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

Total Questions: 100

1. A real number x is such that the sum of the number and four times its square is the least. What is that number?

Correct Answer: (b) -0.125
Solution:

2. The difference of the square of two natural numbers m and n (m > n) is 72. How many pairs of natural numbers will satisfy?

Correct Answer: (a) 3
Solution:Given that, the difference of the
square two natural numbers m and n (m > n) 72.
According to the problem,
m² - n² = 72
(m - n)(m + n) = 72
∴ (m, n) be (19, 17), (11, 7), (9, 3)
There 3 pairs of natural number satisfy.
∴ (a) is the correct option.

3. Let N be a 5-digit number. When N is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of N?

Correct Answer: (b) 99956
Solution:N is the number divided by 6, 12, 15, 24
it leaves remainder 2, 8, 11, 20, respectively.
Difference is (6 - 2) = 4 (12-8) = 4
and (15-11) 4 and (24 - 20) = 4
LCM of (6, 12, 15, 24) = 120
Largest number of 5 digits is.
N = 99999
99999/120 leaves remainder 39.
Then, 99999 - 39 = 99960
and 99960 - 4 = 99956
∴ 99956 is the required number

4. What is the remainder when 111²²² + 222³³³ + 333⁴⁴⁴ is divided by 5?

Correct Answer: (d) 4
Solution:

5. What are the last three digits in the multiplication of

4321012345 × 98766789 ?

Correct Answer: (b) 2, 0, 5
Solution:4321012345 × 98766789 ?
Last 3 digit is 205.

6. p varies directly as (x² + y² + z²) When x = 1, y = 2, z = 3 then p = 70. What is the value of p when x = - 1, y = 1, z = 5 ?

Correct Answer: (c) 135
Solution:(x² + y² + z²) ∝ P
⇒ x² + y² + z² = kP
Where, k is proportionality constant.
According to the problem,
(1)² + (2)² + (3)² = k(70)
k = 14/70 = 1/5
Now, (-1)² + (2)² + (5)² = 1/5(P)
P = 27 × 5
P = 135

7. Let N be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of the following is correct?

Correct Answer: (d) 1200 < N < 1300
Solution:LCM of (6, 12, 15, 18) 180
N be the number multiple of 11 and
leaves remainder 5, when divided
by 6, 12, 15, 18.
∴ Number be like 180k + 5.
and 180k + 5 is the multiple of 11.
∴ Take k = 7
∴ N = 180 × 7 + 5 = 1265
1200 < N < 1300

8. .

Correct Answer: (c) 11
Solution:

9. Train X crosses a man standing on the platform in 24 sec and train Y crosses a man standing on the platform in 18 sec. They cross each other while running in opposite directions in 20 sec. What is the ratio of speed of X to speed of Y?

Correct Answer: (a) 1 : 2
Solution:

10. Let p, q be the roots of the equation x² + mx - n = 0 and m, n be the roots of the equation x² + px - q = 0 (m, n, p, q are non-zero numbers). Which of the following statements is/are correct?

I. m(m + n) = - 1
II. p + q = 1
Select the answer using the code given below.

Correct Answer: (c) Both I and II
Solution: