MAT 2006 Question Paper

As per the given question,

Given, time taken for the angle of depression to change from $$30^{\circ}$$ to $$40^{\circ}$$ is 12 minutes, so BC = 12s and CD = st

where, s = speed and t = time 

In triangle ABC,

tan 45$$^{\circ} = \frac{AB}{12s}$$ (or) AB = st

In triangle ABD,

tan 30$$^{\circ} = \frac{AB}{BD}$$

$$\frac{1}{\sqrt{3}} = \frac{AB}{s(12 + t)}$$

$$\frac{1}{\sqrt{3}} = \frac{st}{s(12 + t)}$$   ($$\because$$ AB = st)

$$12 + t - \sqrt{3}t = 0$$ (or) $$ 12 - 0.73t = 0$$ (or) t = 16.23 (Approx)

Hence, option B is the correct answer.

Total marks obtained by all the students in 1st section = 40 x 50 = 2000.

Total marks obtained by all the students in 2nd section = 35 x 60 = 2100.

Total marks obtained by all students in 3rd section = 45 x 55 = 2475.

Total marks obtained by all students in 4th section = 42 x 45 = 1890.

Overall average of marks per student = Total marks of students of all sections/ Total number of students of all sections

$$\Rightarrow$$ (2000 + 2100 + 2475 + 1890)/ (162) = 8465/162 = 52.25

Hence, option B is the correct answer.

This problem can be solved easily by allegation method. 

Ratio of boys to girls,

$$\Rightarrow$$ 1.2 : 0.8 = 3:2

Hence, option C is the correct answer.

To calculate the maximum possible price of an item, take price of other items as minimum as possible.

Given that exactly 10 products sells for less than 1000 and no product should be less than 420.

So take the price of 10 items as 420 rs each (minimum possible price)

$$\therefore$$ Price of these 10 items will be 420 x 10 = 4200.

Now, remaining 14 items (except the high priced item) should be 1000 rs each (minimum possible price)

$$\therefore$$ Price of these 14 items will be 14 x 1000 = 14,000

And price of 24 items = 4,200 + 14,000 = 18,200 .....(1)

Given average retail price of 25 products = 1,200 (or) Total price of all items = 25 x 1,200 = 30,000

Now, the greatest possible price of the most expensive product is given by,

Total price of all items - minimum possible price of 24 items (i.e equation (1))

$$\Rightarrow$$ 30,000 - 18,200 = 11,800 

Hence, option D is the correct answer.

Let, the amount of water drained be 'x'

Then total amount of water left in sink = x - 6

$$\therefore$$ x + x - 6 = 12

2x - 6 = 12 (or) x = 9L

Hence, option D is the correct answer.

The chemist has 10 litres of solution of which 10% is nitric acid i.e remaining 90% is water.

Now he wants to dilute it to 4% from 10% by adding water.

Let x be the total water added. Then 4% of nitric acid should become 4% of total solution after adding 'x' litres of water.

Now equation can be written as,

1L = 4 % of (10 + x)

25 = 10 + x (or) x = 15L

Hence, option A is the correct answer.

Let, 'x' and '2x' be the number of goods in first shipment and second shipment respectively. 

Defective pieces in 1st shipment is '1%' and 2nd shipment is 4.5%

Total defective pieces is 100 i.e 

1% of x + 4.5% of x = 100

10% of x = 100 (or) x = 1000.

Total ball bearings in the first bearing = 1000

Hence, option C is the correct answer.

Given,

Total population of the district = 2,96,000

Total number of males = 1,66,000 and females = 1,30,000

Total number of literates = 50 % of 2,96,000 = 1,48,000

70% of males are literates i.e 0.07 x 1,66,000 = 1,16,000 are literates

Now, total number of literate woman are,

(Total number of literates) - ( Total number of male literates)

$$\Rightarrow$$ 1,48,000 - 1,16,000 = 31,800.

Hence, option B is the correct answer.

Speed of first train = 180/4 = 45 kmph.

Speed of second train = 180/3 = 60 kmph.

First train covers 45 km in 1 hour and second train covers 60 km in 1 hour. 

Difference between them = (60 - 45)km = 15km.

Hence, option D is the correct answer.