Instructions

For the following questions answer them individually

Question 61

ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

Video Solution

Solution

The values of 1!, 2!, 3!, 4!, 5!, 6! and 7! are 1, 2, 6, 24, 120, 720 and 5040 respectively.

So, the digits must lie from 1 to 6 only to satisfy the conditions.

6 cannot be one of the digits as the at least one digit in the final number is more than 6.

145 = 1! + 4! + 5!

Question 62

The adjoining figure shows a set of concentric squares. If the diagonal of the innermost square is 2 units, and if the distance between the corresponding corners of any two successive squares is 1 unit, find the difference between the areas of the eighth and the seventh squares, counting from the innermost square.

Video Solution

Solution

Diagonal of 8th square will be = 16
Side of 8th square = $$\frac{16}{\sqrt2}$$
Diagonal of 7th square will be = 14
Side of 7th square = $$\frac{14}{\sqrt2}$$

Difference in areas = $$(\frac{16}{\sqrt2})^2 - (\frac{14}{\sqrt2})^2$$ = 30

Question 63

A, B and C are defined as follows.
A=$$( 2.000004) \div ((2.000004)^2+ 4.000008)$$ ;
B = $$(3.000003) \div ((3.000003)^2+9.000009)$$
C= $$(4.000002) \div ((4.000002)^2 + 8.000004)$$
Which of the following is true about the values of the above three expressions?

Video Solution

Solution

Given expressions can be reduced as follows

A = $$\frac{1}{4.000004}$$

B = $$\frac{1}{6.000003}$$

C = $$\frac{1}{6.000002}$$

Among all of them B is smallest.

Question 64

The value of each of a set of coins varies as the square of its diameter, if its thickness remains constant, and it varies as the thickness, if the diameter remains constant. If the diameter of two coins are in the ratio 4 : 3, what should be the ratio of their thickness' be if the value of the first is four times that of the second?

Video Solution

Solution

Value of coin = $$k (2r)^2 t$$ (where k is proportionality constant, 2r is diameter and t is thickness)
So (value of first coin) = 4 (value of second coin)

$$k (2r_1)^2 t_1 = 4 \times (k(2r_2)^2 t_2)$$

or $$\frac{t_1}{t_2} = \frac{9}{4}$$ (As ratio of diameters 2r will be 9:4)

Question 65

In ABC, points P, Q and R are the mid-points of sides AB, BC and CA respectively. If area of ABC is 20 sq. units, find the area of PQR.

Video Solution

Solution

As we know, the triangle joining midpoints of sides will divide it in 4 similar traingles of equal area.

So area will be = $$\frac{20}{4} = 5$$

Question 66

In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter to the longer side?

Video Solution

Solution

Consider rectangle with shorter side as $$y$$, longer side as $$x$$ and diagonal as $$z$$.

So $$(x+y) - z = \frac{x}{2}$$ => x/2+y=z

Where $$z^2=x^2 + y^2$$

So, $$(x/2 +y)^2$$ = z

squaring on both sides and after putting value of z and solving we can get ratio of y to x as 3:4

Instructions

The Weirdo Holiday Resort follows a particular system of holidays for its employees. People are given holidays on the days where the first letter of the day of the week is the same as the first letter of their names. All employees work at the same rate.

Question 67

Raja starts working on February 25(Sunday), 1996, and finishes the job on March 2, 1996. How much time would T and J take to finish the same job if both start on the same day as Raja?

Video Solution

Solution

Raja takes 7 days to complete the job.

Feb 25, 1996 is a Sunday.

If T and J start working on Sunday, they can complete the work by wednesday because T would have worked for 3 days and J would worked for 4 days, thereby matching the number of days worked by Raja.

Hence, they can complete the job in 4 days.

Question 68

Starting on February 25, 1996 (Sunday), if Raja had finished his job on April 2, 1996, when would T and S together likely to have completed the job, had they started on the same day as Raja?

Video Solution

Solution

The number of days taken by Raja to complete the work is 5+31+2 = 38
So, cumulative number of days needed by T and S to complete the work is 38.

Both of them take two days off in a week as S takes of on Saturday and Sunday and T takes off on Tuesday and Thursday.

So, total number of man-working days per week by the duo is 10.
Hence, after three weeks, they finish 30 man working days.
i.e by end of 17th March 1996 (Sunday), 30 man working days are finished.

Both of them work on Monday,
S works on Tuesday
Both of them work on Wednesday
S works on Thursday
Both of them work on Friday and the remaining 8 man working days are also over.

Hence, the required date is 17+5 = 22 March 1996 (Friday)

Instructions

Boston is 4 hr ahead of Frankfurt and 2 hr behind India. X leaves Frankfurt at 6 p.m. on Friday and reaches Boston the next day. After waiting there for 2 hr, he leaves exactly at noon and reaches India at 1 a.m. On his return journey, he takes the same route as before, but halts at Boston for 1hr less than his previous halt there. He then proceeds to Frankfurt.

Question 69

If his journey, including stoppage, is covered at an average speed of 180 mph during the journey from Frankfurt to India, what is the distance between Frankfurt and India?

Video Solution

Solution

X leaves Frankfurt at 6 PM and reaches Boston at 10 AM, which is 6 AM in Frankfurt => 12-hour journey

Leaves Boston at 12 PM ad reaches India at 1 AM, which is 11 PM in Boston => 11-hour journey

=> Total time = 12 + 2 + 11 = 25 hours

Average speed = 180 mph

=> Distance = 25 * 180 = 4500 miles

Question 70

If X had started the return journey from India at 2.55 a.m. on the same day that he reached there, after how much time would he reach Frankfurt?

Video Solution

Solution

X leaves Frankfurt at 6 PM and reaches Boston at 10 AM, which is 6 AM in Frankfurt => 12-hour journey

Leaves Boston at 12 PM ad reaches India at 1 AM, which is 11 PM in Boston => 11-hour journey

=> Total time = 12 + 2 + 11 = 25 hours

Return journey is 1 hour lesser => 25 - 1 = 24 hours.

CAT Content

CAT 1997 Question Paper

ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

A, B and C are defined as follows.
A=$$( 2.000004) \div ((2.000004)^2+ 4.000008)$$ ;
B = $$(3.000003) \div ((3.000003)^2+9.000009)$$
C= $$(4.000002) \div ((4.000002)^2 + 8.000004)$$
Which of the following is true about the values of the above three expressions?

In ABC, points P, Q and R are the mid-points of sides AB, BC and CA respectively. If area of ABC is 20 sq. units, find the area of PQR.

In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter to the longer side?

Raja starts working on February 25(Sunday), 1996, and finishes the job on March 2, 1996. How much time would T and J take to finish the same job if both start on the same day as Raja?

Starting on February 25, 1996 (Sunday), if Raja had finished his job on April 2, 1996, when would T and S together likely to have completed the job, had they started on the same day as Raja?

If his journey, including stoppage, is covered at an average speed of 180 mph during the journey from Frankfurt to India, what is the distance between Frankfurt and India?

If X had started the return journey from India at 2.55 a.m. on the same day that he reached there, after how much time would he reach Frankfurt?

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CAT Content

CAT 1997 Question Paper

ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

A, B and C are defined as follows.A=$$( 2.000004) \div ((2.000004)^2+ 4.000008)$$ ;B = $$(3.000003) \div ((3.000003)^2+9.000009)$$C= $$(4.000002) \div ((4.000002)^2 + 8.000004)$$ Which of the following is true about the values of the above three expressions?

In ABC, points P, Q and R are the mid-points of sides AB, BC and CA respectively. If area of ABC is 20 sq. units, find the area of PQR.

In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter to the longer side?

Raja starts working on February 25(Sunday), 1996, and finishes the job on March 2, 1996. How much time would T and J take to finish the same job if both start on the same day as Raja?

Starting on February 25, 1996 (Sunday), if Raja had finished his job on April 2, 1996, when would T and S together likely to have completed the job, had they started on the same day as Raja?

If his journey, including stoppage, is covered at an average speed of 180 mph during the journey from Frankfurt to India, what is the distance between Frankfurt and India?

If X had started the return journey from India at 2.55 a.m. on the same day that he reached there, after how much time would he reach Frankfurt?

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A, B and C are defined as follows.
A=$$( 2.000004) \div ((2.000004)^2+ 4.000008)$$ ;
B = $$(3.000003) \div ((3.000003)^2+9.000009)$$
C= $$(4.000002) \div ((4.000002)^2 + 8.000004)$$
Which of the following is true about the values of the above three expressions?