SOLVED PAPER 2024 (CDS) (II) (Mathematics)

Total Questions: 100

81. Three amounts x, y, z are such that y is the compound interest on x and z is the compound interest on y. The rate of interest per annum and the time period in years are same. Which one of the following is correct?

Correct Answer: (b) y² = zx
Solution:

82. There are n concentric squares. The area of the innermost square is 1 unit and the distance between corresponding corners of any two consecutive squares is 1 unit. Consider the following statements.

I. The diagonal of the nth square is 2n + √2 - 2

II. The area included between nth square and (n - 1)ᵗʰ square is independent of n.

Which of the statements given above is/are correct?

Correct Answer: (a) Only I
Solution:

83. In a rectangle ABCD, AC is one of the diagonals. If AC + AB = 3AD and AC - AD = 4 units, then what is the area of the triangle?

Correct Answer: (c) 48 square unit
Solution:

84. The area of the circle circumscribing three identical circles touching each other is π(2 + √3)²)/3 square cm. What is the radius of one of the smaller circles?

Correct Answer: (b) 1 cm
Solution:

85. In a triangle ABC, AB = 21cm BC = 20 cm and CA = 13 cm. A perpendicular CD is drawn upon the longest side. What is the area of the triangle BCD?

Correct Answer: (a) 96 square cm
Solution:

86. There are two containers A and B. In container A, the ratio of milk and water is 1 : 3 and in container B, the ratio of milk and water is m: n. If the mixture in the containers A and B are mixed in the ratio 2:3 to get 20 L of a mixture having milk and water in the ratio 3 : 7, then what is the value of m/n ?

Correct Answer: (a) 1/2
Solution:

87. A cone, a hemisphere and a cylinder stand on equal base of radius r and have the same height. If the sum of volumes of cone, the hemisphere and the cylinder is equal to volume of a sphere of radius R, then what is (R³)/(r³) equal to

Correct Answer: (b) 1.5
Solution:

88. If x³ + px² + qx + r is an integer for all integral values of x, then consider the following statements

I. p must be an integer.
II. q must be an integer.
III. r must be an integer.
Which of the statements given above is/are correct?

Correct Answer: (c) All I, II and III
Solution:x³ + px² + qx + r is an integer V integral values of x.
If pq and r any one of them is non-integer,
then x³ + px² + qx + n not integer.
⇒ p, q and r must be integer.
∴ (c) is the correct option.

89. XYZ is a 3-digit number, where X, Y, Z are distinct non-zero digits. The difference between the two 3-digits numbers XYZ and YXZ is 90. How many possible values exist for the sum (X + Y)?

Correct Answer: (b) 8
Solution:xyz is 3-digit number and x, y, z are distinct non-zero digits.
According to the problem
xyz - yxz = 90
⇒ (100x + 10y+Z)-
(100y +10x+Z) = 90
90x - 90y = 90
x - y = 1
2 - 1 = 1
3 - 2 = 1
4 - 3 = 1
5 - 4 = 1
6 - 5 = 1
7 - 6 = 1
8 - 7 = 1
9 - 8 = 1.
Then, possible value of (x + y) = 3, 5, 7, 9, 11, 13, 15, 17
⇒ There are 8 possible value of (x + y).
∴ (b) is the correct option.

90. How many times does the minute hand of a clock coincide with the second hand between 2.01 pm and 4.01 pm on the same day?

Correct Answer: (c) 119
Solution:From 2:01 pm to 4:01 pm is exactly 2 h.
Since, the minute and second hands consider once every minute.
We calculate the number of minutes between 2:01pm and 4:01pm
The total time from 2:01 pm to 4:01 pm is 120 minute.
∴ (120-1) number of times minute hand and second hand of clock coincide.
= 119 times
∴ (c) is the correct option.