SOLVED PAPER 2023 (CDS) (I) (Elementary Mathematics)

Total Questions: 100

71. (Questions 71-80) Each of the question given below follow followed by two statements, select which of the statement is / are logically correct according to the question ?

A number 277XY5 (where X,Y are digits) is divisible by 25. What is the value of X.

Statement I The given number is divisible by 9
Statement II X > 5

Correct Answer: (c) Choose this option if the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone
Solution:From the question, the number
277XY 5 is divisible by 25.
Then, Y = 2 or 7
Statement I
The number is divisible by 9, then
sum of digit = 2 + 7 + 7 + X + Y + 5
(divisible by 9)
= 21 + X + Y (divisible by 9)
∴ X + Y = 6, 15
Statement II x > 5
Now,
∵ Y = 7
∴ X = 8  [From X + Y = 15 ]
Hence, the question can be answered by using both the Statements I and II together, but cannot be answered by using either statement alone.

72. What are the unique values of a, b and c if 2 is a root of the equation ax² + bx + c = 0 ?

Statement I Ratio of c to a is I
Statement II Ratio of b to a is (-5/2)

Correct Answer: (d) Choose this option if the Question cannot be answered even by using both Statements together
Solution:The one root of the given
equation ax² + bx + c = 0 is 2.
∴ x = 2 will be satisfy the equation, then
4a + 2b + c = 0 .....(i)

Statement I a ÷ c = 1 .....(ii)
Statement I a ÷ b = -5 ÷ 2 .....(ii)

From statements I and II, the unique
values of a, b and c cannot be determined.
Hence, option (d) is the correct answer.

73. Is m > n where m, n are non-zero numbers

Statement I  m ÷ n > 1
Statement II  m > 2n

Correct Answer: (d) Choose this option if the Question cannot be answered even by using both Statements together
Solution:Statement I  m ÷ n > 1
Statement II  m > 2n

From Statements I and II,
the answer cannot be determined.
Hence, option (d) is the correct answer.

74. Can a circle be drawn through the points A,B and C

Statement I
AB = 5cm BC = 5cm
CA = 6cm

Statement II
AB = 3cm BC = 4cm
CA = 7cm

Correct Answer: (b) Choose this option if the Question can be answered by either Statement alone
Solution:

75. x and y are consecutive odd integers. Can the value of (x + y) be determined uniquely

Statement I  (x + y)⁴ = 256
Statement II  (x + y)³ < 16

Correct Answer: (c) Choose this option if the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone
Solution:Statement I (x + y)⁴ = 256
⇒ x + y = ± 4
Statement II (x + y)³ < 16

From the Statements I and II
together we can determined the
unique value of (x + y) Hence,
option (c) is the correct answer.

76. Is p² + q² + q odd, where p, q are positive integers

Statement I 2q + q is odd.
Statement II Lq - 2p is odd.

Correct Answer: (d) Choose this option if the Question cannot be answered even by using both Statements together
Solution:Statement I 2q + q is odd.
Statement II Lq - 2p is odd.

From Statements I and II together,
answer of the question cannot be determined.
Hence, option (d) is the correct answer.

77. What is the area of the circle C

Statement I A are of length 7 cm subtends an angle 30° at the centre of C.
Statement II A chord of length 10 cm subtends an angle 90° at the centre of C.

Correct Answer: (b) Choose this option if the Question can be answered by either Statement alone
Solution:

78. Is triangle right angled.

Statement I The length of the line segment joining the mid-points of two sides of Δ is half of the third side of Δ
Statement II The angles of Δ are in the ratio 1 : 2 : 3.

Correct Answer: (a) Choose this option if the Question can be answered by one of the Statements alone but not by the other
Solution:

79. The length of two longer sides of the triangle Delta are 25 cm 24 cm. What is the length of the shortest side.

Statement I The angles of Δ are in the ratio 1 : 2 : 3.
Statement II The length of the perpendicular drawn on the longest side of Δ from its opposite vertex is 6.72 cm.

Correct Answer: (a) Choose this option if the Question can be answered by one of the Statements alone but not by the other
Solution:

80. A chord PQ of the circle C divides it into two segments such that 3 times the area of the major segment is 4 times the area of the minor segment. What is the radius of C?

Correct Answer: (b) Choose this option if the Question can be answered by either Statement alone
Solution: