{A}{2}$ and slope $=-3 R$

Question 3

A submarine $S_{1}$ is parked at a depth of 200 m in an ocean on earth. Assume oceans exist on Mars. At about what depth a submarine $\mathrm{S}_{2}$ has to be parked in an ocean on Mars so that $\mathrm{S}_{2}$ will experience same pressure as that of $\mathrm{S}_{1}$ ? Acceleration due to gravity on Mars is $3.7 \mathrm{ m} / \mathrm{s}^{2}$. (Assume that sea water density on Earth and Mars is same, $\rho=1.03 \times 10^{3} \mathrm{ kg} / \mathrm{m}^{3}$ )

(a) 158 m

(b) 435 m

(d) 616 m

Question 4

In an oscillating system, damping results in dissipation of the stored energy. The following figure shows the variation of displacement x with time t for an oscillating system. Which of the following statements best describes this physical phenomenon.

(a) Oscillatory motion of an object without damping

(b) Oscillatory motion of an object with damping such that time measurement was started when the system was at the mean position.

(d) Oscillatory motion of an object with damping such that time measurement was started when the system had maximum potential energy.

Question 5

In the adjacent circuit, the galvanometer $G$ does not show any deflection. If $R=2 \Omega$, the current drawn from the cell is

(a) 1 A

(b) 9 A

(d) $\frac{9}{4} \mathrm{ A}$

Question 6

'Gear' is a mechanical system used to transfer mechanical and rotary motion from one mechanical system to another. As shown in the figure below the driving wheel A drives the driven wheel B without slipping and thus forms the gear system. The wheel A has 16 teeth and B has 24 teeth. Wheel B has a projection (shown by white ring in Fig. 1 and also in the side view of Fig. 2) of radius $\frac{14}{11} \mathrm{ cm}$.

A long massless, inextensible string can be wound / unwound over this circular projection. A mass $m$ is attached to the free end of this long string. If the wheel A makes 6 revolutions per second in the clockwise direction, without slipping, then in $\frac{1}{2}$ second the potential energy of the mass $m$ in CGS unit

(a) increases by $32 m g$

(b) decreases by 32 mg

(d) decreases by 16 mg

Question 7

Canopus is the second brightest star in the night sky. It is about 300 light years away. The energy is produced inside the star through nuclear reactions. If we receive $5.0 \times 10^{-8} \mathrm{ W} / \mathrm{m}^{2}$ energy from Canopus, how much mass does it lose per second?

(a) $1.70 \times 10^{-6} \mathrm{ kg}$

(b) $1.91 \times 10^{9} \mathrm{ kg}$

(d) $6.34 \times 10^{31} \mathrm{ kg}$

Question 8

An average human adult radiates about 100 W energy mainly in infra-red region of the electromagnetic spectrum. 50 persons are sitting in a hall with an air conditioning system which is $50 %$ efficient. How much electricity must be used to maintain temperature of the hall at $25^{\circ} \mathrm{C}$ for 4 hours?

(a) 5 units

(b) 10 units

(d) 40 units

ANY NUMBER OF OPTIONS 4, 3, 2 or 1 MAY BE CORRECT IN THE FOLLOWING QUESTIONS.

Question 9

According to Einstein's theory, light can be assumed to be in the form of a large number of discrete energy packets called 'photons'. In case of light of frequency $v$, each photon carries energy $E=h v$. In a certain surgical procedure a surgeon uses LASER beam of wavelength 650 nm in pulses of 30.0 ms duration. The average power of each pulse is 0.6 W . Here h is Planck's constant. Then

(a) the frequency of this LASER photon is $4.6 \times 10^{14} \mathrm{ Hz}$

(b) the energy in each pulse is $1.1 \times 10^{17} \mathrm{eV}$

(d) number of photons in each pulse is $5.9 \times 10^{16}$

Question 10

In the following circuit, $\mathrm{R}_{1}=6 \Omega, \mathrm{R}_{2}=12 \Omega, \mathrm{ V}=16 \mathrm{ V}$. The currents $\mathrm{I}_{1}$ and $\mathrm{I}_{2}$ flow through the resistances $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ respectively

(a) power generated across $\mathrm{R}_{1}$ is 42.6 watt

(b) the ratio of $\frac{I_{1}}{I_{2}}=2$

(d) as $R_{2}=2 R_{1}$, the voltage across $R_{2}$ will be twice the voltage across $\mathrm{R}_{1}$

Question 11

A glass plate of uniform thickness $t$ and refractive index $\mu$ is as shown in the diagram. AB is the incident ray and FG is the emergent ray. The angles of incidence and refraction are $i$ and $r$ respectively. The perpendicular distance $\mathrm{FC}=x$ between the incident and the emergent rays is called the lateral shift. Then

(a) $x=t\left(\sin i-\frac{\cos i \sin r}{\cos r}\right)$

(b) $x$ depends on refractive index $\mu$

(d) Maximum value of $x=t$ when $i$ is close to $90^{\circ}$