CDS Exam (I)- 2024- Elementary Mathematics Defence Question Paper

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CDS Exam (I)- 2024- Elementary Mathematics

CDS Exam (I)- 2024- Elementary Mathematics Defence Questions

11.
If the equation \( x \cos \theta = x^{2} + p \) has a real solution for every \( \theta \) where \( 0 \leq \theta \leq \frac{\pi}{4} \), then which one of the following is correct?
A.
\( p = \frac{1}{8} \)
B.
\( p \leq \frac{1}{8} \)
C.
\( p \geq \frac{1}{8} \)
D.
\( p \leq \frac{1}{4} \)
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ANSWER :
B. \( p \leq \frac{1}{8} \)
12.
What is the difference between the greatest value and the least value of \( \cos^{2} \theta + 3 \sin^{2} \theta + 2 \)?
A.
4
B.
3
C.
2
D.
1
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ANSWER :
C. 2
13.
ABC is a right-angled triangle, right-angled at B such that AB = 6 cm and BC = 8 cm. What is the perimeter of the square inscribed in the triangle ABC with maximum area?
A.
24/7 cm
B.
96/7 cm
C.
24 cm
D.
32 cm
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ANSWER :
B. 96/7 cm
14.
What is the greatest value of k for which \( 2x^{2} - 4x + k = 0 \) has real roots?
A.
1
B.
2
C.
3
D.
4
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ANSWER :
B. 2
15.
Consider the following data:
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0.1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct?
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
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ANSWER :
B. 2 and 3 only
16.
What is the digit at the 100th place of number \( (225)^{40} \)?
A.
6
B.
5
C.
4
D.
2
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ANSWER :
A. 6
17.
If a, b, c, d are natural numbers, then how many possible remainders are there when \( 1^{a} + 2^{b} + 3^{c} + 4^{d} \) is divided by 10?
A.
3
B.
4
C.
5
D.
6
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ANSWER :
C. 5
18.
If n is a natural number, then what is the sum of all distinct remainders of \( 4^{n} + 6^{n} + 9^{n} + 11^{n} \) when divided by 10 for various values of n?
A.
3
B.
4
C.
6
D.
7
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ANSWER :
B. 4
19.
When the number
(12345678910111213 ... 99100) is divided by 16, what will be the remainder?
A.
15
B.
12
C.
4
D.
3
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ANSWER :
B. 12
20.
A, B, C, D can complete a work in 3, 6, 9, 12 hours respectively. Further, only one person can work at a time in each hour and nobody can work for two consecutive hours. It is not necessary to engage all. What is the minimum number of hours that they will take to finish the work?
A.
36/25
B.
12/5
C.
4
D.
2
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ANSWER :
C. 4
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