IPMAT Indore · Paper 1

Correct option A·9

Only the last digit of each base matters, so look at $3^{85}$, $5^{37}$ and $7^{96}$.

Unit digits of powers of 3 cycle as 3, 9, 7, 1 and powers of 7 as 7, 9, 3, 1 (cycle length 4).

$85 = 4 \times 21 + 1$, so $3^{85}$ ends in 3. $96$ is a multiple of 4, so $7^{96}$ ends in 1. Any power of 5 ends in 5.

Unit digit $= 3 - 5 + 1 = -1$. The whole number is positive (the $987^{96}$ term is by far the biggest), so a $-1$ means a unit digit of $10 - 1 = 9$.

Correct option B·$3 - x$

Dividing by a quadratic leaves a linear remainder, so write $p(x) = (x-1)(x-2)\,q(x) + ax + b$.

Put $x = 1$: $a + b = 2$. Put $x = 2$: $2a + b = 1$.

Subtracting gives $a = -1$, so $b = 3$. The remainder is $3 - x$.

Correct option A·96

Divisible by 6, 7 and 9 means divisible by $\text{LCM}(6, 7, 9) = 126$.

Filter with 9 first: the digits of 4530 add to 12, so the two new digits must add to 6 or 15. That rules out 54 (sum 9).

Now test 7 on the rest: $453096 = 126 \times 3596$ works, while $453078$ and $453042$ leave a remainder when divided by 7.

Correct option C·$[-15, 9]$

Write everything in terms of $\cos x$: $3\sin^2 x = 3 - 3\cos^2 x$.

So $p = 3 - 3\cos^2 x + 12\cos x - 3 = 12c - 3c^2$, where $c = \cos x \in [-1, 1]$.

$f(c) = -3c^2 + 12c$ has its vertex at $c = 2$, outside $[-1, 1]$, so $f$ keeps increasing across the whole interval.

Minimum $= f(-1) = -15$, maximum $= f(1) = 9$. So $p \in [-15, 9]$.

Correct option B·1130

Let $x$ go into A and $10000 - x$ into B.

A: 5% per half-year for two half-years, value $x(1.05)^2 = 1.1025x$. B: value $1.12(10000 - x)$.

$1.12(10000 - x) - 1.1025x = 2310 \Rightarrow 2.2225x = 8890 \Rightarrow x = 4000$.

Interest: A earns $4000 \times 0.1025 = 410$, B earns $6000 \times 0.12 = 720$. Total $= 1130$.

Correct option C·2520

The odd places are the 1st, 3rd and 5th positions, and they must hold odd digits (1, 3, 5, 7, 9).

Fill them first: $5 \times 4 \times 3 = 60$ ways. The first digit is odd, so it can never be 0.

The 2nd and 4th places can take any of the remaining $10 - 3 = 7$ digits: $7 \times 6 = 42$ ways.

Total $= 60 \times 42 = 2520$.

Correct option D·$\dfrac{5}{2}$

Simplify each piece: $\log_7 4 = 2\log_7 2$, $\log_7 25 = 2\log_7 5$, $\log_7 8 - \log_7 4 = \log_7 2$, and $\log_7 5 - \log_7 2 = \log_7 \frac{5}{2}$.

So $a = \dfrac{2\log_7 2 \cdot \log_7 \frac{5}{2}}{2\log_7 5 \cdot \log_7 2} = \dfrac{\log_7 \frac{5}{2}}{\log_7 5} = \log_5 \frac{5}{2}$.

Hence $5^a = 5^{\log_5 (5/2)} = \frac{5}{2}$.

Correct option C·210

Let apples $= a$, bananas $= a + 10$, oranges $= r$.

Fruit sold: $0.7a + 0.6(a + 10) + 0.5r = 0.55(2a + 10 + r)$, which simplifies to $0.05r = 0.2a + 0.5$, i.e. $r = 4a + 10$.

Fruit is sold whole, so $0.7a$ must be a whole number: $a$ is a multiple of 10.

Also $2a + 10$ must be a multiple of 11. For $a = 10, 20, 30, 40, 50$ we get $30, 50, 70, 90, 110$, so the first fit is $a = 50$.

Minimum oranges $= 4(50) + 10 = 210$.

Correct option B·under the weather

"Under the weather" is the fixed idiom for feeling slightly ill, which suits someone with a fever skipping a party.

Idioms can't be rebuilt with a different preposition: "beneath", "beside" and "outside the weather" are not English expressions, however close they look.

Correct option A·off-the-cuff

"Off-the-cuff" means said on the spot, without preparation, which is exactly what a rough estimate is.

"Off-the-mark" means inaccurate, so the speaker would be calling their own figure wrong. "Off-the-grid" means cut off from public utilities or society, and "off-the-mike" is not the idiom.

The clue is "rough estimate": the speaker is giving a quick, unrehearsed figure.

Correct option D·loses his keys

"All the time" signals a habit, and the simple present describes habits: "he loses his keys all the time".

"Has loosed" is wrong twice: wrong tense, and "loose" (not tight) is a different word from "lose" (misplace). A repeats the spelling trap with "loosing", B ("has loses") is ungrammatical, and C ("was losing") clashes with the present "is very careless".

Correct option C·They were all soaking wet and their mother told them to ring their clothes first and then put them out to dry.

In C the mother means squeezing the water out of wet clothes. That verb is "wring", not "ring": "wring their clothes".

The others are correct: a "ring" of forces around the house (a circle), "ring them up" (phone them), and a "ring" worn on the finger.

Correct option C·to have it appraised

To "appraise" something is to have an expert estimate its value, which is what you do with jewellery of unknown worth.

"Apprised" is the look-alike trap, but "apprise" means to inform someone, so "have it apprised" makes no sense. "Appreciated" means valued or risen in value, not assessed, and "get an approval" misses the point.

Memory hook: apPRAISE = assess the price; apPRISE = inform.

Correct option D·silver bullet

A "silver bullet" is a single, simple fix that solves a complex problem completely. The sentence says renewables are key but not enough on their own, so "not a silver bullet" fits exactly.

"Silver lining" (a hopeful side to a bad situation) is the tempting trap, but it doesn't fit the contrast with "key solution". A "red herring" is a distraction and a "dead ringer" is a look-alike.

Correct option B·Had the President known that his allies would let him down

The main clause "he would have handled" is about the past, so the if-clause needs the past perfect (third conditional).

"Had the President known" is the inverted form of "If the President had known" and pairs correctly with "would have handled".

A looks similar, but there "had" is the main verb: it means "If the President had knowledge", a past simple (second conditional) form. The past perfect would be "had had knowledge", so A does not pair with "would have handled".

C ("could know") and D ("knew ... can") also miss the past perfect and mix tenses.

Correct option C·that she pay for the meal.

"Insist" takes "insist on doing" or "insist that someone do" (the subjunctive, with the bare verb). It never takes "insist to do".

So "she always insisted that she pay for the meal" is correct, even though "pay" looks odd without an s.

A ("had paid") and B ("will be paying") break the tense pattern, and D ("insisted on and paid") leaves "on" hanging with no object.

Correct option B·compromise; impair; counter

Stress harms the immune system, worsens well-being, and doctors want to fight those effects, so the blanks run: negative, negative, opposing.

"Compromise" the immune system, "impair" well-being, "counter" the effects fits all three.

A fails on "elevate" (stress doesn't raise well-being), C on "improve", and D on "preserve": each wrong option has one blank that flips the meaning.

Correct option B·precedent; mitigating; incidence

"Without precedent in history" is the standard phrase for something never seen before. "Containing and mitigating the spread" pairs two ways of limiting it, and "reducing the incidence" means lowering the number of new cases.

D is the trap: "without parallel" works, but "reducing the outbreak of the disease" is unidiomatic. A's "without similarity" is not English, and C's "eradicating the spread" doesn't sit with "containing".

Correct option D·ubiquitous; immortalised

The first sentence lists alleyways, cafes, warehouses and barbershops, so the portraits are everywhere: "ubiquitous".

The figures are "held up as symbols of a glorious past", so the portraits preserve them for ever: "immortalised".

A's "rare" contradicts that long list of places, and "confounded" or "dispersed" don't describe faces kept alive on a wall.

Correct option C·She interviewed many teens to collaborate her thesis on their mental health issues.

In C the writer means interviews that support or confirm a thesis. That word is "corroborate", not "collaborate" (to work together).

The other sentences use "collaborate" correctly: refusing to cooperate, siding with an occupying power, and a policeman working with film makers.