Match
PGDBA 2017 Question Paper
In each of the questions a word has been used in sentences in four different ways. Choose the option corresponding to the sentence in which the usage of the word is incorrect or inappropriate
Solution
In sentence A, the word match means betrothal or engagement and the usage is correct.
In sentence B, the word match is the noun which is the object used to light a fire.
In sentence C, the word match means rivals and the usage is correct yet again.
However, in sentence D, the word match though implies equal in quality, the usage is incorrect, and the correct usage would be "This movie sequel is no match to the original."
Therefore the correct option is D.
Accede
Solution
The word "accede" can be used in two contexts, the first being one where it means to agree to a demand or a request and the second instance where it means to assume office or position.
In option A, the word correctly used as the manager can accede to a request.
In option B, again the usage is correct as it can be difficult to accede to a proposal.
In option D, the usage is appropriate as assuming the throne can be rephrased as acceding to the throne.
But in option C, the usage is incorrect as here the word "demands" means requirements or needs and the correct usage would be "meet" or "fulfil".
So the correct choice is C.
For the following questions answer them individually
Arrange the sentences in the most logical order to form a coherent paragraph. From the given options (a, b, c, d) choose the most appropriate option.
(i) It would secure a 25% increase in overall revenue; and devoted but cash-strapped supporters would have more opportunities to watch their team.
(ii) The Football Supporters Federation maintains that, under government regulations about spectator density, safe-standing sections would allow 1.8 people to occupy the same space as one seated match-goer.
(iii) The willingness of the Premier League to consider reintroducing terraces has less to do with reminiscing, however, than with pragmatism.
(iv) If the Football Supporters Federation's is correct, then both clubs and fans would stand to gain since the teams could offer a reduction on the price of standing tickets.
Solution
Sentence (iii) is an opening statement about the idea of Premier League to reintroduce terraces.
Sentence (ii) is the statement of the Football Supporters Federation in favour of (iii).
Sentence (iv) states that if (iii) is correct, then it would benefit both clubs and the fans.
Sentence (i) elaborates how (iv) will help each party.
Therefore the right order is (iii)(ii)(iv)(i).
Arrange the sentences in the most logical order to form a coherent paragraph. From the given options (a, b, c, d) choose the most appropriate option.
(i) In an integrated market one country might specialise in a high-wage industry with increasing returns to scale and others in areas in which wages are lower.
(ii) New models of trade do not imply that close economic integration should cause incomes to converge.
(iii) As freer trade expands the size of the market, producers with initial size advantages outcompete rivals.
(iv) Firms and places are often subject to economies of scale: they become more productive as they grow larger.
Solution
(ii) puts forward a statement about new models of trade and how they do not imply that close economic integration should cause incomes to converge. (iv) gives an example for (ii). (i) talks about the implications of an integrated market whereas (iii) provide the effects of free trade.
Therefore the correct order is (ii)(iv)(i)(iii).
Arrange the sentences in the most logical order to form a coherent paragraph. From the given options (a, b, c, d) choose the most appropriate option.
(i) Taken together, these elements enable developers to discover and build on what works, to jettison what does not work, and, when necessary; to "fail fast"—before they have expended significant resources or large amounts of time on a project.
(ii) Over the past few decades, the business world has seen the emergence of several process and product improvement platforms.
(iii) Both of those platforms emphasize experimentation and rapid iteration, strong feedback loops that facilitate early and continuous engagement with end users, and the use of minimally designed prototypes to test products or processes.
(iv) Examples include human-centred design, a product innovation method developed by the design firm IDEO, and lean experimentation, an entrepreneurship method that originated in Silicon Valley.
Solution
(ii) puts forward a statement about the emergence of new concepts in business in the last many years. (iv) gives two examples of the innovations mentioned in (ii). (iii) presents the similarities between the two cases discussed earlier, whereas (i) tells us the applications both these innovations have when taken together.
Hence the correct order is (ii)(iv)(iii)(i).
Read the passage and answer the questions that follow:
Passage I
There are two main kinds of development agency: the one which trace to introduce specific changes and is mainly interested in material development: and the other which is primarily interested in people. On the whole the first wants to "get things done"; the other to develop the people's own abilities for leadership, wise judgement and co-oprative action. For agencies of the second kind, the material result is less important than the way it is achieved.
Agencies and workers, who themselves decide the specific form development should take, assume, of course, that they know better than the people what the people need. Most social development workers and technical officers have worked on this assumption in the past, and although they were often right they were not always right, for they sometimes made the mistake of assuming that what was good within their own culture was certain to be good in other cultures too. Missionaries, for instance, insisted on their converts wearing clothes because they were used to them themselves, and they established schools with syllabuses that suited the missionaries' own countries, rather than the countries where the schools were built.
