# CAT Model Paper 3 Questions and Answers with Explanation Part 5

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Q: 26. Ravi has recently bought an android phone. It has a screen lock which operates in the following way: there are 9 dots arranged in a square formation. Unfortunately, after setting the lock, he has forgotten it. The only thing he remembers is that to unlock, one has to tap 4 dots in the correct sequence.

What is the probability that he will open the lock in the 37^{th} trial given that he does not repeat any failed trials?

(A)

(B)

(C)

(D) None of these

Ans: D

Solution:

The total number of sequences possible is .

The probability of succeeding at the first trial is .

In the nth trial (assuming the first trials are unsuccessful) , he will not try out the options tried out in the previous trials. Hence, the number of sequences from which he chooses one at random is 3024 – (n - 1) or n. Therefore, the probability of success in the nth trial alone is The probability of success in the nth trial having failed in the previous trials is .

Hence, the probability of success in any trial is

Option (D)

Q: 27. How many A. P. s of 3 terms can be formed from the set of the 1^{st} 40 natural numbers?

(A)

(B)

(C)

(D) None of these

Ans: C

Solution:

All A. P. s must have integral common difference (given that all terms are natural numbers) .

Every triplet will give us 2 series, one increasing and the other decreasing e. g. &

Number of series with common difference 1, e. g. , (1^{st} term can be anything from 1 to 38)

Number of series with common difference (1^{st} term can be anything from 1 to 36)

Number of series with common difference (1^{st} term can be anything from 1 to 34)

Number of series with common difference 19 (highest common difference)

So, total number of series = .

Q: 28. Mamata borrowed a total of ₹ from two persons to be repaid at the end of two years, with the interest being compounded annually. She repaid ₹ 31,100 more to the first person when compared to the second at the end of two years. The first person՚s rate of interest was 10 percentage points more than that of second. Instead, if Mamata borrowed equal amounts from them at the same rates, she would have paid ₹ more to the first. Find actually how much was borrowed from the first person.

(A)

(B)

(C)

(D)

Ans: C

Solution:

Let the interest rate charged by first and second person be

respectively.

When Mamata borrows equal amounts from the two persons at these rates, the difference in n amount repaid is due to difference in interest calculated and equal to , i.e.. ,

Solving the equation above, we get r

Now let us consider that the loans taken from the first and second persons were x and respectively.

We are given that the difference between on x and

Formulating the equation as above and solving, we get .

Q: 29. Triangle is isosceles and has sides units. A line is so drawn such that A lies on PR and B lies on QR and is distinct from Q and it divides the triangle into two parts having equal area as well as equal perimeter. If units. Find AR.

(A) units

(B) units

(C) units

(D) units

Ans: B

Solution

Consider the figure given below

We have, let then . Perimeter of triangle units. Further, since the perimeters of ABR and APQB are the same and AB is common to both, . Since units.

Now area of triangle area of triangle ARB (given that the areas of and are equal) .

Thus or,

Solving, we get that , then B would coincide with Q which contradicts the condition given in the problem. Hence as given in option (B) .

Q: 30. Let us consider a set of distinct positive numbers. From that set, all possible distinct combinations of 99 numbers are selected and their averages computed as is the average of the complete set of averages so obtained. If Y be the average of the original set of 100 numbers, then which of the following is definitely true regarding the relationship between X and Y?

(A)

(B)

(C)

(D) Depends on the 100 numbers selected.

Ans: C

Solution:

Let be any number present within the original set of numbers. Out of the possible combinations, this number definitely belongs to distinct combinations. Thus if the sum of the original numbers is S, then the sum of the sums of each of the individual sets of numbers would be . Then, it is easy to see that , where Si is sum of the combination of 99 numbers. Dividing both sides by 99, we obtain , ; which implies that the average of the complete set of averages obtained would also be the average of the original numbers. Hence as given in option (3) .

Alternate explanation for those who find the verbal reasoning difficult: If Ai is the average of all numbers excluding the number (Ni) and Y is the original average of the complete set of numbers,

or

=

= Y