IPMAT Rohtak pattern paper 1

Solution

Equal roots need the discriminant to be zero: k² − 4ac = 0.

k² = 4 × 9 × 64 = 2304, so k = 48.

Solution

Expand along the first row.

= 0(25) − (4)(22) + (4)(17) = −20.

Solution

LCM(6, 9, 15) = 90; the number has the form 90k + 4.

The smallest such number with 3 digits is 184.

Solution

P(no head) = (12)³ = 18.

P(at least one head) = 1 − 18 = 78.

Solution

By alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper).

= (79 − 51) : (51 − 30) = 28 : 21 = 4 : 3.

Solution

Treat the vowels AO as one block: arrange 4 consonants and the block = 120 ways.

Arrange the vowels inside the block: 2 ways.

Total = 240.

Solution

Let each article cost Rs. 1. Selling 14 articles brings Rs. 30, so SP per article = 157.

Profit = 114 27%.

Solution

The total falls by 6 × 2 = 12 kg.

New person = 41 − 12 = 29 kg.

Solution

The roots are −2 and 7, so the new roots are −3 and 6.

Sum = 3, product = −18: the equation is x² − 3x − 18 = 0.

Solution

Interest = 1 × principal over 25 years.

Rate = 1 × 10025 = 4%.

Solution

Equal roots need the discriminant to be zero: k² − 4ac = 0.

k² = 4 × 1 × 36 = 144, so k = 12.

Solution

Fix one person to remove rotations; arrange the rest.

(6 − 1)! = 120.

Solution

tan 30° = 75/d, so d = 75/tan 30°.

tan 30° = 1/√3, so d = 75√3 m.

Solution

Adding all three: 2(x + y + z) = 38, so x + y + z = 19.

z = 19 − 18 = 1, x = 19 − 8 = 11, y = 19 − 12 = 7.

xyz = 11 × 7 × 1 = 77.

Solution

Write each log with a common base: log_p q = log q / log p.

The product is (log 6 / log 2) × (log 7 / log 6) × (log 4 / log 7) = log 4 / log 2.

That is log₂ 4 = 2.

Solution

LCM(6, 8, 12) = 24; the number has the form 24k + 2.

The smallest such number with 3 digits is 122.

Solution

Each remainder is one less than the divisor, so n + 1 is a multiple of each.

LCM = 60, so n + 1 is a multiple of 60.

The required number is 179.

Solution

The perpendicular from the centre bisects the chord: half-chord = √(13² − 5²) = 12.

Chord = 2 × 12 = 24 cm.

Solution

Hypotenuse = 10. The hypotenuse is a diameter of the circumcircle.

R = 102 = 5 cm.

Solution

Write each log with a common base: log_p q = log q / log p.

The product is (log 6 / log 3) × (log 13 / log 6) × (log 81 / log 13) = log 81 / log 3.

That is log₃ 81 = 4.

Solution

C(12, 3) = 220 triples in all.

Collinear triples make no triangle: C(3, 3) = 1.

Triangles = 219.

Solution

n(A ∪ B ∪ C) = 12 + 38 + 14 − 7 − 5 − 5 + 2 = 49.

Exactly two = Σ(pairs) − 3(all three) = 17 − 6 = 11.

Solution

Total subsets = 2⁵ = 32.

Remove the empty set and the 5 single-element subsets: 32 − 1 − 5 = 26.

Solution

Area of path = outer rectangle − inner rectangle.

= 1984 − 1800 = 184 sq m.

Solution

2^{x + 2} = 4 × 2^x, so the left side is (4 + 1) × 2^x = 5 × 2^x.

2^x = 40 / 5 = 8 = 2³, so x = 3.

Solution

540 = 2² × 3³ × 5.

Sum of factors = (1 + 2 + 2²) × (1 + 3 + 3² + 3³) × (1 + 5) = 1680.

Solution

The radius to the point of contact is perpendicular to the tangent.

Tangent = √(20² − 12²) = 16 cm.

Solution

36 = 6², so log₆ 36 = 2.

Similarly the other two logs are 2 and 3.

Value = 2 − 2 + 3 = 3.

Solution

g(3) = 8, so f(g(3)) = 62.

f(3) = 7, so g(f(3)) = 16.

Difference = 62 − (16) = 46.

Solution

g(4) = 20, so f(g(4)) = 397.

f(4) = 13, so g(f(4)) = 56.

Difference = 397 − (56) = 341.

Solution

V = πr^2h = (227) × 14² × 6 = 3696.

Solution

Multiply the factors: 2320 × 65 = 6950.

Net change = 38% (increase).

Solution

Area = (12)|x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|.

= (12)|0(5) + −5(−6) + 4(1)| = 17.

Solution

a + b = 26 and ab = 25.

They are the roots of t² − 26t + 25 = 0: 1 and 25.

The larger is 25.

Solution

n(A ∪ B ∪ C) = 44 + 19 + 35 − 7 − 10 − 9 + 2 = 74.

Exactly two = Σ(pairs) − 3(all three) = 26 − 6 = 20.

Solution

g(0) = −1, so f(g(0)) = −1.

f(0) = −2, so g(f(0)) = −7.

Difference = −1 − (−7) = 6.

Solution

The total rises by 13 × 1 = 13 kg.

New person = 44 + 13 = 57 kg.

Solution

The remainder equals the value at x = −1: −7 + k = −3.

k = 4.

Solution

Let the number be 10a + b. Reversing changes it by 9(a − b), so |a − b| = 4.

The reversed number is larger, so the tens digit is the smaller one: a − b = −4.

With a + b = 10: a = 3, b = 7. The number is 37.

Solution

Let each article cost Rs. 1. Selling 27 articles brings Rs. 30, so SP per article = 109.

Profit = 11 19%.

Solution

It loses 4 minutes in 24 hours, so in 36 hours it loses 4 × 3624 = 6 minutes.

Solution

Check every group of 4: 15 groups in all.

The groups that meet every condition are: PRSU, QRTU, RSTU.

So 3 teams are possible.

Solution

Each term is 3 times the previous one.

? = 27 × 3 = 81.

Solution

The pattern is n, n², n³: 6-36-216.

For 4: 4-16-64.

Solution

GAD → FCI: the letters are reversed and moved 2 places forward.

So HEO → QGJ.

Solution

Days from 1 January to 18 December = 352 (leap year).

352 ÷ 7 leaves 2, so the day is Sunday + 2 = Tuesday.

Solution

'blue' is common to the first two sentences, so its code is the one common to their codes: ta.

'dark' is common to the second and third: ka.

The remaining word in the second sentence is 'eat', so its code is ru.

Solution

Each letter of MILK moves 3 places forward to give PLON.

Doing the same to SUN: VXQ.

Solution

The letters are swapped in pairs: NA TI ON → AN IT NO.

So MIRROR → IMRRRO.

Solution

Start from F: B is F's son. G is B's brother. C is G's son.

So C is F's grandson.

Solution

Each term is 2 times the previous one.

? = 6 × 2 = 12.

Solution

Differences: 5, 7, 9, 11, 13, increasing by 2.

Next = 68 + 15 = 83.

Solution

EAGLE, ORANGE, OFFICE begin with a vowel.

MOUSE begins with a consonant.

Solution

100 years have 76 ordinary + 24 leap years = 76 + 48 = 124 odd days → 5 odd days. So 100 → 5, 200 → 3, 300 → 1, 400 → 0 (the 400th year is a leap year).

So 200 years have 3 odd days.

Solution

For a mirror image, subtract the time from 11:60.

11:60 − 10:15 = 1:45.

Solution

Days from 1 January to 12 April = 101.

101 ÷ 7 leaves 3, so the day is Friday + 3 = Monday.

Solution

Letters in odd positions move 1 forward and letters in even positions move 1 back: FOREST → GNSDTS.

So PLANET → QKBMFS.

Solution

The hour hand turns 30° per hour, that is 0.5° per minute.

285 minutes × 0.5° = 142.5°.

Solution

'No walls being stones is a possibility.' does not follow: the statements force 'some walls are stones'.

'No bricks being stones is a possibility.' follows: a diagram where no bricks are stones is consistent with the statements.

Solution

Check every group of 4: 35 groups in all.

The groups that meet every condition are: PQRS, PQRT, PQST, PQSU, PQTU, PQUV, QRST, QSTU, QSUV.

So 9 teams are possible.

Solution

Draw the family tree: Ravi + Kavita → Nikhil (married to Tanvi) → Ajay, Mohan.

Answer: brother.

Solution

Each letter moves 3 places backward.

M − 3 = J.

Solution

Within a group letters rise by 2; each group starts 3 letters after the previous one.

Next: SUW.

Solution

8-81, 5-36, 7-64 follow (n + 1)².

3-19 does not: (n + 1)² for 3 gives 16.

Solution

Check every group of 3: 20 groups in all.

The groups that meet every condition are: PQT, PRT, PRU, PST, QRT, QRU, QST, RTU.

So 8 teams are possible.

Solution

40 m North → 40 m East → 60 m South → 25 m West.

Net movement: 15 m East and 20 m South.

So Varun is South-East of the start.

Solution

The others are kinds of memory.

Solution

Students who passed = 12 + 32 − 1 = 43.

Class strength = 43 + 2 + 4 = 49.

Solution

Facing the centre, a person's right is the next seat anticlockwise and the left is the next seat clockwise.

The clues fix the clockwise order as Q → V → W → U → S → P (and back to Q).

So the answer is V.

Solution

Total = 29 + 20 − 1 = 48.

Solution

The gaps grow by one: +1, +2, +3, +4.

Next gap +5: P → U.

Solution

Combining the chain gives G > C, so 'G ≥ C' is definitely true.

Solution

Decode: K % P # B % E means K = P ≤ B = E.

Combining the statements gives P ≤ B, so 'P = B' is not definitely true.

Combining the statements gives P ≤ B, so 'P > B' is not definitely true.

Solution

Start from C: A is C's brother. M is A's daughter. K is M's brother.

So K is C's nephew.

Solution

50 m North → 50 m West → 10 m North.

Net movement: 50 m West and 60 m North.

So Priya is North-West of the start.

Solution

Only one schedule fits all the clues: Mon – Q, Tue – S, Wed – R, Thu – P, Fri – V.

So Q lectures on Monday.

Solution

In the evening the sun is in the West, so shadows fall towards the East.

Facing North, the East is to his right.

Solution

'Some teachers are not singers.' follows: no possible diagram allows 'all teachers are singers', so it must be true.

'Some teachers are doctors.' follows: no possible diagram allows 'no teachers are doctors', so it must be true.

Solution

With >, =, ≥: D > T = S ≥ E, which gives D > E, so 'D ≥ E' holds.

With ≥, =, ≤: D ≥ T = S ≤ E does not fix 'D ≥ E'.

With ≤, ≥, =: D ≤ T ≥ S = E does not fix 'D ≥ E'.

With >, ≤, >: D > T ≤ S > E does not fix 'D ≥ E'.

Solution

I alone allows (2, 1), (4, 2), (6, 3), (8, 4), (10, 5), …: the answer is always Yes.

II alone allows (2, 1), (3, 1), (3, 2), (4, 1), (4, 2), …: the answer is always Yes.

Answer: Each statement alone is sufficient.

Solution

Pernicious describes something that causes harm in a gradual or hidden way.

Solution

Illegible: that which cannot be read.

Solution

'Explain' takes 'something to someone', not an indirect object without 'to'.

Solution

Past perfect → 'had been + past participle'.

Solution

'Cut corners' means: do something cheaply or carelessly to save time or money.

Solution

Hostile means unfriendly; amicable means friendly.

Solution

'Keep one's head above water' means: manage to survive, especially financially.

Solution

After 'one', use 'one's': 'one's promises'.

Solution

The order is 3-4-2-1:

3. Rivers in the Himalaya are fed by both monsoon rain and melting glaciers.

4. In the dry months before the monsoon, meltwater keeps them flowing.

2. As glaciers shrink, this summer flow is expected first to rise and then to fall.

1. Farmers downstream who depend on that water will need to plan for the change.

The first sentence introduces the topic; each following sentence picks up a word or idea from the one before it.

Solution

Profits rising sharply have soared. Soured would mean turned bad.

Solution

A nonchalant person appears calm and casually unconcerned.

Solution

'Neither' is singular, so 'is' is right. No error.

Solution

The order is 2-3-4-1:

2. Vaccines train the immune system without causing the disease itself.

3. They do this by presenting a harmless part or a weakened form of the germ.

4. The body responds by producing antibodies and memory cells.

1. If the real germ arrives later, the body recognises it and responds quickly.

The first sentence introduces the topic; each following sentence picks up a word or idea from the one before it.

Solution

Millennium: a period of a thousand years.

Solution

'In the same boat' means: in the same difficult situation.

Solution

'Scissors' is a plural noun and takes 'are'.

Solution

To expand is to grow larger; to contract is to grow smaller.

Solution

Conciliatory words or actions are meant to end anger or a quarrel.

Solution

The subject is 'the teacher' (singular); 'along with her students' does not change it. No error.

Solution

Vague means not clearly expressed; precise means exact.

Solution

A point in time with an action continuing to now uses the present perfect (continuous) with 'since'.

Solution

The order is 3-4-2-1:

3. When a new product is launched, the price is one of the hardest decisions.

4. Set it too high, and few people try the product.

2. Set it too low, and buyers may assume it is of poor quality.

1. Many firms therefore test several prices in small markets before choosing one.

The first sentence introduces the topic; each following sentence picks up a word or idea from the one before it.

Solution

Evidence too weak to convict is flimsy.

Solution

Belligerent means aggressive or ready to fight.

Solution

The verb agrees with the first subject, 'Rahul', when 'as well as' joins them.

Solution

The relative pronoun 'who' refers to 'people', so the verb is plural.

Solution

An imperative becomes 'Let + object + be + past participle'.

Solution

A lenient teacher punishes mildly; a strict one does not tolerate rule-breaking.

Solution

A negative imperative becomes 'Let + object + not be + past participle' (or 'Let not … be').

Solution

'Discuss' takes a direct object: 'We discussed the plan'.

Solution

Somnambulist: a person who walks in their sleep.

Solution

'The tip of the iceberg' means: a small visible part of a much larger problem.

Solution

The adjective safe is needed after 'is no longer'.

Solution

Mint: a place where coins are made.

Solution

With two objects, either can become the subject; 'given' is the past participle.

Solution

The second paragraph describes him talking to farmers for weeks before proposing a channel.

Solution

This is stated directly in the second paragraph.

Solution

The passage says his life is 'drawn here from several careers of the period'.

Solution

The engineer argued that engineering is as much about listening as about calculation.

Solution

The writer presents his method and views favourably and calls them a lesson for modern planners.