[Download PDF] CAT 2001 Question Paper Paper with Solutions

Instructions

For the following questions answer them individually

Question 11

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, 3 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?

Video Solution

Solution

B is the south gate. A is the house and BC = 9 km.

In triangle ABC and triangle AMO

ABC = OMA = 90 degree

BAC = OAM = Common angle

Triangle ABC is similar to triangle AMO.

$$\frac{(2r+3)}{\sqrt{(r+3)^2-r^2}} = \frac{9}{r}$$

r = 9/2

Question 12

In the given diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the areas of CEF and that of the rectangle?

Video Solution

Solution

Let the length of AB be 3X and the length of AD (and BC) be Y.

As the length of AB = 3X, AE = EF = FB =X

So, the area of the rectangle ABCD is length * breadth = 3X * Y = 3XY

The area of triangle CEF = 1/2 * base * height = 1/2 * EF * BC = 1/2*X*Y

So, required ratio = 1/2 : 3 = 1:6

Question 13

A can complete a piece of work in 4 days. B takes double the time taken by A, C takes double that of B, and D takes double that of C to complete the same task. They are paired in groups of two each. One pair takes two-thirds the time needed by the second pair to complete the work. Which is the first pair?

Video Solution

Solution

A takes 4 days to complete the work.
So, B takes 8 days to complete the same work.
C takes 16 days to complete the work.
D takes 32 days to complete the same work.

In order to measure, let the total work be of 64 units. Hence, the speed of working of each of the four persons is given below.

A - 16 units/hr
B - 8 units/hr
C - 4 units/hr
D - 2 units/hr

From the given options, we need to find two pairs in such a way that their speeds are in the ratio 3:2. Note that A+D=18 while B+C=12 and the ratio is 3:2

Hence, the first pair is A and D and the second pair is B and C

Question 14

In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?

Video Solution

Solution

Let the 4 digit no. be xyzw.
According to given conditions we have x+y=z+w --- Eq 1 , x+w=z --- Eq 2,y+w=2x+2z --- Eq 3
Eq 2 - Eq 3 : x-y = -2x-z --- Eq 4
Eq 1+ Eq 4 :2x = -2x+w
4x=w --- Eq 5
Substitute w = 4x inEq2
5x = z
Substitute w = 4x inEq3
y+4x=2x+10x
y = 8x
Now the minimum value x can take is 1 so z =5 and the no. is 1854, which satisfies all the conditions.
Hence option A .

Question 15

Two men X and Y started working for a certain company at similar jobs on January 1, 1950. X asked for an initial monthly salary of Rs. 300 with an annual increment of Rs. 30. Y asked for an initial monthly salary of Rs. 200 with a rise of Rs. 15 every 6 months. Assume that the arrangements remained unaltered till December 31, 1959. Salary is paid on the last day of the month. What is the total amount paid to them as salary during the period?

Video Solution

Solution

January 1, 1950 to December 31, 1959 is a period of 10 years or 20 half years.
The person X after 1st year gets Rs. 300 in next year he gets Rs. 330 and so on.
So his earning is in AP with 10 300+330+360+...

Similarly earning of Y is in AP with 20 terms 200+215+230+245.... .

So, the total earnings of X equals 12*(300+330+....10 terms) = 52200
The total earnings of Y equals 6*(200+215+230+...20 terms) = 41100

So, the total earnings of the two equals 52200+41100 = 93300

Question 16

Anita had to do a multiplication. In stead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

Video Solution

Solution

Let the number be X.
From the given information, 53x - 35x = 540 => 18x = 540 => x = 30

So, new product = 35*30 = 1050

Question 17

A college has raised 75% of the amount it needs for a new building by receiving an average donation of Rs. 600 from the people already solicited. The people already solicited represent 60% of the people the college will ask for donations. If the college is to raise exactly the amount needed for the new building, what should be the average donation from the remaining people to be solicited?

Video Solution

Solution

Let there be total 100 people whom the college will ask for donation. Out of these 60 people have already given average donation of 600 Rs. Thus total amount generated by 60 people is 36000. This is 75% of total amount required . so the amount remaining is 12000 which should be generated from remaining 40 people. So average amount needed is 12000/40 = 300

Question 18

x and y are real numbers satisfying the conditions 2 < x < 3 and - 8 < y < -7. Which of the following expressions will have the least value?

Video Solution

Solution

$$xy^2$$ will have it's least value when y=-7 and x=2 and equals 98.
So $$xy^2>98$$

$$x^2y$$ will have it's least value when y=-8 and x=3 and equals -72.
So, $$x^2y > -72$$

$$5xy$$ will have it's least value when y=-8 and x=3 and equals -120
So, $$5xy > -120$$

So, of the three expressions, the least possible value is that of 5xy

Question 19

$$m$$ is the smallest positive integer such that for any integer $$n \geq m$$, the quantity $$n^3 - 7n^2 + 11n - 5$$ is positive. What is the value of $$m$$?

Video Solution

Solution

$$n^3 - 7n^2 + 11n - 5 = (n-1)(n^2 - 6n +5) = (n-1)(n-1)(n-5)$$
This is positive for n > 5
So, m = 6

Question 20

A ladder leans against a vertical wall. The top of the ladder is 8 m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?

Video Solution

Solution

When the ladder is moved 2 m away from the wall, the top of the ladder rests against the foot of the wall. So, the hypotenuse is 2 m more than the second side of the right triangle. The other non-hypotenuse side is 8 m. So, $$8^2 + x^2 = (x+2)^2$$

=> x = 15 m and length of the ladder = x+2 = 17 m

CAT Content

CAT 2001 Question Paper

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, 3 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?

In the given diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the areas of CEF and that of the rectangle?

In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?

Anita had to do a multiplication. In stead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

x and y are real numbers satisfying the conditions 2 < x < 3 and - 8 < y < -7. Which of the following expressions will have the least value?

$$m$$ is the smallest positive integer such that for any integer $$n \geq m$$, the quantity $$n^3 - 7n^2 + 11n - 5$$ is positive. What is the value of $$m$$?

A ladder leans against a vertical wall. The top of the ladder is 8 m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?

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CAT Content

CAT 2001 Question Paper

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, 3 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?

In the given diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the areas of CEF and that of the rectangle?

In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?

Anita had to do a multiplication. In stead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

x and y are real numbers satisfying the conditions 2 < x < 3 and - 8 < y < -7. Which of the following expressions will have the least value?

$$m$$ is the smallest positive integer such that for any integer $$n \geq m$$, the quantity $$n^3 - 7n^2 + 11n - 5$$ is positive. What is the value of $$m$$?

A ladder leans against a vertical wall. The top of the ladder is 8 m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?

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