Previous Year CAT Questions on Number System PDF PDF

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Previous Year CAT Questions on Number System PDF

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Instructions For the following questions answer them individually Question 1 If N = (11 p+7)(7 q−2)(5 r+1)(3 s) is a perfect cube, where value of p + q + r + s is :

A

5

B

6

C

7

D

8

E

9

p, q, r and s are positive integers, then the smallest

Answer: E Explanation: It has been given that N = (11p+7 )(7q−2 )(5r+1 )(3s ) is a perfect cube. All the factors given are prime. Therefore, the power of each number should be a multiple of 3 or 0.

p, q, r and s are positive integers. Therefore, only the power of the expressions in which some number is subtracted from these variables or these variables are subtracted from some number can be made 0.

11p+7: This expression must be made a perfect cube. The nearest perfect cube is take is 9 − 7

119. Therefore, the least value that p can

= 2.

7q−2 The least value that q can take is 2. If q = 2, then the value of the expression preventing the product from becoming a perfect cube.

7q−2 will become 70 = 1, without

5r+1: The least value that

r can take is 2.

3s ): The least value that

s can take is 3.

Therefore, the least value of the expression Therefore, option E is the right answer.

p + q + r + s is 2 + 2 + 2 + 3 = 9.

Question 2 If

1067 − 87 is written as an integer in base 10 notation, what is the sum of digits in that integer? ·

A

683

B

489

C

583

D

589 Answer: D

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Explanation:

1067 − 87 = 9999....99913 (total 67 digits) Sum of digits = 65 ∗ 9 + 1 + 3 = 589 Question 3 6407522209 3600840049

2−

=

A

0.666039

B

0.666029

C

0.666009

D

None of the above Answer: A

Explanation: 6407522209

2 − 3600840049 = 2 − = 2 − 1.3339610 = 0.666039

80047 60007

Therefore, option A is the right answer.

Take 3 Free CAT Mocks (With Solutions) Question 4 1

The sum of 1 − 6

A B

1

1

1

5 18 )+ .... is

2 3 2 3 2 3

C

D

1

+ (6 × 4) − (6 × 4 ×

3 2

Answer: D Explanation: We can see that the magnitude in each succeeding term is less than that of preceding term. Hence, we can say that for S =

1−

1 6

1

1

1

1

+ (6 × 4) − (6 × 4 ×

5 18 )+ ...

The value will lie between (5/6, 1). We can check with option choices. 2

5

Option A: 3 < 6 . Hence, this can't be the answer. Option B:

2 3 = 1.155 > 2 3

Option C:

1. Hence, this can't be the answer. 5

= 0.8164 < 6 . Hence, this can't be the answer.

3

Option D: 2 = 0.866. This lies between (5/6, 1).Hence, this is the correct answer. Question 5 A rod is cut into 3 equal parts. The resulting portions are then cut into 12, 18 and 32 equal parts, respectively. If each of the resulting portions have integer length, the minimum length of the rod is

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A

6912 units

B

864 units

C

288 units

D

240 units Answer: B

Explanation: The rod is cut into 3 equal parts thus the length of the rod will be a multiple of 3. Each part is then cut into

18 = 2 ∗

32

12 = 22 ∗ 3

25

and 32 = parts and thus, each part of rod has to be a multiple of Thus, the rod will be a multiple of 288 ∗ 3 = 864 Thus, the minimum length of the rod is 864 units. Hence, option B is the correct answer.

25 ∗ 32 = 288

Question 6 If the product of

n positive integers is nn, then their sum is

A

a negative integer

B

equal to n

C

n+ n

D

never less than

1

n2

Answer: D Explanation: We know that for a given product the sum of all the numbers will be the least when all the numbers are equal Thus, the sum of all the number in

nn will be least when

nn = n ∗ n ∗ n ∗ ........... ∗ (ntimes) Thus, their sum = n + n + ........ + (ntimes) = n2 Hence, the sum can never be less than n2 Hence, option D is the correct answer.

CAT Previous Papers PDF Question 7 If the product of the integers a, b, c and d is 3094 and if 1 < a < b < c < d, what is the product of b and c?

A

26

B

91

C

133

D

221 Answer: B

Explanation: 3094 = 2*7*13*17 Given, 1 < a < b < c < d

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Thus, a = 2, b = 7, c = 13 and d = 17 Hence, b*c = 7*13 = 91 Hence, option B is the correct answer. Question 8 A sum of Rs.1400 is divided amongst A, B, C and D such that A’s share : B’s share = B’s share : C’s share 3

= C’s share = D’s share = 4 how much is C’s share?

A

Rs.72

B

Rs.288

C

Rs.216

D

Rs.384 Answer: D

Explanation: A:B = B:C = C:D = 3:4 Let C be 3x and D be 4x B : C = 3 : 4 and C = 3x So, B = 9x/4 A : B = 3 : 4 and B = 9x/4 So, A = 27x/16 Therefore, A : B : C : D is 27x/16 : 9x/4 : 3x : 4x Also, A + B + C + D = Rs. 1400 On solving, we get x = 128 Therefore, C's share = 3x = Rs. 384 Hence, option D is the correct answer. Question 9 Z is the product of first 31 natural numbers. If X = Z + 1, then the numbers of primes among X + 1, X + 2 ..., X + 29, X + 30 is

A

30

B

2

C

Cannot be determined

D

None of the above Answer: D

Explanation: It is given that Z = 31! X = 31! + 1 X+1 = 31!+2 this is divisible by 2 X+2 = 31!+3 this is divisible by 3 X+3 = 31!+4 this is divisible by 4 . . . . X+29 = 31!+30 this is divisible by 30 X+30 = 31!+31 this is divisible by 31 Hence, none of X + 1, X + 2, ..., X + 29, X + 30 is a prime number. Hence, option D is the correct answer.

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Question 10 In the following series, what numbers should replace the question marks? -1, 0, 1, 0, 2, 4, 1, 6, 9, 2, 12, 16, ? ? ?

A

11, 18, 27

B

-1, 0, 3

C

3, 20, 25

D

Cannot be ascertained Answer: C

Explanation: The given series is a combination of 3 series:1st containing all 3n+1 terms i.e. 1st, 4th, 7th and so on terms:-1, 0, 1, 2 and thus, the next number will be 3. 2nd containing all 3n+2 terms i.e. 2nd, 5th, 8th and so on terms:0, 2, 6, 12 as we can see 2n is being added to each term to get the next term and thus the next term will be 20. 3rd containing all 3n terms i.e. 3rd, 6th, 9th and so on terms:1, 4, 9, 16 an we can see these are squares of the natural numbers and thus, the next term will be 25. Hence, ?,?,? will be 3, 20, 25 Hence, option C is the correct answer.

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