Arithmetic (UPSC) Part- VI

Total Questions: 29

1. A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct? [2023-II]

(a) R = S
(b) R > S
(c) R < S
(d) R ≤ S

Correct Answer: (c) R < S

Solution:

2. How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers? [2023-II]

(a) 6
(b) 7
(c) 8
(d) 9

Correct Answer: (d) 9

Solution:1186, remainder = 31

then, Number = 1186 — 31 = 1155

1155=3x5x7x 11 :
Factor of 1155 = 3, 5, 7, 11, 15, 21, 33, 35, 55, 73, 105, 385, 231, 165, 1155
No. factor’s > 31, gives remainder of 31 when divided to 1186.
Hence, required numbers are 9 and those are 33, 35, 73, 105,
385, 231, 165, 1155

3. Let pp, qq and rr be 2-digit numbers where p < q < r. If pp + qq + rr = tt0, where tt0 is a 3-digit number ending with zero, consider the following statements: [2023-II]

The number of possible values of p is 5.
The number of possible values of q is 6.

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Correct Answer: (c) Both 1 and 2

Solution:p < q < r

∴ 1 ≤ p ≤ 7 ; 2 ≤ q ≤ 8 and 3 ≤ r ≤ q

Now, pp + qq + rr = tto

i.e. (p+q+r)=10 to 20

So, possible triplet of (p, q, r) = (1, 2, 7), (1, 3, 6),
(1,4, 5); 2, 3, 5), (3, 8, 9), (4, 7, 9), (5, 6, 9) and
(5, 7, 8),
Possible value of P are 1, 2, 3, 4 and 5.
Possible value of q are 2, 3, 4, 6, 7 and 8.
Hence, both statements are true.

4. There are three traffic signals. Each signal changes colour from green to red and then from red to green. The first signal takes 25 seconds, the second signal takes 39 seconds and the third signal takes 60 seconds to change the colour from green to red. The durations for green and red colours are same. At 2:00 p.m, they together turn green. At what time will they change to green next, simultaneously? [2023-II]

(a) 4:00 p.m.
(b) 4:10 p.m.
(c) 4:20 p.m.
(d) 4:30 p.m.

Correct Answer: (b) 4:10 p.m.

Solution:L.C.M. of 25, 39, 60

= 3x4 x 5 x 5 x 13 = 3900 seconds

= 65 minutes.
All three signals changes from green to red simultaneously after
65 minutes. And again red to green after another 65 minutes.
:. All three light changes to green next simultaneously at 2:00 PM + (65 + 65) minutes

= 4:10 PM.

5. Question: Is p greater than q? [2023-II]

Statement-1: p × q is greater than zero.
Statement-2: p² is greater than q².
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question cannot be answered even by using both the Statements together.

Correct Answer: (d) The Question cannot be answered even by using both the Statements together.

Solution:Suppose, P = 2 and q = 1
p × q = 2 × 1 = 2 > 0
and p² = (2)² = 4 > (q)² = 1
Again, p = -2 and q = -1
p × q = -2 ×-1 = 2 > 0
p² = (-2)² = 4 > (q)² = (-1)² = 1
∴ p < q
Hence, we cannot answer the question using both statements

6. Question: Is (p + q - r) greater than (p - q + r), where p, q and r are integers? [2023-II]

Statement-1: (p - q) is positive.
Statement-2: (p - r) is negative.
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question cannot be answered even by using both the Statements together.

Correct Answer: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

Solution:p + q - r > (p - q + r) ⇒ q > r
S1: p - q > 0 ⇒ p > q
p - q + r > r
S2: p - r < 0
⇒ p < r
From S1 and S2: q < p < r
0 < p - q < r ⇒ p - q + r < 2r
Again, q < p < r
p - r < 0
q + p - r < q ⇒ (p - q - r) < q
As q < r, ∴ (p + q - r) < (p - q + r)
Thus (p + q - r) is not greater than (p - q + r)

7. Consider a 3-digit number. [2023-II]

Question: What is the number?
Statement - 1: The sum of the digits of the number is equal to the product of the digits.
Statement - 2: The number is divisible by the sum of the digits of the number.
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question cannot be answered even by using both the Statements together.

Correct Answer: (d) The Question cannot be answered even by using both the Statements together.

Solution:Consider a number with its 3-digits are 1, 2 and 3
Statement 1: Sum of its digit = 1 + 2 + 3 = 6
Product of its digit = 1 x 2 x 3 = 6
Now numbers are 123, 132, 213, 231, 312 and 321
Statement 2: Numbers are divisible by the sum of the digits of the number.
Sum of its digits = 1 + 2 + 3 = 6
Any number, that is divisible by 6, must be divisible by 2 and 3
∴ Numbers are 132 or 312
Hence, we cannot find exact number

8. In a party, 75 persons took tea, 60 persons took coffee and 15 persons took both tea and coffee. No one taking milk takes tea. Each person takes, at least one drink. [2023-II]

Question: How many persons attended the party?
Statement-1: 50 persons took milk.
Statement-2: Number of persons who attended the party is five times the number of persons who took milk only.
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question cannot be answered even by using both the Statements together.

Correct Answer: (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

Solution:n(Tea) = 75, n(coffee) = 60, n(Tea ∩ coffee) = 15
n(Tea ∪ coffee) = 75 + 60 — 15 = 120
From question, no one taking milk, takes tea but they can take coffee.
Statement 2:
Let n persons took milk only
Then, 120 + n = 5n
∴ n =120/4 = 30
∴ total number of persons attended the party
= 5 x 30 = 150
Hence, Statement II is sufficient to answer.

9. For five children with ages a < b < c < d < e; any two successive ages differ by 2 years. [2023-II]

Question: What is the age of the youngest child?
Statement-1: The age of the eldest is 3 times the youngest.
Statement-2: The average age of the children is 8 years.
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question cannot be answered even by using both the Statements together.

Correct Answer: (b) The Question can be answered by using either Statement alone.

Solution:S1: Let youngest age is x year
Then eldest age is 3x year.

∴ x + 8 = 3x ⇒ x = 4 years

S2: Average Age = 8

x + (x + 2) + (x + 4) + (x + 6) + (x + 8)/5 = 8
x + 4 = 8 ⇒ x = 4 years

10. What is the remainder if 2¹⁹² is divided by 6? [2023-II]

(a) 0
(b) 1
(c) 2
(d) 4

Correct Answer: (d) 4

Solution:2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
Periodicity of ‘2’ is ‘4’
192 ÷ 4 = 48

(Unit digit)

Now, when we divide 8, 16, 32, 64, 128, 256, ........... by 6, we get remainders 2, 4, 2, 4, 2, 4, .........

When we divide odd exponent of 2 by 6,
i.e. 23 or 25 or 27 ÷ 6; remainder is 2

Again, when we divide even exponent of 2 by 6,
i.e. 24 or 26 or 28 ÷ 6, remainder is 4

2¹⁹² ÷ 6 = Remainders = 4

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