Agencies and their workers tend to be more careful nowadays, but experts and specialists trained in Western ways still often make mistakes in cultures other than their own. Agencies everywhere are now realizing that they are risking failure if they assume that their own ideas are right in environments and cultures other than their own. The East African Groundnut Scheme failed because it did not take the local conditions of soil and climate sufficiently into account. The West African Anchau Rural Development Scheme illustrates, less spectacularly, the result of failing to consider the human factor when working in a different culture.
This Scheme was started in 1937 to eradicate sleeping sickness from a part of the Zaria province of the Northern Region of Nigeria. The people in charge made a detailed survey of the area, made detailed studies of the farming conditions in sample hamlets and made a careful census of the people. Indeed, they scientifically examined in minute detail every aspect of the situation that seemed to them important. But it failed because people were thought of as being there "to be done good to" in the mass, but they were not envisaged as persons, each with one's own small world of hopes and fears, who might in some way be consulted.
In the passage "development agency" refers to
Solution
In the passage, the word "agency" does not refer to a business or organization providing a particular service or a group of people working for such a firm. Here it means a plan of action for producing results or in other words, an agenda for development.
Therefore the answer we are looking for is A.
According to the author, development agencies who want to "get things done" are
Solution
The author mentions about two types of development agencies. The first type, which wants to "get things done," focuses on making specific changes to improve the area, like building infrastructure or improving the economy.
Now, these agencies target for tangible results, such as economic progress. So, the correct answer is that they are "interested in economic outcomes and progress for the area."
The West African Anchau Rural Development Scheme failed because
Solution
The author states that the West African Anchau Rural Development Scheme failed due to a lack of consideration for the human factor when operating in a different culture.The passage highlights that the individuals in charge treated the locals as "people to be done good to," without considering them as individuals with their own hopes, fears, and opinions. They did not involve or consult the local people in the decision-making process, which contributed to the scheme's failure. Therefore, the answer is C.
Option A: While the passage does mention that Western experts sometimes fail to consider local conditions, particularly in cultures different from their own, it doesn't say that the Anchau Rural Development Scheme failed solely because of this.
Option B: The passage doesn't say that the men heading the project "forgot the big picture." The issue wasn't about excessive detail but rather about neglecting the human factor and failing to consult the local community.
Option D: The passage mentions that experts sometimes assume they know what's best for people. However, the primary reason for the failure of the Anchau Scheme was not the assumption that they knew better, but rather the failure to engage the local population in the process.
In this passage the main point that the author wants to make is that
Solution
In the passage, the author is trying to make the point that the idea of
development varies in different cultures. What may be development for
one society may be detrimental in another community.
The option that best conveys this idea is option C.
Read the passage and answer the questions that follow:
Passage II
Humans are pretty inventive creatures. That might be cause for optimism about the future of global change. We've found solutions to lots of problems in the past. And with a much larger and better-educated population than the world has ever seen — the supply of good ideas can only increase. So innovation will figure out a way to sustainable futures.
But what is innovation? The media and companies routinely equate innovation with shiny new gadgets. In the same spirit, politicians charged with managing economies frequently talk as if all innovation is good. The history of almost any technology, however — from farming to applied nuclear physics — reveals a mixture of good and bad.
The study of the concept of innovation, and of whether it can be steered, is a relatively recent academic effort. There are three ways that scholars have thought about innovation. The first was basically linear: science begets invention that begets innovation. Physics, for instance, gives us lasers, which give us —eventually — compact discs. Result: Growth! Prosperity! Rising living standards for all! From this perspective, it's assumed that science is the basis for long-term growth, and that innovation largely involves commercialisation of scientific discoveries. There is a role for the state, but only in funding the research. The rest can be left to the private sector.
By the 1970s, economists interested in technology and some policy-makers were talking about something more complicated: national systems of innovation competing with each other. Such "systems" included measures to promote transfer of technology out of the lab, especially by building links between centres of discovery and technologists and entrepreneurs.
The key failing of these two approaches is that they treat less desirable outcomes of innovation as externalities and are blind to the possibility that they may call for radically different technological priorities. The environmental effects of energy and materials-intensive industries may turn, out to be more destructive than we can handle.
Radical system change is a third way to think about innovation. Technological trajectories aren't pre-ordained: Some paths arc chosen at the expense of others. And that's harder because it needs more than incremental change. The near future is about transformation. The more complex historical and social understanding of innovation now emerging leads to a richer concept of infrastructure, as part of a system with social and technical elements interwoven.
An emphasis on the new, the experimental, the innovative - and on promoting social and technical solutions to global problems must overcome the sheer inertia of the systems we have already built - and are often still extending. Aiming for transformation leads to another take on creative destruction. It isn't enough to promote innovation as creation, the existing system has to be destabilized as well. System shifts of the radical kind envisaged will call for creation of a new infrastructure. But that won't do the job unless the old systems are deliberately removed on roughly the same time-scale. Achieving that will call for a lot more thought about how to if not destroy the old systems, at least set about dismantling them.
From the passage, we can conclude that the author believes
Solution
In the passage, the author focuses on innovation, and innovation cannot be equated to science.
He also mentions that we should also consider the less desirable outcomes of innovation.
The idea that "science begets invention that begets innovation" is not that of the author, but instead, it was used by scholars who thought about innovation.
The idea of the author is made clear in the last paragraph. He is of the opinion that innovation can be made plausible only by dismantling, if not completely destroying, the existing systems. So creative destruction is one way of describing innovation.
Therefore the correct option is C.
According to the author, the first two approaches related to the study of innovation are inadequate because
Solution
The author explicitly states the reason for the inadequacy of the first two approaches in the following paragraph:
"The key failing of these two approaches is that they treat less desirable outcomes of innovation as externalities and are blind to the possibility that they may call for radically different technological priorities. The environmental effects of energy and materials-intensive industries may turn, out to be more destructive than we can handle."
Hence the correct option is A.
The key difference between the first and second approaches related to the study of innovation is
Solution
In the first approach, the government plays a minimal role by just funding the research. On the other hand, the second approach presents the idea of national systems of innovation which compete against one another. So the difference between the two approaches is the varying emphasis on the role of government and policy in innovation.
Hence the correct option is D.
Which of the following statements best describe the author's view on innovation?
Solution
The author’s view on innovation is multifaceted and evolves throughout the passage. Initially, the author expresses optimism, suggesting in the first paragraph that human inventiveness and a growing, educated population could lead to innovation solving global challenges, implying that "innovation will figure out a way to sustainable futures" (option a).
However, as the passage progresses, the author critiques simplistic views of innovation, discussing its historical mix of good and bad outcomes and exploring three scholarly perspectives: linear innovation, national systems, and radical system change. The third perspective, emphasized toward the end, highlights the need for transformative change beyond incremental innovation, noting that existing systems must be destabilized and dismantled to enable sustainable futures (aligning with option d, and to some extent b).
Despite this complexity, the question asks for the statement that "best describes" the author’s view. The author’s foundational optimism in the opening—linking innovation to sustainable futures—sets the tone, while later arguments refine how this might be achieved (through radical change and destabilization).
Option a captures this overarching belief in innovation’s potential most directly, even as the author qualifies it with conditions later. Options b and d, while reflecting key aspects of the radical change perspective, focus on specific mechanisms rather than the broader view, and c aligns with only the outdated linear model the author critiques.
By the expression, "The media and companies routinely equate, innovation with Shiny new gadgets" , the author is
Solution
When the author says "The media and companies routinely equate, innovation with Shiny new gadgets", he is unfairly critical of the media-commercial approach of media and companies.
So the correct option is D.
According to the author, radical systems change is primarily about
Solution
The author discusses radical system change as involving more than incremental change; it requires transformation. This transformation is linked to the creation of new infrastructure, which must be accompanied by the deliberate dismantling of old systems to achieve meaningful progress. This is captured in Option C.
Option A: Although negative outcomes of innovation, such as environmental impacts, are significant, radical system change is not just about addressing these concerns.
Option B: This idea is part of the discussion on how innovation is viewed in the third approach and not the primary focus of radical system change.
Option D: While experimentation is mentioned as part of promoting innovation, radical system change involves more than merely trying new things; it entails fundamentally transforming systems, which requires more than just experimentation.
Answer the questions based on the following information.
Examinations were held during the two weeks of January — Sunday the 3rd to Saturday the 16th. There was one examination each for the six subjects namely, Sociology, Psychology, Economics, Political Science, Anthropology and Biology. There was no more than one examination on any day. No examinations were held on Saturdays, Sundays and on January 5th, which was a national holiday. Exactly three examinations were held in each week. The Psychology examination was held before the Economics examination, and the Political Science examination was held the day after the Biology examination. The Economics and the Political Science examinations were held on the same day of the week. Similarly, the Sociology and the Psychology examinations were held on the same day of the week. There were no examinations for three days between the Sociology examination and the examination prior to it. The Biology and the Anthropology examinations were held on a Tuesday and a Thursday respectively.
On which of the following set of dates were there no examinations?
On which of the following dates was the Biology examination held?
Which examination was held on $$4^{th}$$ January?
The number of days (including weekends and holidays, if any) between the Psychology and the Anthropology examination is
Which examinations were held in the first week?
Answer the questions based on the following information.
The following table gives the urban population of a country and the percentages of total population in rural and urban areas as recorded in the 10-years censuses during 1901-81.
Table 1. Urban and rural population 1901-1981
The percentage increase in total population of the country between 1901 and 1981 is
Solution
From the table ,
Total population in 1901 = Rural population in 1901 + Urban population in 1901
=225+25 = 250
Total population in 1981 = Rural population in 1981 + Urban population in 1981
=450+150 = 600
Percentage increase in population from 1901 to 1981 is $$\frac{600-250}{250}$$ * 100
=350/250 *100
=140%
Hence C is the correct answer.
The percentage increase in density of population in the urban areas between 1951 and 1981 is
Solution
Since area is not known density of urban areas cannot be determined.
Hence D is the correct answer.
The largest rate of increase in urban population in a decade during 1901-1981 occurred in
Solution
Let's solve the options one by one .
Option A : 1971-81 $$\frac{150-100}{100}$$ = 0.5
Option B : 1961-71 $$\frac{100-79}{79}$$ = 0.266
Option C : 1951-61 $$\frac{79-62}{62}$$ = 0.274
Option D : 1941-1951 $$\frac{62-44}{44}$$ = 0.41
Among the above values,the largest rate of increase in urban population occurred during 1971-81
Hence A is the correct answer.
The rate of urban population growth per year over the period 1901-81 is about
Solution
from the table
Urban population in 1901 = 25 million
Urban population in 1981 = 150 million
Change in urban population = 150-25 = 125 million in 80 years
Therefore, the rate of urban population growth per year over the period 1901-81 is about = 125/80 = 1.5625 million per year
Hence B is the correct answer.
The smallest rate of increase in urban population in a decade during 1921-1961 occurred in
Solution
Let's solve the options one by one ,
Option A :Rate of increase of urban population during 1951-61 = $$\dfrac{79-62}{62}$$ = 0.274
Option B :Rate of increase of urban population during 1941-51 = $$\dfrac{62-44}{44}$$ = 0.409
Option C :Rate of increase of urban population during 1931-41 = $$\dfrac{44-33}{33}$$ = 0.33
Option D :Rate of increase of urban population during 1921-31 = $$\dfrac{33-28}{28}$$ = 0.1785
Among the above values smallest rate of increase was during 1921-31
Hence D is the correct answer.
For the following questions answer them individually
If $$a \in R$$, then the equation $$x^2 + x + a = 0$$ and $$x^2 + ax + 1 = 0$$ have a common real root for
Solution
Let '$$\alpha$$' be the common root which satisfies both the equations.
$$\alpha^{2}$$ + $$\alpha$$ + a = 0 and $$\alpha^{2}$$ + a$$\alpha$$ + 1 = 0
On subtracting the equations,$$\alpha\left(1-a\right)=1-a$$
$$\alpha$$ = 1
Substitute the value of $$\alpha$$ in any of the equations, we get
a=-2
C is the correct answer.
A man standing x metres north of A tower finds the angle of elevation of as top to be 30° .He then starts walking towards the tower. After walking a distance of x/2 metres, he turns east and walks another x/2 metres. Then he turns south and walks another x/2 metres. The angle of elevation of the top of the tower from his new position is
Let the equations of two circles $$C_1$$ and $$C_2$$ be given by $$x^2 + y^2 - 4x - 4y + 6 = 0$$ and $$x^2 + y^2 - 10x - 10y + k = 0$$ respectively, where $$k$$ is a constant. Suppose that $$C_1$$ and $$C_2$$ have exactly two common tangents. Then possible values of $$k$$ are
Solution
Centre and radius of a circle $$x^2 + y^2 +2gx +2fy + c = 0$$ is (-g,-h) and $$\sqrt{g^{2}+h^{2}-c}$$
Centre and the radius of $$C_1$$ is (2,2) and $$\sqrt{2}$$
Centre and the radius of $$C_2$$ is (5,5) and $$\sqrt{50-k}$$
Distance between the centres of the two circles = $$\sqrt{18}$$
According to the properties of the circle
If two circles have two common tangents then ,
|Difference of the radii| < Distance between the centres of the two circles < |Sum of the radii|
$$\sqrt{50-k}$$ - $$\sqrt{2}$$ < $$\sqrt{18}$$ < $$\sqrt{50-k}$$ + $$\sqrt{2}$$
Let's look at the options one by one ,
Option A : let k= 1 |7-$$\sqrt{2}$$| < 3$$\sqrt{2}$$ < |7+$$\sqrt{2}$$| -------------------> Does not satisfy
Option B : let k= 25 |5-$$\sqrt{2}$$| < 3$$\sqrt{2}$$ < |5+$$\sqrt{2}$$| -------------------> Satisfied
Option C : let k= 49 |1-$$\sqrt{2}$$| < 3$$\sqrt{2}$$ < |1+$$\sqrt{2}$$| -------------------> Does not satisfy
Hence B is the correct answer.
Consider the function
$$f(x) = \begin{cases}2x -1 & if & x < -1\\x^2 + 1 & if & -1\leq x \leq 1\\x + 1 & if & x > 1.\end{cases}$$
Then
Solution
F(x)=
$$\lim_{x \rightarrow -1^{-}}$$ (2x-1) = 2(-1)-1 = -3
$$\lim_{x \rightarrow -1^{+}}$$ $$(x^{2} + 1)$$ = 1+1 = 2
Since the left-hand limit and right-hand limit are not equal,the function is discontinuous at -1.
$$\lim_{x \rightarrow 1^{-}}$$ $$(x^{2} + 1)$$ = 1+1 = 2
$$\lim_{x \rightarrow -1^{+}}$$ x+1 = 1+1 = 2
Since the left-hand limit and right-hand limit are equal, the function is continuous at 1.
So the function is continuous everywhere except at -1.
C is the correct answer.
The sum of the first $$50$$ terms of the series: $$3 + 7 + 13 + 21 + 31 + 43 +...$$ is
Solution
$$3 + 7 + 13 + 21 + 31 + 43 +...$$
The difference of two consecutive terms is 4,6,8,10
Since the difference of two consecutive terms is in AP, the general term will be a quadratic expression in n
So the general term is t(n) = $$an^{2}+bn+c$$
t(1)= a+b+c = 3
t(2) = 4a+2b+c = 7
t(3) = 9a+3b+c = 13
we get the values of a ,b,c as 1,1,1 respectively
General term = $$1n^{2}+n+1$$
$$\sum1n^{2}+n+1$$ = n($$\frac{n^{2}+3n+5}{3}$$)
Substitute n = 50
=50*885
Hence D is the correct answer.
If
$$A_n = \frac{1.2.3 + 2.3.4 + 3.4.5 + .... upto n terms}{n(1.2 + 2.3 + 3.4 + .... upto n terms)}$$
then $$\lim_{n \rightarrow \infty} A_n$$ is
Solution
$$\frac{Numerator}{n}$$ = $$\Sigma\ n\left(n+1\right)\left(n+2\right)\ =\ \frac{\left(\Sigma\ n^3+3\Sigma n^2+2\Sigma\ n\ \right)}{n}$$
$$\frac{n^2\left(n+1\right)^2}{4n}+\frac{3n\left(n+1\right)\left(2n+1\right)}{6n}+\frac{2n\left(n+1\right)}{2n}$$
Denominator
$$\Sigma\ n\left(n+1\right)\ =\ \Sigma\ n^2+n\ =\ \frac{n\left(n+1\right)\left(2n+1\right)}{6}+\frac{n\left(n+1\right)}{2}$$
Numerator/Denominator = $$\frac{3}{4}\left[1+\frac{3}{n}\right]$$
When n -> $$\infty\ ,\ \frac{1}{n}->\ 0$$
= 3/4
The function $$f: R \rightarrow R$$, defined by $$f(x) = x^3 - 3x^2 + 6x - 5$$, is
The number of distinct words that can be formed using all the letters except vowels of the word 'PROBABILITY' is
Solution
The letters except the vowels are P, R, B, B, L, T, Y.
The number of words that can be formed = 7!/2 =2520
A is the correct answer.
The area enclosed between the curves $$y =2x^2$$ and $$y = 6$$ is
Solution
$$y =2x^2$$ and $$y = 6$$
6 = $$2x^2$$
x = -$$\sqrt{3}$$ and $$\sqrt{3}$$
$$\int_{-\sqrt{3}}^{\sqrt{3}} 2 *x^2$$
Since x^2 is an even function
$$\int_{-\sqrt{3}}^{\sqrt{3}} 2 *x^2$$
= 4$$\int_{0}^{\sqrt{3}} x^2$$
= 4*$$\frac{x^3}{3}$$ where x varies from 0 to $$\sqrt{3}$$
= $$4\sqrt3$$.
Now this is the area between the parabola and the x-axis. The area between the line y=6 and x-axis is $$6\times\ 2\sqrt{\ 3}$$
Hence, the area between the curves is $$8\sqrt{\ 3}$$.
The value of $$lim_{x \rightarrow 0} \frac{\sin(x^2)}{x \sin x}$$ is
Solution
We know that $$lim_{x \rightarrow 0} \frac{\sin x}{x}$$ is 1
=$$\lim_{x\rightarrow\ 0} \frac{sin x^2}{x^2}*\frac{x}{sin x}$$
= 1
The value of $$\frac{30_{C_1}}{2} + \frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30}$$ is
Solution
We know that $$(1+x)^n$$ = $$^nC_0*1^n+^nC_1*1^{n-1}*x......................................^nC_n*x^n$$
On Integrating, we get
$$\frac{(1+x)^{n+1}}{n+1}$$ = $$\frac{^nC_0*1^n}{1}+\frac{^nC_1*1^{n-1}*x}{2}......................................\frac{^nC_n*x^n}{n+1}$$
When x = 1
$$\frac{2^{31}}{31}$$ = $$\frac{30_{C_0}}{1}$$ +$$\frac{30_{C_1}}{2} + \frac{30_{C_2}}{3} +\frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30} + \frac{30_{C_{30}}}{31}$$ -- Eq 1
When x = -1
0= $$ \frac{30_{C_0}}{1}$$ -$$\frac{30_{C_1}}{2} + \frac{30_{C_2}}{3} -\frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30} - \frac{30_{C_{30}}}{31}$$ -- Eq 2
Adding Eq 1 and 2, we get
$$\frac{2^{31}}{31}$$ = 2($$\frac{30_{C_1}}{2} + \frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30}$$)
($$\frac{30_{C_1}}{2} + \frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30}$$) = $$\frac{2^{30}}{31}$$
In the quadrilateral $$ABCD$$ below, $$\angle DAB$$ = 90° and $$AB = 24$$cm , $$BC = 40$$cm, $$CD = 50$$cm and $$AD = 18$$cm(The diagram is not drawn to scale) Find the area of the quadrilateral
Solution
Join BD
Area of ABCD = area of ABD + area of BCD
In ABD $$AB^2\ +\ AD^2\ =BD^2$$
Therefore $$24^{2\ }+\ 18^{2\ }=\ BD^2$$ , on solving BD= 30
also $$30^{2\ }+\ 40^2\ =50^2$$ which implies BCD is also a right-angled triangle, right angled at B.
Area of ABD = $$\dfrac{1}{2}\times\ 24\times\ 18$$ = 216
Area of BCD = $$\dfrac{1}{2}\times\ 30\times\ 40$$= 600
Area of ABCD = 816
Let $$x = \frac{\pi}{40}$$. Then the value of $$\cot x \cot 2x \cot 3x ....\cot 19x $$ is
Solution
The expression will be $$\cot\left(\frac{\pi}{40}\right)\cot\left(\frac{2\pi}{40}\right)....\cot\left(\frac{19\pi}{40}\right)$$
Now $$\cot A=\tan\left(\frac{\pi}{2}-A\right)$$
Thus $$\cot\frac{\pi}{40}=\tan\left(\frac{\pi}{2}-\frac{\pi}{40}\right)=\tan\frac{19\pi}{40}$$
Also, (cotA)(tanA)=1.
Hence, all the terms other than $$\cot\frac{10\pi}{40}$$ pair up and become 1.
Now $$\cot\frac{10\pi}{40}=\cot\frac{\pi}{4}=1$$
Hence, the value of the expression will be 1.
Consider the function: $$f(x) = \mid {2 - \mid x - 1\mid}\mid$$ for all $$x \in R$$. Then the value of $$f'(-2) + f'(0) + f'(2) + f'(4)$$ is
Solution
$$\mid x - 1\mid$$
=$$\begin{cases}x-1 & x \geq 1\\1-x & x < 1 \end{cases}$$
x < 1 f(x)= $$\mid x + 1\mid$$
x $$\geq$$ 1 f(x) = $$\mid 3-x\mid$$
x < -1 f(x) = -x-1
x $$\geq$$ -1 f(x) = x+1
x < 3 f(x) = x-3
x $$\geq$$ 3 f(x) =3-x
Calculate the values of
f'(-2) = -1, f'(0) = 1,f'(2) = 1, f'(4) = -1
$$f'(-2) + f'(0) + f'(2) + f'(4)$$ = 0
Hence B is the correct answer.
Let
$$P = \begin{bmatrix}a & b & 0\\-1 & 2 & 1\\2 & -3 & -2 \end{bmatrix}$$
with $$det(P) = -2$$. Then the minor $$M_{22}$$ of $$P$$ is
Solution
$$det(P)$$ = a(2*-2)-(-3*1) + (-b)[(-1*-2)-(2*1)]+0[(-1*-3)-(2*2)]
=a(-1)-b(0)
=-a
But given $$det(P)$$ = -2
Hence a = 2
Minor of $$M_{22}$$ = $$det\begin{bmatrix}a & 0 \\2 & -2 \end{bmatrix}$$
det of $$\begin{bmatrix}2 & 0 \\2 & -2 \end{bmatrix}$$
= -4
Hence A is the correct answer.
If $$\alpha$$ and $$\beta$$ are two roots of the equation $$x^2 + x + 1 = 0$$, then the value of $$\alpha^{2017} + \beta^{2017}$$ is
Solution
Consider, $$x^3-1$$ = (x-1)($$x^2 + x + 1 $$)=0
It will have roots as 1, $$\alpha\ $$ and $$\beta\ $$
Now, since both $$\alpha\ $$ and $$\beta\ $$ satisfy $$x^3-1=0$$, Hence,$$\alpha^{3}-1=0$$ => $$\alpha^3=1$$
and $$\beta^3-1=0$$ => $$\beta^3=1$$
Hence $$\alpha^{2017}+\beta^{2017}$$ = $$\alpha^{3\times\ 672+1}+\beta^{3\times\ 672+1}$$ = $$\alpha\ +\beta$$
Now, the sum of roots of the equation $$x^3-1$$=0 is zero.
Hence, $$1+\alpha\ +\beta\ =0=>\alpha+\beta\ =-1$$
The number of different solutions $$(x,y,z)$$ of the equation $$x + y + z = 10$$, where $$x, y$$ and $$z$$ are positive integers, is
Solution
Assume x=a+1, y=b+1, z=c+1
a+1+b+1+c+1=10,
a+b+c=7
The number of non-negative solutions for a, b and c will give the postive integral solution for x,y,z which is $$^{7+3-1}C_{3-1}$$ = $$\ \frac{\ 9\times\ 8}{2}$$ = 36
In the $$xy$$-plane, the equation $$x^2 - y^2 = 2y + 1$$ represents a
Solution
We have, $$x^2 - y^2 = 2y + 1$$
=> $$x^2 = y^2 + 2y + 1$$
=> $$x^2 - (y+1)^2 $$=0
=> (x-y-1)(x+y+1)=0
Here we have a pair of straight lines: x-y-1=0 and x+y+1=0.
There are 100 students in a class. in an examination, 50 of them failed in Mathematics, 45 failed in Physics and 40 failed in Biology. 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. The number of students failing in all the three subjects is
Solution
Assume the number of students who failed in exactly 1 subject is x
The number of students who failed in exactly 2 subjects is y
The number of students who failed in exactly 3 subjects is z
Now, x+2y+3z=50+45+40=135
x+y+z=100-1=99
On subtracting the two equations, we get,
y+2z=36
It is given that y=32
Hence, 2z=36-32=4 =>z=2
The point $$R (4,10)$$ lies on the curve $$C: y = x^2 - 6x + 18.$$ The tangent and normal to $$C$$ at $$R$$ meets the Y-axis at points $$P$$ and $$Q$$ respectively. A circle passes through the points $$P,Q$$ and $$R$$. The radius of this circle is
Solution
$$\ \frac{\ dy}{dx}\ =\ 2x-6$$
Slope of the tangent at R(4, 10) = 2
A normal will be perpendicular to tangent, so the slope of the normal = $$\ \frac{\ -1}{2}$$
Equation of the tangent at R(4, 10) =
y-10 = 2(x-4)
y=2x+2
The above line intersects Y-axis at P (0,2)
Equation of the normal at R(4, 10) =
y-10 = $$\ \frac{\ -1}{2}$$ (x-4)
2y=-x+24
The above line intersects Y-axis at Q (0,12)
Let the equation of the circle be $$x^2+y^2+2gx+2fy+c=0$$ with centre (-g, -f)
Now we have to find the equation of the circle passing through P (0, 2), Q(0, 12), R(4, 10)
On substituting these values in the equation of the circle, we get g = 0, f = -7, c = 24
Radii of the circle = $$\sqrt{\ \left(g^2+f^2-c\right)}$$
= 5
An equilateral triangle, having each side as a , has its corners cut away so as to form a regular hexagon. The area of the hexagon is
Solution
To make a regular hexagon, three equilateral triangles of side$$\ \frac{\ a}{3}$$ will be cut from the corners to make a regular hexagon of side a.
