IPMAT Indore 2020 · Question Paper & Solutions

The quotient $q$ and remainder $r$ satisfy $q+r=10$ and $qr=24$, so $\{q,r\}=\{4,6\}$.

The remainder must be less than the divisor 5, so $r=4$ and $q=6$.

$$\text{Dividend}=5\times 6+4=34$$

The circle has centre $(0,0)$ and radius $1$. Distance of $(-4,3)$ from the centre is $\sqrt{16+9}=5$.

The point lies outside the circle, so the shortest distance is $5-1=4$.

The series is geometric: $\frac{1}{4}+\frac{1}{8}+\cdots =\frac{1/4}{1-1/2}=\frac{1}{2}$.

${\log }_{\sqrt{5}}\frac{1}{2}=2{\log }_5\frac{1}{2}=-2{\log }_{5}2$, and $0.04=5^{-2}$.

$$0.04^{-2{\log }_{5}2}=5^{(-2)(-2{\log }_{5}2)}=5^{4{\log }_{5}2}=2^4=16$$

The matrix as printed is garbled (it shows four columns with $b^2$ in the first row); the intended determinant is the standard one
$$\left|\begin{array}{ccc}a& a^2& a^3-1\\ b& b^2& b^3-1\\ c& c^2& c^3-1\end{array}\right|=0.$$

Split the last column:
$$\left|\begin{array}{ccc}a& a^2& a^3\\ b& b^2& b^3\\ c& c^2& c^3\end{array}\right|-\left|\begin{array}{ccc}a& a^2& 1\\ b& b^2& 1\\ c& c^2& 1\end{array}\right|=abcV-V=(abc-1)V,$$
where $V=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=(a-b)(b-c)(c-a)$ (both determinants reduce to $V$ after moving columns, with an even number of swaps).

Since $a,b,c$ are distinct, $V\ne 0$, so $abc-1=0$, i.e. $abc=1$. The value $a=3$ does not change this.

$f(x)=|x-(-2)|+|x-3|+|x-5|$ is a sum of distances from $x$ to $-2,3,5$. Such a sum is minimised at the median point, $x=3$.

$$f(3)=0+5+2=7$$

Let a pen cost $p$ and a pencil $q$ (positive integers).

Second visit: 4 pens and 1 pencil, so $4p+q=112$, i.e. $q=112-4p$.

First visit: $2k$ pens and $3k$ pencils, so $k(2p+3q)=86$. Substituting, $2p+3q=336-10p$, which must be a divisor of 86: $1,2,43$ or $86$.

• $336-10p=86\Rightarrow p=25$ (integer), $q=12$, $k=1$.
• $43$, $2$, $1$ give non-integer $p$.

So $p=25$ and the pens on the second visit cost $4\times 25=\text{Rs }100$.

The thousands digit cannot be 0, so the units digit is 0 and the thousands digit is $7$.

The hundreds digit $h$ and tens digit $t$ satisfy $h-t=7$ and $ht=18$. Then $t(t+7)=18\Rightarrow t=2$, $h=9$.

The number is 7920.

Anyone passing all three also passes each pair, so the number passing all three is at most 20.

Check that 20 is achievable: let 20 pass all three and nobody pass exactly two. Then only M $=50-20=30$, only P $=30-20=10$, only C $=40-20=20$.

Total $=20+30+10+20=80$, which fits the class exactly and meets every pair limit. Maximum $=20$.

When the faster runner finishes 3000 m, the slower one has run $3000\times \frac{2}{3}=2000$ m.

The lead gained is $1000$ m $=\frac{1000}{300}=3\frac{1}{3}$ laps. Each full lap of lead is one overtaking, so the winner passes the other 3 times.

Choose $k$ of the identical objects (only one way for each $k$) and $4-k$ of the 9 distinct ones:

$$\sum _{k=0}^4\left(\genfrac{}{}{0ex}{}{9}{4-k}\right)=\left(\genfrac{}{}{0ex}{}{9}{4}\right)+\left(\genfrac{}{}{0ex}{}{9}{3}\right)+\left(\genfrac{}{}{0ex}{}{9}{2}\right)+\left(\genfrac{}{}{0ex}{}{9}{1}\right)+\left(\genfrac{}{}{0ex}{}{9}{0}\right)=126+84+36+9+1=256$$

$10^{19}=2^{19}{5}^{19}$ has $20\times 20=400$ factors $2^a{5}^b$ ($0\le a,b\le 19$).

