Arithmetic (UPSC) Part- VI

Total Questions: 29

21. A Question is given followed by two Statements I and II. Consider the Question and the Statements. There are three distinct prime numbers whose sum is a prime number. [2024-II]

Question:
What are those three numbers?

Statement-I: Their sum is less than 23
Statement-II: One of the numbers is 5.

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:Given that the average marks in a class are 60.

From the question there are 3 prime numbers whose sum is a prime number.

Here the sum will be an odd number as it is prime and the 3 prime numbers also should be odd:

As, odd + odd + 2 = even (2 is the only even prime number).
So 2 is not in the 3 prime numbers.

There it should be odd + odd + odd which gives an odd number.
Let’s take the first statement; the sum of 3 prime numbers is
less than 23,
i.e., the 3 numbers is definitely less than 23

Taking prime numbers less than 23— 2, 3, 5, 7, 11, 13, 17

Trial 1: 3 + 5 + 7 = 15; not a prime

Trial 2: 3 + 5 + 11 = 19; is a prime also less than 23

Trial 3: 3 + 5 + 13 = 21; not a prime

Trial 4 : 3 + 5 + 17 = 25; neither prime nor less than 23
Therefore only possible input is 3, 5 and 11 as continuing trial only proves that if the sum is a prime then it will be greater than 3; This contradicts Statement 1

Statement 2 states that one of the integer is 5 which is not
needed as we will know that from the trials.

22. A Question is given followed by two Statements I and II. Consider the Question and the Statements. [2024-II]

Question: Is (x + y) an integer?
Statement-I: (2x + y) is an integer
Statement-II: (x + 2y) is an integer
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:To find if (x + y) is an integer

From statements 1 and 2 it is clear that (2x + y) and (x + 2y) is
an integer.

Let’s give x = 1/3 and y = 1/3

Then, 2x + y = 2/3 + 1/3 = 3/3 = 1; Statement 1 is valid for
given values of x and y

Now, x + 2y = 1/3 + 2/3 = 3/3 = 1; Statement 2 is valid for given
values of x and y

The Question cannot be answered even by using both the Statements together.

23. A Question is given followed by two Statements I and II. Consider the Question and the Statements. A person buys three articles p, q and r for ₹50. The price of the article q is ₹16 which is the least. [2024-II]

Question:
What is the price of the article p?

Statement-I: The cost of p is not more than that of r.
Statement-II: The cost of r is not more than that of p.
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:q = 16 (which is least)
∴ p, r > q = 16
Then, p + r = 50 - 16 = 34
S1: 16 < p ≤ r
∴ p = 17, r = 17
S2: For 16 < r ≤ p
r = p = 17

24. 3²⁵ +2²⁷ is divisible by [2024-II]

(a) 3
(b) 7
(c) 10
(d) 11

Correct Answer: (c) 10

Solution:(c) 32⁵ + 2²⁷

(25)⁵ + (2)²⁷ = 25² + 227 = 24 (2 + 23) = 24 x 10
Hence, number is divisible by 10.

25. What are the values of m and n, where m and n are natural numbers? [2024-II]

Statement-I:
m + n > mn and m > n
Statement-II:
The product of m and n is 24.
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:From I and II, we get
From II: m × n = 24
(m₁ n) = (1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2),
and (24, 1).
From II and I: m > n₁ (m, n) = (24, 1), (12, 2), (8, 3)
and (6, 4) m + n > m.n ⇒ (24 + 1) > 24 x 1
25 > 24
(m₁ n) = (24, 1)

26. A person buys three articles P, Q and R for ₹3,330. If P costs 25% more than R and R costs 20% more than Q, then what is the cost of P? [2024-II]

(a) ₹1,000
(b) ₹1,200
(c) ₹1,250
(d) ₹1,350

Correct Answer: (d) ₹1,350

Solution:Let Q = 100

R = 100 × 120/100 = 120 and P = 120 × 125/100 = 150

Now, Total cost (100 + 120 + 150) then P = 150

∴ When total cost 3330 then P = 150 × 3330/370 = 1350

27. If the sum of the two-digit numbers AB and CD is the three-digit number 1CE, where the letters A, B, C, D, E denote distinct digits, then what is the value of A? [2024-II]

(a) 9
(b) 8
(c) 7
(d) Cannot be determined due to insufficient data

Correct Answer: (a) 9

Solution:AB
CD

—

1CE
Maximum carry forward by (B + D) is one.
So, C can not be greater than one
To get E = 10, (B, D) = (2, 8), (3, 7), (4, 6), (6, 4), (7, 3) and (8, 2)
Then, A = 9

Example: 92
18
110

28. Three numbers x, y, z are selected from the set of the first seven natural numbers such that x > 2y > 3z. How many such distinct triplets (x, y, z) are possible? [2024-II]

(a) One triplet
(b) Two triplets
(c) Three triplets
(d) Four triplets

Correct Answer: (d) Four triplets

Solution:There are four triplets satisfying the given conditions :
(5, 2, 1), (6, 2, 1), (7, 2, 1) and (7, 3, 1).

29. The total cost of 4 oranges, 6 mangoes and 8 apples is equal to twice the total cost of 1 orange, 2 mangoes and 5 apples. Consider the following statements: [2024-II]

The total cost of 3 oranges, 5 mangoes and 9 apples is equal to the total cost of 4 oranges, 6 mangoes and 8 apples.
The total cost of one orange and one mango is equal to the cost of one apple.

Which of the statements given above 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:Let the costs of oranges, mangoes and apples be Q, M and A.
According to the question,
4Q + 6M + 8A = 2 (Q + 2M + 5A) or
4Q + 6M + 8A = 2Q + 4M + 10A or
2Q + 2M = 2A or
Q + M = A
Statement 1: 3Q + 5M + 9A = 4Q + 6M + 8A is correct
or A = Q + M It is already proved.
Statement 2: Q + M = A It is also correct. Hence, both statement
1 & 2 are true.

« Previous 1 23