CAT 2017 – Slot 2 – Quantitative Ability – If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4)

Q. 1: If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

log (2^a. 3^b. 5^c) = [log (2^2.3^3.5) + log (2^6.3.5^7) + log (2.3^2.5^4)]/3

3 * log (2^a. 3^b. 5^c) = log (2^9.3^6.5^12)

log (2^a. 3^b. 5^c)^3 = log (2^9.3^6.5^12)

log (2^3a. 3^3b. 5^3c) = log (2^9.3^6.5^12)

3a = 9

a=3

Answer: 3

 

Q. 2: Let a1, a2, a3, a4, a5 be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with 2a3.
If the sum of the numbers in the new sequence is 450, then a5 is

5 consecutive odd numbers are a1 , a2 , a3 , a4 , a5.

5 consecutive even numbers are 2a3 – 8, 2a3 – 6, 2a3 – 4, 2a3 – 2, 2a3

Sum of these 5 numbers = 10a3 – 20 = 450

a3 = 47 and a5 = 51.

 

Q. 3:How many different pairs (a, b) of positive integers are there such that a ≤ b and 1/a + 1/b = 1/9

9(a + b) = ab

ab – 9a – 9b + 81 = 81

(a – 9) (b – 9) = 81 = 34

As a, b > 0 and a ≤ b, there are only 3 ordered pairs, given by a – 9 = 1, 3 or 9 and correspondingly b – 9 = 81, 27, 9.

We have to make sure that we satisfy the condition, a≤b

These are the following pairs of (a,b) that satisfy the condition

(a,b) = (10,90), (12, 36), (18,18)

Answer: 3

 

Q. 4: In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens?

Number of pens that Amal gets = a+1

Number of pens that Bimal gets = b+2

Number of pens that Kamal gets = k+3

So, (a+1) + (b+2) + (k+3) = 8

a + b + k = 2

So, we get (2 + 3 – 1)C(3-1) = 6 [Based on the formula, (n+r-1)C(r-1)]

Answer: 6

Q. 5: How many four digit numbers, which are divisible by 6, can be formed using the digits 0, 2, 3, 4, 6, such that no digit is used more than once and 0 does not occur in the left-most position?

Correct Answer:- 50

Explanation:- The sum of the digits must be a multiple of 3. We can use (A) 2,4,0,3 or (B) 2,4,0,6 or (C) 2,4,3,6

(A) _ _ _ 0 (6 numbers)

_ _ _ 2 (4 numbers)

_ _ _ 4 (4 numbers)

(B) _ _ _ 0 (6 numbers)

_ _ _ 2 (4 numbers)

_ _ _ 4 (4 numbers)

_ _ _ 6 (4 numbers)

(C) 2, 4, 3, 6 (18 numbers, with even digit in the units place) There are a total of 50 numbers.

 
Checkout Other Questions of CAT 2017 Slot 2 Paper:

Verbal Ability :              |   Q.01- Q.06  |  Q.07- Q.12  |  Q.13- Q.18  |  Q.19- Q.21  |  Q.22- Q.24  |  Q.25- Q.29  |  Q.30 – Q.34  |

Logical Reasoning :    |   Q.01- Q.04 Q.05- Q.08  |  Q.09- Q.12  |  Q.13- Q.16  |  Q.17- Q.20 Q.21- Q.24  |  Q.25 – Q.28  |   Q.29 – Q.32  |

Quantitative Aptitude: |   Q.01- Q.05  |   Q.06- Q.10Q.11- Q.15  |  Q.16- Q.20  |  Q.21- Q.25  |   Q.26- Q.30  |  Q.31 – Q.34  |

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