Hence, this hexagonal can be further divided into 6 equilateral triangles of side $$\ \frac{\ a}{3}$$.
Hence the area of the hexagon = $$\ 6\cdot\ \frac{\ \sqrt{\ 3}}{4}$$*($$\ \ \frac{\ a}{3}$$)$$^2$$ = $$\frac{\sqrt3 a^2}{6}$$
A is the answer.
Let $$f(x) = a_0 + a_1 \mid x\mid + a_2 \mid x\mid^2 + a_3 \mid x\mid^3$$, where $$a_0, a_1, a_2$$ and $$a_3$$ are constants. Which of the following statements is correct?
Solution
For the function to be differentiable at x=0, f'(x) =$$\lim\ h\longrightarrow\ 0$$ $$\ \frac{\ f\left(x+h\right)-f\left(x\right)}{h}$$
=> f'(0)=$$\lim\ h\longrightarrow\ 0$$ $$\ \frac{\ f\left(h\right)-f\left(0\right)}{h}$$
Now using the value of f(x), we get f'(0) =$$\lim\ h\longrightarrow\ 0$$ $$\ \frac{\ a_0+a_1\mid h\mid+a_2\mid h\mid^2+a_3\mid h\mid^3-a_0}{h}$$
Now, f'(0)= $$\lim\ h\longrightarrow\ 0$$ $$\ \frac{a_1\mid h\mid+a_2\mid h\mid^2+a_3\mid h\mid^3}{h}$$
For h<0, f'(0)=$$-\ a_1+a_2h-a_3h^2$$
For h>0, f'(0)=$$\ a_1+a_2h+a_3h^2$$
So for $$h\longrightarrow\ 0$$, the two values will only be equal if $$a_1$$ = 0.
Hence, C is the answer.
If $$P = \begin{bmatrix}a & b & c\\x & y & z\\p & q & r \end{bmatrix}$$ and $$Q = \begin{bmatrix}-x & a & -p \\y & -b & q\\z & -c & r \end{bmatrix}$$ then
Solution
Taking -1 common from both 2nd and 3rd row of Q, we get
Q = (-1)(-1)$$\begin{bmatrix}-x & a & -p \\-y & b & -q\\-z & c & -r \end{bmatrix}$$
= $$\begin{bmatrix}-x & a & -p \\-y & b & -q\\-z & c & -r \end{bmatrix}$$
Now taking -1 common from 1st and 3rd column of Q, we get
Q= (-1)(-1)($$\begin{bmatrix}x & a & p \\y & b & q\\z & c & r \end{bmatrix}$$
= ($$\begin{bmatrix}x & a & p \\y & b & q\\z & c & r \end{bmatrix}$$
After interchanging the 1st and second column, Q becomes the the transpose of P.
Since the row have been exchange once and the determinant of the transpose matrix is the same, hence det(P)=(-1)$$^1$$*det(Q)
=> $$det(P) = -det(Q)$$
D is the answer.
Let S = {1,2,...,100}. The number of nonempty subsets T of S such that the, product of numbers in T is even is
Solution
The number of ways to select non-empty set out of S = 100C1+100C2+100C3+..........100C100 = $$2^{100}$$-1
Similarly, the number of ways to select non-empty set from (say) P= {1,3,5,7,9,...................99} = $$2^{50}$$-1
Hence, the required number of set = $$2^{100}$$-1 - ($$2^{50}$$-1) = $$2^{50}\left(2^{50}-1\right)$$
What is the sum of the interior angles at the vertices of a 5-pointed star as shown below? The star need not have sides of the same length.
Solution
In triangle AFJ, $$x_{1}$$+180-a+180-b=180
=> $$x_{1}$$+180=a+b
Similarly, $$x_{2}$$+180=a+e
$$x_{3}$$+180=d+e
$$x_{4}$$+180=c+d
$$x_{5}$$+180=b+c
Adding all the equations,
$$x_{1}$$+$$x_{2}$$+$$x_{3}$$+$$x_{4}$$+$$x_{5}$$+180*5= 2(a+b+c+d+e)
=> Sum of interior angles = 2*(a+b+c+d+e)-900 = 2*540-900=180 (Sum of interior angles of a pentagon = 540)
