SOLVED PAPER 2024 (CDS) (I) (Mathematics)

Total Questions: 100

61. (Questions 61-70) Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions :

Let a, b, c and d be positive integers.

Question: Which one of a, b, c, d is closest to the product abcd ?

Statement I a > b > c

Statement II c is not the smallest.

Correct Answer: (c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
Solution:Statement I a > b > c
Here, d is not fixed. So, we can't conclude any decision to closest of the product abcd.
Statement II c is not the smallest the answered can't be get by the statement Il alone.
From Statements I and II,
a > b > c > d
So, a will be closest of abcd.
∴ Both the statements together can be used to answer the questions.

62. Answer each item using the following instructions :

Let mn = k, where m and n are prime numbers and k is an even number.

Question: What is the value of mn - n + 1 ?

Statement I m > n

Statement II One of the numbers is 2.

Correct Answer: (d) If the Question cannot be answered even by using both Statements together.
Solution:Given, mx = k ∀ m, x ∈ Prime numbers
∀ k ∈ Even number
⇒ m or n = 2

Statement I
m>n ⇒ It is not tell any variable value of m, n.
So, on that basis, we can't find the value of (mx - n + 1)
Statement II
One of the numbers is 2.
When m or x = 2 we can't get the value of other variable.
⇒ On that basis, we can't find the value of (mn - n + 1)
So, the question can't be answered even by using both statement together.

63. Answer each item using the following instructions :

Question: If p is a positive integer, then what is the remainder when pⁿ is divided by p + 1 ?

Statement I n is even.

Statement II p is even.

Correct Answer: (a) If the Question can be answered by one of the Statement alone, but not by the other.
Solution:Statement I
On dividing pⁿ to (p + 1)
p ≡ -1 (mod p + 1)
⇒ (p)ⁿ ≡ (-1)ⁿ (mod p + 1 ) ......(i)
x = 2k (let) where k ∈ N
⇒ (p)²ᵏ ≡ 1 (mod p + 1 )
⇒ (p²ᵏ -1) ≡ 0 (mod p + 1 )
⇒ (pᵏ - 1)(pᵏ + 1) ≡ 0 (mod p + 1 )

When k = 1
(p - 1) (p+1) ≡ 0 (mod p + 1)
⇒ p ≡ 1 mod p + 1)
∴ Remainder = 1

When, k = 2
(p² - 1) (p² +1) ≡ 0 (mod p + 1)
⇒ (p-1) (p² + 1) ≡ 0 (mod p + 1)
⇒ p³ + p - p² -1 ≡ 0 (mod p + 1)
⇒ p(p² - p + 1) ≡ 1 (mod p + 1)
∴ Remainder = 1 .... so on

So, it is very clearly see that we can find the remainder at ne even number.

Statement II Clearly when p is even/odd then we can't see any effect to getting the remainder. It is an useful statement.

So, only one statement is enough to solve the question.

64. Answer each item using the following instructions :

Correct Answer: (a) If the Question can be answered by one of the Statement alone, but not by the other.
Solution:

65. Answer each item using the following instructions :

Let a, b and c be sides of a triangle ABC.

Question: Is the triangle equilateral ?

Statement I a² + b² + c² = (ab + bc + ca)

Statement II 3a² + 3b² + 4c² = 2ab + 4bc + 4ca

Correct Answer: (b) If the Question can be answered by either Statement alone.
Solution:

66. Answer each item using the following instructions :

Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.

Question: Is P > Q ?

Statement I x : y = 2 : 1

Statement II The angle between the two adjacent sides of the parallelogram is 60°.

Correct Answer: (b) If the Question can be answered by either Statement alone.
Solution:

67. Answer each item using the following instructions :

A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD.

Question: What is the perimeter of the quadrilateral?

Statement I AB + DC = 10cm

Statement II AD + BC = 10cm

Correct Answer: (b) If the Question can be answered by either Statement alone.
Solution:

68. Answer each item using the following instructions :

Question: What is the ratio of the lengths of diagonals of a rhombus ?

Statement I One diagonal of the rhombus is equal to its side.

Statement II The longer diagonal of the rhombus is equal to √3 times its side.

Correct Answer: (b) If the Question can be answered by either Statement alone.
Solution:

69. Answer each item using the following instructions :

The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units.

Question: What are the values of r and R?

Statement I r is an integer.

Statement II R is an integer.

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

70. Answer each item using the following instructions :

P, Q, R, S are the mid points of sides AB, BC, CD, DA, respectively of a quadrilateral ABCD.

Question: What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?

Statement I Area of the quadrilateral ABCD is 100 square unit.

Statement II Area of the quadrilateral PQRS is 50 square unit.

Correct Answer: (b) If the Question can be answered by either Statement alone.
Solution: