IIFT 23rd Dec 2021 Slot 1

$$6-9\log_8\left(\frac{4}{x}\right)^{\frac{1}{3}}-8\left(\log_{256}x\right)^{\frac{2}{3}}-\left(\log_{2\ }x^8\right)^{\frac{1}{3}}=0$$

$$6-\log_24+\log_2\ x-2\left(\log_2x\right)^{\frac{2}{3}}-2\left(\log_2x\right)^{\frac{1}{3}}=0$$

Let $$\left(\log_2x\right)^{\frac{1}{3}}$$ be t.

$$4+t^3-2t^2-2t=0$$

or, $$\left(t-2\right)\left(t^2-2\right)=0$$

so, $$t=2$$ or

$$\log_2\ x=8$$ 

Height of triangle = $$\left(10\sqrt{3}\right)\times\ \left(\frac{\sqrt{3}}{2}\right)=15$$

Distance between the centroid of the triangle and the vertex= $$15\times\ \frac{2}{3}=10$$

Since in-radius is one-third of the height of the triangle

therefore, length of the line from vertex to the point where small circle and big circle touch each other = $$2\times3=6$$

Radius of the bigger circle = 10 - 6 = 4cm

Ratio of areas will be square of the ration of radius which is 4:1.

Since the question asks for a possible location of the ground, the most efficient way to solve such questions is to go by the options.

Option A:

Distance of point $$\left(-1.25+200\sqrt{2},\ 8.75\right)$$ from lines y-x=10 and x+y=15/2

Distance from y-x=10 is $$\frac{\left|1.25-200\sqrt{2}+8.75-10\right|}{\sqrt{\ 1}+\sqrt{\ 1}}=200$$

Distance from x+y=15/2 is $$\frac{\left|-1.25+200\sqrt{2}+8.75-7.5\right|}{\sqrt{\ 1}+\sqrt{\ 1}}=200$$

Since, in both cases, we are getting a distance equal to 200 units(equidistant from both roads), this is the correct option.

Thus, the correct option is A.

Let rate of consumption per hour be R

Therefore, $$R=k\left(V^2\right)$$

When speed= 40km/hr R= 1200kg/hr

So, When speed = V

R= $$\left(\frac{1200}{1600}\right)\times\ V^2$$

Cost of coal consumed per hour = $$\left(\frac{1200}{1600}\right)\times\ V^2\times\ \frac{16}{100}=\frac{12}{100}V^2$$

All other expenses = Rs 13/ hr

Therefore operating cost per hour = Rs.($$13+\frac{12}{100}V^2$$)

Operation cos in paise per hour = $$1300+12V^2$$

Let the amount of olive oil in the jar be 1000 units.

After Lata drew 20% and replaced it with groundnut oil, the amount of olive oil in the mixture will be 800 units, and that of groundnut oil will be 200 units.

Now, Usha will draw 10% of the volume; thus, the amount of olive oil left will be $$800\times\ 0.9\ =\ 720$$. Similarly, the amount of groundnut oil will be 180 units, and the amount of mustard oil will be 100 units.

Again after Lata's draw, the amount of olive oil will become $$720\times\ 0.8=576$$. Similarly, the amount of mustard oil will become 80 units, and the amount of groundnut oil will become  $$180\times\ 0.8+200=344$$ units.

Lastly, Usha will again draw 10% of the volume. Thus, the volume of olive oil will become $$576\times\ 0.9=518.4$$. Similarly, the amount of groundnut oil will become 309.6, and the amount of mustard oil will become $$80\times\ 0.9+100\ =\ 172$$

Thus, the ratio will be 5184 : 3096 : 1720

The correct option is B.

Alternative solution:

Percentage of olive oil in the mixture = $$100\times\ \left(\frac{8}{10}\right)\times\ \left(\frac{9}{10}\right)\times\ \left(\frac{8}{10}\right)\times\ \left(\frac{9}{10}\right)=51.84\%$$. 

Option B: Don't guess is in itself a full sentence and should be followed by a full stop. Thus, this is grammatically incorrect.

Option C: Since comma should follow sank, this is also grammatically incorrect.

Options A and D are grammatically correct sentences.

Thus, the correct option is A.

A procrustean bed: a scheme or pattern into which someone or something is arbitrarily forced.

Paper is more patient than Man:  it easier to be honest in writing than to speak openly with other people.

Scrape acquaintance with: Make an effort to become familiar with someone, especially for one's own benefit.

Bare Bodkin: a mere dagger.

Thus, the correct combination is given in option D.

The correct option is D.

Options A and C have sentences with wrong punctuation, and hence,

The correct option is B.

A. A small area of ground that is slightly higher than the ground around it: Hummock

B. A waterbody separated from a large water body by a reef: Lagoon

C. A narrow water body that connects 2 large water bodies: Straits

D. Small island: Islet

E. Ring-shaped coral reef or island: Atoll

F. A land bridge that connects two larger land masses: Isthmus

G. A group of islands: Archipelago

H. Rocky islands: Skerry

I. River island: Ait

Since only option D has the right sequence,

The correct option is D.

A. A small area of ground that is slightly higher than the ground around it: Hummock

B. A waterbody separated from a large water body by a reef: Lagoon

C. A narrow water body that connects 2 large water bodies: Straits

D. Small island: Islet

E. Ring-shaped coral reef or island: Atoll

F. A land bridge that connects two larger land masses: Isthmus

G. A group of islands: Archipelago

H. Rocky islands: Skerry

I. River island: Ait

Since only B has the correct sequence,

The correct option is B.