A multiple of $10^{15}$ needs $15\le a,b\le 19$: $5\times 5=25$ factors.

$$P=\frac{25}{400}=\frac{1}{16}$$

Let the sides be $n-1,n,n+1$. The triangle is acute iff the square of the largest side is less than the sum of the other two squares:
$$(n+1{)}^2<(n-1{)}^2+n^2⟺n^2-4n>0⟺n>4.$$

Perimeter $3n\le 100\Rightarrow n\le 33$. So $n=5,6,\dots ,33$: 29 triangles.

If the line meets the axes at $(p,0)$ and $(0,q)$, the midpoint is $(p/2,q/2)=(-5,4)$, so $p=-10$, $q=8$.

$$\frac{x}{-10}+\frac{y}{8}=1\Rightarrow -4x+5y=40\Rightarrow 10y-8x=80$$

Use complementary and supplementary angles:

• $\cos \frac{3\pi }{8}=\sin \frac{\pi }{8}$, so ${\cos }^2\frac{\pi }{8}+{\cos }^2\frac{3\pi }{8}=1$.
• $\cos \frac{5\pi }{8}=-\cos \frac{3\pi }{8}$ and $\cos \frac{7\pi }{8}=-\cos \frac{\pi }{8}$, so the last two terms also sum to $1$.

$$\text{Sum}=1+1=2$$

Numerically: $0.8536+0.1464+0.1464+0.8536=2$. The printed expression evaluates to 2 (option C), not 1. The official key gives A (1), which does not match any reading of the printed question; for comparison, the sum of fourth powers would be $\frac{3}{2}$, also not 1.

The even terms are $\frac{1}{2^2}+\frac{1}{4^2}+\cdots =\frac{1}{4}\sum \frac{1}{n^2}=\frac{1}{4}\cdot \frac{{\pi }^2}{6}$.

$$\text{Odd terms}=\frac{{\pi }^2}{6}-\frac{{\pi }^2}{24}=\frac{3{\pi }^2}{24}=\frac{{\pi }^2}{8}$$

Use Bayes' theorem. He reports "five" if it is a five and he tells the truth, or if it is not a five and he lies.

• $P(\text{five and truthful})=\frac{1}{6}\cdot \frac{4}{5}=\frac{4}{30}$
• $P(\text{not five and lies})=\frac{5}{6}\cdot \frac{1}{5}=\frac{5}{30}$

$$P(\text{five}\mid \text{reported five})=\frac{4/30}{4/30+5/30}=\frac{4}{9}$$

${\log }_{5}y=0\Rightarrow y=1$, so ${\log }_8(x^2-1)=1\Rightarrow x^2-1=8\Rightarrow x=\pm 3$.

Among the options, $x=3$.

Note $(1+x)(1-x+x^2)=1+x^3$. For $-1<x<1$, $1+x>0$, so
$$1-x+x^2-\frac{1}{1+x}=\frac{1+x^3-1}{1+x}=\frac{x^3}{1+x}.$$

• $0<x<1$: $x^3>0$, so $\frac{1}{1+x}<1-x+x^2$. Statement (a) is true, (b) false.
• $-1<x<0$: $x^3<0$, so $\frac{1}{1+x}>1-x+x^2$. Statement (d) is true, (c) false.

The true pair is the 1st and 4th statements: (i) and (iv).

Water $=0.3(50)+0.2(80)=15+16=31$ L in $130$ L.

Adding $w$ litres of water: $\frac{31+w}{130+w}=\frac{1}{4}\Rightarrow 124+4w=130+w\Rightarrow w=2$.

Let the first worker take $t$ hours. Then the third takes $2t$ and the second $t+1$.

$$\frac{1}{t}+\frac{1}{2t}+\frac{1}{t+1}=1\Rightarrow \frac{3}{2t}+\frac{1}{t+1}=1\Rightarrow 3(t+1)+2t=2t(t+1)$$
$$2t^2-3t-3=0\Rightarrow t=\frac{3+\sqrt{33}}{4}(t>0).$$

Average time $=\frac{t+(t+1)+2t}{3}=\frac{4t+1}{3}=\frac{(3+\sqrt{33})+1}{3}=\frac{\sqrt{33}+4}{3}$.

Count numbers from 1 to 140 divisible by none of 2, 3, 5.

In every block of 30 there are $30\cdot \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{4}{5}=8$ such numbers. $140=4\times 30+20$ gives $4\times 8=32$.

In 121–140 (same pattern as 1–20): $1,7,11,13,17,19$, i.e. 6 more.

Total $=32+6=38$.

$f(3)=9+{\log }_{3}3=10$, so $g(3)=2(3)+10=16$.

Ordered fillings: $6\cdot 5\cdot 4\cdot 3=360$.

For $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ to be singular, $ad=bc$ with all four distinct. The only equal products of two disjoint pairs are $1\cdot 6=2\cdot 3$ and $2\cdot 6=3\cdot 4$.

For each such pair of pairs: choose which pair is the diagonal (2 ways) and order each pair (2 × 2), giving 8 matrices. Total $2\times 8=16$.

$$P=\frac{16}{360}=\frac{2}{45}$$

Let Ashok invest $x$ for 12 months and Bharat $x/2$ for $m$ months.

$$\frac{12x}{(x/2)m}=\frac{3}{1}\Rightarrow \frac{24}{m}=3\Rightarrow m=8.$$

Bharat was in for 8 months, so he joined after $12-8=4$ months.

Total $=6\times 64=384$. Arrange marks $a_1<a_2<\cdots <a_6$; we want $a_5-a_2$ as large as possible.

• Make $a_2$ small: $a_1=40$, $a_2=41$.
• The remaining four marks sum to $384-81=303$, and one of them is 70. Keep $a_3$ as small as possible: $a_3=42$, $a_4=70$.
• Then $a_5+a_6=303-42-70=191$ with $a_6>a_5$, so the largest $a_5=95$ ($a_6=96$).

Placing 70 as $a_5$ or $a_6$ only makes $a_5$ smaller. Maximum difference $=95-41=54$ (confirmed by exhaustive check).

The table's first day column is printed as "Sunday"; since the week's five trading days and the questions refer to Monday, it is read as Monday.

She buys at Monday's buy price and sells at Friday's sell price. Return per rupee:

• Dabur: $461/462-1=-0.22\%$
• Marico: $372/346-1=+7.51\%$
• HUL: $1966/1933-1=+1.71\%$
• ITC: $253/238-1=+6.30\%$
• Britannia: $3140/3046-1=+3.09\%$

Marico gives the highest return (about Rs 2.7 lakh on 10,404 shares, versus about Rs 2.27 lakh for ITC).

The table's first day column is printed as "Sunday"; since the week's five trading days and the questions refer to Monday, it is read as Monday.

HUL: buy at 1933 (Monday), sell at 1979 (Wednesday). Britannia: buy at 3115 (Wednesday), sell at 3140 (Friday).

$$\text{Growth factor}=\frac{1979}{1933}\times \frac{3140}{3115}=1.02380\times 1.00803\approx 1.0320$$

Return $\approx 3.20\%$ (the same to two decimals when whole shares are bought).

The table's first day column is printed as "Sunday"; since the week's five trading days and the questions refer to Monday, it is read as Monday.

Spread for each stock (absolute difference between buy and sell), in order Dabur, Marico, HUL, ITC, Britannia:

• Monday: $2+1+2+1+2=8$
• Tuesday: $3+19+3+1+1=27$ (Marico is printed as sell 355, buy 336)
• Wednesday: $1+3+2+5+5=16$
• Thursday: $3+3+4+4+2=16$
• Friday: $1+2+3+3+4=13$

With five stocks every day, the lowest total means the lowest average: Monday ($8/5=1.6$).

The table's first day column is printed as "Sunday"; since the week's five trading days and the questions refer to Monday, it is read as Monday.

Bought at Tuesday's buy price 3101: value $31,01,000$, fee $0.001\times 3,101,000=3101$.

Sold at Thursday's sell price 3025: value $30,25,000$, fee $3025$.

Total brokerage $=3101+3025=\text{Rs }6,126$.

The table's first day column is printed as "Sunday"; since the week's five trading days and the questions refer to Monday, it is read as Monday.

Bought at Monday's buy price 238. Selling prices on later days: Tuesday 238, Wednesday 246, Thursday 221, Friday 253.

The highest sell price is Friday's 253, so sell on Friday.

The passage says the address reflects the mood of his walk and shows that his eager interest in every aspect of the natural world had made him a "citizen of the universe". The address expresses a love of nature not tied to any one place, which option C captures.

• A: nothing suggests he lacked a permanent address; the address is symbolic.
• B and D: the passage does not say he wanted to travel or live outdoors all his life.

Muir writes that he usually had no great difficulty in finding a loaf of bread in the scattered clearings of the farmers. That is option A.

• B: he says he often went without supper or breakfast, so he did not eat regularly.
• C: sleeping without blankets is described as a hardship, not something easy.
• D: when the wolves howled he got up to build up his fire; he did not chase them.

Muir describes himself as feeding on God's inexhaustible "spiritual beauty bread" as he wandered, and speaks of the deep peace of the wilderness. Nature fed him spiritually, so A fits.

• B and C: the passage does not mention becoming a better botanist or collecting rare plants.
• D: "free as the winds" describes his freedom; it does not say he enjoyed the breeze.

The passage says the eye injury was less serious than he first thought. Within weeks he found he was "not very much disabled" for reading and botanizing. Later he wrote that his eye was neither better nor worse and that a cloud remained over it, but he often did not notice it while looking at wide landscapes. So his sight was affected, but he carried on. That is option C.

• A: the eye did not heal; it was "not better, nor worse".
• B: he only said he had little heart to open his good eye right after the accident, not that he could not open it during his travels.
• D (the key): he had spent weeks studying maps *before* the accident to plan his route. The passage does not present that as an effect of the injury. The injury only delayed his start.

Our reading supports C. The official key gives D.

To *botanize* means to study plants, especially where they grow in the wild. Muir uses the word for field work: he "botanized one week on the prairie". Option B is the dictionary sense.

• A and D add note-taking or classification, which the word does not require.
• C ties it to "his exploration", which is too narrow; he also botanized close to home while his eye healed.

Muir says a cloud is over his *eye*, not over the landscape. He adds that when gazing over wide landscapes he often does not notice it. So B is false, and it is the answer.

• A: he botanized on the Illinois prairies on his way home, which is true.
• C: he says all plants are more precious to him than before, which is true (read "commuted" as "committed").
• D: the volume comes from his notebook, which is true.

A *nest egg* is money saved over time for the future. That fits "built carefully over the years" and "lent all of it".

"Nests and eggs", "bird's nest" and "nestled eggs" are literal phrases or not idioms at all.

*Cut to the chase* means come straight to the point. That matches "declare outright" after being vague.

"Be curt to the face", "curb the farce" and "cut globe" are not idioms with that meaning.

*Steal someone's thunder* means taking the credit or attention that belongs to someone else, often by using their idea.

"Steal your credit" is not idiomatic; the usual phrase is "take credit". "Steal your crown" and "steal your gloating" do not fit.

*Bring someone up to speed* means give them the latest information, and it takes "on". The result reads "going to bring you up to speed on the latest developments", which is idiomatic and grammatical.

• A: "taking you through on" is ungrammatical.
• B: "going to share on" lacks an object and misuses "on".
• D: "going to be explaining on" is ungrammatical.

"Despite statistics to prove" sets up a contrast: he refused to accept the evidence. *In denial* means exactly that.

• A: "in affirmation" would agree with the statistics, so there is no contrast.
• C and D: "in protest" and "in shock" do not contrast with the proof.

*Throw light on* means explain or clarify, which is what a historian would be asked to do.

"Shed meaning on", "turn light on" and "focus meaning on" are not standard collocations. The idiom is "shed/throw light on".

Option B is direct and idiomatic: "studying the effects of X on Y".

• A and C are wordy and awkward ("what the effects ... would have", "what are the effects in").
• D reverses the cause and effect: morale does not affect salaries here.

The opening phrase "Besides offering physiological rewards" must describe what offers the rewards, which is karate. So the main clause must begin with *karate*. Only D does this.

• A starts with "one".
• B starts with "the body".
• C starts with "when karate is practiced", so the subject of the main clause is still "the body".

All three leave the modifier dangling.

*Not only ... but also* must join parallel parts: "not only on the plight of dogs, but also on that (the plight) of bulls". B keeps both halves parallel and uses "that" for the singular "plight".

• A misplaces "not only" and uses the plural "those" for "plight".
• C drops the "on" that "raised awareness" needs.
• D has "but also" with no "not only".

The blank sits between "new germs ... cause previously unknown diseases" and "They become ill more easily". C links the two: people have no immunity to a new germ, so *they* (the local people) fall ill more easily.

• A: going back and starting an epidemic has nothing to do with this.
• B: pollution is off topic.
• D: "enhancing conditions" contradicts the harm described.

The paragraph stresses a young man from a village using simple digital tools to show his creative side. D draws the natural conclusion: people with limited resources can now express their creativity.

• A adds an unsupported comparison with television, and "very popular ... than" is ungrammatical.
• B and C bring in making money, which the paragraph never discusses.

• Blank 1: coral reefs are famously *fragile* ecosystems.
• Blank 2: a *fascinating* paradox is a natural collocation.
• Blank 3: "*prolific* and productive" pairs two similar ideas, which suits the contrast with nutrient-poor water.

The other options fail on at least one blank:

• A: "common" is wrong, since reefs are not among the most common ecosystems.
• C: reefs are shallow, not "deep".
• D: an "unrecognised paradox" contradicts its being well known, and "wealthy communities" is a poor fit.

The contrast with war zones shows the first sentence names a blessing. It is a *privilege* to live without *bigotry*, while life in war zones is filled with *terror*.

• A: "coincidence ... adventures" is the wrong tone.
• C: "It is a lovely live" is ungrammatical.
• D: "punishment" reverses the meaning.

*Shape and guide* a team are natural leadership verbs, and disruptive technologies change the *competitive landscape*, which is a standard collocation.

• A: a "sustainable" landscape does not "shift dramatically" because of disruption.
• C: "drive a team" is weaker, and "tectonic landscape" is not a business collocation.
• D: "involve a team" is odd, and "green landscape" is irrelevant.

He wants to be elected *next year*, so he must have failed this time: *failing* to get elected. He then *returns* to his old job to improve his *chances* next time.

• A: "happening to get elected" contradicts the retry, and "quit to his job" is ungrammatical.
• C: "upgrade to his job" is odd, and elections have no "certainty".
• D: "give up to his job" is ungrammatical, and "increase his luck" is unidiomatic.

A collective fund is made of *pooled* accounts. The second blank needs a verb, because the sentence otherwise has none: the institution *groups* assets to *develop* one larger portfolio.

• A: "discounts" and "conceive" do not fit.
• B: "collected" cannot serve as the main verb after "The financial institution", and "simulate" is wrong.
• C: "covered" and "synthesise" are wrong.

*Urbane* means suave and courteous, and describes a person's manner. D needs *urban* (relating to a city).

A, B and C all describe people's refined manners correctly.

*Ingenious* means clever and inventive. The man who believed false promises was naive, which is *ingenuous*. C confuses the two words.

A (a clever solution), B (an inventive cook) and D (clever at finding excuses) use "ingenious" correctly.

An *ordinance* is a law or decree. Weapons and ammunition are *ordnance*. So C should say "ordnance".

A, B and D use "ordinance" correctly for a council, royal or city regulation.

*All together* means everyone in one place or at once. *Altogether* means entirely. B means "not entirely a bad thing", so it needs "altogether".

A, C and D all refer to people being together, so "all together" is correct there.

The noun is *after-effects*. "Affect" is a verb (or, in psychology, a noun meaning emotion). C should read "after-effects".

A and B use the verb *affect* correctly. In D, *affected* means artificial or put-on, which is also correct.

• 2 opens by defining an indie film.
• 4 follows with "such films", which refers back to indie films, and says they are low-budget and made by small companies.
• 3 adds "also", giving a further limitation (limited release).
• 1 turns with "However, they stand out..." after the list of limitations.

Order: 2-4-3-1. Option B (2412) ends with 2 and cannot work, since 2 is the definition. C puts "also" (3) before the first limitation.

• 3 is a general opening: there is more to human health than meets the eye.
• 4 follows with "This is particularly important in the Indian context", where only physical well-being is seen.
• 1 then names the hidden problem: mental illness still struggles for recognition.
• 2 backs it with "The data" on 197 million people.

Order: 3-4-1-2. Options C and D open with 2 or 4, whose "The data" and "This" need something before them. Option A puts the data before the problem it supports.

• 3 states the main idea: the planet's ability to cope is fraying.
• 5 follows with "This is what emerged at a recent conference", giving the source.
• 2 says what the papers at that conference showed: impacts worse than understood.
• 4 gives "For example", with sea-level rise and ocean warming.
• 1 draws the conclusion: major polluters have a moral obligation.

Order: 3-5-2-4-1. Option C (34521) puts the example (4) before the evidence it illustrates. A and B open with 2 or 1, which rely on context.