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

Total Questions: 100

91. (Question 91-100) Consider the following for the next ten items that follow.

Is m > n if m, n are real numbers?
Statement I
m = (1 - p) (p² + p + 1)
and n = (p + 1) (p² - p + 1)

Statement II
m = pn

Correct Answer: Mark option (d) if the question cannot be answered even by using both the statements together.
Solution:I. m = (1 - p) (p² + p + 1)
= 1 - p³
n = (p + 1) (p² - p + 1)
= p³ + 1 = 1 + p ^ 3
Now,
If p > 0 ⇒ m > n
and p < 0 ⇒ m < n
II. m = pn
If p > 0 ⇒ m > n and p < 0
⇒ m < n

92. Find the value of other root.

Correct Answer: Mark option (b) if the question can be answered by using either statement alone.
Solution:

93. What is the 3-digit number which is divisible by 10?

Statement I
If the digits in hundred's place and ten's place of the number are interchanged, the resulting number is diminished by 180.

Statement II
If the digit in hundred's place is halved and digit in ten's place and unit place of the number are interchanged, the resulting number is diminished by 336.

Correct Answer: Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone. Translate
Solution:Let the number digit be xy0.
I. (100x + 10y) - (100y + 10x) = 180
⇒ 90x - 90y = 180
⇒ x - y = 2 .....(i)

II. (100x + 10y) - (50x + y) = 336
⇒ 50x + 9y = 336 .....(ii)
From Equations (i) and (ii), we get
50(y + 2) + 9y = 336
⇒ 50y + 100 + 9y = 336
⇒ 59y = 236
⇒ Y = 4
∴ x = y + 2 = 6
So, number is 640.

94. Are x, y and z equal, where x, y, z are real numbers?

Statement I
x² + y² + z² - xy - yz - zx = 0

Statement II
x³ + y³ + z³ - 3xyz = 0

Correct Answer: Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Solution:I. x² + y² + z² - xy - yz - zx = 0
⇒ 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0
⇒ (x - y)² + (y - z)² + (z - x)² = 0
⇒ x - y = y - z = z - x = 0
⇒ x = y = z

II. x³ + y³ + z³- 3xyz = 0
⇒ (x + y + z) ( x² + y² + z² - xy - yz - zx ) = 0
⇒ x + y + z = 0
or x² + y² + z² - xy - yz - zx = 0
⇒ x + y + z = 0
or x = y = z

95. What is the ratio x : y : z equal to if x, y, z ≠ 0 ?

Correct Answer: Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone.
Solution:I. (x + z) ÷ y = z ÷ x
⇒ x² + xz =y dot z .....(i)

II. (z - y) ÷ x = x ÷ z
⇒ z² - yz = x²  .....(ii)
From Equations (i) and (ii), we get
z² - yz + xz = yz
⇒ z² + xz = 2yz
⇒ z + x = 2y  .....(iii)

Now,
(x + z) ÷ y = z ÷ x [from Equations (iii)]
⇒ (2y) ÷ y = z ÷ x
⇒ z = 2x
∴ z + x = 2y
⇒ 2x + x = 2y
⇒ 3x=2y
⇒ y = (3 ÷ 2)x
∴ x : y : z = x : (3÷2)x : 2x
= 2 : 3 : 4

96. What is the sum of two natural numbers?

Statement I
LCM of the two numbers is 144.

Statement II
One of the numbers is 72.

Correct Answer: Mark option (d) if the question cannot be answered even by using both the statements together.
Solution:If LCM and one number is given, then we can't find the other number.
So, sum of numbers can't be determined.

97. Is average of the largest and the smallest of 4 given numbers greater than the average of the 4 numbers ?

Statement I
The difference between the largest and the second largest numbers is less than the difference between the second smallest and the smallest of the numbers.

Statement II
The difference between the largest and the smallest numbers is greater than the difference between the second largest and second smallest of the numbers.

Correct Answer: Mark option (d) if the question cannot be answered even by using both the statements together.
Solution:


98. Is (a - b + c) > (a + b - c), where a, b and care real numbers ?

Statement I b is negative.

Statement II c is negative.

Correct Answer: Mark option (d) if the question cannot be answered even by using both the statements together.
Solution:a - b + c > a + b - c
⇒ 2c > 2b ⇒ c > b

∴ This does not implies that
either b is negative or c is negative.

99. What is the cost of 15 pens, 21 pencils and 18 note books?

Statement I
The cost of 7 pens, 6 pencils and 5 note books is ₹ 200.

Statement II
The cost of 3 pens, 8 pencils and 7 note books is ₹ 210.

Correct Answer: Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone.
Solution:We have,
7 pen + 6 pencil + 5 notebook = ₹ 200
3 pen + 8 pencil + 7 notebook = ₹ 210
∴ 10 pen + 14 pencil + 12 notebook = ₹ 410
⇒ 5 pen + 7 pencil + 6 notebook = ₹ (410 ÷ 2)
⇒15 pen + 21 pencil + 18 notebook = ₹ (410 × 3) ÷ 2
= ₹ 615

100. What is the area of the triangle inscribed in a semi-circle with the diameter as the base ?

Statement I
The diameter of semi-circle is 20 cm.

Statement II
Two shorter sides of the triangle are 12 cm and 16 cm.

Correct Answer: Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Solution: