SOLVED PAPER 2022 (CDS) (II) (Elementary Mathematics)

Total Questions: 100

91. Solved

Correct Answer: (b)
Solution:

92. Solved

Correct Answer: (d) a + b + c + d = 0
Solution:

93. If (a³ + b³) is proportional to (a² - b²) then (a² - ab + b²) is proportional to

Correct Answer: (a) (a - b)
Solution:Given, a³ +b³ ∝ (a² - b²)
a³ + b³ = k(a² - b²)
(a + b)³ - 3ab(a + b) = k(a² - b²)
(a + b) (a² - ab + b²) = k(a² - b²)
a² - ab + b² = k(a - b)
∵ a² - ab + b² is proportional to (a - b).

94. Solved

Correct Answer: (c) 10
Solution:

95. What is the HCF of (x⁸ - y⁸) and (x⁷ - y⁷ + x⁵ y² - x² y⁵) ?

Correct Answer: (a) (x² + y²)
Solution:Here,
x⁸ - y⁸ = (x⁴ + y⁴) (x² + y²)
(x + y) (x - y)
= x⁷ - y⁷ + x⁵ y² - x² y⁵
= x⁵ (x² + y²) - y⁵ (x² + y²)
= (x⁵ - y⁵) (x² + y²)
Hence, HCF = x² + y²

96. If the sum of the square of the roots of the equation x² - 14x + k = 0 is 100, then what is the value of k?

Correct Answer: (b) 48
Solution:Let α, β be the roots of
the equation.
α² + β² = 100
⇒ (α² + β²) - αβ = 100
⇒ 196 - 2k = 100
⇒ 96 = 2k
∴ k = 48

97. If (x + k) is the HCF of x² + px + q and x² + qx + p where p ≠ q then what is the value of k?

Correct Answer: (a) -1
Solution:Given equation, x² + px + q = 0
x² + qx + p = 0
Let k is a root of equation,
k² - pk + q = 0
k² - qk + p = 0
⇒ (q - p)k + (q - p) = 0
k = - 1

98. If three times the greater of two numbers is divided by the smaller number, the quotient will be 6 and the remainder will be 6. If five times the smaller number is divided by the greater number, the quotient will be 2 and the remainder will be 3. What is the difference between the numbers?

Correct Answer: (b) 9
Solution:Let the two numbers be x and y
and x > y
According to the question,
3x = 6y + 6
x = 2(y + 1)
x = 2y + 2  ...(i)

Now, 5y = 2x + 3
2x = 5y - 3  ...(ii)
In Equations (i) × 2 we get
2x = 4y + 4 and put the value of
2x in
Equations (ii)
4y + 4 = 5y - 3
y = 7
Put the value of y in Equations (i)
x = 2y + 2 = 2 × 7 + 2 = 16
Now, x - y = 16 - 7 = 9

99. Consider the following statements:

1. (ab + bc + ca) is a factor of a² (b - c)³ + b² (c - a)³ + c² (a - b)³.
2. (a + b + c) is a factor of a² (b - c)³ + b² (c - a)³ + c² (a - b)³.

Which of the statements given above is/are correct?

Correct Answer: (a) Only 1
Solution:

100. Solved

Correct Answer: (d)
Solution: