[Download PDF] MAT 2007 Question Paper Paper with Solutions

Instructions

For the following questions answer them individually

Question 71

Three pipes are made of different shapes. The cross-sections of the pipes are an equilateral triangle, a hexagon and a circle. The perimeter of each of these cross-sections is equal. The flow through the pipes is proportional to the area of cross section. If it takes 8 minutes for the triangular pipe to fill up the tank, what will be the difference in the times taken by the hexagonal and circular pipes?

Video Solution

Solution

Let the sides of triangle be a, that of hexagon be b and circle be c.

Given:

3a = 6b = 2πr

Given: flow rate is proportional to area, so, Flow rate(F) = k Area

Area of triangle = √3/4 a^2

Area of hexagon = 3√3/8 a^2

Area of circle = πr^2 = 9a^2/4π

If triangular pipe takes 480 seconds, hexagonal and circular pipe will take 320 and 290 seconds.

Required difference = 30 seconds = 0.5 minutes.

Question 72

The Qutab Minar casts a shadow 150 m long at the same time when the Vikas Minar casts a shadow 120 m long on the ground. If the height of the Vikas Minar is 80 m, find the height of the Qutab Minar.

Video Solution

Solution

The triangles formed by Qutub and Vikas minar will be similar, so

Let x be the height of the Qutub minar

150/120 = x/80

On solving, we get

x = 100 m

Question 73

Due to global recession starting in January, Ram’s monthly salary of Rs. 8000 was cut by 10%. The monthly expenses, which were Rs. 6000, increased at the rate of 5% per month. Since which month will he have no savings if the recession lasted for a year ?

Video Solution

Solution

Jan:

Rams's monthly salary in Jan = 0.9 * 8000 = 7200

Expenses = 6000

Feb:

Expenses = 6000*105/100 = 6300

March:

Expenses = 6300*105/100 = 6615

April:

Expenses = 6615*105/100 = 6945.75

May:

Expenses = 6945.75*105/100 > 7200

So, The month starting will be May in which he have no savings if the recession lasted for a year.

Question 74

An oil refinery takes 100 litres of crude oil as input and after refining for 1 hour gives certain amount of output oil X litres. This can be sold in the market at a profit of Rs 30 per litre. If this oil is further refined for $$\frac{1}{2}$$ hour it gives oil Y litres. This can be sold at a profit of Rs. 50 per litre. Output and input ratio at both the stages is 90%. The maximum amount that can be earned from 1000 litres of crude input is

Video Solution

Solution

Case 1:

Input = 1000L

Output = 1000*.9 = 900L

Profit = 900*30 = 27000

Case 2:

Input = 900L

Output = 900*.9 = 810L

Profit = 810*50 = 40500 rupees.

Question 75

A mixture (40 litres) contains tonic and water in the ratio 3 : 1 to make the ratio 2 : 1, how much additional amount of water is required?

Video Solution

Solution

Mixture of tonic and water:

Tonic = 30 l

Water = 10l

Now,

quantity of water = 10 + x (let x be added)

10 + x = 1/3 * (40 + x)

Solving for x, we get

x = 5 litres.

Question 76

If 11,109,999 is divided by 1111, then what is the remainder?

Video Solution

Solution

Since we can see that 1111*1000

= 1111000 , which is greater than 11109999, so we will see for a lesser quotient.

On simple dividing, we get quotient is 9999 and remainder is 1110.

Question 77

If an angle of a triangle remains unchanged but each of its two including side is doubled, then by what factor does the area get multiplied ?

Video Solution

Solution

Let the area of triangle be A.

A = 1/2*b*c *sin A (b,c are sides which have included angle A)

b,c is doubled and A remains same

A' (new area) = 1/2*2b*2c*sinA

A' = 2bcsinA = 4A

New Area becomes 4 times of previous one.

Question 78

A number when divided by 765 leaves a remainder 42. What will be the remainder if the number is divided by 17 ?

Video Solution

Solution

Let the number be N.

N = 765k + 42

Let k = 1, so N = 807

When 807 is divided by 17, remainder = 8

Question 79

If m and n are natural numbers such that $$ 2^{m}-2^{n}=960$$, what is the value of m ?

Video Solution

Solution

Since 2^10 is 1024, so we will look for value of m to be greater and equal to 10.

Let m = 10, then

1024-960 = 64, [equation given]

Since 64 is of the form 2^m, then

so n = 6, satisfies the equation.

Hence A

Question 80

The radius of a cylindrical cistern is 10 metres and its height is 15 metres. Initially the cistern is empty, we start filling the cistern with water through a pipe whose diameter is 50 cm, water is coming out of the pipe with a velocity of 5 m/s. How many minutes will it take in filling the cistern with water?

Video Solution

Solution

Since Flow rate = Volume = Area * height

Volume of cylinder = Area of pipe * velocity * time

time = πr^2h/πR^2 * v

t = π* 10^2*15/π (0.25)^2*5

t = 4800 seconds = 80 minutes

CAT Content

MAT 2007 Question Paper

The Qutab Minar casts a shadow 150 m long at the same time when the Vikas Minar casts a shadow 120 m long on the ground. If the height of the Vikas Minar is 80 m, find the height of the Qutab Minar.

Due to global recession starting in January, Ram’s monthly salary of Rs. 8000 was cut by 10%. The monthly expenses, which were Rs. 6000, increased at the rate of 5% per month. Since which month will he have no savings if the recession lasted for a year ?

A mixture (40 litres) contains tonic and water in the ratio 3 : 1 to make the ratio 2 : 1, how much additional amount of water is required?

If 11,109,999 is divided by 1111, then what is the remainder?

If an angle of a triangle remains unchanged but each of its two including side is doubled, then by what factor does the area get multiplied ?

A number when divided by 765 leaves a remainder 42. What will be the remainder if the number is divided by 17 ?

If m and n are natural numbers such that $$ 2^{m}-2^{n}=960$$, what is the value of m ?

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