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# EW phase transition and baryogenesis

There are three Sakharov conditions of baryogenesis:

1. CP symmetry violation.

2. Baryon number conservation violation.

3. Thermodynamical equilibrium violation.

In SM the first one is satisfied by CKM phases, the second one is satisfied by instanton-like transitions for zero temperature and sphaleron-like transitions for non-zero temperatures. As for the third one, I've thought that EW phase transition provides violation of equilibrium, but after reading Gorbunov and Rubakov book "Introduction to the Theory of the Early Universe: Hot Big Bang Theory" I've started to doubt. They write that this transition is of second type one; thus it doesn't provide sufficient thermodynamical equilibrium violation.

So can EW phase transition provide baryogenesis?

Within the standard model: It cannot!

Even though there is CP violation in the SM (as you state), the amount is not enough to give the ratio of

$\frac{n_B}{n_\gamma} \sim 10^{-10} \ .$

Furthermore, the electroweak phase transition (EWPT) in the SM is not even a second order transition, but merely a crossover.

In order to render the transition (strongly) first order and to account for enough CP violation, new physics beyond the standard model is needed. Supersymmetry (SUSY) is just one of the theories in the literature. One simple extension of the SM is the inert doublet model (IDM) which has a first order EWPT, but unfortunately no additional CP violation.

Although we have not yet found signs of new physics at colliders, the EWPT is still a hot candidate for baryogenesis.


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# Mathematical theory of aesthetics

The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric figures aesthetically pleasing and others less so. Naturally, some would mention the properties of elegance, symmetry and surprise but I think these constitute basic ideas and not a well-developed thesis.

In this spirit, I would like to know whether there are any references to mathematicians who have developed a mathematical theory of aesthetics as well as algorithms(if possible) for discovering aesthetically pleasing mathematical structures.

To give precise examples of mathematical objects which are generally considered aesthetic, I would include:

1. Mandelbrot set
2. Golden ratio
3. Short proofs of seemingly-complex statements(ex. Proofs from the Book)

I think the last example is particularly useful as Jürgen Schmidhuber, a famous computer scientist and AI researcher, has attempted to derive a measure of beauty using Kolmogorov Complexity in his series of articles titled 'Low Complexity Art'. Meanwhile, I find the following research directions initiated by computer scientists particularly fruitful:

Note: From a scientific perspective, researchers on linguistic and cultural evolution such as Pierre Oudeyer have identified phenomena which are both diverse and universal. Diversity is what makes our cultures different and universality enables geographically-isolated cultures to understand one another. In particular, many aesthetics have emerged independently in geographically isolated cultures especially in cultures which developed in similar environments. Basically, I believe that if we take into account what scientists have learned from the fields of cultural and linguistic evolution, embodied cognition, and natural selection I think we could find an accurate mathematical basis for aesthetics which would also be scientifically relevant.

• I'd be very skeptical of any such theory, on the grounds that what is aesthetically pleasing is highly subjective and culture-specific. A well-supported theory of the type you request would prove me wrong, but an inability to identify such a theory would confirm my view. Mar 22, 2018 at 23:27
• @NikWeaver What I find interesting is that we observe both diversity, which you mention, and universality. Universality in the sense that many aesthetics emerge independently in geographically isolated cultures. Work on cultural evolution and linguistic evolution by Pierre Oudeyer and others supports this perspective. Mar 22, 2018 at 23:35
• @Qfwfq I understand but disagree with the question being put on hold as off-topic. The way I see it is that the question is on a very fruitful meta-level: what guides mathematicians in their research, and in the case of aesthetics, has any systematic, scientific, serious and /or objective study been made of how this aesthetic guidance works? This to me is a very interesting research-level question, since this is exactly what determines which type of solutions and even questions I consider when tackling research math. Below research level, the question doesn't even truly crop up I would say. Mar 25, 2018 at 16:19
• (...) so okay, perhaps I read a too broad scope in the question, which was not there. So I'll vote to reopen, considering also there are already 6 answers. [by the way, it's not me who decides about opening/closing questions here: I just happened to have cast my vote!] Mar 26, 2018 at 2:54
• There is some papers on arXiv that uses deep learning to assess aesthetics of artistic images, such as this one: arxiv.org/abs/1707.03981 Maybe these works can be adapted to also assess the aesthetics of mathematical objects... Mar 27, 2018 at 20:01

George D Birkhoff, Aesthetic Measure, 1933

An attempt to bring the basic formal side of art within the purview of simple mathematical formula defining aesthetic measure. Contents: the basic formula; polygonal forms; ornaments and tilings; vases; diatonic chords; diatonic harmony; melody; musical quality in poetry; earlier aesthetic theories; art and aesthetics. Over 20 plates and illustrations.

• I like the sound of this book. It appears very ambitious in scope. Mar 23, 2018 at 0:18

A search for aesthetic* in the title at MathSciNet yields 100 hits, e.g.,

MR3751155 Lähdesmäki, Tuuli – Aesthetics of geometry and the problem of representation in monument sculpture. Aesthetics of interdisciplinarity: art and mathematics, 275–290, Birkhäuser/Springer, Cham, 2017.

MR3751140 Cohen, Mark Daniel – The geometric expansion of the aesthetic sense. Aesthetics of interdisciplinarity: art and mathematics, 29–43, Birkhäuser/Springer, Cham, 2017.

MR3644156 Pimm, David; Sinclair, Nathalie – Explaining beauty in mathematics: an aesthetic theory of mathematics [book review of MR3156013]. Math. Intelligencer 39 (2017), no. 1, 79–81.

MR3623974 Kao, Yueying; He, Ran; Huang, Kaiqi – Deep aesthetic quality assessment with semantic information. IEEE Trans. Image Process. 26 (2017), no. 3, 1482–1495.

One name and two books come to mind: Joseph Schillinger, and his two books, The Mathematical Basis of the Arts and The Schillinger System of Musical Composition.

The Mathematical Basis of the Arts is a work that aims at generalizing the concepts present in art pieces, in general, from a geometrical point of view, and how this affects the human perception mechanism. I cannot comment too much about this book because I have read it but haven't really study from it. Schillinger's works are clear, but the (natural) language he uses, and the strange mathematical context/notation make it so that some time and practice are needed to really absorb his ideas. A quick look at the table of contents should wet anyones appetite.

The Schillinger System of Musical compositon is as much a philosopical corpus about aesthetics based on geometrical/psycological/physiological arguments, as a theory of composition. It is the only work that I am aware of that has a philosophical theory of melody. That is, it throughly studies what makes a melody what it is, and not just how to write a melody. The principles that he presents and develops apply to art in general, and not only to music, however, he presents his ideas in the context of musical theory.

I should also say that his is not so much a mathematical theory of art/music/aesthetics as a general theory of aesthetics. It's just that he uses the concepts and language of mathematics to present those general concepts. In doing so, many people come to believe that he is developing a mathematical theory of art/music. He's not. As he points out, he is developing a scientific theory of how we create and perceive art, and thus, effectively, he's developing a formal theory of aesthetics, since beauty is always in the eye of the beholder.

Schillinger's work is from an epoch that believed that human intellect trumped statistical analysis. And that is how it is developed. From a rational derivation from a few basic (universal?) principles, and not from a brute-force approach to find regularities in works of art.

• The very first words of chapter one of The Mathematical Basis of the Arts are as far as can be from what I consider a balanced viewpoint. In fact, being both visual artist and mathematician, I can only vehemently disagree. Such arrogance about the perceived superiority of science over art comes, I think, from a deep misunderstanding of the essential character and interplay of both disciplines. Mar 25, 2018 at 16:10

Being both a professional visual artist and mathematician, I feel obliged to attempt an answer.

There are to me very strong similarities, common mechanisms, overlaps, correspondences, between artistic and scientific aesthetics.

Nonetheless, I usually have trouble explaining that these correspondences go beyond what I consider a more superficial level, which is the level of nice polygons, polyhedra, symmetrical objects,... in other words the typical mathematical art' that many scientists associate with aesthetically pleasing.

Birkhoff's book is interesting, but to me falls short of addressing the essential complexity of aesthetics, in any discipline. By 'essential complexity' I mean that -in my perhaps not so humble opinion- one cannot approach understanding aesthetics by simplifying to a clearly less complex setting.

Also in mathematics, I have seen people disagree on the beauty of certain proofs or theories. It seems to co-depend on which kind of patterns we can or like to discern...

But I do think that aesthetics guides us in mathematics, and that we all know the gratification of discovering 'beauty'. What is less underscored, is that the discovery of 'non-beauty' can be just as fruitful for furthering our mathematical universes. This I see as a strong parallel with art. Another such parallel is the way in which we associate patterns with meanings', interpretations, observations,...

We not only discover patterns, but we create them too. Irregularity and asymmetry are as much a part of beauty as regularity or symmetry. Even very imperfect creation has its own aesthetic appeal... and mathematics is a very creative science.

Well, that's my 2 cts worth attempt...I admit Birkhoff gave it a lot more work and attention, and his book is therefore the better enjoyable :-)

I think you can do no better than Proofs from THE BOOK, a collection of mathematical beauties:

Aigner, Martin, and Günter M. Ziegler. Proofs from THE BOOK. Springer, 2014. (Springer link.)

There is a nice recent interview of Günter in Quanta Magazine, where he says:

"We’ve always shied away from trying to define what is a perfect proof. And I think that’s not only shyness, but actually, there is no definition and no uniform criterion. Of course, there are all these components of a beautiful proof. It can’t be too long; it has to be clear; there has to be a special idea; it might connect things that usually one wouldn’t think of as having any connection.

For some theorems, there are different perfect proofs for different types of readers. I mean, what is a proof? A proof, in the end, is something that convinces the reader of things being true. And whether the proof is understandable and beautiful depends not only on the proof but also on the reader: What do you know? What do you like? What do you find obvious?"

• This is just about beautiful proofs, not about beauty generally. Mar 23, 2018 at 0:30
• Beautiful proofs may nevertheless provide important insights into the nature of mathematical beauty. Mar 23, 2018 at 8:32
• It should be noted that not everyone, by far, agrees with the aesthetic decisions of those authors. Not in any hostile sense, but just disagreeing. Mar 25, 2018 at 22:16
• In this context, a recent paper from The Mathematical Intelligencer described a brief psychological experiment to investigate if people recognize aesthetic aspects of mathematical arguments: link.springer.com/article/10.1007%2Fs00283-018-09857-5 Dec 20, 2018 at 12:50

Chai Wah Wu from IBM’s TJ Watson Research Center has built a machine-learning algorithm using data from OEIS to learn to identify if a sequence will be interesting or not:

[1805.07431] Can machine learning identify interesting mathematics? An exploration using empirically observed laws

Abstract: "We explore the possibility of using machine learning to identify interesting mathematical structures by using certain quantities that serve as fingerprints. In particular, we extract features from integer sequences using two empirical laws: Benford's law and Taylor's law and experiment with various classifiers to identify whether a sequence is nice, important, multiplicative, easy to compute or related to primes or palindromes."

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Wavelet-based analysis of ground deformation coupling satellite acquisitions (Sentinel-1, SMOS) and data from shallow and deep wells in Southwestern France

Abstract

Acquisitions of the Sentinel-1 satellite are processed and comprehensively analyzed to investigate the ground displacement during a three-year period above a double gas storage site (Lussagnet and Izaute) in Southwestern France. Despite quite low vertical displacements (between 4 and 8 mm) compared to the noise level, the cyclic motion reflects the seasonal variations due to charge and discharge during summer and winter periods, respectively. We can simulate the ground deformation at both storage sites by a simple mechanical model. However, ground movements of low-magnitude may be also induced by natural factors, such as the temperature or the soil moisture. Using a wavelet-based analysis, we show there is a soil expansion in the Lussagnet zone that contrasts both in phase and period with the seasonal deformation and that is linked to the surface soil moisture measured by the SMOS satellite. This other displacement is consistent with the water infiltration in the unsaturated zone followed by the swelling of a clay layer. This work reveals the combination of two different processes driving the ground displacement with the same order of magnitude (about 6 mm), namely the pressure variation of a deep gas reservoir and the swelling/shrinking of the shallow subsurface.

Introduction

The Underground Gas Storages (UGS) are designed to address different needs that include a strategic gas reserve, a regulation of the gas supply, meeting seasonal peak heating and electricity demand and balancing the intermittent supply of renewable energy. Gas is stored from spring to autumn when the demand is lower and withdrawn during the winter period from October to April, when the demand is higher. UGS development required an appropriate site selection based on subsurface characterization, a suitable performance analysis, based on a fully integrated geological, fluid-dynamic and geo-mechanical approach, and finally a monitoring over the entire life of the storage. An integrated monitoring should include a network of microseismic sensors, observation wells to follow the reservoir as well as overburden pressures and a technique to measure ground surface displacements over wide areas. Standard ground surface monitoring techniques provide information on a very limited number of points within an area, both in the cases of geotechnical monitoring (clinometers, extensometers…) or GPS. Allowing a higher density of measurement points, the ground monitoring using Differential Synthetic Aperture Radar interferometry (DInSAR) has been intensively developed in the last two decades1,2 and validated against ground-based measurements3,4. Land subsidence related to groundwater extraction or uplift caused by the recharge of aquifers of large cities are some applications of the DInSAR technique, e.g. the subsidence in Mexico5,6 or the uplift in Brussels7 and in the London administrative area8. Conversely, the monitoring of gas storage using DInSAR has received scarce attention in scientific literature9,10. A major limiting factor to this purpose was the non-availability of both spatially and temporally high-resolution Synthetic Aperture Radar (SAR) dataset. Indeed, it has been shown that DInSAR techniques are more performant on regular motions than on displacements that have non-linear behavior with respect to time (i.e with strongly varying rates) as such displacements require finer temporal sampling to be better characterized4. This last issue was particularly sensitive with several past space missions for which few acquisitions per year were available (e.g. about 10 per year for Envisat). That was a limitation on the precision on the characterization of an annual cyclic motion. The launch of the Sentinel-1 mission by the European Space Agency (ESA) changed drastically the availability of SAR data by regular acquisition11. The mission consists of two satellites, respectively Sentinel-1A launched in April 2014, followed by Sentinel-1B in April 2016. Combining Sentinel-1A/1B, SAR images are acquired in interferometric mode every 6 days since October 2016. Another limiting factor is the low amplitude of deformation associated to gas storage in deep geological layers, the displacement to be measured is in this case of the same order of DInSAR measurement precision (typically a few millimeters)9,10.

Due to the potential substantial damage to buildings and infrastructure, the quantification of the clay swelling potential of expansive soils is a major concern for prevention plans. In France, the shrink/swell risk is the second most important cause of financial compensation from insurance companies behind the flood risk. In 2010, the French Geological Survey (BRGM) published a 1:50 000 swelling-risk map of France. This map indexed the territory as (i) no, (ii) low, (iii) moderate, or (iv) high risk. In the Aquitaine Basin of interest for this study, there is a low or a moderate swelling risk. At this resolution, the heterogeneity of the mineralogical composition of the sedimentary formations is not considered12,13,14. Therefore, some predictions of this swelling-risk map may be locally inaccurate or even wrong, according to the lithological characteristics of sedimentary formation at the given location. One way to improve this map consists in monitoring the soil moisture variations and ground movements by the instrumentation of experimental sites15. Another way is to use remote sensing satellite or aerial photography. Until now, the use of DInSAR was not operational for the mapping of swelling clays, mainly because of the non-availability of both spatially and temporally high-resolution and high-quality SAR dataset suitable to the very high variability of such surface deformation phenomena. Twenty years later, the question “Can we map swelling clays with remote sensing?” asked by Van der Meer in 199916 is still a topical issue.

This work investigates therefore two main questions: (1) Are the variations detected in DInSAR measurements strongly related to the underground gas storage operations at Lussagnet and Izaute sites?; (2) Can the DInSAR processing be used to assess locally the shrink/swell hazard as given by the 1:50 000 geological map? The results section of this paper is organized around these two questions.

Data and Methods

Geographical and Geological Setting

The studied area is a double storage site lying between both departments “Les Landes” and “Le Gers”, situated in the Aquitaine basin in Southwestern France, about 100 km north of the Pyrenees mountain (Fig. 1). The total reservoir structure is an anticline with two culminations, which are only approximately 10 km apart, the Lussagnet reservoir at the west side and the Izaute at the east. The highest point of the top of Lussagnet reservoir is located at a depth of about 550 m below ground level (mbgl) with a thickness of 40 m; the top of Izaute is at approximatively 510 mbgl17. Both reservoirs were deposited during the Eocen epoch, and are composed of unconsolidated sandstones, called “infra-mollassic sand”, with some interlayered claystones18. The hydrodynamic parameters of these reservoirs are variable. The total mean porosity of the sandstones varies from 20% to 35%, and their average permeability is from 1 to 10 Darcy. The Lussagnet and Izaute gas storage total capacity are respectively equal to 2.9 and 3.0 1012 m3 (referred to normal conditions) and only the working gas, respectively 1.4 and 1.5 1012 m3, is stored and withdrawn during the exploitation cycle. The remaining part, called cushion gas, supplies pressure support and prevents surface installations from excessive water production.

Concerning the surface layer, the shrink/swell hazard with 4 levels (null, low, moderate and high) has been evaluated in both departments “Les Landes” and “Le Gers” using different criteria, including the surface geological map at the scale 1:50 000 and other geotechnical and mineralogical criteria19,20. In particular, the Tortonien clay surface layer is characterized by a moderate shrink/swell hazard and the Serravallien sand surface layer by a low hazard. Just around both gas exploitation wells (L1 and I1), there is a moderate shrink/swell hazard at the Lussagnet site and a low hazard at the Izaute site (Fig. 1).

Sentinel-1 DInSAR processing

SqueeSAR® is a proprietary multi-interferogram technique which provides high precision measurements of ground displacement by processing multi-temporal satellite radar images acquired over the same area21,22. This technique allows the measurement of surface displacements by exploiting both point-wise coherent Permanent Scatterers (i.e. the PS) and partially coherent Distributed Scatterers (DS) (see Supplementary information for details). A dataset of 115 SAR images acquired by Sentinel-1A/B (C-band, IW mode) has been processed with the SqueeSAR® technique providing 1931 measurement points including 1303 PS and 628 DS over the area of interest. The detected displacement is characterized by a time series sampled every 12 days until 1st October 2016 and 6 days thereafter. The displacement measurements are carried out along the sensor’s Line of Sight (LOS), which is the sensor-to-target direction. DInSAR measures the projection of real motion along the LOS and provides 1D measurements. Those measurements are differential in space and time. They are spatially related to a reference point, and temporally to the date of the first available satellite acquisition. This reference point is steady through time and has been chosen around the 790 m/NGF isobath and outside the Tortonian clay surface layer (Fig. 1c). The combination of data results obtained from at least two different datasets with different acquisition geometries (ascending and descending), acquired over the same area in the same period, allows the estimation of 2D measurements, along the vertical and East-West directions. It is important to note that horizontal motion along the North-South direction cannot be measured with DInSAR techniques due to the low sensitivity to this component as this direction approximately corresponds to the flying direction of the satellites. This methodology requires that the same target is identified from both the ascending and the descending geometries. A projection of the LOS measurement along the vertical direction provides the real vertical displacement only in case the horizontal component of the movement is negligible. The random temporal changes on the surface of the Earth can reduce the signal-to-noise ratio (SNR), which is characterized through the interferometric coherence index given by the SqueeSAR® technique.

Wavelet-based analysis of DInSAR time series

DInSAR processing chains usually compute displacements as the superposition of linear and non-linear terms. Linear terms exhibit an infinite period along time, in the frequency domain. Non-linear component can exhibit different periods with higher frequency patterns at different time intervals, including seasonal fluctuations. The continuous wavelet transform (CWT) is especially suited to extract features from low signal-to-noise ratio time-series23. CWT expands time-series records into time/frequency space and can therefore identify localized intermittent periodicities24. The time-series input data must be equally spaced in time, i.e. evenly sampled with a fixed time interval. Additionally, two individual CWTs can be combined by using the cross wavelet transform (XWT) tool, if the relationship between two different time series is of interest. XWT is computed by multiplying the CWT of one time-series by the complex conjugate of the CWT of the second time-series. XWT image is the 2-D representation of the absolute value and the phase of the complex number in the time-frequency space. The absolute value of the XWT will be high in the time-frequency areas where both CWTs display high values, so this helps identify common time patterns in the two data sets. The phase of the XWT indicates the time lag between the two time-series. Consequently, this tool is very useful for exploring seasonal patterns which might have a time-lag (shown by the phase of the XWT) between the cause and the effect. XWT tool permits the recognition of common power and relative phase in time-frequency space, along with assessing confidence levels against red noise backgrounds23. For many geophysical phenomena, an appropriate background spectrum is either white noise (with a flat wavelet spectrum) or red noise (increasing power with decreasing frequency). These spectra are used to estimate the significance of a peak in the wavelet power spectrum. The proposed methodology here is to use CWT and XWT for which there are freely available Matlab codes (see Acknowledgements) to analyze comprehensively the DInSAR time-series vs. potential triggering factors time series (e.g. bottomhole pressure, rainfall, surface soil moisture, piezometric level). First, linear and non-linear components of the LOS displacement time series are separated: the linear component is computed by means of a linear least squares fitting and the non-linear component as the difference between the displacement time-series and the previously calculated linear component. Although SAR satellites have a regular revisit interval (6–12 days for Sentinel-1), some images may be missing or excluded from processing (Supplementary Fig. S2). After the separation between linear and non-linear components, the missing values of the non-linear component are linearly interpolated using a constant time interval of 12 days. Finally, we re-sample the time-series of potential triggering factors using the same time interval. For the bottomhole pressure or the piezometric level, we down-sample using the revisiting time period of Sentinel-1A (12 days) because they present generally a shorter time sampling (typically 1 day). For the rainfall, we calculate the accumulation during the Sentinel-1A period (12 days). For the surface soil moisture, we calculate an average value for each 12-day period using the Level 3 products of the Soil Moisture Ocean Salinity (SMOS) satellite as explained in the next section.

Surface Soil Moisture extracted from SMOS Level 3 product

The first satellite mission to focus primarily on the collection of soil moisture data was the SMOS satellite (see Supplementary Information). We use here the term Surface Soil Moisture (SSM) to refer to the volumetric soil moisture in the first few centimeters (0–5 cm) of the soil. The SMOS Level 3 SSM products are downloaded through the website of the Centre Aval de Traitement des Données SMOS (CATDS, https://www.catds.fr/). The data are presented over the Equal-Area Scalable Earth (EASE grid 2)25 with a sampling of about 25 km x 25 km and the studied area is included in one grid cell (Fig. 1). The CATDS also provides a 10-day product that contains median, minimum and maximum values of soil moisture over 10 days. We use these 10-day SMOS-CATDS SSM products for descending overpasses between 18 October 2014 and 26 October 2017 to calculate an average value for each Sentinel-1 period (12 days).

Results and Discussion

LOS displacement observations around gas exploitations wells

In order to estimate the noise level in the DInSAR time series, we choose a reference zone far away from L1 and I1 wells around the 790 m/NGF isobath (Figs 1c and 2a). Supposing a sinusoid shape, the calculated noise level is about 2.66 mm (see Supplementary Information). That shows the power of DInSAR analyses to detect displacements in the studied zone with magnitudes of as small as a few millimeters.

To investigate the DInSAR products in both gas exploitation zones, eight and fourteen DInSAR measurement points are respectively selected at Lussagnet site (4 PS and 4 DS) and Izaute site (11 PS and 3 DS) (Fig. 1d,e). This selection is based on three criteria: (i) the highest interferometric coherence index as calculated by the SqueeSAR® technique (at least higher than 0.95); (ii) the proximity to the gas exploitation wells (less than 300 m from L1 and 1000 m from I1); (iii) a location outside buildings or surface infrastructures. We calculate the average of all the time-series using a constant time interval of 12 days (Fig. 2b). LUG-M8 and IZA-M14 time profiles are the average values of the LOS non-linear displacement in the Lussagnet and Izaute gas exploitation zone, respectively (see method section). During the three-year period, the average peak-to-peak LOS displacement is 5.8 mm ±1.3 mm in the Lussagnet case (Fig. 3a) and 5.1 mm ±0.6 mm in the Izaute case (Fig. 4a).

Additive decomposition of the DInSAR time series using the reservoir pressure and the surface temperature

DInSAR time-series may be decomposed with the addition of a trend component T, a seasonal component S and an irregular residual component. In our case, the seasonal part S itself may be split into one bottomhole pressure component P and one surface temperature component ST (see Supplementary Information). The key result of this additive model is that the magnitude of the ST component is at least one order lower than the magnitude of P component (0.3 mm vs. 4 mm, see Supplementary Fig. S3).

Vertical displacement due to the reservoir pressure variations

There is qualitatively a good correlation between the mean LOS displacement and the bottomhole pressure at both sites during the three-year period (Figs 3 and 4). The correlation coefficient is 0.77 in the Lussagnet case and 0.73 in the Izaute case. It is worth estimating a possible surface deformation due to the pressure variations in the reservoir and comparing quantitatively with the DInSAR surface observations9. In order to know briefly its amplitude, we adopt a linear elastic approach26,27. The predicted deformation pattern is quite independent, meaning that the deformation amplitude is close to zero at the middle of the two reservoirs (see Supplementary Fig. S4). During each injection/production cycle, the simulated vertical displacement attains the value of 8 mm at L1 well and 4 mm at I1 well. Around L1 and I1 wells, the simulated horizontal displacement is less than 2 mm (Supplementary Fig. S4e). This value is less than the estimated noise level in the reference zone (2.7 mm). Therefore, we can neglect the contribution of horizontal displacement around L1 and I1 wells and transform the displacement from LOS to the vertical using $${u}_{z}=LOS/\cos \,\theta$$. Considering the local incidence angle of Sentinel-1 sensor (θ = 37.97°), we find a vertical displacement $${u}_{z}$$ of 7.4 mm ±1.7 mm around the L1 well and 6.4 mm ±0.8 mm around the I1 well. This vertical displacement is fairly well estimated by the model in the Lussagnet case (8 mm) and lightly underestimated in the Izaute case (4 mm). Using a sensitivity analysis, we find that this simple model is consistent with DInSAR data around both gas exploitation wells (see Supplementary information).

Nonetheless, despite a general consistency, some features in the Lussagnet case can not be explained only by the reservoir pressure variations. In 2015, the maximum of pressure change in the reservoir was 24.6 bar with a minimum of LOS displacement of 4.1 mm, whereas in 2016 (Fig. 3a) the maximum of LOS displacement was 7.3 mm with a minimum of pressure change of 18.7 bar. More particularly, there is an average uplift of about 1.2 mm during a 2-month period during the first semester of year 2016, while there is a decrease of the reservoir pressure of 6 bar during the same period (Fig. 3a). Our hypothesis is that this difference is due to another process driving the ground deformation, namely the clay swelling in the Lussagnet zone (see the Tortonien clay layer in Fig. 1d). The vertical ground deformation due to the gas storage calculated by the mechanical model at the monitoring wells is less than the estimated noise level (2.7 mm) (Supplementary Fig. S4c). Therefore, it seems sound to investigate the LOS displacement at the monitoring wells and we compare first the LOS displacements at the monitoring wells and the piezometric levels (Figs 3 and 4). As predicted by the model, no obvious correlation is observed between these two time series. In order to extract features from these low signal-to-noise ratio time-series, we use now the wavelet-based analysis (see method section 3). The proposed methodology is to study the time-frequency relationships between the DInSAR displacements at the monitoring wells and three potential triggering factors (Fig. 5a and Supplementary Fig. S1): (i) the rainfall at the Mont-de-Marsan Weather Station; (ii) the soil Moisture acquired by the SMOS satellite in a 25 km grid cell around the studied area; (iii) the shallow phreatic groundwater piezometric level measured at Latrille.

Wavelet-based analysis of LOS displacement at the monitoring wells in the Lussagnet zone

From the analysis of the continuous wavelet transforms (CWT) of the LOS time-series (Fig. 6a), some power signals with a significant level against red noise can be recognized with a period of 4 to 8 months at L3 and L2 and with a period of about 4 months at L4 during the year 2016. CWT of the LOS time series at the three monitoring wells (L2, L3, L4) is now compared to the CWT of three natural triggering factors time series (Fig. 6). From XWT with the rainfall, there is a high common power with a significant level against red noise during the first semester of year 2016 with a period of 4 months and 8 months at L3 and L2 (Fig. 6a,b). A similar high common power with the rainfall can be recognized for the LOS displacement at L4 (Fig. 6c). During a 4-month episode, the rainfall is in phase with the LOS displacement at L3 (arrow pointing right) and is leading the LOS displacement at L2 (arrow pointing down). Using SSM instead of the rainfall, XWT gives a similar result with the period of 4 months at L3 and L2. Concerning the piezometric level at Latrille, there is no higher frequencies than the one-year cycle (Supplementary Fig. S5). In conclusion, the cross analysis with the rainfall and SSM reveal the role of a 4-month episode (between end of December 2015 and end of April 2016) on the ground deformation at the three monitoring wells (L3, L2, L4).

Wavelet-based analysis of LOS displacement (LUG-M8) in the Lussagnet gas exploitation zone

There is a clear 1-year cycle using the CWT of LUG-M8 (Fig. 7a) and a strong in-phase relation (arrows pointing right) with the bottomhole pressure using XWT (Fig. 7b). There is also a period of about 4 months but with a much lower magnitude than at the monitoring wells (Fig. 7a). Using XWT, there is the same common high power with the rainfall and SSM during the 4-month episode (Fig. 7c,d) and the rainfall is leading LUG-M8 (arrows pointing down).

Modeling of the LOS displacement in the Lussagnet zone during the 4-month episode

We compare now in Fig. 5c the times series of the LOS displacement at the monitoring wells (L2, L3 and L4) with the rainfall and the SSM during the 4-month episode (between end of December 2015 and end of April 2016). The LOS displacement increase is 4.4 mm and 5.4 mm at L2 and L4, respectively, and the decrease is about 5 mm at L3. This LOS displacement is in phase with SSM at L3 and is leading the LOS displacement at L2 and L4. A possible explanation is that the observed soil expansion is due to the clay swelling and that the time lag is due the infiltration time of water in the unsaturated zone from the surface to the clay layer. Using the available geological logs from the BRGM subsurface database (BSS), we find indeed there is a clay layer at the surface at L3 and at the same depth (3 m) at L2 and L1 (Fig. 8). We try now to simulate the LOS displacement near L1 (called BKB8Z5Y) supposing that the displacement due to the clay swelling is the same at L1 and L2 (because of the same depth of the clay layer). We test a sinusoidal regression model SR(t) with a one-year period to simulate the seasonal displacement due to the gas exploitation:

$$SR(t)={A}_{max}\,\cos ((t+\phi )\times 2\pi /T)$$
(1)

where t is the time in days (relative to 18 October 2014), Amax the amplitude (2 ± 0.32 mm), φ the phase delay (32 days) and T the period (365 days). We calculate another model by adding to SR(t) the LOS displacement at L2:

$$M(t)=SR(t)+L2(t)$$
(2)

This second model fits indeed quite well the LOS displacement at L1 during the 4-month episode (Fig. 2c). In conclusion, the swelling/shrinking cycle takes about 4 months with a LOS displacement of about 5 mm, corresponding to a vertical displacement of about 6 mm (if we neglect the horizontal displacement). To support this conclusion, we use now the same wavelet-based method to analyze the LOS displacement in the Izaute zone.

Wavelet-based analysis of LOS displacement in the Izaute zone

Using CWT, some power signals during the first semester of the year 2016 with a significant level against red noise can be recognized with a period of 4 and 8 months at I4 (Fig. 6d), with a period of 4 months at I2 (Fig. 6e) and with a period of 12 months at I5 (Fig. 6f). From the cross analysis with the rainfall and the SSM, we find a high common power during the first semester of year 2016 with a significant level against red noise with a period of 4 months at I4 and I2 (Fig. 6d,e). During this episode, the rainfall is leading the LOS displacement at I4 by about one month (arrow pointing straight down, Fig. 6d) and is in anti-phase with the LOS displacement at I2 (arrow pointing left, Fig. 6e) and also at I5 but with a much lower magnitude (Fig. 6f). From the analysis of the CWT of the InSAR time-series, we can observe a clear 1-year cycle (12-month period) for the Izaute exploitation zone (IZA-M14) during the three years (Fig. 7a). Interestingly, XWT reveals that the rainfall is also leading during this episode the LOS displacement IZA-M14 but with a very low magnitude compared to the Lussagnet case (Fig. 7c). The presence of small-scale lenses of clays in the Izaute exploitation zone (not indicated by the 1:50 000 geological map) may explain this last result. In conclusion, XWT with the rainfall and the SSM reveals the role of a 4-month episode for the LOS displacement around two monitoring wells in the Izaute zone (I4 and I2).

LOS displacement during the 4-month episode in the Izaute zone

We compare now the time series of the LOS displacements near the Izaute monitoring wells during this 4-month episode. The LOS displacement at I5 is leading the LOS displacement at I4, which is leading the LOS displacement at I2 (Fig. 5d). There is also another uplift at I5 corresponding to the uplift at I2. There is no significant LOS displacement at I3. As indicated by the phase of the XWT (see previous section), the rainfall and the SSM is leading the LOS displacement at I4 by about one month and the LOS displacement at I2 by about two months. The LOS displacement at I4 displacement is very similar in phase with the LOS displacement at L2 (with a leading phase for L2). All these XWT results are consistent with the depth of the clay layer given by the geological logs from the BSS database (Fig. 8): 3 m at L2, 4 m at I4 and 5 m at I2. The complex behaviour at I5 (Fig. 5d) may be explained by the addition of one thin clay layer at the surface and another one at 5 m in the clayey sand (Fig. 8). The lack of ground deformation at I3 is consistent with its location in a sand formation (Fig. 1e).

A new method coupling satellites acquisitions and the BSS database to improve the global French shrink/swell hazard map

Locating the areas of low-magnitude soil vertical expansion would be useful for predicting areas with a high risk of shrinking in the event of a long drought. The most complex task is to extract the signal linked to the clay swelling (typically 5 mm) from the noise in the DInSAR time series. The two limiting factors to this purpose is the quality of the SAR dataset (e.g. low quality due to a dense vegetation) and the availability of logs in the BSS near the DInSAR measurement points. Because of their influence on the soil moisture, the buildings should also be excluded when selecting the measurement points. We show here first that the wavelet-based analysis is a powerful tool to quantify the correlation (amplitude and phase) between the LOS displacement and the Surface Soil Moisture acquired using both satellites (Sentinel-1 and SMOS). As a second step, the geological logs extracted from the BSS database spatially close to DInSAR scatterers are used to test the phase consistency. This method may be generalized using a global SAR dataset to improve the French shrink/swell hazard map.

Conclusion

By combining high temporal Sentinel-1 interferometric products (12 days) and observations from a network of shallow and deep wells, we performed a first integrated monitoring of a double gas storage site during three years. We observed a vertical ground deformation between 4 and 8 mm during the summer and winter periods due to the gas exploitation at both gas storages sites. A comprehensive analysis of the ground deformation at the Lussagnet site shows also a transient surface deformation that contrasts both in phase and period with the cyclic deformation due to gas exploitation, but with a similar vertical amplitude (about 6 mm). This last result is consistent with the swelling of a clay layer described in the geological logs extracted from the BSS database. In these terms, the high spatial-temporal resolution of the satellite Sentinel-1 reveals the combination of two different processes driving the ground displacement with the same order of magnitude, namely the pressure variation of a deep reservoir and the swelling/shrinking of the shallow subsurface. The process linked to the gas storage differs from the swelling/shrinking process in that it impacts a larger spatial scale (see Supplementary Fig. S4), hence it is homogeneous at the level of buildings or other surface infrastructures. This 3-year monitoring coupling satellites acquisitions and ground-based data is a first step and could be followed in the future by the aggregation of heterogeneous data over longer time periods. As a perspective of this work, the same method coupling both satellite acquisitions (Sentinel-1 and SMOS) and the BSS database can be generalised using a global SAR dataset to improve the French shrink/swell hazard map.

Data Availability

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

References

1. 1.

Massonnet, D. & Feigl, K. L. Radar interferometry and its application to changes in the Earth’s surface. Rev. Geophys. 36, 441–500 (1998).

2. 2.

Hanssen, R. F. In Radar interferometry: data interpretation and error analysis (Springer Science & Business Media, 2001).

3. 3.

Adam, N., Parizzi, A., Eineder, M. & Crosetto, M. Practical persistent scatterer processing validation in the course of the Terrafirma project. J. Appl. Geophys. 69, 59–65 (2009).

4. 4.

Raucoules, D. et al. Validation and intercomparison of Persistent Scatterers Interferometry: PSIC4 project results. J. Appl. Geophys. 68, 335–347 (2009).

