NDA & NA Exam (I)- 2018 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2018 - Mathematics

NDA & NA Exam (I)- 2018 - Mathematics Defence Questions

11.
If \( x + \log_{15}(1 + 3^{x}) = x \log_{15} 5 + \log_{15} 12 \), where x is an integer, then what is x equal to?
A.
−3
B.
2
C.
1
D.
3
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ANSWER :
C. 1
12.
How many four-digit numbers divisible by 10 can be formed using 1, 5, 0, 6, 7 without repetition of digits?
A.
24
B.
36
C.
44
D.
64
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ANSWER :
A. 24
13.
Consider the following information for the next two (02) items that follow:
In a class, 54 students are good in Hindi only, 63 students are good in Mathematics only and 41 students are good in English only. There are 18 students who are good in both Hindi and Mathematics. 10 students are good in all three subjects.

What is the number of students who are good in either Hindi or Mathematics but not in English?
A.
99
B.
107
C.
125
D.
130
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ANSWER :
C. 125
14.
What is the number of students who are good in Hindi and Mathematics but not in English?
A.
18
B.
12
C.
10
D.
8
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ANSWER :
D. 8
15.
If \( \alpha \) and \( \beta \) are different complex numbers with \( |\alpha| = 1 \), then what is \( \left|\frac{\alpha - \beta}{1 - \bar{\alpha}\beta}\right| \) equal to?
A.
\( |\beta| \)
B.
2
C.
1
D.
0
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ANSWER :
C. 1
16.
The equation \( |1 - x| + x^{2} = 5 \) has
A.
a rational root and an irrational root
B.
two rational roots
C.
two irrational roots
D.
no real roots
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ANSWER :
A. a rational root and an irrational root
17.
The binary number expression of the decimal number 31 is
A.
1111
B.
10111
C.
11011
D.
11111
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ANSWER :
D. 11111
18.
What is \( i^{1000} + i^{1001} + i^{1002} + i^{1003} \) equal to (where \( i = \sqrt{-1} \))?
A.
0
B.
i
C.
−i
D.
1
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ANSWER :
A. 0
19.
What is \( \frac{1}{\log_{2} N} + \frac{1}{\log_{3} N} + \frac{1}{\log_{4} N} + \ldots + \frac{1}{\log_{100} N} \) equal to (N ≠ 1)?
A.
\( \frac{1}{\log_{100!} N} \)
B.
\( \frac{1}{\log_{99!} N} \)
C.
\( \frac{99}{\log_{100!} N} \)
D.
\( \frac{99}{\log_{99!} N} \)
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ANSWER :
A. \( \frac{1}{\log_{100!} N} \)
20.
The modulus-amplitude form of \( \sqrt{3} + i \), where \( i = \sqrt{-1} \), is
A.
\( 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) \)
B.
\( 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) \)
C.
\( 4\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) \)
D.
\( 4\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) \)
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ANSWER :
B. \( 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) \)
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