IPMAT Indore pattern paper 1

Solution

(43)π × 18³ = π × 2² × h.

h = 4 × 5832/(3 × 4) = 1944 cm.

Solution

A sum of distances |x − p| is least when x is at the median of the points.

The points are −8, −6, 2, 3; the minimum is reached for any x between −6 and 2.

Value = 2 + 0 + 8 + 9 = 19.

Solution

Put y = |x|: y² − 8y + 12 = 0, so y = 2 or 6.

Then x = ±6 or ±2: four real roots.

Sum of squares = 2(36 + 4) = 80.

Solution

A number is divisible by 11 when the alternating sum of its digits is divisible by 11.

Only x = 9 works: 10989 ÷ 11 = 999.

Solution

A matrix is singular when its determinant is 0.

The determinant works out to −36k + 36.

−36k + 36 = 0 gives k = 1.

Solution

Distance = |a x0 + b y0 + c|/√(a² + b²) = |3(−4) + 4(3) − 10|/√(9 + 16).

= |−10|/5 = 2.

Solution

Complete the squares: (x − 1)² + (y + 1)² = 1 + 1 + 62 = 64.

Radius = √64 = 8.

Solution

The log is defined only for x − 3 > 0, that is x > 3.

The base 12 is less than 1, so the inequality reverses: x − 3 < (12)⁻³ = 8.

So 3 < x < 11, which holds for 7 integers (4 to 10).

Solution

Rationalise each term: 1/(√k + √(k+1)) = √(k+1) − √k.

The sum telescopes: (√2 − √1) + (√3 − √2) + … + (√81 − √80) = √81 − 1.

√81 − 1 = 9 − 1 = 8.

Solution

The points with |x| + |y| = k number 4k for k ≥ 1, plus the origin for k = 0.

Total = 1 + 4(1 + 2 + … + 3) = 1 + 4 × 6 = 25.

Solution

Alphabetical order of the letters: A, C, T.

Words that come before it: 1 × 2! = 2 (position 1: letters before C among ACT).

Rank = 2 + 1 = 3.

Solution

Multiplying a 2 × 2 matrix by 3 multiplies each of its 2 rows by 3.

det(3A) = 3² × −2 = −18.

Solution

Each period of 3 years multiplies the sum by 2.

8 = 2³, so it takes 3 × 3 = 9 years.

Solution

Inclusion–exclusion: 333 + 200 + 142 − 66 − 28 − 47 + 9 = 543 are divisible by at least one.

Divisible by none: 1000 − 543 = 457.

Solution

The square root needs (x + 2)(x − 5) ≥ 0, true for x ≤ −2 or x ≥ 5.

The integers strictly between −2 and 5 are excluded: 6 of them.

Solution

The roots are −9 and −6, so the new roots are −8 and −5.

Sum = −13, product = 40: the equation is x² + 13x + 40 = 0.

Solution

If the roots are integers m and n, then mn = 18 and k = −(m + n).

Factor pairs of 18: (1, 18), (2, 9), (3, 6). Each pair can be both positive or both negative.

The sums are ±19, ±11, ±9: 6 distinct values of k.

Solution

d = (10 − 24)/(4 − 11) = 2; first term a = 10 − 3 × 2 = 4.

S₁₃ = (132)[2 × 4 + 12 × 2] = 208.

Solution

n(A ∪ B ∪ C) = 52 + 49 + 28 − 13 − 18 − 10 + 6 = 94.

None = 102 − 94 = 8.

Solution

Each path is an arrangement of 7 R's and 3 U's.

C(10, 7) = 120.

Solution

Relative speed = 18 km/h = 5 m/s.

Distance = 210 + 130 = 340 m.

Time = 340 ÷ 5 = 68 s.

Solution

81 = 3⁴, so 81³⁴ = 3³ = 27.

8 = 2³, so 8¹³ = 2¹ = 2.

Result: 27 ÷ 2 = 272.

Solution

The log is defined only for x − 5 > 0, that is x > 5.

The base 12 is less than 1, so the inequality reverses: x − 5 < (12)⁻⁴ = 16.

So 5 < x < 21, which holds for 15 integers (6 to 20).

Solution

Rationalise each term: 1/(√k + √(k+1)) = √(k+1) − √k.

The sum telescopes: (√2 − √1) + (√3 − √2) + … + (√121 − √120) = √121 − 1.

√121 − 1 = 11 − 1 = 10.

Solution

Area ∝ r²: (54)² = 2516.

Increase = 56.25%.

Solution

By AM ≥ GM: x + 25/x ≥ 2√(25) = 10.

Equality holds when x = 25/x, that is x = 5.

Minimum value = 10.

Solution

5 sin θ − 12 cos θ has minimum −√(25 + 144) = −13.

Minimum = 15 − 13 = 2.

Solution

Take the job as 84 units: A does 7 and B 12 units a day.

Each 2-day pair finishes 19 units. Continue day by day until 84 units are done.

The job is done in 9 112 days.

Solution

Let a book cost x and a notebook cost y: 4x + 5y = 141 and 5x + 3y = 147.

Eliminating y: x = (141 × 3 − 147 × 5) / (4 × 3 − 5 × 5) = −312/-13 = 24.

Then y = 9.

Solution

Multiply the factors: 75 × 1720 = 119100.

Net change = 19% (increase).

Solution

150 = 2 × 3 × 5².

Number of factors = (1 + 1) × (1 + 1) × (2 + 1) = 12.

Solution

sin θ = 817.

cosec θ = 178, cot θ = 158; difference = 14.

Solution

r² = 819 = 9, so r = 3; first term = 99 = 1.

S₇ = 1((3)⁷ − 1)/(3 − 1) = 1093.

Solution

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

Tangent = √(25² − 7²) = 24 cm.

Solution

16 ≡ 5 (mod 11).

5⁵ ≡ 1 (mod 11), so powers repeat every 5.

242 = 5 × 48 + 2, so 16²⁴² ≡ 5² ≡ 3 (mod 11).

Solution

f(10) = f(1) + 1(1 + 2 + … + 9).

1 + 2 + … + 9 = 45.

f(10) = 4 + 45 = 49.

Solution

P(both miss) = 35 × 34 = 920.

P(hit) = 1 − 920 = 1120.

Solution

A matrix is singular when its determinant is 0.

The determinant works out to −6k + 6.

−6k + 6 = 0 gives k = 1.

Solution

log 2⁵⁴ = 54 × log 2 = 54 × 0.3010 = 16.2540.

A number whose log has integer part k has k + 1 digits.

The integer part is 16, so 2⁵⁴ has 17 digits.

Solution

√(3 − |x|) needs |x| ≤ 3, that is −3 ≤ x ≤ 3.

1/√(x + 2) needs x + 2 > 0, that is x > −2.

Together: −2 < x ≤ 3, which holds for 5 integers.

Solution

A² = 1362413.

Sum of entries = 56.

Solution

Treat the vowels EE as one block: arrange 4 consonants and the block = 20 ways.

Arrange the vowels inside the block: 1 ways.

Total = 20.

Solution

A only: 25 × 12; B only: 12 × 35.

Sum = 12.

Solution

Each time, the milk is multiplied by (40 − 20)/40 = 12.

Milk left = 40 × (12)³ = 5 litres.

Solution

For 2 years, CI − SI = P(r/100)² = 8000 × (5100)² = 20.

Solution

For two people use the comparative with 'the': 'the taller of the two'.

Solution

'Affect' is usually a verb. The noun needed in the last sentence is 'effect'.

Solution

Separate, distinct categories are 'discrete'. 'Discreet' means careful not to attract attention.

Solution

'Bury the hatchet' means: make peace and end a quarrel.

Solution

A period of time uses 'for' with the present perfect.

Solution

'Prefer' takes 'to': 'prefer tea to coffee'.

Solution

To prohibit is to forbid; to permit is to allow.

Solution

Record crowds every day suggest a very great success: resounding.

Solution

The order is 3-2-4-1:

3. Online reviews have become a powerful influence on what people buy.

2. Most shoppers read them, and many trust them as much as advice from friends.

4. This trust has created an incentive for sellers to post fake reviews.

1. Platforms now use software to detect suspicious patterns, such as many five-star reviews written on the same day.

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

Solution

Scrupulous means very careful to do what is right or accurate.

Solution

Versatile means able to do many different things; limited is the opposite.

Solution

'Unless' means 'if not', so no extra 'not'; the present tense is used for a future condition.

Solution

The order is 3-2-1-4:

3. The printing press made books far cheaper than before.

2. As a result, many more people could own and read them.

1. This spread of reading helped new ideas travel quickly across countries.

4. Some historians argue that it changed religion, science and politics within a century.

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

Solution

Waste is discharged (released) into water. Discard means throw away; dispense means distribute.

Solution

Abundant means existing in large quantities; scarce means in short supply.

Solution

A candid remark is honest and direct.

Solution

'News' is singular: 'was broadcast'.

Solution

Something about to happen is 'imminent'. 'Eminent' means famous and respected.

Solution

Immaculate means perfectly clean; soiled means dirty.

Solution

Obsolete means no longer used because something newer exists.

Solution

Ubiquitous means found everywhere.

Solution

A false appearance is an 'illusion'. An allusion is an indirect reference.

Solution

'Hit the sack' means: go to bed.

Solution

'Up in arms' means: very angry and protesting.

Solution

'Both' is plural: 'are good singers'.

Solution

'Play second fiddle' means: take a less important role than someone else.

Solution

To go forward is to 'proceed'. 'Precede' means to come before.

Solution

The order is 3-4-2-1:

3. A good presentation begins long before the speaker stands up.

4. It begins with a clear answer to one question: what should the audience remember?

2. Every slide and every example is then chosen to support that single point.

1. Anything that does not serve it, however interesting, is left out.

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

Solution

'Police' is plural: 'The police have arrested'.

Solution

Zeal is great energy and enthusiasm; apathy is a lack of interest.

Solution

Unreal conditions use the subjunctive 'were' for all persons.

Solution

The order is 1-4-2-3:

1. A sleep-deprived student often believes she is coping well.

4. Tests, however, show that her reaction time and memory have both declined.

2. The problem is that tiredness also weakens our ability to judge our own performance.

3. That is why the effects of poor sleep are so often underestimated.

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

Solution

'Although' sets up a contrast: it seemed hard (daunting) but was finished quickly.

Solution

An eloquent speaker expresses ideas fluently and persuasively.

Solution

'Return' already means 'come back'; 'back' is redundant.

Solution

The passage argues that brief, unstructured time can aid original thinking, and that filling every gap may remove it.

Solution

It is the dull task given to participants before the creative test.

Solution

The third paragraph separates short unstructured time (possibly useful) from long, enforced boredom (linked to stress).

Solution

Enforced boredom is boredom that people cannot escape, which is imposed on them.

Solution

The second paragraph gives this explanation.

Solution

The first paragraph defines sticky prices as those that stay the same for months despite cost changes.

Solution

The passage never mentions government rules; the other three are stated in the second paragraph.

Solution

It is how some firms deal with higher costs while keeping the price fixed.

Solution

The final paragraph links slow price adjustment to the delayed effect of policy, which is why banks look ahead.

Solution

The writer explains an economic idea calmly, with reasons and consequences.