5. 5.

Sowter, A. et al. Mexico City land subsidence in 2014–2015 with Sentinel-1 IW TOPS: Results using the Intermittent SBAS (ISBAS) technique. International Journal of Applied Earth Observation and Geoinformation 52, 230–242 (2016).

6. 6.

Strozzi, T. &Wegmuller, U. Land subsidence in Mexico City mapped by ERS differential SAR interferometry (Geoscience and Remote Sensing Symposium, 1999. IGARSS'99 Proceedings. IEEE 1999 International Ser. 4, IEEE, 1999).

7. 7.

Declercq, P. et al. A Study of Ground Movements in Brussels (Belgium) Monitored by Persistent Scatterer Interferometry over a 25-Year Period. Geosciences 7, 115 (2017).

8. 8.

Bonì, R. et al. A methodology to detect and characterize uplift phenomena in urban areas using Sentinel-1. data. Remote Sensing 10, 607 (2018).

9. 9.

Teatini, P. et al. Geomechanical response to seasonal gas storage in depleted reservoirs: A case study in the Po River basin, Italy. J. Geophys. Res. 116 (2011).

10. 10.

Rohmer, J. & Raucoules, D. On the applicability of Persistent Scatterers Interferometry (PSI) analysis for long term CO2 storage monitoring. Engineering Geology 147–148, 137–148 (2012).

11. 11.

Salvi, S. et al. The Sentinel-1 mission for the improvement of the scientific understanding and the operational monitoring of the seismic cycle. Remote Sensing of Environment 120, 164–174 (2012).

12. 12.

Dufréchou, G., Hohmann, A., Bourguignon, A. & Grandjean, G. Targeting and mapping expansive soils (Loiret, France): geometrical analysis of laboratory soil spectra in the short-wave infrared domain (1100–2500 nm). Bulletin de la Société géologique de France 187, 169–181 (2016).

13. 13.

Bourguignon, A., Delpont, G., Chevrel, S. & Chabrillat, S. Detection and mapping of shrink-swell clays in SW France, using ASTER imagery. Geological Society, London, Special Publications 283, 117 (2007).

14. 14.

Truche, C. Characterization and quantification of clay minerals in expansive soils by infrared spectrometry on laboratory and field scales, PhD dissertation (Université Paul Sabatier - Toulouse III, 2010).

15. 15.

Vincent, M., Le Roy, S., Dubus, I. & Surdyk, N. Experimental monitoring of water content and vertical displacements in clayey soils exposed to shrinking and swelling. Revue Française de Géotechnique, 45–58 (2007).

16. 16.

van der Meer, F. Can we map swelling clays with remote sensing? International Journal of Applied Earth Observation and Geoinformation 1, 27–35 (1999).

17. 17.

Douez, O. Response of a multi-layered aquifer system to the paleoclimatic variations and the anthropic pressure (approach to coupled hydrodynamical, thermal and geochemical modelling), PhD dissertation (Université Michel de Montaigne, Bordeaux, 2007).

18. 18.

Cavelier, C., Friès, G., Lagarigue, J. L. & Capdeville, J. P. Early Cenozoic prograding sedimentation in southern Aquitaine: a model. Géologie de la France 4, 69–79 (1997).

19. 19.

Norie, A., Capdeville, J. P. & Vincent, M. Cartographie de l’aléa retrait-gonflement des sols argileux dans le département du Gers. BRGM/RP-51251-FR (2001).

20. 20.

Karnay, G. Cartographie de l’aléa retrait-gonflement des sols argileux dans le département des Landes. BRGM/RP-56583-FR (2008).

21. 21.

Ferretti, A. et al. A new algorithm for processing interferometric data-stacks: SqueeSAR. IEEE Trans. Geosci. Remote Sens. 49, 3460–3470 (2011).

22. 22.

Raspini, F. et al. Continuous, semi-automatic monitoring of ground deformation using Sentinel-1 satellites. Scientific Reports 8, 7253 (2018).

23. 23.

Torrence, C. & Compo, G. P. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61–78 (1998).

24. 24.

Grinsted, A., Moore, J. C. & Jevrejeva, S. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear processes in geophysics 11, 561–566 (2004).

25. 25.

Brodzik, M. J., Billingsley, B., Haran, T., Raup, B. & Savoie, M. H. EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets. ISPRS International Journal of Geo-Information 1 (2012).

26. 26.

Geertsma, J. Land subsidence above compacting oil and gas reservoirs. J. Pet. Technol. 25, 734–744 (1973).

27. 27.

Segall, P. Induced stresses due to fluid extraction from axisymmetric reservoirs. Pure Appl. Geophys. 139, 535–560 (1992).

Acknowledgements

ENTREPOSE-GEOSTOCK is the sponsor of the Web-RDS project funded by the program CITEPH (Collaborative Technological Innovation in Energy Domains, http://www.citeph.fr/en/). The reservoir data was provided by the gas storage operator TEREGA (https://www.terega.fr/en/). Acquisitions of Sentinel-1 satellite were provided by the European Space Agency (ESA, https://sentinel.esa.int/web/sentinel/sentinel-data-access) and data of SMOS satellite by the French ground segment for the Level 3 data (CATDS, https://www.catds.fr/). The MATLAB wavelet toolbox is provided by A. Grinsted (http://www.glaciology.net/wavelet-coherence) and the wavelet software by C. Torrence and G. Compo (http://paos.colorado.edu/research/wavelets/).

Author information

Authors

Contributions

A.B. wrote the main manuscript, prepared the Figures 1–8, performed the SMOS data and the BSS database processing, the wavelet-based analysis and provided the overall interpretation. H.A. performed the mechanical modeling, prepared the Figure S4 and the Table S2. F.V. performed the geological data processing. F.N.K. performed the overall SAR data processing, performed the additive decomposition of the DInSAR time series, prepared the Figure S2 and the Table S1. A.F. developed the algorithm for the additive decomposition. D.R. contributed to the interpretation of the SAR data and the mechanical modeling. P.C. performed the gas exploitation data processing. C.M. conceived, supervised the CITEPH Web-RDS project and contributed to the overall interpretation. All authors contributed to the text and reviewed the manuscript.

Corresponding author

Correspondence to André Burnol.

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Burnol, A., Aochi, H., Raucoules, D. et al. Wavelet-based analysis of ground deformation coupling satellite acquisitions (Sentinel-1, SMOS) and data from shallow and deep wells in Southwestern France. Sci Rep 9, 8812 (2019). https://doi.org/10.1038/s41598-019-45302-z

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text:
# Why does Gauss' Law apply to any shape given that it's a closed surface?

I've read some answers on stack exchange for this question, but I don't find them particularly clear. I am convinced of the case for a sphere, since the area of a sphere is used in the derivation for the law.

So maybe relating all other areas to the area of a sphere would provide an intuitive 'proof'?

I was uncertain, however, about approximating an infinitesimal curved surface to a flat surface.

• The electric flux through an open in general surface $\,\rm S\,$ in empty space due to a point electric charge $\,q\,$ is $$\Phi_{\rm S}\boldsymbol=\dfrac{\Theta}{4\pi}\dfrac{q}{\epsilon_{0}}$$ where $\,\Theta\,$ the solid angle by which the charge $\,q\,$ $''$ sees $''$ the surface $\,\rm S$. In the case of closed surfaces, like a sphere, we have a complete solid angle $\,\Theta\boldsymbol=4\pi\,$ so $$\Phi_{\text{closed }\rm S}\boldsymbol=\dfrac{q}{\epsilon_{0}}$$ Sep 25 at 10:23

Gauss law deals with electric flux and electric flux is related to number of field lines cutting or passing through a surface. And a charge create definite number of field lines, thus if any closed surface is enclosing the charge every field lines emmited or created by the charge will pass through the closed surface. So, whatever the closed surface be, the number of field lines passing through the surface will always be the same.

In the picture you included, there are two distinct elements. One is Gauss’s law proper $$\oint \vec E\cdot d\vec A=\frac{q}{\epsilon_0}\, ,$$ which is always true and works for any closed surface.

The other part deals with using Gauss’s law to recover the field, i.e. the step $$E=kq /r^2$$. This is only possible in some circumstances where the surface and the field have high levels of symmetry.

In the example of your picture, imagine first the source charge is outside the Gaussian sphere. Then by Gauss’s law $$\oint \vec E\cdot d\vec A=0$$ since there is no charge inside the Gaussian surface. Of course, this does not imply there is no field. In other words, $$\oint \vec E\cdot d\vec A=0$$ does not imply $$\vec E=0$$.

Next imagine the Gaussian sphere is replaced by a long rectangle containing the charge. There is now charge inside the rectangular Gaussian rectangle but I can make the area of the rectangle whatever I want so it would seem I could get any field I want. In other words, $$\oint \vec E\cdot d\vec A=q/\epsilon_0\ne E A$$ always.

So while $$\oint \vec E\cdot d\vec A=q/\epsilon_0$$ is always true irrespective of the Gaussian surface, it is only when $$\oint \vec E\cdot d\vec A=E A$$ (which only holds for specific surfaces placed in certain ways around the source) that Gauss’s law is useful in recovering the field.

This is because every shape can be changed into other more symmetric shape. We can convert it to sphere and thus derive Gauss' law.


text:
# Solved papers for CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2007

### done CET - Karnataka Engineering Solved Paper-2007

• question_answer1) All components of the electromagnetic spectrum in vacuum have the same

A) energy

B) velocity

C) wavelength

D) frequency

• question_answer2) Which one of the following graph represents the variation of maximum kinetic energy $({{E}_{k}})$ of the emitted electrons with frequency v in photoelectric effect correctly?

A)

B)

C)

D)

• question_answer3) A and B are two metals with threshold frequencies $1.8\times {{10}^{14}}Hz$ and $2.2\times {{10}^{14}}Hz.$Two identical photons of energy 0.825 eV each are incident on them. Then photoelectrons are emitted by (Take$h=6.6\times {{10}^{-34}}J-s$)

A) B alone

B) A alone

C) neither A nor B

D) both A and B

• question_answer4) The ionization energy of $L{{i}^{2+}}$ is equal to

A) $9hcR$

B) $6hcR$

C) $2hcR$

D) $hcR$

• question_answer5) In the Wheatstone's network given, $P=10\Omega$, $Q=20\,\Omega ,\,\,R=15\,\Omega ,\,\,S=30\,\Omega$ the current passing through the battery (of negligible internal resistance) is

A) 0.36 A

B) zero

C) 0.18 A

D) 0.72 A

• question_answer6) Electrons in a certain energy level $n={{n}_{1}}$, can emit 3 spectral lines. When they are in another energy level, $n={{n}_{2}}$, they can emit 6 spectral lines. The orbital speed of the electrons in the orbits are in the ratio

A) $4:3$

B) $3:4$

C) $2:1$

D) $1:2$

• question_answer7) A circular coil carrying a certain current produces a magnetic field ${{B}_{0}}$ at its centre. The coil is now rewound so as to have 3 turns and the same current is passed through it. The new magnetic field at the centre is

A) $\frac{{{B}_{0}}}{9}$

B) $9\,{{B}_{0}}$

C) $\frac{{{B}_{0}}}{3}$

D) $3\,{{B}_{0}}$

• question_answer8) A proton and a deuteron with the same initial kinetic energy enter a magnetic field in a direction perpendicular to the direction of the field. The ratio of the radii of the circular trajectories described by them is

A) $1:4$

B) $1:\sqrt{2}$

C) $1:1$

D) $1:2$

• question_answer9) Two tangent galvanometers A and B have coils of radii 8 cm and 16 cm respectively and resistance $8\Omega$ each. They are connected in parallel with a cell of emf 4 V and negligible internal resistance. The deflections produced in the tangent galvanometers A and B are ${{30}^{o}}$and ${{60}^{o}}$ respectively. If A has 2 turns, then B must have

A) 18 turns

B) 12 turns

C) 6 turns

D) 2 turns

• question_answer10) A charged particle is moving in a magnetic field of strength B perpendicular to the direction of the field. If q and m denote the charge and mass of the particle respectively, then the frequency of rotation of the particle is

A) $f=\frac{qB}{2\,\pi m}$

B) $f=\frac{qB}{2\,\pi {{m}^{2}}}$

C) $f=\frac{2\,{{\pi }^{2}}m}{qB}$

D) $f=\frac{2\,\pi \,m}{qB}$

• question_answer11) Two identical capacitors each of capacitance $5\mu F$ are charged to potentials 2 kV and 1 kV respectively. Their -ve ends are connected together. When the +ve ends are also connected together, the loss of energy of the system is

