8 IPMAT Indore 2019
8.1 Quantitative Ability (SA)
The sum of the interior angles of a convex \(n\)-sided polygon is less than \(2019^{\circ}\). The maximum possible value of \(n\) is
Suppose that \(a, b, c\) are real numbers greater than 1. Then the value of \(\frac{1}{1+\log_{a^{2}}c^{\frac{c}{a}}} + \frac{1}{1+\log_{b^{2}}c^{\frac{a}{b}}} + \frac{1}{1+\log_{c^{2}}a^{\frac{b}{c}}}\) is
A real-valued function \(f\) satisfies the relation \(f(x)f(y) = f(2xy+3) + 3f(x+y) - 3f(y) + 6y,\) for all real numbers \(x\) and \(y\). Then the value of \(f(8)\) is
Let \(A, B, C\) be three \(4 \times 4\) matrices such that \(\det(A) = 5\), \(\det(B) = -3\), and \(\det(C) = \frac{1}{2}\). Then \(\det(2AB^{-1}C^{3}B^{T})\) is
If \(A\) is a \(3 \times 3\) non-zero matrix such that \(A^{2} = 0\), then the determinant of \([(1+A)^{2} - 50A]\) is equal to
Three friends divided some apples in the ratio \(3:5:7\) among themselves. After consuming 16 apples they found that the remaining number of apples with them was equal to the largest number of apples received by one of them at the beginning. The total number of apples these friends initially had was
A shopkeeper reduces the price of a pen by 25% as a result of which the sales quantity increased by 20%. If the revenue made by the shopkeeper decreases by \(x\%\), then \(x=\)?
For all real values of \(x\), \(\frac{3x^{2}-6x+12}{x^{2}+2x+4}\) lies between 1 and \(k\), and does not take any value above \(k\). Then \(k=\) ?
The maximum distance between the point \((-5, 0)\) and a point on the circle \(x^{2}+y^{2}=4\) is
If \(x, y, z\) are positive real numbers such that \(x^{12}=y^{16}=z^{24}\) and the three quantities \(3\log_{y}x,\;4\log_{z}y,\;n\log_{x}z\) are in arithmetic progression, then the value of \(n\) is
The number of pairs \((x, y)\) satisfying the equations \(\sin x + \sin y = \sin(x+y)\) and \(|x|+|y|=1\) is
The circle \(x^{2}+y^{2}-6x-10y+k=0\) does not touch or intersect the coordinate axes. If the point \((1, 4)\) does not lie outside the circle, and the range of \(k\) is \((a, b)\), then \(a+b\) is
If a \(3 \times 3\) matrix is filled with \(+1\)’s and \(-1\)’s such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is
Let the set \(P = \{2,3,4,\ldots,25\}\). For each \(k \in P\) define \(Q(k) = \{x \in P \mid x>k \text{ and } k \text{ divides } x\}\). Then the number of elements in the set \(P - \bigcup_{k=2}^{25}Q(k)\) is
The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has the shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (assume uniform thickness and no loss of material), is
If \(|x|<100\) and \(|y|<100\), then the number of integer solutions of (x, y) satisfying the equation \(4x+7y=3\) is
The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is
Assume that all positive integers are written down consecutively from left to right as in 1234567891011…… The \(6389^{th}\) digit in this sequence is
The number of pairs of integers whose sums are equal to their products is
You have been asked to select a positive integer N which is less than 1000, such that it is either a multiple of 4, or a multiple of 6, or an odd multiple of 9. The number of such numbers is
8.2 Quantitative Ability (MCQ)
If the compound interest earned on a certain sum for 2 years is twice the amount of simple interest for 2 years, then the rate of interest per annum is ____ percent
The maximum value of the natural number n for which \(21^n\) divides 50! is
The remainder when \((29^{29})^{29}\) is divided by 9 is
Placing which of the following two digits at the right end of 4530 makes the resultant six digit number divisible by 6,7 and 9?
In a school 70% of the boys like cricket and 50% like football. If x% like both Cricket and Football, then
In a class of 65 students 40 like cricket, 25 like football and 20 like hockey. 10 students like both cricket and football, 8 students like football and hockey and 5 students like all three sports. If all the students like at least one sport, then the number of students who like both cricket and hockey is
If \(x\in(a,b)\) satisfies the inequality, \(\frac{x-3}{x^{2}+3x+2}\) then the largest possible value of b - a is
If a, b, c are real numbers \(a^{2}+b^{2}+c^{2}=1,\) then the set of values ab + bc + ca can take is:
The inequality \(\log\_{2}\frac{3x-1}{2-x}<1\) holds true for
The set of values of x which satisfy the inequality \(0.7^{2x^{2}-3x+4}<0.343\) is
A chord is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals
Points P, Q, R and S are taken on sides AB, BC, CD and DA of square ABCD respectively, so that \(AP:PB=BQ:QC=CR:RD=DS:SA=1:n\). Then the ratio of the area of PQRS to the area of ABCD is
On a circular path of radius 6 m a boy starts from a point A on the circumference and walks along a chord AB of length 3 m. He then walks along another chord BC of length 2 m to reach point C. The point B lies on the minor arc AC. The distance between point C from point A is
The area enclosed by the curve \(2|x|+3|y|=6\) is
Two points on a ground are 1 m apart. If a cow moves in the field in such a way that its distance from the two points is always in ratio 3:2 then
Given that \(cos x+cos y=1\) the range of \(sin x-sin y\) is
If \(sin\theta+cos\theta=m\) then \(sin^{6}\theta+cos^{6}\theta\) equals
If \(\begin{bmatrix} 2 & -0.5 \ -1 & x \end{bmatrix}^{-1} = \begin{bmatrix} 1 & 1 \ 2 & 4 \end{bmatrix}\), then the value of \(x\) is
The function \(f(x) = \frac{x^{3} + 5x^{2} - 8x}{3}\) is
For \(a>b>c>0,\) the minimum value of the function \(f(x)=|x-a|+|x-b|+|x-c|\) is
Let α, β be the roots of \(x^{2}-x+p=0\) and γ, δ be the roots of \(x^{2}-4x+q=0\) where p and q are integers. If α, β, γ, δ are in geometric progression then \(p+q\) is
If \((1 + x - 2x^{2})^{6} = A_{0} + \sum_{r=1}^{12} A_{r}x^{r}\), then the value of \(A_{2} + A_{4} + A_{6} + \cdots + A_{12}\) is
The number of terms common to both the arithmetic progressions 2,5,8,11,…., 179 and 3,5,7,9,….., 101 is
From a pack of 52 cards, we draw one by one, without replacement. If \(f(n)\) is the probability that an Ace will appear at the \(n^{\text{th}}\) turn, then
A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is
In a given village there are only three sizes of families: families with 2 members, families with 4 members and families with 6 members. The proportion of families with 2,4 and 6 members are roughly equal. A poll is conducted in this village wherein a person is chosen at random and asked about his/her family size. The average family size computed by sampling 1000 such persons from the village would be closest to
The value of \((\log_{3}30)^{-1} + (\log_{4}900)^{-1} + (\log_{5}30)^{-1}\) is
The inequality \(\log_{a} f(x) < \log_{a} g(x)\) implies that
Three cubes with integer edge lengths are given. It is known that the sum of their surface areas is 564 \(cm^{2}\). Then the possible values of the sum of their volumes are
Determine the greatest number among the following four numbers
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0,0), (0,31), and (31,0) is
Two small insects, which are x metres apart, take u minutes to pass each other when they are flying towards each other, and v minutes to meet each other when they are flying in the same direction. Then, the ratio of the speed of the slower insect to that of the faster insect is
An alloy P has copper and zinc in the proportion of 5:2 (by weight), while another alloy Q has the same metals in the proportion of 3:4 (by weight). If these two alloys are mixed in the proportion of a:b (by weight), a new alloy R is formed, which has equal contents of copper and zinc. Then, the proportion of copper and zinc in the alloy S, formed by mixing the two alloys P and Q in the proportion of b:a (by weight) is
How many different numbers can be formed by using only the digits 1 and 3 which are smaller than 3000000?
There are numbers \(a_{1},a_{2},a_{3},...,a_{n}\) each of them being +1 or -1. If it is known that \(a_{1}a_{2}+a_{2}a_{3}+...+a_{n}a_{1}=0\) then
Instructions [56-60]:
Analyze the given data for exports and imports of rubber in Rs. crores from 2006 to 2017 and answer the questions based on the analysis.
Average annual exports for the given period (2006-2017) was approximately
The percentage decline in exports during the period 2006-2011 is more than the percentage decline in exports during 2012-2017 by approximately
The maximum difference between imports and exports is
Balance of trade is defined as imports subtracted from exports (= exports - imports). Which of the following blocks of three years has witnessed the largest average negative balance of trade?
The percentage increase in imports over the previous year is maximum during
8.3 Verbal Ability
Instructions [61-66]:
Read the following passage and choose the answer that is closest to each of the questions that are based on the passage.
Supposing half a dozen or a dozen men were cast ashore from a wreck on an uninhabited island and left to their own resources, one of course, according to his capacity, would be set to one business and one to another; the strongest to dig and to cut wood, and to build huts for the rest: the most dexterous to make shoes out of bark and coats out of skins; the best educated to look for iron or lead in the rocks, and to plan the channels for the irrigation of the fields. But though their labours were thus naturally severed, that small group of shipwrecked men would understand well enough that the speediest progress was to be made by helping each other-not by opposing each other, and they would know that this help could only be properly given so long as they were frank and open in their relations, and the difficulties which each lay under properly explained to the rest. So that any appearance of secrecy or separateness in the actions of any of them would instantly, and justly, be looked upon with suspicion by the rest, as the sign of some selfish or foolish proceeding on the part of the individual. If, for instance, the scientific man were found to have gone out at night, unknown to the rest, to alter the sluices, the others would think, and in all probability rightly think, that he wanted to get the best supply of water to his own field; and if the shoemaker refused to show them where the bark grew which he made the sandals of, they would naturally think, and in all probability rightly think, that he didn’t want them to see how much there was of it, and that he meant to ask from them more corn and potatoes in exchange for his sandals than the trouble of making them deserved. And thus, although each man would have a portion of time to himself in which he was allowed to do what he chose without let or inquiry - so long as he was working in that particular business which he had undertaken for the common benefit, any secrecy on his part would be immediately supposed to mean mischief; and would require to be accounted for, or put an end to: and this all the more because, whatever the work might be, certainly there would be difficulties about it which, when once they were well explained, might be more or less done away with by the help of the rest; so that assuredly every one of them would advance with his labour not only more happily, but more profitably and quickly, by having no secrets, and by frankly bestowing, and frankly receiving, such help as lay in his way to get or to give.
When a dozen men are cast away on an imaginary island, the best educated would look for metals in rocks because
The author states that any appearance of secrecy or separateness would instantly and justly be looked upon with suspicion. From this statement we may infer that
The instance of the shoemaker who refuses to show his source and asks for more corn and potatoes, is an example of
According to the author, whatever one’s work might be
The author’s belief is that for progress to happen
The writer makes a hypothesis, which can be related to
Instructions [67-72]:
Read the following passage and choose the answer that is closest to each of the questions that are based on the passage.
The perennial debate over gender differences threatens to remain inconclusive. Stereotypes pertaining to male superiority and female submissiveness could be traced to earlier ages where assigned roles were needed as survival measures. But, can we today see a swing away from these stereotypes, or have they established a stranglehold on our perceptions? In this gendered world, we continue to live with notions that one’s gender determines one’s skills and preferences, from toys and colours to career choices. So the girl child will be presented with a Barbie doll, while the boy child will receive a Lego set. Does that mean that our brains are different? This myth has been exploded by a British professor of cognitive neuroimaging. Her research attempts to establish how these stereotypes mould our ideas of ourselves. She examines how science has been misinterpreted or misused to ask the wrong questions, instead of challenging the status quo. She urges us to move beyond a binary view of people’s brains and instead to see these as highly individualised, profoundly adaptable, and full of unbounded potential. Her conclusive ¦ndings establish that no brain differences can be found that are solely gender related. In other words, modern neuroscientists have identi¦ed no decisive category-de¦ning differences between the brains of men and women. As a result of these ¦ndings we owe it to ourselves to dump the myths and look at ourselves afresh. We need to recognise that the male and female brain debate is a distraction, besides being based on inaccuracies. It is possibly harmful too, because it can be used as a hook to justify saying there is no point in girls doing science because they do not have a science brain; or compelling boys to opt for science because their brains are shaped for that subject. It can also condemn boys for being emotional, as this is seen as a feminine trait. And, most dangerous of all, to proclaim that boys, not girls, are meant to lead.
The research of a British professor of cognitive neuroimaging has succeeded in establishing that
By referring to the world as “gendered” the writer wants to convey that
One of the dangers in adopting a binary view of the human brain is that it can
The writer of this passage wants to emphasise the need to
The writer of this passage wants to emphasise the need to
The antonym for “unbounded” (Para 2) is
Instructions [73-78]:
Complete the following sentences by choosing the most appropriate phrase from the options given below.
Although he is recovering from his illness, he has to follow certain diet restrictions. He cannot eat junk food. Please do not pity him and ____
The problems may be difficult, but all you have to do is ____ as long as you can.
The standards set by the examination board are so high that it would be difficult for poorly prepared students to ____
After all the alliances and arithmetic, the party is likely to ____ a majority in the assembly election.
I’ll have to ____ because I don’t know how Sheila’s parents are going to react to this offer.
If you had been more alert, this golden opportunity would not have ____
Instructions [79-84]:
In each of the following sentences the incorrect part of the sentence is underlined. Choose an alternative from the four given options so that the sentence is rendered correct.
The place where her father disappeared and the reason why he did are unknown to her
Which team has the best record, yours or theirs?
After sensing a problem with the factory workers, the personnel officer demanded to know who the union leaders had contacted to conduct the petition drive.
His wrongdoing was completely exposed, but not once he apologised for his actions
The Municipal Council can no longer wash its hands off its responsibilities.
The letter states that one can avail the service offered by the company till June 20, 2019
Instructions [85-87]:
Infrastructure, in the form of paved surfaces, disrupts water absorption and lowers water retention. This leads to disastrous levels of flooding which diminishes the biodiversity and impoverishes the people of the region. ____ Land should be used mindfully to prevent water logging during heavy rains.
Few look forward to old age and all that it brings in its wake - deteriorating health, loss of vigour, restricted mobility, increasing dependence on others, not to mention a sense of foreboding and anxiety. Yet, one has to learn to cope with the onset of old age. Firstly, it is imperative to prepare to accept old age in spite of the restrictions or limitations it imposes on one’s mobility. ____ Equally important is the need to adopt a positive attitude towards life. Above all, peace of mind, is the efficacious balm that brings equanimity to one’s life. We must resign ourselves to growing old, and in the process let us try to make life as fulfilling and meaningful as possible.
The Arab Spring is widely believed to have stemmed from dissatisfaction with the rule of local governments, though some have speculated that wide gaps in income levels may have had a hand as well. Issues such as political corruption, human rights violations, unemployment, and educated but dissatisfied youth may have been responsible as well.
Instructions [88-92]:
One who is ____ gets on with his job in spite of obstacles, while the one who is ____ hardly shows any progress. The latter spends all his time ____ about his troubles.
The ____ of multiculturalism, in times of war or economic ____ tribalism is what causes those in power to confine groups of people with different ____ into ghettos or in communes on the margins of their cities.
That the artiste went about systematically to get traditional ____ back into the mainstream and ____ a textile culture for dance is to be celebrated.
Instructions [91-92]:
In response to my friend’s request, I decided to write her a letter, which I hoped would be honest and practical, while also serving as a ____ of sorts for my own feminist thinking. This book is a ____ that letter, with some details changed.
Quantum Physics really begins to point to this discovery. It says that you can’t have a Universe without mind ____ into it, and that the mind is actually ____ the very thing that is being ____.
Instructions [93-97]:
One of the statements below contains a word used incorrectly. Choose the option which has the incorrect or inappropriate usage of the word.
Instructions [98-100]:
The sentences given below, when properly sequenced, form a coherent paragraph. Each sentence is labelled with a number. Decide on the most logical order and enter the sequence of numbers in the space provided.
- He just harvested the wild grains.
- The hunter-gatherer went from place to place in search of food.
- As the crops began to give better yields, this reduced his need to go in search of animals and wild plants.
- This was followed by an attempt to grow food by scattering the spare grains.
- People here are one injury away from starvation, one misspoken word away from detainment or death.
- Soon, however, she notices the lack of access to basic medical care or education.
- Life in a rural Kashmiri village seems idyllic to Shalini at first, as she’s befriending lovely people and admiring majestic natural scenery, especially in contrast to the cacophony of urban Mumbai.
- Moreover, the ever-present political disruptions mean that life in Kashmir is far from a Shangri-La utopia.
- The study, published in the Lancet recently, revealed that people living in democratic countries live longer than those who don’t; they also have less of a chance of dying from heart disease, strokes, and even road accidents.
- Incredible as it may sound, we are now told that democracy is not just good for the soul, it is good for the body too.
- Without pressure from voters or foreign-aid agencies, dictators have less incentive to finance more expensive prevention and treatment of heart disease, cancers, and other chronic illnesses.
- The study suggests that elections and the health of the people are increasingly inseparable.
- A study spanning 170 countries found a strong correlation between health and the most form of government.