A) 160 J

B) zero

C) 5 J

D) 1.25 J

• question_answer12) A parallel plate capacitor with air as the dielectric has capacitance C. A slab of dielectric constant K and having the same thickness as the separation between the plates is introduced so as to fill one-fourth of the capacitor as shown in the figure. The new capacitance will be

A) $\left( K+3 \right)\frac{C}{4}$

B) $\left( K+2 \right)\frac{C}{4}$

C) $\left( K+1 \right)\frac{C}{4}$

D) $\frac{KC}{4}$

• question_answer13) A current of 5 A is passing through a metallic wire of cross-sectional area $4\times {{10}^{-6}}{{m}^{2}}$. If the density of charge carriers of the wire is$5\times {{10}^{26}}{{m}^{-3}}$, the drift velocity of the electrons will be

A) $1\times {{10}^{2}}m{{s}^{-1}}$

B) $1.56\times {{10}^{-2}}m{{s}^{-1}}$

C) $1.56\times {{10}^{-3}}m{{s}^{-1}}$

D) $1\times {{10}^{-2}}m{{s}^{-1}}$

• question_answer14) Two bulbs rated 25 W-220 V and 100 W-220 V are connected in series to a 440 V supply. The,

A) 100 W bulb fuses

B) 25 W bulb fuses

C) both the bulbs fuse

D) neither of the bulb fuses

• question_answer15) The current passing through the ideal ammeter in the circuit given below is

A) 1.25 A

B) 1 A

C) 0.75 A

D) 0.5 A

• question_answer16) A and D are two infinitely long straight parallel conductors. C is another straight conductor of length 1m kept parallel to A and B as shown in the figure. Then the force experienced by C is

A) towards A equal to $0.6\times {{10}^{-5}}N$

B) towards B equal to $5.4\times {{10}^{-5}}N$

C) towards A equal to $5.4\times {{10}^{-5}}N$

D) towards B equal to $0.6\times {{10}^{-5}}N$

• question_answer17) An electric bulb has a rated power of 50 W s: 100 V. If it is used on an AC source 200 V 50 Hz, a choke has to be used in series with is This choke should have an inductance of

A) 0.1 mH

B) 1 mH

C) 0.1 H

D) 1.1 H

• question_answer18) An inductance of $\left( \frac{200}{\pi } \right)\,mH$, a capacitance of $\left( \frac{{{10}^{-3}}}{\pi } \right)F$and a resistance of $10\,\Omega$ are connected in series with an AC source 220 V, 50 Hz. The phase angle of the circuit is

A) $\frac{\pi }{6}$

B) $\frac{\pi }{4}$

C) $\frac{\pi }{2}$

D) $\frac{\pi }{3}$

• question_answer19) A step-down transformer reduces the voltage of a transmission line from 2200 V to 220 V. The power delivered by it is 880 W and its efficiency is 88%. The input current is

A) 4.65 A

B) 0.045 A

C) 0.45 A

D) 4.65 A

• question_answer20) Current in a coil changes from 4 A to zero in 0.1 s and the emf induced is 100 V. The self inductance of the coil is

A) 0.25 H

B) 0.4 H

C) 2.5 H

D) 4 H

• question_answer21) The electromagnetic theory of light failed to explain

A) photoelectric effect

B) polarization

C) diffraction

D) interference

• question_answer22) Light from two coherent sources of the same amplitude A and wavelength $\lambda$ illuminates the screen. The intensity of the central maximum is ${{I}_{0}}$. If the sources were incoherent, the intensity at the same point will be

A) $4{{I}_{0}}$

B) $2{{I}_{0}}$

C) ${{I}_{0}}$

D) $\frac{{{I}_{0}}}{2}$

• question_answer23) In Young's double slit experiment with sodium vapour lamp of wavelength 589 nm and the slits 0.589 mm apart, the half angular width of the central maximum is

A) ${{\sin }^{-1}}\left( 0.01 \right)$

B) ${{\sin }^{-1}}\left( 0.0001 \right)$

C) ${{\sin }^{-1}}\left( 0.001 \right)$

D) ${{\sin }^{-1}}\left( 0.1 \right)$

• question_answer24) A single slit Fraunhofer diffraction pattern is formed with white light. For what wavelength of light the third secondary maximum in the diffraction pattern coincides with the second secondary maximum in the pattern for red light of wavelength $6500\overset{o}{\mathop{A}}\,$?

A) $4400\overset{o}{\mathop{A}}\,$

B) $4100\overset{o}{\mathop{A}}\,$

C) $4642.8\overset{o}{\mathop{A}}\,$

D) $9100\overset{o}{\mathop{A}}\,$

• question_answer25) The head lights of a jeep are 1.2 m apart. If the pupil of the eye of an observer has a diameter of mm and light of wavelength 5896 ${{A}^{o}}$is used, what should be the maximum distance of the jeep from the observer if the two head lights are just separated?

A) 33.9 km

B) 33.9 m

C) 3.34 m

D) 3.39 m

• question_answer26) The de-Broglie wavelength of a proton (charge $=1.6\times {{10}^{-19}}C$, mass $=1.6\times {{10}^{-27}}kg$) accelerated through a potential difference of 1 kV is

A) $600\,\overset{o}{\mathop{A}}\,$

B) $0.9~{{10}^{-12}}m$

C) $7\,\overset{o}{\mathop{A}}\,$

D) 0.9 nm

• question_answer27) A radioactive element forms its own isotope after 3 consecutive disintegrations. The particles emitted are

A) $3\,\beta$-particles

B) $2\,\beta$-particles and $1\,\alpha$-particle

C) $2\,\beta$-particles and $1\,\gamma$-particle

D) $2\,\alpha$-particles and $1\,\beta$-particle

• question_answer28) A radioactive substance contains 10000 nuclei and its half-life period is 20 days. The number of nuclei present at the end of 10 days is

A) 7070

B) 9000

C) 8000

D) 7500

• question_answer29) In Raman effect, Stokes? lines are spectral lines having

A) frequency greater than that of the original line

B) wavelength equal to that of the original line

C) wavelength less than that of the original line

D) wavelength greater than that of the original line

• question_answer30) The principle of LASER action involves

A) amplification of particular frequency emitted by the system

B) population inversion

C) stimulated emission

D) all of the above

• question_answer31) A ray of light is travelling from glass to air. (refractive index of glass = 1.5). The angle of incidence is ${{50}^{o}}$. The deviation of the ray is

A) ${{0}^{o}}$

B) ${{80}^{o}}$

C) ${{50}^{0}}-{{\sin }^{-1}}\left[ \frac{\sin {{50}^{0}}}{1.5} \right]$

D) ${{\sin }^{-1}}\left[ \frac{\sin {{50}^{0}}}{1.5} \right]-{{50}^{0}}$

• question_answer32) A vessel of height 2d is half-filled with a liquid of refractive index $\sqrt{2}$ and the other half with a liquid of refractive index n (the given liquids are immiscible). Then the apparent depth of the inner surface of the bottom of the vessel (neglecting the thickness of the bottom of the vessel) will be

A) $\frac{n}{d(n+\sqrt{2})}$

B) $\frac{d\,(n+\sqrt{2})}{n\sqrt{2}}$

C) $\frac{\sqrt{2}n}{d\,(n+\sqrt{2})}$

D) $\frac{nd}{d+\sqrt{2}n}$

• question_answer33) A ray of light is incident normally on one face of a right angled isosceles prism. It then grazes the hypotenuse. The refractive index of the material of the prism is

A) 1.33

B) 1.414

C) 1.5

D) 1.732

• question_answer34) Two thin equiconvex lenses each of focal length 0.2 m are placed coaxially with their optic centres 0.5 m apart. Then the focal length of the combination is

A) -0.4 m

B) 0.4 m

C) - 0.1 m

D) 0.1 m

• question_answer35) A prism of a certain angle deviates the red and blue rays by ${{8}^{o}}$ and ${{12}^{o}}$ respectively. Another prism of the same angle deviates the red and blue rays by ${{10}^{o}}$ and ${{14}^{o}}$ respectively. The prisms are small angled and made of different materials. The dispersive powers of the materials of the prisms are in the ratio

A) $5:6$

B) $9:11$

C) $6:5$

D) $11:9$

• question_answer36) When the angle of incidence is ${{60}^{o}}$ on the surface of a glass slab, it is found that the reflected ray is completely polarised. The velocity of light in glass is

A) $\sqrt{2}\times {{10}^{8}}m{{s}^{-1}}$

B) $\sqrt{3}\times {{10}^{8}}m{{s}^{-1}}$

C) $2\times {{10}^{8}}m{{s}^{-1}}$

D) $3\times {{10}^{8}}m{{s}^{-1}}$

• question_answer37) A 20 cm length of a certain solution causes right handed rotation of ${{38}^{o}}$. A 30 cm length of another solution causes left handed rotation of ${{24}^{o}}$. The optical rotation caused by 30 cm length of a mixture of the above solutions in the volume ratio $1:2$ is

A) left handed rotation of ${{14}^{o}}$

B) right handed rotation of ${{14}^{o}}$

C) left handed rotation of ${{3}^{o}}$

D) right handed rotation of ${{3}^{o}}$

• question_answer38) Two identical charges repel each other with a force equal to 10 mg wt when they are 0.6 m a part in air $(g=10\text{ }m{{s}^{-2}})$. The value of each charge is

A) 2 mC

B) $2\times {{10}^{-7}}C$

C) 2 nC

D) $2\,\mu C$

• question_answer39) The potential of the electric Held produced by point charge at any point $(x,y,z)$ is given by$V=3{{x}^{2}}+5$, where $x,y$ are in metres and V is m volts. The intensity of the electric Held at $(-2,1,0)$ is

A) $+17\text{ }V{{m}^{-1}}$

B) $-17\text{ }V{{m}^{-1}}$

C) $+12\text{ }V{{m}^{-1}}$

D) $-12\text{ }V{{m}^{-1}}$

• question_answer40) The potential of a large liquid drop when eight liquid drops are combined is 20 V. Then the potential of each single drop was

A) 10 V

B) 7.5 V

C) 5 V

D) 2.5 V

• question_answer41) The dimensional formula for impulse is

A) $[ML{{T}^{-1}}]$

B) $[M{{L}^{-1}}T]$

C) $[{{M}^{-1}}L{{T}^{-1}}]$

D) $[M{{L}^{-1}}{{T}^{-1}}]$

• question_answer42) The maximum height attained by a projectile when thrown at an angle $\theta$ with the horizontal is found to be half the horizontal range. Then 0 is equal to

A) ${{\tan }^{-1}}\left( 2 \right)$

B) $\frac{\pi }{6}$

C) $\frac{\pi }{4}$

D) ${{\tan }^{-1}}\left( \frac{1}{2} \right)$

• question_answer43) A shell of mass 20 kg at rest explodes into two fragments whose masses are in the ratio $2:3$. The smaller fragment moves with a velocity of$6\,\,m{{s}^{-1}}$. The kinetic energy of the larger fragment is

A) 96 J

B) 216 J

C) 144 J

D) 360 J

• question_answer44) Water rises in plant fibres due to

A) capillarity

B) viscosity

C) fluid pressure

D) osmosis

• question_answer45) The acceleration due to gravity becomes $\left( \frac{g}{2} \right)$ (g = acceleration due to gravity on the surface of the earth) at a height equal to

A) 4 R

B) $\frac{R}{4}$

C) 2 R

D) $\frac{R}{2}$

• question_answer46) The cylindrical tube of a spray pump has a cross-section of $8\text{ }c{{m}^{2}}$, one end of which has 40 fine holes each of area ${{10}^{-8}}{{m}^{2}}$. If the liquid flows inside the tube with a speed of$0.15\text{ }m\text{ }mi{{n}^{-1}}$, the speed with which the liquid is ejected through the holes is

A) $50\text{ }m{{s}^{-1}}$

B) $5\text{ }m{{s}^{-1}}$

C) $0.05\text{ }m{{s}^{-1}}$

D) $0.5\text{ }m{{s}^{-1}}$

• question_answer47) During an adiabatic process, the cube of the pressure is found to be inversely proportional to the fourth power of the volume. Then the ratio of specific heats is

A) 1

B) 1.33

C) 1.67

D) 1.4

• question_answer48) Two identical rods AC and CB made of two different metals having thermal conductivities in the ratio $2:3$ are kept in contact with each other at the end C as shown in the figure. A is at ${{100}^{o}}C$ and B is at ${{25}^{o}}C$. Then the junction C is at

A) ${{55}^{o}}C$

B) ${{60}^{o}}C$

C) ${{75}^{o}}C$

D) ${{50}^{o}}C$

• question_answer49) 310 J of heat is required to raise the temperature of 2 moles of an ideal gas at constant pressure from ${{25}^{o}}C$ to ${{35}^{o}}C$. The amount of heat required to raise the temperature of the gas through the same range at constant volume is

A) 384 J

B) 144 J

C) 276 J

D) 452 J

• question_answer50) A Carnot's engine operates with source at ${{127}^{o}}C$ and sink at ${{27}^{o}}C$. If the source supplies 40 kJ of heat energy, the work done by the engine is

A) 30 kJ

B) 10 kJ

C) 4 kJ

D) 1 kJ

• question_answer51) The maximum particle velocity in a wave motion is half the wave velocity. Then the amplitude of the wave is equal to

A) $\frac{\lambda }{4\pi }$

B) $\frac{2\lambda }{\pi }$

C) $\frac{\lambda }{2\pi }$

D) $\lambda$

• question_answer52) The ratio of the velocity of sound in hydrogen$\left( \gamma =7/5 \right)$to that in helium $\left( \gamma =\frac{5}{3} \right)$ at the same temperature is

A) $\sqrt{\frac{5}{42}}$

B) $\sqrt{\frac{5}{21}}$

C) $\frac{\sqrt{42}}{5}$

D) $\frac{\sqrt{21}}{5}$

• question_answer53) An engine moving towards a wall with a velocity $50\text{ }m{{s}^{-1}}$ emits a note of 1.2 kHz. Speed of sound in air is $350\text{ }m{{s}^{-1}}$. The frequency of the note after reflection from the wall as heard by the driver of the engine is

A) 2.4 kHz

B) 0.24 kHz

C) 1.6 kHz

D) 1.2 kHz

• question_answer54) A glass tube is open at both the ends. A tuning fork of frequency $f$ resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now the air column inside the tube is in unison with another fork of frequency $f'$. Then

A) $f'=f$

B) $f'=4f$

C) $f'=2f$

D) $f'=\frac{f}{2}$

• question_answer55) The surface temperature of the sun which has maximum energy emission at 500 nm is 6000 K. The temperature of a star which has maximum energy emission at 400 nm will be

A) 8500 K

B) 4500 K

C) 7500 K

D) 6500 K

• question_answer56) The volume of a nucleus is directly proportional to

A) A

B) ${{A}^{3}}$

C) $\sqrt{A}$

D) ${{A}^{1/3}}$ (where A = mass number of the nucleus)

B) a baryon

C) a nucleon

D) a lepton

• question_answer58) Minority carriers in a p-type semiconductor are

A) free electrons

B) holes

C) neither holes nor free electrons

D) both holes and free electrons

• question_answer59) In a reverse biased diode when the applied voltage changes by 1 V, the current is found to change by $0.5\,\mu A$ The reverse bias resistance of the diode is

A) $2\times {{10}^{5}}\Omega$

B) $2\times {{10}^{6}}\Omega$

C) $200\,\Omega$

D) $2\,\Omega$

• question_answer60) The truth table given below is for CA and B are the inputs, Y is the output)

 A B Y 0 0 1 0 1 1 1 0 1 1 1 0

A) NOR

B) AND

C) XOR

D) NAND

• question_answer61) The number of unidentate ligands in the complex ion is called

A) oxidation number

B) primary valency

C) coordination number

D) EAN

• question_answer62) $2S{{O}_{2}}(g)+{{O}_{2}}(g)$ is an example for

A) neutralisation reaction

B) homogeneous catalysis

C) heterogeneous catalysis

D) irreversible reaction

• question_answer63) The amino acid which is not optically active is

A) lactic acid

B) serine

C) alanine

D) glycine

• question_answer64) For a stable molecule, the value of bond order must be

A) there is no relationship between stability and bond order

B) zero

C) positive

D) negative

• question_answer65) Which one of the following is a second order reaction?

A) ${{H}_{2}}+B{{r}_{2}}\xrightarrow{{}}2HBr$

B) $N{{H}_{4}}N{{O}_{3}}\xrightarrow{{}}{{N}_{2}}+3{{H}_{2}}O$

C) ${{H}_{2}}+C{{l}_{2}}\xrightarrow{sunlight}2HCl$

D) $C{{H}_{3}}COOCH+NaOH\xrightarrow{{}}C{{H}_{3}}COONa$$+{{H}_{2}}0$

A) ethanol + methahol

B) rectified spirit + methanol + naphtha

C) undistilled ethanol

D) rectified spirit

• question_answer67) During the formation of a chemical bond

A) electron-electron repulsion becomes more than the nucleus-electron attraction

B) energy of the system does not change

C) energy increases

D) energy decreases

• question_answer68) One mole of oxygen at 273 K and one mole of sulphur dioxide at 546 K are taken in two separate containers, then,

A) kinetic energy of ${{O}_{2}}>$ kinetic energy of$S{{O}_{2}}$

B) kinetic energy of ${{O}_{2}}<$ kinetic energy of$S{{O}_{2}}$

C) kinetic energy of both are equal

D) None of the above

• question_answer69) $+I$ effect is shown by

A) $-C{{H}_{3}}$

B) $-Br$

C) $-Cl$

D) $-N{{O}_{2}}$

• question_answer70) Formation of coloured solution is possible when metal ion in the compound contains

A) paired electrons

B) lone pair of electrons

C) unpaired electrons

D) none of the above

• question_answer71) Benzene reacts with chlorine in sunlight to give a final product

A) $CC{{l}_{4}}$

B) ${{C}_{6}}{{H}_{6}}C{{l}_{6}}$

C) ${{C}_{6}}C{{l}_{6}}$

D) ${{C}_{6}}{{H}_{5}}Cl$

• question_answer72) In the periodic table metals usually used as catalysts belong to

A) $f-$block

B) d-block

C) p-block

D) s-block

• question_answer73) Dalton's law of partial pressure is applicable to which one of the following systems?

A) $N{{H}_{3}}+HCl$

B) $NO+{{O}_{2}}$

C) ${{H}_{2}}+C{{l}_{2}}$

D) $CO+{{H}_{2}}$

• question_answer74) The general formula of a cycloalkane is

A) ${{C}_{n}}\,\,{{H}_{n}}$

B) ${{C}_{n}}\,\,{{H}_{2n}}$

C) ${{C}_{n}}\,\,{{H}_{2n-n}}$

D) ${{C}_{n}}\,\,{{H}_{2n\,+\,2}}$

• question_answer75) In acetylene molecule, between the carbon atoms there are

A) three pi bonds

B) one sigma and two pi bonds

C) two sigma and one pi bonds

D) three sigma bonds

• question_answer76) Which of the following is an intensive property?

A) temperature

B) viscosity

C) surface tension

D) all of these

• question_answer77) Hofmann's bromamide reaction is to convert

A)  acid to alcohol

B)  alcohol to acid

C)  amide to amine

D)  amine to amide

• question_answer78) IUPAC name of $N{{a}_{3}}[Co{{(N{{O}_{2}})}_{6}}]$ is

A) sodium hexanitrito cobaltate (II)

B) sodium hexanitro cobaltate (III)

C) sodium hexanitrito cobaltate (III)

D) sodium cobaltmitrite

• question_answer79) In equilibrium state the value of $\Delta G$ is

A) zero

B) negative

C) positive

D) may be negative or positive

• question_answer80) How many chiral carbon atoms are present in 2,3, 4-trichloropentane?

A) 4

B) 1

C) 2

D) 3

• question_answer81) Which one of the following shows functional isomerism?

A) ${{C}_{2}}{{H}_{4}}$

B) ${{C}_{3}}{{H}_{6}}$

C) ${{C}_{2}}{{H}_{5}}OH$

D) $C{{H}_{2}}C{{l}_{2}}$

• question_answer82) In the ionic equation $-BiO_{3}^{-}+6{{H}^{+}}+x{{e}^{-}}\xrightarrow{{}}$$B{{i}^{3+}}+3{{H}_{2}}O$, the values of $x$ is

A) 6

B) 2

C) 4

D) 3

• question_answer83) Molarity of a given orthophosphoric acid solution is 3 M. It's normality is

A) 9 N

B) 0.3 N

C) 3 N

D) 1 N

• question_answer84) Acidified sodium fusion extract on addition of ferric chloride solution gives blood red colouration which confirms the presence of

A) S and $Cl$

B) N and S

C) N

D) S

• question_answer85) A body of mass 10 mg is moving with a velocity of $100\,\,m{{s}^{-1}}$. The wavelength of de-Broglie wave associated with it would be $(h=6.63\times {{10}^{-34}}Js)$

A) $6.63\times {{10}^{-35}}m$

B) $6.63\times {{10}^{-34}}m$

C) $6.63\times {{10}^{-31}}m$

D) $6.63\times {{10}^{-37}}m$

• question_answer86) Angle strain in cyclopropane is

A) ${{24}^{o}}44'$

B)  ${{9}^{o}}44'$

C) $44'$

D)  $-{{5}^{o}}16'$

• question_answer87) The number of antibonding electron pairs in $O_{2}^{2-}$ molecular ion on the basis of molecular orbital theory is (Atomic number of 0 is 18.)

A) 5

B) 4

C) 3

D) 2

• question_answer88) Hydroxyl ion concentration of ${{10}^{-2}}M\,HCl$ is

A) $1\times {{10}^{1}}\,mol\,d{{m}^{-3}}$

B) $1\times {{10}^{-12}}\,mol\,d{{m}^{-3}}$

C) $1\times {{10}^{-1}}\,mol\,d{{m}^{-3}}$

D) $1\times {{10}^{-14}}\,mol\,d{{m}^{-3}}$

• question_answer89) Geometrical isomerism is shown by

A) $-C-C-$

B) $\rangle C=C\langle$

C) $-C\equiv C-$

D) None of these

• question_answer90) The oxidation state of iron in ${{K}_{4}}[Fe{{(CN)}_{6}}]$is

A) 1

B) 4

C) 3

D) 2

• question_answer91) During the extraction of gold the following reactions take place $Au+C{{N}^{-}}+{{H}_{2}}O\xrightarrow{{{O}_{2}}}[X]$$[X]+Zn\xrightarrow{{}}[Y]+Au$X and Y are respectively

A) ${{[Au{{(CN)}_{2}}]}^{-}}$ and ${{[Zn{{(CN)}_{6}}]}^{4-}}$

B) ${{[Au{{(CN)}_{4}}]}^{2-}}$ and ${{[Zn{{(CN)}_{4}}]}^{2-}}$

C) ${{[Au{{(CN)}_{4}}]}^{3-}}$ and ${{[Zn{{(CN)}_{4}}]}^{2-}}$

D) ${{[Au{{(CN)}_{2}}]}^{-}}$ and ${{[Zn{{(CN)}_{4}}]}^{2-}}$

• question_answer92) The number of gram molecules of chlorine in $6.02\times {{10}^{25}}$ hydrogen chloride molecules is

A) 10

B) 100

C) 50

D) 5

• question_answer93) Graphite is a soft solid lubricant extremely difficult to melt. The reason for this anomalous behaviour is that graphite

A) is an allotropic form of carbon

B) is a non-crystalline substance

C) has carbon atoms arranged in large plates of rings of strongly bond carbon atoms with weak interplate bonds

D) has molecules of variable molecular masses like polymers

A) antipyretic

B) analgesic

C) both (a) and (b)

D) antimalarial

• question_answer95) Which one of the following has maximum number of atoms of oxygen?

A) 2 g of carbon monoxide

B) 2 g of carbon dioxide

C) 2 g of sulphur dioxide

D) 2 g of water

• question_answer96) $M{{g}^{2+}}$ is isoelectronic with

A) $C{{u}^{2+}}$

B) $Z{{n}^{2+}}$

C) $N{{a}^{+}}$

D) $C{{a}^{2+}}$

• question_answer97) Gram molecular volume of oxygen at STP is

A) $3200\,c{{m}^{3}}$

B) $5600\,c{{m}^{3}}$

C) $22400\,c{{m}^{3}}$

D) $11200\,c{{m}^{3}}$

• question_answer98) Presence of halogen in organic compounds can be detected using

A) Leibigstest

B) Duma's test

C) Kjeldahltest

D) Beilstien?s test

• question_answer99) The electronic configuration of $C{{r}^{3+}}$ is

A) $[Ar]\,3{{d}^{4}}4{{s}^{2}}$

B) $[Ar]\,3{{d}^{3}}4{{s}^{0}}$

C) $[Ar]\,3{{d}^{2}}4{{s}^{1}}$

D) $[Ar]\,3{{d}^{5}}4{{s}^{1}}$

• question_answer100) What is the equivalent weight of $SnC{{l}_{2}}$ in the following reaction$SnC{{l}_{2}}+C{{l}_{2}}\xrightarrow{{}}SnC{{l}_{4}}?$

A) 95

B) 45

C) 60

D) 30

• question_answer101) Which of the following forms a colourless solution in aqueous medium?

A) $C{{r}^{3+}}$

B) ${{V}^{3+}}$

C) $S{{c}^{3+}}$

D) $T{{i}^{3+}}$

• question_answer102) When a sulphur sol is evaporated sulphur is obtained. On mixing with water sulphur sol is not formed. The sol is

A) lyophilic

B) reversible

C) hydrophobic

D) hydrophilic

• question_answer103) An alkyl halide reacts with alcoholic ammonia in a sealed tube, the product formed will be

A) a primary amine

B) a secondary amine

C) a tertiary amine

D) a mixture of all the three

• question_answer104) When cone. ${{H}_{2}}S{{O}_{4}}$ is heated with ${{P}_{2}}{{O}_{5}}$ the acid is converted into

A) sulphur trioxide

B) sulphur dioxide

C) sulphur

D) a mixture of sulphur dioxide and sulphur trioxide

• question_answer105) Entropy of the universe is

A) constant

B) zero

C) continuously decreasing

D) continuously increasing

• question_answer106) Which one of the following salts on being dissolved in water gives $pH>7$ at ${{25}^{o}}C$?

A) $KCN$

B) $KN{{O}_{3}}$

C) $N{{H}_{4}}Cl$

D) $N{{H}_{4}}CN$

• question_answer107) The reagent used in Clemmensen?s reduction is

A) conc.${{H}_{2}}S{{O}_{4}}$

B) Zn?Hg/conc.$HCl$

C) $aqKOH~$

D) ale. $KOH$

• question_answer108) When $KBr$ is dissolved in water, ${{K}^{+}}$ ions are

A) hydrated

B) hydrolysed

C) reduced

D) oxidized

• question_answer109) The noble gas mixture is cooled in a coconut bulb at 173 K. The gases that are not adsorbed are

A) Ne and Xe

B) He and Xe

C) Ar and Kr

D) He and Ne

• question_answer110) The volume of 10 N and 4 N $HCl$ required to make 1 L of 7 N $HCl$are

A) 0.50 L of $10\text{ }N\,HCl$ and 0.50 L of 4 N $HCl$

B) 0.60 L of $10\text{ }N\,HCl$ and 0.40 L of 4 N $HCl$

C) 0.80 L of $10\text{ }N\,HCl$ and 0.20 L of 4 N $HCl$

D) 0.75 L of $10\text{ }N\,HCl$ and 0.25 L of 4 N $HCl$

• question_answer111) A metal present in vitamin ${{B}_{12}}$ is

A) aluminium

B) zinc

C) iron

D) cobalt

• question_answer112) An oxide of the element contains 20% ${{O}_{2}}$ by weight. Calculate the equivalent weight of the element.

A) 8

B) 16

C) 32

D) 12

• question_answer113) Maximum number of molecules of $C{{H}_{3}}I$ that can react with a molecule of $C{{H}_{3}}N{{H}_{2}}$ are

A) 3

B) 4

C) 2

D) 1

• question_answer114) The relation between $\Delta \,H$ and $\Delta \,U$ is

A) $\Delta \,H=\Delta U+RT$

B) $\Delta \,H=\Delta U-\Delta nRT$

C) $\Delta \,H=\Delta U+\Delta nRT$

D) $\Delta \,U=\Delta H+\Delta nRT$

• question_answer115) Identify the ore not cotaining iron.

A) limonite

B) siderite

C) carnallite

D) chalcopyrites

• question_answer116) In which of the following process, maximum increase in entropy is observed?

A) Melting of ice

B) - Sublimation of naphthalene

C) Condensation of water

D) Dissolution of salt in water

• question_answer117) Decomposition of benzene diazonium chloride by using $C{{u}_{2}}C{{l}_{2}}/HCl$ to form chlorobenzene is

A)  Raschig's reaction

B)  Sandmeyer's reaction

C)  Kolbe's reaction

D)  Cannizaro's reaction

• question_answer118) Which complex cannot ionise in solution?

A) $[CoC{{l}_{3}}{{(N{{H}_{3}})}_{3}}]$

B) ${{K}_{4}}[Fe{{(CN)}_{6}}]$

C) ${{K}_{2}}[Pt({{F}_{6}})]$

D) $[Pt{{(N{{H}_{3}})}_{6}}]C{{l}_{4}}$

• question_answer119) Considering the reaction $C(s)+{{O}_{2}}(g)$ $\xrightarrow{{}}C{{O}_{2}}(g)+393.5\,kJ$ the signs of $\Delta H,\,\,\Delta S$ and $\Delta G$ respectively are

A) $+,-,-$

B) $-,+,+$

C) $-,-,-$

D) $-,+,-$

• question_answer120) The product formed when hydroxylamine condenses with a carbonyl compound is called

A) hydrazide

B) oxime

C) hydrazine

D) hydrazine

• question_answer121) If $\vec{a}=2\hat{i}+3\hat{j}-k,$ $\vec{b}=\hat{i}+2\hat{j}-5\hat{k},$ $\vec{c}=3\hat{i}+5\hat{j}-\hat{k},$Then a vector perpendicular to $\vec{a}$and in the plane containing $\overrightarrow{b}$ and $\overrightarrow{c}$ is

A) $-17\hat{i}+21\hat{j}-97\hat{k}$

B) $17\hat{i}+21\hat{j}-123\hat{k}$

C) $-17\hat{i}-21\hat{j}+97\hat{k}$

D) $-17\hat{i}-21\hat{j}-97\hat{k}$

• question_answer122) $\overrightarrow{OA}$ and $\overrightarrow{BO}$ are two vectors of magnitudes 5 and 6 respectively. If Z $\angle BOA={{60}^{o}},$then $\overrightarrow{OA}.\overrightarrow{OB}$is equal to

A) $0$

B) $15$

C) $-15$

D) $15\sqrt{3}$

• question_answer123) A vector perpendicular to the plane containing the points $A(1,-1,2),$ $B(2,0,-1),$ $C(0,2,1)$ is

A) $4\hat{i}+8\hat{j}-4\hat{k}$

B) $8\hat{i}+4\hat{j}+4\hat{k}$

C) $3\hat{i}+\hat{j}+2\hat{k}$

D) $\hat{i}+\hat{j}-\hat{k}$

• question_answer124) $\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+....\frac{1}{(3n-1)(3n+2)}$is equal to

A) $\frac{n}{6n-4}$

B) $\frac{n}{6n+3}$

C) $\frac{n}{6n+4}$

D) $\frac{n+1}{6n+4}$

• question_answer125) The ninth term of the expansion ${{\left( 3x-\frac{1}{2x} \right)}^{8}}$ is

A) $\frac{1}{512{{x}^{9}}}$

B) $\frac{-1}{512{{x}^{9}}}$

C) $\frac{-1}{256.{{x}^{8}}}$

D) $\frac{1}{256.{{x}^{8}}}$

• question_answer126) The solution of is ${{\tan }^{-1}}x+2{{\cot }^{-1}}x=\frac{2\pi }{3}$is

A) $-\frac{1}{\sqrt{3}}$

B) $\frac{1}{\sqrt{3}}$

C) $-\sqrt{3}$

D) $\sqrt{3}$

• question_answer127) ${{\sin }^{2}}{{17.5}^{o}}+{{\sin }^{2}}{{72.5}^{o}}$ is equal to

A) ${{\cos }^{2}}{{90}^{o}}$

B) ${{\tan }^{2}}{{45}^{o}}$

C) $co{{s}^{2}}{{30}^{o}}$

D) ${{\sin }^{2}}{{45}^{o}}$

• question_answer128) The conjugate of the complex number $\frac{{{(1+i)}^{2}}}{1-i}$is

A) $1-i$

B) $1+i$

C) $-1+i$

D) $-1-i$

• question_answer129) ABC is a triangle with $\angle A={{30}^{o}},$ $BC=10\text{ }cm$. The area of the circum circle of the triangle is

A) $100\pi \,sq\,m$

B) $5\,sq\,cm$

C) $25\,sq\,cm$

D) $\frac{100\pi }{3}\,sq\,cm$

• question_answer130) If $\sin \,3\theta =\sin \theta ,$how many solutions exist such that $-2\pi <\theta <2\pi$?

A) $8$

B) $9$

C) $5$

D) $7$

• question_answer131) A graph G has ?m' vertices of odd degree and ?n' vertices of even degree. Then which of the following statements is necessarily true?

A) $m+n$is an odd number

B) $m+n$ is an even number

C) $n+1$is an even number

D) $m+1$is an odd number

• question_answer132) If P is any point on the ellipse $\frac{{{x}^{2}}}{36}+\frac{{{y}^{2}}}{16}=1,$ and S and S' are the foci, then $PS+PS'$is equal to

A) $4$

B) $8$

C) $10$

D) $12$

• question_answer133) The value of $\sin \left[ 2{{\cos }^{-1}}\frac{\sqrt{5}}{3} \right]$ is

A) $\frac{\sqrt{5}}{3}$

B) $\frac{2\sqrt{5}}{3}$

C) $\frac{4\sqrt{5}}{9}$

D) $\frac{2\sqrt{5}}{9}$

• question_answer134) If $\frac{{{x}^{2}}}{36}-\frac{{{y}^{2}}}{{{k}^{2}}}=1$ is a hyperbola, then which of the following statements can by true?

A) $(-3,1)$ lies on the hyperbola

B) $(3,1)$lies on the hyperbola

C) $(10,4)$lies on the hyperbola

D) $(5,2)$ lies on the hyperbola

• question_answer135) The focus of the parabola $y=2{{x}^{2}}+x$is

A) $(0,0)$

B) $\left( \frac{1}{2},\frac{1}{4} \right)$

C) $\left( -\frac{1}{4},0 \right)$

D) $\left( -\frac{1}{4},\frac{1}{8} \right)$

• question_answer136) If $A\left[ \begin{matrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \\ \end{matrix} \right],10B=\left[ \begin{matrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \\ \end{matrix} \right]$ and B is the inverse of A, then the value of $\alpha$ is

A) $2$

B) $0$

C) $5$

D) $4$

• question_answer137) If $A=\left[ \begin{matrix} 0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x \\ \end{matrix} \right]$ is singular, then the possible values of x are

A) $0,12,-12$

B) $0,1,-1$

C) $0,4,-4$

D) $0,5,-5$

• question_answer138) If $A=\left[ \begin{matrix} 1 & -2 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \\ \end{matrix} \right]$ then A . adj is equal to

A) $\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \\ \end{matrix} \right]$

B) $\left[ \begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \\ \end{matrix} \right]$

C) $\left[ \begin{matrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \\ \end{matrix} \right]$

D) $\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right]$

• question_answer139) If $f:R\to R$ is defined by $f(x)=|x|,$then

A) ${{f}^{-1}}(x)=-x$

B) ${{f}^{-1}}(x)=\frac{1}{|x|}$

C) the function ${{f}^{-1}}(x)$ does not exist

D) ${{f}^{-1}}(x)=\frac{1}{x}$

• question_answer140) The value of $\left| \begin{matrix} x & p & q \\ p & x & q \\ p & q & x \\ \end{matrix} \right|$ is

A) $x(x-p)(x-q)$

B) $(x-p)(x-q)(x+p+q)$

C) $(p-q)(x-q)(x-p)$

D) $pq(x-p)(x-q)$

• question_answer141) The imaginary part of ${{i}^{i}}$is

A) $0$

B) $1$

C) $2$

D) $-1$

• question_answer142) The amplitude of ${{(1+i)}^{5}}$ is

A) $\frac{3\pi }{4}$

B) $\frac{-3\pi }{4}$

C) $\frac{-5\pi }{4}$

D) $\frac{5\pi }{4}$

• question_answer143) ABC is triangle. G is the centroid. D is the mid point of BC. If $A=(2,3)$ and $G=(7,5),$ then the point D is

A) $\left( \frac{9}{2},4 \right)$

B) $\left( \frac{19}{2},6 \right)$

C) $\left( \frac{11}{2},\frac{11}{2} \right)$

D) $\left( 8,\frac{13}{2} \right)$

• question_answer144) $\underset{x\to 1}{\mathop{\lim }}\,\,\frac{\tan ({{x}^{2}}-1)}{x-1}$is equal to

A) $2$

B) $\frac{1}{2}$

C) $-2$

D) $\frac{-1}{2}$

• question_answer145) If $y={{2}^{\log x}},$ then $\frac{dy}{dx}$ is

A) $\frac{{{2}^{\log x}}}{\log 2}$

B) ${{2}^{\log x}}.\log 2$

C) $\frac{{{2}^{\log x}}}{x}$

D) $\frac{{{2}^{\log x}}.\log 2}{x}$

• question_answer146) ${{7}^{2{{\log }_{7}}5}}$is equal to

A) ${{\log }_{7}}35$

B) $5$

C) $25$

D) ${{\log }_{7}}25$

• question_answer147) In the group $(G,\,\,{{\otimes }_{15}}),$ where $G=\{3,6,9,12\},$${{\otimes }_{15}}$is multiplication modulo 15, the identity element is

A) $3$

B) $6$

C) $12$

D) $9$

• question_answer148) A group $(G,*)$ has 10 elements. The minimum number of elements of G, which are their own inverses is

A) $2$

B) $1$

C) $9$

D) $0$

• question_answer149) If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}+\vec{b}|$$=|\vec{a}-\vec{b}|,$ then the angle between $\vec{a}$ and $\vec{b}$is

A) ${{120}^{o}}$

B) ${{60}^{o}}$

C) ${{90}^{o}}$

D) ${{30}^{o}}$

• question_answer150) $\frac{3{{x}^{2}}+1}{{{x}^{2}}-6x+8}$is equal to

A) $3+\frac{49}{2(x-4)}-\frac{13}{2(x-2)}$

B) $\frac{49}{2(x-4)}-\frac{13}{2(x-2)}$

C) $\frac{-49}{2(x-4)}-\frac{13}{2(x-2)}$

D) $\frac{49}{2(x-4)}+\frac{13}{2(x-2)}$

• question_answer151) The number of common tangents to the circles ${{x}^{2}}+{{y}^{2}}=4$and ${{x}^{2}}+{{y}^{2}}-6x-8y-24=0$is,

A) $3$

B) $4$

C) $2$

D) $1$

• question_answer152) If $3x+y+k=0$ is a tangent to the circle ${{x}^{2}}+{{y}^{2}}=10,$ the values of k are,

A) $\pm 7$

B) $\pm 5$

C) $\pm 10$

D) $\pm 9$

• question_answer153) The negation of the proposition "If 2 is prime, then 3 is odd" is

A) if 2 is not prime, then 3 is not odd

B) 2 is prime and 3 is. not odd

C) 2 is not prime and 3 is odd

D) if 2 is not prime, then 3 is odd

• question_answer154) The equation of two circles which touch the Y-axis at $(0,3)$ and make an intercept of 8 unit on X-axis are

A) ${{x}^{2}}+{{y}^{2}}\pm 10x-6y+9=0$

B) ${{x}^{2}}+{{y}^{2}}\pm 6x-10y+9=0$

C) ${{x}^{2}}+{{y}^{2}}-8x\pm 10y+9=0$

D) ${{x}^{2}}+{{y}^{2}}+10x\pm 6y+9=0$

• question_answer155) The orthocenter of the triangle with vertices $A(0,0),$ $B\left( 0,\frac{3}{2} \right),$ $C(-5,0)$is

A) $\left( \frac{5}{2},\frac{3}{4} \right)$

B) $\left( \frac{-5}{2},\frac{3}{4} \right)$

C) $\left( -5,\frac{3}{2} \right)$

D) $(0,0)$

• question_answer156) ${{x}^{2}}+{{y}^{2}}-6x-6y+4=0,$ ${{x}^{2}}+{{y}^{2}}-2x$$-4y+3=0,$ ${{x}^{2}}+{{y}^{2}}+2kx+2y+1=0$. If the radical centre of the above three circles exists, then which of the following cannot be the value off k?

A) $2$

B) $1$

C) $5$

D) $4$

• question_answer157) If the circles ${{x}^{2}}+{{y}^{2}}-2x-2y-7=0$and ${{x}^{2}}+{{y}^{2}}+4x+2y+k=0$ cut orthogonally, then the length of the common chord of the circles is

A) $\frac{12}{\sqrt{13}}$

B) $2$

C) $5$

D) $8$

• question_answer158) The coordinates of the foot of the perpendicular drawn from the point $(3,4)$ on the line $2x+y-7=0$ is

A) $\left( \frac{9}{5},\frac{17}{5} \right)$

B) $(1,5)$

C) $(5,1)$

D) $(1,-5)$

• question_answer159) The area enclosed by the pair of the lines $xy=0,$ the line $x-4=0$and $y+5=0$ is

A) $20\,sq\,unit$

B) $10\,sq\,unit$

C) $\frac{5}{4}\,sq\,unit$

D) $0\,sq\,unit$

• question_answer160) If the area of the auxiliary circle of the ellipse $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ $(a>b)$ is twice the area of the ellipse, then the eccentricity of the ellipse is

A) $\frac{1}{\sqrt{2}}$

B) $\frac{\sqrt{3}}{2}$

C) $\frac{1}{\sqrt{3}}$

D) $\frac{1}{2}$

• question_answer161) The range in which $y=-{{x}^{2}}+6x-3$ is increasing is

A) $x<3$

B) $x>3$

C) $7<x<8$

D) $5<x<6$

• question_answer162) The value of the integral$\int_{0}^{\pi /2}{({{\sin }^{100}}x-{{\cos }^{100}}x)dx}$is

A) $\frac{1}{100}$

B) $\frac{100!}{{{(100)}^{100}}}$

C) $\frac{\pi }{100}$

D) $0$

• question_answer163) OA and OB are two roads enclosing an angle of${{120}^{o}}$. X and y start from ?O' at the same time. X travels along OA with a speed of 4 km/h and Y travels along OB with a speed of 3 km/h. The rate at which the shortest distance between X and V is increasing after $1\text{ }h$is

A) $\sqrt{37}km/h$

B) $37m/h$

C) $13km/h$

D) $\sqrt{13}km/h$

• question_answer164) If $k\int_{0}^{1}{x}.f(3x)dx=\int_{0}^{3}{t.f(t)dt,}$ then the value of k is

A) $9$

B) $3$

C) $\frac{1}{9}$

D) $\frac{1}{3}$

• question_answer165) The value of $\int{\frac{1}{1+\cos \,8\,x}}dx$is

A) $\frac{\tan \,2x}{8}+c$

B) $\frac{\tan \,8x}{8}+c$

C) $\frac{\tan \,4x}{4}+c$

D) $\frac{\tan \,4x}{8}+c$

• question_answer166) If ${{\sec }^{-1}}\left( \frac{1+x}{1-y} \right)=a,$then $\frac{dy}{dx}$ is

A) $\frac{y-1}{x+1}$

B) $\frac{y+1}{x-1}$

C) $\frac{x-1}{y-1}$

D) $\frac{x-1}{y+1}$

• question_answer167) If $y={{\cos }^{2}}\frac{3x}{2}-{{\sin }^{2}}\frac{3x}{2},$ then $\frac{{{d}^{2}}y}{d{{x}^{2}}}$is

A) $-3\sqrt{1-{{y}^{2}}}$

B) $9y$

C) $-9y$

D) $3\sqrt{1-{{y}^{2}}}$

• question_answer168) If the function $f(x)=\left\{ \begin{matrix} \frac{1-\cos \,\,x}{{{x}^{2}}} & for & x\ne 0 \\ k & for & x=0 \\ \end{matrix} \right.$is continuous at $x=0,$ then the value of k is

A) $1$

B) $0$

C) $\frac{1}{2}$

D) $-1$

• question_answer169) If $1,\,\omega ,\,{{\omega }^{2}}$ are the cube roots of unity then $(1+\omega )(1+{{\omega }^{2}})(1+{{\omega }^{4}})(1+{{\omega }^{8}})$is equal to

A) $1$

B) $0$

C) ${{\omega }^{2}}$

D) $\omega$

• question_answer170) If ${{x}^{x}}={{y}^{y}},$then $\frac{dy}{dx}$is

A) $-\frac{y}{x}$

B) $-\frac{x}{y}$

C) $1+\log \left( \frac{x}{y} \right)$

D) $\frac{1+\log x}{1+\log y}$

• question_answer171) The value of $\int{{{e}^{x}}({{x}^{5}}+5{{x}^{4}}}+1).dx$is

A) ${{e}^{x}}.{{x}^{5}}+c$

B) ${{e}^{x}}.{{x}^{5}}+{{e}^{x}}+c$

C) ${{e}^{x+1}}.{{x}^{5}}+c$

D) $5{{x}^{4}}.{{e}^{x}}+c$

• question_answer172) The value of $\int{\frac{{{x}^{2}}+1}{{{x}^{2}}-1}}dx$ is

A) $\log \left( \frac{x-1}{x+1} \right)+c$

B) $\log \left( \frac{x+1}{x-1} \right)+c$

C) $x+\log \left( \frac{x-1}{x+1} \right)+c$

D) $\log ({{x}^{2}}-1)+c$

• question_answer173) The area bounded by the curve $x=4-{{y}^{2}}$and the Y-axis is

A) $16\text{ }sq\text{ }unit$

B) $32\text{ }sq\text{ }unit$

C) $\frac{32}{3}\text{ }sq\text{ }unit~~$

D) $\frac{16}{3}sq\text{ }unit$

• question_answer174) The differential equation of the family of straight lines whose slope is equal to y-intercept is

A) $(x+1)\frac{dy}{dx}-y=0$

B) $(x+1)\frac{dy}{dx}+y=0$

C) $\frac{dy}{dx}=\frac{x-1}{y-1}$

D) $\frac{dy}{dx}=\frac{x+1}{y+1}$

• question_answer175) The order and degree of the differential equation ${{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{5}} \right]}^{\frac{1}{3}}}=\frac{{{d}^{2}}y}{d{{x}^{2}}}$ are respectively

A) $1,5$

B) $2,1$

C) $2,5$

D) $2,3$

• question_answer176) The point on the curve ${{y}^{2}}=x,$the tangent at which makes an angle ${{45}^{o}}$ with -X-axis is

A) $\left( \frac{1}{4},\frac{1}{2} \right)$

B) $\left( \frac{1}{2},\frac{1}{4} \right)$

C) $\left( \frac{1}{2},\frac{-1}{2} \right)$

D) $\left( \frac{1}{2},\frac{1}{2} \right)$

• question_answer177) The length of the sub tangent to the curve ${{x}^{2}}{{y}^{2}}={{a}^{4}}$ at $(-a,a)$ is

A) $\frac{a}{2}$

B) $2a$

C) $a$

D) $\frac{a}{3}$

• question_answer178) The number of positive divisors of $252$ is

A) $9$

B) $5$

C) $18$

D) $10$

• question_answer179) The remainder obtained when ${{5}^{124}}$is divided by $124$ is

A) $5$

B) $0$

C) $2$

D) $1$

• question_answer180) Which of the following is not a group with respect to the given operation?

A) The set of even integers under addition

B) The set, of odd integers under addition

C) $\{0\}$ under addition

D) $\{1,-1\}$under multiplication


text:
Universality of Composition Operators and Applications to Holomorphic Dynamics

Research paper by Andreas Jung

Indexed on: 03 Mar '16Published on: 03 Mar '16Published in: Mathematics - Complex Variables

Abstract

By investigating which level of universality composition operators \$C_f\$ can have, where the symbol \$f\$ is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of \$f\$, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.


text:
Characteristic Resistance

The characteristic resistance of a solar cell is the output resistance of the solar cell at its maximum power point. If the resistance of the load is equal to the characteristic resistance of the solar cell, then the maximum power is transferred to the load and the solar cell operates at its maximum power point. It is a useful parameter in solar cell analysis, particularly when examining the impact of parasitic loss mechanisms. The characteristic resistance is shown in the figure below.

The characteristic resistance of a solar cell is the inverse of the slope of the line, shown in the figure above, which, after Green [1], can be given as:

It can alternately be given as an approximation where: